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Robustness Analysis of Simultaneous Stabilization and itsApplications in Flight Control
by
Yasaman Saeedi
A thesis submitted in conformity with the requirementsfor the degree of Master of Applied Science
Graduate Department of Aerospace Science and EngineeringUniversity of Toronto
Copyright c© 2011 by Yasaman Saeedi
Abstract
Robustness Analysis of Simultaneous Stabilization and its Applications in Flight
Control
Yasaman Saeedi
Master of Applied Science
Graduate Department of Aerospace Science and Engineering
University of Toronto
2011
Simultaneous stabilization is an important problem in the design of robust controllers.
It is the problem of designing a single feedback controller which will simultaneously
stabilize every member of a finite collection of liner time-invariant systems. This provides
simplicity and reliability which is desirable in aerospace applications. It can be used as
a back up control system in sophisticated airplanes, or an inexpensive primary one for
small aircraft. In this work the robustness of the simultaneous stabilization problem,
known as the Robust Simultaneous Stabilization (RSS) problem, is addressed. First, an
optimization methodology for finding a solution to the Simultaneous Stabilization (SS)
problem is proposed. Next, in order to provide simultaneous stability while maximizing
the stability robustness bounds, a multiple-robustness optimization design methodology
for the RSS problem is presented. The two proposed design methodologies are then
compared in terms of robustness of the designed controller.
ii
Acknowledgements
I would like to express my extreme gratitude to my supervisor, Dr. Hugh H.T. Liu,
for providing me with the opportunity to pursue my interests in this research field. His
insight and perspective on the topic was of great value to me and none of this would
be possible without his extreme support and guidance throughout the past few years. I
would also like to thank him for his kind understanding and support while I was going
through a difficult time in the past few months.
I would also like to thank my co-supervisor, Dr. Ruben Perez, for his much valued
support and insight. His knowledge on the topic of simultaneous stabilization was of
great value to me, he always made time to answer my questions and provide guidance
and suggestions, and I definitely owe much of what I know to him.
I would like to thank all my friends in the FSC lab, all the past and present members.
They made my experience at UTIAS enthusiating and inspiring and their appreciated
friendship and support is what I will take away from this.
Last, but not least, I would like to thank my family for their never ending love and
support, and for always being there by my side. I also truly want to thank my circle of
friends for their spiritual support and friendship whenever I needed it.
iii
Contents
1 Introduction 1
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Motivation & Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Thesis Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Simultaneous Stabilization 6
2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Necessary and Sufficient Condition . . . . . . . . . . . . . . . . . . . . . 7
2.3 Simultaneous Stabilization by Linear State Feedback Control . . . . . . . 8
2.4 Optimal Stabilization via Linear State Feedback Control . . . . . . . . . 12
2.5 Bi-Level Decomposition-Based Strategy . . . . . . . . . . . . . . . . . . . 15
2.5.1 Decomposition formulation . . . . . . . . . . . . . . . . . . . . . . 15
2.5.2 Decomposed equivalent of necessary and sufficient conditions . . . 17
2.6 Parameter Optimization Approach . . . . . . . . . . . . . . . . . . . . . 17
2.6.1 Cost Function Definitions . . . . . . . . . . . . . . . . . . . . . . 18
2.6.2 Multiple Objective Design . . . . . . . . . . . . . . . . . . . . . . 19
2.7 Proposed SS Optimization Methodology . . . . . . . . . . . . . . . . . . 20
3 Robustness Analysis 22
3.1 Kharitonov’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Extreme Point Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2.1 Stability as a Nonsingularity Problem via the ‘Kronecker Lyapunov
Matrix’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
iv
3.2.2 Necessary and Sufficient Vertex Solution for Robust Stability . . . 26
3.3 Stability Robustness Bounds . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4 Robustness Analysis of a Numerical Example . . . . . . . . . . . . . . . 29
