E. ROZENWASSER_Sensitivity of Automatic Control Systems_2000

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Preface

The idea of publishing this book in English came to the authors whilereviewing the generalizing monograph Theory of Sensitivity in Dynamic

Systems by Prof. M. Eslami (Springer-Verlag, 1994), which deals mostlywith sensitivity of linear systems. That monograph covers a wide scope of sensitivity theory problems and contains a bibliography consisting of morethan 2500 titles on sensitivity theory and related topics. Nevertheless,the authors were surprised to find that it does not contain many topicsof sensitivity theory that are important from both theoretical and appliedviewpoints. These problems are considered in the present monograph thatis brought to attention of the reader.

By sensitivity of control systems one usually means dependence of their

properties on parameters variation. The aggregate of principles and meth-ods related to sensitivity investigation forms sensitivity theory.

Sensitivity problems in one or another form have been touched on intheory of errors in computational mathematics and computer science, intheory of electrical and electronic networks, as well as in disturbance theoryin classical mechanics, and so on.

Sensitivity theory became an independent scientific branch of cyberneticsand control theory in the sixties. This was connected in major part withquick development of adaptive (self-tuning) systems that were constructed

for effective operation under parametric disturbances. Lately, sensitivitytheory methods were widely used for solving various theoretical and appliedproblems, viz.: analysis and synthesis, identification, adjustment, monitor-ing, testing, tolerance distribution, and so on. Step by step, methods of sensitivity theory have become a universal tool for investigating controlsystems. This has stimulated many works on employing methods of sen-sitivity theory to systems of various nature (technical, biological, social,economical, and so on). Nevertheless, a general basis for solving sensitivityproblems for control systems has not been exposed systematically. Onlya relatively small number of works consider the general theoretical and

mathematical foundation of sensitivity investigation and elucidate qualita-

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tive and quantitative connections between sensitivity theory and classicalsections of mathematics and control theory. Many techniques of sensitivityinvestigation used in applications have not received due justification.

It appears that many results obtained by the Russian scientific school

are totally unknown to English-speaking readers. We refer specifically tothe general concept of mathematical substantiation of sensitivity theory. Inthis book the problem of sensitivity investigation is considered as a stabilityproblem in an augmented state space that contains, in addition to initialstate variables, changing parameters of the system. This concept makes itpossible to employ profound results obtained by the Russian mathematicalschool on the basis of Lyapunov’s methods to problems of sensitivity theory.For sensitivity investigation for systems given by block-diagrams, the theoryof parameter-dependent generalized functions is used. All this taken in theaggregate leads to the possibility to obtain constructive solutions for a

number of problems that are only mentioned briefly in the monograph byProf. M. Eslami or are not considered at all. The following problems canbe placed in this group:

1. Sensitivity investigation of nonlinear non-stationary systems of gen-eral form, including discontinuous systems

2. Investigation of sensitivity with respect to initial conditions andparameters of exogenous disturbances

3. Investigation of sensitivity with respect to singular parameters, for

which sensitivity investigation based on the first approximation isnot correct

4. Sensitivity investigation for boundary-value problems, sensitivity of autonomous and non-autonomous oscillating processes

5. Estimation of a norm of additional motion over finite and infinitetime intervals on the basis of Lyapunov’s functions

6. A detailed investigation of sensitivity invariants

7. Sensitivity investigation for mathematical programming and calcu-lus of variations problems

8. Statement and solution of a number of applied direct and inverseproblems of sensitivity theory

The goal of the present book is to compensate, in a sense, for the afore-mentioned deficiency, and to expose, as rigorous and completely as possible,foundation of sensitivity theory as an independent and clear-cut branch of control theory that has, at the same time, organic relations with numerous

adjacent disciplines. Such a concept of the book did not allow to us present




many specific but often fairly attractive investigation methods. Neverthe-less, the authors hope that the book will contribute to strengthening thegeneral basis for such investigations and for solving a number of new prob-lems.

In this book, the problems of sensitivity theory of real systems and theirmathematical models are formulated from general positions, and connec-tions between sensitivity and choice of technical parameters of control sys-tem have been established. Much attention is paid to theoretical substanti-ation of sensitivity investigation methods for finite-dimensional continuous-time systems. For such systems, relations between sensitivity problems andclassical stability problems is established, rigorous mathematical conditionsare given for applicability of the first approximation, and some critical casesare considered where the use of the first approximation gives qualitativelywrong results.

For the finite-dimensional case, a general theory of sensitivity investiga-tion of boundary-value problems is presented. Elements of the above theoryare employed for sensitivity analysis of solutions of nonlinear programmingand variational calculus, as well as for analysis of sensitivity of oscillatingprocesses.

Sensitivity investigation methods for discontinuous systems, includingthose given by operator models, are considered from general positions. It isshown that, in the latter case, theory of parameter-dependent generalizedfunctions provides a natural method for solving sensitivity problems.

Much attention is paid to substantiation and generalization of sensitiv-ity investigation methods for non-time characteristics of control systems,viz.: transfer functions, frequency responses, zeros and poles of transferfunctions, eigenvalues and eigenvectors of systems matrices and integral es-timates. Methods for obtaining indirect characteristics of sensitivity func-tions are also given.

As in network theory, sensitivity invariants are introduced, i.e., spe-cial functional dependencies between sensitivity functions, that facilitatetheir determination and investigation. Methods and techniques are givenfor derivation of linear sensitivity invariants for time-domain, frequency-

domain and other characteristics of control systems.Generalization of some applications of sensitivity theory is presented from

the viewpoint of direct and inverse problems.

The present monograph is addressed to scientific workers and engineersspecializing in the field of theoretical investigation, design, testing, adjust-ment, and exploitation of various types of control systems and their units.It can also be useful for graduate and post-graduate students working incorresponding fields.

The methods of sensitivity investigation presented in the book may ap-

pear quite useful in various applications other than control problems, e.g.,




in stability theory, oscillation theory, motion dynamics, mathematical eco-nomics and many others.

Generally speaking, it should be noticed that the primary sections of the monograph by Prof. M. Eslami and the present book actually do not

intersect, so that these works complement each other. The authors hopethat their book can contribute to English-speaking literature on sensitivitytheory and will provide for an instrument for solving new theoretical andapplied problems.

The bibliography given in the book does not pretend to be complete andcontains mostly monographs, major textbooks, proceeding of conferencesand symposia on sensitivity theory, as well as works immediately used inthe text.

Chapters 1–4 are written by E. N. Rosenwasser, Chapters 5–8 by R. M.Yusupov. The general idea and concept of the monograph belong to both

the authors. The material of Section 6.5 was written together with Yu. V.Popov, and Section 8.1.6 together with V. V. Drozhin. The authors aregrateful to Prof. P. D. Krut’ko for his useful comments on improving themonograph.

The authors are particularly grateful to Dr. K. Polyakov who translatedthe book into English, and to post-graduate student Yu. Putintseva fortechnical preparation of the manuscript.

Special thanks are due to Prof. M. Eslami who contributed much toappearance of the book in English. We are also grateful to Mrs. NoraKonopka from CRC Press for exellent organization of the publishing process

despite of large geographic distance between Russia and the USA.Y. Rosenwasser and R. Yusupov

20th March 1999




Contents

1 Parametric Models

1.1 State Variables and Control Systems Parameters1.1.1 General Principles of Control

1.1.2 Directed Action Elements

1.1.3 State Variables and Parameters

1.1.4 Sensitivity and Problem of Technical Parameters

Selection

1.2 Parametric Models of Control Systems

1.2.1 Mathematical Model of Control System

1.2.2 Parametric System Model

1.2.3 Determining Sets of Parameters

1.2.4 Problem of Sensitivity of Parametric Model

1.3 Sensitivity Functions and Applications

1.3.1 Sensitivity Functions

1.3.2 Main and Additional Motions

1.3.3 Analysis of First Approximation

1.3.4 Statement of Optimization Problems with Sensitivity

Requirements

2 Finite-Dimensional Continuous Systems

2.1 Finite-Dimensional Continuous Systems Depending on a

Parameter

2.1.1 Mathematical Description

2.1.2 Parametric Dependence of Solutions on Finite Time

Intervals

2.1.3 Calculation of Derivatives by Parameters

2.1.4 Parametric Models and General Sensitivity Equations

2.1.5 Sensitivity Equations of Higher Orders

2.1.6 Multiparameter Case

2.1.7 Analytical Representation of Single-Parameter

Family of Solutions

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2.1.8 Equations of Additional Motion

2.1.9 Estimation of First Approximation Error

2.2 Second Lyapunov’s Method in Sensitivity Theory

2.2.1 Norms of Finite-Dimensional Vectors and Matrices

2.2.2 Functions of Constant and Definite Sign

2.2.3 Time-Dependent Functions of Constant and Definite

Sign

2.2.4 Lyapunov’s Principle

2.2.5 Norm of Additional Motion

2.2.6 Parametric Stability

2.2.7 General Investigation Method

2.2.8 Sensitivity of Linear System

2.3 Sensitivity on Infinite Time Intervals

2.3.1 Statement of the Problem

2.3.2 Auxiliary Theorem

2.3.3 Sufficient Conditions of Applicability of First

Approximation

2.3.4 Classification of Special Cases

2.4 Sensitivity of Self-Oscillating Systems in Time Domain

2.4.1 Self-Oscillating Modes of Nonlinear Systems

2.4.2 Linear Differential Equations with Periodic Coefficients

2.4.3 General Properties of Sensitivity Equations

2.4.4 Sensitivity Functions Variation over Self-Oscillation

Period

2.4.5 Sensitivity Functions for Periodicity Characteristics

2.4.6 Practical Method for Calculating Sensitivity Functions

2.4.7 Application to Van der Paul Equation

2.5 Sensitivity of Non-Autonomous Systems

2.5.1 Linear Oscillatory Systems

2.5.2 Sensitivity of Linear Oscillatory System

2.5.3 Sensitivity of Nonlinear Oscillatory System2.6 Sensitivity of Solutions of Boundary-Value Problems

2.6.1 Boundary-Value Problems Depending on Parameter

2.6.2 Sensitivity Investigation for Boundary-Value Problems

2.6.3 Implicit Functions Theorems

2.6.4 Sensitivity Functions of Solution of Boundary-Value

Problems

2.6.5 Sensitivity of Non-Autonomous Oscillatory System

2.6.6 Sensitivity of Self-Oscillatory System

2.6.7 Boundary Conditions for Sensitivity Functions

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3 Finite-Dimensional Discontinuous Systems

3.1 Sensitivity Equations for Finite-Dimensional Discontinuous

Systems

3.1.1 Time-Domain Description

3.1.2 Time-Domain Description of Relay Systems3.1.3 Parametric Model and Sensitivity Function of

Discontinuous System

3.1.4 General Sensitivity Equations for Discontinuous

Systems

3.1.5 Case of Continuous Solutions

3.2 Sensitivity Equations for Relay Systems

3.2.1 General Equations of Relay Systems

3.2.2 Sensitivity Equations for Systems with Ideal Relay

3.2.3 Systems with Logical Elements

3.2.4 Relay System with Variable Delay3.2.5 Relay Extremal System

3.2.6 System with Pulse-Frequency Modulation of First Kind

3.3 Sensitivity Equations for Pulse and Relay-Pulse Systems

3.3.1 Pulse and Relay-Pulse Operators

3.3.2 Sensitivity Equations of Pulse-Amplitude Systems

3.3.3 Sensitivity of Pulse-Amplitude Systems with Respect

to Sampling Period

3.3.4 Sensitivity Equations of Systems with Pulse-Width

Modulation

4 Discontinuous Systems Given by Operator Models

4.1 Operator Parametric Models of Control Systems

4.1.1 Operator Models of Control Systems

4.1.2 Operator of Directed Action Element

4.1.3 Families of Operators

4.1.4 Parametric Properties of Operators

4.1.5 Parametric Families of Linear Operators

4.1.6 Transfer Functions and Frequency Responses of

Linear Operators

4.1.7 Parametric Operator Model of System

4.2 Operator Models of Discontinuous Systems

4.2.1 Generalized Functions

4.2.2 Differentiation of Generalized Functions

4.2.3 Multiplication of Generalized Functions

4.2.4 Operator Equation of Open-Loop Linear System

4.2.5 Operator Equation of Closed-Loop Linear System

4.3 Sensitivity of Operator Models

4.3.1 Generalized Functions Depending on Parameter

4.3.2 Generalized Differentiation by Parameter

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4.3.3 Sensitivity Equations

4.3.4 Sensitivity Equations for Multivariable Systems

4.3.5 Higher-Order Sensitivity Equations

4.4 Sensitivity Equations for Relay and Pulse Systems

4.4.1 Single-Loop Relay Systems4.4.2 Pulse-Amplitude Systems

4.4.3 Pulse-Width Systems

4.4.4 Pulse-Frequency Systems

5 Non-Time Characteristics

5.1 Sensitivity of Transfer Function and Frequency Responses of

Linear Systems

5.1.1 Sensitivity of Transfer Function

5.1.2 Sensitivity of Frequency Responses

5.1.3 Relations between Sensitivity Functions of FrequencyCharacteristics

5.1.4 Universal Algorithm for Determination of Sensitivity

Functions for Frequency Characteristics

5.1.5 Sensitivity Functions for Frequency Characteristics of

Minimal-Phase Systems

5.1.6 Relations between Sensitivity Functions of Time and

Frequency Characteristics

5.1.7 Relations between Sensitivity Functions of Open-

Loop and Closed-Loop Systems

5.1.8 Sensitivity of Frequency-Domain Quality Indices5.2 Sensitivity of Poles and Zeros

5.2.1 General Case

5.2.2 Sensitivity of the Roots of a Polynomial

5.2.3 Sensitivity of Poles and Zeros for Open-Loop and

Closed-Loop Systems

5.2.4 Relations between Sensitivity of Transfer Function

and that of Poles and Zeros

5.3 Sensitivity of Eigenvalues and Eigenvectors of Linear Time-

Invariant Systems

5.3.1 Eigenvalues and Eigenvectors of Matrices

5.3.2 Sensitivity of Eigenvalues

5.3.3 Sensitivity of Real and Imaginary Parts of Complex

Eigenvalues

5.3.4 Sensitivity of Eigenvectors

5.3.5 Sensitivity Coefficients and Vectors of Higher Orders

5.3.6 Sensitivity of Trace and Determinant of Matrix

5.4 Sensitivity of Integral Quality Indices

5.4.1 Integral Estimates

5.4.2 Sensitivity of Integral Estimate I 0

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5.4.3 Sensitivity of Quadratic Estimates. Transformation

of Differential Equations

5.4.4 Sensitivity of Quadratic Estimates. Laplace

Transform Method

5.4.5 Sensitivity Coefficients of Integral Estimates forDiscontinuous Control Systems

5.5 Indirect Characteristics of Sensitivity Functions

5.5.1 Preliminaries

5.5.2 Precision Indices

5.5.3 Integral Estimates of Sensitivity Functions

5.5.4 Envelope of Sensitivity Function

6 Sensitivity Invariants

6.1 Sensitivity Invariants of Time Characteristics

6.1.1 Sensitivity Invariants

6.1.2 Existence of Sensitivity Invariants

6.1.3 Sensitivity Invariants of Single-Input–Single-Output

Systems

6.1.4 Sensitivity Invariants of SISO Nonlinear Systems

6.1.5 Sensitivity Invariants of Multivariable Systems

6.1.6 Sensitivity Invariants of Weight Function

6.2 Root and Transfer Function Sensitivity Invariants

6.2.1 Root Sensitivity Invariants

6.2.2 Sensitivity Invariants of Transfer Functions

6.3 Sensitivity Invariants of Frequency Responses

6.3.1 First Form of Sensitivity Invariants of Frequency

Responses

6.3.2 Second Form of Sensitivity Invariants of Frequency

Responses

6.3.3 Relations between Sensitivity Invariants of Time and

Frequency Characteristics

6.4 Sensitivity Invariants of Integral Estimates

6.4.1 First Form of Sensitivity Invariants

6.4.2 Second Form of Sensitivity Invariants

6.5 Sensitivity Invariants for Gyroscopic Systems

6.5.1 Motion Equations and Transfer Functions

6.5.2 Sensitivity Invariants of Amplitude Frequency

Response

6.5.3 Sensitivity Invariants of Integral Estimates

6.5.4 Sensitivity Invariants of Damping Coefficient

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7 Sensitivity of Mathematical Programming Problems

7.1 Sensitivity of Linear Programming Problems

7.1.1 Actuality of Investigation of Optimal Control

Sensitivity

7.1.2 Linear Programming7.1.3 Qualitative Geometric Sensitivity Analysis

7.1.4 Quantitative Sensitivity Analysis

7.2 Sensitivity of Optimal Solution to Nonlinear Programming

Problems

7.2.1 Unconstrained Nonlinear Programming

7.2.2 Nonlinear Programming with Equality Constraints

7.2.3 Sensitivity Coefficients in Economic Problems

7.2.4 Nonlinear Programming with Weak Equality

Constraints

7.2.5 Sensitivity of Convex Programming Problems7.3 Sensitivity of Simplest Variational Problems

7.3.1 Simplest Variational Problems

7.3.2 Existence Conditions for Sensitivity Function


7.4 Sensitivity of Variational Problems

7.4.1 Variational Problem with Movable Bounds

7.4.2 Existence Conditions for Sensitivity Functions


7.4.4 Case Study

7.4.5 Variational Problem with Corner Points

7.5 Sensitivity of Conditional Extremum Problems

7.5.1 Variational Problems on Conditional Extremum

7.5.2 Lagrange Problem

7.5.3 Variational Problem with Differential Constraints

7.5.4 Sensitivity of Isoperimetric Problem

8 Applied Sensitivity Problems

8.1 Direct and Inverse Problems of Sensitivity Theory

8.1.1 Classification of Basic Applied Sensitivity Problems

8.1.2 Direct Problems

8.1.3 Inverse Problems and their Incorrectness

8.1.4 Solution Methods for Inverse Problems

8.1.5 Methods for Improving Stability of Inverse Problems

8.1.6 Investigation of Convergence of Iterative Process

8.2 Identification of Dynamic Systems

8.2.1 Definition of Identification

8.2.2 Basic Algorithm of Parametric Identification Using

Sensitivity Functions

8.2.3 Identifiability and Observability




8.3 Distribution of Parameters Tolerance

8.3.1 Preliminaries

8.3.2 Tolerance Calculation Problem

8.3.3 Initial Mathematical Models

8.3.4 Tolerance Calculation by Equal Influence Principle8.3.5 On Tolerance Distribution with Account for Economic

Factors

8.3.6 On Requirements for Measuring Equipment

8.4 Synthesis of Insensitive Systems

8.4.1 Quality Indices and Constraints

8.4.2 Problems of Insensitive Systems Design

8.4.3 On Design of Systems with Bounded Sensitivity

8.4.4 On Design of Optimal Insensitive Systems

8.5 Numerical Solution of Sensitivity Equations

8.5.1 General Structure of Numerical Integration Error8.5.2 Formula for Integration of Sensitivity Equations

8.5.3 Estimates of Solution Errors

8.5.4 Estimates of Integration Error for a First-Order System

8.5.5 Results of Numerical Calculation

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Chapter 1

Parametric Models of Control Systems and Statement of the Problem in Sensitivity Theory

1.1 State Variables and Control Systems Parameters

1.1.1 TGeneral Principles of Control

The notion of control always implies a plant to which the control isapplied, and a target that is to be attained using this control. In practice,

control process is realized by means of specially formed exogenous actionsapplied to the plant. These actions are intended to change some propertiesof the latter. This idea is illustrated in Figure 1.1, where O is a multivariableplant, Y is a vector of generalized coordinates that define properties of theplant, and U is a vector of control actions.

Figure 1.1

Block-diagram of control process

Denote by M Y the set of values of Y for which properties (state) of theplant satisfy the requirements, and by RY the set of values of Y that can

really take place during plant operation. Then, the target of the control




can be defined by the relation

RY ⊂ M Y . (1.1)

If equation (1.1) is satisfied for the plant without any interference fromoutside, control process becomes superfluous. Control is necessary if theplant is affected by factors (we will call them disturbances ) that lead toviolation of (1.1)1.

In practice control action U is realized by a system called controller thatis, in the general case, functionally connected with the state of the plantand disturbances applied to it. As a result, we obtain a more detailedcontrol scheme (Figure 1.2), where P is a controller and V o and V p are

Figure 1.2

General block-diagram of control

disturbances applied to the plant and control, respectively. According toFigure 1.2, the controller P is to provide equation (1.1) in the presence of disturbances V o applied to the plant and V p infringing controller actions.The block-diagram shown in Figure 1.2 is the most general and, therefore,the least substantial. Schemes of actual control systems, and still moretheir realizations, are very diversified and are determined by a number of factors. The main factors are: physical nature of the plant; the choice of the

vector Y , which gives the state of the plant; the set M Y of admissible statesof the plant; availability of information about variations of plant propertiesduring the control process; availability of information about vectors V o andV p, etc.

If the information about the plant and corresponding disturbances is suf-ficiently complete, we can, as a rule, employ the classic block-diagram of combined control shown in Figure 1.3, where N V o , N V p , and N Y are sen-sors of corresponding values, and U is an amplifier-transducer of controller

1

Hereinafter such a notation is applied to equations.




Figure 1.3

Block-diagram of combined control

signals. In such systems, sufficient information about the plant and vectorV o and V p allows us, as a rule, to choose a fixed controller that provides foradmissible performance of the system as a whole.

Other situations arise when there is no sufficient information about theplant and vectors V o and V p, or it can substantially vary during systemoperation. In this case, it can hardly be supposed that a controller with a

fixed structure will successfully cope with its functions in all possible situ-ations. Therefore, it is necessary to change controller properties dependingon available information about the plant and disturbance during systemoperation. Then, the generalized block-diagram takes the form shown inFigure 1.4, where I o, I V o and I V p are identifiers of properties of the plantand vectors V o and V p, respectively.

1.1.2 Directed Action Elements

The block-diagrams shown above illustrate general principles of controland bear generalized character. In practice, any control system can berepresented in the form of a detailed block-diagram, which is determinedby system elements and their physical nature, as well as by their interaction.

A widely used method of structural representation of control systemsconsists in splitting the system onto directed action elements that inter-act on the basis of three basic kinds of connections: series, parallel, andfeedback.

By directed action element we usually mean a physical plant having a

vector input X and vector output Y of an arbitrary nature. It is assumed




Figure 1.4

Block-diagram of control system

that the input X affects the output Y in some sense, but there are noreverse influence of Y on X . The relation between Y and X can be writtenin a symbolic form as

Y = L(X, V ), (1.2)

where it is additionally assumed that the output Y is affected by somedisturbances acting on the element. The block-diagram corresponding to(1.2) is shown in Figure 1.5.

Figure 1.5

Directed action element

Assume that the system under consideration can be represented as an




assembly of directed action elements

Y i = Li(X i, V i), i = 1, . . . , n , (1.3)

connected by some links. These links in the general case can also be repre-sented as directed action elements defined by symbolic relations

X i = S i(Y 1,...,Y n, V i), i = 1, . . . , s , (1.4)

where V i are disturbances acting on the links.

Equations (1.3) and (1.4) taken in the aggregate reflect performance in-terrelations of separate elements of the system and simultaneously deter-mine some structural representation of the system. Thus, an arbitrary

structure constructed from directed action elements and directed links gen-erates a set of symbolic equations (1.3) and (1.4). On the other hand, anyset of equations of the form (1.3)–(1.4) can be associated with a controlsystem of a definite structure. In some cases, Li and S i vary themselvesduring operation. This circumstance can be taken into account by augment-ing (1.3)–(1.4) by some additional equations. Consider, as an example, ageneral system of symbolic equations describing the performance of theadaptive control system shown in Figure 1.4. According to Figure 1.4, forthe plant we have

Y = L0(U, V 0), (1.5)

which indicates a link between state vector of the plant, control action anddisturbances acting on the plant. In the general case, for the controller wehave

U = Lp(Y , V 0, V p, V p), Lp = Lp(L0, V 0, V p), (1.6)

These equations determine control action applied to the plant. The argu-ment L0 in the second equation in (1.6) indicates the fact that controller

performance should, in the general case, change when plant performancechanges. Therefore, the second equation characterizes the possibility of controller adaptation. Moreover, equations

L0 = L0(L0), V 0 = V 0(V 0, L0), V p = V p(V p) (1.7)

describe identification processes for the plant and disturbances applied tothe system.

Note that the material given in this section do not claim to be rigorous

and complete and has a descriptive character.




1.1.3 State Variables and Parameters

Assume that the system under investigation is split into directed actionelements according to some principle, and its structure (1.3)–(1.4) has beenfixed. Then, the values X i and Y i taken in the aggregate will be calledcharacteristics of state variables of the system associated with the structuredefined by (1.3)–(1.4). The variables V i and V i characterize exogenousdisturbances applied to the system.

It should be emphasized that the same system can be described by varioussets of state variables depending on specification of the structural represen-tation (1.3)–(1.4). As a rule, a relatively small number of controllable andmeasurable values which gives basic information on system performance arechosen as state variables. Moreover, usually only some exogenous distur-bances that affects the system substantially are taken into account.

On the other hand, any set of state variables and exogenous disturbances,whichever wide, does not describe completely all properties of the system.This is explained by the fact that properties and performance of actualsystems depends on many extra factors (often uncontrollable), which specifyfeatures of any individual exemplar of the system and operating conditions.Various factors of this kind will hereinafter be called parameters .

The whole set of parameters that determine properties of a technicalsystem can usually be divided into two following groups: 1) technical pa-rameters, and 2) parameters of environment and operating conditions.

By technical parameters we shall mean values that determine the differ-

ence between individual exemplars of the system under the same operatingconditions. By environmental parameters we mean the difference betweenoperating conditions of individual exemplars of the system. For example,image quality of two TV sets of the same type that work in the same roomand use the same power supply will be different due to different technicalparameters. But two practically identical TV sets can show different imagequality in various operating conditions, for instance, for different tempera-ture or air humidity in the room.

Using the system symbolic description given by (1.5)–(1.6), technicalparameters are various factors affecting properties of the relations L

0and

Lp for individual exemplars of the system. Operating condition parametersvariation can manifest themselves in changing properties of the disturbancesV o and V p and in variation of initial conditions.

Technical parameters of a system include sizes of various elements andprecision of their manufacture, physical properties of materials as well assome other values determining its functional implementation. Parametersof operating conditions depend on physical nature of system elements andcan be diversified. In principal, parameters can assume constant values foreach realization of the system or can be variable. Hereinafter in the book

we consider mostly the first case.




1.1.4 Sensitivity and Problem of Technical ParametersSelection

Denote by αi, β i, γ i and δ i vectors of parameters characterizing sys-

tem properties. Then, equations (1.3)–(1.4) can be represented in a moredetailed form

Y i = Li[X i, αi, V i(γ i)], X i = S i[Y 1,...,Y n, β i, V i(δ i)]. (1.8)

If relations Li and S i are fixed, properties of the system are determinedby the choice of technical parameters αi and β i and parameters of operatingconditions γ i and δ i.

Denote by αT the set of all technical parameters of the system (1.8), and

by αE the set of all parameters of operating conditions. Moreover, combineall state variables Y i and X i into a single vector Y . Then, under otherequal conditions, finally we obtain

Y = Y (αT , αE ), (1.9)

where Y is the set of state variables of the system.

Thus, variation of parameters αT and αE causes variation of state vari-ables that reflect essential system properties.

The ability of a system to change its properties under variation of pa-rameters αT and αE is called (parametric) sensitivity .

As follows from (1.9), the conditions of normal operation for the sys-tem (1.1) can be represented in the form

RY (αT , αE ) ⊂ M Y , (1.10)

Therefore, to assure system workability, technical parameters must be cho-

sen taking into account parameters of operation conditions.Let possible operation conditions (value of the vector αE ) be defined by

αE ∈ M E , (1.11)

where M E is a given set. Then, the traditional problem statement of thechoice of technical parameters αT reduces to obtaining vectors αT such that

RY (αT , αE ) ⊂ M Y for any αE ∈ M E . (1.12)




The set of values of technical parameters for which conditions (1.12) holdwill be called workability area and denoted by M p. Thus, if the condition

αT ∈ M p (1.13)

holds, this means that conditions (1.12) hold as well.

Let α1T and α2T be two fixed values of the technical parameters vectorsatisfying the workability condition (1.13). From the viewpoint of satis-fying conditions (1.12) we may equivalently adopt either α1T or α2T asdesign values. Nevertheless, accounting for system sensitivity with respectto technical parameters, this conclusion can be erroneous. For various tech-nological and physical reasons actually we will always have

αT = α1T + ∆α1 , αT = α2T + ∆α2 ,

where vectors δα1 and δα2 characterize inevitable parameters variationand belong to some sets N 1 and N 2. This means that choosing αT =α1T or αT = α2T as initial (nominal) value (Figure 1.6), we actually will

Figure 1.6

Workability area and parameters variation

have areas Q1(α1T ) and Q2(α2T ) in parameter space that can be far from

equivalent. Figure 1.6 demonstrates that the choice αT = α1T ensures




workability conditions despite inevitable parameters variation, while thechoice αT = α2T does not. Therefore, a realistic statement of the problemof setting technical parameters must definitely take into account parametricsensitivity of the system. It should be emphasized that the above reasoning

does not assume that the variations δα1 and δα2 are small, and bears ageneral meaning.

1.2 Parametric Models of Control Systems

1.2.1 Mathematical Model of Control System

One of the most important problems arising in design of a control systemconsists in predicating its probable properties. The main method of suchprediction is the use of physical and mathematical models. Hereinafter,if it is not specified, by models of systems and elements we will meanmathematical ones.

Mathematical model of a directed action element (1.3) has the form

Y iM = LiM (X iM , V iM ), (1.14)

where LiM is a computational algorithm or formula, and X iM and V iM are

arguments. Moreover, it is assumed that the argument X iM reproduces ina sense the input X i of the element, while V iM and Y iM reflect exogenousdisturbances V i and the output Y i, respectively.

Thus, the model of a directed action element includes a model of therelation Li from (1.3), models of input and output values X i and Y i, re-spectively, and that of disturbances V i.

In a similar way, actual connections (1.4) can be associated with mathe-matical models

X iM = S iM (Y 1M ,...,Y nM , V iM ). (1.15)

Relations (1.14) and (1.15) taken in the aggregate specify a mathemat-ical model of an initial system, which reproduces in a sense the structureof the system. Moreover, the set of the values X iM and Y iM simulatescorresponding state variables.

1.2.2 Parametric System Model

Evidently, a sufficiently complete mathematical model of an element (ora system) must reflect its basic properties, including possible dependence

on parameters. Formally, dependence of properties of a directed action




element (1.3)–(1.4) on some independent parameters can be taken into ac-count by introducing some auxiliary variables into relations (1.14)–(1.15).Then, similarly to (1.8), instead of (1.14)–(1.15) we will have

Y iM = LiM [X iM , αiM , V iM (γ iM )],X iM = S iM [Y 1M ,...,Y nM , β iM , V iM (δ iM )],

(1.16)

where αiM , β iM , γ iM and δ iM are vecors of auxiliary variables (parameters).It is assumed that the variables αiM , β iM , γ iM and δ iM are independent of LiM , S iM , X iM , Y iM , V iM and V iM .

Relations (1.16) will be called parametric model of the system (1.3)–(1.4).As will be shown later, in practice there is much more freedom in construct-ing parametric models. In special cases, model parameters may include

some characteristics of exogenous disturbances, operation conditions, aswell as various variables that have no direct physical meaning but are in-cluded in parametric model (1.16), for example, coefficients of correspond-ing differential equations. Therefore, the term “parameter” has differentmeaning for initial actual system and its model. For an actual system,as a rule, parameter is a value that has a definite physical sense and re-flects its objective properties. On the other hand, by a parameter of amathematical model we can mean any independent variable included in itsparametric model (1.16). This more general notion of parameter makes itpossible to consider a wider spectrum of problems, because it is possible

to investigate the influence on the system of many diversified factors ig-noring their physical nature. Nevertheless, model parameters are usuallychosen, when possible, in some correspondence with actual parameters andoperation conditions of the system under investigation.

Hereinafter we consider, in fact, only problems connected with investiga-tion of sensitivity of mathematical models. Therefore, the subscript “M ”with corresponding arguments is omitted in (1.16) and similar relations.

1.2.3 Determining Sets of Parameters

Combining all the parameters included in (1.16) into a single vector α,equation (1.16) can be written in the form

Y i = Li(X i, α), X k = S k(Y 1,...,Y n, α),i = 1, . . . , n, k = 1, . . . , s .

(1.17)

Thus, any parametric model is a system of equations that determinesa chosen set of state variables and depends on a set of independent vari-

ables (parameters). In principle, depending on the nature of the problem




at hand, some parameters may be fixed while others be considered as vari-able. By determining set of parameters for a given problem we will meana chosen finite set of parameters of the associated parametric models thatare considered as independent variables.

Let, for example, the motion of a real pendulum be described by thefollowing mathematical model:

J y + 2ny + k2y = q 1x1(t) + q 2x2(t), (1.18)

where y is the angle of deviation of the pendulum from the vertical; J , nand k are constants (product of inertia, damping coefficient, and hangerrigidity); q 1 and q 2 are constants, and x1(t) and x2(t) are given functions.In the model (1.18), the value y is a state variable, and J , n, k, q 1 and q 2

are parameters. In the general case, motion of the pendulum for t ≥ 0 canbe described on the basis of the model (1.18) by the following relation:

y(t) = y(t, y0, y0,J,n,k,q 1, q 2), (1.19)

where y0 and ˙ y 0 are initial deviation and initial angular velocity, which arealso model parameters.

Assume that initial conditions and constructive parameters are consid-ered to be constant, and let us investigate the influence of various exogenousdisturbances belonging to a given class. Then, the determining set of pa-

rameters consists of q 1 and q 2, and the dependence

y = y(t, q 1, q 2). (1.20)

will be of interest.

If constructive parameters and exogenous disturbances can be consideredas fixed, then initial conditions will be determining, so that

y = y(t, y0(0), y0(0)). (1.21)

In the general case, the choice of a determining set of parameters corre-sponds, from the mathematical viewpoint, to the choice of some family of nonlinearly similar mathematical models. From the physical viewpoint, itcorresponds to the choice of a family of actual systems with similar phys-ical properties and operation conditions. For example, the choice of someset of constructive parameters to be determining corresponds to some setof systems with similar constructive implementation. If the determiningset of parameters characterizes influence of disturbances, we consider a set

of identical systems functioning under different operation conditions. In




principle, a successful choice of a determining set of parameters makes itpossible to describe mathematically and investigate diversified propertiesof actual physical systems.

Let us define an important property of a determining set of parameters,

called completeness .

DEFINITION 1.1 A determining set of parameters α is called com-plete if it uniquely determines all state variables of the model under inves-tigation.

For a system defined by (1.17) that there are the following unique de-pendencies:

Y i = Y i(t, α), X k = X k(t, α). (1.22)

For example, for the system of equations for pendulum, the set of pa-rameters appearing in the right side of (1.19) is complete. If some of theseparameters are fixed and assume constant values, the set of all remainingparameters is complete. It should be noted that the complete set of pa-rameters is not unique. For example, if we pass to a new set of parametersin (1.17) using a bi-unique functional relation, the new set of parametersobtained in this way is, evidently, also complete. In practice, the ques-tion about completeness of a chosen set of parameters is not always simpleand is connected with existence and uniqueness of solution of equations

constituting the mathematical model.

1.2.4 Problem of Sensitivity of Parametric Model

Let Equation (1.17) determine a chosen parametric model of the system,and let α be a complete determining set of parameters. Assume that M α isthe set of admissible values of parameters belonging to α. Denote by M Y the set of all laws of variation of the vector consisting of variables Y i and X i,which satisfy admissible operation conditions. Then, in analogy with (1.10),the condition of proper choice of parameter vector can be represented in

the formRY (t, α) ∈ M Y , α ∈ M α. (1.23)

It should be noted that in this section we consider the sensitivity problemfor a parametric model of the system. Evidently, it is implicitly assumedthat the mathematical model corresponds to the actual system. It is nec-essary to bear in mind that model parameters are considered here indepen-dently of their physical meaning. Therefore, variation of model parameterscan be associated either with variation in technical parameters of initial sys-tem or with variation in operating conditions. It is convenient to represent

the conditions (1.23) of normal operation of the system in another form.




Let J = (J 1, . . . , J m) be a set of some quality indices, which are functionalsdefined over the set of state variables. If M J is the set of admissible valuesof the vector J with components J 1, . . . , J m, instead of (1.23) we can write

J (α) ∈ M J , α ∈ M α. (1.24)

Relation (1.23) includes the case when the value of one of the indices,say J 1, is chosen to be maximal, so that

J 1 = J 1max (1.25)

The simplest problem connected with system analysis reduces to finding atleast one vector α for which (1.23) or (1.24) hold. Taking system parame-

ters equal to α, we could expect that the system will operate satisfactorilyunder corresponding design conditions, provided that the correspondencebetween the system and its parametric model is sufficiently good. Never-theless, this by no means ensures that the system will perform satisfactorilyunder real operation conditions, because in practice, the set of parametersα differs from the design parameters for various physical and technologicalreasons, and this leads to a deviation of operation conditions from nominalconditions. This property of a real system manifests itself in changing mod-el state variables Y i and X i under parameters variation. This property of a model will be called sensitivity with respect to fluctuations of the chosen

determining group of parameters .If the problem is stated correctly, model sensitivity simulates, in a sense,

sensitivity of the initial system with respect to variation of technical andoperational parameters.

In most cases, it is required that the system (and, accordingly, mod-el) have low sensitivity (insensitivity) with respect to parameters varia-tion. From the practical viewpoint, if there is necessary correspondencebetween the system and model, this reduces to the requirement that condi-tions (1.23) or (1.24) hold not only for the design set of parameters α = α,but for all (or almost all) sets of parameters that can take place. Analysisof system sensitivity on the basis of its model reduces just to establishingthis fact.

In principle, requirements for system sensitivity must be taken into ac-count as early as during its design. System structure and elements opera-tors should be chosen in such a way that conditions (1.23) or (1.24) holdover the whole range of possible parameters variation. This approach leadsto the problem of system design taking into account requirements for itssensitivity (insensitivity).

It can easily be seen that the sensitivity problem formulated above is

ideologically close to that of stability when possible system parameter vari-




ations are considered as disturbing factors. As will be shown below, there isa definite formal interconnection between the sensitivity and stability prob-lems. Nevertheless, from the practical viewpoint, sensitivity problems aremuch more diversified and have richer subject matter than routine stability

problems. Sensitivity investigation makes it possible to track interdepen-dence between quality indices and most diversified circumstances connectedwith peculiarities of system design, implementation, and operation.

1.3 Sensitivity Functions and Applications

1.3.1 Sensitivity FunctionsThe simplest (from the ideological viewpoint) method of sensitivity anal-

ysis consists in numerical investigation of system parametric model over thewhole range of variation of the determining set of parameters. In practice,as a rule, the use of this approach appears to be inexpedient or impossibledue to the huge amounts of required computations and obtained results.

The main investigation method in sensitivity theory consists in using so-called sensitivity functions. Let us introduce corresponding definitions. Letα1, . . . , αm be a set of parameters constituting a complete set α. Moreover,

let state variables Y i, (i = 1, . . . , n) and quality indices J 1, . . . , J s be single-valued functions of parameters α1, . . . , αm, i.e.,

Y i(t, α) = Y i(t, α1,...,αm), i = 1(1)n, (1.26)

and

J i(α) = J i(α1,...,αm), i = 1(1)s. (1.27)

The whole set of state variables is denoted by Y i, (i = 1, . . . , n) for simplic-ity.

DEFINITION 1.2 The following partial derivatives

∂Y i(t, α)

∂αk

,J i(t, α)

∂αk

(1.28)

are called first-order sensitivity functions of the values Y i and J i with respect

to corresponding parameters.




Thus, first-order sensitivity functions are derivatives of various state vari-ables and quality indices with respect to parameters of the determininggroup.

Note that in automatic control literature first-order sensitivity functions

are often called simply “sensitivity functions.” We also use this terminology.

DEFINITION 1.3 Partial k-th order derivatives of the values Y i and J i with respect to arguments α1, . . . , αm

∂ kY i

∂αk11 ...∂αkm

m

,∂ kJ i

∂αk11 ...∂αkm

m

, k1 + ... + km = k, (1.29)

will be called k-th order sensitivity functions with respect to corresponding parameters combinations.

Obviously, sensitivity functions of state variables Y i(i, α) depend on tand parameters α1, . . . , αm, while sensitivity functions of quality indicesdepend only on parameters α1, . . . , αm. It is assumed that the derivativesin (1.28) and (1.29) do exist. Definition 1.3 assumes also that sensitivityfunctions are independent on the order of differentiating by the respectiveparameters. For this to be true, it suffices to assume that [114] in a localityO of the point α(α1, . . . , αm) there exist all possible derivatives of the (k-

1)-th order as well as mixed derivatives of the k-th order, and all of thesederivatives are continuous.

As will be shown in the next chapters, sensitivity functions of various or-ders are solutions of special equations that can be obtained directly from aknown parametric model of a system. These equations are called sensitivity equations . The aggregate of initial mathematical model and auxiliary rela-tions determining sensitivity functions is usually called sensitivity model of the system under consideration .

1.3.2 Main and Additional Motions

Let α1, . . . , αm be a fixed set of parameters. This set α will be called themain or basic set . The chosen set α generates a set of state variables

Y i = Y i(t, α), (1.30)

which will be called main or basic motion . Basic motion is associated withbase quality indices

J i = J i(α). (1.31)




Varying the values of parameters

αi = αi + µi (1.32)

we obtain a new motion

Y i = Y i(t, α1 + µ1,..., αm + µm) = Y i(t, α + µ), (1.33)

which is associated with new values of quality indices

J i = J i(α1 + µ1, ..., αm + µm) = J i(α + µ). (1.34)

DEFINITION 1.4 The vector ∆Y i defined by the relation

∆Y i = Y i(t, α + µ)− Y i(t, α), (1.35)

will be called additional motion due to variation of parameters α1, . . . , αm.

Additional motion ∆Y i and corresponding increment of quality indices

∆J i = J i(α + µ)− J i(α) (1.36)

characterize variation of system properties, which are of interest to an in-vestigator, caused by variation of corresponding parameters. Therefore,investigation of additional motions and their relations to properties of ini-tial system is, in fact, the primary problem in sensitivity analysis.

Let us establish relations between sensitivity functions and additionalmotion. With this aim in view, we use properties of higher-order differen-tials for functions of several variables [114].

Let y = Y (x1, . . . , xr) be a function of variables x1, . . . , xr having con-tinuous partial derivatives of the k-th order with respect to all sets of ar-guments. Then the following expressions are called k-th order differentials :

d(k)Y =

∂

∂x1dx1 + ... +

∂

∂xrdxr

k

Y, (1.37)

where ∂/∂xi denotes differentiation with respect to the corresponding vari-able. To expand (1.37) we formally raise the expression inside the bracketsto the power and convolute the product of the operators according to therule

∂

∂xi

∂

∂xj=

∂ 2

∂xi∂xj, (1.38)




and then apply the obtained operator to the function Y .

As follows from (1.37) and (1.38), the k-th differential is a form of the k-thpower with respect to the values dx1, . . . , d xr. Assuming that all sensitivityfunctions up to the k-th order do exist and are continuous at the point α,

we can write Taylor’s formula with remainder term

∆Y (t, α) = Y (t, α + µ)− Y (t, α) =

= dY (t, α) + 12!d

(2)Y (t, α) + ... + 1n!d

(n)Y (t, α)+

+ 1(n+1)!d

(n+1)Y (t, α + θµ).

(1.39)

Notice that the variable t is a constant parameter in (1.39). Ignoring

the remainder term, we obtain the following approximate expression foradditional motion:

∆(k)Y (t, α =ki=1

1

i!d(i)Y (t, α)

α=α

, (1.40)

It will be called the k-th approximation for additional motion or, equiva-lently, approximation to within (k+1)-th order of smallness . The value

∆kY = 1k! d(k)Y (1.41)

will be called the k-th order correction of additional motion .

As follows from (1.37) and (1.41), the k-th order correction is given by

∆kY =1

k!

∂

∂α1µ1 + ... +

∂

∂αm

µm

kY (t, α)

α=α

. (1.42)

Hence, the k-th order correction is a linear form of k-th order sensitivity

functions calculated for the base parameter values, as well as a k-th orderform with respect to parameter increase µ1, . . . , µm. Note that if differen-tiability conditions hold, we can write equations for respective incrementsof quality indices similar to (1.39) and (1.40). By Taylor’s formula we have

∆J (t, α) = J (t, α + µ)− J (t, α) = dJ (t, α)+

+ 12!d

(2)J (t, α) + ... + 1n!d

(n)J (t, α)+

+1

(n+1)!d(n+1)

J (t, α + θµ).

(1.43)




Then, the k-th approximation takes the form

∆(k)J (α) =k

i=11

i!dJ (α)

α=α. (1.44)

1.3.3 Analysis of First Approximation

According to (1.37) and (1.40), the first approximation for additionalmotion can be written in the form

∆(1)Y (t, α) = U 1(t, α)µ1 + ... + U m(t, α)µm, (1.45)

where

U i = ∂Y (t, α)∂αi

α=α

, i = 1(1)m, (1.46)

are corresponding sensitivity functions.

As follows from (1.45), the first approximation is linear with respectto increments of parameters and sensitivity functions, therefore it is veryconvenient for analysis and is often used as an approximate representationof additional motion:

∆Y (t, α) ≈ ∆(1)Y (t, α). (1.47)

Evidently, correctness of (1.47) must be substantiated in any special caseby theoretical or experimental results. Yet another important property of the first approximation that is also instrumental in applied investigationsis its invariance with respect to functional transformations of the space of parameters.

Let

αi = αi(β 1, . . . , β q), i = 1(1)m, (1.48)

where β 1, . . . , β q are new parameters constituting a vector β , and, generallyspeaking, q = m. The right sides of (1.48) are assumed to be continuouslydifferentiable functions of all their arguments. Then, the first approxi-mation (1.45) preserves its form if by µ1, . . . , µm we mean the followingdifferentials (increments):

µi =

qj=1

∂αi

∂β jν j , i = 1(1)m, (1.49)

where ν 1, . . . , ν q are increments of parameters β 1, . . . , β q.




Thus, having a set of sensitivity functions with respect to a complete setof parameters α and using (1.45) and (1.49), it is possible to investigate theinfluence on additional motion of arbitrary parameters connected with αby (1.48). Therefore, it is expedient to choose the set α with the minimal

number of elements, because in this case, the number of required sensitivityfunctions (1.46) will be minimal.

Notice that higher-order approximations, as distinct from the first ap-proximation, are invariant with respect to transformation (1.48) if and on-ly if this transformation is linear. If the approximate equality (1.47) holdswith a sufficient precision, using (1.47) we can fairly simply analyze vari-ous properties of additional motion. For example, evaluating (1.47) with anorm, we obtain

∆

(1)

Y (t, α)≤

m

i=1

U i(t, α)

µi, (1.50)

which gives an approximate estimate of the norm of additional motion forany time instant.

Let now µ1, . . . , µm be dependent stochastic values with mathematicalexpectations M i and correlation matrix K = kij, i, j = 1, . . . , m, wherekii = di are variances of corresponding parameters. Then, mathematicalexpectation M [∆Y ] of additional motion is approximately equal to

M [∆Y ] ≈ M

∆(1)Y

=mi=1

U i(t, α)M i. (1.51)

The correlation matrix of the first approximation can also be easily cal-culated. With this aim in view, subtract (1.51) from (1.45):

∆(1)Y 0(t, α) =m

i=1

U i(t, α)µ0i = Z, (1.52)

Where the zero superscript denotes a centered component.

Transposing (1.52), we obtain

Z T (t) =

mi=1

U T i (t, α)µ0i , (1.53)

where superscript “T ” denotes the transpose operator.




Taking the product of (1.52) and (1.53), we obtain

Z (t1)Z T (t2) =m

i=1

m

j=1 U i(t1, α)U T j (t2, α)µ0i µ0

j =

=m

i=1m

j=1 U ij(t, α)µ0i µ

0j ,

(1.54)

where

U ij(t1, t2, α) = U i(t, α)U j(t, α). (1.55)

Averaging (1.54), we obtain

K Z (t, α) =mi=1

mj=1

U ij(t1, t2, α)kij , (1.56)

where K z(t1, t2, α) is the correlation matrix of the first approximation,which approximately determines the correlation matrix of additional plantmotion.

In a special case when random parameters µ1, . . . , µm are independent,formula (1.56) gets simplified:

K Z (t1, t2, α) =

mi=1

U ii(t1, t2, α)di, (1.57)

where di = K ii is the variance of parameter αi.

Denoting trace of the matrix K z(t,t, α) by Dz(t, α), i.e.,

DZ (t, α) = tr K Z (t,t, α), (1.58)

from (1.56) we obtain

DZ (t, α) = tr

mi=1

mj=1

U ij(t,t, α)kij . (1.59)

Here DZ (t, α) denote the sum of variances of components of the vec-tor ∆(1)Y 0(t, α), which give approximately the variances of correspondingcomponents of the vector of additional motion.

1.3.4 Statement of Optimization Problems withSensitivity Requirements

In the problems formulated above it was implicitly assumed that area

M α of possible values of parameters vector α is known. Nevertheless, in




many cases, such an assumption is not substantiated, and the investigatormust find a set of values of some parameters Rα that ensure conditions of normal operation (1.23) or (1.24). Solution of such a problem may precedeoptimization, when it is required to choose a vector α0 from the set Rα

such that condition (1.25) holds.Nevertheless, in many practical cases, spending a great deal of time and

resources on obtaining the results of the optimization problem appears to beinexpedient, because obtained optimal system is too sensitive to parametersvariation and, therefore, is practically unworkable. A method to overcomethis difficulty consists of a statement of the optimization problems takinginto account sensitivity requirements. Consider a general statement of theproblem.

According to (1.26) and (1.28), a classical optimization problem can beformulated as follows: it is required to find a vector α0 ∈ Rα, where Rα is

a given set of parameters, such that

J 1(α0) ≥ J (α), α ∈ Rα, Y (t, α0) ∈ M Y . (1.60)

Solution of the problem ensures optimality of the system only for α = α0.In reality we will always have

α = α0 + ∆α, (1.61)

where ∆α is parameters fluctuation belonging to a region ∆Rα, whichdepends on α0 in the general case.

Then, taking into account the necessity of satisfying (1.23), we must addconditions (1.60) by

Y (t, α + ∆α) ∈ M Y where ∆α ∈ ∆Rα. (1.62)

Instead of (1.62) we may also write

Y (t, α0) + ∆Y (t, α) ∈ M Y where ∆α ∈ ∆Rα, (1.63)

where ∆Y (t, α) = Y (t, α0+∆α)−Y (t, α0) is the relevant additional motion.

Conditions (1.60) and (1.63) determine the statement of the optimizationproblem with restrictions imposed on additional motions. In the frame-work of the exposed approach, this problem can be formulated in terms of sensitivity functions. Indeed, approximating additional motion by an ap-proximation of the form (1.40), instead of (1.63) we obtain restrictions onsensitivity functions:

Y (t, α0) + ∆(k)

Y (t, α) ∈ M Y while ∆α ∈ ∆Rα, (1.64)




Adding corresponding sensitivity equations to these relations, we for-mally obtain a classical optimization problem in augmented space of statevariables and their sensitivity functions. Note that similar situations areoften characteristic of various problems of adaptation and learning. Indeed,

as was shown by Tsypkin [118], these problems can in most cases be re-duced to the problem of parametric optimization under uncertainty, whena system optimizes automatically a functional of the form

J (α) =

Q(Y, α)P (Y )dY ,

where Q(Y, α) is a functional defined over the set of state variables, whichdepends on parameters vector α, and P (Y ) is, generally speaking, unknowndistribution density. In the general case, even successful organization of the adaptation process, i.e., automatic adjustment of parameters α, doesnot ensure satisfactory properties of the system if the requirements (1.63)are not met. Therefore, strictly speaking, such restrictions must be takeninto account in the adaptation procedure itself. Note the solution of thesynthesis problem accounting for restrictions on additional motion appearsto be cumbersome and is connected with many difficulties. Nevertheless,sensitivity analysis of an optimal system is an obligatory design stage.




Chapter 2

Sensitivity of Finite-Dimensional Continuous Systems

2.1 Finite-Dimensional Continuous SystemsDepending on a Parameter

2.1.1 Mathematical Description

The most general mathematical description of the class of systems underconsideration is given by systems of ordinary differential equations in thenormal Cauchy form:

dy1dt

= f 1(y1, . . . , yn, t),

. . . . . . . . . . . . . . . . . . . . .dyndt

= f n(y1, . . . , yn, t),

(2.1)

where y1, . . . , yn are state variables, f 1, . . . , f n are nonlinear (in the generalcase) functions, and t is the argument hereinafter called time .

Introducing column vectors Y and F with components y1, . . . , yn andf 1

, . . . , f n

, respectively, the system (2.1) can be written as a vector equation

dY

dt= F (Y, t). (2.2)

In order to describe, at least approximately, real physical process, (2.2)should satisfy some requirements caused by the necessity of similarity be-tween physical process and its mathematical model. First of all, (2.2) shouldsatisfy the requirement of existence and uniqueness of the solution for giveninitial conditions. Sufficient conditions of existence and uniqueness of the

solution are given by the following theorem.




THEOREM 2.1 EXISTENCE THEOREM [77]

Assume that in some open area Γ of the space of n+1 variables y1, . . . , yn, tthe right sides of (2.1) are defined and continuous, as well as the following partial derivatives:

∂f i∂yj

(y1, . . . , yn, t), i, j = 1, . . . , n . (2.3)

Then, for any point y10, . . . , yn0, t0 in the area Γ there exists a unique so-lution with these initial conditions yi(t0) = yi0, i = 1, . . . , n defined over some interval of the argument t that includes the point t0.

Hereinafter we consider systems of Equations (2.1) and (2.2) defined forall t ≥ t0, where t0 is a constant. Physically, this means that the mathe-

matical model under investigation describes initial systems on any intervalof its existence. Nevertheless, it should be noted that the existence theo-rem does not guarantee that the interval ts < t < te for which the solutionis defined, is unilaterally or bilaterally infinite. Therefore, in order to besure that the corresponding solutions are defined for all t ≥ t0, it is nec-essary to impose some additional restrictions on Equation (2.1) [77, 101].Further, it is always assumed that these requirements are satisfied, Equa-tions (2.1) and (2.2) under consideration satisfy conditions of existence anduniqueness, and the solution can be continued for t → ∞.

2.1.2 Parametric Dependence of Solutions on Finite TimeIntervals

According to general principles described in Chapter 1, the presence of various parameters changing the properties of the initial system can bemathematically described by introducing free variables (parameters) intoequations of its mathematical model. Therefore, we can write a parametricmodel of a finite-dimensional continuous system in the following expandedform:

dyidt = f i(y1, . . . , yn, t , α1, . . . , αm), i = 1, . . . , n . (2.4)

Introducing parameter vector α, similarly to (2.2) we can write the vectorequation

dY

dt= F (Y,t,α). (2.5)

Further, we implicitly assume that for any α ∈ Rα, where Rα is a regionof possible variation of the vector α, Equations (2.4) satisfy conditions of existence and uniqueness of solution, and this solution can be continued for

t → ∞.




One of the main tasks of sensitivity theory is investigation of properties of solutions of Equations (2.5) under variations of α. Some of these propertiesare defined by general theorems of differential equations theory. Below wegive without proofs some basic theorems needed later [77].

THEOREM 2.2 CONTINUITY WITH RESPECT TO PARAMETERS

Let in an open area Γ of n+m+1 variables y1, . . . , yn, α1, . . . , αm, t the right sides of equations (2.4) be continuous together with the partial derivatives

∂f i∂yj

(Y,t,α). (2.6)

Let also

Y = Y (t, α) (2.7)

be solutions of (2.5) in Γ satisfying the initial condition

Y (t, α0) = Y 0, (2.8)

where t0, Y 0 and α0 belongs to Γ.

Then, if the condition Y = Y (t, α) is defined on the interval t1 ≤ t ≤ t2,there is a positive number ρ such that for |α − α0| < ρ the solution (2.7) is defined on the same interval t1

≤t

≤t2, and the function Y = Y (t, α) is

continuous with respect to t and α for |α − α0| < ρ and t1 ≤ t ≤ t2.

Here and in the following the notation | · | denotes a norm.

THEOREM 2.3 DIFFERENTIABILITY WITH RESPECT TOPARAMETERS

If the right sides of (2.4) have continuous partial derivatives in area Γwith respect to variables y1, . . . , yn, α1, . . . , αm up to order s (inclusive)and the solution (2.7) satisfies conditions (2.8), for

|α

−α0

|< ρ and

t1 ≤ t ≤ t2 (where ρ is a sufficiently small positive number) this solution has continuous partial derivatives with respect to α1, . . . , αm up to order s(inclusive).

It should be noted that in these theorems the initial moment t0 and initialcondition Y (t0) are assumed invariant.

On the basis of these theorems we can consider the dependence of thesolutions of (2.4) on the initial data, which are included as parameters intothe general parametric model of the finite-dimensional system (2.5). In the

general case, the solution of (2.2) depends on initial moment t0 and initial




value Y 0 = Y (t0), so that it can be written as

Y = Y (t, t0, Y 0). (2.9)

Moreover, the following identity holds:

Y (t0, t0, Y 0) ≡ Y 0. (2.10)

Let

Y = Z + Y 0, t = τ + t0. (2.11)

Considering (2.11) as a change of variables, from (2.5) we easily obtain

dZ

dτ = F (Z + Y 0, τ + t0, α) = G(Z , τ, Y 0, α). (2.12)

The right side of (2.12) explicitly depends on the parameters Y 0 and t0,while initial conditions Z (0) = 0 do not. Therefore, we can apply the abovetheorems on continuity and differentiability with respect to parametersto (2.12). Returning to variables Y and t and using some additionalrelations [77], it is possible to prove the following theorem.

THEOREM 2.4 CONTINUITY AND DIFFERENTIABILITY WITH

RESPECT TO INITIAL DATAIn an open area Γ of variables y1, . . . , yn, t let the right sides of (2.4)be continuous together with the partial derivatives (2.3). If t0, Y 0 is an arbitrary point of the area Γ, and the general solution (2.9) satisfying initial conditions (2.10) is defined on the interval t1 ≤ t ≤ t2, there is a constant ρ such that for

t1 ≤ t ≤ t2, | τ − t0 | ρ, | Z 0 − Y 0 |≤ ρ

the solution Y (t,τ,Z 0

) is continuous with respect to t, τ and Z 0

and has continuous partial derivatives with respect to components z1, . . . , zn of the vector Z 0, and with respect to parameter τ . Moreover, there are continuous partial derivatives

∂ 2Y (t,τ,Z 0)

∂τ∂zi,

independent of the order of differentiation.

Hereinafter, when necessary, the conditions of the above theorems are

assumed to be valid.




It is noteworthy that the theorems formulated in this section relate toa finite time interval. This circumstance restricts the applicability area of these theorems, because it may happen that the time interval appearingin a theorem is less than the interval on which performance of the system

is investigated. To remove this restriction, one needs to obtain theoremson continuous dependence on parameters on infinite time intervals. Thisproblem is analyzed in the following sections.

2.1.3 Calculation of Derivatives by Parameters

If the conditions of differentiability with respect to parameters formulatedabove hold, on the basis of (2.4) one can obtain a system of differentialequations determining the derivative of the solution with respect toparameters. This fact is formulated below as a theorem.

THEOREM 2.5

Let conditions of the integral theorem on differentiability with respect toparameters hold, and let t0 and Y 0 be independent of α. Then, derivatives of solutions with respect to parameters are defined by differential equations

d

dt

∂ys

∂αj

=n

i=1∂f s∂yi

∂yi

∂αj

+∂f s∂αj

,

s = 1, . . . , n, j = 1, . . . , m ,

(2.13)

with initial conditions

∂ys

∂αj

(t = t0) = 0, s = 1, . . . , n, j = 1, . . . , m . (2.14)

It can be easily seen that Equations (2.13) are obtained from (2.4) by

means of formal differentiating with respect to αj .From this theorem we can derive equations for determining derivatives

with respect to initial conditions y10, . . . , yn0. These derivatives satisfy theequations

d

dt

∂ys∂y0j

=n

i=1

∂f s∂yi

∂yi

∂y0j, s = 1, . . . , n, j = 1, . . . , m , (2.15)

which are obtained from (2.13) by excluding the second term in the right

side. Equations (2.15) are often called equations in variations . The question




about calculation of initial conditions will be considered in the next sectionfrom a more general viewpoint.

Equations (2.13) will be called sensitivity equations with respect toparameter αj .

2.1.4 Parametric Models and General SensitivityEquations

The general solution (2.9) for equation (2.5) depends on n + m + 1scalar parameters t0, y01, . . . , y0n, α1, . . . , αm, which are a complete groupof parameters (cf. Sec. 2.3 of Chapter 1), because they determine a uniquesolution of (2.5).

In the general case, an arbitrary family of parameters µ1, . . . , µk iscomplete if there are unique relations

t0 = t0(µ1, . . . , µk), Y 0 = Y 0(µ1, . . . , µk),α = α(µ1, . . . , µk).

(2.16)

In this case, k parameters µ1, . . . , µk uniquely specify initial conditionsand vector α(µ), i.e., uniquely specify the corresponding solution Y =Y (t, µ). Therefore, hereinafter we shall say that (2.16) specifies the k-parametric family of solutions of (2.15). Let us note that the last relationin (2.16),

α = α(µ) (2.17)

defines a change of variable in the space of parameters. Substituting (2.17)into (2.5), we obtain

dY

dt= F (Y,t,α(µ)) = F (Y,t,µ). (2.18)

Moreover, with (2.16) and (2.17) initial conditions can be represented inthe form

t0 = t0(µ), Y 0 = Y 0(µ). (2.19)

In the special case when there exists only one scalar parameter µ = α,so that (2.5) has the form

dY

dt= f (Y,t,α), (2.20)

parameter α constitutes a complete group and, therefore, determines asingle-parameter family of solution, if initial conditions depend uniquely onα:

t0 = t0(α), Y 0 = Y 0(α). (2.21)




Now let us formulate a general theorem on differentiability with respectto a parameter, which is a direct result of the general theorems given in theprevious section.

THEOREM 2.6

If the function (2.21) is continuously differentiable with respect to α, the solution Y (t, α) satisfying the condition

Y (t0(α), α) = Y 0(α), (2.22)

is continuously differentiable with respect to α on any closed interval t for which the solution Y (t, α) belongs to the area Γ, in which the right side of (2.20) is continuously differentiable with respect to Y and α. The derivative

U (t, α) =∂Y (t, α)

∂α(2.23)

is determined by the equation

dU

dt=

∂F

∂Y

Y =Y (t,α)

U +∂F

∂α

Y =Y (t,α)

(2.24)

and initial conditions

U (t0) =dY 0dα

− Y 0dt0dα

=dY 0dα

− F (Y 0(α), t0(α), α)dt0dα

. (2.25)

In (2.24) the derivative ∂F/∂Y of the vector F by vector Y is a square matrix:

∂F

∂Y = q ik =

∂f i∂yk

, i, k = 1, . . . , n . (2.26)

PROOF With due account for (2.21), Equation (2.20) can be reducedto the integral equation

Y (t, α) =

t t0(α)

F (Y (t, α), t , α)dt + Y 0(α). (2.27)

In the conditions of the theorem hold, equation (2.27) can be

differentiated with respect to parameter α, and interchanging of









y(t0, α) = y0(α), . . . , y(n−1)(t0, α) = y(n−1)0 (α), t0 = t0(α), (2.36)

the question on differentiability of equations with respect to the parameterand calculation of the corresponding derivatives is solved directly on thebasis of the theorems given above.

Indeed, assuming

y = y1, y = y2, . . . , y(n−1) = yn,

from (2.35) we obtain the equations

dy1dt

= y2,

dy2dt

= y3,

. . .

dyn

dt= f (t, y1, . . . , yn, α),

(2.37)

and initial conditions reduce to (2.21). Therefore, the following theoremholds.

THEOREM 2.7

If the functions t0(α), y(i)0 (α), i = 0, . . . , n − 1 are continuously

differentiable with respect to α, the solution y(t, α) of (2.35) satisfying the initial conditions (2.36) is continuously differentiable with respect to α on any closed interval of t in the area Γ, where the

function f (t , y , . . . , y(n−1), α) is continuously differentiable with respect toy , . . . , y(n−1), α. The derivatives with respect to time y(t, α), . . . , yn−1)(t, α)are continuously differentiable with respect to α on the same interval of t.

Using (2.24) and (2.25), it is easy to find that the derivative

u(t, α) =∂y(t, α)

∂α(2.38)

must satisfy the equation

u(n) =n−1

s=0

∂f

∂y(s)

y(i)=y(i)(t,α)

u(s) +∂f

∂αy(i)=y(i)(t,α)

(2.39)





u(s)0 =

dy(s)0

dα− dt0

αy(s+1)0 , s = 0, . . . , n − 2,

u(n−1)0 =

dy(n−1)0

dα− dt0

dαf (t0, y0, . . . , y

(n−1)0 , α).

(2.40)

The sensitivity equation (2.39) is obtained by formal differentiating of theinitial equation (2.35) with respect to parameter.

2.1.5 Sensitivity Equations of Higher Orders

Using the reasoning of the previous section, we can establish the

conditions of existence of higher order sensitivity functions and findequations that determine them.

Consider, as an example, the sensitivity function

U (2)(t, α) =∂ 2y(t, α)

∂α2. (2.41)

Let initial conditions (2.21) define a family of solutions Y (t, α) of equation(2.20). Then, as was proved above, the first-order sensitivity functions

(2.23) define, for various α, a single-parameter family of solutions of sensitivity equations (2.24) given by initial conditions (2.25). Moreover,existence of second-order sensitivity functions for initial equation (2.20)appears to be equivalent to existence of first-order sensitivity functions for(2.24). Applying Theorem 2.7 to (2.24), we obtain that if the right sideof (2.24) is continuously differentiable with respect to Y and α, and thefunctions t0(α) and U 0(α) are continuously differentiable with respect toα, the function (2.41) does exist and is continuous on the correspondingintervals of t. It is the solution of the differential equation obtained bydifferentiation of (2.24) with respect to α:

dU (2)

dt=

∂F

∂Y U (2) +

∂ 2F

∂Y 2(U ) + 2

∂ 2F

∂ Y ∂ αU +

∂ 2F

∂α2

Y =Y (t,α)

, (2.42)

where the second term in the right side is an aggregate of terms representingsecond-order derivatives with respect to state variables.

To find the initial conditions we use (2.25). Obviously,

U

(2)

0 =

dU 0

dα −˙

U 0

dt0

dα . (2.43)




By (2.25) we have

dU 0dα

=d2Y 0

dα− dY 0

dα

dt0dα

− Y 0d2t0dα2

. (2.44)

Since

Y 0dα

=∂F 0∂Y 0

dY 0dα

+∂F 0∂t0

dt0dα

+∂F 0∂α

=

∂F 0∂Y 0

Y 0 +∂F 0∂t0

+

dt0dα

+∂F 0∂Y 0

U 0 +∂F 0∂α

= Y 0dt0dα

+ U 0,

F 0 = F (Y 0, t0, α),

(2.45)

equation (2.43) can be written in the form

U (2)0 =

d2Y 0dα2

− 2U 0dt0dα

− Y 0d2t0dα2

− Y 0

dt0dα

2

. (2.46)

Then, using Theorem 2.7, we arrive at the conclusion that the sensitivityfunction (2.41) exists, in the general case, if the right sides of the scalarequations in 2.20 have

1. continuous partial derivatives with respect to Y and α up to thesecond order;

2. continuous second-order partial derivatives with respect to Y , α,and t where differentiation by t is performed only once;

3. functions (2.21) have continuous second-order derivatives withrespect to α.

Let us note that if the starting point t0 is fixed, the above relations getgreatly simplified. Indeed, if t0 = const, equations (2.43) and (2.46) yield

U (2)0 =

dU 0dα

=d2Y 0dα2

. (2.47)

Thus, for t0 = const the above sufficient conditions get weaker and reduceto the following:

1. the right side of (2.20) must be twice continuously differentiable with

respect to Y and α;




2. the function Y 0(α) must be twice continuously differentiable withrespect to α.

It is easy to verify that this reasoning can be directly extended onto

sensitivity functions of an arbitrary order. A general result is formulatedbelow as a theorem.

THEOREM 2.8

Continuous n-th order sensitivity functions

U (n)(t, α) =∂ nY (t, α)

∂αn(2.48)

of a single-parameter family of solutions defined by initial conditions (2.21)

exist if

1. the right sides of the scalar equations forming vector equation (2.20)have continuous partial derivatives with respect to Y , α, and t p ton-th order, where differentiation with respect to t is performed at most n − 1 times, and

2. initial conditions (2.21) are also n times differentiable with respect to α.

Under these conditions, the associated sensitivity equations can be obtained by differentiating initial equation (2.20) by the parameter α n times, and the corresponding initial conditions are derived by successive employment of (2.25).

If the starting point t0 is independent of the parameter α, existenceconditions for sensitivity functions get simplified, because the condition of differentiability of the vector F (Y,t,α) by t becomes redundant.

Sensitivity equation of the n-th order can be written in the following

general formdU (n)

dt=

dnF (Y,t,α)

dαn, (2.49)

where d/∂α is the so-called operator of the complete partial derivative with respect to α.

Let us note some general properties of equations (2.49).

1. As follows from (2.49), the general form of sensitivity equationsis independent of initial conditions and is determined only by the

dependence of the right side of (2.20) on α. Therefore, Equation




(2.49) can be called sensitivity equation of the n-th order with respect to parameter α.

2. Sensitivity equation of the n-th order is linear with respect to

sensitivity function U (n)

(t, α) and can be represented in the form

dU (n)

dt=

∂F

∂Y

Y =Y (t,α)

U (n) + Rn(Y,U,U (2), . . . , U (n−1), t , α). (2.50)

3. The homogeneous part of sensitivity equations is the same for anyorder and coincides with equations in variations (2.15).

2.1.6 Multiparameter Case

If equations of the system under investigation depend on a vectorparameter α = (α1, . . . , αm) and have the form (2.5), it is natural toconsider all possible sensitivity functions of the n-th order:

U (l1,...,lm)(t, α) =∂Y (t, α)

∂αi11 . . . ∂ αlm

m

l1 + l2 + . . . + lm = n. (2.51)

Sufficient conditions of existence of continuous sensitivity functions (2.51)

can be derived on the basis of general theorems given above. UsingTheorem 2.7 several times, we can arrive at the following conclusion:

PROPOSITION 2.1

Let the family of solutions under consideration be given by Equations (2.5)and initial conditions

t0 = t0(α), Y 0 = Y 0(α), (2.52)

where α is a vector parameter. Then, continuous sensitivity functions (2.51) exist and are independent of the order of differentiation by various parameters if the following conditions hold:

• the right side of (2.5) admits n continuous partial derivatives by the variables Y and α and n continuous partial derivatives with respect to Y , α, and t, where differentiating by t is performed at most n − 1times;

• functions (2.52) have continuous n-th order partial derivatives with

respect to all parameters.




The corresponding sensitivity equations can be obtained directly by differentiating the initial equation by the parameters, while initial conditions are derived by successive usage of (2.25).

It is noteworthy that in special cases, for instance, when initial conditionsare independent of some parameters, the above sufficient conditions can beweakened.

A general principle of obtaining sufficient conditions of existence of thecorresponding functions can be formulated as follows.

PROPOSITION 2.2

Let initial equation (2.5) and initial conditions (2.52) be given. Then,sufficient conditions of existence of sensitivity functions (2.51) can be

derived in the following way. First, differentiate successively Equation (2.5)with respect to parameters the desired number of times (it is assumed that this is possible). As a result, we obtain an equation of the form

dU (l1,...,lm)

dt= R

U (l1,...,lm), . . . , U , Y , t , α

, (2.53)

which is linear with respect to each sensitivity function Y (l1,...,lm).Moreover, employing formally Relation (2.25), we obtain the following relation

U (l1,...,lm)(t0) = G(Y 0, t0, α). (2.54)

Then, if the right sides of (2.53) and (2.54) are continuous, the associated sensitivity function exists and is independent of the order of differentiation with respect to various parameters. Equation (2.53) will be the desired sensitivity equation, while Relation (2.54) gives the necessary initial conditions.

Example 2.1

Let us have a scalar equation

dy

dt= f (y,t,β ), t0 = t0(α), y0 = y0(β ). (2.55)

It is required to find sufficient conditions of existence of the sensitivityfunction uαβ = ∂ 2y(t,α,β )/∂α∂β . Differentiating formally the initialequation with respect to α, we obtain

duα

dt=

∂f

∂yuα, uα =

∂y(t,α,β )

∂α. (2.56)




Using (2.25), we derive the initial conditions

uα(t0) = −f (y0, t0, β )dt0dα

. (2.57)

Then, differentiating (2.56) formally with respect to β yields

duαβ

dt=

∂f

∂yuαβ +

∂ 2f

∂y2uαuβ +

∂ 2f

∂y∂β uαβ , (2.58)

where the function uβ = ∂y/∂β satisfies the equation

duβ

dt=

∂f

∂yuβ

+∂f

∂β (2.59)


uβ(t0) =dy0dβ

. (2.60)

Using (2.25) and (2.57), we obtain

U αβ(t0) =

duα(t0)

dβ = −∂f (y0, t0, β )

∂y0

dy0

dβ +

∂f (y0, t0, β )

∂β dt0

dα . (2.61)

As follows from the above reasoning, if the derivatives

∂ 2f

∂y2,

∂ 2f

∂y∂β

are continuous and so are the functions

∂f ∂β

, ∂y0∂β

, ∂t0,∂α

,

the function uαβ exists and is independent of the order of differentiationby α and β . Corresponding sensitivity equation has the form (2.58), and

initial conditions are given by (2.61).

For investigation of multiparameter problems it is often convenient toemploy vector differentiation. Such an approach is especially handy while

considering first-order sensitivity functions. If Equation (2.5) is given and




conditions of existence of first-order sensitivity functions hold, the followingsensitivity matrix can be introduced:

U (t, α) =∂Y

∂α= ∂yi(t, α)

∂αk , i = 1, . . . , n, k = 1, . . . , m .

This matrix will be the solution of the sensitivity equation similar to (2.24):

dU

dt=

∂F

∂Y

Y =Y (t,α)

U +∂F

∂α

Y =Y (t,α)

(2.62)


U (t0) = dY 0(α)dα − Y 0 dt0dα ,

where matrices dY 0dα

and ∂F ∂α

and vector dt0dα

are determined by the relations

∂F

∂α=

∂f i∂αk

,dY 0dα

=

∂y0i∂αk

,dtT 0dα

=

dt0dα1

, . . . ,dt0dαk

, (2.63)

i = 1, . . . , n, k = 1, . . . , m .

In the special case when the components of the initial conditionsvector y0 are taken as parameters, from (2.63) we obtain the followingnonhomogeneous equation in variations:

dU

dt=

∂F

∂Y U, (2.64)

with initial conditions of the form

U (t0) = E, (2.65)

where E denotes the identity matrix of the corresponding dimension.

2.1.7 Analytical Representation of Single-ParameterFamily of Solutions

In this section we discuss the problem of expanding of parametric familiesof solutions into power series in parameters values, which was considered

in Chapter 1 in a general form.




First, consider the case of a single-parameter family of solutions Y (t, α)defined by Equation (2.20) and initial conditions (2.21). Assume that theconditions of existence and continuity of sensitivity functions up to n +1-thorder inclusive hold in an area Γ containing the interval J α of parameter

values α1 ≤ α ≤ α2. Let α0 be a parameter value belonging to the intervalJ α so that α1 ≤ α0 ≤ α2. Then, using Lagrange’s formula for differentialcalculus, for all µ such that α1 ≤ α0 + µ ≤ α2 and corresponding t we canwrite

Y (t, α0 + µ) = Y (t, α0) + U (t, α0)µ + . . . +1

n!U (n)(t, α0)µn

+1

(n + 1)!U (n+1)(t, α0 + ϑµ)µn+1, 0 < ϑ < 1.

(2.66)

According to (2.49), the sensitivity fnctions U (i) in (2.66) are determinedby the differential equations

dU (i)

dt=

di

dαiF (Y,t,α) (2.67)


U (i)(t0

) =dU (i−1)(t0)

dα −U (i−1)(t

0)

dt0

dα(2.68)

Ignoring the remainder term in (2.66), we obtain an approximation of then-th order

Y (t, α0 + µ) ≈ Y (t, α0) + U (t, α0)µ + . . . +1

n!U (t, α0)µn. (2.69)

According to Section 3.2 of Chapter 1, the difference

∆Y (t, µ) = Y (t, α0 + µ) − Y (t, α0) (2.70)

represents additional motion caused by the variation of parameter α. From(2.69) we have n-th approximation for the additional motion

∆(n)Y (t, µ) ≈ U (t, α0)µ + . . . +1

n!U (n)(t, α0)µ(n). (2.71)

Similar expansions can be written for the multiparameter case as well. If

all sensitivity functions up to n + 1-th order exist and are continuous, we




can write

∆Y (t, µ) = dY (t, α0) +1

2!d2Y (t, α0) + . . .

+ 1n!

dnY (t, α0) + 1(n + 1)!

dn+1Y (t, α0 + ϑ, µ), (2.72)

where α0 is the nominal (base) value of the parameter vector α, µ is thevector of parameter perturbation, and the argument t is considered as aparameter.

Ignoring the remainder term, we obtain the n-th approximation foradditional motion:

∆(n)Y (t, µ) ≈ dY (t, α0) + 12!

d(2)Y (t, α0) + . . . + 1n!

d(n)Y (t, α0). (2.73)

Formulas (2.72) and (2.73) have already been given for the general casein Chapter 1. Nevertheless, for finite-dimensional systems we presentedsufficient conditions for existence of sensitivity functions and generalequations for their determination. Moreover, the relation between theerror of the n-th approximation and sensitivity functions of n + 1-th orderhas been established. For example, if sufficient conditions of existence of

continuous second-order sensitivity functions hold, for sufficiently small |µ|,where | · | denotes a norm of vector, the first approximation has the form

∆Y (t, µ) ≈ dY (t, α) =mi=1

U i(t, α0)µi, (2.74)

which is very useful for investigation of general properties of additionalmotion.

2.1.8 Equations of Additional Motion

Let Y (t, α) be a single-parameter family of solutions of Equation (2.20),corresponding to initial conditions (2.21) in a closed interval J α : α1 ≤α ≤ α2. Let α = α0 be some fixed base value of the parameter from theinterval J α, and Y (t, α0) be the corresponding solution. If the parametervalue differs from the base one, we obtain the solution

Y (t, µ) = Y (t, α0 + µ), (2.75)






3. A definite family of solutions Y (t, α) is associated with a single-parameter family of additional motions, which are solutions of Equation (2.79) and satisfy the initial conditions

Z (t0, µ) = Y (t0, α0 + µ) − Y (t0, α0),t0 = t0(α0 + µ) = t0(µ).

(2.83)

4. If Γ is an area in the space of variables Y , t, and α, in which Equation(2.20) is defined, the corresponding domain of the additional motionequation is defined by the change of variables (2.81). In this case,parameter µ will be inside the interval J α so that

α1

−α0

≤µ

≤α2

−α0 (2.84)

including the point µ = 0.

In a similar way, we can derive equations determining deviation of additional motion from a chosen approximation. For simplicity, let usconsider the first approximation. Let in (2.79)

Z (t, µ) = U (t, α0)µ + V (t, µ), (2.85)

where

V (t, µ) = ∆Y (t, µ) − ∆(1)Y (t, µ). (2.86)

According to (2.80), the right side of (2.79) can be represented in the form

F (Z,t,µ) =∂F

∂Y

Y =Y (t,α0)

α=α0

Z +∂F

∂α

Y =Y (t,α)

α=α0

µ + G(Z,t,µ). (2.87)

Considering (2.85) as a change of variable and taking (2.87), we obtain

dV

dt= G(V + U (t, α0)µ,t,µ). (2.88)

2.1.9 Estimation of First Approximation Error

Using sensitivity equations, it is possible, generally speaking, to obtainsome estimates of the error of representation of additional motion in theform of a power series in parameter value. Consider the single-parametercase. Restrict ourselves by investigation of the first approximation error,

which is most important from the practical viewpoint.




According to (2.74), the first approximation for additional motion is givenby

U (t, α0)µ = Z (1)(t, µ) (2.89)

As before, denote additional motion by

∆Y (t, µ) = Y (t, α0 + µ) − Y (t, α0), (2.90)

Then, by (2.66) we find the error between precise and approximate formulasfor additional motion as

Z (t, µ) − Z (1)(t, µ) =1

2!U (2)(t, α + ϑµ)µ2, 0 < ϑ < 1. (2.91)

Estimating both sides of (2.91) with a norm, it is easy to obtainZ (t, µ) − Z (1)(t, µ) ≤ 1

2!maxϑ

U (2)(t, α + ϑµ) µ2 , 0 ≤ ϑ ≤ 1. (2.92)

To estimate the value

∆(1) =1

2maxϑ

U (2)(t, α + ϑµ) µ2

, 0 ≤ ϑ ≤ 1, (2.93)

we use sensitivity equations. Let us describe a procedure of such estimation.

Let Equation (2.20)

dY

dt= F (Y,t,α)

and initial conditions (2.21)

t0 = t0(α), Y 0 = Y 0(α)

determine the family of solutions Y (t, α) in a closed set Γ of variables Y , t,and α. Assume that in this area there exist continuous sensitivity functionsof the first and second order, U (t, α) and U (2)(t, α), respectively, satisfyingthe equations

dU

dt=

∂F

∂Y U +

∂F

∂α, U 0 = U (t0) =

Y 0dα

− Y 0dt0dα

, (2.94)

dU (2)

dt=

∂F

∂Y U (2) + R2(Y,U,t,α),

U

(2)

(t0) =

dU 0

dα −˙

U 0

dt0

dα = U

(2)

0 .

(2.95)






From (2.101) follows the desired estimate of the second-order sensitivityfunction

U (2)

≤

c1 +

b1a

ea(t−t0) − b1a

, t, α ∈ Γ. (2.103)

Let us return to Equation (2.93). Considering it in the closed area Γ andusing (2.103), we obtain

∆(1) ≤ 1

2max

α0+µ,t∈Γ

c1 +

b1a

ea(t−t0) − b1

a

µ2

. (2.104)

Similarly we can derive, for a fixed set Γ, an error estimate forapproximation of an arbitrary order. For this purpose Equation (2.72)

can be used, which yieldsZ (t, µ) − Z (n)(t, µ) ≤ 1

(n + 1)!max

θU (n+1)(t, α0 + ϑµ)

µn+1. (2.105)

The value of the right side of (2.105) can be estimated on the basisof sensitivity equations using the recurrent technique described above.In principle, this estimation method can be used without essentialmodifications in multiparameter problems. Another estimation technique,based on Lyapunov’s method, is presented in the next section.

2.2 Second Lyapunov’s Method in Sensitivity Theory

2.2.1 Norms of Finite-Dimensional Vectors and Matrices

DEFINITION 2.1 A norm of a real column vector Y with real components y1, . . . , yn is a number |Y | satisfying the following conditions:

•| Y |> 0 where Y = O, | O |= 0, (2.106)

where O is the zero column vector;

•

| cY |=| c || Y |, (2.107)




where c is a scalar numeric multiplier;

•

|Y 1 + Y 2

|≤|Y 1

|+

|Y 2

|. (2.108)

Relation (2.108) is called triangle inequality . Inequality

| Y |≤ r, r = const (2.109)

defines a n-dimensional ball Γr with radius r in n-dimensional space.As follows from Axioms (2.106)–(2.108), “the ball” is a convex centrallysymmetric body that contains vector −Y together with Y , and vectorcY 1 + (1

−c)Y 2 (where 0

≤c

≤1) together with Y 1 and Y 2. It can be

shown [111], that conversely, any centrally symmetric body V generates anorm |Y |V defined by the following relations:

| Y |V = inf t,1

tY ∈ V. (2.110)

The surface

| Y |= 1 (2.111)

is called n-dimensional unit sphere . Geometric sense of unit sphere can be

various for various choices of initial norm | · |. For example, assuming

| Y |=| Y 1 |= maxi

| yi |, (2.112)

we find that the set of vectors satisfying Condition (2.111), forms the surfaceof a unit cube

−1 ≤ y1 ≤ 1, . . . , −1 ≤ yn ≤ 1. (2.113)

Such a norm is called cubic . If we take

| Y |=| Y |3=

ni=1

y2i , (2.114)

Equation (2.111) defines a unit sphere, therefore the norm (2.114) is calledspherical .

For different r’s, equations

| Y |= r (2.115)




define the set of spheres S r embedded into each other, which fill all space.

DEFINITION 2.2 Operator norm of a finite-dimensional square matrix

A = aik, i, k = 1, . . . , n , (2.116)

associated with a given vector norm | · | is a number |A| such that

| A |= max|Y |=1

| AY | (2.117)

It can be shown that for the vector norm (2.112) the associated operatornorm is defined by the relation

| A |1= maxi

nk=1

| aik | . (2.118)

For the spherical norm (2.114) the operator norm of a matrix A is given by

| A |3=√

λ, (2.119)

where λ is the maximal eigenvalue of the matrix AT A, where “T ” denotesthe transpose of a matrix.

Hereinafter by a “norm of matrix” we will implicitly mean some operatornorm. An arbitrary matrix norm satisfy the conditions

| A |> 0 where A = O, | O |= 0 (2.120)

where O is a zero matrix and

| cA |=| c | A |, c = const, (2.121)

|A + B

|≤|A

|+

|B

|, (2.122)

| AB |≤| A || B | . (2.123)

Norms of finite-dimensional vectors and matrices possess an importantequivalency property . Let | · |1 and | · |2 be arbitrary norm of vector. Then,there are positive constants µ1 and µ2 independent of Y such that

µ1 | Y |1≤| Y 2 |≤ µ2 | Y |1 . (2.124)

A similar relation holds for two arbitrary norms of finite-dimensional ma-

trix.




2.2.2 Functions of Constant and Definite Sign

DEFINITION 2.3 A single-valued continuous function v(Y ) in vari-ables y1, . . . , yn is called function of constant sign in a simply connected area Γ of the space Y , if the following conditions hold:

v(O) = 0, v(Y ) ≥ 0, Y ∈ Γ (2.125)

DEFINITION 2.4 A function v(Y ) is called positive definite in an area Γ, if

v(O) = 0, v(Y ) > 0 where Y = 0, Y ∈ Γ, (2.126)

DEFINITION 2.5 A function v(Y ) is called negative definite in an area Γ, if the function −v(Y ) is positive definite.

Below we consider some general properties of functions of definite signsthat are needed further.

PROPOSITION 2.3

Let a continuous and single-valued function v(Y ) be positive definite in a

closed area Γ. Let a number l be defined by the relation

l = minY ⊂S Γ

v(Y ) = l(Γ), (2.127)

where S r is the boundary of Γ. Then, equation

v(Y ) = c = const < l(Γ) (2.128)

determines a family of closed surfaces embedded to each other, which belong

to Γ and contain the origin. Moreover, for c1 < c2 < Γ the surface v(Y ) =c1 is inside the surface v(Y ) = c2.

PROOF Draw an arbitrary continuous curve L from the origin to theboundary S r. Along this curve, the function v(Y ) takes values in the in-terval from 0 to l ≥ l(Γ). Therefore, at some point of the curve L thefunction v(Y ) will take any value c < l(Γ), i.e., the curve L intersects allthe surfaces (2.128), and all these surfaces are closed. Since the functionv(Y ) is single-valued, the surfaces (2.128) do not intersect. Moreover, if

c1 < c2 < l(Γ), the value v = c1 is encountered, along the curve L, earlier




than v = c2, i.e., the surface v(Y ) = c1 is inside the surface v(Y ) = c2.

PROPOSITION 2.4

Let a continuous and single-valued function v(Y ) be positive definite inside a sphere S r (2.115). Let a number l(r) be given by the relation

l(r) = min|Y |=r

v(Y ). (2.129)

Then, equation

v(Y ) = c = const (2.130)

where c < l(r), defines a set of closed surfaces embedded into each other,

which are inside S r and cover the origin. Moreover, for c1 < c2 < l(r) the surface v(Y ) = c1 is inside the surface v(Y ) = c2.

PROOF The proof of this proposition follows from Proposition 2.3, if we take area Γ in the form of the ball Γr. For c = 0 the surface (2.130)

degenerates into the origin.

PROPOSITION 2.5

Let a surface w(Y ) = ρ be closed and contain the origin, and let a number

η(ρ) be defined by η(ρ) = min

ω=ρ| Y |, (2.131)

where | · | is a norm of vector. Then, the sphere |Y | = r < η(ρ) is inside the surface w(Y ) = ρ.

PROOF This proposition follows also from Proposition 2.3, becauseany norm |Y | is a positive definite function of its arguments.

DEFINITION 2.6 A function v(Y ) is called infinitely large if for any positive number a there is r(a) such that outside the sphere |Y | = r(a) we have v(Y ) > a.

In [6] it is shown that for an infinitely large positive definite function v(Y )the surfaces (2.130) fill all space and are closed for all c. Indeed, taking asufficiently large r in (2.129), we can ensure the inequality c < l(r), whichproves the claim.

From the aforesaid, it follows that there is a definite link between norms

and positive definite functions. Any norm is an infinitely large positive




definite function. The inverse proposition is, generally speaking, false. Aninfinitely large positive definite function v(Y ) defines a norm in the spaceof variables y1, . . . , yn if its level surfaces (2.130) are convex and centrallysymmetric.

2.2.3 Time-Dependent Functions of Constant and DefiniteSign

DEFINITION 2.7 A continuous and single-valued function v(Y, t),where t is a scalar parameter, is called positive definite on the interval I t(t1 ≤ t ≤ t2) in an area Γ of the space Y if

v(O, t) = 0, v(Y, t) ≥ v(Y ), Y ∈ Γ, t ∈ I t, (2.132)

where v(Y ) is a positive definite function in Γ independent of t.

Time dependent positive definite functions admit geometric descriptionsimilar to that given in the previous section.

Let S r be a sphere with radius r in the area Γ, and let a number l(r) bedefined by (2.129). Consider the equation

v(Y, t) = c < l(r). (2.133)

For various t’s Equation (2.133) defines a closed surface in the space of variables Y covering the origin, which is deformed with a change of t. Letus show that for all t ∈ I t the surface (2.133) is inside the surface v(Y ) = c,and, therefore, is inside the sphere S r. Indeed, if v(Y ) = c, due to (2.132)v(Y , t) ≥ c and the point Y is outside the surface v(Y, t) = c or on it.

Thus, from the condition

v(Y , t)≤

c < l(r), t∈

I t, (2.134)

it follows that the point Y is inside a closed area of the space of variablesY bounded by the surface v(Y ) = c.

DEFINITION 2.8 If, instead of (2.132), weaker conditions hold:

v(O, t) = 0, v(Y, t) ≥ 0, Y ∈ Γ, t ∈ I t, (2.135)

the function v(Y, t) is called positive function of constant sign.




DEFINITION 2.9 A function v(Y, t) is called negative definite (or neg-ative function of constant sign), if the function −v(Y, t) is positive definite (or positive function of constant sign, respectively).

2.2.4 Lyapunov’s Principle

Let us have a system of differential equations

dyidt

= f i(y1, . . . , yn, t), (2.136)

where the right sides are continuous with respect to all arguments andsatisfy conditions of existence and uniqueness of solution inside a sphere S ron the interval I t : t1

≤t

≤t2. It is assumed than Equation (2.136) has a

trivial solution

yi = 0, i = 1, . . . , n , (2.137)

for which it is necessary and sufficient that

f i(0, . . . , 0, t) = 0, i = 1, . . . , n . (2.138)

Let v(Y, t) = v(y1, . . . , yn, t) be a positive definite function in Y ∈ S r andt∈

I t, that is continuously differentiable with respect to all arguments. Asis known, the function

v(Y, t) =dv

dt=

∂v

∂t+

ni=1

∂v

∂yi

f i(Y, t), (2.139)

is called the complete derivative of the function v(Y, t) due to Equation(2.136). If we rewrite (2.136) in a vector form as

dY

dt = F (Y, t), (2.140)

the complete derivatives can be represented in a more compact form:

v =dv

dt=

∂v

∂t+

∂v

∂Y · F

, (2.141)

where (·) denotes scalar product.

Analyzing properties of the functions v(Y, t) and v(Y, t) often makes it

possible to obtain important information about properties of solutions of




(2.140) without integrating the latter. For the present book, the followingproposition, called Lyapunov’s principle , is very important.

PROPOSITION 2.6

[LYAPUNOV’S PRINCIPLE] Let for Y ∈ Γr and t ∈ I t a function v(Y, t)be positive definite,

and let the derivative satisfy the relations

v(Y, t) ≤ 0, v(0, t) = 0, (2.142)

i.e., the function v(Y, t) is negative function of constant sign. Let also v(Y )be the function appearing in (2.132), and

l(r) = min|Y |=r

v(Y ). (2.143)

Then, for any solution Y (t) of Equation (2.140) such that Y (t0) = Y 0,Y 0 ∈ Γr, t0 ∈ I t, from the condition

v(Y 0, t0) = c < l(r) (2.144)

follow the inequality

v(Y (t), t) ≤ c, t0 ≤ t ≤ t2, (2.145)

and for all t0 ≤ t ≤ t2 the solution Y (t) is inside a bounded area v(Y ) ≤ c.

The function v(Y, t) will hereinafter be called Lyapunov’s function .

The above proposition given without a proof is a basis of numerous inves-tigations in stability theory and adjacent fields [5, 53, 62]. It is noteworthythat Relation (2.145) is a specific estimate of the norm of solutions, becauseit follows that solution Y (t) for all t0

≤t≤

t2 is inside a bounded area inthe sphere S r.

If condition (2.132) holds for all t ≥ t0, Relation (2.145) holds also for allt ≥ t0. In this case, the trivial solution (2.137) is stable by Lyapunov [5, 62],i.e., for any > 0 there is a number η() > 0 such that from the condition|Y 0| < η() follows that |Y (t)| < for all t ≥ t0. Given a number ( < r),the number η() can be found in the following way.

1. Given , find

l() = min|Y |=

v(Y ). (2.146)




2. With l(), find

η() = minv(Y,t0)=l()

| Y | . (2.147)

The values l() and η() depend on the choice of the norm | · |. Geometricconstructions associated with (2.146) and (2.147) are given in Figure 2.1.

Figure 2.1

Geometric interpretation

It can be easily seen that actually the relation |Y (t)| < will followalready from the condition v(Y 0, t0) < l().

2.2.5 Norm of Additional MotionWe will demonstrate the possibility of applying Lyapunov’s principle for

evaluation of the norm of additional motion. Let the equation of additionalmotion have the form (2.79):

dZ

dt= F (Z,t,µ) (2.148)

and has the trivial solution Z = 0, µ = 0 by construction.

THEOREM 2.9

Let a function v(Z,µ,t) satisfy the following conditions for all µ ∈ I µ :−µ ≤ µ ≤ µ, µ > 0, t ∈ I t inside the ball Γr of the space Z :

v(O, 0, t) = 0, v(Z,µ,t) ≥ v(Z ), (2.149)

where v(Z ) is a positive definite function, and

˙v =∂ v

∂t+ ∂ v

∂Z ·F ≤

0. (2.150)




Then, if the number l(r) is chosen according to (2.129), from

v(Z 0, µ , t0) = c < l(r), Z 0 ∈ Γr, t0 ∈ I t, µ ∈ I µ, (2.151)

follows the inequality

v(Z ) ≤ v(Z,µ,t) ≤ c, t ∈ I t. (2.152)

PROOF Consider the function

ω(Z,µ,t) = v(Z,µ,t) +aµ2

2, a = const > 0. (2.153)

As follows from (2.149), the function (2.153) is positive definite in a cylin-drical area Γ of n + 1-dimensional space of variables (Z, µ) given by

Z ∈ Γr, −µ ≤ µ ≤ µ. (2.154)

The area (2.154) can be considered as a “unit sphere” in the space of n + 1variables (Z, µ), where the norm of the n + 1-dimensional vector (Z, µ) isdefined by

| (Z, µ) |= max

| Z |r

,| µ |

µ

, (2.155)

where |Z | is a chosen norm of the vector Z in the n-dimensional space of variables z1, . . . , zn.

Write Equation (2.148) as a system of n + 1-th order

dµ

dt= 0,

dZ

dt= F (Z,µ,t)

(2.156)

and calculate the derivative dw/dt of the function (2.153) with (2.156).Obviously,

dω

dt= ˙v. (2.157)

As follows from (2.150), the derivative (2.157) is negative with constantsign. Then, from Lyapunov’s principle, it follows that if the number l ischosen according to

l = min|(Z,µ)|=1

v(Z ) +aµ2

2 , (2.158)




due to (2.144) and (2.145) we obtain

v(Z ) ≤ v(Z (t), µ, t) ≤ v(Z 0, µ, t0), (2.159)

if (Z 0, µ, t0) ≤ l. (2.160)

Notice that the number a > 0 in (2.153) can be taken as small as nec-essary. Therefore, for a → 0 condition (2.160) is changed by a weakerone:

v(Z 0, µ, t0) < l(r), (2.161)

wherel(r) = min

|Z |=rv(Z ),

It is noteworthy that the above theorem does not depend on a specificform of Equation (2.148).

2.2.6 Parametric Stability

If the conditions of Theorem 2.9 hold for all t ≥ t0, for the system(2.156) there is a function of constant sign from n +1 variables (2.153), thederivative of which is a negative function with constant sign. Hence, the

trivial solution Z = 0, µ = 0 of the system (2.156) is Lyapunov stable, i.e.,if |(Z, µ)| is any norm of the n + 1-dimensional vector (Z, µ), for any > 0there is a number η() > 0 such that the condition |(Z 0, µ)| < η() yields|(Z (t), µ)| < for all t ≥ t0. If this property holds, the initial equation(2.148) will be called parametrically stable .

Thus, the solution Z = 0, µ = 0 is parametrically stable by definition, if the trivial solution of the system (2.156) is Lyapunov stable.

Using equivalency of norms of a finite-dimensional vectors, it can beeasily proven that parametric stability is independent of the choice of norm

|(Z, µ)

|. If parametric stability takes place for one norm

|(Z, µ)

|1, the same

is valid for another norm |(Z, µ)|2.Let us note that parametric stability of Equation (2.148) follows Lya-

punov stability of its trivial solution Z = 0 for µ = 0, i.e., Lyapunovstability of the zero solution of the equation

Z = F (Z, 0, t). (2.162)




The inverse proposition is, in the general case, false. Indeed, let a norm|(Z, µ)| be defined by (2.155). Then, if parametric stability takes place in anarea (2.154), from the condition |(Z 0, µ)| < η() it follows that |(Z (t), µ)| < for all t ≥ t0. Hence, for Equation (2.148) with µ = 0 from |Z 0| < rη()

follows that |Z (t)| < r for all t ≥ t0, i.e., Lyapunov stability takes place.Let us give an example demonstrating that parametric stability of Equa-

tion (2.148) does not follow even from asymptotic stability of the trivialsolution of (2.142).

Example 2.2

Let the equation of additional motion (2.148) have the form

dzdt = µz + az3, t0 > 0, a = const < 0. (2.163)

It can be easily verified by direct calculations that for µ = 0 we haveasymptotic Lyapunov stability. On the other hand, for any µ > 0 we haveinstability. Indeed, for µ > 0 the linear approximation

dz

dt= µz

is unstable, then, by Lyapunov’s theorem on stability by the first approxi-mation, the trivial solution of Equation (2.163) is unstable.

Note that from the physical viewpoint the notion of parametric stability,as applied to Equation (2.148), is much more meaningful than Lyapunovstability, i.e., stability with respect to initial data. According to Chap-ter 1, variation of parameter µ in (2.148) can be caused by variation of design parameters and operating conditions, or by exogenous disturbances.Therefore, it is natural that to ensure parametric stability it is necessary

to meet much stronger restrictions than to provide Lyapunov stability forµ = 0.

2.2.7 General Investigation Method

As was shown by examples, Theorem 2.9 in many cases does not pro-vide constructive results for estimation of the norm of additional motionsand investigation of parametric stability. In the present section we deriveanother, more general, theorem, which allows one to extend the class of

problems under consideration.




THEOREM 2.10

Let for Equation (2.148) exist a function v(Z,µ,t) that satisfies the fol-lowing conditions for all t ∈ I t, µ ∈ I µ inside a ball Γr in the space Z :

1.v(Z,µ,t) ≥ v(Z ), v(0, µ, t) = 0, (2.164)

where v(Z ) is a positive definite function.

2. If the number l(r) is defined by (2.129), for any c < l(r) there exists µ(c) > 0 such that

˙vv=c=

∂ v

∂t+

∂ v

∂Z · F

v=c

< 0 where | µ |< µ(c). (2.165)

Then, from the conditions

v(Z 0, µ , t0) < c < l(r), | µ |< µ(c) (2.166)

it follows that

v(Z (t), µ, t) < c < l(r), t ∈ I t, (2.167)

and the solution Z (t, µ) is inside a bounded area v(Z ) = c for all

|µ

|< µ(c),

t ∈ I t.

PROOF Let for t = t0

v(Z 0, µ, t0) < c < l(r), ˙v(Z 0, µ, t0) < 0, | µ |< µ(c), (2.168)

Then, due to continuity, at least on a finite interval t0 ≤ t ≤ T ≤ t2,

v(Z (t), µ , t) < c. (2.169)

If for t = T Relation (2.169) violates for the first time, then

v(Z (t), µ, T ) = c, ˙v(Z (t), µ, T ) ≥ 0, (2.170)

which contradicts (2.165). This contradiction proves that Relation (2.167)holds for the whole interval t0 ≤ t ≤ t2.

Notice that for t0 ≤ t ≤ t2 the solution Z (t, µ) is inside the surfacev(Z ) = c, because, due to (2.164), the surfaces v(Z,µ,t) < c are inside the

surface v(Z ) = c.




Let us formulate a number of important corollaries of this claim.

1. If conditions of Theorem 2.10 hold for all t ≥ t0, instead of (2.167)we have the inequality

v(Z (t), µ, t) < c, t ≥ t0. (2.171)

It can be shown [89], that in this case Equation (2.148) is paramet-rically stable. Thus, the fact that conditions of the theorem holdfor all t ≥ t0 is a sufficient condition for parametric stability.

2. The above theorem can also be used for evaluation of the error of the n-th approximation for additional motion, considering the n-th

order error equation instead of (2.148).

2.2.8 Sensitivity of Linear System

As an example of application of Theorem 2.10, consider the problem of evaluation of perturbation of additional motion of a linear system underparameter variation.

Let the vector equation of additional motion have the form

dY

dt = AY + µBY + µF (t), (2.172)

where A and B are square matrices, and F (t) is a vector bounded on anyfinite time interval.

Assume that all roots of the characteristic equation

det | A − λE |= 0 (2.173)

have negative real parts. Then, by the known Lyapunov’s theorem [6, 62],

for any negative definite quadratic form

ω = Y T GY, (2.174)

where G is a constant symmetric matrix, there is a positive definite form

v = Y T HY, (2.175)

such that

HA + AT

H = G. (2.176)




Calculate the derivative dv/dt of the function (2.175) using Equation(2.172). Obviously,

dv

dt=

dY T

dtHY + Y T H

dY

dt. (2.177)

Transposing (2.172), we have

dY T

dt= Y T AT + µY T BT + µF T (t). (2.178)

Substituting (2.172) and (2.178) into (2.177), with due account for (2.176)we obtain

dv

dt= ω(Y ) + µ Y T BT H + HBY + µ

F T (t)HY + Y T HF (t)

. (2.179)

Let us show that in this case the condition (2.165) holds for any finitetime interval, and give a practical method to estimate µ(c).

Rewrite the derivative dv/dt in the form

dv

dt= Y T P Y + µQT Y, (2.180)

where, according to (2.179),

P = G + µ(BT H + HB), QT = 2F T H. (2.181)

Obviously,

maxv=c

dv

dt≤ max

v=cY T P Y + | µ | max

v=cQT Y. (2.182)

Each term in the right side can easily be calculated. Since v is a positivedefinite form, we have [114]

maxv=c

Y T P Y = λmaxc, (2.183)

where λmax is the largest root of the following equation:

det(P − λH ) = det(G + µ(BT H − HB) − λH ) = 0. (2.184)

To find the second term, consider the following Lagrange function:

L = QT

Y − λ(v − c) = QT

Y − λ(Y T

HY − c). (2.185)




Extremum condition for this function has the form

2λH Y = Q, (2.186)

Hence,

Y =1

2λH −1Q. (2.187)

Since v = c, we have

Y T H Y =1

4λ2QT

H −1

T HH −1Q = c,

i.e., the Lagrange multiplier is given by

λ =1

2√

c

QT H −1Q, (2.188)

because (H −1)T = H −1 due to symmetry.

Then, from (2.186) it follows that

2λY H Y = (QT Y )max = 2λc,

i.e., QT Y

max = √ c

QT H −1Q. (2.189)

Using (2.183), (2.189), and (2.181), from (2.182) we obtain

dv

dt

v=c

≤ λmaxc + 2 | µ | √ c√

F τ HF . (2.190)

If, in the time interval under consideration, we have

| F (t) |= ni=1

f 2i (t) ≤ h, (2.191)

then, evaluating the radicand in (2.190) as for derivation of (2.183), wehave

F T (t)HF (t) ≤ ν maxh2, (2.192)

where ν max is the largest roots of the equation

det(H − νE ) = 0. (2.193)




As follows from (2.190),

dv

dt

v=c

≤ λmax(µ)c + 2 | µ | h√

ν maxc. (2.194)

For µ → 0 we have

λmax(µ) → λmax < 0, (2.195)

where λ = λmax(0) is the largest roots of Equation (2.184) for µ = 0.

Therefore, the required value µ(c) does exist and its estimate can bedetermined from the inequality

λmax [µ(c)]√

c + 2µ(c)h√

ν max < 0. (2.196)

Using Theorem 2.10 for the case at hand and noting that the functionv is infinitely large, it is easy to find that for any c > 0 and any solutionY (t, µ) of Equation (2.172) with v(Y 0) = c and |µ| < µ(c) we have

v(Y (t)) ≤ v(Y 0) = c. (2.197)

The above calculations get greatly simplified if B = 0. In this case,this approach, using Lyapunov’s method, gives an estimate of disturbanceinfluence onto a linear system.

Example 2.3

Consider the simplest first-order equation

dy

dt= −ay + µby + µf (t), | f (t) |≤ h, (2.198)

where a > 0, b, and h are constants. Let

v(y) =

1

2 y

2

. (2.199)

The derivatives with due account for Equation (2.188) have the form

dv

dt= −ay2 + µby2 + µf (t)y. (2.200)

If v = c, from (2.200) we obtain

dv

dtv=c ≤

√ 2c

|µ

| b√

2c + h −2ac. (2.201)




If b√

2c + h > 0, we can take

µ(c) =a√

2c

h + b√

2c(2.202)

and for y20 < 2c, b√

2c + h > 0, |mu| < µ(c)

y2(µ, t) < 2c. (2.203)

If b√

2c + h ≤ 0, Relation (2.203) holds for any µ.

2.3 Sensitivity on Infinite Time Intervals

2.3.1 Statement of the Problem

Using the classical theorems given in Section 2.1, one can justify thevalidity of approximate representation of additional motion via sensitiv-ity functions (2.73) and (2.74) on finite sufficiently small time intervals.Though it is possible, in principle, to estimate such time intervals, the pro-cedure is very difficult in practice. Therefore, for applied problems, it isuseful to derive conditions under which Formulas (2.73) and (2.74) remainvalid for all t ≥ t0, where t0 is a constant. If this is the case, it is notnecessary to estimate the scope of corresponding time intervals. Below wepresent a solution of this problem based on Theorem 2.10 [89].

Let the vector equation of additional motion have the form (2.79):

dZ

dt= F (Z,t,µ), (2.204)

where initial solution Y = Y (t, α) is associated with the trivial solution of Equation (2.204)

Z = 0, µ = 0. (2.205)

The initial family of solutions Y (t, α) is associated with a family Z (t, µ) of additional motions with initial conditions (2.83)

Z (t0, µ) = Y (t0, α0 + µ) − Y (t0, α0),

t0 = t0(α0 + µ) = t0(µ).(2.206)

By construction, the family of solutions (2.206) includes the trivial solution

(2.205), therefore, we must have Z (t0, 0) = 0.




Let

U (t) = U (t, α0) =∂Y (t, α)

∂α

α=α0

(2.207)

be the sensitivity function of the family of solutions under consideration.A widely employed applied method of investigating additional motion

consists in approximate substitution of additional motion by the first ap-proximation

Z (t, µ) ≈ U (t, α0)µ. (2.208)

As follows from Section 2.1.7, under conditions of existence of the second-order sensitivity functions for a sufficiently small |µ| Relation (2.208) holdsfor any fixed t with an arbitrary precision.

Indeed, let

Z (t, µ) = U (t)µ + V (t, µ). (2.209)

As follows from (2.66), under conditions of existence of the second ordersensitivity functions we have

V (t, µ) =1

2U (2)(t, α0 + ϑµ)µ2, 0 < ϑ < 1, (2.210)

Therefore, for any fixed t,

limµ→0

| V (t, µ) |= 0. (2.211)

Nevertheless, validity of the approximate equality (2.208) for any fixed tdoes not guarantee its applicability for all t ≥ t0, because it may happenthat admissible value |µ| tends to zero as t increases.

In order that the representation (2.208) be valid for all t ≥ t0, it issufficient that Relation (2.211) holds uniformly with respect to t for allt ≥ t0, i.e., that for any > 0 there exists µ() > 0 such that

| V (t, µ) |< where | µ |< µ(), t0 ≤ t < ∞. (2.212)

If conditions (2.212) hold, from (2.209) we have, uniformly with respectto t,

| Z (t, µ) − U (t)µ |< while | µ |< µ(), t ≥ t0. (2.213)

Thus, the problem of finding sufficient conditions of applicability of theapproximate representation on an infinite time interval can be reduced to

obtaining sufficient conditions of validity of (2.212) and (2.213). As will be




shown below, solution of this problem is connected with parametric stabilityof some auxiliary system of equations.

The above problem can be generalized onto the case of approximate n-thorder representation of additional motions.

Assume that there exist continuous sensitivity functions up to n + 1-thorder inclusive. Let, with account for (2.66) and (2.71),

Z (t, µ) − ∆(n)Y (t, µ) = V (n)(t, µ). (2.214)

The approximate representation

Z (t, µ) ≈ ∆(n)Y (t, µ), t ≥ t0, (2.215)

holds if for any > 0 there is µn() > 0 such that

| V (n)(t, µ) |< where 0 <| µ |< µn(), t ≥ t0. (2.216)

2.3.2 Auxiliary Theorem

THEOREM 2.11

Let us have a system of differential equations of n + 1-th order of the form

dµ

dt= 0,

dZ

dt= A(t)Z + G(Z,t,µ),

(2.217)

satisfying the following conditions:

1. the matrix A(t) is bounded on the half-axis

| A(t) |≤ d1 = const, t ≤ t0, (2.218)

2. Cauchy’s matrix H (t, τ ) for the homogeneous equation

dY

dt= A(t)Y (2.219)

satisfies the estimate

| H (t, τ ) ≤ d2e−ν 1(t−τ )

, t0 ≤ τ ≤ t < ∞, (2.220)




where d1 and ν 1 are positive constants;

3. the nonlinear function G(Z,t,µ) in an area of the space Z, µ cover-ing the origin is continuous with respect to all arguments and satis-

fies the estimate

| G(Z,t,µ) |≤ d3 | µ |ν 2 (| Z | +d4), (2.221)

where d3, d4, and ν 2 are positive constants;

Then, Equations (2.217) have a trivial solution Z = 0, µ = 0 and this solution is parametrically stable, i.e., for any > 0 there is η() > 0 such that from the condition |Z 0, µ| < ν () follows |Z (t), µ| < .

PROOF The existence of the trivial solution Z = 0, µ = 0 followsimmediately from (2.217), because from (2.221) we have G(Z,t, 0) = 0.Then, we prove parametric stability, i.e., Lyapunov stability of the trivialsolution Z = 0, µ = 0.

As follows from the conditions (2.218) and (2.220) [5, 62], there exists aquadratic form

v(Z, t) = Z T H (t)Z (2.222)

(where H (t) is a symmetric time-dependent square matrix) such that

dvdt

=∂v(Z, t)

∂t+

∂v(Z, t)

∂Z · A(t)Z

= −Z T Z = − | Z |2, (2.223)

where

| Z |=√

Z T Z. (2.224)

Moreover, there are constants 0 < k1 < k2 such that

k1Z T Z ≤ v(Z, t) ≤ k2Z T Z, t ≥ t0. (2.225)

Conditions (2.225) mean that the quadratic form v(Z, t) is positive definiteand bounded with respect to t for t ≥ t0. As follows from (2.225),

c

k2≤ Z T Z =| Z |2≤ c

k1where v(Z, t) = c. (2.226)

Then, we calculate the derivative dv/dt of the positive definite functionv(Z, t) given by (2.217). Obviously,

dv

dt=

∂v

∂t+ ∂v

∂Z ·A(t)Z + ∂v

∂Z ·G(Z,t,µ) . (2.227)






2.3.3 Sufficient Conditions of Applicability of FirstApproximation

Assume that the right side of the vector equation of additional motion(2.204) is continuously differentiable with respect to Z, µ for all t

≥t0 in

some area Γ covering the origin. Then, Equation (2.204) can be representedin the form

dZ

dt= A(t)Z + B(t)µ + G(Z,t,µ), (2.236)

where

A(t) =∂ F

∂Z

Z =0,µ=0

, B(t) =∂ F

∂µ

Z =0,µ=0

, (2.237)

and the function G(Z,t,µ) contains an aggregate of nonlinear terms withrespect to Z, µ.

Let Z (t, µ) be a single-parameter family of additional motions which are,for sufficiently small |µ|, inside the area Γ. Due to the above assumptions,there exists a continuous sensitivity function

U (t) =∂Z (t, µ)

∂µ µ=0, (2.238)

that is a solution of the following sensitivity equation

dU

dt= A(t)U + B(t) (2.239)


U (t0) =dZ 0(µ)

dµ µ=0

, (2.240)

where the starting moment t0 is assumed to be fixed, and the function(2.230) to be continuous.

THEOREM 2.12

Assume that for Equation (2.236) there are positive constants di, i =1, . . . , 4, and = ν j, j = 1, 2 such that

| A(t) |< d1, t ≥ t0 (2.241)




| H (t, τ ) ≤ d2e−ν 1(t−τ ), t ≥ τ ≥ t0, (2.242)

| B(t) | < d3, t ≥ t0, (2.243)

| G(Z,t,µ) | ≤ d4 | Z, µ |1+ν 2 , t ≥ t0. (2.244)

Then, for any > 0 there is µ() > 0 such that for |µ| < µ() we have

| Z (t, µ) − U (t)µ |≤| µ | , t ≥ t0. (2.245)

PROOF First, we note that from the conditions (2.241)–(2.243) therefollows the boundedness of all solutions of the sensitivity equation (2.239)for all t ≥ t0 [62]. Thus,

|U (t)

|≤d = const, t

≥t0. (2.246)

Now let

Z (t, µ) =1

µ(Z (t, µ) − U (t)µ). (2.247)

in (2.236). As a result, we obtain an equation with respect to Z

dZ

dt= A(t)Z + K (Z , t , µ), (2.248)

where

K (Z , t , µ) = G(µ(Z + U (t)), t , µ)µ−1. (2.249)

Taking a definite norm in n-th dimensional space |Z | and defining a normin the space Z, µ as

| Z, µ | + | Z | + | µ |, (2.250)

from the conditions (2.244), (2.246), and (2.249) we obtain

K (Z,t ,µ) ≤ d4| µ |

µZ + U (t)

, µ

1+ν 2

≤ d4 | µ |ν 2 (| Z | +d + 1).(2.251)

Then, (2.251) yields

K (Z,t, 0) = 0 (2.252)

and Equation (2.248) has a trivial solution Z = 0, µ = 0. Let us show

that this solution is parametrically stable. Indeed, as follows from (2.241),




(2.242), and (2.251), in the given case all the conditions set in Theorem 2.11hold.

Thus, for any > 0 there is η() > 0 such that if

| Z 0, µ |< η(), (2.253)

then

| Z (t), µ |< , t ≥ t0. (2.254)

Using (2.247) and (2.250), we find that if

1

| µ | | Z 0 − U 0µ | + | µ |< η(), (2.255)

then 1

| µ | | Z (t) − U (t)µ | + | µ |< , t ≥ t0, (2.256)

and, obviously,

| Z (t) − U (t)µ |<| µ | . (2.257)

Nevertheless, using (2.240), we can write the condition (2.255) in theform

1

|µ

| Z 0 − dZ 0

dµµ + | µ |< η(). (2.258)

The left side of (2.258) tends to zero as |µ| → 0, therefore, for any η)there is µ(η) > 0 such that Relation (2.255) holds for all |µ| < µ(η()). For

such µ we have (2.257).

Let us analyze the sufficient conditions of applicability of the first ap-proximation given by this theorem.

Conditions (2.241) and (2.243) mean uniform boundedness of the matrixA(t) and vector B(t) appearing in the sensitivity equation (2.239) with re-

spect to t for all t ≥ t0. Its is important that due to (2.237) the matrix A(t)is defined only by the base solution Y (t, α0), while the vector B(t) dependson the choice of parameter α in initial equations. Therefore, including thebase solution Y (t, α0) in various single-parameter families of solutions, wewill obtain the same matrices A(t), but, generally speaking, different vec-tors B(t). Hence, for the same base motion the condition (2.243) can betrue or false, depending on the choice of parameters.

As is known, the condition (2.242) imposed on Cauchy’s matrix ensuresasymptotic Lyapunov stability of the base solution for µ = 0(α = α0).Since the matrix H (t, τ ) is uniquely determined by the matrix A(t), the

fulfillment of the condition (2.242) depends only on the base solution and




is independent of the choice of a single-parameter family of solutions inwhich the base solution is included.

It is know that if the matrix A(t) in (2.239) is constant,

H (t, τ ) = eA(t−τ ) (2.259)

and the condition (2.242) reduces to the condition Re λρ < 0, ρ = 1, . . . , n,where λρ are the roots of the characteristic equation

det(A − λE ) = 0. (2.260)

Condition (2.244) means that nonlinear terms in the equations of ad-ditional motion are bounded with respect to t and small with respect to

µ, Z .Note that the conditions (2.241)–(2.243) can, in principle, be checked

immediately by the sensitivity equation. To check the validity of (2.244)it is necessary to analyze nonlinear terms in the equations of additionalmotion.

The above theorem can be used also for justification of the possibility toapply n-th approximation Z (n)(t, µ) (2.71) on large time intervals. Withthis aim in view, introduce the variable

Z (n)(t, µ) =1

µn

Z (t, µ) − Z (n)(t, µ)

, (2.261)

Then, we obtain an equation of the form (2.248). The fulfillment of theconditions of the theorem will, in this case, guarantee applicability of thecorresponding approximation.

2.3.4 Classification of Special Cases

The case when all the conditions of Theorem 2.12 hold will be called

nonsingular or regular . Various cases when these conditions are violated willbe called singular . Using the analysis of the theorem given in Section 2.3.3,singular cases can be divided into two types.

The first type includes problems for which one of the conditions (2.241)–(2.243) is violated. These conditions are characterized by the fact that theirvalidity can be checked immediately by the sensitivity equations (2.239).

Special cases of the second type are connected with violation of the con-dition (2.244) and depend on the form of the terms in the right side of equations of additional motion having order higher than one with respect

to Z, µ.




It can be shown by examples that Relation (2.208) can become invalidif any of the conditions (2.241)–(2.243) is violated. For instance, this mayhappen if the conditions (2.241)–(2.242) hold while (2.243) and (2.244) donot. In this case, the base solution is stable by Lyapunov, but the approxi-

mate Formula (2.208) is invalid for sufficiently large time intervals. Param-eters for which the conditions (2.241)–(2.242) hold, but (2.243)–(2.244) arefalse, will be called singular .

Let us show by examples the possibility of existence of singular param-eters, when the base motion is asymptotically stable, but Formula (2.208)does not give uniform approximation on the whole interval t ≥ t0.

Example 2.4

Let the equation of additional motion have the form

dz

dt+ z = sin µt, t ≥ 0. (2.262)

The base solution is the trivial one:

z = 0, µ = 0. (2.263)

Consider the family of periodic solutions of Equation (2.262)

zp(t, µ) = 1µ2 + 1

sin µt − µµ2 + 1

cos µt, (2.264)

that transforms to the trivial one for µ = 0.

The sensitivity equation (2.239) in this case has the form

du

dt+ u = t, (2.265)

so that

A(t) = −1, B(t) = t. (2.266)

Moreover, it is easy to see that

H (t, τ ) = e−(t−τ ). (2.267)

Thus, in the case under consideration, the conditions (2.241) and (2.242)hold, but (2.243) does not. Condition (2.244) is also violated. Let usshow that Formula (2.208) in this case does not give an approximation of

additional motion for sufficiently large time intervals.






2.4 Sensitivity of Self-Oscillating Systems in TimeDomain

2.4.1 Self-Oscillating Modes of Nonlinear Systems

Let us have an autonomous nonlinear system

dY

dt= F (Y, α), (2.274)

where Y and F are vectors, and α is a scalar parameter. Assume that thevector F is continuously differentiable with respect to all arguments.

In many problems corresponding to real physical phenomena, Equation(2.274) has a family of isolated periodic solutions Y p(t, α) depending on α:

Y p(t + T (α), α) = Y p(t, α), (2.275)

where T = T (α) is the period of self-oscillation depending on the parameter.

As is known, such solutions are called self-oscillatory . If the family of solutions (2.275) exists, it is associated with unknown initial conditions

Y p(0, α) = Y 0(α). (2.276)

Let α = α0 be a parameter value (α1 ≤ α ≤ α2) that is associated witha specific self-oscillating mode Y p0(t) with period T 0 = T (α0), so that

Y p0(t + T 0) = Y p0(t). (2.277)

It is very important for applications to investigate the dependence of self-oscillating modes on the parameter α. In the present section we consider thepossibilities of analyzing these relations on the basis of sensitivity theory.

Such an approach is connected with the proper account for a number of essential peculiarities that need be investigated specially. Let us enumeratesome of them.

1. The problem of determining self-oscillating mode on the basis of Equations (2.274) and the periodicity condition (2.274) is a bound-ary problem. In any special case it can have different number of solutions or, as a special case, have no solutions at all. Therefore,hereinafter we consider a specific isolated family of self-oscillating,

modes assuming that it does exist.




2. Under the above assumptions, we postulate the existence of a single-parameter family of self-oscillating modes defined by the initial con-ditions (2.276). Nevertheless, it is incorrect, in the general case, toassume that the function Y 0(α) is differentiable with respect to α,

hence, it is incorrect to assume that there exist sensitivity functionseven of the first order. Nevertheless, in the present paragraph, itwill be assumed that the initial conditions Y 0(α) and the period of self-oscillation T (α) are continuously differentiable functions of theparameter. For many applied problems such an assumption is jus-tified. Some general ideas on this topic will be presented below in theparagraph dealing with sensitivity of boundary problems.

3. If the vector Y 0(α) is differentiable with respect to α, there is asensitivity function

U p(t, α) = ∂Y p(t, α)∂α

. (2.278)

For α = α0 the corresponding sensitivity equation has the form

dU

dt= A(t)U + B(t), (2.279)

where

A(t) = ∂F ∂Y

Y =Y p(T,α0)

, B(t) = ∂F ∂α

Y =Y p(T,α0)

. (2.280)

Since the vector Y p(t, α0) is periodic by construction, the matrixA(t) and vector B(t) are also periodic with period T 0.

Thus, the sensitivity equation of a self-oscillating mode is a linearnonhomogeneous vector equation with a periodic coefficient matrixand periodic fundamental term.

For further investigation of the properties of the problem under consider-ation we need some results from the theory of linear differential equationswith periodic coefficients that are given in the next paragraph.

2.4.2 Linear Differential Equations with PeriodicCoefficients

Let us have a homogeneous equation

dU

dt = A(t)U, A(t) = A(t + T ). (2.281)




Denote by H (t) the square matrix satisfying Equation (2.281) and initialconditionH (0) = E , where E is an identity matrix of the correspondingdimensions. It can be shown [42, 132], that the matrix H (t) can be writtenin the form

H (t) = D(t)eNt, (2.282)

where the matrix D(t) is nonsingular and satisfies the conditions

D(t) = D(t + T ), D(0) = E, (2.283)

and the constant matrix N is given by

N =1

T ln H (T ) =

1

T ln M. (2.284)

The matrix

M = H (T ) = eNT (2.285)

is called the monodromy matrix . From (2.283) and (2.284) we have

H (t + T = D(t + T )eN (t+T ) = H (t)eNT = H (t)M. (2.286)

From (2.282) we can derive a general expression for the Cauchy’s matrix

for Equation (2.281)

H (t, τ ) = H (t)H −1(τ ) = D(t)eN (t−τ )D(τ ). (2.287)

By the change of variable

U = D(t)V (2.288)

Equation (2.281) transforms into the following equations with constant

coefficients dV

dt= NV. (2.289)

Using the variable V in the nonhomogeneous equation (2.279), we obtain

dV

dt= NV + B(t), (2.290)

where

B(t) = D−1

(t)B(t) (2.291)






It can be shown, so that all solutions of Equation (2.281) tend exponen-tially to zero as t → ∞, it is necessary and sufficient that the followingcondition hold for the roots of Equation (2.297):

Re λi < 0, i = 1, . . . , n . (2.298)

According to (2.295), this is equivalent to the requirement that all multi-plicators be inside the unit circle, i.e.,

| ρi |< 1, ρ = 1, . . . , n . (2.299)

2.4.3 General Properties of Sensitivity Equations

Let us show that the homogeneous part of sensitivity equations, i.e. equa-

tions in variations of self-oscillating mode,

dU

dt= AU,

has a periodic solution. Indeed, for α = α0 and Y (t) = Y p(t, α0), from(2.274) we identically have

dY p(t, α0)

dt≡ F (Y p(t, α0), α0). (2.300)

Differentiating the identity (2.300) with respect to t, we obtain

dX

dt=

∂F

∂Y

Y =Y p(t,α0)

X = A(t)X, (2.301)

where

X =dY p(t, α0)

dt. (2.302)

A comparison of (2.302) and (2.279) demonstrates that the equation invariation (2.281) has a periodic solution (2.302) that is associated with amultiplicator equal to 1.

Hereinafter, we assume that all multiplicators of the equation in variation(2.281), except for one equal to 1, are inside the unit circle. Then, by theknown theorem by Andronov and Vitt [42, 62], the self-oscillating modecorresponding to α = α0 is Lyapunov stable (not asymptotically).

In this case, the fundamental matrix H (t) of the equation in variation(2.281) can be represented in the form of a block matrix as

H (t) = Y p(t, α0) G(t) P, (2.303)






with a periodic matrix Q(t) with period 2T 0, that transforms Equation(2.281) into the system of equations

dz1

dt= q 1(t),

dzidt

=n

s=2

piszs + piz1 + q i(t), i = 2, . . . , n ,

(2.312)

where the functions q j(t), ( j = 1, . . . , n) are periodic with respect to t; pi

and pis, (i = 2, . . . , n, s = 2, . . . , n) are constants. Moreover, the charac-teristic equation of the second group of equations in (2.312) has all rootswith negative real parts.

Let

q 1(t) = q 10 + q 1(t), q 10 =1

2T 0

2T 0 0

q 1(t)dt. (2.313)

Hereinafter we assume that q 10 = 0. Then, from the first equation in (2.312)we obtain a particular solution

z1(t) = q 10t +

t

0

q 1(t)dt. (2.314)

The second term in (2.314) is periodic with respect to t. Substituting(2.314) into the remaining equations in (2.312), it can be easily seen thatthese equations have a solution of the form (2.309) with constant R and S .

Returning to the initial data, we prove the claim.

REMARK 2.7 As will be shown below, actually the vectors R(t) and

S (t) appearing in (2.309) have period T 0.

Since the sensitivity equations (2.279) have a particular solution of theform (2.309), their general solution can be represented as

U (t) = H (t)U + R(t)t + S (t), (2.315)

where U is an arbitrary constant vector. Since for t = 0 we have

U (0) = U + S (0), (2.316)




a general solution of (2.315) can be expressed in terms of initial data as

U (t, U 0) = H (t)(U 0 − S (0)) + R(t)t + S (t). (2.317)

From (2.317), (2.310) and (2.305) it follows that if R(t) ≡ 0, all the solu-tions of the sensitivity equations are not bounded. Hence, for the problemat hand the first approximation (2.208) does not give a result on sufficient-ly large time intervals. Indeed, for α sufficiently close to α0, additionalmotions associated with self-oscillating modes are bounded and cannot beapproximated by expressions of the form (2.315) on large time intervals.

Note that in the given case the condition (2.242) of Theorem 2.12 re-stricting the Cauchy’s matrix H (t, τ ) does not hold. Indeed, if the condition(2.242) holds for τ = 0, for t ≥ 0 we obtain

| H (t) |≤ d2e−ν 1t. (2.318)

But, as follows from (2.305)–(2.308), the condition (2.318) cannot be sat-isfied in the given case.

Thus, investigating sensitivity of self-oscillating modes we always haveto deal with a special case when the condition (2.242) of Theorem 2.12 isviolated. Nevertheless, we will also show that, despite of this fact, the firstorder sensitivity functions give important information on properties of the

system.Let us specify an explicit form of the general solution (2.317). With this

aim in view, we substitute (2.317) into (2.279), and, taking into accountthe properties of H (t, τ ), obtain

dR(t)

dtt + R(t) +

dS (t)

dt= AR(t)t + AS (t) + B(t), (2.319)

Then, comparing the coefficients of t, we find the following equation:

dR

dt= A(t)R, (2.320)

i.e., the vector R(t) is a 2T -periodic solution of Equation (2.281). Hence,

R(T ) = kY p(t, α0), R(t) = R(t + T ), (2.321)

where k is a constant, because Equation (2.281) has no other bounded

solutions.




Thus, from (2.321) and (2.317) we have

U (t, U 0) = H (t)(U 0 − S (0) + ktY p(t, α0) + S (t). (2.322)

2.4.4 Sensitivity Functions Variation over Self-OscillationPeriod

Let

∆U (t, U 0) = U (t + T, U 0) − U (t, U 0) (2.323)

be the variance of an arbitrary solution U (t, U 0) of Equation (2.281) overthe period of self-oscillation. Let us note a number of important propertiesof the variance ∆U (t, U 0) and the general solution (2.322).