3.4.1 Extreme Point Solution Application . . . . . . . . . . . . . . . . . 30
3.4.2 Kharitonov’s Theorem Application . . . . . . . . . . . . . . . . . 34
3.4.3 Stability Robustness Bound Application . . . . . . . . . . . . . . 35
4 Robust Simultaneous Stabilization Problem 38
4.1 An Extended Decomposition-Based Strategy for the RSS Problem . . . . 39
4.2 Multi-Objective Optimization . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2.1 Formulation & the Concept of Pareto Optimality . . . . . . . . . 42
4.2.2 Weighted-Sum Method . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2.3 Multiple Robustness Optimization . . . . . . . . . . . . . . . . . . 46
5 Linear Simulation: An F4-C Flight Control Case Study 49
5.1 Introduction of the Test Case . . . . . . . . . . . . . . . . . . . . . . . . 49
5.2 Robustness Investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.2.1 Perturbations Due to CLα uncertainties . . . . . . . . . . . . . . . 52
5.2.2 Robustness Optimization . . . . . . . . . . . . . . . . . . . . . . . 59
6 Non-Linear Simulation: A CRJ-200 Flight Control Case Study 67
6.1 Modelling of the CRJ-200 . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.1.1 Nonlinear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.1.2 Linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.2 Introduction of the Test Case . . . . . . . . . . . . . . . . . . . . . . . . 72
6.3 Results: Ordinary and Gust-Encountered Flight . . . . . . . . . . . . . . 75
6.4 Robustness Investigation & Optimization . . . . . . . . . . . . . . . . . . 77
7 Conclusion and Future Developments 86
7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
7.2 Future Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
v
Bibliography 90
vi
List of Tables
3.1 F4-E flight operating conditions . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 Allowed perturbation resulting in a stable matrix family . . . . . . . . . 33
3.3 Allowed perturbation resulting in an unstable matrix family . . . . . . . 33
3.4 Kharitonov’s polynomials coefficients . . . . . . . . . . . . . . . . . . . . 35
3.5 Stability robustness bounds . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.1 F4-C flight operating conditions . . . . . . . . . . . . . . . . . . . . . . . 50
5.2 Simultaneous Stabilization solution . . . . . . . . . . . . . . . . . . . . . 51
5.3 Closed-loop system eigenvalues . . . . . . . . . . . . . . . . . . . . . . . 51
5.4 F4-C characteristics at different flight conditions . . . . . . . . . . . . . . 55
5.5 Maximum allowable deviation in ∆CLα . . . . . . . . . . . . . . . . . . . 56
5.6 Kharitonov’s polynomials coefficients . . . . . . . . . . . . . . . . . . . . 57
5.7 Solutions to the SS and RSS problems . . . . . . . . . . . . . . . . . . . 60
5.8 Effect of relaxing the robustness on |∆CLα|max . . . . . . . . . . . . . . . 61
5.9 Results from different RSS optimization methodologies and objective func-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.10 A comparison of the closed-loop eigenvalues for different optimization
methodologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.1 CRJ-200 flight operating conditions . . . . . . . . . . . . . . . . . . . . . 74
6.2 Open-loop system eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . 75
6.3 Closed-loop system eigenvalues . . . . . . . . . . . . . . . . . . . . . . . 76
6.4 Maximum allowable deviation in ai,j . . . . . . . . . . . . . . . . . . . . 81
6.5 Effect of relaxing the robustness on |∆ai,j|max . . . . . . . . . . . . . . . 83
vii
6.6 A comparison of the closed loop eigenvalues for different optimization
methodologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
viii
List of Figures
2.1 Multidisciplinary optimization and simultaneous stabilization problem [27] 16
3.1 response to initial condition of the perturbed nominal plant . . . . . . . . 34
3.2 response to initial condition of the perturbed nominal plant . . . . . . . . 36
4.1 Robust Simultaneous Stabilization solution approach . . . . . . . . . . . 41
4.2 Pareto set in a convex objective space . . . . . . . . . . . . . . . . . . . . 44
4.3 Pareto Set in a non-convex objective space . . . . . . . . . . . . . . . . . 44
4.4 Pareto sets in a weighted-sum optimization problem . . . . . . . . . . . . 45
5.1 Response to initial condition . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.2 Kharitonov’s robust stability graphical check for different flight conditions 58