PROPOSITION 2.8

For any U 0 the function ∆U (t, U 0) is a solution of the equation in variations (2.281).

PROOF Indeed, substituting t + T for t in (2.279), we obtain

dU (t + T )

dt= A(t)U (t + T ) + B(t). (2.324)

Subtracting (2.279) from (2.324) yields

d∆U

dt= A(t)∆U.

PROPOSITION 2.9

In fact, the vector S(t) in (2.322) has period T , i.e., S (t) = S (t + T ).

PROOF Let us calculate the variance ∆U (t, U 0), using (2.322):

∆U (t, U 0) = [H (t + T ) − H (t)](U 0 − S (0))

+ kT Y p(t, α0) + [S (t + T ) − S (t)].(2.325)

The left side and the first two terms in the right side satisfy Equation(2.281), therefore, we have also

d[S (t + T )

−S (t)]

dt = A(t)[S (t + T ) − S (t)]. (2.326)




Thus, the 2T -periodic function S (t + T ) − S (t) is a solution of the homo-geneous

equation (2.281). Hence, as in (2.281), we obtain

S (t + T ) − S (t) = lY p(t, α0) = lY p0(t), l = const. (2.327)

Substituting t + T for t in (2.327) yields

S (t) − S (t + T ) = lY p0(t). (2.328)

From (2.327) and (2.328) it follows that l = 0.

Next, calculate the constant k in (2.322). With this aim in view, consider

the identity

Y p(t + T (α), α) = Y p(t, α). (2.329)

Differentiating with respect to α, from (2.329) we obtain for α = α0

∆U p(t) = U (t + T 0) − U p(t) = −Y p0(t)dT

dα, (2.330)

where U p(t) is the sensitivity function of the self-oscillating mode, and

∆U p(t) is its variance over the period. As follows from (2.330), the varianceof the sensitivity function of the self-oscillating mode is periodic with periodT = T 0 = T (α0).

On the other hand, from (2.325) we have, bearing in mind that S (t+T ) =S (t),

∆U (t, U 0) = [H (t + T ) − H (t)](U 0 − S (0)) + kT Y 0p(t). (2.331)

From (2.331) it follows that the variance ∆U (t, α0) is periodic only if

U 0 = U p0 = S (0). (2.332)

In this case

∆U (t, U 0) = ∆U p(t) = kT Y 0p(t). (2.333)

Comparing (2.333) with (2.330), we find

k = −1

T

dT

dα . (2.334)




Returning back to Relation (2.322) and using (2.334), we find that ageneral solution of the sensitivity equation has the form

U (t, U 0) = H (t)(U 0−

S (0))−

Y 0p(t)t

T

dT

dα+ S (t). (2.335)

and the sensitivity function of self-oscillating mode is

U p(t) = −Y 0p(t)t

T

dT

dα+ S (t). (2.336)

Accordingly, the variance of an arbitrary solution of the sensitivity equationcan be transformed, by means of (2.331) and (2.305), to the form

U (t, U 0) = [H 2(t + T ) − H 2(t)](U 0 − S (0)) − Y 0p(t) dT dα

, (2.337)

Hence, for U 0 = S (0) we obtain (2.330).

Using the above relations, we will derive the differential equation deter-mining S (t). Substitute (2.335) into the sensitivity equation (2.279). Then,accounting for the fact that H (t) is a solution of the homogeneous equation,we obtain

dS

dt = A(t)S + B(t) +˙

Y 0p(t)

1

T

dT

dα , S (t + T ) = S (t). (2.338)

Let us also find the initial conditions determining the vector S (t). Asfollows from (2.336), for the sensitivity function of self-oscillating mode wehave

S (0) = U p(0) = U p0 =dY p(t, α)

dα

α=α0

. (2.339)

2.4.5 Sensitivity Functions for Periodicity CharacteristicsUsing the results of the preceding paragraphs, it is possible to propose a

technique of determination of the sensitivity functions for the main charac-teristics of self-oscillations.

First, consider the problem of determining the sensitivity function forthe period of self-oscillations, i.e., calculating the derivative dT

dα. With this

aim in view, we note that (2.308) and (2.337) yield

∆U (t, U 0) + Y 0p(t)dT

dα < d(1 + e−bT )e−bt, (2.340)




Therefore, independently of U 0, we have

limt→∞

∆U (t, U 0) + Y 0p(t)

dT

dα

= 0. (2.341)

As a special case, for each component of the vector relation (2.341) wehave

limt→∞

∆ui(t, U 0) + ypi(t)dT

dα

= 0. (2.342)

Let tk, k = 0, 1, 2, . . ., be any infinitely increasing sequence of observationmoments such that ypi(tk) = 0. Then, for sufficiently large k we have theapproximate equality

∆ui(tk, U 0) ≈ −ypi(tk) dT dα

, (2.343)

Hence,

dT

dα≈ −∆ui(tk, U 0)

ypi(tk)(2.344)

The approximate equality tends to a strict one as tk → ∞.

Next, consider the problem of determining the sensitivity function foramplitude of self-oscillation. We will call the value

Ai(α) = max0≤t≤T

| ypi(t, α) | . (2.345)

amplitude of the i-th phase coordinate.

Let tim(α) = ti(α) + mT (α), where m = 0, 1, 1, . . . and 0 ≤ ti(α) < T (α),be a sequence of argument values for which

Ai(α) =| ypi(tim(α), α) | . (2.346)

Below we give a method of evaluating the value dAi/dα which, accordingto (2.346), coincides with the derivative

d

dαypi(tim(α), α), (2.347)

or differs from the latter only in sign.

Since ypi(t, α) is a smooth function of its arguments,

dypi

dt (tim(α), α) = 0. (2.348)




Using (2.348), we obtain

d

dαypi(tim(α), α) =

∂

∂αypi(tim, α) = upi(tim, α), (2.349)

i.e., the values of the sensitivity function upi(t, α) at the moments t =tim(α) coincide with the sensitivity function for amplitude of self-oscillationup to the sign.

Transforming the vector formula (2.336) to the coordinate form, weobtain

upi(t) = −ypi(t)t

T

dT

dα+ si(t), (2.350)

With due account for (2.348) and (2.349) we obtain

d

dαypi(tim(α), α) = si(tim(α)), (2.351)

and, according to (2.345),

dAi

dα= sign ypi(tim(α), α)upi(tim(α))

= sign ypi(tim(α), α)si(tim(α)).(2.352)

Let us show that the value upi(tim, α) can be expressed in terms of thevalues of the variance of the sensitivity functions ∆upi(t, α). With this aimin view, we note that, acording to (2.305) and (2.306),

H (t)(U 0 − S (0)) =

= [Y p(t, α0) On,n−1 + On,1 G(t)] P (U 0 − S (0)).(2.353)

Assuming that

P = pik, i, k = 1, . . . , n; U 0 − S (0) = uoi − s0i, u = 1, . . . , n ,(2.354)

and using (2.306), we can easily verify that

H 1(t)P (U 0 − S (0)) = Y p(t, α0)f 0, (2.355)

where

f 0 =n

i=1

p1i(ui0

−si0) (2.356)




is a scalar constant depending on initial conditions. Taking (2.356), we canwrite Relation (2.335) in the form

U (t, U 0) = H 2(t)(U 0 − S (0)) + Y p(t)

f 0 −t

T

dT

dα

+ S (t). (2.357)

Using (2.348), (2.351) and (2.352), for t = tim(α) we obtain for the i-thcomponent of the vector equation (2.357)

ui(ti(α) + mT,U 0) =

= [H 2(ti(α) + mT )(U 0 − S (0))]i +dAi

dαsign ypi(ti(α)).

(2.358)

From (2.358) and (2.308) it follows that there exists the limit

limm→∞

ui(ti(α) + mT,U 0) =dAi

dαsign ypi(ti(α)). (2.359)

Indeed, since

∆U (ti(α) + mT,U 0) =

= U (ti(α) + (m + 1)T, U 0) − U (ti(α) + m T , U 0), (2.360)

we have

ui(ti(α) + mT,U 0) = ui(ti(α), α) +m−1s=0

∆ui(ti(α) + sT,U o). (2.361)

Equations (2.361) and (2.359) give the desired relation

dAi

dαsign yp(ti) == ui(ti) + lim

m→∞

m−1s=0

∆ui(ti + sT,U o)

, (2.362)

where ti = ti(α).

From (2.308) and (2.337) it follows that the series in the right side of (2.362) converges absolutely and the tendency to zero has exponential

character.




2.4.6 Practical Method for Calculating SensitivityFunctions

On the basis of the relations given above it is possible to develop a prac-tical method of calculating sensitivity functions for self-oscillating mode.

Consider an initial system of equations

Y = F (Y, α), Y 0(α) = Y p0(α), Y (t) = Y (t + T ) (2.363)

for a fixed parameter value α = α0, where Y p0(α) represents initial con-ditions of self-oscillating mode, and T is the corresponding period of self-oscillations.

Simultaneously, consider the system of equations

dV dt

= A(t)V + k(t)B(t), V 0 = 0, (2.364)

where A(t) and B(t) are the same as in the sensitivity equations (2.279),and the function k(t) is defined by

k(t) =

1, 0 ≤ t < T 0, t > T.

(2.365)

For 0

≤t < T Equation (2.364) coincides with the sensitivity equation,

while for t > T it coincides with the corresponding homogeneous system.The solution of Equation (2.365) with chosen initial conditions can be pre-sented on the interval 0 ≤ t < t in the following explicit form:

V (t) =

t 0

H (t)H −1(τ )B(τ )dτ, 0 ≤ t < T. (2.366)

For t = T we have

V (T ) =

T 0

H (T )H −1(τ )B(τ )dτ. (2.367)

Solving the homogeneous equation (2.281) with initial conditions (2.367),we obtain a solution of Equation (2.364) for t ≥ T as

V (t) = H (t, T )

T

0

H (T )H −1(τ )B(τ )dτ. (2.368)




According to (2.287),

H (t, T ) = H (t)H −1(T ), (2.369)

Hence, Equation (2.368) yields

V (t) = H (t)

T 0

H −1(τ )B(τ )dτ. (2.370)

Therefore,

V (0) =)

T

0

H −1(τ )B(τ )dτ. (2.371)

On the other hand, consider a partial solution of the sensitivity equations(2.366) for all t ≥ 0. Its variance for the period of oscillation is given by

∆V (t) = V (t + T ) − V (t) = H (t)

T 0

H (T )H −1(τ )B(τ )dτ. (2.372)

Due to the results of Section 2.4.4, the function (2.372) is a solution of thehomogeneous equation (2.281). Then, assuming that t = 0, from (2.371)we have

∆V (0) = H (t)

T 0

H −1(τ )B(τ )dτ, (2.373)

i.e., ∆V (t) = V (t + T ). thus, the solution V (t) of Equation (2.364) fort ≥ T coincides with the variance of the sensitivity function (2.366) for the

period of self-oscillation. Hence, due to (2.340), as t increases, the function(2.372) tends exponentially to the periodic function

−Y n(t)dT

dα. (2.374)

Therefore, calculating the function V (t) for sufficiently large t > T (whereit can be considered as a periodic function with a given accuracy), in coor-dinate form we obtain

vi(t) ≈ −ypi(t)

dT

dα , (2.375)




i.e.,

dT

dα≈ − vi(t)

ypi(t). (2.376)

Moreover, fixing, for each i, the moments tim = ti+mT , when the term of self-oscillating mode ypi has the maximal absolute value, and using (2.362),for sufficiently large m we will have the approximation

dAi

dαsign ypi(ti) ≈ vi(ti) +

ms=1

vi(ti + sT ). (2.377)

Since the series converges exponentially, in practice it is possible to use asmall number of terms in (2.377).

REMARK 2.8 For integration of (2.364) we can choose arbitrary initialconditions V (0) = V ). Let V (t) be the corresponding solution. Then, to

obtain the variance ∆V (t) Equation (2.364) is to be solved for t ≥ T withinitial conditions

∆V (T ) = V (T ) − V 0. (2.378)

Therefore, initial conditions taken in (2.364) are the most convenient ones,

because there is no need to recalculate initial conditions for t = T .

REMARK 2.9 Using this technique, there is no need to calculate pre-cisely initial conditions of self-oscillating mode Y 0(α) = Y p0(α). Choosingarbitrary initial conditions (sufficiently near to Y p0(α)), Equation (2.363)should be solved until the solution becomes periodic with a given accuracy.Then, Equation (2.363) is increased by (2.364), and the above procedure

must be repeated.

2.4.7 Application to Van der Paul Equation

A known van der Paul equation [63, 103] can be written in the form of asystem of two equations as

y1 = y2,y2 = −µ(y21 − y2) − y1.

(2.379)

Differentiating by µ, we obtain the sensitivity equation

u1 = u2,

u2 = −µ(2y1u1 − u2) − u1 − (y21 − y2).

(2.380)




An auxiliary system of equations (2.364) for this case has the form

v1 = v2,

v2 = −µ(2y1v1 − v2) − v1 − k(t)(y

2

1 − y2).

(2.381)

where the function k(t) is defined by (2.365).

As was demonstrated by simulation [14], for µ = 1 Equation (2.379) hasa periodic solution with initial conditions y10 ≈ 2.01, y20 = −0.903 · 10−3

and period T = 6.66 sec. Integrating together (2.379) and (2.381) withinitial conditions y10 = 2.01, y20 = −0.9 · 10−3, v10 = 0, and v20 = 0,we determine simultaneously u1(tj) = u1(t1) and

m−1k=0 ∆u1(t1 + kT , 0).

This integration process continues until the solution of (2.379) and (2.381)

becomes periodic. Then, using Formulas (2.376) and (2.377), we obtaindT/dµ = 0.84 and dA1/dµ = 0.0104. In a similar way we obtained dT/dµand dA1/dµ for µ = 2, 3, 4. Figures 2.2 and 2.3 demonstrate the curves

Figure 2.2

Curves T and dT/dµ versus µ

T , dT/dµ, A1, and dA1/dµ versus µ. The phenomenon of increasing thesensitivity function is illustrated by Figure 2.4. Figure 2.5 demonstratesthe variation ∆u1(t, U 0 = 0) and ∆u1(t, U 0 = 1). These curves show that,in fact, after time interval t = T the function ∆u1(t, U 0) becomes periodic

independently of initial conditions. Moreover, it can be easily seen from




Figure 2.3

Curves A1 and dA1/dµ versus µ

the same figure that after the interval t = T

∆u1(t, U 0) ≈ −dT

dµy1 = −dT

dµy2.

2.5 Sensitivity of Non-Autonomous Systems

2.5.1 Linear Oscillatory Systems

Consider vector equations of the form

dY

dt= A(τ )Y + B(τ ), τ = ωt, (2.382)

where ω is the frequency and is considered as a parameter. The matrixA(τ ) = A(ωt) and vector B(τ ) = B(ωt) are assumed to be continuouslydifferentiable and periodic with respect to τ . Hereinafter, without loss of generality, we assume that

A(τ + 2π) = A(τ ), B(τ + 2π) = B(τ ). (2.383)




Figure 2.4

Sensitivity function variation

Let also the homogeneous system

dY

dt= A(τ )Y (2.384)

be asymptotically stable in the considered frequency interval ω1 ≤ ω ≤ ω2

so that

H (t)H −1(ν )

< ce−λ(t−ν ), 0 ≤ τ ≤ ν < ∞ (2.385)

where c and λ are positive constants independent of ω.

As follows from (2.385) [63], the system under consideration for all ω1 ≤ω ≤ ω2 has a unique steady-state mode Y p(t) = Y p(t + 2π

ω) defined by

Y p(t) =

T 0

Φ(t, t − ν )B(ων )dν (2.386)

where the matrix Φ(t, τ ) is called pulse frequency response (PFR) of the




Figure 2.5

Variations of ∆u1(t, U 0)

system at hand and is defined by the following relations [86, 87]:

Φ(t, ν ) =

H (t)(E − M )−1H −1(ν ), 0 < ν < t < T,H (t)(E − M )−1MH −1(ν ), 0 < t < ν < T.

(2.387)

Since (2.386) is a particular solution of Equation (2.382), its generalsolution can be represented in the form

Y (t) = H (t)Y + Y p(t), (2.388)

where Y is an arbitrary constant vector. For further analysis, the generalsolution (2.388) can be written as

Y (t) = H (t)(Y 0 − Y p0) + Y p(t), (2.389)

where Y 0 = Y (0), Y p0 = Y p(0).

Consider, together with (2.382), the following equation:

dY

dt= A(τ )Y + B(τ ) + tB1(τ ), τ = ωt, (2.390)

where B1(τ ) = B1(τ + 2π).




We will search for a forced steady-state solution of Equation (2.390) of the form

Y (t) = Y 1(t) + tY 2(t), Y i(t) = Y i t +2π

ω , i = 1, 2. (2.391)

Substituting (2.391) into (2.390) and equating the coefficients with t in theleft and right sides, we obtain

dY 1dt

= A(τ )Y 1 + B(τ ) − Y 2,

dY 2dt

= A(τ )Y 2 + B1(τ ).

(2.392)

From (2.386) and the second equation in (2.392) it follows that

Y 2(t) =

T 0

Φ(t, t − ν )B1(ων )dν. (2.393)

Using Formula (2.386) to the first equation in (2.392), we obtain

Y 1(t) =T

0

Φ(t, t

−ν )[B(ων )

−Y 2(ν )]dν

= Y p(t) −T 0

Φ(t, t − ν )Y 2(τ )]dν

(2.394)

Similarly to (2.389), a general solution of Equation (2.390) can be repre-sented in the form

Y (t) = H (t)(Y 0 − Y 10) + Y 1(t) + tY 2(t). (2.395)

2.5.2 Sensitivity of Linear Oscillatory System

Differentiating Equation (2.382) with respect to ω, we obtain the equa-tions of the first-order sensitivity functions for steady-state mode:

dU

dt= A(τ )U + tA1(τ )Y p(t) + tB1(τ ), (2.396)

where

A1(τ ) =dA

dτ τ =ωt

, B1(τ ) =dB

dτ τ =ωt

(2.397)




are periodic with period 2π.

Using Formula (2.395), we find that a general solution of the sensitivityequation (2.396) has the form

U (t) = H (t)(U 0 − U 10) + U 1(t) + tU 2(t), (2.398)

where

U 2(t) =T 0

Φ(t, t − ν )[A1(ων )Y p(ν ) + B1(ων )]dν,

U 1(t) = −T 0

Φ(t, t − ν )Y 2(ν )dν.

(2.399)

The functions U 1(t) and U 2(t) are periodic solutions of the followingequations:

dU 1dt

= A(τ )U 1 − U 2,

dU 2dt

= A(τ )U 2 + A1(τ )Y p(t) + B1(τ ).

(2.400)

Since from (2.385) for ν = 0 it follows that

|H (t)

|< ce−λt, t

≥0, (2.401)

and the functions U 1(t) and U 2(t) are periodic, from Formula (2.398) wecan find that all solutions of the sensitivity equation (2.396) are bounded.At the same time, from (2.389) it follows that all solutions of the initialsystem (2.382) are bounded for t > 0. Therefore, all additional motions of the system (2.382) in the frequency range under consideration are bounded.Hence, in this case, the approximation

∆Y p(t) ≈ U p(t)∆ω (2.402)

where U p(t) is the sensitivity function of steady-state mode, is not correctfor sufficiently large time intervals.

Thus, in this problem we encountered a special case when the first-ordersensitivity functions cannot be used for approximation of additional motionover sufficiently large time intervals. This corresponds to the general The-orem 2.12, because in this case the condition (2.243) is obviously violated.

Nevertheless, as will be shown below, the bounded functions U 1(t) andU 2(t) can be used for solving sensitivity problems. A simulation block-diagram that makes it possible to obtain the functions (2.399) without

considering infinite processes is given in Figure 2.6.




Figure 2.6

Simulation block-diagram

Our main goal is now to obtain the sensitivity function for amplitude of forced oscillation (2.386). As in the previous paragraph, by amplitude Ai

of the component ypi of forced oscillation we will mean absolute deviation

from zero for a period .Substituting t + T = t + 2π/ω for t in (2.386), with due account for

periodicity of these functions we obtain

dU (t + T )

dt= A(τ )U (t + T ) + (t + T )A1(τ )Y p(t) + (t + T )B1(τ ). (2.403)

Subtracting (2.396) from (2.403), we find

d∆U dt = A(τ )∆U + T [A1(τ )Y p(t) + B1(τ )], (2.404)

where

∆U = U (t + T ) − U (t). (2.405)

On the other hand, since we have the identity

Y p(t, ω) = Y p(t + T, ω), (2.406)

differentiating with respect to ω we obtain

∂Y p(t, ω)

∂ω=

∂Y p(t + T, ω)

∂ω+ Y p(t)

dT

dω, (2.407)

or, equivalently,

∆U p(t) = U p(t + T ) − U p(t) =˙

Y p(t)

dT

dω =˙

Y p(t)

2π

ω2 . (2.408)




Hence, the variation of the sensitivity function U p(t) over a period is aperiodic function.

On the other hand, for any solution of the sensitivity equation (2.409)Equation (2.398) yields

∆U (t) = U (t + T ) − U (t)

= [H (t + T ) − H (t)](U 0 − U 10) + tU 2(t)

= h(t)(M − E )(U 0 − U 10) + T U 2(t),

(2.409)

where M is the monodromy matrix (2.285). As follows from (2.409), thefunction ∆U (t) is periodic only for U 0 = U 10. In this case

∆U p(t) = T U 2(t). (2.410)

Comparing (2.410) with (2.408), we find

U 2(t) =1

ωY p(t). (2.411)

Assuming U 0 = U p0 = U 10 in (2.398) and taking account of (2.411), weobtain

U p(t) = U 1(t) +t

ωY p(t). (2.412)

Let now Ai be defined as in (2.345). Then, similarly to (2.349) we have

dAi

dω= sign ypi(ti)upi(ti), (2.413)

where the time instant ti is defined by the condition Ai = |ypi(ti)|.Since ypi = 0, from (2.413) and (2.412) comes that

dAidω

= sign ypi(ti)u1i(ti), (2.414)

From the above relations it follows the following simple technique of con-structing the value dAi/dω. Consider the following system of vector equa-tions for arbitrary initial conditions.

dY

dt= A(τ )Y + B(τ ),

U 1

dt = A(τ )˜U 1 −

1

ω

dY

dt .

(2.415)






2.5.3 Sensitivity of Nonlinear Oscillatory System

The above approach can be used in the general case for a nonlinear non-autonomous oscillatory system. Consider the vector equation

dY

dt= F (Y, τ ), τ = ωt, (2.423)

where F is a continuousy differentiable function of arguments Y, τ such that

F (Y, τ ) = F (Y, τ + 2π). (2.424)

Assume that Equation (2.423) has a family of periodic steady state solutions

Y p(t, ω) = Y p

t +

2π

ω, ω

(2.425)

in some interval of frequencies ω1 ≤ ω ≤ ω2. Assuming that the solution(2.425) is differentiable with respect to ω, from (2.423) we can obtain thesensitivity equation of the steady-state mode. Differentiating (2.423) with

respect to ω gives the sensitivity equation

dU

dt= A(t)U + tB1(t), (2.426)

where

A(t) =∂F

∂Y Y = Y p(t)

τ = ωt

, B1(t) =∂F

∂τ Y = Y p(t)

τ = ωt

, (2.427)

By construction, A(t) = A(t + T ), B1(t) = B1(t + T ). Hereinafter weassume that in the chosen frequency range the condition (2.385) holds,so that the solution (2.425) is asymptotically stable by Lyapunov for allω1 ≤ ω ≤ ω2. As follows from (2.399), a general solution of the sensitivityequation (2.426) has the form

U (t) = H (t)(U 0 − U 10) + U 1(t) + tU 2(t), (2.428)




where U 1(t) and U 2(t) are periodic solutions of the following system of equations:

dU 1dt

= A(t)U 1 − U 2,

dU 2dt

= A(t)U 2 + B1(t).

(2.429)

Nevertheless, in the given case we can greatly simplify the solution, becauseit appears that

U 2(t) =1

ωY p(ti) =

1

ωF (Y p(t), ωt). (2.430)

This formula is proved by step-by-step repetition of the transformationsused for derivation of Relation (2.411). Moreover, similarly to (2.414),

dAi

dω= sign ypi(ti)u1i(ti), (2.431)

where ti(0 ≤ ti < T ) is the moment when the function ypi(t) takes themaximal absolute value. Therefore, to calculate the sensitivity functionsof amplitudes of steady-state mode components it suffices to obtain thefunction U 1(t). For this purpose we can use the first equation in (2.429),which, with account for (2.430), appears as

dU 1

dt = A(t)U 1 −1

ω F (Y p(t), ωt). (2.432)

A practical method of calculating the function U 1 can be described as fol-lows. Consider the system of equations

dY

dt= F (Y,ωt),

U 1dt

= A(t)U 1 − 1

ωF (Y,ωt).

(2.433)

If we choose initial conditions sufficiently near to Y p(0), due to asymptoticstability the solution Y (t) will tend to Y p(t), and, correspondingly, theright side of the second equation (2.433) will converge to the right side of (2.432). Therefore, the desired signal will establish at the output of theblock diagram shown in Figure 2.7.

Example 2.8

As an example, we consider Duffing’s equation

y + αy = βy3

= N cos ωt, (2.434)




Figure 2.7

Determination of U 1

which is equivalent to the following system of equations

dy1dt

= y2,

dy2dt

= −βy31 − αy2 + N cos ωt.

(2.435)

Differentiating (2.435) with respect to ω yields the sensitivity equation

du1dt

= u2,

du2

dt = −3βy21u1 − αu2 − tN sin ωt.

(2.436)

while the second equation in (2.433) can be written in an expanded formas

dv1dt

= v2 − 1

ωy2,

dv2dt

= −3βy21v1 − αv2 +1

ω

βy31 + αy2 − N cos ωt

.

(2.437)

where the notation is changed in order to avoid confusion.Returning to the variable y and referring to (2.437) and (2.435), we can

rewrite the system of equations (2.433) in the form

Y + αy + βy3 = N cos ωt,

v + αv + 3βy2v = −α

ωy − 2

ωy.

(2.438)

The steady-state periodic mode v = vp(t) makes it possible to calculate

the sensitivity function for amplitude of the periodic solution dA/dω. If for




t = ti the value yp(t) has the maximal absolute value, the value vp(ti) gives

the desired sensitivity function up to sign.

2.6 Sensitivity of Solutions of Boundary-ValueProblems

2.6.1 Boundary-Value Problems Depending on Parameter

The methods of constructing sensitivity functions developed in the pre-ceding paragraphs require that initial conditions (2.21) are known, i.e., in

fact we dealt with the sensitivity of Cauchy’s problem. In that case, forcontinuous first-order sensitivity functions of a single-parameter family of solutions of Equation (2.20) to exist it is sufficient that the vector F (Y,t,α)be continuously differentiable with respect to Y, α, and the initial conditions(2.21) be continuously differentiable with respect to α. The existence con-ditions for higher-order sensitivity functions were formulated on the basisof the same proposition.

Nevertheless, in many applied problems, initial conditions of the fam-ily of the solutions under consideration are not given and, conversely, are

unknown to be found. Such a situation arises, for example, for variousboundary-value problems. In such cases, the question of existence of corre-sponding family of solutions and, the more so, of differentiability of initialconditions with respect to parameter, needs to be investigated specially.

Let us give examples of some general important problems in control the-ory that can be reduced to boundary-value problems for systems of ordinarydifferential equations.

1. Determination of non-autonomous periodic oscillations . Let us havea system of equations of the form (2.20)

dY dt

= F (Y,t,α), (2.439)

and let F (Y,t,α) = F (Y, t + T, α). The problem of determining periodicsolutions with period T reduces to determining solutions Y = Y p(t) forwhich the following boundary conditions hold:

Y p(0) = Y p(T ). (2.440)

If in some interval of parameter value α1 ≤ α ≤ α2 there is a desired




periodic solution satisfying the initial conditions Y p(0) = Y p0(α), then, thevector Y p0(α) is not always differentiable with respect to α [63].

2. Determination of periodic oscillation of autonomous systems . Let ushave an autonomous system

dY

dt= F (Y, α), (2.441)

and it is required to find its periodic modes Y p(t) = Y p(t + T ) (self-oscillations). In this case, the boundary conditions (2.440) hold as well.Nevertheless, as distinct from the previous case, period T of desired periodicsolutions is an unknown value depending on the parameter, i.e., T = T (α).The, as is known [63, 103], one of the components of the initial conditionsvector can be assumed. Therefore, based on Equation (2.441) and bound-

ary conditions (2.440) it is required to find n values, viz. n − 1 componentsof the vector Y p0 and period T .

Let us, for example, have the following second-order equation

y + f (y, y, α) = 0, (2.442)

which is equivalent to the system of equations

dy1

dt

= y2,

dy2dt

= −f (y1, y2, α).(2.443)

Let y1p(t) = y1p(t + T ) and y2p(t) = y2p(t + T ) be periodic solutions of Equations (2.443). Since the function yp(t) = y1p(t) has equal values fort = 0 and t = T , there is a moment t(0 ≤ t < T ), when y2p(t) = yp(t) = 0.Therefore, the desired initial conditions can be found in the form

y1(0) = y10, y2(0) = 0. (2.444)

Now let y1(t, y10, y20, α) and y2(t, y10, y20, α) be a general solution of thesystem of equations (2.443) such that

y1(0, y10, y20, α) = y10, y2(0, y10, y20, α) = y20. (2.445)

Then, with account for (2.444), the periodicity conditions (2.440) of thedesired solution can be written in the form

y1(T, y10, 0, α) − y10 = 0, y2(T, y10, 0, α) = 0. (2.446)




In principle, the desired values y10(α) and T (α) can be obtained from(2.446).

3. Boundary-value problems of various types arise in variation calculusand optimal control problems. For example, for the problem of minimizing

the simplest functional

J =

T (α) 0

Φ(y, y,t ,α)dt, (2.447)

depending on a parameter, the desired extremals are determined by Euler’sequations

d

dt

∂ Φ

∂ y− ∂ Φ

∂y= 0, (2.448)

satisfying the boundary conditions, where y1(α) and y2(α)

y(0) = y1(α), y(T (α)) = y2(α), (2.449)

are given functions. In problems with free endpoints, the functions y1(α),y2(α), and T (α) can be unknown.

Generalizing the above examples, let us formulate a general scheme of two-point boundary-value problem. It is assumed that there are a vectorequation (2.439) and boundary conditions in the form

G(Y 0, Y 1, t0, t1, α) = G(Y (t0), Y (t1), t0, t1, α) = 0, (2.450)

where G is a vector functional, t0 = t0(α) and t1 = t1(α) are functions inα, which are, in the general case, unknown and determined during solutionof the boundary-value problem. Moreover, we will assume that the totalnumber of scalar equations defined by the boundary conditions (2.450) isequal to the number of variables of the boundary-value problem.

If boundary conditions relate the values of the desired solution in morethan two points, such a boundary problem is called multipoint . For in-stance, the boundary conditions for a three-point problem have, similarlyto (2.450), the form

G(Y (t0), Y (t1), Y (t2), t0, t1, t2, α) = 0. (2.451)

2.6.2 Sensitivity Investigation for Boundary-ValueProblems

If Equations (2.439) and boundary conditions (2.450) define a single-

parameter family of solutions Y k(t) = Y k(t, α) in an interval of parameter




values α1 ≤ α ≤ α2, the problem of calculation of the following sensitivityfunction arises naturally:

U = U k =∂Y k(t, α)

∂α. (2.452)

In the general case, the question of existence of a family of solutions Y k(t, α)and, the more so, of existence of sensitivity functions (2.452) is very difficult.On the other hand, as follows from Theorem 2.1, a continuous sensitivityfunction (2.452) does exist is the vector F (Y,t,α) is continuous with respectto all its argument, continuously differentiable with respect to Y and α, andif there exist continuous derivatives dY k(t0)/dα and dt0/dα. Hereinafterwe assume that the vector F (Y,t,α) satisfies these requirements. Then,existence conditions of the sensitivity function U k reduce to conditions of

existence and differentiability for the vector of initial conditions Y k0(α).In a special case when the starting point is fixed, the sensitivity functionU k exists and is differentiable if the vector of initial conditions Y k0(α) iscontinuously differentiable.

Then, we show that a boundary-value problem can, in principle, be re-duced to a problem of solving a nonlinear system of equations determiningthe initial conditions vector. For concreteness, at first we assume that thereis a boundary-value problem (2.439), (2.450), where the values t1 and t2are fixed.

LetY = Y (t, Y 0, t0, α), (2.453)

be a general solution of Equation (2.439) such that

Y (t0, Y 0, t0, α) = Y 0. (2.454)

Due to the above assumptions on the properties of the vector F (Y,t,α),the right side of (2.453) is continuously differentiable with respect to all

arguments. As follows from (2.453),

Y (t1) = Y (t1, Y 0, t0, α). (2.455)

Substituting (2.455) into the boundary conditions (2.450), we obtain

G(Y 0, Y (t1, Y 0, t0, α), t0, t1, α) = 0. (2.456)

Thus, if the vector Y 0(α) defines a solution of the boundary-value problem

(2.439), (2.450), it is a solution of the vector equation (2.456). Conversely,




if a vector Y 0(α) is a solution of Equation (2.456), the family

Y (t, α) = Y (t, Y 0(α), t0, α) (2.457)

gives a single-parameter family of solutions of the boundary problem.Hence, the boundary-value problem at hand is equivalent to the problem

of solving the vector equation (2.456).

If the moments t0(α) and t1(α) are known, Equation (2.456) holds, but itmust have a special structure such that it could be possible to find, togetherwith t0(α) and t1(α), all unknown components of the vector Y 0(α). Assumethat for α = α0, α1 ≤ α0 ≤ α2, Equation (2.456) has a solution Y 0(α), t(α),i.e., the initial boundary-value problem has a solution

Y k(t, α0) = Y (t, Y 0(t0, Y 0(α0), t0(α0), α0). (2.458)

Then, for the sensitivity function (2.452) to exist it is sufficient that Equa-tion (2.456) be solvable in a locality of the point α = α0, and the solutionY 0(α), t0(α) be differentiable for α = α0. Sufficient conditions of existenceof such a solution and a method of its determination can be obtained usingimplicit functions theory.

2.6.3 Implicit Functions Theorems

In this paragraph we consider the simplest properties of implicit functionsneeded later [18, 114].

THEOREM 2.13

Let us have a system of equations

f i(α, y1, . . . , yn) = 0, i = 1, . . . , n , (2.459)

where f i are continuous real functions of real arguments, and let the left sides of the equations become zero at the point K (α0, y10, . . . , yn0). Then,if in a locality R of the point K the functions f i have partial derivatives with respect to y10, . . . , yn0, which are continuous at the point K , and the

following functional determinant:

J = det

∂f 1∂y1

. . .∂f 1∂yn

. . . . . . . . .∂f n

∂y1. . .

∂f n

∂yn

(2.460)




is not zero at the point K , the system of equations (2.459) has in a locality of the point K a unique continuous solution

yi = φi(α), i = 1, . . . , n , (2.461)

satisfying the condition

yi(α0) = yi0, i = 1, . . . , n . (2.462)

THEOREM 2.14

If, in addition to the conditions of Theorem 2.13, the left sides of Equa-tions (2.459) are continuously differentiable with respect to y10, . . . , yn0, α,

the functions φi(α) are continuously differentiable in a locality of the point α = α0.

Then, we presented a practical method for calculating the derivativesdφi(α)/dα, provided that they do exist. With this aim in view, we differ-entiate Equations (2.459) with respect to α for α = α0:

n

h=1

∂f i∂yk

· ∂φk

dα= −∂f i

∂α. (2.463)

Solving Equations (2.463) with respect to dφk/dα, we obtain

dφi

dα

α=α0

= −J iJ

(2.464)

where J is the Jacobian (2.460), and the determinants J i have the form

J = det

∂f 1∂y1

. . .∂f 1

∂yi−1

∂f 1∂α

∂f 1∂yi+1

. . .∂f 1∂yn

. . . . . . . . . . . . . . . . . . . . .∂f n∂y1

. . .∂f n

∂yi−1

∂f n∂α

∂f n∂yi+1

. . .∂f n∂yn

. (2.465)

Let us note also that the fact that the Jacobian (2.460) is zero does notmean that the solution (2.461) does not exist or does not possesses requiredproperties. A detailed investigation of a special case when J = 0 is given

in [18].




2.6.4 Sensitivity Functions of Solution of Boundary-ValueProblems

Let us apply the implicit function theorems to investigating Equations(2.456) and assume, for concreteness, that the values t

0and t

1are given

and it is required to find all components of the vector Y 0. This calculationtechnique can be used without modifications for the general case as well.

Assume that the vector G = (g1, . . . , gn) appearing in the boundaryconditions (2.450) is continuously differentiable with respect to all its ar-guments. To calculate the Jacoby matrix

I = det

∂g1∂y10

. . .∂g1

∂yn0

. . . . . . . . .

∂gn∂y10

. . . ∂gn

∂yn0

(2.466)

where g1, . . . , gn and y10, . . . , yn0 are components of the vectors G and Y 0,respectively, we employ the rules of vector differetiation.

Differentiating the left side of (2.456) with respect to the vector Y 0, wehave

I =∂G(Y 0, Y (t, Y 0, t0, α), α)

∂Y 0

+∂G(Y 0, Y (t1, Y 0, t0, α), α)

∂Y 1

∂Y (t1, Y 0, t0, α)

∂Y 0.

(2.467)

But, as follows from (2.64) and (2.65), the matrix

H (t, t0) =∂Y (t, Y 0, t0, α), α)

∂Y 0(2.468)

is a solution of the equation in variations

dH (t, t0)

dt= A(t)H (t, t0) (2.469)

satisfying the initial conditions H (t0, t0) = E , i.e., the matrix H (t, t0) isthe Cauchy’s matrix for the equation in variations. From (2.468) we have

∂Y (t1, Y 0, t0, α)

∂Y 0= H (t

1, t0

) (2.470)




and, correspondingly, the Jacobian of the problem under consideration hasthe form

J (α) = det I = det

∂G

∂Y 0+

∂G

∂Y 1H (t1, t0)

. (2.471)

Then, using implicit functions theorems, we can formulate the followingproposition.

THEOREM 2.15

Assume that the right side of Equation (2.449) and vector G appearing in (2.450) are continuously differentiable with respect to Y 0, Y 1, α. Assume also that for α = α0 the associated boundary problem has a solution Y =Y k(t, α). Then, if the following condition holds:

J (α0) = 0, (2.472)

in a locality of α0 there is a single-parameter family Y (t, α) of solutions of the boundary-value problem, and the initial vector Y k(t0, α) is continuously differentiable with respect to α.

Assuming that the conditions set in Theorem 2.15 hold, let us calculatethe derivative dY k(t0, α)/dα. With this aim in view, we consider, accordingto Section 2.6.2, the equation

G(Y 0(α), Y (t1, Y 0, t0, α), α) = 0. (2.473)

The complete derivative with respect to α is given by

∂G

∂Y 0

dY 0dα

+∂G

∂Y 1

∂Y

∂Y 0

dY 0dα

+∂Y

∂α

+

∂G

∂α= 0, (2.474)

or

J (α)dY 0

dα=

∂G

∂Y 1

∂Y

∂α −∂G

∂α, (2.475)

Hence,

dY k(t0, α)

dα= −

J −1

∂G

∂Y 1

∂Y

∂α+

∂G

∂α

α=α0

. (2.476)

Let us determine the vector

U =∂Y (t1, Y 0(α), t0, α)

∂αα=α0

(2.477)




appearing in (2.476). Since for differentiation in (2.477) the values t0, t1and Y 0 are assumed to be fixed, the function U is a solution of the sensitivityequation for zero initial conditions. If the sensitivity equation has the form

dU dt

= A(t)U + B(t), (2.478)

where the matrix A(t) and vector B(t) are determined for α = α0, y =Y k(t, α0), its solutions for zero initial conditions has the form

U (t, α) =

t t0

H (t, τ )B(τ )dτ, U (t0, α) = 0. (2.479)

Therefore

U =

t1 t0

H (t1, τ )B(τ )dτ. (2.480)

Substituting (2.480) into (2.476), we finally obtain

dY k(t0, α)

dα

=

−J −1

∂G

∂Y 1

t1

t0

H (t1, τ )B(τ )dτ +∂G

∂αα=α0

. (2.481)

2.6.5 Sensitivity of Non-Autonomous Oscillatory System

In this section we apply the above results to the sensitivity investigationof non-autonomous periodic oscillations defined by the periodic equation(2.439) and boundary conditions (2.440) and assuming, at first, that pe-riod T is independent of the parameter. Taking t0 = 0, we can write theboundary conditions in the form (2.456) as

Y 0 − Y (T, Y 0, 0, α) = 0. (2.482)

To calculate the Jacoby matrix we employ (2.467) and (2.470). In the givencase,

∂G

∂Y 0= E,

∂G

∂Y 1= −E. (2.483)

Moreover,

H (t1, t0) = H (T, 0) = H (T ) = M, (2.484)




where m is the monodromy matrix (2.285). Therefore, the Jacoby matrixis

I = E − M. (2.485)

Condition (2.472) takes the form

det(E − M (α0)) = 0. (2.486)

Thus, we proved the following theorem.

THEOREM 2.16

If for α = α0 the periodic system (2.439) has a solution with period T , the

right side of (2.439) is continuously differentiable with respect to Y, α, and the condition (2.486) holds, there is a single-parameter family of periodic solutions Y = Y p(t, α) and the vector of initial conditions Y 0 = Y p(0, alpha)is continuously differentiable with respect to α for α = α0.

Let us note that according to Section 2.4.2, the condition (2.486) meansthat there should be no solutions with period T of the equation in variations(2.281).

Next, using (2.481), we obtain an explicit expression for the derivativedY 0/dα. Referring to (2.483)–(2.485) and using the fact that ∂G/∂α = 0,we have

dY 0dα

= (E − M )−1T 0

H (T )H −1(τ )B(τ ) dτ |α=α0

=

(E − M )−1 − E T 0

H −1(τ )B(τ ) dτ |α=α0.

(2.487)

By writing the solution of the sensitivity equation (2.478) with initial con-ditions (2.487), we obtain the sensitivity function of forced oscillation:

U p(t) = H (t)dY 0dα

+

t 0

H (t, τ )B(τ )dτ. (2.488)

It can be easily verified that U p(0) = U p(T ).

Then, let period T be a known function of the parameter, i.e., T = T (α).

For example, if α = ω is the frequency of periodic excitation, then T =




2π/T . In this case, instead of (2.482), we have

Y 0 − Y (T (α), Y 0, 0, α) = 0. (2.489)

Since the function T = T (α) is assumed to be known, the Jacoby matrix of Equations (2.489) coincides with (2.485), and the condition (2.486) remainsvalid. Hence, the assumptions made in Section 2.5.3 ensure the existenceof sensitivity functions of additional mode, and this serves as a justificationfor the method presented in Section 2.5.

If (2.486) holds, differentiating (2.489) with respect to α we have

dY 0dα

− ∂Y

∂Y 0

dY 0dα

− ∂Y

∂α− Y (T (α))

dT

dα= 0. (2.490)

If for α = α0 the equation has a solution

Y 0(α0) = Y p0, Y (0) = F (Y p0, 0, α0) ,

from (2.490) we obtain, similarly to (2.487),

dY p0dα

= (E − M (α))−1

M (α)

T

0

H −1(τ )B(τ )dτ + Y p0dT

dα

α=α0

(2.491)

2.6.6 Sensitivity of Self-Oscillatory System

As was noted above, the problem of determining self-oscillatory modes isan example of a boundary-value problem where one of the endpoint t1 =T (α) is unknown. In this case, the boundary conditions (2.440) hold aswell, but, as was said before, one of the components of the vector Y 0, forinstance, y0n, can be assumed known, and the value T is unknown. For

example, let

y0i = y, i = 1, . . . , n − 1, y0n = a = const, T = yn, (2.492)

Let Equation (2.489) for α = α0 have a solution

yi = yi0, i = 1, . . . , n − 1, yn0 = T 0 = T (α0) (2.493)

and, correspondingly, Equation (2.441) have a periodic solution Y p(t) =

Y (t + T 0).




Let us calculate the Jacobian (2.471) of Equations (2.489) at the point(2.493). Rewrite Equation (2.489) in a scalar form

gi = yi

−yi(yn, y1, . . . , yn−1, 0, α) = 0,

gn = a − yn(yn, y1, . . . , yn−1, 0, α) = 0. i = 1, . . . , n − 1 , (2.494)

Hence, with account for (2.493) and (2.484), we have

∂gi∂ yk

α=α0

= 1 − mik(T 0), i, k = 1, . . . , n . − 1, (2.495)

where mik(T 0) are components of the monodromy matrix M (T 0) = M (α0).Moreover,

∂gi

∂ yn

α=α0

= −ypi(T 0) = −ypi(0), i = 1, . . . , n . (2.496)

From the last equation in (2.494) we have

∂gn

∂ yi

= −mni(T 0), i = 1, . . . , n − 1. (2.497)

Thus, the Jacoby matrix has the form

I a(α0) =

1 − m11(T 0) . . . −m1,n−1(T 0) yp1(0)

. . . . . . . . . . . .−mn−1,1(T 0) . . . 1 − mn−1,n−1(T 0) −ypn−1(0)−mn,1(T 0) . . . −mn,n−1(T 0) −ypn(0)

. (2.498)

As a result, we can formulate the following proposition.

THEOREM 2.17

If for α = α0 Equations (2.494) have a solution (2.493) and the determi-nant of the Jacoby matrix (2.498) is not zero, i.e.,

det I a(α0) = 0 (2.499)

there exists a single-parameter family of solutions Y = Y p(t, α), and, more-over, initial conditions Y p(0) = Y p(0, α) and period T = T (α) are continu-

ously differentiable with respect to α.




Let us obtain explicit formulas for calculation of corresponding deriva-tives provided that all the conditions of Theorem 2.17 hold.

Differentiating (2.494) with respect to α, we have

dyidα

=n−1s=1

mis(T 0)dys

dα+ ypi(0)

dyndα

+∂yi

∂α

α=α0

,

0 =n−1s=1

mns(T 0)dysdα

+ ypn(0)dyndα

+∂yn

∂α

α=α0

,

(2.500)

Introducing the vector of unknowns

X 0dy1

dα

, . . . ,dyn

dα ,

we can write Equation (2.501) in a vector form

I a(α0)X 0 = U 0, (2.501)

where I a(α0) is the Jacoby matrix (2.498), and the vector U 0 is given, withaccount for (2.480), by

U 0 =

T 0

H (T )H −1(τ )B(τ )dτ = M (α0)

T 0

H −1(τ )B(τ )dτ. (2.502)

As follows from (2.501),

X 0 = I −1a (α0)U 0. (2.503)

Let us note that ypn = const due to statement of the problem. Then, initialconditions for sensitivity function of self-oscillatory mode have the form

ui0 =dyidα

, i = 1, . . . , n, un0 = 0, (2.504)

where dyi/dα, i = 1, . . . , n − 1 are the first n − 1 components of the vector(2.503). The last component of this vector gives the value dT/dα|α=α0 .

It should be taken into account that the condition (2.486) is never sat-isfied for self-oscillatory mode, because the corresponding equation in vari-ations (2.281) has a periodic solution, and, therefore, the characteristicequation (2.292) has a root ρ = 1. Nevertheless, as follows from the results

of [49], if all the remaining roots of the characteristic equation (2.487) differ






From (2.505) we have

U 1 = H (t1, t0)U 0 +

t1

t0

H (t1, τ )B(τ )dτ. (2.510)

Substituting (2.510) into (2.509) yields

IU 0 = K, (2.511)

where the matrix I is defined by (2.467) and vector K is given by

K = −∂G

∂α− ∂G

∂Y 1

t1 t0

H (t1, τ )B(τ )dτ. (2.512)

Due to (2.472), the matrix I is not singular, and, therefore, Equation (2.511)yields

U 0 = I −1K. (2.513)

Thus, a general expression for the sensitivity function of the boundary-value problem has the form

U (t) = H (t, t0)I −1K +

t1

t0

H (t, τ )B(τ )dτ, t0 ≤ t ≤ t1. (2.514)

If the sensitivity function (2.514) has been obtained in some way, thesolution of initial boundary-value problem for sufficiently small |µ| has theform

Y k(t, α0 + µ) ≈ Y k(t, α0) + U (t)µ. (2.515)




Chapter 3

Sensitivity of Finite-Dimensional Discontinuous Systems

3.1 Sensitivity Equations for Finite-DimensionalDiscontinuous Systems

3.1.1 Time-Domain Description

We will consider systems described by different vector differential equa-tions on different time intervals:

dY dt = F i(Y, t), ti−1 < t < ti, (3.1)

where Y is a vector of unknowns, F i a vector of nonlinear functions. Here-inafter, all vectors F i are assumed to be continuously differentiable withrespect to all arguments.

Moments ti, when one equation (3.1) is changed for another, will becalled the switching moments . It is also assumed that the vector Y isdiscontinuous at the switching moment ti so that

Y +i = Φi(Y −i , ti), (3.2)

where Y +i = Y (ti + 0) and Y −i = Y (ti − 0) are the values of the vector Y before and after the break, and Φ is a vector of nonlinear functions that arecontinuously differentiable with respect to all arguments. Relation (3.2) willbe called break condition . In the simplest case when the solution remainscontinuous at the switching moments, the break condition transforms inthe following continuity condition:

Y +i = Y

−

i = Y i. (3.3)




The switching moments ti can, in the general case, be found from someconditions of the form

f i(Y −i , ti) = 0, (3.4)

that will be called switching conditions . The scalar functions f i in (3.4) areassumed to be continuously differentiable with respect to all arguments.

Having a complete system of equations (3.1)–(3.4) that describe a dis-continuous system in the time domain, it is possible to construct solutionsassociated with various initial data. Let us have an initial condition

Y 0 = Y +0 = Y (t0 + 0). (3.5)

Then, integrating Equations (3.1) successively taking into account breaksand switching conditions we can, at least in principle, obtain a discontinuous

solutionY = Y (t, Y +0 ), Y (t0 + 0, Y +0 ) = Y +0 . (3.6)

Assume that this process of continuing the solution is possible up to themoment te > t0. Let also i , j , k, . . . be a sequence of numbers of equations(3.1) that was to be integrated while continuing the solution. This sequencewill be called type of motion on the interval (t0, te). So type of motion canbe symbolically defined in the form

i, j, k, . . . , t0, tk. (3.7)

Thus, the mathematical description of a discontinuous system in thetime domain used in this chapter includes motion equations (3.1) on theintervals between the switching moments, switching conditions (3.4) andbreak conditions (3.2).

We note that mathematical description of real discontinuous systems in-cludes, except for the above equations, some additional relations defined theorder of changing one equation (3.1) by another. These conditions are ob-tained by preliminary analysis of a mathematical model of the discontinuous

system under consideration. Therefore, the description of a discontinuoussystem given above as an initial one is, in fact, a result of fairly complexpreliminary analysis.

3.1.2 Time-Domain Description of Relay Systems

As an illustration, we consider the simplest relay system [13, 113] de-scribed by the equations

dY

dt = A Y + Hf (σ), σ = J

T

Y, (3.8)




where A is a constant matrix, H and J are constant column vectors, T

denotes the transpose operator, f (σ) is a nonlinear function shown in Fig-ure 3.1 having the following analytical representation

f (σ) = kp sign σ =

kp, if σ > 0,−kp, if σ < 0.

(3.9)

Figure 3.1

Relay characteristic

Hereinafter we consider continuous solutions of Equations (3.8) and (3.9),for which the conditions (3.3) hold for any switching moment.

Equation

g(Y ) = J T Y = σ = 0 (3.10)

defines a plane containing the origin in the phase space of variables Y . Theplane (3.10) divides the phase space into two areas: N + where σ > 0, andN − where σ < 0.

As follows from (3.8) and (3.9), in areas N + and N − we have, respectively,the following systems of equations:

dY

dt = A Y + Hkp, Y ∈ N +, (3.11)dY

dt= A Y − Hkp, Y ∈ N −. (3.12)

Equations (3.11) and (3.12) describe motion along the right and left branchof the nonlinear characteristic (3.9), respectively.

Hereinafter the motion of the system under consideration described byEquations (3.11) and (3.12) will be called normal .

Nevertheless, for complete description of all possible solutions of the relay

system (3.8) in the time domain it is not sufficient to fix Equations (3.11)




and (3.12) and switching conditions (3.10). Let Y 0 = Y (t0) and the initialvalue of the variable σ is such that

σ0 = J T Y 0 < 0. (3.13)

Then, integrating (3.12), we obtain [13, 86]

Y (t) = eA(t−t0)Y 0 − A−1

eA(t−t0) − E

H, (3.14)

where eAt is a matrix exponential, and

σ(t) = J T eA(t−t0)Y 0 − J T A−1 eA(t−t0) − E H. (3.15)

the motion of the system (3.8) is described by relations (3.14) and (3.15)until the corresponding trajectory intersects the switching surface (3.10).The switching moment t1 can be found from the equation

σ(t1) = J T eA(t1−t0)Y 0 − J T A−1

eA(t1−t0) − E

H = 0. (3.16)

Assume that

σ(t1 − 0) > 0. (3.17)

From (3.12) it follows that

σ(t1) = J T A Y (t1) − J T Hkp, (3.18)

and Relation (3.17) can be written in the form

J T

A Y (t1) − ρkp > 0, (3.19)

where the constant ρ is given by

ρ = J T H. (3.20)

Having found the moment t1 from Equation (3.16), we obtain

Y (t1) = e

A(t1−t0)

Y 0 − A

−1 e

A(t1−t0)

− E

H. (3.21)




Taking the value (3.21) as the initial one, we can continue the solutionover the interval t1 < t < t2. Nevertheless, there are several possibilitiesthat will be analyzed separately.

1. Assume that for t > t1 the motion continues in the area σ > 0. Then,

solving Equation (3.11) with initial condition (3.21), we obtain

Y (t) = eA(t−t1)Y (t1) + A−1

eA(t−t1) − E

H, t > t1, (3.22)

and this holds up to the next switching moment. Nevertheless, such anassumption (and, therefore, Formula (3.22)) is not always correct. Indeed,for derivation of (3.22) we assumed that σ(t) > 0 immediately after theswitching moment t1 where Relation (3.16) holds. For this to be true, it isnecessary that, in addition to (3.17), the following inequality hold:

σ(t1 + 0) > 0, (3.23)

which can, with reference to (3.11) and (3.21), be written in the form

J T A Y (t1) + ρkp > 0. (3.24)

If ρ ≥ 0, Equation (3.24) follows from (3.19), and Formula (3.22) holds. If ρ < 0, so that (3.24) holds, it is necessary that

| J T A Y (t1) |>| ρkp | . (3.25)

2. Assume that ρ < 0 and the condition (3.25) is not satisfied. Then,

σ(t1 + 0) < 0, (3.26)

and motion in the area N + (along the right branch of the nonlinear char-acteristic) is impossible. Therefore, we cannot use (3.22) if (3.26) holds.In the case of (3.26) it is assumed [13, 109, 120] that the system performssliding motion along the switching plane (3.10) after the switching momentt1. In this case, system motion is described by the equation

dY

dt= AcY, (3.27)

where

Ac = A −1

ρHJ T A. (3.28)




Taking the vector Y (t) (3.21) as initial condition for the sliding motion, wefind the solution of sliding mode equation:

Y (t) = eAC(t−t0) Y (t1), (3.29)

which lies in the switching plane. The sliding mode exists up to a momentt2 when one of the following equalities holds:

J T A Y (t2) = ±ρ, (3.30)

after that the motion will continue in the area N + if

J T A Y (t2) = −ρ, (3.31)

and in N − in the opposite case, and we can use Equations (3.11) and (3.12)to continue the solution.

As was shown by a more detailed investigation [13], the transition fromthe sliding mode to the area N + or N −, will take place if

J T A2 Y (t2) + J T A H > 0, (3.32)

or

J T A2 Y (t2) − J T A H < 0, (3.33)

respectively. If both the conditions (3.32) and (3.33) are violated, thesystem continues sliding motion.

Repeating the above reasoning, we can, generally speaking, continue con-struction of the solution for any finite time interval. In the general case,the solution will consist of segments with normal and sliding motion.

Thus, for time-domain description of the relay system (3.8) it is necessaryto have three systems of differential equations (3.11), (3.12), and (3.27),

switching conditions (3.10), the relations (3.30), which can be consideredas conditions of switching from sliding mode to normal, the conditions(3.32) and (3.23), and, perhaps, some other additional conditions [13].

In principle, the obtained system of equations makes i possible to continueany solution given for t = t0 over infinite time interval t > t0.

Consider the question on determining various types of motion for thesystem (3.8). Assign numbers 1, 2, and 3 to Equations (3.11), (3.12), and(3.13), respectively. Then, the notation of a type of motion

2, 3, t0, ∞




means that from the moment t0 to a moment t1 the motion is described byEquation (3.12), after that the system operates in a sliding mode describedby (3.27) and remains in this mode for all t > t1.

The motion type

2, 1, 3, t0, ∞

means that the system operates in a sliding mode after two intervals of normal motions. As was shown by Yu. Neimark, the relay system (3.8) canhave motions of any complex types.

3.1.3 Parametric Model and Sensitivity Function of Discontinuous System

To define the parametric model of a discontinuous system we will assumethat initial equations (3.1), switching conditions (3.4), and break conditions(3.2) depend on a scalar parameter α, i.e., equations of the motion havethe form

dY

dt= F i(Y,t,α), ti−1 < t < ti. (3.34)

The switching conditions depend on the parameter

f i(Y −i , ti, α) = 0, (3.35)

and the break conditions can be written in the form

Y +i = Φi(Y −i , ti, α). (3.36)

Assume that initial conditions are given

t0 = t0(α), Y +(t0) = Y +(t0, α). (3.37)

and the system of additional relations determining the system in the time-domain is such that for any α from some set it is possible to find a solutionon an interval t0 ≤ t < t(α). In general, the switching moments ti(α)appearing in (3.34), will be functions of α. As a result, we can construct aset of discontinuous solutions

Y = Y (t, α), t0(α) ≤ t < t(α). (3.38)

DEFINITION 3.1 The set of solutions (3.38) forms a single-parameter

family on the interval J (α): t0(α) ≤ t < t(α) for all α from the interval




α1 ≤ α ≤ α2 if all solutions (3.38) have the same type, i.e., if they are described by the same sequence of Equations (3.34).

DEFINITION 3.2 We will call the sensitivity function of the single-parameter family of solutions (3.38) the derivative

U (t, α) =∂Y (t, α)

∂α, (3.39)

where the operator ∂/∂α denotes the so-called ordinary derivative of the dis-continuous function that is equal to the ordinary derivative at the points of differentiability of Y (t, α) and is not defined at the points of discontinuities (or breaks).

3.1.4 General Sensitivity Equations for DiscontinuousSystems

Let (3.38) be a single-parameter family of solutions of the discontinuoussystem (3.34)–(3.36). Then, for any α from the interval α1 ≤ α ≤ α2 andinitial condition (3.37) there is a moment t1(α) > t0(α) such that

dY

dt= F 1(Y,t,α), t0(α) ≤ t < t1(α). (3.40)

If the functions t0(α) and Y 0(α) are continuously differentiable with re-spect to α and the function F 1(T , t , α) is continuous with respect to allarguments and continuously differentiable with respect to Y , α, then initialconditions (3.37) and Equation (3.40) define a single-parameter solution onthe interval t0 ≤ t < t0, for which the existence conditions formulated inSection 2.1 hold for the first-order sensitivity function. Therefore, we canpropose that on the time interval under consideration the derivative (3.39)exists in the classical sense and is given by

dU dt

= A1(t)U + B1(t), t0(α) ≤ t < t1(α), (3.41)

where

A1(t) =∂F 1(Y,t,α)

∂Y

Y =Y (t,α)

, B1(t) =∂F 1(Y,t,α)

∂α

Y =Y (t,α)

(3.42)


U 0 =

dY 0

dα −˙

Y 0

dt0

dα . (3.43)




Below we will show that under some conditions on any interval ti−1(α) <t < ti(α) where the following condition holds:

dY

dt = F i(Y,t,α), (3.44)

the derivative exists in the classical sense and is given by

dY

dt=

∂F i∂Y

Y =Y (t,α)

U +∂F i∂α

Y =Y (ti,α)

. (3.45)

In practice, to construct the sensitivity function (3.39) it is necessary to

obtain the relations for initial conditions associated with each equation in(3.45).

The final result can be formulated as a theorem.

THEOREM 3.1

Let the functions f j and Φi in Equations (3.34)– (3.36) be continuously differentiable with respect to all arguments, and the functions F i be continuous with respect to all arguments and continuously differentiable with respect to

Y , α. Let also for a chosen α0(α1 ≤ α0 ≤ α2) the following conditions hold:

∂f i∂Y

F −i +∂f −i

∂t= 0, (3.46)

where the superscript − means that the corresponding value is calculated at the moment t = ti − 0. Then, the derivative (3.39) exists for all t = ti(α0)and satisfies the conditions

dU

dt=

∂F i∂Y

U +∂F i∂α

, ti−1(α0) < t < ti(α0), (3.47)

with initial conditions (3.43)

U 0 =dY 0dα

− Y 0dt0dα

.

Moreover, the transition from one equation in (3.47) to another are






respect to ti, from (3.53) we obtain

∂f 1∂Y

dY

dt+

∂f 1∂t

α=α0t1=t1−0

= 0, (3.54)

which coincides with the condition (3.46) for i = 1.

Thus, if the condition (3.46) holds for i = 1, the function t1 = t1(α) iscontinuously differentiable with respect to α. Then, the function

Y −1 = Y (t1 − 0) = Y 1(t1(α), t0(α), Y 0(α), α) (3.55)

is also continuously differentiable with respect to α in a locality of the pointα0.

On the time interval t > t1(α) the solution can be continued by meansof the equation

dY

dt= F 2(Y,t,α), t > t1(α), (3.56)


t1 = t1(α), Y +(t1) = Φ1(Y −1 , t1, α). (3.57)

Owing to the assumption on differentiability of the function Φ1 appearingin (3.57), the function Y +1 = Y +(t1) is continuously differentiable withrespect to α for α = α0. Therefore, on the interval until the next switchingmoment t2 there exists a sensitivity function U (t, α) satisfying the equation

dU

dt=

∂F 2∂Y

U +∂F 2∂α

(3.58)


U +1 = dY +1dα

− F 2(Y +1 , t1, α) dt1dα

. (3.59)

Let us expand Relation (3.59) in detail. First of all, by (3.57) we have

dY +1dα

=∂ Φ−1∂Y

dY −1dα

+∂ Φ−1

∂t

dt−1dα

+∂ Φ−1∂α

(3.60)

But

Y −

1 (α) = Y 1

(t1(α), t0(α), Y 0(α), α), (3.61)




and, differentiating with respect to α we obtain

dY −1

dα

= F 1(Y −1, t1, α)dt1

dα

+ U −1 (t1, α). (3.62)

Differentiating (3.52) with respect to α (it has been proved that this ispossible), with account for (3.62) we obtain

∂f −1∂Y

F −1

dt1dα

+ U −1

+

∂f −1∂t

dt1dα

+∂f −1

∂t= 0. (3.63)

With account for (3.53), from (3.63) we have

dt1dα

= −

∂f −1∂Y

U −1 +∂f −1∂α

∂f −1∂Y

F −1 +∂f −1

∂t

, (3.64)

Substituting (3.60), (3.62), and (3.64) into (3.59), we obtain

U +1 =

−∆F 1 +

∂ Φ−1∂Y

− E

F −1 +

∂ Φ−1∂t

dt1dα

+∂ Φ−1∂Y

U −1 +∂ Φ−1∂α

,

(3.65)

which is equivalent to (3.48). Thus, the claim of the theorem is provedfor i = 1. Obviously, if (3.46) holds, the above reasoning is valid for any

switching monent.

Consider some properties of the obtained relations.

1. Substituting (3.50) into (3.48), after routine transformations weobtain

U +i = P iU

−

i + Qi, (3.66)




where P i and Qi are, respectively, a matrix and column defined by

P i = −1

f −i

−∆F i +

∂ Φ−i∂Y

− E

F −i

+ ∂ Φ−

i

∂t

∂f

−

i

∂Y

T

+ ∂ Φ−

i

∂Y ,

Qi = −1

f −i

−∆F i +

∂ Φ−i∂Y

− E

F −i

+∂ Φ−i

∂t

∂f −i∂α

+∂ Φ−i∂α

,

(3.67)

2. In principle, all the above relations remain valid if sequential equa-tions in (3.1) have different order. In this case the derivatives

∂ Φ−i /∂Y i are rectangular rather than square matrices. The matricesP i in (3.67) will also be rectangular.

3. Relations (3.48) determining breaks of the sensitivity function atthe switching moments are linear with respect to the sensitivityfunctions.

4. In some special cases Relations (3.67) and the break conditions forsensitivity functions (3.48) can be simplified. If the switching mo-ments ti are independent of the parameter, we have dti/dα = 0

and∆U i =

∂ Φ−i∂Y

− E

U −i +

∂ Φ−i∂α

. (3.68)

Therefore, in (3.67) we obtain

P i =∂ Φ−i∂Y

, Qi =∂ Φ−i∂α

. (3.69)

3.1.5 Case of Continuous Solutions

The above equations get simplified if we consider continuous solutions of the sequence of equations (3.1). Indeed, in this case instead of the generalbreak conditions (3.2) we have (3.3), whence

∂ Φ−i∂Y

= E,∂ Φ±i∂α

= 0,∂ Φ±i

∂t= 0, (3.70)

Then, Equation (3.48) yields

∆U i = −∆F i

dti

dα , (3.71)








Writing the relations similar to (3.59) and (3.62) for any i, we have

dY +idα

= U +i + F +idtidα

dY −idα

= U −i + F −idtidα

(3.87)

For the case of continuous solutions,

Y +i = Y −i = Y i(ti, α), (3.88)

and Relations (3.87) yield

dY (ti(α), α)dα

= U −i + F −i dtidα

= U +i + F +i dtidα

. (3.89)

3.2 Sensitivity Equations for Relay Systems

3.2.1 General Equations of Relay Systems

Control systems often include a large number of nonlinear transformatorscalled relay elements . Let us give a general mathematical description of various relay elements as a class of special nonlinear discrete operators [87,120].

In the most general case, an arbitrary nonlinear element with one scalarinput and one scalar output can be considered as a nonlinear operatortransforming a set of input signals x(t) into a set of piecewise-continuousfunctions y(t). It is assumed that for any input signal x(t) of the consideredclass we can define a countable sequence of switching moments

ti = ti[x(t)], (3.90)

and a countable numerical sequence

ki = ki[x(t)]. (3.91)

Then, the output signal is defined by the relations

y(t) = ki where ti−1 < t < ti. (3.92)




The switching conditions (3.90) together with (3.91) and (3.92) define ageneralized discrete element.

Hereinafter, for the aggregate of the relations (3.90)–(3.92), we will usethe notation

y = Ld[x] (3.93)

DEFINITION 3.3 A discrete element described by the relations (3.93)will be called relay element if all switching moments depend on the form of the input signal.

In most applied problems, the switching conditions are given in implicitform. For example, for an ideal relay having the characteristic shown inFigure 3.1, we have

ki = kp sign x(t), ti−1 < t < ti, (3.94)

and the switching moments are defined by the relations

x(ti) = 0. (3.95)

Figure 3.2

Generalized symmetric relay characteristic

In applications, relay elements with the generalized symmetric character-istic shown in Figure 3.2 are widely used. The coefficient kr characterizesthe output signal value, σ0 defines a dead zone, and µ is called the return coefficient . Figure 3.3 demonstrates characteristics of relay elements thatcan be obtained from the generalized one for various values of the returncoefficient µ.

Another widespread type of relay characteristic is one with the variable

dead zone shown in Figure 3.4. In this case, the output signal takes two




Figure 3.3

Relay characteristics for various µ

Figure 3.4

Relay characteristics with variable dead zone

values:

y(t) = ±kp, ti−1 < t < ti, (3.96)

and the switching moments are defined by some functions from the inputsignal

ti = ti[x(t)]. (3.97)

A particular case of such relays are those with the characteristic shownin Figure 3.5 used in relay extremal systems.

In some control systems relay elements with non-symmetric character-istics are used. Examples of such characteristics are shown in Figure 3.6.

Characteristics of relay elements depend on a definite set of parameters.For instance, parameters of generalized elements with the characteristicsshown in Figure 3.2 are the values kr, σ0 and µ. Parameters of a relaywith the characteristic shown in Figure 3.3 are kr and σ. In general, to

characterize dependence of relay element characteristics on a generalized




Figure 3.5

Relay characteristics used in extremal systems

Figure 3.6

Non-symmetric relay characteristics

parameter we will use the notation

y = Lp[x, α]. (3.98)

Now consider equations of relay systems. Hereinafter we investigate sys-

tems that differ from linear ones by the presence of a single relay element.For a wide scope of problems, such systems can be described by equationsof direct control [61, 86]:

dY

dt= A Y + HLp[σ], σ = J T Y, (3.99)

where A is a constant matrix, and H and J are constant column vectors.

If Equations (3.99) depend also on a scalar parameter α, instead of (3.99)




we will have, in general,

dY

dt= A(α) Y + H (α)Lp[σ, α], σ = J T (α) Y. (3.100)

Taking into account that the signal z = Lr(σ, α) is a piecewise constantfunction and assuming

z = q i = const, ti−1 < t < ti, (3.101)

from (3.99) we obtain a sequence of equations

dY

dt= A Y + Hq i, σ = J T Y, (3.102)

that will be called equations of normal motion . As follows from the exampledealing with ideal relay system, in general the sequence of equations of nor-mal motion does not specify completely all possible motions in the system(3.99), because modes with sliding motion are also possible. Therefore, thesequence of equations of normal motions should be added by equations of sliding modes. In general, these equations can be obtained in the followingway [109, 113]. Let us have a surface S in the phase space of variablesy1, . . . , yn defined by

f (y) = f (y1, . . . , yn) = 0, (3.103)

and dividing the space into areas N + where f > 0 and N − where f < 0.In the areas N + and N − we have two systems of differential equations,respectively,

dY

dt= F +(Y, t), Y ∈ N +,

dY

dt= F −(Y, t), Y ∈ N −,

(3.104)

Each of them describes normal motion in the corresponding areas N + andN −. If the phase trajectories of Equations (3.104) at some points of the sur-face S are directed toward each other, the system performs sliding motiongiven by the equations [109, 113]

dY cdt

= λF + + (1 − λ)F −, 0 ≤ λ ≤ 1, (3.105)

where

λ =grad f · F −

grad f · [F − − F +]. (3.106)






For a reverse transition, we have

∆U i = 2Hkpdtidα

(3.114)

Let, for instance, the matrix A and vector H in Equations (3.11) and(3.12) depend on α. Then, before the break, we have the sensitivity equa-tion

dU

dt= A U +

dA

dαY −

dH

dαkp, t ≤ ti − 0. (3.115)

while after the break,

dU

dt

= A U +dA

dα

Y +dH

dα

kp, t ≥ ti + 0. (3.116)

The switching moment is defined by the equation

f = σ(ti) = J T Y (ti) = 0. (3.117)

Moreover, since f = J T in the given case, Formulas (3.77) yield

dtidα

= −1

σ(ti − 0)J T U −i = −

1

σ(ti + 0)J T U +i (3.118)

Using (3.113) and the first relation in (3.118), we have

U +i = P iU −i (3.119)

and

P i = E +2kpρ

σ(ti − 0)HJ T , ρ = J T H, (3.120)

whereσ(ti − 0) = J T Y (ti − 0) = J T A Y (ti) − ρkp. (3.121)

Using the second relation in (3.118), instead of (3.119) we obtain

U −i = RiU +i , (3.122)

where

Ri = E −2kpρ

σ(ti + 0)HJ T . (3.123)




As follows from (3.84), (3.85), (3.120), and (3.123),

det P i =σ(ti + 0)

σ(ti − 0),

det Ri = σ(ti − 0)σ(ti + 0)

,

(3.124)

that can be immediately verified.

Thus, if σ(ti − 0) = 0, i.e.,

J T A Y (ti) = ρ, (3.125)

the matrix P i has no sense. Nevertheless, Relations (3.122) may hold, but

they cannot be used for calculation of U +i , because detRi = 0.

Similarly, if we consider a normal transition from the half-plane σ > 0into the half-plane σ < 0, we have the sequence of equations (3.11) and(3.12). Then, with account for (3.114), we obtain

U +i =

E −

2kpρ

σ(ti − 0)HJ T

U −i ,

U −i =

E +

2kpρ

σ(ti + 0)HJ T

U +i .

(3.126)

Relations (3.119), (3.122), and (3.126) can be, using the signs of σ(ti − 0)and σ(ti + 0), combined in the form

U +i =

E +

2kpρ

| σ(ti − 0) |HJ T

U −i ,

U −i =

E −

2kpρ

| σ(ti + 0) |HJ T

U +i .

(3.127)

Now assume that there is a switch from a normal motion in the half-planeσ < 0 to a sliding mode at the moment ti(α). Then, before the momentti − 0 we have

dY

dt= A Y − Hkp = F −, t ≤ ti − 0, (3.128)

and in the sliding mode

dY

dt= A −

1

ρHJ T AY = F +, t ≥ ti + 0. (3.129)




Associated sensitivity equations have the form

dU

dt= A U +

dA

dαY +

dH

dαkp (3.130)

and

dU

dt=

A −

1

ρHJ T A

U +

dA

dαY −

d

dα

1

ρHJ T A

Y. (3.131)

Let us find relations between the values of the vector U before and afterthe switching moment. In this case

∆F i = F +i − F −i = H

kp − 1ρ

J T A Y (ti)

= H

kp −

σ(ti − 0) + ρkpρ

= −

σ(ti − 0)

ρH,

(3.132)

Therefore, due to (3.71),

∆U i =σ(ti − 0)

ρH

dtidα

. (3.133)

Assuming that σ(ti − 0) = 0, for calculation of dti/dα we can use thefirst relation in (3.118), which yields, together with (3.133),

∆U i = −1

ρHJ T U −i . (3.134)

This relation can be represented in the form (3.66), where

P i = E − 1ρ

HJ T , Qi = 0. (3.135)

The transition from the half-plane σ > 0 to a sliding mode can be analyzedin a similar way.

Then, consider the transition from the sliding mode to the normal one.Let us have the equation of sliding mode (3.129) before the moment ti(α).At the moment ti(α) defined by the conditions (3.31) and (3.32)

J T

A Y (ti) + ρ = 0, J T

A2

Y (ti) + ρ > 0, (3.136)




there is a transition to the normal motion in the half-plane σ > 0, i.e., tothe equation

dY

dt= A Y + Hkp = F +. (3.137)

The sensitivity equations before and after switching are obtained from(3.129) and (3.137) by differentiating with respect to α:

dU

dt=

A −

1

ρHJ T A

U +

dA

dαY −

d

dα

1

ρHJ T A

Y,

t ≤ ti − 0,

dU

dt= A U +

dA

aαY +

dH

dαkp, t ≥ ti + 0.

(3.138)

Next we consider the break conditions. In the case at hand

∆F i = Hkp +1

ρHJ T A Y (ti) = H (kp − 1). (3.139)

The derivative dti/dα can be obtained using the first relation from (3.77)and the switching condition (3.136):

dtidα

= −J T U −i + J T dA

dαY i + J T dH

dαJ T A2Y i + ρ

. (3.140)

Using the general formula (3.66), we have

U +i = P iU −i + Qi, (3.141)

where

P i = E + ν −1

(kp − 1)HJ T

,

Qi = ν −1(kp − 1)HJ T

dA

dαY i +

dH

dα

,

ν = J T A2Y i + ρ.

(3.142)

3.2.3 Systems with Logical Elements

Logical elements are described by a special class of relay operators withseveral input and several outputs, each of them able to assume a number

of discrete values. In the present section we discuss sensitivity of a system




with the simplest logical unit with the scheme given in [74, 78]. Equationsof system motion have the form

dY

dt= AY + HL[σ

1, σ

2], σ

i= J T

iY, i = 1, 2, (3.143)

where A is a constant matrix, H and J are constant vectors, and L is thecharacteristic of the logical unit. Logical control law z = L(σ1, σ2) can bevisually presented on the plane σ1, σ2 (Figure 3.7). The equation of logical

Figure 3.7

Logical unit characteristic

unit can be defined analytically in the form

z =

1, if σ1, σ2 ∈ S 1,−1, if σ1, σ2 ∈ S 2,

0, if σ1, σ2 ∈ S 3,(3.144)

where S 1, S 2, and S 3 are regions marked in Figure 3.7. The characteristic(3.144) can be described in a shorter form using basic logical functions [74].

Consider main types of motions in the system at hand. First of all, inthe regions S i we have normal motions described by the equations

dY 1dt

= A Y + H, σ1, σ2 ∈ S 1, (3.145)

dY

dt= A Y − H, σ1, σ2 ∈ S 2, (3.146)

dY

dt= A Y, σ1, σ2 ∈ S 3, (3.147)

The switching moments when one of Equations (3.145)–(3.147) is changed

for another one will be called normal . Assume that before the moment ti




we had Equation (3.147), and after it Equation (3.145). Assume also thatthe transition from the region S 3 into the region S 2 happens at the pointM (see Figure 3.7), and σ(ti − 0) > 0 and σ(ti + 0) > 0.

We will find the sensitivity equation for this case. For A = A(α), H =

H (α), J 2 = J 2(α), and b = b(α) the equations of the motion have the form

dY

dt= A(α) Y, t ≤ ti − 0,

dY

dt= A(α) Y + H (α), t ≥ ti + 0.

(3.148)

The sensitivity equations on the intervals of normal motion can be ob-tained by differentiating (3.148) with respect to α in the form

dU

dt= A U +

dA

dαY, t ≤ ti − 0,

dU

dt= A U +

dA

dαY +

dH

dα, t ≥ ti + 0.

(3.149)

Next, we calculate the breaks of the sensitivity function. In the case underconsideration,

F +i = A Y i + H, F −i = AY i, ∆F i = H, (3.150)

while the switching condition has the form

σ2(ti) = J T 2 Y i = −b. (3.151)

Using (3.66) and (3.72), and assuming that

σ2(ti − 0) = J T 2 A Y i > 0,

we obtain

U +i = P iU −i + Qi, (3.152)

where

P i = E +1

σ2(t2 − 0)HJ T , Qi =

1

σ2(ti − 0)

dJ T 2dα

Y i +db

dα

. (3.153)

Consider the possibility of transition at the point M to a sliding mode

in the plane (3.151). For such a mode to appear, the following conditions




must be satisfied

σ2(ti − 0) = J T 2 A Y i > 0,

σ2(ti + 0) = J T 2 A Y i + ρ2 < 0, ρ2 = J T 2 H.(3.154)

Assume that the conditions (3.154) are satisfied and the sliding modetakes place. Using (3.147), (3.145), and (3.111) it can be easily seen thatin this case the equation of the sliding mode has the form

dY

dt=

A −

1

ρ2HJ T 2 A

Y. (3.155)

The sensitivity equation in the sliding mode is given by

dU

dt=

A −

1

ρ2HJ T 2 A

U +

d

dα

A −

1

ρ2HJ T A

Y. (3.156)

To calculate the breaks of the sensitivity function we will use (3.71),bearing in mind that

F +i = A Y i −1

ρ2HJ T 2 AiY i, F − = A Y i, (3.157)

while the switching condition still have the form (3.151). After some trans-formations, we obtain Formula (3.152), where

P i = E +1

ρ2HJ T 2 , Q =

1

ρ2

dJ T 2dα

Y i +db

dα

. (3.158)

3.2.4 Relay System with Variable Delay

Consider sensitivity equations for a system including, except for a relaywith a dead zone, a pure delay element [87, 104], where the delay changesat the switching moments. The system equations have the form

dY

dt= A Y + HL[σ, τ 1, τ 2], σ = J T Y, (3.159)

where A is a constant matrix, H and J are constant vectors, and L(σ, τ 1, τ 2)is a characteristic of the complex nonlinear element formed by series con-nection of an element with variable delay EVD and a relay element RE (see

Figure 3.8). Let us assume that the characteristic of the relay element has




Figure 3.8

Relay element with variable delay

the form shown in Figure 3.3 b, and the EVD transforms the input signalaccording to the rule

z = L[u, τ 1, τ 2] =

u(t − τ 1) for u = ±kp,u(t − τ 2) for u = 0,

(3.160)

where τ 1 and τ 2 are nonnegative constants.

We will consider continuous solutions of the system (3.159). Let σ(t) bethe input signal of the relay element, which is a continuous function of t.Denote by ti the switching moments defined by the conditions

σ(ti) = J T Y (ti) = ±σ0. (3.161)

We call the interval I : t1 < t < t2 an interval of the first type I 1 if for t ∈ I

we have |σ(t)| < σ0, and an interval of the second type if for t ∈ I we have|σ(t)| > σ0. Then, Equation (3.160) yields

z(t) =

u(t − τ 1) for t ∈ I 2,u(t − τ 2) for t ∈ I 1.

(3.162)

Hence, the function z(t) has, generally speaking, breaks at the moments

t1i = ti + τ 1, t2i = ti + τ 2. (3.163)

depending on the values of τ 1, τ 2 and the duration of pulses incoming fromthe relay. Therefore, the EVD changes the number and duration of pulsesacting on it.

In general, at the switching moments (3.163) there can be a change of anycombinations of the values kr, −kr, 0. For example, in the situation shownin Figure 3.9, we have a switch from kr to 0 at the moment t = 0, while inFigure 3.10 this moment corresponds to a switch from kr to −kr. Thus, if we do not take into account sliding modes in the switching planes (3.161),

an arbitrary motion of the system under investigation can be described by




Figure 3.9

Switch from kr to 0

Figure 3.10

Switch from kr to −kr

a sequence of differential equations of the form

dY

dt= AY ± Hkp,

dY

dt= AY.

(3.164)

It should be emphasized that any equation in (3.164) can, in principle, besubstituted by any of the two remaining ones. This circumstance is causedby the unit of variable delay.

Now let us construct the sensitivity equations. At first, we assume thatthe matrix A and vectors H and J depends on the parameter α, while thevalues τ 1 and τ 2 are constants.

Then, between the switching moments we will have the following equa-




tions obtained from (3.164) by differentiating with respect to α:

dU

dt= AU +

dA

dαY ±

d

dα(Hkp),

dU

dt= AU +

dA

dαY.

(3.165)

For the switching moments ti of the relay element we have Equation (3.161).Since τ 1 and τ 2 are assumed to be independent of the parameter,

dt1idα

=dt2idα

=dtidα

, (3.166)

Moreover, as before, we have

dtidα

= −J T U −i +

dJ T

dαY i ±

dσ0dα

σ(ti − 0). (3.167)

Using Equations (3.164) before and after the switching moments and For-

mulas (3.71) and (3.167), it is easy to construct the corresponding condi-tions for breaks of the sensitivity function.

Let us take the value α = τ 1 as a parameter, assuming that A, H , J , kr,and σ0 are independent of α. In this case, the sensitivity equation (3.165)reduces to the single equation

dU

dt= A U, (3.168)

The derivatives of the switching moment are given by

dt1idα

=dtidα

+ 1,dt2idα

=dtidα

, (3.169)

Moreover, in this case

dti

dα= −

1

σ(ti − 0)J T U −

i

. (3.170)




3.2.5 Relay Extremal System

Consider the following system of equations:

dY 1dt = A1Y 1 + H 1φ(σ2), σ2 = J T 2 Y 2,

dY 2dt

= A2Y 2 + H 2Lp[σ1], σ1 = J T 1 Y 1,

(3.171)

where A1 and A2 are constant square matrices, H i and J i(i = 1, 2) areconstant vectors of corresponding dimensions, φ(σ2) is an even continuouslydifferentiable function of the argument σ2 having a maximum at σ2 = 0.In applications it is often assumed [66] that

φ(σ2) = −kσ22 , k = const > 0. (3.172)

Hereinafter we will use (3.172) for simplicity. Moreover, Lr[σr] in Equa-tions (3.171) is the relay characteristic shown in Figure 3.4. The value σ0determining the switching moments can be defined in various ways depend-ing on the method of forming the relay switching law.

Thus, if it is assumed that there is a fixed bilateral quantization grid, inFigure 3.4 we can take σ0 = 0, and switching of the relay will happen for

σ1(ti) = J T i Y 1(ti) = σ0, σ1(t1) < 0. (3.173)

If the quantization grid if connected to σ1(t), we have

σ0 = σ1max − χ, (3.174)

where σ1max is the previous maximal value of the coordinate σ1, and ξ is afixed quantization step. Relation (3.174) can be considered as a switching

condition defining the switching moment ti:

σ1(t1) = σ1(tm) − χ, (3.175)

where tm is a moment when extremum is reached so that σ1(tm) = 0.

Let us show that the system of equations (3.171) is equivalent to equa-tions of a single-loop nonlinear system of extremal control. Assumingd/dt = p, from the first equations in (3.171) we have

( pE − A1)Y 1 = H 1φ(σ2), ( pE − A2)Y 2 = H 2L[σ1], (3.176)




Hence,

Y 1 = ( pE − A1)−1H 1φ(σ2), Y 2 = ( pE − A2)−1H 2L[σ1]. (3.177)

Multiplying the first equation by J T 1 and the second one by J T 2 , we obtain

σ1 = ω1( p)φ(σ1), σ2 = ω2( p)L[σ1], (3.178)

where

ωi( p) = J T i ( pE − Ai)−1H i, i = 1, 2, (3.179)

which corresponds to general equations given in [66]. If in this case each of the matrices A1 and A2 has one zero eigenvalue, we have

ωi( p) =1

pωi( p), i = 1, 2, (3.180)

and Equations (3.171) reduce to ones considered in [66].

Then, let us investigate the technique of constructing sensitivity equa-tions for the system (3.171) for normal modes when the second equation of (3.171) reduces to the sequence of equations

dY 2

dt = A2Y 2 ± Hkp. (3.181)

Then, assuming that all constant matrices and vectors appearing in (3.171)depend on the parameter α, we have

dU 1dt

= A1U 1 − 2kσ2HJ T 2 U 2+

+dA1

dαY 1 − 2kσ2H

dJ T 2dα

Y 2 − kσ2dH 1dα

,

dU 2dt

= A2U 2 + dA2

dα± d

dα(H 2kp),

(3.182)

where

U 1 =∂Y 1(t, α)

∂α, U 2 =

∂Y 2(t, α)

∂α, (3.183)

Next, we construct the break conditions for the sensitivity functions.Note that the right side of the first vector equation (3.171) is continuous.Then, from the general formula (3.71) it follows that the sensitivity function

U 1 is continuous.






modulation. We restrict our discussion by bipolar modulation of the firstkind [56]. In this case, the function z(t) can be expressed in the form

z = L[σ] = sign σ for | σ(ti) |> ∆,

0 for | σ(ti) |< ∆, ti < t < ti+1,(3.187)

and the duration of discreteness interval is given by

ti+1 − ti = f [σ(ti)], (3.188)

where f (σ) is a positive even function such that limσ→∞ f (σ) = 0. Here-inafter the function f (σ) will be assumed to be differentiable.

It is noteworthy that, according to the above definition, the system underconsideration belongs to the class of relay systems, because all switching

moments change when the form of the input signal changes. There is somecontradiction with the adopted terminology. In [120] it was also noted thatsystems with pulse frequency modulation are similar to relay systems.

Depending on the values of the controlling function, all possible motionsof the system under consideration are described by the following system of equations derived from (3.99) and (3.187):

dY

dt= A Y ± H,

dY

dt= A Y. (3.189)

The sensitivity equations between the switching moments can be ob-tained, as before, by formal differentiation of Equations (3.189) with re-spect to the parameter. Therefore, we proceed at once to construct thebreak conditions for the sensitivity functions.

Note that in this case Equations (3.189) can follow one after another inan arbitrary order.

Let, for instance,

Y (ti − 0) = A Y (ti) − H = F −i ,

Y (ti + 0) = A Y (ti) = F +i , (3.190)

i.e.,

∆F i = F +i − F −i = H. (3.191)

The expression for the derivative dti/dα can be obtained by direct differ-entiation of (3.188). Obviously, we have

dti+i

dα =

dti

dα +

df i

dσ

dσ(ti)

dα , (3.192)




But, according to (3.178),

dσ(ti)

dα=

d

dα J T Y (ti, α)

=

= J T

Y (ti − 0) dtidα

+ U (ti − 0)

+ dJ T

dαY (ti).

(3.193)

Therefore, Equation (3.192) yields

dti+1dα

= J T U −idf idσ

+

1 + σ−i

df idσ

dtidα

+dJ T

dαY i

df idσ

. (3.194)

Let us note that in this case we have a recurrent relation (3.194) instead

of an explicit expression for the derivative dti/dα. Nevertheless, this doesnot make calculations more difficult, because having initial conditions

t0 = t0(α), Y (t0) = Y 0(α),

we can obtain the derivative dt0/dα and all other derivatives of interest(3.194). It should be noted that the above reasoning remains valid if weadditionally assume that the switching function f also depends on the pa-rameter, i.e.,

f = f (σ, α). (3.195)

In this case, instead of (3.192) we have

dti+1dα

=dtidα

+df idσ

dσ(ti)

dα+

df idα

. (3.196)

3.3 Sensitivity Equations for Pulse and Relay-PulseSystems

3.3.1 Pulse and Relay-Pulse Operators

DEFINITION 3.4 Discrete operator (3.93) given by (3.90) and (3.91)will be called pulse if the sampling moments ti are independent of the input

signal.








Let us also give examples of relay-pulse elements. Consider pulse widthmodulation

y(t) = kp sign x(nT ), nT < t < nT + t1n,

0, nT + t1n < t < (n + 1)T,

(3.209)

with

0 ≤ t1n = g[x(nT )] ≤ T, (3.210)

where g[x] is a given function.

For an arbitrary input signal, the output signal will have the form of astep function assuming the values 0 and ±k. Moreover, two sequences of switching moments can be separated out: tn = nT and tn = nT + t1n. Thefirst one is independent of the form of the input signal, while the second does

depend. Therefore, by the adopted classification, a pulse width modulatingelement is relay-pulse. Thus, the terminology used here differs from theusual one.

Hereinafter we shall consider, for concreteness, a system of equations of the form

dY

dt= AY + HLd[σ], σ = J T Y, (3.211)

where A is a constant matrix, H and J are constant vectors, and Ld[σ] isa discrete (pulse or relay-pulse) element.

3.3.2 Sensitivity Equations of Pulse-Amplitude Systems

As a starting point, we consider the system of equations (3.211), whereLd[σ] is a nonlinear extrapolator without a feedback, so that on the intervalbetween the switching moments of the pulse elements we have the followingsequence of equations:

dY

dt= AY + Hf [σs], sT < t < (s + 1)T, (3.212)

where f (σ) is a single-valued continuously differentiable nonlinear function

σs =

qk=0

αskmk(t), αsk =l

ν =0

β kν αks[(s − ν )T ]. (3.213)

To determine the motion of the system (3.212)–(3.213) for t ≥ 0 it is nec-essary to define the values σ(t) for t = iT (i = 0, −1, . . . , −l).

Denote

σs = σ(sT ) (3.214)




Let us also have a vector of initial conditions Y (0) = Y 0 so that J T Y 0 = σ0.Then, on the interval 0 < t < T we have, according to (3.212) and (3.213),the equation

dY

dt= AY + Hf (σ

0), (3.215)

where

σ0 =l

ν =0

qk=0

β kν σ[−νT ]mk(t). (3.216)

Solving the linear equation (3.215) with the given initial conditions, weobtain

Y (t) = e

At

Y 0 +

t

0

e

A(t−τ )

Hf (σ0(τ )) dτ, (3.217)

Hence, for t = T we have

Y (T ) = eAT

Y 0 +

T 0

e−Aτ Hf (σ0(τ )) dτ

(3.218)

and, correspondingly,

σ(T ) = J T eAT

Y 0 +

T 0

e−Aτ Hf (σ0(τ )) dτ

. (3.219)

It can be easily seen that Relations (3.218) and (3.219) together with thevalues σs for s = −l+1, . . . , 1 completely define the equations of the motionfor T < t < 2T and the corresponding initial conditions.

Continuing this process of constructing the solution, we find that thevalues σ−l, . . . , σ−1 and initial conditions Y 0 completely define the motionof the system under consideration.

Then, consider the construction of the sensitivity equations assumingthat A = A(α), H = H (α), J = J (α), Y = Y 0(α), f = f (σ, α), β kν =β kν (α), σ−l , . . . , σ−1 are known continuously differentiable functions of theparameter. Moreover, we will assume that the modulated functions mk(t) =mk(t, α) are continuously differentiable with respect to α for 0 < t < T .

In this case, the sampling period T is assumed to be constant. In canbe easily shown, using the above process of continuing the solution, thatunder the given assumptions the right sides of all the equations in (3.113)

are continuously differentiable with respect to α. Therefore, on the intervals






For t = T we obtain

Y (T ) = P Y 0, (3.225)

where

P = eAt

+ kA−1

(eAT

− E )HJ T

. (3.226)


Y (nT ) = P nY 0. (3.227)

It is known [13, 117] that if the roots of the characteristic equation

det(λE − P ) = 0 (3.228)

are inside the unit circle, the system (3.223) is asymptotically stable andall its solutions tends to zero (by a norm) as t → ∞.

The condition of asymptotic stability can be written in analytical formas

| λρ |< 1, ρ = 1, . . . , n , (3.229)

where λρ, ρ = 1, . . . , n are the roots of Equation (3.218).

Let us expand the characteristic equation (3.228) in detail. With accountfor (3.226), we have

det(λE − P ) = det(λE − eAT

− kA−1

(eAT

− E )HJ T

) = 0. (3.230)

As follows from (3.230), the coefficients of the characteristic equationsare continuous functions of the parameter T . Since the asymptotic stabil-ity conditions (3.229) are expressed by strict inequalities with respect tothe coefficients of the characteristic equation, we can propose that if theconditions (3.229) hold for some T = T ∗, they hold also for T = T ∗ + ∆T if |∆T | is sufficiently small.

Thus, denoting by Y = Y (t, Y 0, T ) the general solution of Equation(3.223) and assuming that (3.229) holds for T = T ∗, for sufficiently small

|∆T | the additional motions

∆Y = Y (t, Y 0, T ∗ + ∆T ) − Y (t, Y 0, T ∗) (3.231)

are uniformly bounded and tend to zero as t → ∞.

Then, we investigate sensitivity with respect to the parameter T , assum-ing that the matrix A and vectors H and J are independent of α = T .Differentiation of Equations (3.223) with respect to T yields

dU

dt = A U + kHJ

T d

dT Y (nT,T ), (3.232)




where, according to (3.89),

d

dT Y (nT,T ) = U (nT + 0)+

+ nY (nT + 0) = U (nT − 0) + nY (nT − 0).

(3.233)

Then, consider the breaks of the sensitivity function U . In this case, theswitching moments ti(i = 0, 1, . . .) are equal to

ti = iT. (3.234)

Therefore,

dti

dT = i. (3.235)

At the break instant t = ti we have

Y −i = A Y i + kHJ T Y i−1, Y +i − = A Y i + kHJ T Y i, (3.236)

Hence,

∆F i = Y +i − Y −i = kH J T (Y i − Y i−1) = kH (σi − σi−1). (3.237)

Then, using (3.71) and (3.235), we obtain

U +i = U −i + ikHJ T (Y i−1 − Y i), (3.238)

Referring to (3.227), we find the expanded form

U +i = U −i + ikHJ T (E − P )P i−1Y 0. (3.239)

Using the above relations, we can easily construct a general expressionof the sensitivity function. Similarly to (3.227), from (3.232) and (3.233)we find

U [(n + 1)T − 0] = P U (nT + 0) + n Q Y (nT + 0), (3.240)

where P is the matrix given by (3.226), and

Q = kA−1

(eAT

− E )HJ T

. (3.241)




Bearing in mind that

Y (nT + 0) = (A + HJ T )Y (nT ) = (A + HJ T )P nY 0, (3.242)

and using the break conditions (3.239), we obtain the recurrent relation

U +i+1 = P U +i +

(i + 1)kHJ T (E − P )

+ ikA−1

eAT − E

HJ T

A + HJ T

P iY 0,(3.243)

which yield explicit expressions for the values U +i and U −i . Let us note thatthe sequences U +i and U −i tend to a common limit, despite the presence of an infinitely increasing multiplier i in (3.239). This is explained by the factthat under the condition (3.229) the elements of the matrix P i tend to zeroas i → ∞ as a geometric progression, so that we have

limi→∞

iP i = 0. (3.244)

3.3.4 Sensitivity Equations of Systems with Pulse-WidthModulation

Consider the equations

dY

dt= A Y + Hf (σ, t), σ = J T Y, (3.245)

where the controlling function is given by

z = f (σ, t) =

sign σ(iT ) for iT < t < iT + t1i,

0, for iT + t1i < t < (i + 1)T,(3.246)

Here T is a fixed sampling period, while the pulse duration t1i is a functionof the value σi = σ(iT ):

t1i = g(σi),

where g(σi) is a differentiable function satisfying the conditions (3.210).

According to the adopted classification, systems with pulse-width mod-ulation are relay-pulse ones, because we have both switching caused by thepulse element and switching of a relay type at the moments t1i. The latterdepend on the properties of the input signal σ(t).

According to (3.245) and (3.246), between the switching moments wehave

dY

dt= A Y + H sign σi, σi = J T Y i,

iT < t < iT + t1i,

(3.247)












If the conditions of existence and uniqueness of solution of Cauchy’sproblem hold, for a given initial condition Y (0) = Y 0 we have the generalsolution of the form

Y (t) = Y (t, Y 0, X , V ), Y (t) = Y (t0, Y 0, X , V ) ≡ Y 0 (4.4)

For a fixed Y 0, Equation (4.4) defined the corresponding operator L.For given initial conditions, Equation (4.3) defines the operator (4.4) inimplicit form. It is known that only for some special cases we can constructan explicit expression for the operator (4.4).

Example 4.4

Various quality criteria of control systems are also functionals, for instance,an expression of the form

y =

Ω

f (x, v)P xvdxdv, (4.5)

where f (x, v) is a given function, P xv is a simultaneous probability density,

and Q is an area in the space of variables x, v.

4.1.3 Families of Operators

In most cases, it is impossible to describe operation of real elements andsystems by unique relations of the form (4.1). Most often, mathematicaldescription of an element determines a unique relation between the inputand output only under some additional conditions.

Example 4.5

Let the input and output be related by Equation (4.3), but initial conditionsbe not given. In this case, the input and output will be uniquely connectedif some additional conditions are given, viz., initial Y (0) = Y 0 or boundaryones.

If initial conditions are not fixed, Equation (4.4) defines a family (set)of operators depending on a finite number of scalar parameters, namely,components of the vector Y 0.

To describe many elements and systems, we must use more complex

families of operators.












For (4.18) and (4.20) we have, respectively,

Y (t, α) =

t

−∞

H (t,τ,α)X (τ, α)dτ

=

∞ 0

H (t, t − τ, α)X (t − τ, α)dτ .

(4.24)

Y (t, α) =

t −∞

H (t − τ, α)X (τ, α)dτ

=

∞

0

H (τ, α)X (t − τ, α)dτ .

(4.25)

4.1.6 Transfer Functions and Frequency Responses of Linear Operators

A widespread method of description of linear stationary operators con-sists in using transfer functions and frequency responses. In the presentparagraph we develop a general approach for linear nonstationary opera-tors [87, 88].

DEFINITION 4.6 The function

W (λ, t) = e−λtL(Eeλt), (4.26)

where λ is a complex variable and E is the identity matrix, is called the transfer function (matrix) of the linear operator (4.14).

It is assumed that Relation (4.26) is considered in an area Γλ of valueof the argument λ where Equation (4.26) has meaning, i.e., the improper

integral

W (λ, t) =

∞ −∞

H (t, τ )e−λ(t−τ )dτ, (4.27)

obtained from (4.14) and (4.26) converges.After a change of variable, from (4.27) we obtain

W (λ, t) =

∞

−∞

H (t, t − τ )e−λτ dτ =

∞

−∞

G(t, τ )e−λτ dτ, (4.28)




where

G(t, τ ) = H (t, t − τ ), (4.29)

Hence, for a fixed λ, transfer function is defined as the bilateral Laplace

transform for the function G(t, τ ).As follows from general properties of bilateral Laplace transformation

[19, 88], the area of convergence of the integral (4.27), provided that itexists, is a vertical strip S λ:

q 1(t) < Re λ < q 2(t), (4.30)

inside which the transfer function is analytical with respect to λ. thus,by the transfer function of the operator (4.14) we mean a complex-valued

function defined by (4.27) and (4.28) in a strip (4.30).According to (4.17), for a causal operator the second integral in (4.28)yields

W (λ, t) =

∞ 0

G(t, τ )e−λτ dτ, (4.31)

which is, for a fixed t, the routine (right-sided) Laplace transform. The areaof convergence of the integral (4.31) (provided that it exists) is a half-plane

Re λ > q 1(t). (4.32)

Thus, the transfer function of a causal operator defined by (4.31) is an-alytical with respect to α in the half-plane (4.32).

Now we assume that the strip (4.30) (or the half-plane (4.32)) containsthe imaginary axis, i.e.,

q 1(t) < 0 < q 2(t) (4.33)

or, respectively,

q 1(t) < 0. (4.34)

In the case (4.33) the integral (4.27) has meaning for λ = iω, where ω isany real value, and the following expression:

W (iω,t) =

∞ −∞

H (t, τ )e−iω(t−τ )dτ =

∞ −∞

G(t, τ )e−iωτ dτ, (4.35)

is defined.








a transfer function W (t, λ) in the strip S λ. Then, we will write

Y = W (λ, t)X, Re λ ∈ S λ. (4.45)

Further, in many cases the strip of convergence will not be specified.Nevertheless, one should bear in mind that transfer function is defined,generally speaking, only in a corresponding strip of convergence.

DEFINITION 4.8 Relation (4.45) will be called operator equation of linear element.

Operator equation of an element defines uniquely the corresponding op-erator. Indeed, having a transfer function, by Formula (4.42) we can find

the corresponding Green function H (t, τ ) and operator (4.14). From (4.26)it also follows that if X = Eeλt, Re λ ∈ S λ, then Y = W (t, λ)eλt.

Moreover, using the above relations, it is easy to verify that if

X = eνtX 1, Y = eνtY 1, (4.46)

the values X 1 and Y 1 are related by the operator equation

Y 1 = W (λ + ν, t)X 1. (4.47)

provided that the convergence conditions hold.

If there is a parametric family of operators, the corresponding operatorequation has the form

Y = W (λ,t,α)X. (4.48)

4.1.7 Parametric Operator Model of System

Using the notions introduced above, we can give a formal definition of operator model of a system.

DEFINITION 4.9 We will say that a system is given by an operator model if it is represented in the form of a block-diagram containing linear and nonlinear elements, where linear elements are defined by equations of the form

Y i = W i(λ, t)X i, i = 1, . . . , s , (4.49)

For brevity, we did not mention strip of convergence in (4.49), Neverthe-

less, we have to bear in mind that the operator model including relations




of the form (4.49) has meaning only if all transfer functions W i(t, λ) havea common strip of convergence.

The aggregate of nonlinear elements included in operator model are de-scribed by relations of the form

X i = Li(Y 1, . . . , Y s), i = 1, . . . , m , (4.50)

where Li are nonlinear (possibly discrete) operators.

Thus, according to the above definitions, by operator model of a systemwe hereinafter mean an aggregate of relations (4.49) and (4.50). If thetransfer functions W i(t, λ) depend on a scalar parameter α, instead of (4.49)and (4.50) we will have the following system of equations:

Y i = W i(λ,t,α)X i, X i = Li(Y 1, . . . , Y s, α), (4.51)

DEFINITION 4.10 The system of equations (4.51) is called paramet-ric operator model of the system.

Let us discuss some ideas connected with rigorous investigations of sys-tems given by (4.49) and (4.50). Their main feature is that for many casesthey demand to go outside the limits of classical theory of functions andbasic operations with them. Below we consider some examples.

1. According to (4.16), the notion of the Green function of a linear

operator is connected with investigation of element response to input signalsin the form of Dirac delta functions. But it is known [22, 124] that delta-function is not a function in the classical sense of this term, and the mainformula (4.14) has, strictly speaking, no meaning for the input (4.15) if integral in (4.14) is understood in traditional sense. Thus, the fact of definition of Green functions calls for a generalization of the class of inputsignals and integration operation.

2. Consider a linear stationary element given by an operator equation

y = ω(λ)x, (4.52)

where

ω(λ) =cλ

λ2 − (a + b)λ + ab(4.53)

is the transfer function, and a, b, and c are constants.

For the element (4.52) we can calculate and realize physically the responseto the input signal of the form

x(t) = 0, if t < 0,

1, if t > 0.(4.54)






with their derivatives of any order and is zero outside a bounded region,then

limi→∞

F (φi) = 0. (4.57)

Hereinafter we denote the action of the functional F on a basic function φby < F, φ >.

If f (t) is an ordinary function integrable on any finite interval, we candefine a linear continuous functional F acting according to the formula

F, φ =

∞ −∞

f (t)φ(t)dt. (4.58)

Identifying the function f (t) with the functional F , we find that theset of generalized functions contains, in particular, all integrable functions.An important example of generalized function that cannot be reduced toan ordinary one is a delta-function. Delta-function is a functional actingaccording to the formula

δ, φ = φ(0). (4.59)

Equality (4.58) is convenient to use when F does not reduce to an ordinaryfunction, so that Formula (4.59) takes the following “widely known” form

∞ −∞

δ (t)φ(t)dt = φ(0). (4.60)

Generalized functions always allow summation and multiplication by anumber. Another important operation is a shift operation F (t − t0) de-fined by the relation

F (t − t0), φ = F, φ(t + t0) . (4.61)

For a delta-function we have, according to (4.61),

δ (t − t0), φ = δ, φ(t + t0) = φ(t0). (4.62)

Using the integral form, we obtain

∞

−∞

δ (t − t0)φ(t)dt = φ(t0). (4.63)














or, equivalently,

dny

dtn+ a1(t)

dn−1y

dtn−1+ . . . + an(t)y = b1(t)

dn−1x

dtn−1+ . . . + bn(t)x. (4.93)

Moreover, the following linear relations must hold:

q si = rsi, s = 0, . . . , n− 2, q n−1,i = 0, (4.94)

which relates the breaks of the functions y and x to their derivatives.

Thus, interpreting Equation (4.86) from the viewpoint of theory of gen-eralized functions is equivalent to specifying a linear differential equation(4.93) with the conditions (4.94), which will be called the break conditions .

We note that if y and x in (4.86) are continuous together with the required

number of derivatives, Equation (4.93) coincides with (4.86), i.e., if theinput x(t) is sufficiently smooth, Equations (4.85) and (4.92) coincide.

For illustration we consider the following equation:

λ2y + a1(t)λy + a2(t)y = b0(t)λ2x + b1(t)λx + b2(t)x, (4.95)

assuming that the function x(t) is continuous together with the first andsecond derivatives everywhere except for the moment t = t1. Let us findconditions for which Equation (4.95) has an ordinary discontinuous functiony(t) as a solution.

According to (4.89), we have

λy =Dy

dt=

dy

dt+ ∆yδ (t − t1),

λ2y =D2y

dt2=

d2y

dt2+ ∆yδ (t − t1) + ∆yδ (t − t1),

(4.96)

λx =Dx

dt=

dx

dt+ ∆xδ (t − t1),

λ2x =D2x

dt2=

d2x

dt2+ ∆xδ (t − t1) + ∆xδ (t − t1).

(4.97)

Substituting this equation into (4.95) and regrouping the terms, we obtain

d2y

dt2+ a1(t)

dy

dta2(t)y + [∆y + a1(t)∆y] δ (t − t1)

+ ∆yδ (t − t1) = b0(t)d2x

dt2+ b1(t)

dx

dt+ b2(t)x

+ b0(t)∆xδ (t − t1) + [b0(t)∆x + b1(t)∆x] δ (t − t1).

(4.98)




Assuming that the coefficients a1(t) and b1(t) are continuous for t = t1,and the coefficient is continuous at t = t0 together with the first derivative,by (4.80) and (4.83) we have

a1(t)δ (t − t1) = a1(t1)δ (t − t1),

bi(t)δ (t − t1) = bi(t1)δ (t − t1) (i = 0, 1),

b0(t)δ (t − t1) = b0(t1)δ (t − t1) − b0(t1)δ (t − t1).

(4.99)

Substituting (4.99) into (4.98), and equating the coefficients by δ (t − t1)and δ (t − t1), we find that the following equation holds outside the breakmoment:

d2ydt2

+ a1(t) dydt

+ a2(t)y = b0(t) d2xdt2

+ b1(t) dxdt

+ b2(t)x, (4.100)

and, moreover, at the moment t1 the break conditions hold:

∆y = b0(t1)∆x,

∆y + a1(t1)∆y = b0(t1)∆x + [b1(t1) − b0(t1)]∆x.(4.101)

As follows from (4.101), for b0(t1) = 0 we have ∆y = 0, i.e., the solutiony(t) is continuous.

4.2.5 Operator Equation of Closed-Loop Linear System

In the previous paragraph the breaks of the output signal y(t) and itsderivatives are completely defined by the properties of the input signal x(t).Considering closed-loop systems we have to analyze them together, becauseproperties of the signal x(t) will depend on y(t). Cosnider Equation (4.86)

assuming additionally that

x = f (y), (4.102)

where f (y) is a discontinuous nonlinear function. In this case, the break(switching) moments ti are defined by the properties of the nonlinear el-ement. As before, between the switching moments we have the followingequations:

ω1 d

dt, t y = ω2 d

dt, tx, x = f (y). (4.103)










where ti are the points of discontinuity of ∂f/∂α, and

∆f i =∂f i∂α

(ti, α) −∂f i−1

∂α(ti, α). (4.119)

Expressions for derivatives of higher orders can be found in a similar way.

Now we consider differentiation of a product of generalized functions withrespect to a parameter.

Let a(t, α) be a function differentiable with respect to t and α requirednumber of times, and F (t, α) be a generalized function depending on aparameter. In this case, the product aF is defined, according to (4.77), by

a(t, α)F (t, α), φ = F (t, α), a(t, α)φ = q aφ(α). (4.120)

Using (4.120), it can be easily shown that we have, analogously to (4.135),

a(t, α)δ (s)(t − ti(α)) = (−1)sds

dτ s[a(τ )δ (t − τ )] |t=ti(α) . (4.121)

In virtue of (4.110), the generalized derivative of the product with respectto the parameter

D

dα[a(t, α)F (t, α)] = F 1 (4.122)

is given by the relation

F 1, φ =dq aφ(α)

dα. (4.123)

It can be shown [87] that in this case the following formula of differenti-ation of a product holds:

D

∂α[a(t, α)F (t, α)] =

∂a(t, α)

∂αF (t, α) + a(t, α)

DF (t, α)

∂α. (4.124)

which is similar to the classical one. From (4.124) it follows that

D

∂α[a(t, α)sign(t − t(α))] =

=∂a(t, α)

∂αsign(t − t(α)) − 2a(t(α), α)δ (t − t(α))

dt(α)

dα

(4.125)

If f 1(t, α) and f 2(t, α) are ordinary piecewise continuous functions, For-mula (4.124) in the general case does not hold, and the product is to be

differentiated directly by (4.113).





The use of generalized differentiation with respect to parameter yields aneffective and simple method of constructing sensitivity equations. Let usdemonstrate it by an example of a single-loop nonlinear system describedby the equations

λny + a1(t, α)λn−1y + . . . + an(t, α)y =

= b0(t, α)λn−1x + . . . + bn(t, α)x, x = f (y, α).(4.126)

Assuming λ = D/dt, we consider (4.126) as an equation in generalizedfunctions

ω1(λ,t,α)y = ω2(λ,t,α)x, x = f (y, α). (4.127)

DEFINITION 4.15 Let x = f (y, α) be a discontinuous function, and y be an ordinary function, possibly discontinuous. Then, the sensitivity functions uy and ux are defined as ordinary derivatives

uy =∂y(t, α)

∂α, ux =

∂x(t, α)

∂α. (4.128)

A general method of constructing sensitivity equations can be describedas follows.

1. Both the sides of Equation (4.127) are differentiated with respect tothe parameter. In this operation we employ the commutative property of differentiation by t and α.

2. As a result, we obtain a differential equation with respect to the gen-eralized derivatives Dx/∂α and Dy/∂α. In any special case these equationsmust be derived using the above features of multiplication and differentia-tion of generalized functions.

3. Using relations between ordinary and generalized derivatives

ux = Dx(t, α)∂α

+i

∆x(ti) dtidα

δ (t − ti),

uy =Dy(t, α)

∂α+i

∆y(ti)dtidα

δ (t − ti),(4.129)

we find an equation in generalized functions with respect to desired sensi-tivity functions (4.128).

For instance, if all coefficients of ai(t, α) and bi(t, α) in (4.126) are suf-ficiently smooth, for generalized differentiation with respect to α we may

use the rule (4.124).




As a result, we obtain

ω1(λ,t,α)Dy

∂α+

∂ω1(λ,t,α)

∂αy =

= ω2(λ,t,α) Dx∂α

+ ∂ω2(λ,t,α)∂α

x.

(4.130)

Substituting (4.129) in this equation, we obtain the sensitivity equation

ω1uy − ω1

i

∆yidti

dαδ (t − ti) +

∂ω1

∂αy =

= ω2ux − ω2

i

∆xi

dti

dαδ (t − ti) +

∂ω2

∂αx.

(4.131)

Transforming Equation (4.131), we note that

ω1(λ,t,α)i

∆yidtidα

δ (t − ti) =

=k

αk(t, α)i

∆yidtidα

δ (n−k)(t − ti)

=i

∆yidtidα

ak(t, α)δ (n−k)(t − ti).

(4.132)

Using Formula (4.132), we find

ω1(λ,t,α)i

∆yidtidα

δ (t − ti) =n

ρ=0

i

µρiδ (ρ)(t − ti). (4.133)

where µρi are constants. Similarly, we can obtain

ω2(λ,t,α)i

∆xi dtidα δ (t − ti) =

n−1ρ=0

i

ν ρiδ (ρ)(t − ti). (4.134)

Then, let us define, as usual, the following generalized derivatives:

Dsuy

dts=

dsuy

dts+

s−1ρ=0

i

∆u(ρ)yi δ (s−ρ−1)(t − ti),

Dsux

dts

=ds

dts

df

dα

+s−1

ρ=0

i

∆df

dα(ρ)

i

δ (s−ρ−1)(t − ti)

(4.135)




and substitute Relations (4.133)-(4.135) into the sensitivity equation. Asa result, we find ordinary differential equations determining the functionsuy on the intervals between the switching moments, and the correspondingbreak conditions for the function uy. We will not present the transforma-

tions in the general case due to their awkwardness, but these calculationsusually are not cumbersome in practical cases.

As an example of applying the developed method, we construct the sensi-tivity equations for the system (4.95), (4.102), assuming that the coefficientsai and bi depend on the parameter α, while the nonlinear function does notdepend on the parameter explicitly:

x = f i(y), ti < t < ti+1,

so that the parameter α does not appear in the function f i explicitly. Usinggeneralized differentiation, from (4.95) we have

λ2Dy

∂α+ a1λ

Dy

∂α+ a2

Dy

∂α+

∂a1∂α

λy +∂a2∂α

y

= b0λ2Dx

∂α+ b1λ

Dx

∂α+ b2

Dx

∂α+

∂b0∂α

λ2x +∂b1∂α

λx +∂b0∂α

x.

(4.136)

Moreover,

Dy

∂ = uy −

i

∆yidtidα

δ (t − ti),

λDy

∂α=

D

∂t

Dy

∂α=

duy

dt+i

∆uyiδ (t − ti)

−i

∆yidtidα

δ (t − ti),

λ2Dy

∂α=

D2

∂t2Dy

∂α=

d2uy

dt2+

i∆uyiδ (t − ti)

+

∆uyi δ (t − ti) −i

∆yi dtidα

δ (t − ti).

(4.137)

Similarly,

Dx

∂ = g −

i

∆f idtidα

δ (t − ti),

λDx

∂α=

dg

dt+ ∆giδ (t − ti) −

i

∆f idtidα

δ (t − ti),










Substituting (4.153) into (4.152), we obtain an equation for uy:

d2uy

dt2+ sign(t − τ )

duy

dt+ a2uy =

= 2y duy

dtsign (t − τ ) + 2yuy sign(t − τ ),

(4.154)

and the break conditions at the moment t = τ :

∆uy = uy(τ + 0) − uy(τ − 0) = −2y2(τ ),

∆uy − [y(τ + 0) + y(τ − 0)] = 2y(τ )[uy(τ + 0) + uy(τ − 0)].(4.155)

4.3.4 Sensitivity Equations for Multivariable Systems

Using the relations of the preceding paragraphs, a standard procedurefor constructing sensitivity equations can be proposed for multivariablesystems containing discontinuous nonlinearities.

First, let equations of the system have the form

W 1(λ, α)Y = W 2(λ, α)F (Y, α), (4.156)

where W i(λ, α)(i = 1, 2) are polynomial matrices in generalized differenti-

ation operator λ = D/dt, Y is a vector of unknowns, F is a discontinuousvector of nonlinear functions, and α is a scalar parameter.

In this case, the operators W i(λ, α) are commutative with operator of generalized differentiation D/∂α. Therefore, differentiating (4.156) withrespect to α yields

W 1(λ, α)DY

∂α+

∂

∂αW 1(λ, α)Y =

= W 2(λ, α)DF

∂α+

∂

∂αW 2(λ, α)F ,

(4.157)

where

DY

∂α= U −

i

∆Y idtidα

δ (t − ti),

DF

∂α=

dF

dα−i

∆F idtidα

δ (t − ti),(4.158)

and

U =

∂Y (t, α)

∂α ,

dF

dα =

∂F

∂Y U +

∂F

∂α . (4.159)




The derivatives and switching moments appearing in (4.158) can be ob-tained, in the general case, by the rules of differentiation for implicit func-tions. For example, if the break conditions have the form

gi[Y (ti − 0), ti, α] = 0, (4.160)

by (3.50) we have

dtidα

= −

∂g−i∂Y

U −i +∂g−i∂α

∂g−i∂Y

Y −i +∂g−i

∂t

. (4.161)

As in the scalar case, initial equations and sensitivity equations can bereduced to a sequence of ordinary differential equations linked with one

another by the break conditions. Let us have

W 1(λ, α) =n

k=0

Akλn−k, W 2(λ, α) =n

k=1

Bkλn−k, (4.162)

in (4.156). Then, using the same reasoning as in the scalar case, we obtain

W 1d

dt, α = W 2

d

dt, αF. (4.163)

During the transition from one equation (4.163) to another we have thebreak conditions

A0∆Y k = 0, A0∆Y k + A1∆Y k = B1∆F k, . . . , (4.164)

where ∆Y k and ∆F k are the breaks of the vectors Y and F at the switchingmoments tk.

Analogously, we can obtain, on the basis of (4.157), ordinary equations

and break conditions for the sensitivity functions.To illustrate the above theory, let us obtain the sensitivity equation fora system given in the normal Cauchy’s form. Let

dY

dt= F (Y,t,α), F (Y,t,α) = F i(Y,t,α), ti < t < ti+1. (4.165)

In this case, at the moments t = ti solutions of Equations (4.165) areconnected by means of conditions (3.2):

Y (ti + 0) = Y +i = Φ[Y (ti − 0), ti(α)] = Φ(Y

−

i , ti, α). (4.166)








For construction of the sensitivity equation it is necessary to consider twopossibilities. If all coefficients Ai(t, α) and Bi(t, α) are sufficiently smooth,the products

Ai(t, α)

DsY

dts , Bi(t, α)

DsY

dts (4.182)

are differentiable with respect to α using the classical formulas of differen-tiation:

D

∂α

Ai(t, α)

DsY

dts

=

∂Ai

∂α

DsY

dts+ Ai(t, α)

Ds

dtsDY

∂α. (4.183)

For further calculation, the terms Ai(t, α)δ (s)(t − ti) should be expanded

according to (4.69) and (4.70).Another situation takes place when the coefficients Ai(t, α) and Bi(t, α)

are discontinuous. Such being the case, the theory of generalized functionsis applicable only if delta-function and its derivatives in (4.182) are multi-plied by functions having a required number of derivatives. Nevertheless,products of ordinary functions having breaks at the same points are allow-able, as was demonstrated in the scalar case. We will not present detailedcalculations, because they are clear from the examples given above.

4.3.5 Higher-Order Sensitivity Equations

DEFINITION 4.16 Sensitivity function of a vector function Y (t, α)is defined as the following ordinary derivative:

U =∂Y

∂α=

DY

∂α+i

∆Y idtidα

δ (t − ti), (4.184)

where t = ti are the points of discontinuity of the function Y (t, α).

DEFINITION 4.17 Sensitivity functions of higher orders of the func-tion Y (t, α) are defined as ordinary higher derivatives

U (s) =∂ sY

∂αs, s ≥ 1. (4.185)

Relations between sensitivity functions of higher orders and correspond-

ing generalized derivatives can be established immediately. For instance,




applying generalized differentiation with respect to α to (4.184), we have

DU

dα=

D2Y

∂α2+

i

d

dα ∆Y idtidα δ (t − ti)

−i

∆Y i

dtidα

2

δ (t − ti),(4.186)

Then, using the fact that

DU

∂α=

∂U

∂α−

∆U idtidα

δ (t − dti),∂U

∂α= U (2), (4.187)

we find

U (2) =D2Y

∂α2+i

d

dα

∆Y i

dtidα

δ (t − ti)

+i

∆U idtidα

δ (t − ti) −i

∆Y i

dtidα

2

δ (t − ti).

(4.188)

In some problems it is required to find differential equations for sensitivityfunctions of higher orders. In principle, this problem causes no difficulties.Indeed, from (4.185) it follows that

U (s) =∂U (s−1)

∂α. (4.189)

Therefore, the sensitivity function of the s-th order is the first-order sen-sitivity function for U (s−1). Having the differential equation for the sensi-tivity function U (s−1) and constructing the associated sensitivity equations,we obtain the required sensitivity equations of higher orders. To illustratethis idea, we construct the second-order sensitivity equations for the systemgiven by (4.165) and (4.166).

Notice that the sensitivity equation (4.176) can be combined with thebreak conditions (4.177) in the form

DU

dt=

dF

∂α+i

∆U iδ (t − ti). (4.190)

In principle, Equation (4.190) does not differ from (4.169). Using




generalized differentiation with respect to α, we have

D

dt

DU

dα=

D

∂α

dF

∂α

+

id∆U i

dαδ (t − ti) −

i∆U i

dtidα

δ (t − ti). (4.191)

Moreover, similarly to (4.172),

D

dt

DU

∂α=

DU (2)

dα−i

∆U idtidα

δ (t − ti). (4.192)

Since

D

∂αdF

∂α

=

d2F

∂α2 −i ∆

dF

∂αi

dti

dα δ (t − ti), (4.193)

using the same reasoning as for derivation of the relations (4.176)-(4.178),from (4.191) we obtain an equation for the second-order sensitivity function

dU (2)

dt=

d2F

∂α2, (4.194)

and, moreover,

∆U (2)i = −∆

dF ∂α

i

dtidα

+ ddα

∆U i, (4.195)

where ∆U i are determined by Formula (4.177).

In a similar way we can find initial conditions for sensitivity functionsof higher orders. For example, using Formula (2.25), we find, for the casewhen the sensitivity equations for the system (4.165)–(4.166) have the form(4.176)–(4.177),

U (2)

0 =

dU 0dα − U 0

dt0dα =

dU 0dα −

dF

∂α

t=t0

dt0dα , (4.196)

Moreover, by (2.25) we have

U 0 =dY 0dα

− f (Y 0, t0, α)dt0dα

, (4.197)

and

dF

∂α

t=t0

= ∂F

∂Y

t=t0+0

U 0 + ∂F

∂α

t=t0+0

. (4.198)










Solving successively Equations (4.214) and “sewing” them with thehelp of the conditions (4.215), we obtain the sensitivity function and itsderivatives.

But the procedure described above is fairly awkward. Moreover, in prac-

tical problems we often need only the sensitivity function itself, but not itsderivatives. In such cases it is more convenient to use the second calculationtechnique described in Section 3.3.

With this aim in view, we rewrite (4.212) in the form

ω1(λ)u = q (t) + ω2(λ)i

u(ti − 0)

|y(ti − 0)|δ (t − ti), (4.216)

where

q (t) =∂ω2

∂αx −

∂ω1

∂αy (4.217)

is a known generalized function.If u(t) is a fixed solution of the equation

ω1(λ)u = q (t) (4.218)

(it is, according to the given assumptions, an ordinary function), then thesolution of (4.216) can be represented in the form

u(t) = u(t) + 2kpi

u−

iy−i g(t − ti), (4.219)

where g(t) is the weight function of the linear stationary system with thetransfer function

ω(λ) =ω2(λ)

ω1(λ). (4.220)

It is known [70] that if the transfer function (4.220) can be expanded intopartial fractions as

ω(λ) =m

i

cim

(λ − λi)m , (4.221)

then

g(t) =m

i

cim(m − 1)!

eλittm−1. (4.222)

Expression (4.219) makes it possible to immediately construct the sensi-tivity function in time domain. Indeed, let t0 be the starting point and t1be the first switching moment. Then, from (4.219) it follows that

u(t) = u(t), t0 < t ≤ t1 − 0, (4.223)






Then, obviously,

x(t) = (−1)ikp, ti < t < ti+1, (4.229)

Hence,

∂x

∂kp= r(t) = (−1)i, ti < t < ti+1. (4.230)

Since the parameter kr is not included in the condition of the switchingsurface, the value dti/dkr is, as before, given by Relation (4.210). Repeatingthe above calculations, we find

Dx

∂kp = r(t) = 2kpi

u−iy−i δ (t − ti). (4.231)

Assume that initial conditions are independent of the parameter on thefamily of solutions under consideration. Then, the sensitivity function canbe calculated by (4.219), where u(t) is the solution of the equation

ω1(λ)u = ω2(λ)r(t) (4.232)

with zero initial conditions.As is known [70], u(t) is defined by

u =

t t0

g(t − τ )r(τ )dτ, (4.233)

where g(t) is the weight function (4.222). Substituting the explicit formulafor r(t), we obtain

u(t) =n−1i=0

(−1)iti+1 ti

g(t − τ )dτ + (−1)nt

tn

g(t − τ )dτ,

tn < t < tn+1.

(4.234)

Using (4.234) and (4.219), as well as the recursive procedure describedabove, we can construct the required sensitivity function.

Next, let us illustrate the method of constructing the sensitivity equation

with respect to the parameter σ0. Using generalized differentiation with




respect to σ0 in (4.199), we obtain, similarly to (4.228),

ω1(λ)u = ω2(λ)Dx0∂σ0

. (4.235)

Since the parameter σ0 is not included in the nonlinear function explicitly,we have

Dx

∂σ0= −

i

∆xi

dtidσ0

δ (t − ti). (4.236)

Then, considering the break conditions (4.206) as an implicit equationdetermining the function ti(σ0), we find

dtidσ0

= −u−i ± 1

y−i= −

u+i ± 1

y+i, (4.237)

and Equation (4.235) takes the form

ω1(λ)u = ω2(λ)

2kp

i

u−i + (−1)iy−i

δ (t − ti)

. (4.238)

In this case, q (t) = 0 (see (4.217)), and, assuming that the initial condi-tions are independent of σ0, we have u(t) = 0, which immediately yields

u = 2kpi

u−i + (−1)iy−i g(t − ti). (4.239)

As before, it is assumed for concreteness that the first switching after t0occurs from the upper branch of the relay characteristic to the lower one.

In particular, from (4.239) it follows that

u(t) = 0, t0 < t < t1. (4.240)

From the physical viewpoint this result is clear. Indeed, since initialconditions are assumed to be independent of σ0, it is obvious that varia-tion of this parameter can affect the solution only after the first switchingmoment t1.

Let us construct, for the case at hand, the equation for the second-order

sensitivity function.






f (y, t) is the characteristic of the sampling element:

x = f (y, t) =

y(nT ), nT < t < nT + t1,

0, nT + t1 < t < (n + 1)T.(4.246)

First, we assume that the sampling period T is independent of the pa-rameter. Using the break conditions (4.164), it is easy to find that in thiscase the solution y(t) is continuous, therefore,

Dy

dt=

dy

dt,

Dy

∂α=

∂y

∂α= u. (4.247)

From continuity of y(t) it follows that the function (4.246) has disconti-nuities only at the moments

ti = iT, ti = ti + t1, (4.248)

which are independent of the parameter α.

Therefore, according to (4.205), in the given case we have

Dx

∂α=

∂x

∂α. (4.249)

Using generalized differentiation with respect to α is (4.245) and taking

(4.247) and (4.249) into account, we obtain

ω1(λ, α)u = ω2(λ, α)∂x

∂α−

∂ω1

∂αy +

∂ω2

∂αx. (4.250)

Let us calculate the ordinary derivative dx/dα. With this aim in view, wenotice that the sampling period is independent of the parameter, therefore

∂y(nT,α)

∂α= u(nT ), (4.251)

Hence,

∂x

∂α=

u(nT ), nT < t < nT + t1,

0, nT + t1 < t < t(n + 1)T.(4.252)

Equation (4.252) does not, in principle, differ from (4.246). Thus, in thegiven case, the sensitivity equation (4.250) is the equation of a pulse systemsimilar to the initial one, but acted on by the exogenous disturbance

q (t) = −

∂ω1

∂α y +

∂ω2

∂α x, (4.253)




Since x is a piecewise continuous function, we have

∂ω2

∂αx =

n

i=1dbidα

λn−ix; (4.254)

Due to the above results,

λx =Dx

dt=

∆x(ti)δ (t − ti) +

∆x(ti)δ (t − ti),

λ2x =

∆x(ti)δ (t − ti) +

∆x(ti)δ (t − ti),

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(4.255)

where

∆x(ti) = y(ti) = y(iT ), ∆x(ti) = −y(ti) = −y(iT ), (4.256)

As in the previous case, actual determination of the sensitivity functionfrom Equation (4.250) can be performed in two ways: either by a transi-tion to a sequence of ordinary equations connected by the correspondingbreak conditions, or by direct solution of an equation of the form (4.236)with a known function q (t). In this case, using the break equations, it iseasy to show that the sensitivity function is continuous in the case under

consideration.Then, consider the sampling period T as a parameter and assume that

the polynomials wi(λ) are independent of T . In this case the switchingmoments ti and ti depend on the parameter, because

dtidT

= i,dtidT

= i. (4.257)

We have (4.247), as before, but, as distinct from (4.249),

Dx

∂T =

∂x

∂T −i

∆x(ti)iδ (t − ti) −i

∆x(ti)iδ (t − ti). (4.258)

Therefore, generalized differentiation of (4.245) with respect to T yields

ω1(λ)u = ω2(λ)

∂x

∂T −

i ∆x(ti) iδ (t − ti)

−i

∆x(ti) iδ (t − ti).(4.259)




Let us calculate the ordinary derivative ∂x/∂T . For this purpose, weassume that the function y(nT ) depends on T in two ways: as a functionof the current moment tn = nT and as a function of the parameter T on allpreceding stages of motion. Therefore, the value y(nT ) = y(nT,T ) must

actually appear in (4.246), and differentiation yields

dy(nT,T )

dT = n

dy

dt(nT + 0) + u(nT + 0)

= ndy

dt(nT − 0) + u(nT − 0).