5.3 Maximum Eigenvalue of the closed-loop system vs. CLα . . . . . . . . . . 59
5.4 Maximum Eigenvalue of the closed-loop system vs. CLα . . . . . . . . . . 63
5.5 Response to initial condition: SS problem . . . . . . . . . . . . . . . . . . 65
5.6 Response to initial condition: Multiple Robustness Optimization solution 66
5.7 Response to initial condition: Decomposition-Based Strategy solution . . 66
6.1 The Flight Training Device (FTD) Facility . . . . . . . . . . . . . . . . . 68
6.2 Pitch angle tracker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.3 Time history of the states at different flight conditions subject to a 5-deg
step input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.4 Time history of the states at different flight conditions subject to a 5-deg
step input, when encountered with gust . . . . . . . . . . . . . . . . . . . 79
ix
6.5 Time history of the states at different flight conditions subject to a 5-deg
step input, under the RSS problem solution . . . . . . . . . . . . . . . . 85
x
Chapter 1
Introduction
1.1 Overview
Simultaneous stabilization is an open and important problem in the design of robust
controllers. It is the problem of designing a single feedback controller which will simulta-
neously stabilize every member of a finite collection of liner time-invariant systems. The
simultaneous stabilization problem is defined as follows: Given n proper, linear time-
invariant plants P1(s), P2(s), ..., Pn(s), does there exist a single controller, C(s), such
that the closed loop (unitary feedback) system is internally stable for each of the given
plants? Also, under what conditions can such a controller be found? This has been a
problem of interest for many years now and various techniques have been proposed to
solve it.
The simultaneous stabilization problem of finding a single controller, which stabilizes
a finite set of different plants, is of practical interest. One motivation comes from the
stability requirements of a system operating in different modes. A common application is
the desire to control a system under normal operating conditions as well as under several
different failure modes, for example an industrial plant that has to operate in different
modes due to the possible sensor or actuator failure. A system may also have time-varying
parameters or several different normal modes of operation. For example, the dynamics of
the aircraft vary greatly with its altitude and speed. Hence, the attitude response of an
aircraft would be represented by a different mathematical model depending on different
1
Chapter 1. Introduction 2
flight conditions across the flight envelope. Moreover, ensuring the stable operation
of a non-linear system at several different steady states may be desirable in aerospace
applications. A linearized model of such a non-linear system operating at different points
may have time-varying parameters and the dynamics of the system will certainly change.
Thus, for slowly changing plants, a linear controller could potentially stabilize a nonlinear
system if it were to simultaneously stabilize the plants linearized about several different
points of operation. A single stabilizing controller provides simplicity and reliability, as
well, which is of outmost necessity in aerospace applications. It can be used as a back-up
control system in sophisticated airplanes, as well as an inexpensive primary one for small
aircraft.
Simultaneous stabilization is also a subtopic of robust control. Robust stabiliza-
tion simultaneously stabilizes a continuous range of plants, whose parameters lie within
predefined regions, subject to possible performance constraints. The major distinction
between robust control and simultaneous stabilization is in the number of plants they
are attempting to stabilize. Robust stabilization contends with an infinite (uncountable)
number of plants, whereas simultaneous stabilization deals only with a finite number
of plants. Nevertheless, the simultaneous stabilization of a finite number of systems is
difficult. Unlike robust stabilization in which the continuum of plants must not vary
too far from a nominal plant, there may be no assumptions on the interrelatedness of
the finite number of distinct plants. To date, there is a complete, tractable solution to
the simultaneous stabilization problem only when there are no more than two plants to
simultaneously stabilize.