(4.260)

Using the last equality, we find

∂x

∂T = n

dy

dt

(nT − 0) + u(nT − 0), nT < t < nT + t1,

0, nT + t1 < t < t(n + 1)T.(4.261)

Let us show a possible way of actual construction of the sensitivity func-tion by Equation (4.259). Let a function h(t, tn, tn) be a solution of theequation

ω1(λ)y = ω2(λ)x (4.262)

for the case when

x(t) = 1 for tn < t < tn,

0 for t < tn, t < tn, (4.263)

and zero initial conditions. Obviously,

h(t, tn, tn) =

0 for t < tn,t

tn

g(t − τ )dτ for tn < t < tn,

t

tn

g(t − τ )dτ for t > tn

,

(4.264)

where g(t) is a weight function associated with the transfer function (4.220).

Obviously, taking into account (4.256) and (4.261), a solution of Equation(4.258) for zero initial conditions corresponding to the moment t0 has theform

u(t) =i

h(t, ti, ti)

u−i + iy−i

−i

[g(t − ti) − g(t − ti)] yi.(4.265)






where T is a fixed sampling period, and pulse duration t1n ≤ T is a functionof the value yn = y(nT ), so that

t1n = ψ(yn), (4.272)

where ψ(y) is a smooth function.

According to the assumptions made above, the function y(t) is contin-uous, therefore Equation (4.247) holds. The controlling function f (y, t) ispiecewise constant and the height of pulses is independent of the parameter.Therefore, for any α we have

Dx

∂α= −

i

∆xi

dtidα

δ (t − ti), (4.273)

where ti are the corresponding switching moments.In this case, as before, we have two sequences of switching moments:

ti = iT, ti = iT + t1i, (4.274)

and, accordingly,

∆xi = ∆x(ti) = sign yn, ∆xi(ti) = ∆xi = − sign yn. (4.275)

Therefore, the general sensitivity equation of a pulse-width system canbe obtained from (4.245) by generalized differentiation with respect to α:

ω1(λ, α)u = −ω2(λ, α)

i

sign yi

δ (t − ti)

dtidα

− δ (t − ti)

dtidα

−

∂ω1

dαy +

∂ω2

∂αx.

(4.276)

Equation (4.276) gives a number of special cases important in applica-

tions. Assume that the sampling period is independent of the parameter.Then,

dtidα

= 0 (4.277)

and the sensitivity function remains continuous at the moments ti = iT ,while the function y(iT ) = y(iT,α) = yi is continuously differentiable withrespect to α. Therefore,

dt1i

dα=

dψ(yi)

dyi·

∂y(iT,α)

∂α=

dψ

dy(ti)u(ti). (4.278)






expressed in terms of the weight function (4.222) as

u(t) =

ii sign yig(t − ti)

+i

i + dψ

dy(ti)

iy−i + u−i

sign yig(t − ti), (4.284)

From this equation the desired solution can be obtained recursively.

4.4.4 Pulse-Frequency Systems

Sensitivity investigation for pulse-frequency systems is connected with anumber of characteristic features mentioned in Chapter 3.

Ignoring, for simplicity, the dead zone of the sampling element (3.187),we can describe a wide class of pulse-frequency modulators by relations of the form

x = f (y, t) = sign yi, (4.285)

where the pulse duration is defined by

ti+1 − ti = T i = ψ(yi), (4.286)

where ψ(y) is a bounded, even positive, function that will be assumed tobe continuously differentiable.

Let t0 be a starting moment that does not coincide with switching mo-ments ti(i = 1, 2, . . .). Then, as for derivation of Equation (4.276), weobtain the sensitivity equation of pulse-frequency system in the form

ω1(λ, α)u = −ω2(λ, α)i

∆xi

dtidα

δ (t − ti) +∂ω2

∂αx +

∂ω1

∂αy, (4.287)

where ∆xi = 2sign yi for sign yi = −sign yi−1, and ∆xi = 0 for sign yi =sign yi−1.

Therefore, actually only pulses corresponding to sign changes of the con-trolling function are taken in (4.287). According to (4.286), the switchingmoments are given by the recurrent relation

ti = ti−1 + ψ(yi−1). (4.288)

Consider the most general problem when, together with w1 and w2, the

function ψ also depends on the parameter. Note that the function y(t) is

© 2000 by CRC PressLLC



continuous. Therefore, as before, at the switching moments the function

yi = y[ti(α), α]

is differentiable with respect to α disregarding the fact that the sensitivityfunction can be discontinuous.

As follows from the aforesaid,

dyidα

= y−idtidα

+ u−i = y+idtidα

+ u+i . (4.289)

Differentiating (4.288) with respect to α, we find

dtidα

= dti−1dα

+ ∂ψ∂y

(ti) dyi−1dα

+ ∂ψ∂α

(ti), (4.290)

where any expression obtained from (4.289) can be used at the place of dyi−1/dα. Let us note that, as distinct from the previous problems wherewe had finite formulas determining the derivatives from the switching mo-ments, in this case we obtain only recurrent relations (4.290).

Nevertheless, using (4.290) and the sensitivity equations (4.287), forknown initial conditions we can successively find all values of interest.

Let, for instance,∂ω1

∂α=

∂ω2

∂α= 0 (4.291)

From (4.287) we have

ω1(λ) = −ω2(λ)i

∆xi

dtidα

δ (t − ti). (4.292)

Assume also that initial conditions for the sensitivity equation (4.292)are zero, and the moment t0 is independent of the parameter. In this case

u(t) = 0, t0 ≤ t ≤ t1 − 0, (4.293)

where t1 is the first switching moment.

Then, for sign y1 = −sign y0 we have

u = −2sign y1

dt1

dα g(t − t1), t2 − 0 ≥ t ≥ t1 + 0. (4.294)




The switching moment t1(α) is obtained from the equation

t1 = t0 + ψ(y0, α), (4.295)

Hence,dt1dα

=∂ψ(y0, α)

∂α(4.296)

This uniquely determines the solution u(t) on the interval

t1 + 0 ≤ t ≤ t2 − 0.

After that, calculation process can be continued, because we can find thederivative dt2/dα using known dt1/dα and the sensitivity equation, andso on.




Chapter 5

Sensitivity of Non-Time Characteristics of Control Systems

5.1 Sensitivity of Transfer Function and FrequencyResponses of Linear Systems

5.1.1 Sensitivity of Transfer Function

DEFINITION 5.1 For a function x(t) in a real variable t the following function in complex variable s:

Z (s) = L [z(t)] =

∞ 0

z(t)e−stdt. (5.1)

is called the Laplace image (transform).

It is assumed that the convergence conditions formulated in Section 4.1.6hold for the integral (5.1).

Let us have a parametric family of solutions z(t, α), and let for any α ∈

[α1, α2] there exist the transform

Z (s, α) =

∞ 0

z(t, α)e−stdt.

THEOREM 5.1 [114]

Assume that the function z(t, α) = z(t, α)e−st does exist and is continuous with respect to t for t ≥ 0 and α ∈ [α1, α2]. Moreover, let its derivative

∂ z(t, α)/∂α be, for the specified arguments, continuous with respect to both




the arguments. Assume that the integral (5.1) converges for all α ∈ [α1, α2],and the integral

∞

0

∂ z(t, α)

∂αdt

converges uniformly with respect to α in the same interval. Then, for any α ∈ [α1, α2] we have

∂Z (s, α)

∂α=

∞ 0

∂ z(t, α)

∂αdt. (5.2)

From Formula (5.3) we have

∂Z (s, α)

∂α=

∞ 0

∂z(t, α)

∂αe−stdt,

or∂Z

∂α= L

∂z(t, α)

∂α

.

As follows from Theorem 5.1, Laplace transformation is commutativewith respect to the variable t and differentiating by the parameter α. It

is known that transfer function w( p) of a causal single-input-single-outputlinear system is the image of the weight function h(t), so that

ω(s) =

∞ 0

h(t)e−stdt.

The weight function h(t) is defined as the response of the system to theunit delta-function acting on its input. The weight function corresponds tothe sensitivity function u(t) = ∂h(t, α)/dα. By Theorem 5.1, we can write

L

∂h(t, α)

∂α

=

∂

∂αω(s)

DEFINITION 5.2 The derivative ∂w(s)/∂α will be called the sensi-tivity function of the transfer function.

Denote

S

α

ω (s) =

∂ω(s)

∂α .






∂ ln ω(s)

∂α=

∂ ln ω0(s)

∂α1 + ωfb(s)ω0(s)

, (5.6)

Equations (5.5)–(5.6) establish the relations between sensitivity functions

of closed-loop and open-loop systems.

5.1.2 Sensitivity of Frequency Responses

DEFINITION 5.3 Substitute jω for s in the transfer function w(s).The function thus obtained is called the complex frequency response of the system.

In polar coordinates the function w( jω) has the form

w( jω) = A(ω)ejφ(ω), (5.7)

where A(ω) = |w( jω)| is the amplitude frequency response , and φ(ω) =argw( jω) is the phase frequency response. Moreover, the complex functionw( jω) can be written in the form

w( jω) = P (ω) + jQ(ω) (5.8)

where P (ω) = Re w( jω) and Q(ω) = Im w( jω).

DEFINITION 5.4 The function P (ω) is called the real frequency re-sponse, and the function Q(ω) the imaginary frequency response.

The functions A(ω), φ(ω), P (ω), and Q(ω) are connected by the followingrelations:

P (ω) = A(ω)cos φ(ω), Q(ω) = A(ω)sin φ(ω),

A(ω) =

P 2(ω) + Q2(ω), φ(ω) = arctanQ(ω)

P (ω).

(5.9)

If transfer function of a system has the form

w(s) =

gi=0

bisg−i

n

i=0

aisn−i




the following relations hold:

A(ω) = a2(ω) + b2(ω)

c2(ω) + d2(ω), (5.10)

φ(ω) = arctanb(ω)c(ω) − a(ω)d(ω)

a(ω)c(ω) + b(ω)d(ω), (5.11)

P (ω) =a(ω)c(ω) + b(ω)d(ω)

c2(ω) + d2(ω), (5.12)

Q(ω) =b(ω)c(ω) − a(ω)d(ω)

c2(ω) + d2(ω), (5.13)

where

a(ω) =i

(−1)ibg−2iω2i,

b(ω) =i

(−1)ibg−2i−1ω2i+1,

c(ω) =i

(−1)ian−2iω2i,

d(ω) =i

(−1)ian−2i−1ω2i+1.

(5.14)

DEFINITION 5.5 If all poles and zeros of a system are located in the left half-plane of the complex plane, such a system is called minimal-phase.

For minimal-phase systems there are bi-unique relations between thefunctions P (ω) and Q(ω), as well as between A(ω) and φ(ω) [100]:

P (ω) = −1

π

∞

−∞

Q(τ )

τ − ω

dτ, (5.15)

Q(ω) =1

π

∞ −∞

P (τ )

τ − ωdτ, (5.16)

ln A(ω) = −1

π

∞

−∞

φ(τ )

τ − ωdτ, (5.17)










is continuous with respect to t and α. Then, there are sensitivity functions∂P (ω)/∂α and ∂Q(ω)/∂α given by

∂P (ω)

∂α =

∞ 0

∂h(t, α)

∂α cos ωtdω,

∂Q(ω)

∂α= −

∞ 0

∂h(t, α)

∂αsin ωtdω.

Other frequency characteristics and indices considered in the previousparagraph are functionally connected with real and imaginary frequencyresponses by relations of the forms (5.9), (5.22), (5.28) and (5.29). Sincethe right sides of these relations have derivatives with respect to P and Q,there are all corresponding sensitivity functions and coefficients. This factmakes it possible to differentiate frequency responses under considerationwith respect to parameter α.

Let us differentiate Relation (5.20) with respect to α. We obtain

∂w( jω)

∂α=

∂A(ω)

∂α+ jA(ω)

∂φ(ω)

∂α

ejφ(ω) = w( jω)

∂ ln A(ω)

∂α+ j

∂φ(ω)

∂α

or

∂ ln w( jω)

∂α=

∂ ln A(ω)

∂α+ j

∂φ(ω)

∂α. (5.30)

The above equations define the relations between the sensitivity func-tion of the complex phase-amplitude frequency response with sensitivityfunctions of the amplitude and phase responses.

The sensitivity function ∂w( jω)/∂α can be represented in polar coordi-nates as

∂w( jω)

∂α = H (ω)e

jψ(ω)

,

where

H (ω) =

∂A(ω)

∂α

2+ A2(ω)

∂φ(ω)

∂α

2,

ψ(ω) = arctan

∂A(ω)

∂αsin φ(ω) + A(ω)

∂φ(ω)

∂αcos φ(ω)

∂A(ω)

∂α cos φ(ω) − A(ω)

∂φ(ω)

∂α sin φ(ω)

,






Analogously, from (5.9) we can find the following relations:

∂P (ω)

∂α=

∂A(ω)

∂αcos φ(ω) − A(ω)

∂φ(ω)

∂αsin φ(ω),

∂ ln P (ω)∂α

= ∂ ln A(ω)∂α

− ∂φ(ω)∂α

tan φ(ω);

∂Q(ω)

∂α=

∂A(ω)

∂αsin φ(ω) + A(ω)

∂φ(ω)

∂αcos φ(ω),

∂ ln Q(ω)

∂α=

∂ ln A(ω)

∂α+

∂φ(ω)

∂αcot φ(ω);

∂A(ω)

∂α=

∂P (ω)

∂αcos φ(ω) +

∂Q(ω)

∂αsin φ(ω),

∂φ(ω)∂α

=

∂ ln Q(ω)∂α

− ∂ ln P (ω)∂α

cos φ(ω)sin φ(ω),

(5.31)

5.1.4 Universal Algorithm for Determination of Sensitivity Functions for Frequency Characteristics

Any of the aforementioned frequency responses can be considered a com-posite function with respect to the parameter α, i.e.,

f (ω) = f [ω, a0(α), a1(α), . . . , an(α), b0(α), b1(α), . . . , bg(α)],

where f (ω) is a special frequency response, and ai(α) and bi(α) are coeffi-cients of the transfer function.

Then, according to the rules of differentiation for composite functions,for the desired sensitivity function we have

∂f

∂α=

n

i=0∂f

∂ai

dai

dα+

g

i=0∂f

∂bi

dbidα

. (5.32)

In this formula the form of the terms ∂f/∂ai and ∂f/∂bi is defined onlyby the structure of the transfer function (i.e., by the values n and s). Ex-pressions for these functions can be constructed in advance in the form of general universal formulas or in the form of tables for various values of nand s [69].

Consider, for example, the amplitude frequency response A(ω) given by(5.11). Since

∂A2(ω)

∂α = 2A(ω)

∂A(ω)

∂α






∂A2(ω)

∂bk= (−1)

g−k−12

2b(ω)ωg−k

c2(ω) + d2(ω),

g − k = 2i + 1, i = 0, 1, . . .

(5.37)

Multiplying the values ∂A2

(ω)/∂ai and ∂A2

(ω)/∂bj just obtained by1/2A(ω), we find the coefficients ∂A(ω)/∂ai and ∂A(ω)/∂bj , respectively.The multipliers dai/dα and dbj/dα are determined by the dependence of the transfer function on the parameter and can be found specially for eachconcrete system. Thus, the sensitivity function ∂A(ω)/∂α can be found inthree stages.

1. First, by (5.34)–(5.37) the values ∂A2(ω)/∂ai and ∂A2(ω)/∂bj areobtained.

2. Then, using (5.33), by a known amplitude frequency response A(ω)we calculate ∂A(ω)/∂ai and ∂A(ω)/∂bj .

3. Then, the desired sensitivity functions are obtained after evaluationof dai/dα and dbj/dα.

To find the sensitivity function ∂φ(ω)/∂α we employ the same technique.First we find the corresponding derivatives of the argument z in the function(5.12):

z(ω) =b(ω)c(ω) − a(ω)d(ω)

a(ω)c(ω) + b(ω)d(ω)

.

Then, by the formula

φ(ω)

∂α=

1

1 + z2(ω)

n

i=0

∂z(ω)

∂ai

dai

dα+

gi=0

∂z(ω)

∂bi

dbidα

we obtain the desired sensitivity function.The values ∂z(ω)/∂ai and ∂z(ω)/∂bj are determined by formulas similar

to (5.34)–(5.37).

The sensitivity of the real P (ω) and imaginary Q(ω) frequency responseswill be found by the following formulas:

P (ω)

∂α=

ni=0

∂P (ω)

∂ai

dai

dα+

gi=0

∂P (ω)

∂bi

dbidα

,

Q(ω)

∂α=

ni=0

∂Q(ω)

∂ai

dai

dα+

gi=0

∂Q(ω)

∂bi

dbidα

.

For the cofactors ∂P (ω)/∂ai, ∂P (ω)/∂bj, ∂Q(ω)/∂ai, and ∂Q(ω)/∂bj we

can also obtain universal formulas for different m and n.




If we have the transfer function (5.23), Equation (5.32) for the sensitivityfunction gets simplified. Let in the transfer function (5.23) the orders of thepolynomials b2(s) and a2(s) be equal to g2 and n2, respectively. Then, thefirst sum in the right side of (5.32) includes n2 or n2 + 1 terms, while the

second sum contains g2 or g2 + 1 terms. Moreover, the following relationholds:

∂f

∂ ln α=

N i

∂f

∂ ln ai+

M i

∂f

∂ ln bi,

where N and M are the numbers of terms in the corresponding sums.As an example, we consider a system with transfer function

w(s) =1 + T 1s

1 + T 2s.

For this transfer function we have

a(ω) = b1 = 1, b(ω) = b0ω = T 1ω,c(ω) = a1 = 1, d(ω) = a0ω = T 2ω,

A(ω) =

1 + T 21 ω2

1 + T 22 ω2, φ(ω) = arctan

(T 1 − T 2)ω

1 + T 1T 2ω2,

P (ω) =1 + T 1T 2ω2

1 + T 22

ω2, Q(ω) =

(T 1 − T 2)ω

1 + T 22

ω2.

z(ω) =(T 1 − T 2)ω

1 + T 1T 2ω2.

To find ∂A2(ω)/∂T 1 = ∂A2(ω)/∂b0 we employ Formula (5.37), becauseg = 1, k = 0, and g − k = 1. Then,

∂A2(ω)

∂T 1=

2T 1ω2

1 + T 22 ω2.

Since∂A(ω)

∂T 1= ∂A(ω)

∂b0= 1

2A(ω)∂A2(ω)

∂T 1,

we have∂A(ω)

∂T 1=

T 1ω2 (1 + T 21 ω2) (1 + T 22 ω2)

.

The value ∂A2(ω)/∂T 2 = ∂A2(ω)/∂a0 can be found by (5.35), becausen − k = 1:

∂A2(ω)

∂T 2= −

2T 2ω2A2(ω)

1 + T 22 ω2

.












cient) of a root pi root sensitivity:

dpidα = −

∂D(s, α)

∂α∂D(s, α)

∂s

s=pi,α=α0

= −

n

j=0∂D

∂cj

dcjdα

∂D

∂s

s=pi,α=α0

(5.45)

Since∂D

∂ck= pn−k

i , k = 1, . . . , n

∂D

∂pi=

n−1

k=1kcn−k p

k−1i + c0npn−1i ,

we have

dpi∂α

= −

n−1k=1

kcn−k pk−1i + c0npn−1i

−1 nk=1

pn−1i

dckdα

. (5.46)

Formula (5.46) makes it possible to find root sensitivity coefficients forreal as well as complex roots.

As an example, we consider sensitivity coefficients of the roots of the

following characteristic equation of an oscillatory unit:

D(s) = s2 + 2ξω0s + ω20 = 0.

The roots of this equations are

p1,2 = −ξω0 ± jω

1 − ξ 2,

Hence,

∂p1,2

∂ω0= −ξ ± j

1 − ξ 2;

∂p1,2∂ξ

= −ω ∓ jω0ξ 1 − ξ 2

.

To employ Formula (5.46), we preliminarily find

da2dω0

= 2ω0,da2dξ

= 0,

da1

dω0= 2ξ,

da1

dξ = 2ω0.








Example 5.1

Let

w(s) =s + b

s + a

.

Then,

w(s) =s + b

2

s +

a + b

2

,

p1 = −a, q 1 = −b, z1 = −a + b

2.

Obviously,

∂p1∂a

= −1;∂z1∂a

= −1

2.

To find ∂z1/∂α we use Formula (5.49):

∂z1

∂a=

1

− a + b2

+ a

∂p1

∂a 1

− a + b2

+ a

+−

a + b

2+ b

−a + b

2+ a

·1

−a + b

2+ b

−1

= −1

2.

5.2.4 Relations between Sensitivity of Transfer Functionand that of Poles and Zeros

Consider the transfer function

w(s) = k

mi=1

(s − q i)

n

i=1

(s − pi)

. (5.50)






DEFINITION 5.10 The matrix λE − A, where λ is an independent variable, is called the characteristic matrix. Its determinant

∆(λ) = det(λE − A) = λn + a1λn−1 + . . . + an (5.52)

is called the characteristic polynomial of the matrix A, and the roots of the characteristic polynomial are called eigenvalues.

Each eigenvalue λi is associated with an eigenvector X i satisfying theequality

A X i = λi X i. (5.53)

It is known that each eigenvector is determined up to a constant factor.Therefore, in specific problems it is necessary to perform normalization.

Obviously, eigenvalues of the transposed matrix AT coincide with theeigenvalues of the matrix A. Nevertheless, the eigenvectors are, in thegeneral case, different. Denote by Y i the eigenvectors of the matrix AT , sothat

AT Y i = λiY i. (5.54)

The eigenvectors of the initial and transposed matrices are orthogonal, i.e.,

(X i, Y j) = 0, i = j. (5.55)

For an integer k, eigenvalues of the matrix Ak are equal to λki (i =

1, . . . , n).

DEFINITION 5.11 The sum of diagonal elements n

i=1 aij of a ma-trix A is called the trace of the matrix A and is denoted by tr A.

DEFINITION 5.12 A matrix A is called similar to a matrix B if there is a nonsingular matrix H such that A = H −1BH .

Similar matrices have equal characteristic polynomials. Hence followsequality of the corresponding eigenvalues, determinants, and traces,because

tr A =n

i=1

λi = −a1, det A − (−1)nan =n

i=1

λi. (5.56)




5.3.2 Sensitivity of Eigenvalues

Let a matrix A for ∆α ∈ (−, ) have different eigenvalues λ1, . . . , λn,and corresponding eigenvectors X 1, . . . , X n. Consider the matrix A(∆α) =A + ∆αB, where B is an arbitrary real matrix of the same dimensions,and ∆α is a small parameter. The eigenvalues of the new matrix A + ∆αBare denoted by λi(∆α), and the corresponding eigenvectors by X i(∆α) (i =1, . . . , n). From the theory of perturbations of linear operators and matricesit is known [18, 110, 111] that the values λi(∆α) and X i(∆α) (i = 1, . . . , n)are continuous differentiable functions in the parameter ∆α. Moreover,A(0) = A, λi(0) = λi, and X i(0) = X i. In this case, the following powerseries converge:

λi(∆α) = λi +

∞j=1

1

j!∂ jλi(∆α)

∂ ∆jα

∆α=0 ∆j

α, (5.57)

X i(∆α) = X i +∞j=1

1

j!

∂ jX i(∆α)

∂ ∆jα

∆α=0

∆jα. (5.58)

DEFINITION 5.13 The following values β ij and γ ij given by

β ij =

∂ jλi(∆α)

∂ ∆jα

∆α=0

, γ ij =

∂ jX i(∆α)

∂ ∆jα

∆α=0

,

will be called sensitivity coefficient and sensitivity vector of the j-th order,respectively.

For the eigenvalue (5.57) and eigenvector (5.58) we have

A(∆α) X i(∆α) = λi(∆α) X i(∆α).

Differentiating the last equations with respect to the parameter α yields

∂A(∆α)

∂ ∆αX i(∆α) + A(∆α)

∂X i(∆α)

∂ ∆α

= β i1X i(∆α) + λi(∆α)∂X i(∆α)

∂ ∆α

(5.59)

Assuming ∆α = 0, let us transform the obtained sensitivity equation using




multiplication by the eigenvalue Y i of the matrix AT so that

(BX i, Y i) +

∂X i∂ ∆α

, AT Y i − λiY i

=

∂λi

∂ ∆α(X i, Y i).

As a result, using (5.55), we obtain

β i1 =(BX i, Y i)

(X i, Y i)(5.60)

To determine the first-order sensitivity coefficient of an eigenvalue byFormula (5.60) we need the derivative of the matrix A∆α with respect tothe parameter ∆α as well as the corresponding eigenvectors X i and Y i. Allthese values are determined for ∆α = 0.

Let us consider some special cases of Formula (5.60). Assume that anelement of the matrix A, say aks, is variable. Then, ∆α = ∆aks, and thematrix B has the only nonzero element bks. Then,

β i1 =∂λi

∂ ∆bks=

xisyikn

j=1

xijyij

, (5.61)

where xij and yij are the corresponding components of the vectors X i and

Y i. For a diagonal element of the matrix akk we have

β i1 =∂λi

∂ ∆akk=

xikyikn

j=1

xijyij

, (5.62)

If the matrix A is symmetric, the initial and transposed matrices coincide,therefore

∂λi

∂ ∆α=

(BX i, X i)

(X i, X i)(5.63)

∂λi

∂ ∆aks=

xisxikn

j=1

x2ij

, (5.64)

∂λi

∂ ∆akk=

x2ikn

j=1

x2ij

, (5.65)




If the eigenvectors of the matrices A and AT are orthonormalized, i.e.,

(X i, Y j) = δ ij , (5.66)

where δ ij is the Kronecker symbol, the formulas for the sensitivity coeffi-cients of the eigenvalue get simplified so that their denominators are equalto unity.

Let

C = Ak

and the matrix A have eigenvalues λ1, . . . , λn. Denote the eigenvectors of

the matrix C by λ(C )i . Then, as was noted in Section 5.1,

λ(c)i = λk

i .

Moreover,

∂λ(c)i

∂α= kλk−1

i

∂λi

∂α(5.67)

or

∂ ln λ(c)i

∂α= k

∂ ln λi

∂α(5.68)

5.3.3 Sensitivity of Real and Imaginary Parts of ComplexEigenvalues

In the preceding paragraph we presented general expressions for first-order sensitivity coefficients. These formulas are applicable for investiga-tion of real as well as complex eigenvalues. Now we will derive formulasallowing us to evaluate sensitivity of the real and imaginary part of a com-plex eigenvalue separately. With this aim in view, we consider the followingpair of complex-conjugate eigenvalues:

λk = µk + jν k,λk+1 = µk − jν k, (5.69)

They are associated with complex-conjugate eigenvectors

X k = Gk + jH k; X k+1 = Gk − jH k. (5.70)

The eigenvalues λk and λk+1 and the eigenvectors X k and X k+1 satisfythe equations

A X k = λkX k,

A X k+1 = λk+1X k+1,




Therefore,

A Gk = µkGk − ν kH k,A H k = µkH k + ν kGk,

(5.71)

The corresponding eigenvectors of the transposed matrix AT will be de-noted by

Y k = Qk + jV k, Y k+1 = Qk − jV k.

In analogy with (5.54), for the matrix AT we have

AT Qk = µkQk − ν kV k,AT V k = µkV k + ν kQk,

(5.72)

Let the matrix A and, correspondingly, all the components of Equations(5.71), be variable., i.e., depending on a parameter ∆α. Let us differ-entiate Equation (5.71) with respect to this parameter. For convenience,hereinafter we denote partial derivatives with respect to the parameter byindex α, for instance,

Aα =∂A(∆α)

∂ ∆α, µkα =

∂µk(∆α)

∂ ∆α.

Then, after differentiation, we obtain the following sensitivity equations:

AαGk + AGkα = µkαGk + µkGkα − ν kαH k − ν kH kα,AαH k + AH kα = µkαH k + µkH kα + ν kαGk + ν kGkα.

(5.73)

Determining the scalar product of each of the relations in (5.73) and thevectors Qk and V k, for ∆α = 0 we find

(AαGk, Qk) + (AGkα, Qk) = µkα(Gk, Qk) + µk(Gkα, Qk)

− ν kα(H k, Qk) − ν k(H kα, Qk),

(AαGk, V k) + (AGkα, V k) = µkα(Gk, V k) + µk(Gkα, V k)

− ν kα(H k, V k) − ν k(H kα, V k),

(AαH k, Qk) + (AH kα, Qk) = µkα(H k, Qk) + µk(H kα, Qk)

+ ν kα(Gk, Qk) − ν k(Gkα, Qk),

(AαH k, V k) + (AH kα, V k) = µkα(H k, V k) + µk(H kα, V k)

+ ν kα(Gk, V k) − ν k(Gkα, V k).

(5.74)




In this system, let us subtract term-wise the last equation from the firstone,

thus obtaining

[(Gk, Qk) − (H k, Qk)]µkα − [(H k, Qk) + (Gk, V k)]ν kα= (AαGk, Qk) − (AαH k, V k).

(5.75)

Summation and further transformations of the second and third equa-tions yield

[(Gk, V k) + (H k, Qk)]µkα + [(Gk, Qk) − (H k, V k)]ν kα= (AαGk, V k) + (AαH k, Qk).

(5.76)

Equations (5.75) and (5.76) represent a system of linear nonhomogeneousequations with respect to the sensitivity coefficients µkα and ν kα. Thedeterminant of this system is given by

∆ = [(Gk, Qk) − (H k, V k)]2 + [(Gk, Qk) − (H k, Qk)]2

and is a square of the absolute value of the following scalar product:

(X k, Y k) = (Gk + jH k, Qk + jV k)

Due to this reason, ∆ = 0. For normalized eigenvectors we have ∆ = 1.

5.3.4 Sensitivity of Eigenvectors

Consider Equation (5.59). For ∆α = 0 we find the scalar term-wiseproduct of this equation and a vector Y g such that i = g:

(Aγ i1, Y g) + (AαX i, Y g) = (λiγ i1, Y g) + (β i1X i, Y g).

The last equations can be transformed to the form

(AαX i, Y g) + (Y i1, AT Y g − λiY g) = β i1(X i, Y g).

or

(AαX i, Y g) + (γ i1, Y g)(λg − λi) = β i1(X i, Y g).

Hence,

(γ i1, Y g) =(AαX i, Y g) − β i1(X i, Y g)

λi − λg

, λi = λg. (5.77)




Since (X i, Y g) = 0 for i = g, we obtain

(γ i1, Y g) =(AαX i, Y g)

λi − λg, i = g. (5.78)

Equation (5.78) defines relations between the coordinates of the sensitiv-ity vector γ i1 and the elements of the vectors X i and Y g, elements of thematrix A and eigenvalues λi, λg, i = g. This equation can be written in thefollowing coordinate form:

nj=1

ygjγ i1j = dig, i = g, (5.79)

wheredig =

(AαX i, Y g)

λi − λg.

To determine the components γ i1j, j = 1, . . . , n we need n such equa-tions. For different g (g = 1, . . . , n) such that g = i we can construct n − 1ones. The remaining equation can be obtained as follows. Introduce a basisconsisting of the eigenvectors X 1, . . . , X n in the space under consideration(Euclidean n-dimensional real or complex). Then,

γ i1 =n

j=1

γ i1jX j. (5.80)

Introduce a normalization condition for the vector X i(∆α):

(X i(∆α), X i(∆α)) = 1,

Hence,

(γ i1, X i) = 0

or, with with account for orthogonality of eigenvectors and (5.80),

γ i1j = 0.

For the case when the eigenvectors of the matrix A are used as an or-thogonal basis, the remaining components can be found using the relations

γ i1g = (γ i1, Y g) =

n

j=1

γ i1jX j, Y g

.




Then, with account for (5.78), we finally obtain

γ i1g =(AαX j, Y g)

λi − λg, i = g,

γ i1j = 0.

(5.81)

5.3.5 Sensitivity Coefficients and Vectors of Higher Orders

For k ≥ 2, the values β ik and γ ik can be obtained by correspondingdifferentiation of the left and right sides of the sensitivity equations forcoefficients and vectors of lower orders. Consider the sensitivity equation(5.59). Differentiating term-wise with respect to the parameter ∆α, wehave

AααX i + 2AαX iα + AX iαα = λiααX i + 2λiαX iα + λiX iαα, (5.82)

where

Aαα =∂ 2A

∂ ∆2α, X iαα =

∂ 2X i∂ ∆2α

.

Scalar multiplication of the terms of this equation by the vector Y i yields

(AααX i, Y i) + 2(X iα, AT Y i − λiαY i)+ (X iαα, AT Y i − λiY i) = λiαα(X i, Y i),

Hence, the sensitivity coefficient of the second order is given by

λiαα = β i2 =(AααX i, Y i) + 2(X iα, AT

αY i − λiαY i)

(X i, Y i). (5.83)

For the case when the matrix A depends linearly on the variable parameter∆α, we have

Aαα = 0

and

β i2 = 2(X iα, AT

αY i − λiαY i)

(X i, Y i). (5.84)

Then, we find the scalar product of the left and right sides of the sensi-tivity equation (5.82) by the vector Y g, g = i:

(AααX i, Y g) + 2(X iα, AT αY g − λiαY g)

+ (X iαα, AT

Y g − λiY g) = λiαα(X i, Y g),

(5.85)




Take into account the fact that

AT Y g = λsY g.

Then, Equation (5.85) yields

(X iαα, Y g) =2(X iα, AT

αY g − λiαY g) + (Aαα, X i, Y g) − λiαα(X i, Y g),

λi − λg

If A(∆α) + A + ∆α B, (X i, Y g) = 0, i = g, then

(X igg , Y g) =2(X iα, AT

αY g − λiαY g)

λi − λg, i = g. (5.86)

In a similar way we can obtain formulas for sensitivity coefficients andvectors of any order.

5.3.6 Sensitivity of Trace and Determinant of Matrix

Consider sensitivity of the trace of a matrix A:

∂ tr A

∂α =

ni=1

∂aii

∂α

Or, with account for (5.56),

∂ tr A

∂α=

ni=1

∂λi

∂α

The determinant of the matrix A is given by

det A =n

i=1

λi.

The sensitivity coefficient of the determinant is given by

∂ det A

∂α=

ni=1

∂λi

∂α

nj=1

j=1

λj = det An

i=1

∂ ln A

∂α(5.87)




or

∂ Indet A

∂α=

ni=1

∂ ln λi

∂α. (5.88)

5.4 Sensitivity of Integral Quality Indices

5.4.1 Integral Estimates

For indirect evaluation of the quality of transients, various integrals fromsystem phase coordinates, their derivatives and other combinations of all

of them are widely used in automatic control theory.For a system described by the equation

ni=0

aiy(n−i)(t) =

gi=0

bix(g−i)(t), (5.89)

integral estimates that can be represented in the following generalized form

I =

∞ 0

f (y, y, . . . , y(n)

, t)dt

are commonly used.

For different types of integrands there are well-known integral estimates

I 0 =

∞ 0

y(t)dt, I 1 =

∞ 0

|y(t)|dt, I (i)2 =

∞ 0

y(i)(t)

2dt,

i = 0, . . . , n , and so on.

For a stable system described by the equation

Y = A Y,

generalized estimates of the form

I =

∞

0

Y T V Y dt,




are usually used, where V is a constant matrix.

Using the integral estimation method, the above integrals are chosen sothat they can be represented in the simplest way via system parameters.

5.4.2 Sensitivity of Integral Estimate I 0

For evaluation of sensitivity for the integral estimate I 0 we have

∂I 0∂α

=∂

∂α

∞ 0

y(t)dt =

∞ 0

u(t)dt,

where

u(t) =∂y(t)

∂α.

The Laplace transform of the sensitivity function u(t) is equal to

L[u(t)] =

∞ 0

e−stu(t)dt.

Hence,

∂I 0∂α

= L[u(t)]|s=0 .

To find the image L[u(t)], we can use the formula linking the image of the sensitivity function with transfer function of initial system and sensi-tivity model. To obtain ∂I 0/∂α we can also use the sensitivity equation.Consider, for example, a system described by the equations

ni=0

aiy(n−i) = 0, y(i)(0) = y

(i)0 , i = 0, . . . , n − 1. (5.90)

From (5.90) we obtain the sensitivity equation in the form

ni=0

aiy(n−i) = −

ni=0

∂ai

∂αy(n−i),

u(i)

(0) = 0, i = 0, . . . , n − 1.

(5.91)




Integrating both sides of Equation (5.91) from 0 to ∞, we find

n

i=0

ai

∞

0

u(n−i)dt = −n

i=0

∂ai

∂α

∞

0

y(n−i)dt;

Since the initial system is stable, we have y(i)(t) = u(i)(t) = 0 as t → ∞.Therefore,

an

∞ 0

u(t)dt =n−1i=0

∂ai

∂αy(n−i)(0) −

∂an

∂α

∞ 0

y(t)dt,

Hence,

∂I 0∂α

=1

an

n−1i=0

∂ai

∂αy(n−i)(0) −

∂ ln an

∂αI 0 (5.92)

or

∂ ln I 0∂α

=1

anI 0

n−1i=0

∂ai

∂αy(n−i)(0) −

∂ ln an

∂α. (5.93)

5.4.3 Sensitivity of Quadratic Estimates. Transformationof Differential Equations

The sensitivity coefficients of quadratic estimates

∂I (i)2

∂α= 2

∞ 0

y(i)u(i)dt

can be found in many ways. Let us illustrate one of these methods by anexample of a system described by the following second-order equation:

a0y + a1y + a2y = 0, y(0) = y0, y(0) = y0, (5.94)

The sensitivity equation has the form

a0u” + a1u + a2u = −∂a0∂α

y” −∂a1∂α

y −∂a2∂α

y,

u(0) = 0, u

(0) = 0.

(5.95)




Multiplying Equation (5.94) successively by u and u, we find

a0yu + a1yu + a2yu = 0, (5.96)

a0yu + a1yu + a2yu = 0. (5.97)

Similarly, multiply Equation (5.95) by y and y:

a0uy + a1uy + a2uy = −2

i=0

∂ai

∂αy(2−i)y, (5.98)

a0uy + a1uy + a2uy = −2

i=0∂ai

∂αy(2−i)y. (5.99)

Then, we add (5.96) to (5.98) and (5.97) to (5.99). Thus,

a0(yu + uy) + a1(yu + uy) + 2a2yu = −2

i=0

∂ai

∂αy(2−i)y,

a0(yu + uy) + 2a1yu + a2(yu + uy) = −2

i=0

∂ai

∂αy(2−i)y.

Integrating both sides from 0 to ∞, we obtain

a0

∞ o

(yu + uy) dt + a1

∞ o

(yu + uy) dt + 2a2

∞ o

yudt

= −2

i=0

∂ai

∂α

∞ o

y(2−i)y dt,

a0

∞

o

(yu + uy) dt + 2a1

∞

o

yu dt + a2

∞

o

(yu + uy) dt

= −2

i=0

∂ai

∂α

∞ o

y(2−i)y dt.

For the individual integrals we have

∞

0

(yu + uy) dt = −2

∞

0

yu dt = −∂

∂α

∞

0

y2 dt = −∂

∂αI (1)2 ,




∞ 0

(yu + uy) dt = 0,

∞ 0

yy dt = y0y0 − I (1)2 ,

∞ 0

yy dt = −1

2 y20 ,

∞ 0

(yu + yu) dt = 0,

∞ 0

(yu + uy) dt = 0,

∞ 0

yy dt = −1

2(y0)2.

As a result, we obtain the following system of linear algebraic equationswith respect to sensitivity coefficients:

−a0∂I

(1)

2∂α + a2∂I 2∂α = −

∂a0∂α

y0y0 − I

(1)2

+

1

2

∂a1∂α y20 −

∂a2∂α I 2,

a1∂I

(1)2

∂α=

1

2

∂a0∂α

(y0)2

−∂a1∂α

I (1)2 +

1

2

∂a2∂α

y20,

Hence,

∂I (1)2

∂α=

1

2a1

∂a0∂α

y20 +∂a2∂α

y20 − 2∂a1∂α

I (1)2

, (5.100)

∂I 2

∂α=

a0

2a1a2∂a0

∂αy20 +

∂a2

∂αy20 − 2

∂a1

∂αI (1)2

−1

a2

∂a0∂α

y0y0 −∂a0∂α

I (1)2 −

1

2

∂a1∂α

y20 +∂a2∂α

I (1)2

.

(5.101)

Let a0 = T 2, a1 = 2ξT , and a2 = 1, i.e., we consider an oscillatory unit.Then,

∂I (1)2

∂ξ = −

1

ξ I (1)2 or

∂I (1)2

∂ ln ξ = −I

(1)2 ,

∂I 2

∂ξ = −T 2

I (1)2

ξ + T y20 = T 2

∂I (1)2

∂ξ + T y20 ,

∂I (1)2

∂T =

1

2ξ y20 −

1

T I (1)2 ,

∂I 2∂T

= T

T

y202ξ

+ I (1)2 − 2y0y0

+ y20ξ.

Obviously, if the dependence of the cost function I (i)2 on the parameter

α is given in the form of a composite function

I (1)

2 = f [aj(α), bj(α)],




the coefficient ∂I (i)2 /∂α can be found by direct differentiation of the function

f :

∂I (1)2

∂α=

n

k=0

∂f

∂ak

∂ak

∂α+

g

k=0

∂f

∂bk

∂bk∂α

.

5.4.4 Sensitivity of Quadratic Estimates. LaplaceTransform Method

The sensitivity coefficient of the integral estimate

I (1)2 =

∞ 0

y(i)

2dt

is proportional to the integral

∞ 0

y(i)u(i) dt

The Laplace image of the product y(i)(t)u(i)(t) has the form

L[y(i)(t) u(i)(t)] =

∞ 0

y(i)(t) u(i)(t)e−st dt

Hence,∞ 0

y(i)(t) u(i)(t) dt = L[y(i)(t) u(i)(t)]p=0 (5.102)

In Laplace transformation theory the following theorems has been proved[29].

THEOREM 5.2

If functions f 1(t) and f 2(t) have Laplace images F 1(s) and F 2(s), respec-tively, and F 1(s) = a1(s)/b1(s) is a rational algebraic function having only q simple poles, the following equality holds:

L[f 1(t) f 2(t)] =

q

k=1

a1( pk)

b

1( pk)

F 2(s − pk), (5.103)




where

b1( pk) =db1(s)

ds

s=pk

.

THEOREM 5.3

Let functions f 1(t) and f 2(t) have Laplace images F 1(s) and F 2(s), respec-tively, and F 1(s) be a rational algebraic function having n different poles p1, . . . , pn with multiplicities m1, . . . , mn, respectively, so that

ni=1 mi = q .

Then,

L[f 1(t) f 2(t)] =n

k=1

mkj=1

(−1)mk−j

(mk − j)!Rkj

dmk−jF 2(s)

dsmk−j

s=s−pk

(5.104)

where

Rkj =1

( j − 1)!

dj−1

dsj−1(s − pk)mkF 1(s)

s=pk

.

Let us use the results of the above theorems for calculating sensitivitycoefficients of quadratic integral estimates. Denote the image of a functiony(i)(t) by Y i(s) = ai(s)/bi(s), and that of the function u(i)(t) by U i(s). Letthe rational algebraic function ai(s)/bi(s) have only n simple poles. Then,

∞0

y(i)(t) u(i) dt =

qj=1

ai( pj)

bi( pj)U i(s − pk)

s=0

. (5.105)

In the general case, when Y i(s) may have multiple poles, according to(5.104) we have

∞0

y(i)(t) u(i) dt

=

nk=1

mk

j=1

(−1)mk−j

(mk − j)!Rkj

dmk−j

dsmk−jU i(s)

s=s−pk

.

s=0

, (5.106)

where

Rkj =1

( j − 1)!

dj−1

dsj−1(s − pk)mkY i(s)

s=pk

.

As an example, we consider the system

T 2

+2ξT y + y = 0, y(0) = y0, y(0) = y0.




For simplicity, let T = 1, y0 = 0, and y0 = 1. Then,

Y (s) =1

s2 + 2ξs + 1, b(s) = 2(s + ξ ),

U (s) = −2s

(s2 + 2ξs + 1)2,

∂I 2∂ξ

= 2

p1

( p21 + 2ξp1 + 1)2+

p2( p22 + 2ξp2 + 1)2

,

where

p1,2 = −ξ ± ξ 2 − 1.

Substituting the values p1 and p2, we obtain

∂I 2∂ξ

= −1

4ξ 2.

The integral estimate I V can be written in the following scalar form

I V

=n

i=1

n

j=1

∞

0

aij

yiyj

dt.

The corresponding sensitivity coefficient is defined by

∂I V ∂α

=n

i=1

nj=1

∞ 0

∂aij

∂αyiyj + aij(uiyj + yiuj)

dt.

If the coefficients aij are independent of α, we have

∂I V ∂α

=n

i=1

nj=1

aij

∞ 0

(uiyj + yiuj) dt.

As before, determination of the sensitivity coefficient reduces to calculationof the integrals

∞

0

uiyj dt = L[ui(t) yj(t)]|s=0 .




5.4.5 Sensitivity Coefficients of Integral Estimates forDiscontinuous Control Systems

Consider the generalized integral estimate of the form

I =

∞ t0(α)

Q(y, α) dt

where Q(Y, α) is a function of the variables Y and α having continuousderivatives ∂Q/∂Y and ∂Q/∂α on the intervals (ti−1, ti), (tg, ∞), (i =1, . . . , g).

Let Y (t) be a solution of a discontinuous system. Then, we may write

I =g

i=1

ti(α) ti−1(α)

Q(y, α) dt +∞

tg(α)

Q(y, α) dt,

where ti(α) are the switching moments.

Then, for sensitivity coefficient we have

∂I

∂α

=

g

i=1

ti(α)

ti−1(α)

∂Q

∂X

U +∂Q

∂α dt +

∞

tg(α)

∂Q

∂X

U +∂U

∂α dt

+

gi=1

Q(ti, Y −i )

dtidα

− Q(ti−1, Y +i−1)dti−1

dα

− Q(X +g , tg)

dtgdα

,

or,

∂I

∂α=

gi=1

ti ti−1

∂Q

∂Y U +

∂Q

∂α

dt +

∞ tg

∂Q

∂Y U +

∂Q

∂α

dt

+

gi=1

Q(ti, Y −i ) − Q(ti−1, Y +i )

dtidα

− Q(t0, X +0 ) dt0dα

.

(5.107)

If the solution of initial system is continuous, i.e., Y −i = Y +i , we have

∂I

∂α=

gi=1

ti ti−1

∂Q

∂Y U +

∂Q

∂α

dt +

∞ tg

∂Q

∂Y U +

∂Q

∂α

dt

− Q(t0, X 0)

dt0

dα .

(5.108)




5.5 Indirect Characteristics of Sensitivity Functions

5.5.1 PreliminariesFor solution of a number of analysis and design problems for control sys-

tems with account for low sensitivity it is necessary to compare varioussensitivity functions. In the general case, such a comparison appears tobe cumbersome. In this connection, it is required to introduce some indi-rect indices reflecting various properties of sensitivity functions and makingit possible to perform convenient comparative analysis of these functions.This can be done in analogy with indices used for evaluating quality of transients. A large number of quality indices used in automatic controltheory can be divided into the following four groups: indices of precision,stability gain, speed of acting, and composite indices. Many of them, es-pecially precision indices, can be used for estimation of the properties of sensitivity functions.

5.5.2 Precision Indices

For evaluating precision of control systems the error values in varioustypical operating conditions are used. In general, error with respect tosteady-state (forced) process in an asymptotically stable system is defined

by [121]z∞(t) − x(t) − y∞(t), (5.109)

where x(t) is an input action and y∞(t) is the steady-state process, so that

y∞(t) =

t −∞

h(t − τ )x(τ ) dτ =

t −∞

h(τ )x(t − τ ) dτ.

It is assumed that the system is to reproduce a signal x(t) that is inde-pendent of α. With due acount of variations ∆α of the parameter α, forthe steady-state process y∞(t) we have

y∞(t, α0) =

∞ 0

[h(τ, α0) + ∆h(τ )]x(t − τ ) dτ − ∆y∞(t).

For small ∆α,

∆y∞

(t) = u∞

(t)∆α,




∆y∞(t) =

∞ 0

µ(τ )x(t − τ ) dτ ∆α,

where µ(t) = ∂h(t)/∂α is the sensitivity function of the weight function.

As a result, for sensitivity function of the steady-state process we obtain

u∞(t) =

∞ 0

µ(τ )x(t − τ ) dτ. (5.110)

Accordingly, for the sensitivity function of the error with respect to thesteady-state process we have

∂z∞(t)

∂α = −u∞

(t).

Let there exist an expansion of the function x(t − τ ) into Taylor’s seriesin a locality of the point t, i.e.,

x(t − τ ) =∞i=0

(−1)iτ i

i!x(i)(t). (5.111)

Substituting (5.111) into (5.110), we find

u∞(t) =∞i=0

dix(i)(t)

1

i!,

where

di = (−1)i∞ 0

µ(τ )τ i dτ

is the i-th order moment of the sensitivity function µ(t).Obviously, the sensitivity function ∂z∞(t)/∂α is described by theexpression

∂z∞(t)

∂α= −

∞i=0

dix(i)(t)

i!

As is known [8, 121], the steady-state error z∞(t) is given by

z∞(t) =∞

i=0

cix(i)(t)

i!

,




where ci, i = 0, 1, . . . are the error coefficients. In this case we findthat −di = ∂ci/∂α are the sensitivity functions (coefficients) of the errorcoefficients.

As a result,

∂ci∂α

= (−1)i+1∞ 0

µ(τ )τ i dτ.

The sensitivity coefficients ∂ci/∂α can also be calculated by the sensitiv-ity function u(s) of the transfer function w(s):

u(s) =∂w(s)

∂α=

∞

0

µ(τ )e−sτ dτ.

It can be easily shown that

∂ci∂α

= −diu(s)

dsi

s=0

, i = 0, 1, . . .

Consider some examples. Let transfer function of a system have the form

w(s) =b(s)

a(s)=

mi=0

bi pi

ni=0

ai pi

.

Then, for a constant input signal x(t) = x0, that is independent of α, wehave

y∞ =bman

x0

and

u∞(t) =∂

∂α

bman

x0

=

bmx0an

∂ ln bm

∂α−

∂ ln an

∂α

.

Obviously, if the coefficients bs and an are independent of the parameterα, we have u∞(t) = 0. Similar results can be obtained if we consider theimage of the sensitivity function

u(s) =

∂w(s)

∂α x(s).




If transfer function of the system as the form

w(s) =w0(s)

1 + w1(s),

where w1(s) if the transfer function of the open-loop system, then

y∞(t) =w0(0)

1 + kx0,

where k is the total gain of the open-loop system. Then,

u∞ =1

1 + k ∂w0(0)

∂α−

∂ ln(1 + k)

∂α x0.

If the gain k is independent of the parameter α, we have

u∞(t) =1

1 + k

∂w0(0)

∂αx0.

From the last equations it follows that sensitivity of the steady-statevalue of the variable y(t) decreases as the open-loop system gain increases.

5.5.3 Integral Estimates of Sensitivity Functions

These estimates belong to composite indices. In analogy with qual-ity indices for transient, we can introduce integral estimates of sensitivityfunctions of the form

J =

∞ 0

φ(u, u , . . . , u(n), t) dt.

Consider determination of some of these integral estimates in detail.Since

∂I 0∂α

=∂

∂α

∞ 0

y(t) dt =

∞ 0

u(t) dt

we have

J 0 =

∞

0

u(t) dt =∂I 0∂α

.




Methods of determination of the coefficient ∂I 0/∂α were presented inSection 5.4. Below we demonstrate a technique of determinating theestimates

J

(i)

2 =

∞

0

u

(i)

(t)2

dt

by an example of a second-order system

a0y + a1y + a2y = 0, y(i)(0) = y(i)0 (i = 0, 1),

a0u + a1u + a2u = −s

i=0

∂ai

∂αy(2−i),

(5.112)

u(0) = u(0) = 0. (5.113)

Multiplying Equation (5.113) successively by u, u, u and integrating theseexpressions term-wise from 0 to ∞, we obtain

a0

∞ 0

uu dt + a1

∞ 0

uu dt + a2

∞ 0

u2 dt = −2

i=0

∂ai

∂α

∞ 0

y(2−i)udt,

a0

∞

0

uu dt + a1

∞

0

u2 dt + a2

∞

0

uu dt = −2

i=0∂ai

∂α

∞

0

y(2−i)udt,

a0

∞ 0

u2 dt + a1

∞ 0

uu dt + a2

∞ 0

uudt = −2

i=0

∂ai

∂α

∞ 0

y(2−i)udt.

For individual integrals we find

∞ 0

uu dt = −J (1)2 ,

∞ 0

uu dt = 0,

∞ 0

uu dt = 0.

With due account for the above relations, we obtain the following alge-braic system

−a0J (1)2 + a2J 2 = −

2i=0

∂ai

∂α

∞ 0

y(2−i)udt. (5.114)

a1J (1)2 = −

2

i=0

∂ai

∂α

∞

0

y(2−i)udt. (5.115)




a0J (1)2 − a2J

(1)2 = −

2i=0

∂ai

∂α

∞ 0

y(2−i)udt. (5.116)

which yields the desired estimates J 2

, J (1)

2, and J

(2)

2. The terms in the

right sides, including the sensitivity coefficient ∂J (i)2 /∂α, are obtained by

(5.105) and (5.106). The determinant of the system (5.114)–(5.116)

∆ = a0a1a2

is not zero.

Consider a system described by the following two first-order equations:

y1 = a11y1 + a12y2,

y2 = a21y1 + a22y2.

They correspond to the sensitivity equations

u1 = a11u1 + a12u2 + f 1,u2 = a21u1 + a22u2 + f 2,

where

f i =∂ai1

∂α

y1 +∂ai2

∂α

y2, i = 1, 2.

Multiplying both of them by uj( j = 1, 2) and integrating the result from0 to ∞, we obtain

∞ 0

u1u1 dt = a11

∞ 0

u21 dt + a12

∞ 0

u1u2, dt +

∞ 0

f 1u1,dt, (5.117)

∞ 0

u2u1 dt = a21

∞ 0

u21, dt + a22

∞ 0

u1u2, dt +

∞ 0

f 2u1,dt, (5.118)

∞ 0

u1u2 dt = a11

∞ 0

u1u2 dt + a12

∞ 0

u22, dt +

∞ 0

f 1u2,dt, (5.119)

∞

0

u2u2 dt = a21

∞

0

u1u2, dt + a22

∞

0

u22, dt +

∞

0

f 2u2,dt, (5.120)




In this system,

∞ 0

uiui dt = 0, i = 1, 2.

Moreover,

∞ 0

u1u2 dt = u1u2

∞0

−

∞ 0

u1u2 dt.

Hence,

∞

0

u1u2 dt +

∞

0

u1u2 dt = 0. (5.121)

The algebraic equations (5.117)–(5.121) contain five unknowns:

∞ 0

u21 dt,

∞ 0

u22 dt,

∞ 0

u1u2 dt,

∞ 0

u1u2 dt,

∞ 0

u1u2 dt.

The determinant of the system is

∆ = 2(a12a21 − a11a22).

Since the system is stable, the determinant is not zero. It can be easilyshown that the techniques of determining integral estimates of sensitivityfunctions presented above can be generalized onto linear systems of anyorder. To find the estimate J 2, we can also employ the Reilly formula [100]:

∞ 0

[y(t)]2 dt =1

2π

∞ −∞

|ψy( jω)|2 dω,

where ψ( jω) is the frequency spectrum for y(t).

As regards the sensitivity function u(t), we have

∞

0

u2(t) dt =1

2π

∞

−∞

|ψu( jω)|2

dω, (5.122)




5.5.4 Envelope of Sensitivity Function

For many systems the sensitivity functions are oscillatory processes. Thisfact makes comparison of various sensitivity functions much more difficult.Envelopes of sensitivity functions have some advantages in this respect[107]. Many processes in real systems are described by the following equa-tion of oscillatory unit:

T 2y + 2T ξ y + y = kx.

The weight function of such a system is given by

h(t) =k

T 1 − ξ 2e− ξt

T sint

T

1 − ξ 2.

The sensitivity functions of the weight function with respect to parame-ters T and ξ have the form

uT (t) =∂h(t)

∂ ln T =

k

T 2

1 − ξ 2e− ξt

T

(ξ − T )sin φ − t

1 − ξ 2 cos φ

uξ(t) =∂h(t)

∂ ln ξ = kψe− ξt

T

ξ

1 − ξ 2−

t

T

sin φ − tψ cos φ

,

where

φ = tT

1 − ξ 2 , ψ = ξ

T

1 − ξ 2.

The envelope of a sensitivity function is its absolute value. Denote theenvelopes of the sensitivity functions uT (t) and uξ(t) by AT (t) and Aξ(t),respectively. Then:

AT (t) =k

T 2

1 − ξ 2

e− ξtT

(ξt − T )2 + t2(1 − ξ 2), (5.123)

Aξ(t) =kξ

T

1 − ξ 2e− ξt

T

ξ

1 − ξ 2−

t

T

2

+t2ξ 2

T 2(1 − ξ 2). (5.124)

Let us obtain extremal points of the envelopes AT (t) and Aξ(t). Withthis aim in view, we differentiate these functions with respect to the variablet, and equate the derivatives to zero. For small values of T 2 and ξ 2, we findthat the moments of both the envelopes almost coincide with one anotherand are equal to

tmax ≈T

ξ .




In this case, the maximal values of the envelopes are given by

AT (tmax) =k

T ξ e−1, Aξ(tmax) =

k

T e−1,

i.e.,AT (tmax)

Aξ(tmax)=

1

ξ > 1.




Chapter 6

Sensitivity Invariants of Control Systems

6.1 Sensitivity Invariants of Time Characteristics

6.1.1 Sensitivity Invariants

The form of mathematical model of the system under consideration de-termines restrictions and interconnections between sensitivity functions.These restrictions and connections may have the form of equalities of in-equalities and be holonomic and non-holonomic .

DEFINITION 6.1 Connections given by equations that do not in-clude time derivatives of sensitivity functions, i.e., have the form of al-gebraic relations, are called holonomic. Otherwise, connections are called non-holonomic.

Differential sensitivity equations are natural non-holonomic restrictionsimposed on sensitivity functions. Moreover, there are holonomic restric-tions imposed on sensitivity functions of some types. First, such connec-tions were found by C. Belove [7] for electronic RLC -networks. It was

found that some sums of sensitivity functions of networks system functions(transfer functions, conductivity) are constant values and do not change if the networks are equivalently transformed. Such sums are called sensitivity invariants . In [7], properties of homogeneous functions were used to obtainsensitivity invariants.

DEFINITION 6.2 A function f (x1, . . . , xg) is called homogeneous of the v-th order if for an arbitrary µ the following condition holds:

f (µx1, . . . , µ xg) = µv

f (x1, . . . , xg). (6.1)




Let us differentiate (6.1) with respect to µ:

∂f

∂µx1

∂µx1∂µ

+∂f

∂µx2

∂µx2

∂µ+ . . . +

∂f

∂µxg

∂µxg

∂µ= vµv−1f (x1, . . . , xg).

Dividing the last expressions by f and letting µ = 1, we obtain

gi=1

∂ ln f

∂ ln xi

= v = const. (6.2)

Many characteristics of electronic networks are described with respectto parameters of homogeneous functions. As a result, if f is a networkcharacteristic (transfer function, conductivity, and so on) and x1, . . . , xg areparameters, from (6.2) it follows that that the sum of logarithmic sensitivity

functions over all network parameters is a constant value, i.e., an invariant.

DEFINITION 6.3 In the general case, by sensitivity invariant we mean a functional (algebraic) dependence between sensitivity functions that does not contain state variables and parameters of the initial system.

If the system under consideration is characterized by sensitivity functionsu1(q ), u2(q ), . . . , um(q ), where q is an independent variable (time, frequency,Laplace transform variable, and so on), we denote sensitivity invariant in

the formΩ(u1, u2, . . . , um, q ) = 0.

Hereinafter we will distinguish linear and nonlinear invariants.

DEFINITION 6.4 An invariant described by the relation

m

i=1γ iui(q ) = const

or m

i=1

γ iui(q ) = φ(q ),

where φ(q ) is a function depending only on q , and γ i are weight coefficients,is called linear.

DEFINITION 6.5 Invariants described by nonlinear functions Ω are

called nonlinear.




In this book we consider systems described by ordinary differential equa-tions or finite algebraic relations. For them the following three types of invariants can exist:

Ω1m

∂y∂α1, ∂y∂α2

, . . . , ∂y∂αm, q

= 0, (6.3)

Ωn1

∂y1∂α

,∂y2∂α

, . . . ,∂yn

∂α, q

= 0, (6.4)

Ωnm

∂y1∂α1

,∂y2∂α2

, . . . ,∂yn

∂αm

, q

= 0. (6.5)

The first dependence contains the sensitivity function of a single systemvariable with respect to m parameters, the second one includes the sensi-tivity function of n variables with respect to a single parameters, while thethird one describes the sensitivity function of n variables with respect tom parameters. It is noteworthy that, depending on the initial problem andthe form of argument in (6.3)–(6.5), ordinary as well as semi-logarithmicand logarithmic sensitivity functions can be arguments.

Below in this chapter we consider mainly invariants with respect to semi-logarithmic sensitivity functions. The presence of sensitivity invariants addssome specifics to many investigation problems for dynamic systems, in-

cluding automatic control systems. Thus, sensitivity invariants can restrictindependent variation of sensitivity functions in the design of optimal in-sensitive systems. Simultaneously, in many cases the use of invariants cansimplify sensitivity analysis of control systems.

6.1.2 Existence of Sensitivity Invariants

First, we consider the dynamic system of the general form

yi = f i(y1, . . . , yn, α1, . . . , αm, t) i = 1, . . . , n ,

associated with sensitivity models

uij =n

k=1

∂f i∂yk

ukj +∂f i

∂ ln αj

, i = 1, . . . , n, j = 1, . . . , m . (6.6)

Let there be an invariant:

Ωnm(t, u11, . . . , unm) = 0. (6.7)




Then, the following identity holds:

dΩnm

dt=

i,j

∂ Ωnm

∂uij

duij

dt+

∂ Ωnm

∂t= 0.

Substitute the expression for uij from (6.6) into the last equation:

i,j

∂ Ωnm

∂uij

n

k=1

∂f i∂yk

ukj +∂f i

∂ ln αj

+

∂ Ωnm

∂t= 0. (6.8)

The obtained relation is a necessary condition for existence of the invari-ant (6.7). From (6.8) we can obtain existence conditions for the invariants

(6.3) for n = 1 and m > 1:

mj=1

∂ Ω1m

∂uj

∂f

∂yuj +

∂f

∂ ln αj

+

∂ Ω1m

∂t= 0 (6.9)

and existence conditions for the invariants (6.4) for n > 1 and m = 1:

n

i=1

∂ Ωn1

∂ui

n

j=1

∂f i

∂yj

uj +∂f i

∂ ln α+

∂ Ωn1

∂ ln t

= 0. (6.10)

Next, let a multivariable and multiparameter system be described byalgebraic relations

f i(y1, . . . , yn, α1, . . . , αm) = 0, i = 1, . . . , n ,

associated with sensitivity equations of the form

nk=1

∂f i∂yk

ukj +∂f i

∂ ln αj

= 0, j = 1, . . . , m, i = 1, . . . , n .

Consider the invariant

Ωnm(U 11, U 12, . . . , U nm) = 0

or

Ωnm(U 1, U 2, . . . , U m) = 0 (6.11)




where

U T j = [u1j , u2j , . . . , unj ], j = 1, . . . , m ,

are sensitivity vectors.

The vector sensitivity function U j is determined from the system

A U j = Bj , j = 1 . . . , m ,

where A = ∂f i/∂y is the Jacoby matrix, and

BT j =

− ∂f 1

∂ ln αj

, . . . ,∂f n

∂ ln αj

.

Then, if the matrix A is nonsingular, i.e., the Jacobian detA is not zero,the sensitivity vector is equal to

U T j = A−1Bj . (6.12)

Substituting (6.12) into (6.11), we obtain the following existence condi-tions for the invariant (6.11):

Ωnm A−1B1, A−1B2, . . . , A−1Bm = 0.

For a single variable and m parameters this condition assumes the form

Ω1m(z1, z2, . . . , zm) = 0,

where

zj = − ∂f

∂ ln αj

∂f

∂y

For n output variables and a single parameter we have

Ωn1(β 1, β 2, . . . , β m) = 0,

where

β i =D(f 1, . . . , f i−1, f i, . . . , f n)

D(y1, . . . , yi−1, α , yi+1, . . . , yn):

D(f 1, f 2, . . . , f n)

D(y1, y2, . . . , yn).