Simultaneous stabilization was first studied more than three decades ago and has re-
ceived considerable attention since. Numerous authors have considered the simultaneous
stabilization problem for different types of systems and controllers. Youla et al. (1974)
[39] provided necessary and sufficient conditions for the problem of strong stabilization
of one plant, but it was Ackermann (1980) [1] who first considered the problem of si-
multaneous stabilization of multiple plants and presented a mathematical formulation
for it. He also proposed a solution using the concept of state feedback. Later, Franklin
and Ackermann (1981) [17] considered the problem of stabilization of the longitudinal
Chapter 1. Introduction 3
mode of an aircraft, and proposed a solution for design of a controller using two gyros
and accelerometer. Ackermann (1984) [3] was then able to propose a more efficient solu-
tion, using only two gyros. Vidyasagar and Viswanadham (1982) [34], Ghosh and Byrnes
(1983) [18], and Kale (1990) [20], studied the problem of designing a controller that would
simultaneously stabilize a collection of multiple-input multiple-output systems described
by transfer functions. It was shown by Saeks and Murray (1982) [32], and Vidyasagar
and Viswanadham (1982) [34] that the simultaneous stabilization of two systems reduces
to the problem of strong stabilization of one plant, considered previously by Youla et
al. (1974) [39]. A set of sufficient conditions was derived by Ghosh and Byrnes (1983)
[18] for simultaneous pole-assignability by dynamic output feedback. The single-input
single-output case was considered in detail by Debowski and Kurylowicz (1986) [13], who
developed a necessary and sufficient condition for the existence of a stabilizing compen-
sator and proposed an algorithm for its design. Petersen (1987) [30] studied the problem
of stabilizing a collection of single-input systems represented by state-space models via
non-linear state feedback control, and obtained a sufficient condition for the existence
of a stabilizing non-linear controller. Schmitendorf and Hollot (1989) [33] also consid-
ered the simultaneous stabilization of a collection of linear single-input systems via linear
state feedback control and obtained a sufficient condition for the existence of a stabiliz-
ing linear state feedback controller, which is later proposed in Wu et al. (1990) [35].
Later, Howitt and Luss (1991) [19] provided a necessary and sufficient condition for the
existence of a linear state feedback controller for the simultaneous stabilization problem.
Both Wu et al. (1990) and Howitt and Luss (1991) obtained such a controller by solving
a non-smooth optimization problem, where the objective was to minimize the function
representing the largest real part of the eigenvalues in order to increase the stability
margin, overlooking the transient behaviour of the system. Chow (1990) [10] uses the
definition of a “multimode” system controllability matrix (simultaneously describing the
controllability of all the systems) to provide a sufficient condition for simultaneously
placing the closed-loop system poles in specific locations. Boyd et al. (1993) [7] derived
a set of linear matrix inequalities and demonstrated that if a single solution exists, si-
multaneous stabilization can be guaranteed. It is then shown in Blondel (1994) [5] that
Chapter 1. Introduction 4
it is not possible to rationally decide whether a set of three or more systems is simulta-
neously stabilizable or not, although available sufficient conditions can be used. Here,
the term “rationally undecidable” means that it is not possible to find a necessary and
sufficient general criterion for simultaneous stability of the systems, involving only the
coefficients of the linear systems, rational or logical operations, and sign test operations.
Finally, Paskota et al. (1994) [23] provided a solution to the problem by solving nonlin-
ear Lienard-Chipart constraints, while Dorato et al. (1995) [14] applied the “quantifier
elimination” computational technique to verify Lienard-Chipart stability constraints.
The problem of simultaneously stabilizing a final set of n plants is equivalent to
strongly stabilizing (i.e. using a stable controller or compensator) a set of n− 1 plants.
When n = 2, a necessary and sufficient condition exists, known as the parity interlac-
ing property. The problem, however, becomes harder when n ≥ 3. As said before,
simultaneous stabilization of more than two plants is rationally undecidable and due to
the problem’s nature, an analytical solution is very difficult to find. However, it can
be shown that while the simultaneous stabilization problem is generally rationally in-
tractable, in most cases, it can be tackled numerically by an optimization algorithm, and
simultaneously stabilizing controllers can be found accordingly.
1.2 Motivation & Contribution
In this work, the robustness of the simultaneous stabilization problem known as the Ro-
bust Simultaneous Stabilization (RSS) problem is addressed. At first, a new optimization
methodology for finding a solution to the Simultaneous Stabilization (SS) problem is pro-
posed, based on the previous work presented in the literature. It is preferable for such
a controller to be able to simultaneously stabilize a set of plants and maintain that sta-
bility when encountered with uncertainties. More specificly in aerospace applications, it
is desirable to control an aircraft under normal flight conditions as well as under several
different failure modes and uncertainties, such as structure failure or sudden upset of the
flight conditions. Such a single stabilizing controller provides a back-up control system
with simplicity and reliability. Hence, in order to provide simultaneous stability for a set
Chapter 1. Introduction 5
of systems while maximizing the stability robustness bounds, a new multiple-robustness
optimization design methodology for the RSS problem is presented based on the concept
of multi-objective modelling. The two proposed design methodologies, i.e. the SS and
the RSS problem solutions, are then compared in terms of robustness of the designed
controller.
1.3 Thesis Layout
This dissertation is organized as following. Chapter 2 addresses the background required
for this thesis, including the formulation of the simultaneous stabilization problem as well
as the necessary and sufficient conditions required for the existance of a solution. This
chapter also provides a summary of several approaches from the literature for finding a
solution, and at the end of the chapter a new optimization methodology is presented.
Chapter 3 discusses the robustness of the simultaneous stabilization problem. Several
robustness analysis approaches, namely the extreme point solution, the Kharitonov’s
theorem, and the stability robustness bounds, are introduced for providing the physical
bounds of allowable perturbations. For comparison purposes, these approaches are then
applied to a numerical example. The Robust Simultaneous Stabilization (RSS) prob-
lem is described in Chapter 4, where the concept of multi-objective modelling is used
for providing a new multiple-robustness optimization design methodology for the RSS
problem. The final design is an improved controller in terms of robustness. Chapters 5
and 6 focus on the presentation and discusssion of the obtained results when the design
methodologies presented in the previous chapters are applied to a linear and a non-linear
flight control case study. Finally, conclusions and future work are made in Chapter 7.
Chapter 2
Simultaneous Stabilization
In this chapter, a brief review of various simultaneous stabilization methods is presented.
First, the problem of simultaneous stabilization is formulated, and a necessary and suf-
ficient condition for solving this problem is stated. Later, several approaches from the
literature for finding the solution to the simultaneous stabilization problem are summa-
rized. Finally, based on the previous work in the field, a new simultaneous stabilization
design methodology is proposed.
2.1 Problem Formulation
As described before, simultaneous stabilization is the problem of finding a single unique
control law that can stabilize a finite set of plants simultaneously. In mathematical terms,
consider a collection of m different systems described by the state-space equations:
xk = Akxk (t) +Bkuk (t) , k = 1, 2, ...,m, (2.1)
yk = Ckxk (t)
where in the case of single-input single-output linear systems shown above, xk ∈ Rn is
the state vector of the kth system, and uk is a scalar control. It is assumed that each
system is controllable. A single feedback gain vector f ∈ Rn is sought such that when
the control
uk(t) = −fTxk(t) (2.2)
6
Chapter 2. Simultaneous Stabilization 7
is implemented, each of the closed-loop systems
xk = (Ak −BkfT )xk, k = 1, 2, ...,m (2.3)
will be stable.
2.2 Necessary and Sufficient Condition
An equivalent to the above statement is that all the eigenvalues of each closed-loop
system must have negative real parts in order for the systems to be stable. Therefore,
there exists a solution to the simultaneous stabilization problem if and only if
minf
I = max︸ ︷︷ ︸1≤i≤n, 1≤k≤m
Re(λi,k)
≤ 0, (2.4)
where λi,k is the ith eigenvalues of the kth closed-loop system given by Eq. (2.3). If
the minimum value of I after the optimization is negative, the control law f which
minimizes the objective function will be a solution to the simultaneous stabilization
problem. However, if the minimum value of I is non negative, it can be concluded that
no solution exists to the simultaneous stabilization problem [19].