The above necessary conditions do not provide for universal technique

of construction of sensitivity invariants. It is quite possible that, similarly




to construction of Lyapunov functions for nonlinear systems, constructionof sensitivity invariants calls for special investigation in each specific case.As was shown by investigations, the most general existence condition forsensitivity invariants is parametric redundancy in the model of the initial

system. As was noted in Section 1.2, in some cases we can transform acomplete set of parameters to another one using a functional transforma-tion.

DEFINITION 6.6 If the number of parameters q of the new set is fewer than the number of parameters m for the first one, we have parametric redundancy.

DEFINITION 6.7 The value p = m−

q is called the order of parametric redundancy.

It is believed that the number of independent sensitivity invariants isequal to p.

Currently, using mostly sufficient conditions, linear invariants of linearsystems have been obtained, and sufficient conditions for existence of linearinvariants of nonlinear systems have been formulated. Some of these resultsare presented below. Since later we deal only with linear invariants, theword “linear” will be omitted for brevity.

6.1.3 Sensitivity Invariants of Single-Input–Single-OutputSystems

Consider an automatic single-input–single-output (SISO) system controlsystem described by the equation

n

i=0aiy(n−i) =

g

i=0biy(g−i),

y(i)(0) = y(i)0 , i = 0, . . . , n− 1, n ≥ g.

(6.13)

We assume that the coefficients of this equation depend on parametersα1, . . . , αm and are differentiable with respect to them. Then, the followingsensitivity equations hold:

ni=0

aiu(n−i)j =

gi=0

∂bi

∂ ln αj

x(g−i) −n

i=0

∂ai

∂ ln αj

y(n−i),

u(k)

j (0) = 0, k = 0, . . . , n− 1, j = 1, . . . , m ,

(6.14)




where

uj (t) =∂y(t)

∂ ln αj

Let us take the sum of the sensitivity equations (6.14) over all j:

ni=0

ai

mj=1

u(n−i)j =

gi=0

x(g−i)m

j=1

∂bi

∂ ln αj

−n

i=0

y(n−i)m

j=1

∂ai

∂ ln αj

. (6.15)

Introduce the notation

ψ(t) =m

j=1

uj (t).

Assume that the coefficients ai and bj are considered as parameters anddenoted by α1, . . . , αm. Then, Equation (6.15) transforms to the form

ni=0

aiψ(n−i) =

gi=0

bix(g−i) −n

i=0

aiy(n−i)

ψ(j)(0) = 0, j = 0, . . . , n

−1,

and, due to (6.13),

ni=0

aiψ(n−i) = 0, ψ(j)(0) = 0, j = 0, . . . , n− 1. (6.16)

The last equation is a homogeneous linear differential equation with zeroinitial conditions with respect to ψ(t). It is known that the solution of suchan equation is equal to zero, i.e.,

ψ(t) =n

i=0

∂y

∂ ln αi

+

gi=0

∂y

∂ ln bi

= 0. (6.17)

The above sum is a sensitivity invariant of the transient y(t) with respectto coefficients of the differential equation of the system. It is noteworthythat the model (6.13) is parametrically redundant, and the order of re-dundancy is 1. Indeed, a complete set of parameters for Equation (6.13)for fixed initial conditions incorporates the coefficients ai, (i = 0, . . . , n),

bi, ( j = 0, . . . , g), i.e., n + g + 2 parameters all together. Using a simple




transformation (after division of both sides of the equation by any nonzerocoefficient, for example, a0), we can obtain a model with a complete definingset of n + m + 1 parameters.

In many systems, for instance, in RLC -networks, the coefficients ai and

bj are polylinear functions of primary parameters α1, . . . , αm. Moreover,the following relations hold:

mj=1

αj∂ai

∂αj

= qai, i = 0, . . . , n ,

mj=1

αj∂bi

∂αj

= qbi, i = 0, . . . , g ,

(6.18)

where q is the number of factors in the terms appearing in the equationsfor ai and bj .

It can be easily shown that the invariant (6.17) can be, with due accountfor (6.18), transformed to the form

ni=0

mj=1

∂y

∂ai

∂ai

∂αj

αj +

gi=0

mj=1

∂y

∂bi

∂bi

∂αj

αj = 0,

Hence, the sensitivity invariant with respect to the parameters α1, . . . , αm

is given bym

j=1

∂y(t)

∂ ln αj

= 0. (6.19)

Example 6.1

For a system with the transfer function w(s) = b0/(a0s + a1) the weightfunction is equal to

h(t) =b0

a0

e−a1

a0t

. (6.20)

For this function, the invariant (6.17) takes the form

∂h(t)

∂ ln a0+

∂h(t)

∂ ln a1+

∂h(t)

∂ ln b1= 0,

because, due to (6.20),

∂h(t)

∂ ln a0= h(t)a1t

a0 −1 ,

∂h(t)

∂ ln b0= h(t),

∂h(t)

∂ ln a1=−

h(t)a1t

a0.




The transient of this system is given by

k(t) =b0a1

1 − e

−a1a0

t

.

The sensitivity invariant of the transient is

∂k(t)

∂ ln a0+

∂k(t)

∂ ln a1+

∂k(t)

∂ ln b1= 0,

where

∂k(t)

∂ ln a0= −b0t

a0e−a1

a0t

,∂k(t)

∂ ln b0= k(t),

∂k(t)

∂ ln a1= − b0

a0

1 − e

−a1a0

t

+b0t

a0e−a1

a0t

,

Example 6.2

Consider a system described by the equation

a0y + a − 1y = b0x, x(t) = eλt,

y(0) = 0.

The transient in this system has the form

y(t) =b0

a1 + λa0

eλt − e

−a1a0

t

.

The sensitivity functions are given by

∂y

∂ ln a0= − λa0

a1 + λa0y − a1b0t

a0(a1 + λa0)e−a1

a0t

,

∂y

∂ ln a1= − a1

a1 + λa0y +

a1b0t

a0(a1 + λa0)e−a1

a0t

,

∂y

∂ ln b0= y.




It can be easily seen that

∂y

∂ ln a0+

∂y

∂ ln a1+

∂y

∂ ln b0= 0.

6.1.4 Sensitivity Invariants of SISO Nonlinear Systems

Let the system under consideration be described by the differentialequation

y = f (y, t, α1, . . . , αm), y(0) = y0,

which gives the following sensitivity equations:

ui − ∂f

∂yui =

∂f

∂ ln αi

, ui(0) = 0, i = 1, . . . , m .

Let us add the sensitivity equations with corresponding weight coeffi-cients. Using the notation

ψ(t) =m

i=1

γ iui(t),

we obtain

ψ − ∂f

∂yψ =

mi=1

γ i∂f

∂ ln αi

, ψ(0) = 0. (6.21)

Relation (6.21) is a non-homogeneous ordinary differential equation withzero initial conditions that is linear with respect to the function ψ(t). As isknown from the theory of differential equations, so that the solution ψ(t)is equal to zero it is sufficient that

mi=1

γ i∂f

∂ ln αi

= 0. (6.22)

Therefore, we have the invariant

Ω1m = ψ(t) =m

i=1

γ i∂y

∂ ln αi

= 0. (6.23)




Let us show that the invariant (6.23) satisfies the condition (6.9). Indeed,substituting the left side of (6.23) into (6.9), we obtain

m

j=1

γ j ∂f

∂yuj +

∂f

∂ ln αj =

∂f

∂y

m

j=1

γ j uj +m

j=1

γ j∂f

∂ ln αj

,

Then, with due account for (6.22) and (6.23), we find

mj=1

γ j

∂f

∂yuj +

∂f

∂ ln αj

= 0.

6.1.5 Sensitivity Invariants of Multivariable Systems

Consider the vector equation

Y = F (Y, α1, . . . , αm, t), Y (0) = Y 0, (6.24)

which leads to the following sensitivity equations:

U i =∂F

∂Y U i +

∂F

∂ ln αi

, U i(0) = 0, i = 1, . . . , m , (6.25)

whereU i =

∂Y

∂ ln αi

,∂F

∂Y =

∂f i∂yj

.

Summation of Equations (6.25) yield

mi=1

γ iU i =∂F

∂Y

mi=1

γ iU i +

mi=1

γ i∂F

∂ ln αi

,

mi=1

γ iU i(0) = 0, (6.26)

or

Ψ − ∂F

∂Y Ψ =

mi=1

γ i∂F

∂ ln αi

, Ψ(0) = 0, (6.27)

where

Ψ =

mi=1

γ iU i =

ψ1

. . .ψn

,

ψk =

mi=1

γ iuki, k = 1, . . . , n .

Introduce the notions of vector and scalar invariants for the system (6.24).




DEFINITION 6.8 An invariant of the form

m

i=1

γ iU i = 0, (6.28)

will be called vector invariant.

DEFINITION 6.9 The invariant for a component of the vector Ψ is called scalar and has the form

m

i=1

γ iuki = 0.

As follows from (6.26), a sufficient condition of existence of the vectorinvariant reduces to the fact that the right side of the differential equationwith respect to Ψ must be identically zero:

mi=1

γ i∂F

∂ ln αi

= 0. (6.29)

The necessary existence condition (6.8) for the sensitivity variant (6.28)can be written in the following vector form:

mi=1

∂ Ωnm

∂U i

∂F

∂Y U i +

∂F

∂αi

= 0.

Then, it can be easily shown that the linear invariant (6.28) satisfies this

condition.

Represent Equation (6.26) in the form

Ψ − A(t) Ψ = B(t), Ψ(0) = 0,

where

A(t) =∂F

∂Y

, B(t) =m

i=1

γ i∂F

∂ ln αi

.




Thus, we obtained a system of differential equations with variable coef-ficients. A solution of this equation is defined by the formula

Ψ(t) =

t 0

Φ(t)Φ−1(τ ) B(τ ) dτ, (6.30)

where Φ(t) is the transition matrix of the system.

From the solution (6.30) it follows that the equality

G(t) =

t

0

Φ(t)Φ−1(τ ) B(τ ) dτ = 0, t > 0. (6.31)

is a necessary existence condition for the vector invariant. As a specificcase, Equation (6.29) follows from this result.

For existence of scalar invariant ψk(t) it is necessary that the correspond-ing component of the vector G(t) be identically zero:

Gk(t) = 0. (6.32)

An expanded expression for the left side of the condition (6.31) can befound either directly from (6.31) or with the help of special methods of invariance investigation for systems with variable parameters [57], for in-stance, using the method of equating operators.

Let the initial system (6.24) be linear with constant coefficients, so that

Y = A Y + R(t), Y (0) = Y 0, (6.33)

where A = aij. Then, we have

Ψ − A Ψ =m

i=1

γ i∂A

∂ ln αi

Y

or, in the operator form,

(sE

−A) Ψ =

m

i=1

γ i∂A

∂ ln αi

Y.




Then, the existence condition for the invariant

ψj =m

i=1γ iuji = 0

reduces to formal equality of the following determinant to zero:

∆j =

s − a11 −a12 . . . −a1,j−1 b1 −a1,j+1 . . . −a1n

. . . . . . . . . . . . . . . . . . . . . . . .−an1 −an2 . . . −an,j−1 bn −an,j+1 . . . s − ann

(6.34)

where

b1 =m

i=1

n

k=1

γ i∂agk

∂ ln αi

yk, g = 1, 2, . . . , n .

Let parameters α1, . . . , αm be the elements of the matrix A and γ i =1 (i = 1, . . . , m), where n2 = m. Then, it is easy to verify that

mi=1

γ i∂A

∂ ln αi

= A

andm

i=1

nk=1

γ i ∂aqk∂ ln αi

yk =

nk=1

aqk yk.

Therefore, the condition (6.34) takes the form

∆j =

s − a11 −a12 . . . −a1,j−1 d1 −a1,j+1 . . . −a1n

. . . . . . . . . . . . . . . . . . . . . . . .−an1 −an2 . . . −an,j−1 dn −an,j+1 . . . s − ann

(6.35)

where

dg =n

k=1

agkyk, g = 1, 2, . . . , n .

It is noteworthy that the causality conditions are often very strict for theequalities (6.34) and (6.35). Consider the following linear system withconstant coefficients:

Y = A Y, Y (0) = 0,

where A is a matrix with simple structure. Then, there exists a decomposed

system yi = λiyi, (i = 1, . . . , n), which is associated with the sensitivity




equation

uik = λiuik +∂λi

∂ ln αk

yi, i = 1, . . . , n, k = 1, . . . , m

where

uik(0) = 0.

After summation over all k’s, we obtain

ψ = λiψi + yi

mk=1

∂λi

∂ ln αk

, ψ(0) = 0,

i = 1, . . . , n ,

(6.36)

where

ψi =

mk=1

uik =

mk=1

∂yi

∂ ln αk

.

Equation (6.36) has a specific feature in that it combines two invariants,namely, invariant of root sensitivity and that of sensitivity of time charac-teristics. From this equation it follows that if

mk=1

∂λi

∂ ln αk

= 0,

i.e., if there is an invariant of root sensitivity, we have

mk=1

∂yi

∂ ln αk

= 0,

i.e., there is a sensitivity invariant for transients.

6.1.6 Sensitivity Invariants of Weight Function

Then, consider sensitivity of the weight function for the system describedby Equation (6.13). Assume that its transfer function is free of multiplepoles. Then, the weight function is given by [112]

h(t) =n

i=1

ciepit, (6.37)




where

ci = k

gj=1

( pi − q ) j)

j=1

j=i

( pi − pj ),

k =b0

a0

and pi and q i denote a pole and a zero of the transfer function, respectively.

Differentiating Equation (6.37) with respect to the parameter αk, weobtain

∂h(t)

∂ ln αk

=n

i=1

ciepit∂ ln ci

∂ ln αk

+ t∂pi

∂ ln αk , k = 1, . . . , m .

With due account of the expression for ci, after summation over all k’swe obtain

mk=1

∂h(t)

∂ ln αk

=n

i=1

ciepit

t

mk=1

∂pi

∂ ln αk

+m

k=1

∂ ln k

∂ ln αk

+

+

g

j=11

pi − q j

m

k=1∂pi

∂ ln αk

−m

k=1∂q j

∂ ln αk

−j=1

j=i

1

pi − q j

m

k=1

∂pi

∂ ln αk

−m

k=1

∂pj

∂ ln αk

.

(6.38)

Let us analyze the sumsm

k=1. First, we consider the case when thecoefficients ai and bj of the transfer function are considered as parameters.Then,

mk=1

∂pi

∂ ln αk =

nj=0

∂pi

∂ ln aj +

gj=0

∂pi

∂ ln bj ,m

k=1

∂q i∂ ln αk

=n

j=0

∂q i∂ ln aj

+

gj=0

∂q i∂ ln bj

.

It is easily seen that

∂pi

∂ ln aj

= − pn−ji aj

∂a(s)

∂s

s=pi

,∂q i

∂ ln bj

= − q g−ji aj

∂b(s)

∂s

s=pi

,




Therefore, there are the following root sensitivity invariants:

n

j=0∂pi

∂ ln aj

= 0,

g

j=0∂q i

∂ ln bj

= 0.

As a result, Equation (6.38) takes the form

nj=0

∂h(t)

∂ ln aj

gj=0

∂h(t)

∂ ln bj

=n

i=1

ciepit

n

j=0

∂ ln k

∂ ln aj

+

gj=0

∂ ln k

∂ ln bj

= h(t)

∂ ln k

∂ ln a0+

∂ ln k

∂ ln b0

= 0,

Hence, the sensitivity invariant of the weight function with respect to thecoefficients of the transfer function is given by

nj=0

∂h(t)

∂ ln aj

+

gj=0

∂h(t)

∂ ln bj

= 0.

If (6.18) holds, there is the invariant

mj=0

∂h(t)∂ ln αj

= 0.

6.2 Root and Transfer Function Sensitivity Invariants

6.2.1 Root Sensitivity Invariants

Consider the polynomial

D(s) =n

i=0

ais(n−i), a0 = 0,

Then, its roots p1, . . . , pn are zeros or poles of the system transfer function,and depend on the parameters α1, . . . , αm, so that pi = pi(α1, . . . , αm), i =

1, . . . , n.




The sum of the roots of the polynomial D(s) is equal to

n

i=1 pi = −a1a0

. (6.39)

Let us differentiate the last equation with respect to a parameter αk:

ni=1

∂pi

∂αk

= −∂a1∂αk

a0 − ∂a0∂αk

a1

a20. (6.40)

Assume that the coefficients a0 and a1 and their ratio a1/a0 are inde-pendent of the parameter αk so that

∂a1∂αk

= 0,∂a0∂αk

= 0, (or∂

∂α

a1a0

= 0).

Then,n

i=1

∂pi

∂αk

= 0 orn

i=1

∂pi

∂ ln αk

= 0. (6.41)

Thus, the sum of the sensitivity coefficients of the roots of the polynomial

D(s) with respect to the parameter αk is zero, i.e., is an invariant.Let the coefficients a0, a1, . . . , an are polylinear functions of the parame-

ters α1, . . . , αm. Then, the following relation of the form (6.18) holds:

mk=1

αk∂ai

∂αk

= qai, i = 0, . . . , n . (6.42)

These relations are, in fact, sufficient conditions for homogeneity of thefunction D(s). Multiplying both sides of Equation (6.40) by αk and takingthe sum over all k’s, we obtain

mk=1

ni=1

∂pi

∂ ln αk

= − 1

a20

a0

mk=1

∂a1∂ ln αk

− a1

mk=1

∂a0∂ ln αk

.

With due account for (6.42), from the last relation we find

m

k=1

n

i=1

∂pi

∂ ln αk

=

−

1

a2

0

(a0a1

−a1a0) .




Thus, we have the invariant

m

k=1n

i=1∂pi

∂ ln αk

= 0. (6.43)

Consider the summ

k=1

∂pi

∂ ln αk

for the case of pair-wise different roots. Differentiating the identity

D( pi) = 0

with respect to the parameter αk, we obtain

∂D

∂pi

∂pi

∂αk

+

nj=0

∂D

∂aj

∂aj

∂αk

= 0,

Hence,

∂pi

∂ ln αk

= −

nj=0

pn−ji

∂aj

∂ ln αk

∂D

∂pi

. (6.44)

Then,

mk=1

∂pi

∂ ln αk

= −

nj=0

pn−ji

mk=1

∂aj

∂ ln αk

∂D

∂pi

,

or, with account for (6.42),

mk=1

∂pi

∂ ln αk

= −q

nj=0

pn−ji aj

∂D

∂pi

= −qD( pi)∂D

∂pi

.

As a result, we have the invariant

m

k=1

∂pi

∂ ln αk

= 0. (6.45)




Comparing Equations (6.43) and (6.45), we find that the invariant (6.43)follows from the last one.

Then, let us consider the product of the roots

P =

ni=1

pi.

It is known thatn

i=1

pi = (−1)nan.

Then,

∂P

∂α=

n

i=1

∂pi

∂α

n

j=1

j=i

pj = (−1)n ∂an

∂α

or∂ ln P

∂α=

ni=1

∂ ln pi

∂α=

∂ ln an

∂α.

Thus, we have the invariant

n

i=1

∂ ln pi

∂ ln α

=∂ ln an

∂ ln α

. (6.46)

At last we notice that the conditions (6.42) yield yet another invariant

mk=1

∂ ln ai

∂ ln αk

= q, i = 0, . . . , n . (6.47)

Indeed,m

k=1

∂ ln ai

∂ ln αk

=1

ai

m

k=1

∂ai

∂αk

αk = q = const.

6.2.2 Sensitivity Invariants of Transfer Functions

Consider the transfer function

w(s) =b(s)

a(s)=

gi=0

bisg−i

n

i=0

aisn−i

, g ≤ n,




the coefficients of which ai and bj depend on the parameters α1, . . . , αm.Further we show that if the conditions (6.18) hold, there is the invariant

mk=1

∂ ln w(s)

∂ ln αk = const. (6.48)

Indeed,

∂ ln w(s)

∂ ln αk

=αk

w(s)

n

i=0

∂w(s)

∂ai

∂ai

∂αk

+

gi=0

∂w(s)

∂bi

∂bi

∂αk

=αk

w(s) 1

a(s)

g

i=0

sg−i ∂bi

∂αk

− b(s)

a2(s)

n

i=0

sn−i ∂ai

∂αk =

αk

b(s)

gi=0

sg−i ∂bi

∂αk

− αk

a(s)

ni=0

sn−i ∂ai

∂αk

,

Therefore,

mk=1

∂ ln w(s)

∂ ln αk

=

gi=0

sg−i

b(s)

mk=1

∂bi

∂αk

αk −n

i=0

sn−i

a(s)

mk=1

∂ai

∂αk

αk,

Using (6.18) for the coefficients ai and bj , we find

mk=1

∂ ln w(s)

∂ ln αk

=

gi=0

sg−ibi

b(s)q −

ni=0

q sn−iai

a(s)= 0.

To obtain this invariant, we can employ the following relation betweensensitivity of the transfer function and that of the zeros and poles given inSection 5.2:

∂ ln w(s)

∂ ln αj

=∂ ln k

∂αj

−g

i=1

∂zi

∂αj

1

s − zi

+n

i=1

∂pi

∂αj

1

s − pi

.

Multiplying both sides of this equation by αk and summing up by j, weobtain

m

j=1

∂ ln w(s)

∂ ln αj

=m

j=1

∂ ln k

∂ ln αj −

g

i=1

1

s − zi

m

j=1

∂zi

∂αj

αj +n

i=1

1

s − pi

m

j=1

∂pi

∂α

αj ,




Hence, with account for the invariants (6.45),

m

j=1∂ ln w(s)

∂ ln αj

=m

j=1∂ ln k

∂ ln αj

= const (6.49)

Example 6.3

Let transfer function w(s) have the form

w(s) =1

1 + T s, (6.50)

where T = RC . Represent it in the form

w(s) =h

h + Cs,

where h = 1/R is the conductivity.

Consider the sum

Z (s) =∂ ln w(s)

∂ ln C +

∂ ln w(s)

∂ ln g

Since

∂ ln w(s)

∂ ln C = − Cs

h + Cs,

∂ ln w(s)

∂ ln h= − Cs

h + Cs,

we have Z (s) = 0.

Example 6.4

Let us determine the sensitivity invariant for the transfer function of the

network shown in Figure 6.1. For the transfer function we have

w(s) =1 + T 1s

1 + T 2s,

where T 1 = R1C and T 2 = (R1 + R2)C .

Transformation to the form of a function with polylinear coefficientsyields

w(s) =h1h2 + Ch2s

h1h2 + C (h1 + h2)s.




Figure 6.1

Simplest network


∂ ln w(s)

∂ ln C = − h21Cs

(h1 + Cs)l(s),

∂ ln w(s)

∂ ln h1= − C 2h1s2

(h1 + Cs)l(s),

∂ ln w(s)

∂ ln h2=

Ch1s

l(s), where l(s) = h1h2 + C (h1 + h2)s.

As a result,

∂ ln w(s)

∂ ln C +

∂ ln w(s)

∂ ln h1+

∂ ln w(s)

∂ ln h2= 0

Example 6.5

Consider the active network shown in Figure 6.2 with transfer function

Figure 6.2

Active network




w(s) =k

1 + [C 1(R1 + R2) − (k − 1)C 2R2]s + R1R2C 1C 2s2

=h1h2k

h1h2 + [C 1(h1 + h2) − (k − 1)C 2h1]s + C 1C 2s2,

where h1 = 1/R1 and h2 = 1/R2.The transfer function contains a dimensionless parameter k. To obtain

the invariant (6.49), we need not take sensitivity with respect to the pa-rameter k in the sums [32, 87, 98].

Thus, we obtain∂ ln w(s)

∂ ln h1=

C 1(h1 + C 2s)s

l(s),

∂ ln w(s)∂ ln h2

= [C 1 − (k − 1)C 2]h1s + C 1C 2s2

+ C 1h2sl(s)

,

∂ ln w(s)

∂ ln C 1= −C 1[(h1 + h2)s + C 2s2

l(s),

∂ ln w(s)

∂ ln C 2= −C 2[(1 − k)h1s + C 1s2

l(s),

where

l(s) = h1h2 + [C 1(h1 + h2) − (k − 1)C 2h1]s + C 1C 2s2,

and∂ ln w(s)

∂ ln h1+

∂ ln w(s)

∂ ln h2+

∂ ln w(s)

∂ ln C 1+

∂ ln w(s)

∂ ln C 2= 0.

6.3 Sensitivity Invariants of Frequency Responses

6.3.1 First Form of Sensitivity Invariants of FrequencyResponses

Let us represent a frequency transfer function w( jω) in the form

w( jω) =a(ω) + jb(ω)

c(ω) + jd(ω), (6.51)




where

a(ω) = bg − bg−2ω2 + bg−4ω4 − . . . ,

b(ω) = bg−1ω − bg−3ω3 + bg−5ω5 − . . . ,

c(ω) = an − an2ω2 + an−4ω4 − . . . ,

d(ω) = an−1ω − an−3ω3 + an−5ω5 − . . . ,

(6.52)

Then, it can be easily shown that

w(va0, va1, . . . , v an, vb0, vb1, . . . , v bg, jω)

= w(a0, a1, . . . , an, b0, b1, . . . , bg, jω),(6.53)

where ν is an arbitrary nonzero number. Note that relations of the form(6.53) hold also for other frequency responses, namely, amplitude A(ω),phase φ(ω), real R(ω), and imaginary Q(ω) ones.

Differentiating (6.53) with respect to ν yields

ni=0

∂w( jω)

∂vai

∂vai

∂v+

gi=0

∂w( jω)

∂vbi

∂biv

∂v= 0

Assuming ν = 1 in the last equation, we find

ni=0

∂w( jω)

∂ai

ai +

gi=0

∂w( jω)

∂bi

bi = 0 (6.54)

orn

i=0

∂w( jω)

∂ ln ai

+

gi=0

∂w( jω)

∂ ln bi

= 0 (6.55)

The last relation can be considered as a sensitivity invariant of the fre-

quency transfer function with respect to the coefficients of the transferfunction. This invariant exists for transfer functions of all linear systemswith constant parameters. Consider a system with a frequency transferfunction

w( jω) =b1 + jωb0

a0( jω)2 + jωa1 + a2. (6.56)

Then, we have

∂w( jω)

∂ ln b1=

b1

l( jω),

∂w( jω)

∂ ln b0=

jωb0

l( jω),




∂w( jω)

∂ ln a2= −(b1 + b0 jω)a2

l2( jω),

∂w( jω)

∂ ln a1= −(b1 + b0 jω) jωa1

l2( jω),

∂w( jω)

∂ ln a0

=

−

(b1 + b0 jω)( jω)2a0

l2

( jω)

,

where l( jω) = a0( jω)2 + a1 jω + a2.


∂w( jω)

∂ ln b0+

∂w( jω)

∂ ln b1+

∂w( jω)

∂ ln a0+

∂w( jω)

∂ ln a1+

∂w( jω)

∂ ln a2= 0. (6.57)

The invariant (6.54) can be written for logarithmic sensitivity functionin the form

ni=0

∂ ln w( jω)∂ ln ai

+

gi=0

∂ ln w( jω)∂ ln bi

= 0. (6.58)

Let the conditions (6.18) hold for the coefficients of the transfer function.With account for (6.18), we transform the equality (6.54) to the form

ni=0

∂w( jω)

∂ai

qai +

gi=0

∂w( jω)

∂bi

qbi = 0, (6.59)

orn

i=0

mk=1

∂w( jω)

∂ai

∂ai

∂αk

αk +g

i=0

mk=1

∂w( jω)

∂bi

∂bi

∂αk

αk = 0,

Hence,m

k=0

∂w( jω)

∂ ln αk

= 0. (6.60)

Thus, if the conditions (6.53) and (6.18) hold, the sum of semi-logarithmicsensitivity functions for frequency transfer functions of linear systems withconstant parameters is zero. It is noteworthy that the invariants of theform (6.51), (6.53), and (6.54) can also be obtained from the invariant of transfer function (6.48) by the substitution s = jω.

6.3.2 Second Form of Sensitivity Invariants of FrequencyResponses

Using the following method, we can find yet another sensitivity invariant.Consider, formally, frequency ω as an additional parameter. Assume that

this parameter decreased ν times. To preserve the values a(ω), b(ω), c(ω),




and d(ω), it is necessary to increase the coefficients ai (i = 0, . . . , n) andbj ( j = 0, . . . , g) ν n−i and ν g−j times, respectively. Thus,

a(ω) = bg −

bg−2

v2 ω

v2 + . . . ,

b(ω) = bg−1vω

v

− bg−3v3

ω

v

3+ . . . ,

c(ω) = an − an−2v2ω

v

2+ . . . ,

d(ω) = an−1vω

v

− an−3v3

ω

v

3+ . . . ,

Then,

w

a0vn, a1vn−1, . . . , an, b0vg, b1vg−1, . . . , bm,

jω

v

= w(a0, a1, . . . , an, b0, b1, . . . , bm, jω).

(6.61)

Obviously, Equation (6.61) holds also for amplitude, phase, and real andimaginary frequency responses.

Let us differentiate (6.61) with respect to ν :

ni=0

∂w( jω)

∂v n−iai

(n − i)vn−i−1ai +

gi=0

∂w( jω)

∂vg−ibi

(g − i)vg−i−1bi =∂w( jω)

∂ ω

v

ω

v2.

Letting ν = 1 in the last equation, we find

n

i=0(n− i)

∂w( jω)

∂ ln ai

+

g

i=0(g − i)

∂w( jω)

∂ ln bi

= ω∂w( jω)

∂ω. (6.62)

Thus, we obtained yet another sensitivity invariant with respect to co-efficients of the frequency transfer function. The invariants (6.62) differsfrom those given in Section 3.1 by two features. First, they are weightedsums of the sensitivity functions rather than ordinary sums. Second, thesesums are not constants, but functions in the variable ω.

Example 6.6

Let us find the invariant for the transfer function (6.56). For this example,




Equation (6.62) takes the form

∂w( jω)

∂ ln a0+

∂w( jω)

∂ ln a1+

∂w( jω)

∂ ln b0= ω

∂w( jω)

∂ω. (6.63)

The terms in the left side were derived in the preceding example. Forthe right side we have

w∂w( jω)

∂ω=

j

b0a2 − b1a1 + a0b0ω2

+ 2a0b1ω

(−a0ω2 + a2 + ja1ω)2 ω.

Substituting the values of the corresponding terms, we find that theequality (6.63) holds. Dividing the left and right sides of Equation (6.62)

by w( jω), we obtain the invariant for logarithmic sensitivity functions:

ni=0

(n − i)∂ ln w( jω)

∂ ln ai

+

gi=0

(g − i)∂ ln w( jω)

∂ ln bi

=∂ ln w( jω)

∂ ln ω. (6.64)

Next, we assume that

m

k=1

∂ai

∂αk

αk = (n − i)ai, i = 0, . . . , n ,

mk=1

∂bi

∂αk

αk = (g − i)bi, i = 0, . . . , g .(6.65)

Then,

ni=0

mk=1

∂w( jω)

∂ai

∂ai

∂αk

αk +

gi=0

mk=1

∂w( jω)

∂bi

∂bi

∂αk

αk = ω∂w( jω)

∂ω

orm

k=1

∂w( jω)

∂ ln αk

=∂w( jω)

∂ω. (6.66)

Example 6.7

Consider the system with transfer function (6.50). It is easily seen that the

conditions (6.65) hold for the coefficients of the transfer function. Moreover,




the invariant (6.62) is given by

∂w( jω)

∂C C = ω

∂w( jω)

∂ω, (6.67)

because n = 1 and the terms with coefficients for i = n and i = g are zeroin the sums (6.62). In the invariant (6.67) we have

C ∂w( jω)

∂C = − jhωC

(h + jC ω)2, ω

∂w( jω)

∂ω= − jhωC

(h + jC ω)2.

The invariants obtained above hold for other frequency responses: ampli-tude, phase, real, and imaginary, because the relations (6.53) and (6.61)hold for them. Obviously, linear combinations of some invariants can gen-erate new invariants. Thus, the sum of the invariants (6.55) and (6.62)yields the invariant

ni=0

(n + 1 − i)∂w( jω)

∂ ln ai

+

gi=0

(g + 1 − i)∂w( jω)

∂ ln bi

=∂w( jω)

∂ ln ω, (6.68)

and their difference gives the invariant

ni=0

(n − 1 − i) ∂w( jω)∂ ln ai

+

gi=0

(g − 1 − i) ∂w( jω)∂ ln bi

= ∂w( jω)∂ ln ω

. (6.69)

In general,

ni=0

(n− 1 + q )∂w( jω)

∂ ln ai

+

gi=0

(g − 1 + q )∂w( jω)

∂ ln bi

=∂w( jω)

∂ ln ω, (6.70)

where q = 0,±1,±2, . . ..For (6.56), the invariant (6.68) has the form

3∂w( jω)

∂ ln a0+ 2

∂w( jω)

∂ ln a1+

∂w( jω)

∂ ln a2+ 2

∂w( jω)

∂ ln b0+

∂w( jω)

∂ ln b1=

∂w( jω)

∂ ln ω

Using the above expressions for the summands, this can easily be proved.For the same example, the invariant (6.69) has the form

∂w( jω)

∂ ln a0 −∂w( jω)

∂ ln a2 −∂w( jω)

∂ ln b1 =∂w( jω)

∂ ln ω .




Finally, we will show that the invariant (6.62) has the form of two partialinvariants. Consider the sensitivity functions of the frequency characteristicw( jω) with respect to the coefficients of the transfer function:

∂w( jω)∂ ln ai

= ∂ ∂ ln ai

b( jω)a( jω)

= −b( jω) 1

a2( jω)∂a( jω)∂ ln ai

= −w( jω)

a( jω)

∂a( jω)

∂ ln ai

, i = 0, . . . , n ,

(6.71)

∂w( jω)

∂ ln bi

=1

∂ ln ai

b( jω)

a( jω)

=

1

a( jω)

∂b( jω)

∂ ln bi

, i = 0, . . . , g . (6.72)

The condition (6.61) holds for the functions b( jω) and a( jω). Then, there

are the following sensitivity invariants:

ni=0

(n − i)∂a( jω)

∂ ln ai

=∂a( jω)

∂ ln ω, (6.73)

gi=0

(g − i)∂b( jω)

∂ ln bi

=∂b( jω)

∂ ln ω. (6.74)

Let us multiply both sides of Equations (6.73) and (6.74) by the factors−w( jω)/a( jω) and 1/a( jω), respectively. Then, according to (6.71) and(6.72), we obtain

ni=0

(n − i)∂w( jω)

∂ ln ai

= −w( jω)∂ ln a( jω)

∂ ln ω,

gi=0

(g − i)∂w( jω)

∂ ln bi

=∂ ln b( jω)

∂ ln ωw( jω).

(6.75)

It is easy to seen that the sum of the last two invariants is exactly equal tothe invariant (6.62).

6.3.3 Relations between Sensitivity Invariants of Time andFrequency Characteristics

For a linear system with transfer function w(s) and input signal x(t) andzero initial conditions, the image of the output signal y(t) is equal to

y(s) = w(s)x(s).




In the time-domain, the output signal is given by

y(t) =1

2πj

j∞

−j∞

w( jω)x( jω)ejωt d( jω).

Assuming that integration with respect to the variable jω and differ-entiation by the parameter αk are commutative operations, and x(t) isindependent of αk, we have

∂y(t)

∂αk

αk =1

2πj

j∞ −j∞

∂w( jω)

∂αk

αkx( jω)ejωt d( jω).

If the parameters are the coefficients of the transfer functions,

∂y(t)

∂ai

ai =1

2πj

j∞ −j∞

∂w( jω)

∂ai

aix( jω)ejω t d( jω),

∂y(t)

∂bi

bi =1

2πj

j∞ −j∞

∂w( jω)

∂bi

bix( jω)ejωt d( jω),

(6.76)

Hence,

ni=0

∂y(t)

∂ ln ai

+

gi=0

∂y(t)

∂ ln bi

=1

2πj

j∞ −j∞

x( jω)ejωt

n

i=0

∂w( jω)

∂ ln ai

+

gi=0

∂w( jω)

∂ ln bi

d( jω)

and, due to (6.55),n

i=0

∂y(t)

∂ ln ai

+

gi=0

∂y(t)

∂ ln bi

= 0. (6.77)

If the conditions (6.18) hold, instead of (6.77) we have the followinginvariant with respect to the parameters α1, . . . , αm:

m

i=1

∂y(t)

∂ ln αi

= 0 (6.78)




Multiplying the equations in (6.76) by (n − i) and (s − i), respectively,and summing them up, we can obtain the following sensitivity invariant forthe transient:

ni=0

(n− i) ∂y(t)∂ ln ai

+

gi=0

(g − i) ∂y(t)∂ ln bi

=1

2πj

j∞ −j∞

x( jω)ejω t ∂w( jω)

∂ ln ωd( jω),

(6.79)

which, under the conditions (6.18), takes the form

mi=1

∂y(t)∂ ln αi

= 12πj

j∞ −j∞

x( jω)ejωt ∂w( jω)∂ ln ω

d( jω).

It is known that there are analytic relations between time-domain andfrequency-domain characteristics of linear systems. Thus, weight functioncan be expressed in terms of the frequency responses [100]:

h(t) =2

π

∞

0

P (ω)cos ωtdω,

h(t) = − 2

π

∞ 0

Q(ω)sin ωtdω,

Transfer function is given by

k(t) =2

π

∞

0

P (ω)

ω

sin ωtdω

and so on.

Let us employ these relations to find sensitivity invariants of time-domaincharacteristics of linear systems. Consider, as a special case, the first of theabove relations. Differentiation with respect to a parameter αk yields

∂h(t)

∂αk

=2

π

∞

0

∂P (ω)

∂αk

αk cos ωtdt




Consider the case when coefficients of the transfer function are consideredas parameters. Then,

∂h(t)

∂ai ai =

2

π

∞

0

∂P (ω)

∂ai ai cos ωtdt,

∂h(t)

∂bi

bi =2

π

∞ 0

∂P (ω)

∂bi

bi cos ωtdt.

(6.80)

Summing these expressions up, we obtain

n

i=0

∂h(t)

∂ ln ai

+

g

i=0

∂h(t)

∂ ln bi

=2

π

∞

0

cos ωt n

i=0

∂P (ω)

∂ ln ai

+

g

i=0

∂p(ω)

∂ ln bi dω.

As was shown in Section 3.1, the term in the square brackets is a sensitivityinvariant for frequency responses P (ω) and is identically zero. Therefore,

ni=0

∂h(t)

∂ ln ai

+

gi=0

∂h(t)

∂ ln bi

= 0. (6.81)

Analogously, we obtain the sensitivity invariant of the transient

ni=0

∂k(t)

∂ ln ai

+

gi=0

∂k(t)

∂ ln bi

= 0. (6.82)

Under the conditions (6.18), from the invariants (6.81) and (6.82) withrespect to coefficients of the transfer functions we can find the invariantswith respect to the parameters αj ( j = 1, . . . , m):

mj=1

∂h(t)∂ ln αj

= 0,m

j=1

∂k(t)∂ ln αj

= 0. (6.83)

Next, we multiply the equations in (6.80) by (n − i) and (s − i), re-spectively. After summation and routine transformations, we obtain yetanother sensitivity invariant for the weight function:

n

i=0

∂h(t)

∂ ln ai

(n − i) +

g

i=0

∂h(t)

∂ ln bi

(g − i) =2

π

∞

0

∂P (ω)

∂ ln ωcos ωtdω.




Similarly, for the transient k(t) we have

n

i=0

∂k(t)

∂ ln ai

(n − i) +

g

i=0

∂k(t)

∂ ln bi

(g − i) =2

π

∞

0

∂P (ω)

∂ ln ωcos ωtdω.

Under the conditions (6.65), we find

mi=1

∂h(t)

∂ ln αi

=2

π

∞ 0

∂P (ω)

∂ ln ωcos ωtdω,

mi=1

∂k(t)

∂ ln αi

=2

π

∞ 0

∂P (ω)

∂ ln ωcos ωtdω,

6.4 Sensitivity Invariants of Integral Estimates

6.4.1 First Form of Sensitivity Invariants

Consider the integral estimate

I =

∞ 0

h2(t) dt =1

π

∞ 0

A2(ω) dω, (6.84)

where A(ω) is the amplitude frequency response, and h(t) is the weightfunction.

Let us evaluate sensitivity of the integral estimate (6.84) with respect todeviation of a parameter αi by the value

∂I

∂ ln αi

=∂

∂αi

αi

π

∞ 0

A2(ω) dω

,

Then, assuming that differentiation with respect to the parameter αi andintegration with respect to the variable ω are commutative operations, weobtain

∂I

∂ ln αi

=2

π

∞

0

A(ω)∂A(ω)

∂ ln αi

dω. (6.85)




Let the coefficients ai and bi be considered as the parameters α1, . . . , αm.Then,

∂I

∂ ln ai

=2

π

∞

0

A(ω)∂A(ω)

∂ ln ai

dω,

∂I

∂ ln bi

=2

π

∞ 0

A(ω)∂A(ω)

∂ ln bi

dω,

(6.86)

Summing the last equalities, we have

n

i=0∂I

∂ ln ai

+

g

i=0∂I

∂ ln bi

=2

π

∞ 0

A(ω)

n

i=0∂A(ω)

∂ ln ai

+

g

i=0∂A(ω)

∂ ln bi

dω.

According to the results of the previous paragraph, the term in the squarebrackets is the sensitivity invariant of the amplitude frequency response andis equal to zero. Therefore, we obtain the following sensitivity invariant forthe integral estimate (6.84):

ni=0

∂I

∂ ln ai

+

ni=0

∂I

∂ ln bi

= 0. (6.87)

Then, we find a sensitivity invariant of amplitude frequency response of a system with transfer function

w(s) =k

T s + 1

where b0 = k, a0 = T , and a1 = 1. The amplitude frequency response hasthe form

A(ω) =k

√ 1 + T 2

ω2

,

and the integral characteristic is equal to

I =1

π

∞ 0

A2(ω) dω =k2

2T =

b202a1a0

,

Hence,

∂I

∂ ln a0=−

b20

2a0a1,

∂I

∂ ln a1=−

b20

2a0a1,




∂I

∂ ln b0= − b20

a0a1,

∂I

∂ ln a0+

∂I

∂ ln a1+

∂I

∂ ln b0= 0.

Under the conditions (6.18), the invariant (6.87) takes the form

mi=1

∂I

∂ ln αi

= 0. (6.88)

Then, for the previous example, we have

A(ω) =h√

h2 + C 2ω2, I +

h

2C .

Since

∂I

∂ ln h=

h

2C ,

∂I

∂ ln C = − h

2C ,

we obtain

∂I

∂ ln h+

∂I

∂ ln h= 0.

6.4.2 Second Form of Sensitivity Invariants

Let us multiply relations in (6.86) by (n − i) and (s − i), respectively.After summation we obtain

ni=0

(n − i)∂I

∂ ln ai

+

gi=0

(g − i)∂I

∂ ln bi

=2

π

∞ 0

A(ω)

n

i=0

∂A(ω)

∂ ln ai

(n − i) +

gi=0

∂A(ω)

∂ ln bi

(g − i)

dω,

Hence, using the results of the previous paragraph, we find the invariant

ni=0

(n − i)∂I

∂ ln ai

+n

i=0

(g − i)∂I

∂ ln bi

=2

π

∞ 0

A(ω)∂A(ω)

∂ωωdω. (6.89)

For the example considered in this section, the last expression takes theform

∂I

∂ ln a0=

2

π

∞

0

A(ω)∂A(ω)

∂ ln ωdω =

1

π

∞

0

∂A2(ω)

∂ ln ωdω. (6.90)




Indeed,

∂I

∂ ln a0= − b20

2a0a1= − k2

2T ,

1π

∞ 0

∂A2(ω)∂ ∈ ω

dω = 1π

∞ 0

2k2T 2ω2

(T 2ω2 + 1)2dω = − b20

2a0a1,

Hence, we obtain (6.90).

Assume that the conditions (6.65) hold for the coefficients of the transferfunction. Then, there is the following sensitivity invariant of the integralestimate:

m

i=1

∂I

∂ ln αi

=1

π

∞

0

∂A2(ω)

∂ ln ωdω.

For instance, for

A(ω) =h√

h2 + C 2ω2

this invariant takes the form

∂I

∂ ln C =

1

π

∞ 0

∂A2(ω)

∂ ln ωdω,

where∂I

∂ ln C = − h

2C ,

1

π

∞ 0

∂A2(ω)

∂ ln ωdω = −2C 2h2

π

∞ 0

ω2

(C 2ω2 + h2)2dω = − h

2C .

6.5 Sensitivity Invariants for Gyroscopic Systems

6.5.1 Motion Equations and Transfer Functions

In this section we illustrate possibilities of sensitivity invariants by anexample of investigation of fundamental and forced motion of a three-axis(three-dimensional) gyroscopic stabilizer. Three-dimensional stabilizer is

included in most modern gyroscopic stabilization and inertial navigation




systems. The system of differential equations describing motion of three-dimensional gyrostabilizer is fairly cumbersome. For simplicity, in appliedtheory of gyroscopic systems, weak connections in a three-dimensional gy-rostabilizer are often ignored. Such being the case, only the influence of

rotation angles around the suspender axis and procession angles of hy-drounits are taken into account. In this case, investigation of fundamentalmotion of a three-axis gyrostabilizer reduces to investigation of the scalarstabilization channel, i.e., single-axis stabilizers, with account for significantinterconnections between the channels. Linearized equations of the motionof a single-axis gyrostabilizer has the form [80]

(J ps2 + ns)ρ = Hωx1 + M ρd + M ρc,

(Js + d)ωx1 = M αH

−H sρ

−K αW α(s)ρ

+ (d + ms)ωx0 + M αf sign ωx0 ,

(6.91)

where ρ is the precession angle of the hydrounit, ωx1 is the angle velocityof the platform, ωx0 is the foundation angle velocity, α is the angle of rotation of the platform with respect to the foundation, J p is the torqueof inertia of the gyroscope with respect to the precession axis, J is thetorque of inertia of the platform with moveable elements with respect tothe axis x1, H is the kinetic torque of the gyroscope, n is the specificdamping torque with respect to the precession axis, d is the specific damping

torque of the unloading motor reduced to the stabilization axis, m is thecoefficient that characterizes inertial disturbing torque that acts aroundthe stabilizer axis while running-in the rotor of the unloading unit with areduction gear, M αH and M αf are the torques of exogenous forces and “dry”friction with respect to the stabilization axis, M ρd and M ρc are disturbingand controlling torques with respect to the stabilization axis, K αW α(s)is the transfer function of the unloading channel, and s is the Laplaceoperator.

Equation (6.91) can be written in terms of the coordinates ωx1 and ρ to

be controlled:

W J (s)ωx1 = W g(s)∆ω − W u(s)ρ + W d(s)ωx0 + W ∗d (ωx0),

ρ = W m(s)∆ω, ωy =M ρc

H , ∆ω = ω1 − ωy

(6.92)

where W J (s) = Js + d is the transfer function of the platform,

W g(s) =H 2

J ps + n




is the transfer function with respect to gyroscopic torque,

W m(s) =H

J ps2 + ns

is the measurement transfer function, W u(s) = K αW α(s) is the transferfunction of the unloading channel, W d(s) = ms+d is the disturbance trans-fer function, ∆ω is the stabilization error for the platform angle velocity,and

W ∗d (ωx0) = M αf sign ωx0(t)

is the torque of dry friction with respect to the stabilization axis.

Fundamental motions of the gyrostabilizer are investigated on the basisof Equation (6.92) under the condition ωy = ωx0 = 0. Then, from (6.92)

we obtainW j (s)ωx1 = −W g(s)ωx1 − W u(s)ρ,

ρ = W m(s)ωx1 .(6.93)

From (6.93) we can easily determine the transfer function of the open-loop (with respect to the unloading torque) gyrostabilizer

W 0(s) =W m(s)W u(s)

W J (s) + W g(s)(6.94)

and the closed-loop one

Φ(s) =W m(s)W u(s)

W J (s) + W g(s) + W m(s)W u(s). (6.95)

The transfer function (6.94) can be reduced to the form

W 0(s) =kαW α(s)

H s(1 + 2ξT 0s + T 20 s2), (6.96)

where ω0 = 1/T 0 is the nutation frequency, and ξ is the relative dampingcoefficient of nutational oscillations.

Depending on the form of the functions W m(s) and W u(s), there are thefollowing types of gyrostabilizers.

1. Power stabilizer (n = 0):

W m(s) =H

J ps2W g(s) =

H 2

jps, ξ =

d

2H J p

J .




As a rule, ξ ≤ 0.01, because d H and J m J due to constructivereasons. The gyrostabilizer has a clearly oscillatory transient, i.e.,belongs to the class of low-damped systems.

2. Stabilizer with integrating gyroscope :

W m(s) =H

J ps2 + nsW g(s) =

H 2

J ps + n, ξ =

n

2H

J pJ

.

For a floated gyroscope, ξ > 0.3 . . . 0.5 (often ξ = 1), i.e., the torqueof viscous friction acting with respect to precession angle dampsnutational motion fairly well. For a two-stage gyroscope with largekinetic torque H and forced damping, we have ξ ≤ 0.1, i.e., the

gyrostabilizer falls into the class of low-damped systems.

3. Indicator stabilizer . In such stabilizers, gyroscopic torque with re-spect to stabilization axis may be ignored in comparison with otherones. Therefore, we can assume

W g(s) = 0.

Moreover, the transfer function can have various forms depending on thetype of sensor. As sensors, the following gyroscopes are used: integrat-

ing, differentiating, astatic, doubly integrating. Using the transfer functionΦ(s), we can determine forced motion with respect to the precession andstabilization axis as functions from the exogenous torque M αH and founda-tion angular velocity ωx0 . The form of the forced motion transfer functionis determined by the type of gyrostabilizer. For a power gyrostabilizer withtwo stages, the corresponding transfer function has the form

ωx1(s)

M αH (s)= Φ(s)

J ps2 + ns

HW u(s)=

1

kc(s),

ωx1(s)ωx0(s)

= Φ(s) (ms2

+ ds)(J ps + n)HW u(s)

= L(s).

(6.97)

The most important characteristics of stabilizer forced motion are theangular stabilization rigidity kc(s) and the oscillation damping index L(s).The angular stabilization rigidity determines stabilization errors under theinfluence of various forces and torques

kc(s) = sD(s),

D(s) = W J (s) + W g(s) + W m(s)W u(s).

(6.98)




The oscillation damping index L(s) characterizes the quality of stabiliza-tion under oscillation motion of the foundation. It is equal to the ratio of the amplitude of forced angular oscillation of the platform φx1 to the ampli-tude of platform oscillation with respect to the stabilization axis φx0 . For

a harmonic oscillation motion of the foundation, the oscillation dampingindex is given by

L(s) = Φ(s)ms2 + ds + M αf (c0)

HW u(s)(J ps + n(c0)), (6.99)

where

M αf (c0) =4

π· M 0αf

c0f o

is the coefficient of harmonic linearization of the dry friction torque withrespect to the stabilization axis,

n(c0) = n + 0, 1M 0ρf ·4

π

kα

d· 1

c0 · f k

is the equivalent damping torque with respect to the precession axis, c0is the amplitude of harmonic oscillation motion, f o is the oscillation fre-quency, and M 0αf and M 0ρf are the break-away torques with respect to thestabilization and precession axes, respectively.

6.5.2 Sensitivity Invariants of Amplitude FrequencyResponse

In power stabilizers, non-minimal phase units like phase-inverters areused [80], i.e.,

W u(s) = kα1 − T os

1 + T os. (6.100)

Then, the transfer function (6.94) of the open-loop gyrostabilizer withtwo-stage gyroscopes is given by

W 0(s) =kα(1 − T os)

Hs(1 + 2ξT 0s + T 20 s2)(1 + T os). (6.101)

The amplitude frequency response is obtained from the equation

A20(ω) =

k2α

H 2[4ξ 2T 20 ω2 + (1 − T 20 ω2)2]. (6.102)




Sensitivity of amplitude frequency responses can be conveniently esti-mated by logarithmic sensitivity functions of the form

S A0

αk

(ω) =∂ ln A0(ω)

∂ ln αk

=1

2A20(ω) ·∂A2

0(ω)

∂αk

αk. (6.103)

Differentiating Formula (6.102) according to (6.103), we find the sensitiv-ity functions for all parameters appearing in the transfer function (6.101):

a) S A0

kα(ω) = 1, S A0

T o(ω) = 0,

b) S A0

J p(ω) =

T 20 ω2(1− T 20 ω2)

4ξ 2T 20 ω2 + (1 − T 20 ω2)2,

c) S A0

n (ω) = S A0

ξ (ω) = − 4ξ 2T 20 ω2

4ξ 2T 20 ω2 + (1 − T 20 ω2)2,

d) S A0

J (ω) =T 20 ω2(1 − T 20 ω2 − 4ξ 2)

4ξ 2T 20 ω2 + (1 − T 20 ω2)2,

e) S A0

H (ω) = −1 − 2ω2T 20 (1− T 20 ω2 − 4ξ 2)

4ξ 2T 20 ω2 + (1 − T 20 ω2)2,

f ) S A0

ω0(ω) =

T 20 ω2(4ξ 2 + 2T 20 ω2 − 2)

4ξ 2T 20 ω2 + (1 − T 20 ω2)2.

(6.104)

The process of evaluating the sensitivity function (6.104) of a low-dampedoscillatory system of power gyrostabilizer is numerically unstable in a local-ity of the nutational frequency ω0. Numerical stability can be enhanced byusing sensitivity invariants. Sensitivity invariants for the relative functions(6.104) are given by the sums

a) S A0

kα(ω) + S A0

H (ω) + S A0

J (ω) + S A0

J p(ω) + S A0

n (ω) = 0,

b) S A0

n (ω) + S A0

J p(ω)

−S A0

J (ω) = 0,

c) S A0

H (ω) + 2S A0

J (ω) + S A0

kα(ω) = 0,

d) S A0

ω0(ω) + S A0

ξ (ω) + 2S A0

J p(ω) = 0,

e) S A0

n (ω) − S A0

ξ (ω) = 0,

(6.105)

Equation (6.105) can easily be proved by appropriate summation of thefunctions (6.104). The sensitivity invariants make it possible to simplifythe procedure of sensitivity functions calculation. Indeed, as follows from

the relations b)-d) in (6.105), any two sensitivity functions can be uniquely




determined in terms of two remaining (base) ones. Hence, to calculate thesix sensitivity functions of the response A(ω) of the gyrostabilizer denotedby

S A0

H

(ω), S A0

J

(ω), S A0

J p(ω), S A0

n (ω), S A0

ω0(ω), S A0

ξ

(ω),

it is sufficient to find only two base ones, for example, S A0

n (ω) and S A0

J p(ω),

while the remaining ones can easily be found from the invariant relations(6.105). To enhance numerical stability, sensitivity functions having thesimplest form are to be chosen as the base ones. The calculation procedurefor determining sensitivity functions of amplitude frequency response of open-loop gyrostabilizer is to be performed in the following order. First,the base functions S A0

n (ω) and S A0

J p(ω) are calculated by the relations b)

and c) of (6.104). Then, the remaining sensitivity functions are found from

the following equalities obtained from (6.105):

a) S A0

J (ω) = S A0

n (ω) + S A0

J p(ω),

b) S A0

H (ω) = −1 − 2S A0

J (ω),

c) S A0

ω0(ω) = 2S A0

J p(ω) − S A0

n (ω) = 0.

(6.106)

For the response Ac(ω) =

|Φ( jω)

|of a closed-loop gyrostabilizer the sen-

sitivity invariant is determined by relations similar to (6.105). Nevertheless,base functions are given by the following more complex equations:

S Ac

J p(ω) =

T 20 T oω4

|N ( jω)|2

d(ω)

ωT o− c(ω)

,

S Acn (ω) =

2ξT 0ω2

|N ( jω)|2 [ωT od(ω) + c(ω)] ,

S Ac

kα(ω) = 1 +

kα

H |N ( jω)

|2

[ωT od(ω) − c(ω)] ,

(6.107)

where

Φ( jω) =R( jω)

N ( jω)=

a(ω) + jb(ω)

c(ω) + jd(ω).

Sensitivity functions of gyrostabilizer frequency responses A0(ω) andAc(ω) are connected not only by the invariant relations (6.105) having theform of sums, but also by sensitivity cross-invariants. The latter connection

exposes itself in the fact that on the nutational frequency the sensitivity




functions have the form

S A0

J p(ω0) = S Ac

J p(ω0c) = 0,

S

A0

kα (ω0) = S

A0

H (ω0) = S

A0

ω0 (ω0) = 1,S A0

J (ω0) = S A0

n (ω0) = S A0

ξ (ω0) = −1,

S Ac

kα(ω0c) = S Ac

H (ω0c) = S Acω0c

(ω0c) = q 0,

S Ac

J (ω0c) = S Acn (ω0c) = S Ac

ξc(ω0c) = −q 0.

(6.108)

where q 0 is a constant.

For the frequencies ω0 ± ξω0 and ω0c ± ξ cω0c the sensitivity functionsS H (ω), S ω0

(ω), S J p(ω), and S J (ω) reach their maximal values.

6.5.3 Sensitivity Invariants of Integral Estimates

It is convenient to evaluate sensitivity of the characteristic polynomialD(s) and angle rigidity of stabilization kc(s) by means of sensitivity co-efficient of an integral index of stabilization quality. The use of integralestimates makes it possible to estimate the quality of stabilization withoutsolving differential equations. Integral estimates are expressed in terms of the coefficients of the characteristic polynomial D(s) according to the rules

known in automatic control theory [8].For a gyrostabilizer with transfer function (6.101) the integral quadratic

estimate of the transient process for unit pulse input has the form

I =1

2

a3(a1 − 1)2 + a0(a3a2 − a4a1)

[a1(a3a2 − a4a1) − a23a0], (6.109)

where

a4

=JJ p

H 2T

o, a

3=

H 2

JJ p+

nJ

H 2T

o,

a2 = T o +nJ

H 2, a1 = 1 − jα

H T o, a0 =

kα

H .

Then, we estimate sensitivity of the integral index (6.109) using the ex-pression

∂I

∂ ln αk

=∂I

∂αk

αk =4

i=0

∂I

∂ai

∂ai

∂αk

αk. (6.110)




For a system with the transfer function (6.101) we have, according toFormula (6.110),

a)∂I

∂kα

= a0

∂I

∂a0 −a0

T o

∂I

∂a1,

b)∂I

∂H H = −a0

∂I

∂a0+ a0T o

∂I

∂a1

−4ξT o∂I

∂a2− 2a3

∂I

∂a3− 2a4

∂I

∂a4,

c)∂I

∂J J = 2ξT o

∂I

∂a2+ a3

∂I

∂a3+ a4

∂I

∂a4,

d)∂I

∂J pJ p = T 20

∂I

∂a3+ a4

∂I

∂a4,

e)∂I

∂nn = 2ξT 0

∂I

∂a2+ 2ξT 0T o

∂I

∂a3,

f )∂I

∂T oT o = −a0T o

∂I

∂a1+ +T o

∂I

∂a2+ 2ξT 0T o

∂I

∂a3+ a4

∂I

∂a4.

(6.111)

For a multivariable gyrostabilizer, calculation of the sensitivity coeffi-cients is a difficult problem. The procedure can be simplified by the useof sensitivity invariants. It appears that invariant relations for sensitiv-

ity coefficients of the integral estimate (6.109) and sensitivity function of amplitude-pulse response (6.105) have similar form. For the integral esti-mate (6.109), the sensitivity invariants have the form

a)∂I

∂kα

kα +∂I

∂H H +

∂I

∂J J +

∂I

∂J pJ p +

∂I

∂nn = 0,

b)∂I

∂nn +

∂I

∂J pJ p − ∂I

∂J J = 0,

c)∂I

∂H H + 2

∂I

∂J J +

∂I

∂kα

kα = 0.

(6.112)

Relations (6.112) can easily be proven by summing the sensitivity coeffi-cients (6.111) in an appropriate way.

6.5.4 Sensitivity Invariants of Damping Coefficient

The index of damping of gyrostabilizer foundation oscillations L(ω) canbe shown in the form of the amplitude frequency response of the function

L(s) (see (6.99)). For a gyrostabilizer with the transfer function (6.101),




the square of the damping index L2(ω) is given by

L2(ω) = A2c (ω)

J 2p ω2 + n2(c0)

k2αH 2[(M αf (c0) + mω2)2 + d2ω2]. (6.113)

Since the frequencies of foundation pumping are two-three times less thannutational frequencies, in Equation (6.113) we can assume Ac(ω) = 1 forinvestigation of low-frequency part. Then,

L21(ω) =

J 2p ω2 + n2(c0)

k2αH 2[(M αf (c0) + mω2)2 + d2ω2]. (6.114)

For a constant pumping amplitude c0 we can write the following sensitivityinvariants of the foundation damping index (6.114):

a) S L1

kα(ω) + S L1

H (ω) + S L1

J p(ω) + S L1

n (ω)

+ S L1

M αf (ω) + S L1

m (ω) + S L1

d (ω) = 0,

b) S L1

kα(ω) = S L1

H (ω) = −1

c) S L1

n (ω) + S L1

J p(ω) = 1,

d) S L1

d (ω) + S L1

M αf (ω) + S L1

m (ω) = 1.

(6.115)

For determination of all sensitivity functions of the damping index it isnecessary to find only three base ones:

S L1

n (ω) = S L1

M ρf (ω) =

n2(c0)

J 2p ω2 + n2(c0),

S L1

d (ω) =d2ω2

(M αf − mω

2

)

2

+ d

2

ω

2,

S L1

M αf (ω) =

M αf (M αf − mω2)

(M αf − mω2)2 + d2ω2.

(6.116)

The remaining sensitivity functions are defined, according to the invari-ant relstions (6.115), by the equalities

S L1

J p(ω) = 1 − S L1

n (ω),

S L1

m (ω) = 1 − S L1

d (ω) − S L1

M αf (ω).

(6.117)




For a three-axes gyroscope nine damping coefficients are usually required,and the use of the sensitivity invariants greatly decreases amount of cal-culations. Sensitivity functions of the foundation damping index make itpossible to range parameters of gyrostabilizer by the degree of their influ-

ence on the quality of stabilization and to test stabilizer workability on arolling foundation under extremal testing and operating conditions.

The parameter ranging is performed by the following rule: a parameterαi is more important than a parameter αj on the rolling frequency f k if

S Lαi(f k)

>S Lαj

(f k) ,

or, with account for deviation of the parameters values ∆αi, by the rule

S Lαi(f k)∆αi >

S Lαi(f k)∆αj .

The gyrostabilizer remains workable for extremal conditions if

L(ωk) +m

i=1

∆αiS Lαi(ωk)L(ωk) ≤ L0(ωk),

where L0(ωk) is a given restriction imposed on L(ωk).

If the sensitivity functions of the damping index are calculated by For-mula (6.113), the expression becomes much more involved. In this case thesensitivity invariants makes it possible to simplify the formulas. Thus, forthe index (6.113) we can write

L2(ω) = A23(ω) · L2

1(ω). (6.118)

The sensitivity function of the index L(ω) with respect to a parameterαi is determined by

S Lαi(ω) =

1

2L2(ω)· ∂L2(ω)

∂αi

αi. (6.119)

Substituting (6.118) into (6.119) and performing differentiation, we obtain

S Lαi(ω) = S Ac

αi(ω) + S L1

αi(ω). (6.120)

Equation (6.120) demonstrates the possibility to use sensitivity invariants

for simplification of investigation of complex characteristics. Thus, having




the sensitivity invariants for the terms S Acαi

(ω) and S L1

αi(ω), we can find

sensitivity invariants for the more complex characteristic L(ω) (see (6.113)).With account for (6.105), (6.115) and (6.120), the sensitivity invariants forthe damping coefficient of the form (6.113) are given by

a) S Lkαi(ω) + S LH (ω) + S LJ p

(ω) + S Ln (ω) + S Ac

J (ω) = −1,

b) S Ln (ω) + S LJ p(ω) − S Ac

J (ω) = 1,

d) S LH (ω) + S Lkα(ω) + 2S Ac

J (ω) = −2.

(6.121)

The sensitivity invariants (6.121) make it possible to develop a simplifiedprocess for determining sensitivity functions of a fairly complex dampingindex of foundation oscillation by (6.113) according to the aforesaid rules.This is especially important for investigation of such a complex system as

three-axes gyroscope.




Chapter 7

Sensitivity of Mathematical Programming and Variational Calculus Problems

7.1 Sensitivity of Linear Programming Problems

7.1.1 Actuality of Investigation of Optimal ControlSensitivity

Classical optimization theory is based on the assumption that there issufficient information about a mathematical model of the plant to be opti-

mized. Nevertheless, mathematical models differ from real plants for manyreasons. Moreover, the control itself will be different from the calculatedone, because elements and units realizing it are not ideal. All these reasonslead to violation of optimality conditions and some losses for any specialcase of controlling a plant. For this reason, it is very important to estimatesensitivity of optimal control with respect to variations of parameters de-scribing the plant and control system that can be considered as initial datain optimal control design. Sensitivity analysis is most actual in operationsresearch and system engineering for designing complex schemes, when evensmall uncertainties in initial data and assumptions may lead to great mate-

rial losses. Therefore, sensitivity analysis is currently a mandatory stage of large system investigations [17, 48, 82]. In this chapter we discuss methodsof sensitivity investigation for solutions of mathematical programming andvariational calculus problems based on the use of sensitivity functions.

7.1.2 Linear Programming

Mathematical programming problems when a goal function (functional I )is linear and the set, where extremum of the function is to be found, is given

by a system of linear equalities and inequalities, fall into the class of linear




programming (LP) problems. The generic problem of linear programmingcan be formulated as follows:

Linear programming problem. It is required to find a solution

x10, . . . , xn0 from all nonnegative solutions xj ≥ 0 ( j = 1, . . . , n)satisfying the inequalities

nj=1

aijxj ≤ bi, i = 1, . . . , m ,

such that a linear function of these variables

I = c1x1 + c2x2 + . . . + cnxn

reach the maximal value.

The generic problem can be written in the following vector matrix form

maxI (X ) = C T X , AX ≤ B, X ≥ 0, (7.1)

where

X T = [x1, . . . , xn], C T = [c1, . . . , cn], BT = [b1, . . . , bm],A =

aij

, i = 1, . . . , n .

It is known that any linear programming problem can be reduced to theabove form. If, for instance, it is required to find the minimum of a costfunction I (X ), the generic problem can be obtained by using a new goalfunction I 1(X ) after multiplying the initial function by −1. The linearprogramming problem can be written in the following canonical form:

maxI (X ) = C T X , AX = B, X ≥ 0. (7.2)

Any generic problem can be reduced to the canonical form by introducingadditional variables. For any linear programming problem there is a dualproblem. For example, the dual problem for (7.2) is formulated as follows:

maxJ (X ) = BT Y , AT Y ≥ C, Y ≥ 0. (7.3)

In theory of linear programming it is shown that if X 0 is the optimalsolution of a direct problem and Y 0 is the optimal solution of the associateddual problem, we have

C T

X 0 = BT

Y 0. (7.4)




In linear programming, initial data are determined by components of thevectors C and B and those of the matrix A. First, consider a geometricinterpretation of the influence of initial data variations onto an optimalsolution. Consider the case of two variables x1 and x2.

7.1.3 Qualitative Geometric Sensitivity Analysis

Figure 7.1Geometric interpretation of linear programming problem

Consider Figure 7.1 where the polygon ABDEF is the domain of admis-sible solutions, and the line M N corresponds to the goal function (efficiencyindex). Assume that the optimal solution (for instance, one maximizing theefficiency index) is located in the vertex B. From Figure 7.1 it is knownthat the vertex B corresponds not only to the optimal solution defined bythe line MN . Specific location of the line M N is defined by the vector C .Indeed, the vector C perpendicular to the line M N is the directing vector(gradient) of the goal function I = C T X (see Figure 7.1). The vectors C 1and C 2 will be the directing vectors of the lines AB and BD, respectively.The optimal solution determined by the vertex B will remain unchangedfor variations of the vector C between C 1 and C 2. Denoting the angles of the vectors C , C 1, and C 2 by γ , γ 1, and γ 2, respectively, the solution in thevertex B remains optimal when

γ ∈ (γ 1, γ 2).

If γ = γ 1 (or γ = γ 2) the line of the goal function will be parallel to the




polygon side AB (or BD, respectively). In this case, the LP problem willhave an infinite number of solutions. Further variations of the vector C lead to the case of a single optimal solution, though in another vertex, Aor D, respectively. The new optimal solution also remains valid in a range

of variation of the vector C .In a similar way, we can trace the influence of variation of the vector B

on the optimal solution. Variation of the coefficient bi leads to a linear shiftof the hyperplane

nj=1

aij xj = bi

with respect to the origin. The hyperplane inclination remains the same.In the two-dimensional case shown in Figure 7.1 variation of the coefficientsbi (i = 1, . . . , m) causes parallel shift of the lines AB, BD, DE , and AF .Moreover, in some range of variation of the coefficients bi (i = 1, . . . , m)the optimal solution is determined by intersection of the same sides of thepolygon of admissible solutions. It can be geometrically shown that thissolution remains optimal as far as it is admissible. This result follows fromthe main theorem of linear programming [30, 34]. For geometric interpre-tation of the influence of variations of the vector B on the optimal solutionwe can also consider the dual problem with respect to the initial one.

It is hardly possible to find a vivid representation of the influence of vari-ation of the components of the matrix A onto optimal solution. ConsideringFigure 7.1, we can only propose that local variations of the coefficients aijincluded in the equations of the lines AF , EF , and DE do not changethe optimal solution. These coefficients are not components of the basicmatrix associated with the optimal plan in the vertex B. Three areas of admissible solutions determined by the position of the line AB are shownin Figure 7.2. In the first case shown in Figure 7.2a, the optimality vertexis B, in the second one (see Figure 7.2b) optimum is reached in A1, while

in the third one there is no optimal solution, because the admissible set isempty due to incompatible restrictions.

7.1.4 Quantitative Sensitivity Analysis

Considering initial data as parameters in a linear programming problem,we can consider the influence of their variations on the properties of op-timal solution using the methods developed in the theory of parametricprogramming [25, 30]. Local properties of a solution of a linear program-

ming problem can be estimated by sensitivity coefficients.




Figure 7.2

Areas of admissible solutions

THEOREM 7.1 [33, 64, 72]

Let for nominal values of initial data

A = A0, C = C 0, B = B0

there be unique solutions X 0 and Y 0 of the initial (7.2) and dual (7.3)problems of linear programming. Then, in a small locality of the values A0,

B0, and C 0 the following representation holds:

I max(A0 + ∆A0, B0 + ∆B0, C 0 + ∆C ) = I max(A0, B0, C 0)

+ X T 0 ∆c + Y T 0 ∆B − Y T 0 ∆AX 0 + 0(∆A, ∆B, ∆C ).(7.5)

PROOF Let us represent the optimal value of the goal function in alocality of A0, B0, and C 0 in the form

I max(A0 + ∆A0, B0 + ∆B, C 0 + ∆C )

= I max(A0, B0, C 0) +

∂I

∂C

0

∆C +

∂I

∂B

0

∆B + δ (∆A),

where ∂I

∂C

0

,

∂I

∂B

0

are row sensitivity vectors with respect to variations of the vectors C and

B, respectively, and δ (∆A) is the increment of the goal function owing to




variation of the elements of the matrix A. To determine the sensitivityvector ∂I/∂C it suffices to differentiate the goal function directly, so that

∂I

∂C 0

= ∂ (C T X )

∂C 0

= X T 0 . (7.6)

Thus, the vector of sensitivity with respect to variation of the param-eters of the vector C coincides with the optimal solution of initial linearprogramming problem at the point (A0, B0, C 0). Let the components of thevector C depend on a parameter α, i.e.,

ci = ci(α).

Then, sensitivity of the value I 0 with respect to the parameter variation is

given by the formuladI 0dα

=ni=1

∂I 0∂ci

dcidα

.

To obtain the sensitivity vector (∂I/∂B)0, we consider the correspondingdual problem and use the condition (7.4)

C T X 0 = BT Y 0,

Hence, ∂I

∂B

0

=

∂ (BT Y 0)

∂B

0

= Y T 0 , (7.7)

i.e., the sensitivity vector of the optimal value of the goal function withrespect to variations of the elements of the vector B coincides with theoptimal solution of the dual problem. Assume that we are interested inobtaining the sensitivity of the value I 0 with respect to variations of theparameter α, assuming that the components of the vector B depend on α,i.e.,

bi = bi(α)Obviously,

∂I 0∂α

=mi=1

∂I 0∂bi

dbidα

.

To evaluate sensitivity of the optimal value of the goal function withrespect to variations of the elements of the matrix A, we write restrictionsof the initial problem in the canonical form

(A0 + ∆A)(X 0 + ∆X ) = B0 + ∆B




or, ignoring second-order terms,

A0X = B0 + ∆B + ∆B1,

where

∆B1 = −∆AX 0. (7.8)

As a result, we managed to reduce the variation ∆A to the equivalent,up to second-order terms, variation of the vector B. Then,

I max(A0 + ∆A, B0 + ∆B, C 0 + ∆C )

≈ I max(A0, B0 + ∆B + ∆B1, C 0 + ∆C )

=∂I 0∂C

∆C +∂I 0∂B

(∆B + ∆B1) + I 0

= X T 0 ∆C + Y T 0 ∆B + Y T 0 ∆B1 + I 0

or, with due account for (7.8),

∆I 0 = X T 0 ∆C + Y T 0 ∆B − Y T 0 ∆AX 0. (7.9)

From the latter relations it follows that

δ (∆A) ≈ −Y T 0 ∆AX 0

or, in a coordinate form,

δ (∆A) = −mi=1

nj=1

x0jy0i∆aij ,

Then, for the sensitivity coefficients we find

uij =

∂I

∂aij

0

= −x0jy0i. (7.10)

From the last equation it is evident that the sensitivity coefficient uks iszero provided that any of the elements x0s or y0k is zero. This conclusionmeans that the element aks is a coefficient at non-basis (free) variables ininitial and dual problems. In fact, this proves the hypothesis regarding the

influence of variations of the elements of the matrix A on the solution of a




linear programming problem proposed in the previous paragraph. Let X 1be the vector of basis variables and X 2 be the vector of free variables. De-note X T = [X T 1 X T 2 ]. Then, the equation of restrictions can be representedin the form

A1X 1 + A2X 2 = B,

where A1 and A2 are elements of the block matrix

A = [A1 A2].

Since detA1 = 0, we have

X 1 = A−11 B − A−1

1 A2X 2.

Transformation of the goal function yields

I = C T X = C T 1 X 1 + C T 2 X 2 = C T 1

A−11 B − A−1

1 A2X 2

+ C T 2 X 2

= C T 1 A−11 B − C T 1 A−1

1 A2 − C T 2

X 2.

According to the simplex method of solving linear programming problems,we have X 2 = 0, therefore,

I = C T 1 A−11 B.

Obviously, the sensitivity coefficients of this function to variations of theelements of the matrix A are equal to zero.

All the above reasoning relates to the case when there is a unique solution

to the linear programming problem under consideration. In a special casein which the level line of the goal function is parallel to a side of the area of admissible solutions, the goal function is not differentiable with respect tothe parameters. Then, the sensitivity coefficients are determined accordingto a theorem given in [33] by the formulas

∂I 0

∂c+j= max

x∈DXxj = xjmax,

∂I 0

∂c−j= min

x∈DXxj = xjmin,

∂I 0

∂b+i

= minx∈DY

yi = yimin,∂I 0

∂b−i

= maxy∈DY

yi = ximax,

(7.11)




∂I 0

∂a+ij

=

−xjminyimax, if yimax ≥ 0,−xjmaxyimax, if yimax < 0,

∂I 0

∂a−ij=

−xjmaxyimin, if yimin ≥ 0,−xjminyimin, if yimin < 0.

(7.12)

Here by

∂I 0

∂c+j,

∂I 0

∂b+i,

∂I 0

∂a+ij

we denote the right derivatives, and by

∂I 0

∂c−j,

∂I 0

∂b−i,

∂I 0

∂a−ij

the left partial derivatives of the goal functions with respect to the corre-sponding arguments. According to the conditions of the sets theorem, DX

and DY must be non-empty bounded sets.

Finally, we consider a simplest linear programming problem as an exam-ple.

Example 7.1

Find the maximum of the function

I = cx1 + x2

under the restrictions

x1 + x2 ≤ 3, x1 + x2 ≥ 1,

x1 ≥ 0, x2 ≥ 0

and investigate its sensitivity with respect to the parameter c. The areaof admissible solutions is given in Figure 7.3a. The level line of the goal

function is characterized by the directing vector C with coefficients (c, 1).The type of “rotation” of the goal function line as the parameter c variatesin the interval (−∞, +∞) is shown in Figure 7.3b. Figure 7.3c demonstratesthe dependence of the optimal value of the function on the parameter c.In this figure tan β = 3. In the interval (−∞, 1) the optimal solution isdetermined by the coordinates of the vertex D of the polygon of admissiblesolutions, and

I 0D(0, 3) = 3.




Figure 7.3

Areas of admissible solutions




For c = 1, the level line of the goal function is parallel to the line AD,i.e., we have a special case. For c > 1, the optimal solution is determinedby the coordinates of the vertex A, i.e.,

I 0A(0, 3) = 3c.

The sensitivity coefficient U c = ∂I 0/∂c exists for all values of c exceptfor c = 1, and is determined by the following values:

∂I 0∂c

= x1max = c for c > 1,∂I 0∂c

= x1min = 0 for c < 1.

Figures 7.3d and 7.3f show the curves of the optimal values of x1 and x2on the parameter α = c. It is known [64] that the solution of an arbitrary

finite matrix game can be reduced to a solution of a linear programmingproblem. Such being the case, the above approach can be useful for theinvestigation of the sensitivity of matrix games.

7.2 Sensitivity of Optimal Solution to NonlinearProgramming Problems

7.2.1 Unconstrained Nonlinear Programming

Let a goal function I = I (X ) be defined in n-dimensional Euclidean spaceand differentiable at a point X 0. Then, an extremum exists at the pointX 0 only if all partial derivatives of the first order are zero:

ψi(X ) =∂I

∂xi= 0, i = 1, . . . , n . (7.13)

Moreover, assume that the goal function depends on a non-controllable

parameter α so that

I = I (X, α).

Obviously, the solution X 0 is a function of this parameter

X 0 = X 0(α).

It is required to find the sensitivity coefficients dx0i/dα.

Let functions ψi(X, α) satisfy the conditions of the theorem on differen-

tiability of implicit function, i.e., the functions ψ1, . . . , ψn are defined and




continuous in a locality of the point (X 0, α). Moreover, in this area thereare continuous partial derivatives

∂ψi

∂xj

,∂ψi

∂α

, i, j = 1, . . . , n ,

and the Jacobian

J =D(ψ1, . . . , ψn)

D(x1, . . . , xn).

is nonzero. Then, by the theorem on differentiability of implicit functions[114], there are continuous sensitivity coefficients given by the formulas

dxidα

= −D(ψ1, . . . , ψn)

D(x1, . . . , xi−1, α , xi+1, . . . xn)

D(ψ1, . . . , ψn)

D(x1, . . . xn)

. (7.14)

7.2.2 Nonlinear Programming with Equality Constraints

Consider the following problem. Find an extremum of the goal functionI (X ) under the constraints

f i(X ) = 0, i = 1, . . . m < n. (7.15)

It is assumed that the functions I (X ) and f i(X ) are doubly differentiable.To solve this problem, Lagrange multipliers λ1, . . . , λm and the followingLagrange function are introduced:

L(X, λ1, . . . , λm) = I (X ) −mj=1

λjf j(X ).

Then, necessary extremum conditions take the form

∂L

∂xi= 0,

∂L

∂λj

= 0, i = 1, . . . , n, j = 1, . . . , m ,

or

∂I

∂xi−

mj=1

λj∂f j∂xi

= 0, i = 1, . . . , n ,

f j(X ) = 0, j = 1, . . . , m .

(7.16)




Let the goal function I (X ) and the functions f j ( j = 1, . . . , m) dependon a parameter α. Moreover, let the derivative ∂I/∂α exist. Obviously,the values x1, . . . , xn and λ1, . . . , λm satisfying (7.16) are functions of theparameter α. It is required to find the sensitivity coefficients ∂xi/∂α and

∂λj/∂α.Introduce the notation

∂L

∂xi

= ψi, i = 1, . . . , n ,

∂L

∂λj

= ψn+j , j = 1, . . . , m .

Assume that for α = α0 the following Jacobian is nonzero:

J = D(ψ1, . . . , ψn+m)D(x1, . . . , xn, λ1, . . . , λm)

.

Then, according to the aforesaid theorem on differentiability of implicitfunctions, the desired coefficients ∂xi/∂α and ∂λj/∂α for α = α0 are givenby

∂xi

∂α= −

D(ψ1, . . . , ψn+m)

D(x1, . . . , xi−1, α , xi+1, . . . , xn, λ1, . . . , λm)

J ,

∂λj∂α

= −D(ψ1, . . . , ψn+m)

D(x1, . . . , xn, λ1, . . . , λj−1, α , λj+1, . . . , λm)

J .

(7.17)

It can be easily seen that the values (7.17) are solutions of the follow-ing linear algebraic equations obtained from (7.16) by differentiation withrespect to α:

∂φi

∂α +

n

j=1

∂φi

∂xj

∂xj

∂α −

m

j=1

∂λj

∂α ψji + λj

∂ψji

∂α

+ λj

ns=1

∂ψji

∂xs

∂xs∂α

= 0, i = 1, . . . , n ,

∂f i∂α

+

ni=1

∂f i∂xi

∂xi

∂α= 0, j = 1, . . . , m ,

where

φi = ∂I/∂xi, ψij = ∂f i/∂xj ,




If only the goal function depend on the parameter α, the system fordetermining the sensitivity coefficients takes the form

∂φi

∂α+

n

j=1

∂φi

∂xj

∂xj

∂α −

m

j=1

∂λj

∂αψji + λj

n

s=1

∂ψji

∂xs

∂xs

∂α = 0,

∂f j∂xi

∂xi∂α

= 0, j = 1, . . . , m, i = 1, . . . , n .

Let only the functions f i(X ) (i = 1, . . . , m) depend on the parameter α.Then,

∂I

∂α=

n

i=1∂I

∂xi

∂xi∂α

, (7.18)

ni=1

∂f j∂xi

∂xi∂α

+∂f j∂α

= 0, j = 1, . . . , m . (7.19)

Multiplying (7.19) by λj and adding (7.18) after summation over all j,we obtain

∂I

∂α= −

m

i=1

λi∂f i∂α

+n

i=1

∂I

∂xi

−m

j=1

λj∂f j∂xi

∂xi

∂α.

Since by (7.16) the term in the brackets is zero in the last equation, wehave

∂I

∂α= −

mi=1

λi∂f i∂α

Thus, in the case when only the constraints (7.15) depend on the parame-ter α, the sensitivity of the optimal value of the goal function is determinedby optimal values of Lagrange coefficients and derivatives of the functionsf i with respect to the parameter α for α = α0. In some problems, forexample in economic ones, the constraints have the form

f i(X ) = bi, i = 1, . . . , m , (7.20)

where bi are resource expenses.

Let the coefficient bs play the role of parameter α. Then, Equation (7.20)yields

∂I

∂bs= λs, s = 1, . . . , m , (7.21)




i.e., the sensitivity coefficient of optimal value of the goal function withrespect to the parameter bs is equal to the corresponding optimal Lagrangemultiplier.

In economic problems the value I is interpreted as income or price. Then,

the sensitivity coefficient, i.e., the Lagrange multiplier λi, characterizes vari-ation of maximal income when the i-th resource increases by 1. Equation(7.21) is a generalization of Equation (7.7) obtained for linear programmingproblem with the help of dual variables yi. Note that Lagrange multipli-ers can also be considered as dual variables that can be the basis of a dualproblem for the problem (7.15). Assume that we are interested in obtainingsensitivity of I with respect to variations of the parameter α on which thecoefficients bs depend:

bs = bs(α), s = 1, . . . , m .

Obviously,

dI

dα=

mi=1

∂I

∂bi

dbidα

,

Then, with account for (7.21) we have

dI

dα =

mi=1 λi

dbi

dα .

In the given case, the sensitivity coefficient of the maximal value of thegoal function is defined as the sum of Lagrange multipliers with weightcoefficients equal to corresponding derivative coefficients of the right sidesof constraints (7.20) with respect to the parameter α.

Example 7.2

Consider the following two-dimensional problem

minx1,x2

I (x1, x2)

for

f (x1, x2) = 0.

For this problem,

L = I (x1, x2) − λf (x1, x2)




and necessary minimum conditions have the form

ψ1 =∂L

∂x1=

∂I

∂x1− λ

∂f

∂x1= 0,

ψ2 = ∂L∂x2

= ∂I ∂x2

− λ ∂f ∂x2

= 0,

ψ3 =∂L

∂λ= −f (x1, x2) = 0.

For sensitivity coefficients we have the following expressions:

∂x1

∂α= −J 1

J ,

∂x2

∂α= −J 2

J ,

∂λ

∂α= −J 3

J ,

where

J =D(ψ1, ψ2, ψ3)

D(x1, x2, λ), J 1 =

D(ψ1, ψ2, ψ3)

D(α, x2, λ),

J 2 =D(ψ1, ψ2, ψ3)

D(x1, α, λ), J 3 =

D(ψ1, ψ2, ψ3)

D(x1, x2, α).

As a numerical example, we consider the problem

min(αx21 + x22)

where

f (x1, x2) = x1 − 5 = 0.

In this problem L = αx21 + x22 − λ(x1 − 5), and the necessary minimumconditions have the form

ψ1 = ∂L∂x1

= 2αx1 − λ = 0,

ψ2 =∂L

∂x2= 2x2 = 0,

ψ3 =∂L

∂λ= −x1 + 5 = 0,

Then, the coordinates of the optimal point are

x1 = 5, x2 = 0, λ = 10α. (7.22)




The Jacobians J , J 1, J 2, and J 3 are given by

J =

2α 0 −10 2 0

−1 0 0

= 2, J 1 =

10 0 −10 2 0

0 0 0

= 0,

J 2 =

2α 10 −10 0 0−1 0 0

= 0, J 3 =

2α 0 100 2 01 0 0

= 20.

As result, we find

∂x1

∂α= 0,

∂x2

∂α= 0,

∂λ

∂α= 10.

Obviously, in this illustrative example the desired sensitivity coefficientscan be found by direct differentiation of the coordinates (7.22) of the pointof minimum which are functions of the parameter α.

7.2.3 Sensitivity Coefficients in Economic Problems

Consider the use of sensitivity coefficients in non-classical demand theory[58] connected with description of consumer behavior in the case of varia-tion of product price under the conditions of competitive market. Let theconsumer be described by a continuous doubly differentiable effectivenessfunction (goal function) I (X ) where x1, . . . , xn are volumes of the corre-sponding products. The price of an exemplar of the i-th product is ci. It isrequired to minimize expenses for which a given effectiveness volume I z canbe reached. Thus, we have the following nonlinear programming problemwith an equality constraint:

minni=1

cixi,

I (X ) = I z.

Lagrange function for this problem has the form

L(X, λ) =n

i=1

cixi

−λ(I (X )

−I z).




Using this formula, we can obtain the following necessary extremum con-ditions:

ci − λ∂I

∂xi

= 0,

I (X ) = I z , i = 1, . . . , n .

(7.23)

Assume that it is required to find the sensitivity of optimal productvolumes with respect to variations of a price, say cn, for unchanged valueof I z. Differentiating (7.23) with respect to cn, we obtain

λn

j=1

∂ 2I

∂xi∂xj

∂xj∂cn

+∂I

∂xi

∂λ

∂cn=

0, if i = n,1, if i = n,

nj=1

∂I ∂xj

∂xj∂cn

= 0.

Dividing the first equations in the last relations by λ, we write

RZ = Γ (7.24)

where

R =

0∂I

∂x1. . .

∂I

∂xn

∂I ∂x1

∂ 2I ∂x2

1

. . . ∂ 2I ∂x1∂xn

. . . . . . . . . . . .

∂I

∂xn

∂ 2I

∂x1∂xn

. . .∂ 2I

∂x2n

,

Z T =

1

λ

∂λ

∂cn

∂x1∂cn

. . .∂xn

∂cn

,

ΓT =

0 0 . . . 0 1λ

.

The system (7.24) makes it possible to determine any sensitivity coef-ficient ∂xi/∂cn (i = 1, . . . , n). It can be easily seen that the coefficient∂xn/∂cn, for instance, can be found by the formula

∂xn

∂cn=

1

λ

Rnn

det R, (7.25)

where Rmm is the adjoint matrix to the element ∂ 2

I/∂x2n.




For a minimum we have the following sufficient condition:

Rnn

det R< 0,

Hence, using the fact that λ > 0, we obtain

∂xn∂cn

< 0.

The negative sign of the sensitivity coefficient ∂xn/∂cn shows that thevolume of production of the n-th product decreases as the price cn increases,provided that the efficiency level I z remains the same.

Now we consider the case when the consumer’s income is fixed. Then,mathematically the problem can be formulated as follows:

maxX

I (X ),

ni=1

cixi = cz,

where cz is a fixed income.

The Lagrange function for this problem has the form

L(X, λ) = I (X )− λ

ni=1

cixi − cz

,

and the necessary conditions are

∂I

∂xi

− λci = 0, i = 1, . . . , n ,

ni=1

cixi − cz = 0 (7.26)

Consider, as in the previous example, the influence of the price fluctu-ations cn onto the optimal consumer’s decision. Differentiation of (7.26)with respect to cn yield the corresponding sensitivity equation

n

i=1

∂ 2I

∂xi∂xj

∂xj

∂cn −ci

∂λ

∂cn= 0, if i = n,

λ, if i = n,




ni=1

ci∂xi∂cn

+ xn = 0.

After routine transformations we can write

RZ 1 = P, (7.27)

where

P T = [−λxn 0 . . . 0 λ]; Z T 1 =

− 1

λ

∂λ

∂cn

∂x1∂cn

. . .∂xn∂cn

.

From Equation (7.27) we obtain

∂xi

∂cn=

1

det R(−λxnR1,i+1 + λRn,i+1), (7.28)

where R1,i+1 and Rn,i+1 are the cofactors of the elements (1, i + 1) and(n.i + 1) of the matrix R, respectively.

Then, we investigate the influence of the fluctuations of cz onto the op-timal consumer’s decision (the values of the variables xi). With this aimin view, let us differentiate the necessary conditions (7.26) with respect tothe parameter cz , obtaining the following sensitivity equations:

nj=1

∂ 2I

∂xi∂xj

∂xj∂cz

− ci∂λ

∂cz= 0 i = 1, . . . , n ,

ni=1

ci∂xi

∂cz= 1,

or, in the vector-matrix form,

RZ 2 = B, (7.29)

where

BT = [λ 0 . . . 0].

Therefore,

∂xi

∂cz= λ

R1,i+1

det R. (7.30)




Comparing (7.28) and (7.30), we obtain

∂xi

∂cn= −xn

∂xi

∂cz+ λ

Rn,1+i

det R. (7.31)

This equation is called in economics Slutskii equation [58] and determinesthe influence of non-compensated price fluctuation onto demand for eachproduct. If i = n, the second term in the right side of (7.31) is proportionalto the sensitivity coefficient ∂xn/∂cn obtained in the previous example (see(7.25)). Therefore, for i = n we can write

∂xi∂cn

c=cz

= −xn

∂xn∂cz

cn=cnz

+ k

∂xn

∂cn

I =I z

. (7.32)

The indexes c = cz , cn = cnz and I = I z show under what condition (fixedincome, price, efficiency level) the corresponding sensitivity coefficients arecalculated.

7.2.4 Nonlinear Programming with Weak EqualityConstraints

A typical nonlinear programming problem with weak equality constraintscan be formulated as follows: find extremum (maximum or minimum) of a

goal function I (X ) under the constraints

F (X ) ≤ 0. (7.33)

Then, depending on the specific statement of the problem, some or allcomponents of the vector X may satisfy the non-negativity condition. It isgenerally known that nonlinear programming problems posed in this wayhave no general universal solution algorithm (like the simplex method forlinear programming problems). There are only partial algorithms, deter-

mined by the form of the goal function and constraints, for the simplesttypes of nonlinear programming problems. Further, we consider approachesto sensitivity investigation for some of these problems.

A. Minimization or maximization of a goal function under the constraints

F (X ) = 0, X ≥ 0.

It is assumed that the functions I (X ) and F (X ) are doubly differen-tiable. Assume that the goal function reaches extremum at the point X 0

inside of the boundary of the area of admissible solutions. If the solution is




determined by an internal point, it should satisfy the necessary conditionsobtained with the help of the Lagrange function [116]:

L(X, λ) = I (X ) − ΛT F (X ).

Therefore, at the first stage, all stationary points of the nonnegativeoctant must be found and analyzed. Then, we investigate the boundary of this octant. With this aim in view, we first equate one of the variables tozero and solve the problem with remaining n − 1 variables. Obviously, itis necessary to solve n such problems. Then, we equate each two variablesto zero and solve the problem with remaining n − 1 variables. There willbe n!/(2!(n − 2)!) such problems. Using this technique, the number of variables is reduced to m. The extremum is determined by considering thewhole aggregate of solutions obtained in this very cumbersome procedure.

As a result, the optimal solution is determined by a quite specific model. Itis assumed that the structure of this model remains unchanged under smallfluctuations of the parameters of interest. Then, the sensitivity equationscan be obtained directly from this model on the basis of the results obtainedabove.

B. Minimization or maximization of a goal function under the constraints

f i(X ) ≤ bi, i = 1, . . . , m1,

f j(X )

≥bj , j = m1, . . . , (m1 + m2),

f s(X ) = cs, s = (m1 + m2), . . . , m .

This problem can easily be reduced to the previous one by adding m1+m2

nonnegative variables xqi:

f i(X ) + xqi = bi, i = 1, . . . , m1,

f j(X ) − xqj = bj , j = m1, . . . , (m1 + m2),

f s(X ) = cs, s = (m1 + m2), . . . , m ,

xqi ≥ 0, i = 1, . . . , (m1 + m2).

To find an extremum inside the nonnegative octant of the space of vari-ables xqi the following Lagrange function can be used:

L(X, X q, Λ) = I (X ) −m1i=1

λi[f i(X ) + xqi − bi]

−

m1+m2

j=m1

λi[f j(X )

−xqj

−bj ]

−

m

s=m1+m2

λs[f s(X )

−bs].

(7.34)




Necessary extremum conditions obtained with the use of this functioncontain equalities of the form

∂L

∂xqi=

−λi = 0, i = 1, . . . , m1,

∂L

∂xqj

= −λj = 0, j = m1, . . . , (m1 + m2).(7.35)

Hence, if xqi ≥ 0 at the extremal point, the corresponding constraintsin the form of equalities may be ignored. This is especially importantfor investigating sensitivity of the optimal solution of the initial problem.Thus, under the conditions (7.35), the algorithm of sensitivity investiga-tion is completely reduced to the case of extremal problem with equalityconstraints. If some of xqi (i − 1, . . . , (m1 + m2) are zero at the point of

extremum, the corresponding Lagrange multipliers may differ from zero. Inthe general case, sensitivity equations are constructed for the model asso-ciated with the optimal solution. Moreover, it is assumed that with smallfluctuations of the parameters in question, the structure of the model and,correspondingly, the optimal solution, remain unchanged.

7.2.5 Sensitivity of Convex Programming Problems

In the most general form, the problem of sensitivity analysis for convexprogramming problems is solved by the theorem on marginal values [33, 34].

DEFINITION 7.1 The rates of variation of the goal function and the value of matrix game as a function of parameter fluctuation are called marginal values in linear programming and game theory.

In fact, the marginal values are the corresponding sensitivity coefficients.It seems that the first investigation of marginal values in matrix gamesand linear programming problems were performed in [64]. Further, thoseresults were corrected and generalized onto convex programming problems

in [33, 34].Consider the following convex programming problem:

maxX

I (X, α), (7.36)

f i(X, α) ≥ 0, i = 1, . . . , m , (7.37)

X ∈ DX , α = α0 + ∆α, (7.38)

depending on a parameter ∆α ∈ [0, ].




Let I 0(∆α) be the maximal value of the goal function under the condi-tions (7.37) and (7.38). Introduce the Lagrange function of the above directproblem

L(X, Λ, ∆α) = I (X, ∆α) −mi=1

λif i(X, ∆α) (7.39)

and formulate the dual problem as

minΛ∈Dλ

maxX∈DX

L(X, Λ, ∆α). (7.40)

Then, the theorem on marginal values of the initial problem can be for-mulated as follows [34].

THEOREM 7.2

Let direct and dual problems be solvable for ∆α = 0, and the sets of their solutions, i.e. the domains of DX(0) and DΛ(0), are bounded. Assume alsothat the functions

I (X, ∆α) = f 0, f i(X, ∆α), i = 1, . . . , m ,

are differentiable at the point ∆α = 0 for any X in a locality DX of the

direct problem solution domain DX(0). Moreover, let the derivatives be

such that

f i(X, ∆α) − f i(X, 0)

∆α→ ∂f i(X, 0)

∂ ∆αfor∆α → +0

uniformly with respect to X ∈ DX . Then, the function I 0(∆α) has the right

derivative at the point ∆α = 0 such that

∂I 0∂ ∆α+

= maxX∈DX

minΛ∈DΛ

∂L(X, Λ, ∆α)

∂ ∆α= min

Λ∈DΛ

maxX∈DX

∂L(X, Λ, ∆α)

∂ ∆α,

where

∂L(X, Λ, α)

∂α

0

=

∂I (X, α)

∂α

α=α0

−mi=1

λi

∂f i(X, α)

∂α

α=α0

. (7.41)

Obviously, if only the goal function depends on the parameter ∆α, wehave

∂I 0

∂ ∆α+

= maxX∂I (X, α)

∂αα=α0

. (7.42)




If ∆α is affected only the functions f i (i = 1, . . . , m), then

∂I 0∂ ∆α+

= maxX

minΛ

−

m

i=1

λi∂f i∂α

α=α0

. (7.43)

Finally, if only f j depends on ∆α, we obtain

∂I 0∂ ∆α+

= maxX

minλj

−λj

∂f j∂α

α=α0

. (7.44)

In the case when the sets DX and DY contain the only optimal solution,we have, correspondingly,

∂I 0∂ ∆α+

=

∂I (X, α)

∂α

α=α0

,∂I 0

∂ ∆α+=

mi=1

λi∂f i∂α

α=α0

,

∂I 0∂ ∆α+

= −λj∂f j∂α

α=α0

.

The formulas for left and right derivatives can easily be found apply-ing the procedure of calculating derivative by direction [39, 43] to convex

programming problems.

THEOREM 7.3

Consider the function

φ(X ) = maxY

f (X, Y ).

Assume that the function f (X, Y ) is continuous with respect to its vari-

ables together with ∂f (X, Y )/∂X . Then, the function φ(X ) has the deriva-tive ∂φ/∂S at any point X by any direction S determined by the corre-sponding unit vector S . Moreover,

∂φ(X )

∂S = max

Y

ni=1

∂f

∂xi

si.

It is noteworthy that Theorem 7.3 covers practically all mathematicalprogramming problems considered above. As a special case, from this the-

orem we can deduce theorem on marginal values and the corresponding




formulas for marginal values in linear programming problems. To obtainthis result, it suffices to substitute the Lagrange function of a linear pro-gramming problem into the conditions of Theorem 7.3:

L(X, Y, ∆α) =ni=1

ci(∆α)xi +mj=1

bj(∆α)yj −mi=1

nj=1

aij(∆α)xjyi.

If the derivatives on the left and on the right of the point ∆α = 0 coin-cide, Theorem 7.3 makes it possible to obtain basic relations for sensitivitycoefficients of the form (7.6) and (7.7), and so on.

7.3 Sensitivity of Simplest Variational Problems

7.3.1 Simplest Variational Problems

Let F (t,y, y) be a function having continuous partial derivatives with re-spect to all arguments up to the second-order inclusive. It is required to finda function y(t) having continuous derivative and satisfying the conditions

y(t0) = a, y(t1) = b, (7.45)

such that it ensures a weak extremum of the functional

I (y) =

t1 t0

f (t,y, y) dt (7.46)

It is known that a solution to this problem must satisfy the followingEuler equation

F y − d

dtF y = 0 (7.47)

with the boundary conditions (7.45).

DEFINITION 7.2 Integral curves of the Euler equation are called extremals.

Assume that the functional (7.46) and boundary conditions (7.45) depend




on a parameter α ∈ [α1, α2]:

I (y, α) =

t1(α)

t0(α)

F (t,y, y, α) dt. (7.48)

It is also assumed that all requirements of existence of solution hold forthe simplest variational problem with the functional (7.46) for all values of α ∈ [α1, α2]. Then, we may introduce a single-parameter family of solutionsof the variational problems

y(t, α) = y[t, a(α), b(α), t0(α), t1(α), α]. (7.49)

In the present section we consider techniques for constructing sensitivityequations for determination of the derivative

u(t)∂y(t)

∂α,

called sensitivity function of the simplest variational problem .

7.3.2 Existence Conditions for Sensitivity Function

To derive these conditions, we employ the general approach to sensitivityinvestigation for boundary-value problems developed in Section 2.6.

The Euler equation (7.47) can be written in the form

F yyy + yF yy + F yt − F y = 0 (7.50)

or in the following vector form

Y = Φ(Y,t,α), (7.51)

where

Φ =

y2,

F y − F yt − F yyy2F yy

, y = y1, y = y2, Y =

y1,y2

,

assuming that F yy = 0 for t ∈ [t1, t2] and α = α0.




Assume that for α = α0 there is a solution of Equations (7.50) and(7.51) with the corresponding boundary conditions. Let the vector functionΦ(y, y,t ,α) be continuously differentiable with respect to all arguments.Assume that the functions t0(α) and t1(α) are differentiable with respect

to α. LetY (t) = Y (t, t0, Y 0, α) (7.52)

be a general solution of Equation (7.51) such that

Y (t0, t0, Y 0, α) = Y 0. (7.53)

Moreover,

Y (t1) = Y (t1, t0, Y 0, α). (7.54)

Then, the boundary conditions (7.45) can be reduced to the form

g1(Y 0, t0, α) = y0 − a = 0,

g2(Y 0, t1, α) = y(t1, Y 0, α)− b = 0.(7.55)

Consider the Jacobian

J =D(g1, g2)

D(y0, y0)

= 1 0

∂y(t1

)

∂y0

∂y(t1

)

∂ y0 =

∂y(t1)

∂ y0,

i.e., the Jacobian J is equal to the value of the sensitivity function of thevariable y(t) with respect to initial condition y0 at the moment t = t1. Ac-cording to Theorem 2.17 formulated in Section 2.6, if J = 0, the vector Y 0(α)is continuous and continuously differentiable with respect to the parameterα. And this is sufficient for existence and continuity of the sensitivityfunction ∂y(t)/∂α on the interval [t0, t1]. Denote

z(t) = ∂y(t)∂ y

.

Differentiating Equation (7.47) with respect to the parameter y0, we obtain

F yyz + F yy z − d

dt(F yyz + F yy z) = 0. (7.56)

Obviously, initial conditions for this equation have the form

z(t0) = 0, z(t1) = 1. (7.57)




Linear homogeneous equation of the second order (7.56) coincides withthe Jacoby equation used for investigating sufficient conditions of extremumin calculus of variations. As a special case, if there exists a solution of theJacoby equation that becomes zero for t = t0 and is nonzero in all other

points of the interval (t0, t1), then the Jacoby condition holds, and the arcof the extremal ab can be included in the central field of extremals. Asdistinct from Jacoby condition, the condition z(t1) = 0 is weaker. Forexistence and continuity of the sensitivity function u(t) it is sufficient thatthe sensitivity function z(t) be nonzero at t = t1.


Assume that all the conditions of existence and continuity of sensitivityfunction considered in the previous paragraph hold. Then, the problem

of constructing the sensitivity function for the above variational problemcan be reduced to a boundary-value problem for the sensitivity equationobtained by direct differentiation of Euler equation with respect to theparameter α in the form (7.47) or (7.50). Differentiating (7.47), we obtainthe equation

d

dt(F yyu + F yyu + F yα) − F yy u− F yyu = F yα. (7.58)

Equation (7.58) is a linear non-homogeneous differential equation of thesecond order. Its form coincides with that of the non-homogeneous Jacobyequation. Hereinafter we shall call this relation Euler sensitivity equation .

Equation (7.58) can be rewritten in an explicit form with respect to thesensitivity function u(t):

F yyu +d

dtF yyu +

d

dtF yy − F yy

u− F αy +

d

dtF yα = 0. (7.59)

Represent the boundary conditions for the initial problem in the form

G[Y (t0, α), Y (t1, α), α] = 0, (7.60)

where

g1 = y(t0, α) − a, g2 = y(t1, α) − b, GT = (g1, g2).

Differentiating (7.60) with respect to α, we obtain

∂G

∂Y 0

dY 0

dα+

∂G

∂Y 1

dY 1

dα+

∂G

∂α= 0




or,

dy0dα

=da

dα,

dy1dα

=db

dα,

Butdy(t0)

dα= u(t0) + y0

dt0dα

,dy(t1)

dα= u(t1) + y1

dt1dα

.

Then, we obtain the following boundary conditions for Equation (7.59):

u(t0) =da

dα− y0

dt0dα

, u(t1) =db

dα− y1

dt1dα

. (7.61)

The above approach can be extended onto generalizations of the simplestvariational problem. Thus, if the functional I depends on several function

y1, . . . , yn, we have a system of differential sensitivity equations

nj=1

F yj yj uj +n

j=1

d

dtF yj yj uj +

nj=1

Φiyjuj + Φiα = 0,

i = 1, . . . , n ,

(7.62)

where

ui =dyidα

; Φi =d

dtF yi − F yi .

From Equation (7.58) we can easily derive sensitivity equations for thefollowing typical simplest cases of the Euler equation.

A. Function F is independent of y. The Euler equation has the form

F y(t, y, α) = 0,

The curve described by this equation only passes through the boundarypoints (t0, a) and (t1, b) in exceptional cases. For this equation, the sensi-tivity equation is given by

F yyu + F yα = 0.

B. Function F is independent of y. The Euler equation has the form

d

dtF y = 0,

which is associated with the following sensitivity equation:

F yyu +

d

dt F yyu −d

dt F yα = 0.




C. Function F depends only on y. Euler equation

F yyy = 0

corresponds to the sensitivity equation

F yyu +∂

∂αF yy = 0.

Example 7.3

Given the functional

I (y) =

t1

t0

(y2 + τ 2y2) dt.

construct the sensitivity equation with respect to the function u(t) =dy(t)/dt.

For the given functional we have

F = y2 + τ 2y2, F yy = 2τ 2,d

dtF yy = 0, Φ = 2τ 2y − 2y,

Φy =

−2, Φα = Φτ = 4τ y.

According to (7.59), the sensitivity equation takes the form

τ 2u − u = −2τ y. (7.63)

Assume that the boundary conditions of the initial variational problemare independent of the parameter τ . Then, the boundary conditions of thesensitivity equation (7.63) are zero.

Example 7.4Find a minimum of the functional

I (y) =

2 1

1 + y2

tdt, y1 = y(1) = 0, y2 = y(2) = 1.

A solution of this problem is given by the equation

(y−

c1)2 + t2 =1

c22, c1 = 2, c2 =

1

√ 5,




i.e., (y − 2)2 + t2 = 5.

Let the left boundary condition depend on the parameter α so that

y1(α) = α.

To determine c1 and c2 with such a problem statement, we construct thefollowing system of equations:

(α − c1)2 + 1 =1

c22, (1− c1)2 + 4 =

1

c22,

Hence,

c1 =

4

−α2

2(1− α) , c2 =

1

−α

√ 5− 6α .

and we obtain the solution

y(t, α) = −

5− 6α

(1− α)2− t2 +

2

1− α.

The sensitivity function is determined by direct differentiation of thesolution with respect to α:

u(t) =dy

dα

α=0

= − 2√ 5 − t2

+ 2.

Then, we use the sensitivity equation which has, for this example, theform

d

dt(F yyu) = 0,

or

F yyu = c1

with the boundary conditions

u(t0 = 1) = 1, u(t1 = 2) = 0.

Since

F yy =(5− t2)

32

532 t

,




we have

du = −ct

5

5− t2

32

dt,

Hence,u(t) = −5

√ 5

c1√ 5− t2

+ c2.

Using the boundary conditions, we find

c1 =2

5√

5, c2 = 2

and

u(t) = −2

√ 5 − t2 + 2.

7.4 Sensitivity of Variational Problems

7.4.1 Variational Problem with Movable Bounds

The simplest problem of this kind can be formulated as follows.

Variational problem with movable bounds. Given a point(t0, y0) and a curve y = H (t) find a curve passing through thepoint (t0, y0) and intersecting the curve H (t) with a nonzero anglesuch that the curve ensures a weak extremum to the functional

I (y)

t1 t0

F (t,y, y) dt,

The integral is taken along this curve from the point (t0, y0) tothe point of its intersection with the curve H (t).

It is known that a solution of this problem satisfies the Euler equationand the conditions

y(t0) = a, y(t1) = H (t1) (7.64)




and the transversality condition

F + F y (H t − y)|t=t1 = 0. (7.65)

7.4.2 Existence Conditions for Sensitivity Functions

Let all the conditions set for the simplest variational problem hold. More-over, we assume that the function H (t, α) has continuous second derivativewith respect to t and mixed derivative with respect to t and α. To find ex-istence conditions, we employ the approach developed in Section 2.6. Withthis aim in view, rewrite Equations (7.64) and (7.65) in the form

g1 (y0, t0, α) = 0,

g2 (y0, y0, t0, t1, α) = 0,

f (y0, y0, t0, t1, α) = 0,

where

g1 = y0 − a, g2 = y(t1, t2, y0, y0)− H (t1),

f = F [y(t1, y0, y0, α), y(t1, y0, y0, α), t1, α]

+ F y[y(t1, y0, y0, α), y(t1, y0, y0, α), t1, α](H t1 − y(t1, y0, y0, α)).


J =D(g1, g2, f )

D(y0, y0, t1)=

∂g2∂ y0

∂g2∂t1

∂f

∂ y0

∂f

∂t1

.

If J = 0, the values of y0, y0, and t1 are continuous with respect to theparameter α. As was shown above, this is enough for existence of continuoussensitivity function dy(t)/dα.


It can be shown that under the conditions of existence of sensitivityfunctions the latter are solutions of the Lyapunov equation (7.58) underthe conditions

u(t0) = −y(t0)dt0dα

+dy0dα

, (7.66)

(F t + H tF y + F yH tt)dt1dα

+ (H t − y) (F yyu

+F yyu + F yα) + F yu + F yH tα + F α|t=t1 = 0,

(7.67)




wheredt1dα

= −u(t1) − H α(t1)

y(t1) − H t(t1). (7.68)

Indeed, the differential sensitivity equation can be obtained using thetechnique presented in Section 7.3, provided that the conditions of existenceand continuity of the sensitivity function hold. The boundary condition(7.66) was also derived above. To obtain the conditions (7.67) and (7.68),we differentiate Equations (7.64) and (7.65) with respect to α. This is justified due to the aforesaid conditions. After differentiation we have

F tdt1dα

+ F ydy1dα

+ F ydy1dα

+ F α

+ (H t − y)

F yα + F yy

dy1

dα + F yy

dy1

dα + F yt

dt1

dα

+ F y

H tt

dt1dα

+ H tα − dy1dα

t=t1

= 0,

(7.69)

dy1dα

= H tdt1dα

+ H α|t=t1 , (7.70)

where y1 = y(t1) and y1 = y(t1).

Then, we will find the values dy1/dα and dy1/dα. We have

y(t) =

t t0(α)

y dt + y0,

y(t) =

t t0(α)

φ(t) dt + y0,

(7.71)

where

φ(t) =F y

−F ty

−F yy y

F yy , F yy = 0.


u(t) = −y(t0)dt0dα

+dy0dα

+

t t0

dy

dαdt,

u(t) = −φ(t0)dt0dα

+dy0dα

+

t

t0

dφ

dαdt,




Hence, for t = t1 we have

u(t1) =

−y(t0)

dt0

dα

+dy0

dα

+

t1

t0

dy

dα

dt,

u(t) = −φ(t0)dt0dα

+dy0dα

+

t1 t0

dφ

dαdt,

(7.72)

Differentiating Equation (7.71) with respect to α for t = t1, we obtain

dy1

dα= y(t1)

dt1

dα −y(t0)

dt0

dα+

t1

t0

dy

dαdt +

dy0

dα,

dy1dα

= φ(t1)dt1dα

− φ(t0)dt0dα

+

t1 t0

dφ

dαdt +

dy0dα

.

(7.73)

Relations (7.72) and (7.73) yield

dy1

dα

= y(t1)dt1

dα

+ u(t1),

dy1dα

= φ(t1)dt1dα

+ u(t1).(7.74)

Substituting (7.74) into (7.69) and (7.70), after simple transformations wefind the conditions (7.67) and (7.68). Note that Equation (7.68) coincideswith Formula (3.77) for the derivative of the switching moment with respectto the parameter in sensitivity equations of discontinuous systems.

7.4.4 Case Study

Consider Example 7.3 for the case when y(t0) = 1 and the right end of the extremal can move along the curve y(t) = t + α. First, we determinethe sensitivity function by direct differentiation of the solution y(t, α) of the variational problem with respect to the parameter α. With this aim inview, we find y(t, α), using the problem solution in the form

y(t) = c1 + 1c2

2 −t2. (7.75)




The transversality condition has the form

1 + y1

t1 1 + y21= 0.

From (7.75) we find

y = −t

1

c22− t2

−12

.

Then, the transversality condition can be written as

1− c22t21 = c2t1. (7.76)

At the moment t0 = 0 we have

c22(1− c1)2 = 1. (7.77)

Moreover, from the condition H (t1) = y(t1) we obtain

c1 +

1

c22− t21 = t1 + α. (7.78)

Thus, for determination of the three variables c2, c1, and t1 we may usethe three algebraic equations (7.76), (7.77) and (7.78). As a result, we find

c1 = α, c2 = (1− α)−1, t1 =1√

2(1− α).

Therefore,

y(t, α) = α

− (1

−α)2

−t2.

Then, the sensitivity function appears as

u(t) = 1 − 1√ 1 − t2

. (7.79)

Next we use the sensitivity model (7.58), (7.66), (7.67), and (7.68) to findu(t). As was shown in Section 7.3, the sensitivity equation has the form

F yyu = c,




where

F yy = t−1(1 + y2)−32 .

or, with due account for the expression for y at α = 0,

F yy = t−1(1− t2)32 .

Then,

u =ct

(1− t2)32

,

Hence,

u(t) =c2

√ 1− t2

+ c3.

To find the arbitrary constants c1 and c3, we use the conditions (7.66)and (7.67). From the first equation it follows that

u(0) = c2 + c3 = 0

or,

c2 = −c3. (7.80)

To construct the second equation, we find

F t = −t−2

1 + y2, H t = 1, F y = 0, H tt = 0,

F yy = t−1(1 + y2)−32 , F yy = 0, F yα = 0, H tα = 0, F α = 0.

Then,

−

1 + y2

t2dt1dα

+ (1 − y)1

t(1 + y2)32

u|t=t1 = 0, (7.81)

wheredt1dα

= −u(t1) − 1

y(t1) − 1,

For

α = α0 t1 =1√

2,

we have

y(t1) = −1,

dt1

dα =

u(t1)

−1

2 .




Moreover,

u(t1) = c2(√

2− 1),

u(t1) = 2c2.

Using the last relations, from (7.81) we obtain

c2 = −1,

Hence,

c3 = 1.

Thus,

u(t) = − 1√ 1

−t2

+ 1.

7.4.5 Variational Problem with Corner Points

It is required to find, among all continuous functions y(t) satisfying theboundary conditions

y(t0) = y0, y(t1) = y1, (7.82)

a function that ensures a weak extremum of the functional

I (y) =

t1 t0

F (t, y, y) dt.

It is assumed that admissible curves y(t) may have a break at a pointwith abscissa t∗ such that t0 < t∗ < t1. Obviously, the function F yy(t) maybecome zero for the breakpoint. It is known that each of the two arcs of the broken line satisfies Euler equations. For each arc y1(t) or y2(t) we canwrite

y1(t) = y1(t, c1, c2), y2(t) = y2(t, t∗, c3, c4).

To determine the constants c1, c2. c3, c4, and t∗ it is necessary to haveyet another three relations in addition to the boundary conditions. At thebreakpoint the following Erdman-Weierstrass equation holds

(F − yF y)|t=t∗−0 = (F − yF y)|

t=t∗+0

F y|t=t∗−0 = F y |t=t∗+0(7.83)

The continuity condition for the extremal has the form

y(t∗

− 0) = y(t∗

+ 0). (7.84)




Let the previous problem be parametric so that

F = F (t,y, y, α), y0 = y0(α), y1 = y1(α),

t0 = t0(α), t1 = t1(α),

and let all these functions be continuously differentiable with respect to α.To obtain existence conditions for the sensitivity function, we consider therelations

f 1(y20, y20, t∗, t1) = 0, f 2(y10, y10, t0, y20, y20, t∗) = 0,

f 3(y10, y10, t0, y20, y20, t∗) = 0, f 4(y10, y10, t0, y20, y20, t∗) = 0,

where

f 1 = y(t1) − b,

f 2 = (F − yF y)|t=t∗−0 − (F − yF y)|

t=t∗+0 ,

f 3 = F y|t∗−0 − F y|t∗+0 , f 4 = y1(t∗ − 0) − y20.


J =D(f 1, f 2, f 3, f 4)

D(y10, y20, y20, t∗).

If this Jacobian is nonzero for α = α0, the values y10, y20, and t∗ arecontinuously differentiable functions of the parameter α ina locality of thepoint α = α0. The latter condition is sufficient for existence of the sensi-tivity functions

u1(t) =dy1(t)

dα, u2(t) =

dy2(t)

dα,

To find sensitivity models, we consider two cases.First, assume that F yy = 0 at the breakpoint. Construct a discontinuous

system

dy1dt

= y2,dy2dt

=F y − F yt − F yyy2

F yy= φ (y1, y2, t) . (7.85)

or, in a vector form,

dY

dt = Φ(˜Y , t). (7.86)




Obviously, at the point t∗ the function y2(t) has a break of the first kindso that

∆y2(t∗) = y2(t∗ + 0) − y2(t∗ − 0) = y+2 − y−2 .

Assume that there are left and right derivatives of the function at thispoint, which are defined by the difference

∆φ(t∗) = φ+ − φ−,

where

φ+ = φ(t∗ + 0), φ− = φ(t∗ − 0).

The switching moment t∗ is found from the condition

g(y, y, t∗, α) = F yy(y, y, t∗, α) = 0. (7.87)

The continuity condition for the extremal has the form

y(t∗ + 0) = y(t∗ − 0). (7.88)

Then, according to Theorem 3.1 and the results of Section 7.3, we obtainthe following sensitivity model of the above variational problem with acorner point.

The segments of the extremal on the left and on the right of the point t∗

are denoted by y1(t) and y2(t), respectively, and the corresponding sensi-tivity functions by u1(t) and u2(t). these sensitivity functions are solutionsof equations of the form (7.58) for t0 ≤ t ≤ t∗ and t∗ ≤ t ≤ t1 with theboundary conditions

u1(t0) = −y(t0)dt0dα

+dy0dα

,

u2(t1) =−

y(t1)dt0

dα+

dy1

dα.

(7.89)

The break condition for the sensitivity function is given by

∆u = −∆y2(t∗)dt∗

dα, (7.90)

dt∗

dα= −

F −yyY

T U − + F −yyα

F

−

yyY T

Φ−

+ F

−

yyt

= −

F +yyY

T U + + F +yyα

F

+

yyY T

Φ+

+ F

+

yyt

. (7.91)




The solutions u1(t) = u1(t, c1, c2) and u2(t) = u2(t, c3, c4) depend onfour unknown constants. To find them, we can use the four conditions(7.89)–(7.91).

Then, let us consider the second case, when the function F yy(t) is nonzero

at the breakpoint. Then, the sensitivity functions u1(t) and u2(t) satisfyequations of the form (7.58) on the intervals t0 ≤ t ≤ t∗ and t∗ ≤ t ≤ t1with the boundary condition (7.89). Additional relations for determiningthe constants c1, c2, c3, and c4 can be obtained by differentiating withrespect to the parameter α Erdman-Weierstrass equation (7.83) and thecontinuity condition (7.84).

After differentiation of these conditions we obtain

F yydy

dα+ F yy

dy

dα+ F yt

dt∗

dα+ F yα|t

∗+0t∗−0 = 0,

F α + F ydydα

+ F tdt∗

dα− y

F yy

dydα

+ F yydy

dα+ F yt

dt∗

dα+ F yα

t∗+0

t∗−0

= 0,

dy1dα

t∗−0

=dy2dα

t∗+0

.

(7.92)

Represent the solutions y1(t), y1(t), y2(t), and y2(t) in the form

y1(t) =

t t0

y1(t) dt + y(t0),

y1(t) =

t t0

φ1(t) dt + y1(t0),

y2(t) =

t t∗+0

y2(t) dt + y2(t∗ + 0),

y2(t) =

t t∗+0

φ2(t) dt + y2(t∗ + 0),

Hence,

dy1(t∗ − 0)

dα= y1(t∗ − 0)

dt∗

dα+ u(t∗ − 0), (7.93)

dy1(t∗

−0)

dα = φ1(t

∗

− 0)

dt∗

dα + u(t

∗

− 0), (7.94)




dy2(t∗ + 0)

dα= y2(t∗ + 0)

dt∗

dα+ u(t∗ + 0), (7.95)

dy2(t∗ + 0)

dα = φ2(t

∗

+ 0)

dt∗

dα + u(t

∗

+ 0). (7.96)

Substituting (7.93)–(7.96) into (7.92) and using the fact that

φ(t) =F y − F yt − F yy y

F yy,

we finally obtain

F yyu + F yyu + F ydt∗

dα+ F yα|t

∗+0t∗−0 = 0,

F yu + F tdt∗

dα+ F α − y (F yyu + F yyu + F yα)|t∗+0

t∗−0 = 0,

ydt∗

dα+ u

t∗+0

t∗−0

= 0.

(7.97)

If F yy(t) = 0 for t ∈ (t0, t1), the derivative dt∗/dα is also unknown. It isdetermined together with the coefficients c1, c2, c3, and c4 from the system

of five equations (7.89) and (7.97).

Example 7.5

Find the sensitivity function of the solution of the following variationalproblem with respect to a parameter α:

I (y) =

2

0

(y4 − 6y2) dt,

y(0) = α, y(2) = 0.

For this problem, extremals are given by

y1(t) = c1t + c2, 0 ≤ t < t∗,

y2(t) = c3t + c4, t∗

≤t < 2,

(7.98)




With account for the boundary conditions, they take the form

y1(t) = c1t + α,

y2

(t) = c3

(t−

2).(7.99)

The Erdman-Weierstrass condition and continuity conditions have theform

4y3 − 12y|t∗+0t∗−0 = 0,

−3y4 + 6y2t∗+0t∗−0

= 0,

y1(t∗ − 0) = y2(t∗ + 0)

or, with account for (7.99),

c33 − 3c3 − c31 + 3c1 = 0,

−c43 + 2c23 + c41 − 2c21 = 0,

c1t∗ + α = c3(t∗ − 2),

Hence

c1

=√

3, c3

=−√

3, c2

= α, c4

=−

2c3

= 2√

3,

t∗ = 1α

2√

3.

As a result, we have

y1(t) =√

3t + α, y2(t) = −√

3t + 2√

3.

After all, direct differentiation yields the following sensitivity function

u(t) =

1 for 0 ≤ t < t∗ = 1,0 for t∗ ≤ t ≤ 2.

The sensitivity function has a break at the point t∗:

∆u(t∗) = −1.

Let us employ the sensitivity model including differential sensitivity equa-

tions and the conditions (7.89) and (7.97). For the given problem, the




sensitivity equation has the form

F yyu = c,

Hence, with account for the fact that F yy = const, we have

u(t) = c1t + c2.

Let us express

u1(t) = c1t + c2, 0 ≤ t < t∗,

u2(t) = c3t + c4, t∗ ≤ t ≤ 2.

From the conditions (7.89) we have

u1(0) = 1, u2(2) = 0,

Equation (7.97) yields

[y22(t∗ + 0)− 1]u2(t∗ + 0) = [y21(t∗ − 0)− 1]u1(t∗ − 0),

y2(t∗ + 0)[y22(t∗ + 0) − 1]u2(t∗ + 0)

= y1(t∗ − 0)[y21(t∗ − 0) − 1]u1(t∗ − 0),

y2(t∗ + 0)dt∗

dα+ u2(t∗ + 0) = y21(t∗ − 0)

dt∗

dα+ u1(t∗ − 0).

As a result, we find

c1 = c3 = c4 = 0, c2 = 1,

Hence,

u1(t) = 1, u2(t) = 0.




7.5 Sensitivity of Conditional Extremum Problems

7.5.1 Variational Problems on Conditional Extremum

In these problems it is required to find an extremum of a functional I under some constraints imposed on the functions on which the functionaldepends. Usually bonds of the following three types are considered:

1) f i(t, y1, . . . , yn) = 0, i = 1, . . . , m < n, (7.100)

2) f i(t, y1, . . . , yn, y1, . . . , yn) = 0, (7.101)

3)

t1 t0

Gi(t, y1, . . . , yn, y1, . . . , yn) dt = ci. (7.102)

Bonds of the first type yield a Lagrange problem, while bonds of the thirdtype lead to an isoperimetric problem.

7.5.2 Lagrange Problem

First, we consider the simplest case when it is required to find an ex-tremum of the functional

I =

t1 t0

F (t, y1, y2, y1, y2, α) dt (7.103)

over all corresponding curves

y1 = y1(t), y2 = y2(t), (7.104)

belonging to a surface

f (y1, y2, t) = 0. (7.105)

It is known that the solution of the problem is determined, under some




conditions, by the equations

F y1 + λf y1 −d

dtF y1 = 0,

F y2 + λf y2 − ddt

F y2 = 0,

f (y1, y2, t) = 0,

y1(t0) = y10 = a1, y1(t1) = y11 = b1,

y2(t0) = y20 = a2, y2(t1) = y21 = b2,

(7.106)

where λ is a Lagrange multiplier.

As a result, the values y1(t) and y2(t) are functions of the boundary

values y10, y11, y20, y21 and the Lagrange multiplier λ.Let us rewrite the boundary conditions in the form

g1 = y10 − a1 = 0,

g2 = y20 − a2 = 0,

g3 = y1(t1, y10, y10, y20, y20, λ) − b1 = 0,

g4 = y2(t1, y10, y10, y20, y20, λ) − b2 = 0,

f [y1(t, y10, y10 , y20, y20, λ), y2(t, y10, y10, y20, y20, λ), t] = 0.

(7.107)

Then, the existence condition for sensitivity functions dy1/dα, dy2/dα,and dλ/dα reduces to the requirement that the following Jacobian is nonze-ro:

J =D(g1, g2, g3, g4, f )

D(y10, y20, y10, y20, λ), (7.108)

Due to (7.107), it has the form

J =

∂g3∂ y10

∂g3∂ y20

∂g3∂λ

∂g4∂ y10

∂g4∂ y20

∂g4∂λ

∂f

∂ y10

∂f

∂ y20

∂f

∂λ

.

In general, when the bonds are described by (7.100), the existence con-




dition for sensitivity functions of the Lagrange problem has the form

D(g1, g2, . . . , g2n, f 1, . . . , f m)

D(y10, y20, . . . , yn0, y10, . . . , yn0, λ1, . . . , λm)= 0. (7.109)

If the existence conditions hold for sensitivity functions, the sensitivityequations can be found by direct differentiation of equations of the form(7.58) with respect to the parameter α.

7.5.3 Variational Problem with Differential Constraints

In this case, the existence conditions for sensitivity functions are similarto the conditions of the previous paragraph. Indeed, assume that it isrequired to find an extremum of the functionsl (7.103) under the conditions

f (y1, y2, y1, y2, t) = 0 (7.110)

and the boundary conditions (7.106).

It is known that the functions y1(t) and y2(t) realizing a conditionalextremum of the functional and the multiplier λ must satisfy the equation

Lyi −d

dtLyi = 0, i = 1, 2,

f (y1, y2, y1, y2, t) = 0,

(7.111)

where L = F + λ(t). Obviously,

yi(t) = yi(y10, y20, y10, y20,λ,t, ) i = 1, 2.

Then, we can construct the following system

g1 = y10 − a1 = 0,

g2 = y20 − a2 = 0,

g3 = y1(y10, y20, y10, y20, λ(t1), t1) − b1 = 0,

g4 = y2(y10, y20, y10, y20, λ(t1), t1) − b2 = 0,

f ( y1 , y2, y1, y2, t) = 0,

(7.112)

hence, we obtain a condition of the form (7.109) for m = 1 and n = 2. If theexistence conditions for sensitivity function hold, the sensitivity equationcan be obtained by direct differentiation of Equations (7.58) with respect

to the parameter α.




7.5.4 Sensitivity of Isoperimetric Problem

Consider the simplest isoperimetric problem with the functional

I =

t1 t0

F (y, y,t ,α) dt (7.113)

and constraintt1

t0

D (y, y,t ,α) dt = S. (7.114)

It is known that the isoperimetric problem can easily be reduced to theprevious variational problem with differential constraint by introducing anew variable

z(t) =

t1 t0

D (y, y,t ,α) dt.

and transition to the differential equation

z = D (y, y,t ,α) ,

z(t0) = 0, z(t1) = S. (7.115)

In this case, the existence conditions for the sensitivity functions ∂y/∂αand ∂λ/∂α are checked using the technique developed in the previous para-graph. In the given case the Euler equation takes the form

F y − d

dtF y + λ

Dy − d

dtDy

= 0.

As a result, we obtain

(F yy + λDyy) uα +d

dt(F yy + λDyy) uα

+ (Qy + λH y)uα = −

Qα +∂λ

∂αH

,

where

Q = −F y +

d

dt F y, H = −Dy +

d

dt Dy.




Solution of the sensitivity equation is determined by the relation

uα(t) =∂y(t)

∂α= uα

t, c1, c2,

∂λ

∂α

,

i.e., contains three unknown constants. To find them we must have threeequations. Two of them are derived using the boundary conditions. Thethird one can be obtained as a result of differentiation of the followingisoperimetric condition with respect to the parameter α:

t1 t0

[Dyuα + Dyuα + Dα] dt + D [y(t1), y(t1), t1]dt1dα

− D [y(t0), y(t0), t0]dt0dα

= 0.




Chapter 8

Applied Sensitivity Problems

8.1 Direct and Inverse Problems of Sensitivity Theory

8.1.1 Classification of Basic Applied Sensitivity Problems

The modern stage of development of theoretical and practical methods of control systems design and implementation calls for using methods makingit possible to account for parametric uncertainties (parameter variations).In fact, there is a fairly wide class of problems where the use of sensitivitytheory apparatus is necessary and advantageous. The main problems of this class are the following:

1. precision and stability analysis for parametrically perturbed systems

2. design of insensitive systems

3. identification

4. optimization problems

5. tolerance distribution

6. adjustment, testing, and monitoring of technical systems as well as

their units

Even superficial analysis of the above applied problems of sensitivity theoryshows that sensitivity functions U , additional motion ∆Y and variations of the corresponding parameters ∆α are necessary elements of any problembelonging to this class. The triple of elements (U, ∆Y, α) is used in eachproblem. In most applied problems of sensitivity theory it is required tofind and analyze either additional motion ∆Y or parameters variation ∆α.In both the cases the sensitivity functions are assumed to be known (theyare found using the sensitivity model for the known initial system). In

some problems, estimation of additional motion is combined (alternate)




with obtaining parameters variation.

Using the above approach, most applied problems of sensitivity theorycan be classified into the following three groups:

1. direct problems of sensitivity theory when it is required to investi-gate additional motion ∆Y on the basis of known sensitivity func-tion U and the values (or characteristics) of parameters variation∆α

2. inverse problems of sensitivity theory when parametric influence ∆αis to be estimated using given sensitivity functions U and additionalmotion ∆Y

3. mixed problems that include elements of both direct and inverseproblems

8.1.2 Direct Problems

The solution of direct problems is connected, as a rule, with analysis of additional motion. A general technique of such analysis is given in Sec-tion 1.3, where general ideas about investigation of the first approximationfor additional motion are presented for deterministic and stochastic cases.More detailed information on methods of estimating additional motion of

various classes of dynamic systems by means of sensitivity functions can befound in [67, 94].

8.1.3 Inverse Problems and their Incorrectness

In general, the connection between additional motion ∆Y and parametersvector variation ∆α can be described by the following operator equation

∆Y = Γ∆α, (8.1)

which is a basic equation for solving direct problems of sensitivity theory.Formally, inverse problems are associated with the relation

∆α = Γ−1∆Y, (8.2)

where Γ−1 is the inverse operator for Γ.

The use of Formula (8.2) is difficult, and sometimes even impossible, forapplied problems due to incorrectness of inverse problems. Mathematicalformulation of a correctly posed problem of solving Equation (8.1) can be

given as follows [105].




DEFINITION 8.1 A problem of obtaining the solution ∆α from a space Ωα by initial data ∆Y from a space QY is called correctly posed if the following conditions hold:

1. for any element ∆Y there is a solution ∆α ∈ Ωα

2. the solution is determined uniquely

3. the problem is stable

DEFINITION 8.2 A problem of obtaining a solution ∆α = ψ(∆Y ) is called stable if for any > 0 there is a number ∆() > 0 such that from the inequality

ρY (∆Y 1, ∆Y 2)

≤δ ()

follows

ρα(∆α1, ∆α2) ≤ ,

where

∆α1 = ψ(∆Y 1), ∆α2 = ψ(∆Y 2), ∆Y 1, ∆Y 2 ∈ ΩY , ∆α1,

are corresponding distances in the normed spaces Qα and QY .

In practice, if the stability conditions hold, small errors in initial data∆Y cause small errors in the solution ∆α.

DEFINITION 8.3 Problems for which at least one of the above con-ditions does not hold, are called incorrectly posed.

Using these definitions, let us consider various reasons of incorrectnessof inverse problems in sensitivity theory. For an inverse problem, initialdata are the components of the vector of additional motion ∆Y . In real

conditions they are obtained experimentally and, therefore, with inevitableerrors. Such being the case, for some values of the vector ∆Y Equation (8.1)may not have a solution with respect to ∆α (in the given space Ωα). In thiscase the first condition of problem correctness is violated. If Equation (8.1)is nonlinear with respect to parameters variation ∆α, there can be manyassociated solutions so that the second condition is violated. After all, if asolution exists and is unique, it may be not stable. This manifests itself inthe fact that small errors in initial data (in the vector ∆Y ) cause significanterrors in the solution ∆α. The reason for such “error amplification” is thatthe inverse operator Γ−1 may be discontinuous. Thus, the third correctness

condition is violated.




8.1.4 Solution Methods for Inverse Problems

In practice, much effort of investigators is directed on developing methodsand algorithms for determining approximate solutions for incorrectly posedproblems that are stable with respect to small variations of initial data.

A method of solving incorrect problems called solution by inspection [105]is the most widely used in engineering practice. As applied to the problem(8.1), the method can be described as follows. It is assumed that for anarbitrary element ∆β from an area Ωα of admissible solutions we can findthe value f (∆β ) = Γ∆β , i.e., a direct sensitivity problem is solved. Then,we take as an approximate solution an element ∆β ∈ Ωα such that thedistance between ∆β and ∆Y is minimal, i.e.,

J = ρ ∆Y, f ∆β = min∆β∈Ωα

ρ[∆Y, f (∆β )]. (8.3)

In real problems, the functional (8.3) is formed in the following way.Consider the difference

Z = Y (α)− Y (β ),

where Y (α) is the output signal of a plant (system), Y (β ) is the output sig-nal of plant model, α = α0 + ∆α is an unknown vector of plant parameters,and β is the model parameters vector.

Represent Y (β ) in the form

Y (β ) = Y (β 0) + ∆Y (∆β ).

Then,

Z = Y (α) − Y (β 0)−∆Y (∆β ).

or

Z = ∆Y −∆Y (∆β ) = ∆Y − f (∆β ),

where∆ = Y (α)− Y (β 0).

As a result, the following value can be employed as a distance:

J = ρ[f (∆β ), ∆Y ] = (∆Y − f (∆β ), ∆Y − f (∆β )). (8.4)

In general, for minimization of the functional (8.4), numerical methodsare used. Such being the case, we encounter a fairly difficult problem of

finding extremum of a function of many variables. The problem can be




simplified if we approximate the value Y (β ) by sensitivity functions, forexample, in a locality of the point β 0. Moreover, for a linear approximation

Y (β 0 + ∆β ) = Y (β 0) + U ∆β (8.5)

and corresponding choice of the functional J we can find an analyticalsolution to the inverse problem for each step.

Let the functional J have the form

J =

τ t0

(Y − Y )T D(Y − Y ) dt (8.6)

or,

J =N i=1

Y (ti) − Y (ti)

T D

Y (ti) − Y (ti)

, (8.7)

where D is a weight (usually diagonal) matrix. As a special case, D = E .

Substituting (8.5) into (8.6) and (8.7) yields

J =

τ t0

(∆Y − U ∆β )T D(∆Y − U ∆β ) dt (8.8)

J =N i=1

[∆Y (ti) − U (ti)∆β ]T

D [∆Y (ti)− U (ti)∆β ] . (8.9)

The vector ∆β can be found using the necessary condition of extremumof these functionals. Consider, for example, the functional (8.8). It can berewritten in the form

J =τ

t0

∆Y T D∆Y − 2∆β T U T D∆Y + ∆β T U T DU ∆β

dt. (8.10)

To find the necessary minimum condition for the functional (8.8), wedifferentiate (8.10) with respect to the vector ∆β :

dJ

d∆β = 2

τ

t0 U T DU ∆β − U T DU ∆Y

dt.




As a result, we find the following necessary extremum condition:

C ∆β = P, (8.11)

where

C =

τ t0

U T DU dt, P =

τ t0

U T D∆Y dt.

Columns of the matrix U are the sensitivity vectors

U i =∂ U

∂β i, i = 1, . . . , m .

The matrix C has elements

cij =

τ t0

U T i DU j dt, i, j = 1, . . . , m ,

which will be considered as scalar product of the vector functions U i andU j in m-dimensional linear space and denoted by

cij = (U i, U j).

Then, the matrix C takes the form

C = (U i, U j).

The matrix C in the linear algebraic equation (8.11) is the Gramm matrixof the system of vector functions U 1, . . . , U m. Since det C is the Grammian(Gramm determinant), the matrix C will be non-degenerate if and only if

the sensitivity vectors form a linearly independent system on the interval(t0, τ ). Thus, for a unique solution of the linearized problem of inverse sen-sitivity it is necessary that the vectors U 1, . . . , U m be linearly independentin the observation interval (t0, τ ). In this case,

∆β = C −1P.

Note that the considered procedure of solving inverse problems of sen-sitivity theory by inspection is multistep. Above we described the proce-

dure of a single step. In the general case, on each step we find the value




∆β k = C −1k P k. Then, we obtain the vector β k = β k−1 + γ k∆β k. Thisvector makes it possible to find the matrix C k+1, vector P k+1 and variation∆β k+1 = C −1k+1P k+1. As in many iterative algorithms, the parameter γ k isused for improving convergence of the algorithm. The process is repeated

up to the step N when the following “stop conditions” hold:

|∆β s| ≤ δ, s ≥ N.

As a result, for estimation of the vector β we obtain

β = β 0 +N

i=1∆β i,

and for estimation of desired variations ∆α

∆α = β − α0 = β 0 +N i=1

∆β i − α0.

Usually β 0 = α0. Then,

∆α =

N i=1

∆β i.

Let us show that the condition of linear independence of the sensitivityvector on each iteration step is sufficient to the whole iteration process toconverge. For k + 1 iteration the functional J can be written in the form

J k+1 =

τ

t0 [Y (α)

−Y (β k+1)

T

D Y (α)

−Y (β k+1)

T

dt

=

τ t0

[∆Y k − U k∆β k+1]T

D [∆Y k − U k∆β k+1] dt

or,

J k+1 = J k − 2

τ

t0∆β T k+1U T k D∆Y k dt +

τ

t0∆β T k+1U T k D∆β k+1 dt. (8.12)




The estimate on the (k + 1)-th step is given by

∆β k+1 = τ

t0 U T k DU k dt−1

τ

t0 U T k D∆Y k dt .

Substituting the expression for the estimate in (8.12), we obtain

J k+1 = J k − P T k C −1k P k, (8.13)

where

P k =

τ

t0 U T k D∆Y k dt, C k =

τ

t0 U T k DU k dt.

If the matrix C k is nonsingular, C −1k is a symmetric positive definitematrix. Then, from (8.13) it follows that

J 0 > J 1 > J 2 > .. . > J k > J k+1 > . . . , (8.14)

i.e., estimates of the vector ∆β will improve, according to a chosen qualityindex J , on each iteration step.

8.1.5 Methods for Improving Stability of Inverse Problems

A combined use of solution by inspection and linearization method forsolving inverse problems leads to solving a sequence of linear systems of algebraic equations of the form

A Y = X, (8.15)

where A is the Gramm matrix having the determinant

det A = Γ(U 1, U 2, . . . , U m).

The matrix A is symmetric. If det A = 0, it can be transformed to adiagonal form using appropriate orthogonal transformation, and the system(8.15) will have a decomposed form

λiyi = xi, i = 1, . . . , m (8.16)

where λi are eigenvalues of the matrix A.




Then,

yi =1

λi

xi, i = 1, . . . , m .

If the matrix A is singular, i.e., detA = 0, some of their eigenvalues, forexample, λm−1, . . . , λs where s is the rank of the matrix, will be equal tozero. Such being the case, the inverse operator of the system (8.16) losescontinuity, i.e., the initial problem (8.15) becomes incorrect. Therefore,the solution procedure will be unstable. But nonsingularity of the matrixA is not sufficient for obtaining a solution with desired accuracy, becauseaccuracy depends on its condition number [110, 111]. During numericalcalculation on a computer a nonsingular system can, in fact, appear tobe singular. In applied mathematics there are a number of methods of improving stability of solution of singular and badly conditioned algebraic

systems [105, 111]. Let us present two methods for improving solutionstability for an inverse problem of sensitivity theory based on the peculiarfeatures of the matrix A formed with the help of sensitivity functions.

The idea of the first method boils down to the following. As follows fromthe preceding material, sensitivity functions depend on input signals actingon the plant or system. For solving an inverse problem, there is the freedomof choosing such input signals. This is true, for instance, for problemsof active identification, diagnostics, and controller adjustment. Therefore,there is a possibility to control, via input signals, linear independence of thesensitivity vectors U 1, . . . , U m, and, via this vectors, the Grammian detA.Let us demonstrate this idea by an example of a linear static plant

y(t) =mi=1

αixi(t).

Model of the plant is described by

y(t) =m

i=1 β ixi(t).

For this model, sensitivity functions are given by

ui(t) =∂ y(t)

∂β i= xi(t).

Then,

∆y(t) =m

i=1 ∆αixi(t),




J =

τ t0

∆y(t) −

mi=1

∆αixi(t)

2dt,

Hence, the necessary condition of extremum of the functional J can be

represented in the form

mi=1

∆αi

τ t0

xi(t)xs(t) dt =

τ t0

∆y(t)xs(t) dt, s = 1, . . . , m .

The matrix C for this system is equal to

C = (X i, xj), i, j = 1, . . . , m

where

(xi, xj) =

τ t0

xi(t)xj(t) dt.

In this problem the sensitivity functions coincide with input signals. Letus choose these signals to be pair-wise orthogonal, so that τ

t0

xi(t)xj(t) dt = giδ ij ,

where δ ij is the Kronecker symbol. Then,

C =

g1 0 . . . 0

0 g2 . . . 0

. . . . . . . . . . . .

0 0 . . . gm

and

det C =mi=1

gi = 0.

Analogous techniques have been investigated in detail in the theory of experiment planning.

The second method of improving stability of a solution for inverse prob-lems of sensitivity theory is connected with the following property of theGrammian [28]:

Γ(U 1, U 2, . . . , U m)

≤

m

i=1Γ(U i). (8.17)




In (8.17), which is called Hadamard inequality , equality holds if and onlyif the sensitivity vectors U 1, . . . , U m are pair-wise orthogonal.

Using the properties of Grammian, we can rewrite the Hadamard in-equality in the form

Γ(U 1, . . . , U m) ≤mi=1

(U i, U i) = ∆. (8.18)

Obviously,

∆ = 0. (8.19)

In [105] it was proposed to change an initial equation by a correct equa-

tion that is close, in a sense, to the initial one. To solve Equation (8.15),it was proposed to use, instead of the operator A−1, which is the inverse of A, an operator A−1 defined as follows.

It is known that

A−1 =B

det a=

B

Γ(U 1, . . . , U m), (8.20)

where B is the matrix consisting of the corresponding cofactors for theelements of the matrix A so that bij = Aji . Introduce the operator

A−1 =B

mi=1

(U i, U i)

=B

∆. (8.21)

and consider some its properties. The elements of the matrix B arebounded. Therefore, the operator A−1 is continuous, because ∆ > 0.

If the vectors U 1, . . . , U m are pair-wise orthogonal, the operators A−1

and A−1 coincide. Indeed, in this case

Γ(U 1, . . . , U m) =mi=1

Γ(U i) =mi=1

(U i, U i)

and from (8.20) and (8.21) we have A−1 = A−1.

The solution of the system (8.15) defined by the operator A−1 is givenby

Y = A

−1

X =

B

det A X,




while the pseudo-solution of (8.15) defined by the operator A−1 is

Y = A−1X =B

∆X.

Comparing Y and Y , we find that Y ≤ Y . Equality in the last equationholds only if the sensitivity vectors are pair-wise orthogonal. Deviation of Y from Y can be estimated as

Y − Y =

B

∆X − B

det AX

=

1

∆− 1

det A

BX

=

det A

∆− 1

A−1X ≤

det A

∆− 1

A−1

X .

Since the matrix A is symmetric and positive definite, we have

A−1 =

1

λmin,

where λmin is the least eigenvalue of the matrix A, and 1/λmin is its con-dition number.

Thus,

Y − Y ≤ .det Aδ

− 1 1λmin

X .

If the sensitivity vectors are pair-wise orthogonal, we have

det A = ∆, λmin > 0 and Y − Y = 0,

as was demonstrated above. If the matrix A is singular, we have λmin = 0and the deviation Y from Y is infinitely large.

It can be easily shown that the solution Y is less sensitive to variation of the right side of Equation (8.15) than the solution Y .

Sensitivity of these solutions will be estimated by the matrices.

U =∂Y

∂X , U =

∂ Y

∂X .

It is easy to see that

U = A−1, U = A−1,

U

= A−1 =1

λmin,




U = A−1 =det A

∆A−1 =

ρ

λmin,

where

ρ =det A

∆,

Hence,

U ≤ U .

Thus, the solution Y is more stable with respect to deviation of initialdata than the solution Y . Consider how the convergence condition of thesolution of the inverse problem changes as a result of the transition to thepseudo-solution Y . With this aim in view, we write the functional (8.6) inthe form

J k+1 = J k − 2∆β T

k+1P k + ∆β T

k+1C k∆β k+1. (8.22)

The pseudo-solution is found by

∆β k+1. =Bk

∆k

P k

Substituting it into (8.22), we obtain

J k+1 = J k

−2P T k

BT k

˜∆k

P k + P T kBT k

˜∆k

C kBk

˜∆k

P k

= J k − 2ρkP T k C −1k P k + ρ2kP T k C −1k P k,

where

ρk =det A

∆k

≤ 1.

From analysis of the last equation it follows that convergence of theiterative process remains, though the rate of convergence may decrease.

8.1.6 Investigation of Convergence of Iterative Process

Consider, as a general case, a dynamic system with additional motion∆Y (t) in the form of an n-dimensional function of time and m-dimensionalvector ∆α of variation of parameters α. Let ∆Y (t) be defined by Equation(8.1), and a functional J of the form (8.7) be used. According to the aboveiterative process, the estimate of the vector β of model parameters at the(k+1)-th step is given by

β k+1

= β k + γ (β k)BkP k

det C k, (8.23)




where Bk is the matrix of cofactors of the matrix C k,

C k =N

i=1U T k (ti)D(ti)U k(ti), P k =

N

i=1U T k (ti)D(ti)∆Y k(ti),

or, using pseudo-solution,

β k+1 = β k + γ (β k)BkP k

∆k

. (8.24)

Let us investigate the rate of convergence of the iterative process (8.22).Expanding the vector function Y (t, β ) into Taylor’s series in a locality of the point β k, we find

Y (t, β ) = Y (t, β k) + U k∆β k +1

2K k,

where

K k = ∆β T k∂ 2Y

∂β 2

β=βk

∆β k, F ki =∂ 2Y

∂β 2

β=βk,t=ti

is a three-dimensional matrix. Substituting this formula into (8.24), weobtain

β k+1 = β k + γ (β k)1

∆k

Bk

N i=1

U T k (ti)D(ti)

U k(ti)∆β k +

1

2K k

= β k + γ (β k)

1

∆k

Bk

C k∆β k +

1

2

N i=1

U T k (ti)D(ti)K k

,

Hence, using the formula

C −1k =Bk

det C k,

we obtain

β k+1 = β k + γ (β k)det C k

∆k

∆β k +1

2γ (β k)

Bk

∆k

N i=1

U T k (ti)D(ti)K k.

or,

β k+1

−β = 1

−γ (β k)

det C k

∆k (β k

−β ) +

1

2

γ (β k)Bk

∆k

N

i=1 U T k (ti)D(ti)K k.




As a distance ρ(β k+1, β ) we may use, for instance, Euclidean vector norm.In this case

β k+1

−β ≤ 1− γ (β k)

det C k

∆k β k−

β

+

√ n

2γ (β k)

× Bk∆k

N i=1

[ U k(ti) D(ti) F ki ] β k − β 2.

(8.25)

Equation (8.25) makes it possible to estimate the rate of local convergenceof the iterative process (8.24). As a specific case, from (8.25) it followsthat in the general case convergence is linear. The condition of linearconvergence is defined by the relation

1− γ (β k)det C k

∆k

+√

n

2γ (β k)

Bk∆k

×N i=1

[ U k(ti) D(ti) F ki ] δ β < 1,

(8.26)

where δ β is an infinitely small locality of the point β that includes thevariation ∆β k.

For γ (β k) such that 0 < γ (β k) < 1, the condition of linear convergence of the process (8.24) is completely determined by the convergence conditionof the iterative process (8.23):

det C k >

√ n

2

Bk

N i=1

U k(ti) D(ti) F ki

δ β . (8.27)

To ensure a higher rate of convergence, it is required to satisfy thecondition

γ (β k) → det C k

∆k

when k →∞.

Quadratic convergence is reached only if

γ (β k) =det C k

∆k

and is determined by the condition (8.27) with δ β = 1.




Example 8.1

Consider a dynamic system described by a first-order differential equation

T y(t) + y(t) = kx(t),

where x(t) and y(t) are the input and output signals of the system, respec-tively.

It is required to investigate the rate of convergence for the process of estimating the parameters T and k using sensitivity functions. The sensi-tivity functions uT (t) and uk(t) are obtained by integration of the followingsystem of differential equations:

y =1

T

[kx(t)

−y(t)],

uT =1

T [−y(t) − uT (t)], uk =

1

T [x(t) − uk(t)]

on the interval [t0, τ ] with given initial conditions

uT (t0) = uT 0, uk(t0) = uk0, y(t0) = y0.

The system of normal equations obtained during minimization of the

quadratic functional J has the formc11 c12

c21 c22

∆t

∆k

=

p1

p2

,

where

c11 =

τ

t0 u2T dt, c22 =

τ

t0 u2k(t) dt,

c12 = c21 =

τ t0

uk(t)uT (t) dt, p1 =

τ t0

uT (t)∆y(t) dt,

p2 =

τ t0

uk(t)∆y(t) dt.

It is assumed that there is an initial approximation k0 and T 0 of the

parameters k and T , respectively. Then, the estimates of the parameters k




and T on the (i + 1)-th step are given by

ki+1 = ki + γ c11 p2 − c21 p1

c11c22 − c21c12,

T i+1 = T i + γ c11 p1 − c12 p2c11c22 − c21c12

(8.28)

or, using a pseudo-solution,

ki+1 = ki + γ c11 p2 − c21 p1

c11c22,

T i+1 = T i + γ c11 p1 − c12 p2

c11c22.

(8.29)

The convergence condition of these iterative processes for 0 < γ < 1 canbe written in the form

det C >1

2B

N i=1

U (ti) F i δ β ,

where

B =

c22 c21

c12 c11

, U T (ti) = [uT (ti)uk(ti)],

F i =

∂ 2y(ti)

∂T 2∂ 2y(ti)

∂T∂k

∂ 2y(ti)

∂T∂k

∂ 2y(ti)

∂k2

,

δ β =

1 for solution (8.28),

∆β 1 for solution (8.29).

Tables 8.1 and 8.2 demonstrate the results of numerical solution of thesystems (8.28) and (8.29) for the following initial data:

k = 5, T = 1, k0 = 5, 5, T 0 = 0, 8,y(t0) = uT (t0) = uk(t0) = 0, x(t) = x0 + t, x0 = 5, N = 200.

The system of differential equations was solved by the Runge-Kuttamethod on the interval [0.1, 20] with integration step 0.1. After that, inputand output variables of the model were affected by stochastic values with

zero mathematical expectation and root-mean-square σ = 0.1.




Table 8.1 Results of pseudo-solution

Iteration 1 2 3 4 5

K 5.02773 4.98678 4.98234 4.98181 4.98175

T 0.94442 0.98346 0.98880 0.98957 0.98966

Table 8.2 Results of direct solution of the system

Iteration 1 2 3 4

K 4.97778 4.98164 4.98174 4.98174

T 0.95969 0.98933 0.98966 0.98967

8.2 Identification of Dynamic Systems

8.2.1 Definition of Identification

The identification problem is the most characteristic applied inverse prob-

lem of sensitivity theory.

DEFINITION 8.4 By identification in control theory we mean a pro-cedure of constructing mathematical models of plants (systems) on the basis of observable (measurable) information about input and output signals.

An operator A transforming input signal X (t) into output signal Y (t)is an exhaustive plant characteristic. Using identification, we obtain notthe operator A itself, but its estimate A0, which describes the plant model.

Obviously, the best model is the one for which the operator is, in a sense,close to the plant operator. In real conditions, the proximity is evaluat-ed by a functional J depending on output signals of the plant and model.In many cases, the identification process can be reduced to the followingscheme. There is a plant with measurable input and output signals denotedby X (t) and Y (t), respectively. A plant model is also given over a specifiedclass of operators. The signal X (t) is applied to the inputs of the plant andmodel. The output signal Y (t) of the model is algebraically compared withthe plant output signal Y (t). The difference Y (t) − Y (t) is used for con-structing the proximity functional J . The desired estimate A0 of the model

operator is obtained by minimizing this proximity functional (identification




cost function). A general block-diagram of the above identification processis shown in Figure 8.2. In actual conditions, output and sometimes even

Figure 8.1

Identification process

input signal of the plant is distorted by additive disturbances.

8.2.2 Basic Algorithm of Parametric Identification Using

Sensitivity Functions

In the most general case, the identification problem formulated abovecan be reduced to a nonlinear programming problem that is usually solvedusing numerical methods. Many of these methods employ various gradientalgorithms. As was noted many times, the components of the gradientvector are, in fact, sensitivity functions. Therefore, to find the componentsof the gradient, we can use sensitivity equations. Methods of constructingand solving these equations have been developed in sensitivity theory. Thiscircumstance may well enhance speed and precision of gradient algorithms.

Consider a general scheme of parametric identification of a system de-scribed by the equation

Y = F (Y, α), Y (t0) = Y 0. (8.30)

The model equation has the form

˙Y = F (Y , β ), Y (t0) = Y 0, (8.31)

where β is an analog of vector α of unknown system parameters.




The vector function Y (β 0 + ∆β, t) can be approximated by

Y (β 0 + ∆β 0t) = Y (β 0, t) + U (t)∆β, (8.32)

where

U (t) =∂ Y (β, t)

∂β

∆β=0

if the sensitivity matrix that is the solution of the sensitivity equation

U =∂F (Y , β )

∂ Y U +

∂F (Y , β )

∂β (8.33)

where β = β 0 and U (t0) = 0.Assume that the quality index of identification has the form of the func-

tional (8.6) with D = E . Then, to determine the variation ∆β we obtainthe following algebraic equation

C ∆β = P, (8.34)

where

C =

τ

t0

U

T

(t)U (t) dt

is the identification matrix,

P =

τ t0

U T (t)∆Y (t) dt, ∆Y (t) = Y (t, α)− Y (t, β ),

Here Y (t, α) is the plant output signal, and Y (t, β ) is the solution of the

model equation (8.31) for β = β 0 (model output signal).If C is a nonsingular matrix, we have

∆β = C −1P.

The inequality

det C = 0 (8.35)

is called the identifiability condition . An estimate of the vector α (or that of

its variation ∆α provided that the nominal vector α0 is known) is performed




in N steps (see Section 1.4):

α = β 0 +N

i=1∆β i.

Above, in fact, we described a single iteration step, i.e., ∆β = ∆β 1.Having obtained ∆β 1, we determine basis value of the vector β for thesecond step in the form

β 1 = β 0 + ∆β 1

and so on.

In general, the estimates ∆β 1 are random values. To obtain these esti-mates, linear equations of identification and least-square methods are usedon each iteration step. Therefore, there is a possibility to perform theirstatistical analysis.

8.2.3 Identifiability and Observability

Consider the dynamic system

Y = F (Y,t,α), Z = H (t)Y, Y (t0) = Y 0, (8.36)

where Z is an s-dimensional vector of observable (measurable) coordinates,s ≤ n, and H (t) is the observation matrix of dimensions s× n.

For the given system with D = E the identification cost function has theform

J =

τ t0

(∆Z −G∆β )T (∆Z −G∆β ) dt, (8.37)

where

∆Z = Z −H Y , G = HU.

The necessary minimum condition for this functional is given by

C ∆β = P , (8.38)

where

C =

τ

t0GT Gdt, P =

τ

t0GT ∆Z dt.




Moreover, the identifiability condition is given by the inequality

det C = det

τ

t0U T H T HU dt = 0. (8.39)

Obviously, to meet this requirement it suffices to ensure that the columns(or rows) of the matrix G = HU be linearly independent. Let us write thesensitivity equation (8.33) in the form

U = A(t)U + R(t), U (t0 = 0) = 0, (8.40)

where

A(t) =∂F

∂ Y , R(t) =

∂F

∂β .

Moreover,

∆Z (t) = Ht)U (t)∆β. (8.41)

Multiplying Equation (8.30) term-wise on the right by ∆β and using thenotation

Ω(t) = U (t)∆β, (8.42)

we write

Ω = A(t)Ω + R(t)∆β, Ω(t0) = 0,

δZ = H (t)Ω.(8.43)

Equations (8.43) describe a linear nonstationary system with the vectorof phase coordinates Ω(t) of dimension n and observation vector Z (t) of dimension s ≤ n. Assume that the system is observable, i.e., it is possibleto reconstruct the vector Ω(t) by a known vector Z (t). Then, obviously,the variation ∆β can be, according to (8.42), found from the equation

Ω = U ∆β.

This can be done, for instance, by the least square method or by the methodof pseudo-inverting the matrix U . Consider the observability conditionfor the dynamic system (8.43). With this aim in view, we introduce thetransition matrix (Cauchy matrix) Φ(t, t0 = 0) satisfying the condition

Φ = A(t)Φ, Φ(0) = E. (8.44)

As is known, a linear periodic system with transition matrix Φ(t) of

dimensions n×n and observation matrix H (t) of dimensions s×n is called




fairly observable on the interval (0, τ ) if the matrix

M (τ ) =

τ

t0 ΦT (t)H T (t)H (t)Φ(t) dt (8.45)

is nonsingular.

Let us note that the structure of the matrix M (t) is identical to the struc-ture of the matrix C in Relation (8.38). Assume that the above techniqueemploying sensitivity functions is applied to determine unknown initial con-ditions Y 0. In this case the sensitivity matrix

U (t) =∂ Y

∂ Y 0

satisfies the differential equation

U =∂F

∂ Y U, U (0) = E (8.46)

Comparing (8.44) and (8.46), we obtain that Φ(t) = U (t). Then, theconditions of identifiability (8.35) and observability coincide.

8.3 Distribution of Parameters Tolerance

8.3.1 Preliminaries

System design results in determining nominal (assumed) values of itsparameters that satisfy a given quality index and workability criterion. As

a special case, nominal parameter values can be found by determining anextremum of the quality index. Actual values of system parameters areusually different from the nominal ones.

In general, variations of parameter values from their nominal values in-fluence the quality of system operation. Let operation quality index becharacterized by a value I that is a function of parameters α1, . . . , αm.During system design the nominal parameter values are obtained. Theyare associated with the nominal value of the quality index

I 0 = I (α10, . . . , αm0).




Real systems have, due to the above reasons, other values of the pa-rameters that differ from the nominal ones by ∆α1, . . . , ∆αm, respectively.Then,

I = I (α10 + ∆α1, . . . , αm0 + ∆αm), (8.47)

Hence, the quality index gets an increment

∆I = I (α10 + ∆α1, . . . , αm0 + ∆αm)− I 0. (8.48)

Owing to inevitable parameter perturbation with respect to their nom-inal values and, as a consequence, variation ∆I , at the design stage it isrequired to determine limits of variation of the quality index I or those of itsincrement ∆I . Thus, a system is considered to be operable (see Figure 1.16)if

I (α1, . . . , αm) ∈ M p

or

∆I (∆α1, . . . , ∆αm) ∈ ∆M p, (8.49)

where M w and ∆M w are admissible sets (regions) of the values of thequality index and its variation.

The sets M w and ∆M w characterize tolerances for the quality index andits variation. In general, by tolerances we mean preliminarily specifiedlimits in which the values of parameters and quality criterion determining

operation quality of the system (plant, unit, device) must be kept.Quantitative characteristics of tolerance are the upper xmax and lower

xmin limit deviation from a nominal value, or a tolerance field ∆x, coor-dinates of its mean value x and the half of the tolerance field δ x. Thesevalues are related by

∆x = xmax − xmin, δ x =∆x

2, x =

1

2(xmax + xmin),

xmax = x + δ x, xmin = x− δ x.(8.50)

In technical literature the notion of tolerance is usually equivalent tothe notion of tolerance field ∆x which is, as distinct from other tolerancecharacteristics, a strictly positive value. Other values may be negative orzero.

8.3.2 Tolerance Calculation Problem

Tolerance calculation is a mandatory stage in designing of all types of devices (including radio-electronic, electromechanical, precise mechanical

and so on). The goal of this stage is to coordinate tolerances on quality




criterion DeltaI for the system (unit, device) and those on parameters ∆αi

of the elements. In this case it is important to make a difference betweendirect and inverse problems of determining tolerances.

In a direct problem it is required to find, by given tolerances of the pa-

rameters α1, . . . , αm, the tolerance ∆I in which the value of the qualitycriterion I will be kept. The problem reduces to determining an operator(transformation) L mapping the set (tolerance) ∆α from the parameterspace (α1, . . . , αm) onto the set (tolerance) ∆I in the space of quality cri-terion I :

∆I = L∆α. (8.51)

In an inverse problem, the value of the tolerance of the quality index isgiven. It is required to find tolerances ∆α for system parameters that ensurethat the quality index remains inside the limits of ∆I . Mathematically, theproblem reduces to determining an operator (transformation) Q mappingthe set ∆I onto the set ∆α:

∆α = Q∆I . (8.52)

In fact, the direct problem reduces to analysis of system precision underparameter variation. If an analytical relation (8.47) between the qualitycriterion I and parameters α1, . . . , αm is known, the problem is readilysolved by known methods in both deterministic and stochastic conditions.

Tolerance distribution is a very difficult problem. Currently, there are nogeneral methods for its solution. Some partial approaches to tolerancesetting are given in [4, 32, 45, 71]. The most widespread are methods basedon the use of sensitivity functions and coefficients.

8.3.3 Initial Mathematical Models

When tolerances are calculated using methods of sensitivity theory, itis assumed that deviation of the parameter values from nominal ones aresmall and parameter fluctuations are linear inside the tolerance field. Then,

the variation (8.48) of the quality index I can be represented in the form

∆I =mi=1

ui∆αi, (8.53)

where

ui =∂I

∂αi

∆α=0

is the sensitivity coefficient.




The value ∆αi in (8.53) is the “current” variation given by

∆αi = αi − αi0.

Depending on the relation between αi and αi0, the value ∆α can be pos-itive as well as negative. The signs of the sensitivity coefficients u1, . . . , umare known in advance. For the worst combination of parameters variations,the variation of the quality index I , we can use the following formula

∆I =mi=1

|ui|∆i, (8.54)

where ∆i = ∆αi is the tolerance field for the parameter αi.

Obviously, in this case Equation (8.54) directly determines the tolerancefield for the quality index ∆I , i.e.,

∆I =mi=1

|ui|∆i.

It can be easily shown that a similar formula can be obtained for the half of the tolerance field:

∆I =m

i=1 |ui|δ i (8.55)

Assume that parameter variations ∆α1, . . . , ∆αm are stochastic values.For calculating tolerances it is often possible to determine only mathe-matical expectation and variance of the variation ∆I . Let K ∆α be thecorrelation matrix of the vector ∆α = (∆α1, . . . , ∆αm)T so that

K ∆α =

σ21 K 12 . . . K 1m

K 21 σ22 . . . K 2m

. . . . . . . . . . . .

K m1 K m2 . . . σ2m

.

Then, the mathematical expectation M [∆I ] and variance σ2[∆I ] of thevariation ∆I are given by, respectively,

M [∆I ] =mi=1

uiM [∆αi],

σ2[∆I ] =m

i=1 u2i σ2i + 2i<j uiujK ij .

(8.56)




The values ∆α1, . . . , ∆αm are centered independent stochastic values, i.e.,

M [∆I ] = 0,

σ2

[∆I ] =

mi=1

u2i σ

2i .

(8.57)

8.3.4 Tolerance Calculation by Equal Influence Principle

In the framework of the above models, the problem of tolerance distri-bution has a unique solution only for the case of a single parameter, when,for example,

∆I = |u1|∆1.

Obviously, the tolerance ∆1 should be chosen so that the following con-dition holds:

∆r1 ≥ |u1|∆1,

where ∆r1 is a given value.

For given ∆r1 and sensitivity coefficient, the value of the tolerance ∆1 is

determined by the relation

∆1 ≤ ∆r1

|u1| . (8.58)

If the model is stochastic, for the case of a single parameter we have

M [∆I ] = u1M [∆α1], σ2[∆I ] = u21σ2[∆α1]. (8.59)

Assume that the value ∆α1 has a symmetric distribution Then, the coor-dinate of the center of the field of scattering for the parameter ∆α is givenby

∆α1 = M [∆α1].

Similarly, for the variation ∆I we obtain

∆I = M [∆I ].

As a result, we find

∆α1 =∆I

u1. (8.60)

If ∆I has the normal or close to normal distribution,

P (|∆I −M [∆I ]| < 3σ[∆I ]) = 0, 997,




i.e., it is practically guaranteed that ∆I deviates from M [∆I ] by not morethan 3σ (the rule of three sigma). Then, the field of tolerance can be takenas

∆I = 6σ[∆I ],

Hence, on the basis of (8.59) we determine the tolerance field for the pa-rameter α1:

∆1 ≤ ∆rI

|u1| =6σr[∆I ]

|u1| .

For the case of several parameters, calculation of tolerances can be per-formed by the method of successive approximation, selecting parameterstolerance so that the quality index I remains in a prescribed area.

The equal influence principle yields much simpler algorithm. Using this

principle, it is assumed that the values of all the terms |ui|∆i in (8.54) areequal. Then,

∆I = m|ui|∆i,

Hence,

∆i ≤ ∆I

m|ui| . (8.61)

For a stochastic model (8.57) with normal parameter distribution, theequal influence principle yields the following expression for tolerances:

∆i ≤ ∆rI √

mui, i = 1, . . . , m . (8.62)

8.3.5 On Tolerance Distribution with Account forEconomic Factors

During tolerance calculation it seems to be a natural wish to strengthenthe inequality sign in the relations

∆rI ≤ ∆I =

mi=1

|ui|∆i,

σ2r [∆I ] =≤ σ2[∆I ] =mi=1

u2i σ2i + 2i<j

uiujK ij .

(8.63)

With given values of sensitivity coefficients, this can be achieved onlyby decreasing tolerances ∆i (i = 1, . . . , m), i.e., by enhancing precision andstability of the system. But this approach leads to an increase of the costof the elements and, as a result, the cost of the whole system. Then, there

arises a contradiction between system precision determined by the relations




(8.63) and its cost. This contradiction can be solved by a transition to theoptimization problem of tolerance distribution formulated below.

Optimization problem for tolerance distribution. Find tol-

erances for parameters values α1, . . . , αm for which tolerance forthe quality index does not exceed a given value and the total costof the system is minimal.

For mathematical formulation of the problem the choice of analytical de-pendencies of precision and cost functions is principally important. Equa-tions (8.54)–(8.57) and others similar to them may be used for precisionevaluation. A more involved problem is the choice of the cost functionconnecting the cost of the system with tolerances for its elements [3, 4].

The dependence of the cost of the i-th element on the tolerance ∆i can berepresented in the following general form

ci = ci(∆i) = ci(∆αi). (8.64)

Then, the total cost C of the system is given by

c =m

i=1ci(∆αi). (8.65)

The main difficulties are connected with the choice of the form of thefunctions ci(∆αi). This problem is fairly complicated, because the cost of element production can depend, besides tolerance, on a number of otherfactors. In the present work a detailed discussion of this topic is hard-ly possible. We can only recall a widespread intuitive principle used inthe literature: the more precise, the more expensive. Using this idea, thefollowing cost function was proposed in [4]:

ci = ai +bi

∆sii

. (8.66)

The constants ai, bi, and si in this formula are determined experimentallyif the costs ci for three or more tolerances are known.

Denote the precision function by q :

q = q (∆1, . . . , ∆m).

Then, the problem of tolerance optimization reduces to minimization of




the function c(∆1, . . . , ∆m):

min∆1,...,∆m

c (∆1, . . . , ∆m) (8.67)

under the constraintsq (∆1, . . . , ∆m) ≤ q r. (8.68)

If we let strict equality in (8.68), the above problem can be reduced to aconditional extremum problem solved by means of a generalized Lagrangefunction:

L =mi=1

c1(∆i) + λ[q (∆1, . . . , ∆m)− q r].

In general, it is a nonlinear programming problem with a special goalfunction, which is the sum of several functions, each of them depending on-ly on one variable. Such goal functions are called separable . In the theoryof nonlinear programming [116] there are special approximate methods of solution for separable problems. For a more rigorous approach to formula-tion and solution of the optimization problem, we need to take restrictionsimposed on the variables ∆1, . . . , ∆m into account. First, as was shown inthe first paragraph of this section, these variables are nonnegative, i.e.,

∆i

≥0, i = 1, . . . , m . (8.69)

Second, tolerances on parameters values are determined by the precisionclass of element production. This means that the values ∆i and the func-tions ci(∆i) are discrete, and the tolerances ∆i take values from a finiteadmissible set Γi, i.e.,

∆i ∈ Γi, i = 1, . . . , m . (8.70)

As a result, the optimal choice of tolerances reduces to solving a discretenonlinear programming problem.

In principle, the following dual problem can be formulated and solved:minimize the precision function under restrictions imposed on the costfunction:

min∆1,...,∆m

q (∆1, . . . , ∆m),mi=1

ci(∆i) ≤ cmax,

∆i ≥ 0, ∆i ∈ Γi, i = 1, . . . , m .

Some methods for solution of tolerance optimization problems with ac-

count for the cost function are considered in [3, 4].




8.3.6 On Requirements for Measuring Equipment

The problem of the choice of measuring equipment in the following state-ment can also be reduced to a tolerance distribution problem.

Assume that it is required, in order to determine whether a system isoperable, to estimate variation ∆I of the quality index by measured vari-ations of ∆αi. Measurement errors and the true variation values will bedenoted by δαi and ∆αi, respectively. Then, as a result of measurements,instead of the variation

∆I =mi=1

ui∆αi

we have

∆I =m

i=1(∆αi + δαi) = ∆I + δI ,

δI =mi=1

uiδαi,

(8.71)

where δI is the error of determining variation of the quality index due tomeasurement errors.

When system operability is assessed, the variation δI can lead to anincorrect solution. Indeed, let for a real system the following conditionhold:

|∆I | ≤

∆I r

,

i.e., the system is operable. But, due to measurement error, it may appearthat

|∆I + δI | ≥ ∆I r,

and the system will be declared inoperable.To exclude possible errors, methods of assessing operability must take

account of tolerances on measurement errors. Obviously, the less thesetolerances are, the more “sensitive” is the method. Admissible error formeasuring the variation ∆I may usually be determined, so that

mi=1

uiδαi ≤ δI r.

The hope for strengthening inequality in this relation leads to the neces-sity of using more precise and expensive equipment. Formally, we obtain aproblem similar to the problem of tolerance distribution that can be solvedusing the methods considered above.

Finally, we note that in many cases the values of variations are deter-

mined during indirect measurements as a result of secondary processing




(identification) with the aim of finding estimates of ∆αi. Then, the valuesδαi will be determined by measurements errors as well as by errors of theused processing method.

8.4 Synthesis of Insensitive Systems

8.4.1 Quality Indices and Constraints

As was shown in the preceding chapters, variation of system parametersleads to additional motion. Under uncontrollable parameter variations,additional plant motion is usually undesirable. Therefore, it seems to be

a natural tendency for developing principles of control system design thatwould be able to counteract parametric disturbances. In classical theory,design of control systems is performed with the use of quality criteria andconstraints imposed on system variables. Obviously, for design of controlsystems with account to additional requirements for additional motions, thequality indices or constraints must be correspondingly complicated. Thiscomplication can be reduced, as a special case, to introducing additionalquality indices or constraints. Quality indices that characterize additionalmotion and are functionals from the latter, are introduced similarly toquality indices for transient responses in automatic control theory [8, 79,

100]. For instance, this may be the maximal deviation of additional motion

maxt|∆y(t)|, 0 ≤ t ≤ ∞,

integral quadratic error∞ 0

∆2y(t) dt

and the like.

In the general form, we denote such a quality index by

J = J (∆y, ∆y , . . .). (8.72)

Defining additional motion by

∆y(t) = u(t)∆α,

we can reduce the quality criteria of the form (8.72) to indirect character-

istics of the sensitivity function u(t), some of which have been described in




Section 5.5. Denote these indices by

J = J (u, u , . . .). (8.73)

For system design with account for both the fundamental y(t) and addi-tional ∆y(t) motions, the following additive functionals can be formed:

H 1 = β 1I + β 2J, (8.74)

H 2 = γ 1I + γ 2J, (8.75)

where I is the quality criterion with respect to the fundamental motion,

and β 1, β 2, γ 1, and γ 2 are weight coefficients.The following relations can be used as constraints:

∆y(t) = 0, (8.76)

|∆y(t)| ≤ , (8.77)

u(t) = 0, (8.78)

|u(t)

|= , (8.79)

and so on.

In analogy with the corresponding notions of invariance theory, the rela-tions (8.76) and (8.77) will be called the condition of parametric invariance and condition of parametric invariance up to , respectively.

Equation (8.78) represents, in its turn, the condition of zero sensitivityby the first approximation, and (8.79) is the condition of -sensitivity by thefirst approximation. If necessary, constraints can be imposed on sensitivityfunctions of the second and higher orders, for example,

u(i)(t) =diy(t)

dαi= 0, i = 1, . . . , k , (8.80)

|u(i)(t)| ≤ i, i = 1, . . . , k . (8.81)

These expressions will be called zero and -sensitivity conditions, respec-

tively, up to the k-th order.




8.4.2 Problems of Insensitive Systems Design

We will divide all possible methods of compensating influence of uncon-trollable system parameters variations on its operation quality onto active and passive ones.

Active methods are realized in adaptive (self-adjusting) systems. In thesesystems, parametric disturbances are directly or indirectly evaluated duringsystem normal operation, and this information is then used for correctingcontrol unit algorithm in order to keep a required (optimal) mode of plantoperation. Theory of adaptive systems is currently an independent branchof control theory. These topics are discussed in voluminous literature.

Methods of passive compensation for parametric disturbances are lessdeveloped. By passive methods we mean ones realized by means of appro-priate choice of “fixed” structure and fixed parameters. The structure and

parameters of the system under investigation are obtained during systemdesign with account for indices and constraints introduced in Section 8.4.1.Moreover, several classes of insensitive system design problems can be sep-arated out.

The first class incorporates problems where the insensitivity constraintsare formalized in the form of constraints imposed on additional motion orsensitivity function. As a special case, these restrictions can be expressedby Relations (8.76)–(8.81). Control systems that are designed using thesekinds of restrictions will be called systems with bounded sensitivity . Depend-ing on the specific type of constraints, it is expedient to separate problems

of designing parametrically invariant systems (constraints (8.76)), para-metrically invariant up to (see (8.77)), systems with zero sensitivity (see(8.78)), systems with -sensitivity (see (8.79)), etc.

The second class incorporates problems in which insensitivity require-ments are fulfilled by means of minimizing functionals constructed withaccount for additional motion and sensitivity functions. In the most gen-eral form, these functionals are expressed by Relations (8.72)–(8.75). As arule, this class includes optimization problems where it is required to findparameters or characteristics of optimal “insensitive” control systems.

8.4.3 On Design of Systems with Bounded Sensitivity

General synthesis methods are not known for such systems. For design of parametrically invariant systems and systems with zero sensitivity, the ideasand approaches developed in the invariance theory [57] may be employed.Nevertheless these systems are ideal, as well as invariant systems [121].The latter circumstance usually means that these systems are physicallyunrealizable, because it is practically impossible to ensure the conditionsof parametric invariance and zero sensitivity. On the other hand, systems

with -sensitivity and those parametrically invariant up to can be, in




principle, realized. In the most general form, determination of parametersof these systems can be reduced to a nonlinear programming problem bothin time and frequency domains. In some special cases, already existing orintroduced specially structural redundancy of the system can be used for

synthesis (we mean redundancy only with respect to fundamental motion).In this case, redundant networks and parameters create additional degreesof freedom in the system (the problem becomes ovedetermined). Withaccount for additional motion, the specified redundancy can be used forsatisfying sensitivity requirements.

8.4.4 On Design of Optimal Insensitive Systems

First, let us consider the following optimal control problem [65]. It isnecessary to find the vector functions Y (t)

∈Rn and V (t)

∈Rq on t

∈(t0, τ ) for which the following functional reaches minimum:

I = I (Y, V ) (8.82)

with the differential bonds

Y = F (Y , V , t), (8.83)

restrictions

(Y , V , t) ∈ G (8.84)

and boundary conditions

(Y, t0) ∈ G0, (Y, τ ) ∈ Gτ , (8.85)

where G is an area (set) of an Rn×Rq ×R1, and G0 and Gτ are some setsin Rn ×R1.

Depending on the form of the functional, this problem can be reduced,

with minor refinements, to either constructing an optimal program or de-signing optimal control systems ensuring high-quality motion along the op-timal program. It is possible to assume that all the elements of the aboveproblem depend on a vector parameter α ∈ Rm, i.e.,

I (α) = I (Y , V , α), Y = F (Y , V, t , α),

(Y , V, t , α) ∈ Gα, (Y (α), t0(α)) ∈ G0α, (Y (α), τ (α)) ∈ Gτα.(8.86)

But this model greatly complicates the problem of designing an optimal

insensitive system. Therefore, we consider principal features of the design




procedure for the case when only differential bonds depend on the param-eter vector. Moreover, for simplicity we assume that the vector functionF (Y , V, t , α) is continuous with respect to t and has continuous derivativeswith respect to Y and α. Then, the following problem of optimal control

synthesis can be formulated.It is required to find vector functions Y (t), V (t), and U (t) in Rmn for

t ∈ (t0, τ ) ensuring the minimal value of the functional

J = J (Y , V , U ) (8.87)

with differential bonds

Y = F (Y , V, t , α), (8.88)

U = ∂F ∂Y

U + ∂F ∂α

, (8.89)

under the restrictions

(Y , V , t) ∈ G (8.90)

and boundary conditions

(Y,U,t0) ∈ G0α, (Y , V, τ ) ∈ G0τ . (8.91)

Introduce the vector

Z = [Y T , U T ]T ∈ Rn+mn,

where

U =

U T 1 , U T 2 , . . . , U T m

, U i =∂Y

∂αi

, i = 1, . . . , m .

With respect to the vector Z defined over the augmented phase spaceRn+mn, the elements of the previous problem take the form

J = J (Z, V ), Z = F ∗(Z , V, t),

(Z,V,t) ∈ G, (Z, t0) ∈ G0α, (Z, τ ) ∈ G0τ .(8.92)

Formally, the problem (8.92) of design of an optimal insensitive systemcoincides with the classical optimal control problem (8.82)–(8.85). For itssolution all methods known in optimal control theory can be used. Though

no principal difficulties and problems arise, investigators should bear in




mind some peculiarities of this optimization problem connected with con-struction and use of sensitivity models. The problem under considerationis a boundary-value one as well as any optimal design problem. Therefore,it is necessary to investigate existence conditions for sensitivity function of

the associated boundary-value problem. For this purpose, the methods pre-sented in Section 2.6 can be used. Since initial sensitivity equations are in-cluded in the mathematical model (8.92), to investigate the boundary-valueproblem we need sensitivity functions of the second order. It is known thatcontrollability of the plant at hand is a necessary condition for existence of optimal control. In the problem of optimal insensitive system design theplant incorporates the sensitivity model, i.e., is fairly complicated. Investi-gation of such a plant has its own peculiar features.

Consider the linear systems

Y = A(α)Y + B(α)V (8.93)

with sensitivity equations

U i = A(α)U i +∂A

∂αi

Y +∂B

∂αi

V, i = 1, . . . , m . (8.94)

Analysis of controllability of the system (8.93)–(8.94) for various types of dependence of the matrices A and B on the parameters α1, . . . , αm makes itpossible to obtain the following sufficient uncontrollability conditions [133]:

1. if only the matrix A depends on the vector α, then m > n

2. if only the matrix B depends on α, then m + 1 > q

3. in the general case, m > q + n

If the sensitivity matrix U (t) exists and the system (8.93)–(8.94) is con-trollable, the control action V (t) in the problem of optimal insensitive sys-tem design can be found, as was already noted, by methods developed in

the theory of optimal control.

8.5 Numerical Solution of Sensitivity Equations

8.5.1 General Structure of Numerical Integration Error

General methods of calculation of sensitivity functions of dynamic

systems with lumped parameters were considered in [67, 94]. Therein




an analysis of computer integration methods for sensitivity equations waspresented, possibilities of similar models were elucidated, and experimentaltechniques of determining sensitivity functions were proposed. In thepresent paragraph we present additional information on integration and

evaluation of the error in obtained sensitivity functions. Among computerintegration methods applied to sensitivity equations, the most popularare various methods of direct parallel integration of initial system andsensitivity equations. This means that the following system of equations isintegrated:

Y = F (Y, t), Y 0 = Y (0),

Φ = Γ(U,Y,t), U (0) = U 0,(8.95)

where

Φ =∂F

∂Y U +∂F

∂α .

For simplicity, we assume that the right side of the first equation of thesystem (8.95) is continuous with respect to t and has continuous second-order partial derivatives with respect to y and the parameter α. Let also y0be independent of α. A peculiar feature of the system (8.95) consists in thefact that it is semi-decomposed, i.e., the right side of the first equation doesnot depend on the solution of the second one. Moreover, the right side of thesecond equation is formed, in fact, by linearization of the right side of thefirst equation. These circumstances lead to the conclusion that the problem

of numerical integration of sensitivity equation is specific. This appliesmostly to the character of error increase in integration process. As wasshown above, the equation of the initial system is integrated independentlyof the sensitivity equation. Since numerical integration is approximate, asa result of integration of an equation

y = f (y, t), y(0) = y0

we obtain

y(t) = y(t) + ∆y(t),where ∆y(t) is the numerical integration error caused by transformed, com-putational and methodical errors.

The solution y(t) is substituted into the sensitivity equation

u = φ(u, y, t). (8.96)

As a result, even for correct integration of this equation, there arises theerror ∆u(t) such that

u(t) = u(t) + ∆u(t).




In the first approximation, this error is determined by the differentialequation

∆u =∂φ

∂u∆u +

∂φ

∂y∆y, ∆u(0) = 0,

or

∆u =∂f

∂y∆u +

∂ 2f

∂y2u +

∂ 2f

∂α∂y

∆y, u(0) = 0. (8.97)

From Equation (8.97) it is evident that the value of the error ∆u(t) isdetermined by the second-order derivatives of the right side of the initialequation. But Equation (8.96) is, in its turn, integrated approximatelyusing numerical methods. As a result, we obtain the following estimate of the sensitivity function:

u∗(t) = u(t) + ∆u(t)

or

u∗(t) = u(t) + ∆u(t) + ∆u(t) = u(t) + δu(t),

where

δu(t) = ∆u(t) + ∆u(t).

8.5.2 Formula for Integration of Sensitivity Equations

Assume that the scheme (8.95) is integrated by Euler’s method with astep h. Let us show that in this case the approximate estimates of thesensitivity function are given by the formula

un = hn−1j=1

∂f j∂α

n−j−1i=1

1 + h

∂f i∂y

, (8.98)

where∂f

i∂y =

∂f

∂y y = yi,t = ti

This can be proved by the induction method. By Euler’s method,

un = un−1 + hφ(un−1, yn−1, tn−1).

For n = 1 we have

u1 = u0 + hφ(u0, y0, t0) = h

∂f 0

∂α ,




and, for n = 2,

u2 = u1 + hφ(u1, y1, t1) = h∂f 1∂α

+ h∂f 0∂α

1 +

∂f 1∂y

.

Assume that Equation (8.98) holds for un−1 so that

un−1 = hn−2j=0

∂f j∂α

n−j−2i=1

1 + h

∂f i∂y

.

Then,

un = un−1 + hφ(un−1, yn−1, tn−1)

= un−1

1 + h∂f n−1

∂y

+ h

∂f n−1∂α

= hn−2j=0

∂f i∂α

n−j−2i=1

1 + h

∂f i∂y

×

1 + h∂f n−1

∂y

+ h

∂f n−1∂α

= hn−1j=0

∂f j∂α

n−j−1i=1

1 + h

∂f i∂y

.

8.5.3 Estimates of Solution Errors

It is known that the following estimate is true for the solution y(t) [41]:

|y(tn) − yn| ≤ hM

2N [(1 + hN )n − 1], (8.99)

where y(tn) is the precise solution for t = tn, yn is the approximatevalue found by Euler’s method, and the constants M and N satisfy theinequalities

|f (t, y1)− f (t, y2)| ≤ N |y1 − y2|,

df

dt = ∂f

∂y f +

∂f

∂t ≤ M,

where y1 and y2 are arbitrary values of y(t) from solution domain.

Let us find an estimate similar to (8.99) for the sensitivity function u(t).Consider the difference φ(t, u1)−φ(t, u2). Firstly, we analyze the case whenthe value of y(t) in the function φ(t,y,u) is accurately known. Then,

|φ(t, u1, y) − φ(t, u2, y)|

= ∂f

∂y(y1

−y2) = ∂f

∂y |y1 − y2| ≤

N |y1−

y2|




and

dφ

dt

=

∂ 2f

∂y2f u +

∂ 2f

∂y∂αf +

∂ 2f

∂t∂yu

+∂ 2f

∂t∂α +∂f

∂y ∂f

∂y u +∂f

∂α ≤ P.

In this case, the error estimate for integration of sensitivity equations isgiven by the formula

|u(tn) − un| ≤ hP

2N [(1 + hN )n − 1] . (8.100)

Then, let us estimate the difference φ(t, u1)

−φ(t, u2) with account for

error in determining y(t):

|φ(u1, y + ∆y, t) − φ(u2, y + ∆y, t)|

=

∂f

∂y+

∂ 2f

∂y2∆y

(u1 − u2)

≤

∂f

∂y

|u1 − u2| +

∂ 2f

∂y2

|∆y||u1 − u2|

≤N

|u1

−u2

|+ S max

t

|∆y

||u1

−u2

|,

where S is a constant such that∂ 2f

∂y2

< S.

Let

max |∆y| =hM

2N [(1 + hN )n − 1] .

Then,

|φ(u1, y, t) − φ(u2, y, t)| ≤ N |u1 − u2|

+ S hM

2N [(1 + hN )n − 1] |u1 − u2| = Q|u1 − u2|,

where

Q = N + S

hM

2N [(1 + hN )

n

− 1] .




Obviously, Q > N . As a result, the error of sensitivity function integra-tion can be estimated by the formula

|u(tn) − un| ≤hP

2Q [(1 + hQ)

n

− 1] . (8.101)

The estimate (8.101) is fairly robust and bears only theoretical meaning.Results more acceptable for practice can be obtained from the followingreasoning. Using Euler’s method, approximate values of the solution of thesystem (8.95) can be successively determined by the formulas

yi+1 = yi + hf (yi, ti),

ui+1 = ui + hφ(ui, yi, ti), i = 0, 1, 2, . . .

In the first integration step we have

y1 = y0 + hf (y0, t0),

u1 = u0 + hφ(u0, y0, t0) =∂f 0∂α

,

where

∂f 0∂α

=∂f

∂α

t0,y0

, y1 = y1 + ∆y1, u1 = u1 + ∆u1.

Let ∆y1 and ∆u1 are methodical integration errors. In the second step,

u2 = u1 + hφ(u1, y1, t1)

= h∂ f 0∂α

+ h∂ f 1

∂yu1 +

∂ f 1∂α

=

1 + h∂ f 1∂y

u1 + h

∂ f 1∂α

.

Then, errors in determining y1 and u1 lead to incorrectness in calculating∂f 1/∂y and ∂f 1/∂α and the whole right side. As a result, there appears anadditional error δu2 in determining u2. This error can be estimated by

u2 = 1 + h∂f 1

∂y+ h

∂ 2f 1

∂y2∆y1 (u1 + ∆u1) + h

∂f 1

∂α+ h

∂ 2f 1

∂α∂y∆y1 .




Hence,

u2 ≈

1 + h∂f 1∂y

u1 + h∂f 1

∂α+ h∂ 2f 1

∂y2u1∆y1

+ 1 + h

∂f 1

∂y ∆u1 + h

∂ 2f 1

∂α∂y ∆y1,

Therefore,

δu2 = h∂ 2f 1∂y2

u1∆y1 +

1 + h

∂f 1∂y

∆u1 +

∂ 2f 1∂α∂y

∆y1h.

In general, it can be shown that

ui+1 = ui + hφ(ui, yi, t) = ui + hφ(ui, yi, t) + δui+1, (8.102)

where

δui+1 =

1 + h

∂f i∂y

∆ui + h∆yi

∂ 2f i∂y2

+∂ 2f i

∂α∂y

.

Note that

δyi+1 =

1 + h

∂f j∂y

∆yi.

Thus, the second-order derivatives of the function f (y,t,α) play a signif-icant role in forming a part of the error δui+1. Notice that Relation (8.102)is a discrete analog (difference equation) for the differential equation (8.97)for the error ∆u(t).

8.5.4 Estimates of Integration Error for a First-OrderSystem

As follows from the above, integration errors for sensitivity functionsmay well significantly exceed integration errors for initial systems. Thiscircumstance can be visually demonstrated for linear stationary systems.

Consider the following simplest system

y = αy, y(0) = y0. (8.103)

The sensitivity equation with respect to the parameter α has the form

u = αu + y, u(0) = 0. (8.104)

Solving (8.104) for the variable y, we obtain

u− 2αu + α2

u = 0, u(0) = 0, u(0) = 0. (8.105)




The characteristic equation for (8.105) has the form

s2 − 2αs + α2 = 0,

i.e., there is a multiple root.For the case of a multiple root the solution of the sensitivity equation

(8.105) contains a term proportional to t:

u(t) = c1eαt + c2teαt

or

u(t) = y0teαt,

because c1 = 0 and c2 = y2.

On the other hand, the solution of Equation (8.103) has the form

y(t) = y0eαt.

Euler’s method gives the following approximate solutions:

un = nhy0(1 + αh)n−1,

yn = y0(1 + αh)n.

Then, the errors are given by

u(nh) − un = y0nh[eαnh − (1 + αh)n−1],

u(tn)− un = y0tn[eαtn − (1 + αh)n−1],

y(nh)− yn = y0[eαnh − (1 + αh)n],

y(tn)− yn = y0[eαtn − (1 + αh)n].

It is seen that the integration error of sensitivity equation increases as t.

8.5.5 Results of Numerical Calculation

To justify the results of the previous paragraph, we performed numericalintegration of the sensitivity equations using the methods of Euler andRunge-Kutta with different integration steps h. The initial system wasdescribed by the equation

y = f (y, t) = −1

T y (8.106)




with initial condition y(0) = y0.

Equation (8.106) corresponds to the accurate solution

y(t) = y0e−

tT . (8.107)

The sensitivity equation with respect to the parameter T and the accuratesolution of this equation have the form, respectively,

T u + u = −y(t) = −y0e− tT , u(0) = 0, (8.108)

u(t) =y0t

T 2e− t

T . (8.109)

For integration we assumed

y(0) = 1, T = 1 sec.

Using the simplest Euler’s method

yi+1 = yi + hf (ti, yi), ti = t0 + ih

and Runge-Kutta method of the fourth order

yi+1 = yi +1

6(b1 + 2b2 + 2b3 + b4),

where

b1 = f (yi, ti)h, b2 = f

ti +

h

2, yi +

b12

h,

b3 = f ti +

h

2 , yi +

b2

2 h, b4 = f (ti + h, yi + b3)h.

we determined the following values:

1. the numerical solution y(t) of the initial equation

2. the numerical solution u1(t) of the sensitivity equation for accuratey(t)

3. the numerical solution u2(t) of the sensitivity equation for y(t)




4. the absolute errors

∆u1(t) = u1(t) − u(t), ∆u2(t) = u2(t) − u(t)

in calculating the sensitivity functions

5. the relative errors

δ u1(t) =∆u1(t)

u1(t), δ u2(t) =

∆u2(t)

u2(t)

in calculating sensitivity functions

The results of computation are given in Figures 8.2–8.9. The first fourfigures present the curves obtained by Euler’s integration method, whilethe last four demonstrate the results of applying the Runge-Kutta algo-rithm. For comparison, the curves of accurate solutions are also shown inFigures 8.2, 8.3, 8.6, and 8.7. All the curves in Figure 8.6 practically co-incide. The curves u1(t) and u2(t), ∆u1(t) and ∆u2(t), δ u1(t) and δ u2(t)shown in Figures 8.7–8.9 also converge. Integration by both the methodswas performed for H = 0.05 sec, 0.1 sec and 0.2 sec. The total integrationinterval was 5 sec.

Analysis of the above curves show an evident tendency to increaseof integration errors for sensitivity functions as the integration step hincreases. Advantages of the Runge-Kutta method in comparison with

Euler’s method are also evident. For Euler’s method, the precision of calculating sensitivity function u1(t) is higher than that of u2(t).




Figure 8.2

y(t) and y(t) by Euler’s method




Figure 8.3

u(t) and u(t) by Euler’s method




Figure 8.4

∆u(t) by Euler’s method




Figure 8.5

δ u(t) by Euler’s method




Figure 8.6

y(t) and y(t) by the Runge-Kutta method




Figure 8.7

u(t) and u(t) by the Runge-Kutta method




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E. ROZENWASSER_Sensitivity of Automatic Control Systems_2000