On the other hand, the solution of this problem may be unbounded. This problem
can be solved simply by imposing upper and lower bounds on the magnitudes of the
elements of the feedback gain vector f of the form
−µ ≤ fi ≤ µ i = 1, ..., n, (2.5)
where µ > 0 is positive and finite. Therefore f can not be arbitrarily large. It is also
preferred to choose a feedback gain that yields good performance. A large imaginary
part with respect to the real part of the eigenvalues will result in poor performance due
to insufficient damping. Therefore, in order to limit the minimum damping ratio, it is
desirable to limit the magnitude of the imaginary part with respect to that of the real
part [19]. To impose this constraint, a new parameter η ≥ 0 is introduced and the
following constraint is stated, where αi,k is the value of the real part of the closed-loop
systems’ eigenvalues and βi,k is that of the imaginary part.
η |βi,k| ≤ |αi,k| . (2.6)
Chapter 2. Simultaneous Stabilization 8
To make the objective function continuously differentiable, the above simultaneous
stabilization problem can be translated into an equivalent optimization problem:
minf I = γ < 0
subject to
Re (λi,k) ≤ γ,
η |βi,k| ≤ |αi,k| , i = 1, ..., n, k = 1, ...,m
−µ ≤ fi ≤ µ.
The following sections focus on a number of different approaches taken from the liter-
ature to solve such a problem. Needless to say, the problem of simultaneous stabilization
has been tackled from many different points of view. First, two optimization method-
ologies for solving the simultaneous stabilization problem are presented. Next, a bi-level
decomposition-based strategy and a parameter optimization approach are introduced.
Finally, an alternate optimization methodology is proposed.
2.3 Simultaneous Stabilization by Linear State Feed-
back Control
As a means of determining whether or not a particular simultaneous stabilization problem
has a solution (and thus evaluating the necessary and sufficient condition stated before),
and of constructing a suitable controller when one does indeed exist, an optimization
problem was proposed in [19].
Problem P1. Choose f ∈ Rn to minimize the objective function
I = max︸ ︷︷ ︸1≤i≤n,1≤k≤m
Re (λi,k) . (2.7)
If the minimum value of I after the optimization is negative, the f which minimizes the
objective function will be a solution to the problem. However, if the minimum value of I is
non-negative, it can be concluded that no solution exists to the simultaneous stabilization
problem. Unfortunately, the vast majority of numerical methods for minimization are
Chapter 2. Simultaneous Stabilization 9
suited only to problems with an objective function which is continuously differentiable.
As stated before, this problem can hence be transformed into an equivalent problem
whose objective function is continuously differentiable.
Problem P2. Choose f ∈ Rn and γ ∈ R to minimize the objective function
I = γ, (2.8)
subject to the constraint
Re (λi,k) ≤ γ, i = 1, ..., n, k = 1, ...,m. (2.9)
The eigenvalues λi,k are implicit functions of the feedback gain vector f . This rela-
tionship must be more explicit for computational purposes. Therefore, the eigenvalues
were allowed to be free parameters to be chosen along with f and γ, such that the ob-
jective function I is minimized. It is hence necessary to introduce additional constraints
which will explicitly describe the mathematical relationship between the feedback vector
and the eigenvalues. In order to develop this relationship, consider the kth single-input
single-output system. The vector f and the eigenvalues, λ1,k, λ2,k,..., λn,k of the matrix
Ak −BkfT are related in the following way. Let
pk (s) =n∏i=1
(s− λi,k) = δ1,k + δ2,ks+ ...+ δn,ksn−1 + sn (2.10)
be the characteristic polynomial of the kth closed-loop system. The closed-loop charac-
teristic polynomial coefficients vector ck ∈ Rn, defined by
ck =
δ1,k
δ2,k
...
δn,k
(2.11)
is a function of f since it is dependant on the feedback gain vector. This relationship
takes the form
Gkf + hk = ck, (2.12)
where Gk ∈ Rn×n and hk ∈ Rn can be defined by the following algorithm:
Chapter 2. Simultaneous Stabilization 10
Step 1. Let eTk be the last row of[bk Akbk · · · An−1
k bk
]−1
.
Step 2. Gk = −[ek ATk ek · · ·
(An−1k
)Tek
]−1
.
Step 3. hk = Gk (Ank)T ek.
Although the eigenvalues will be treated as free parameters in the optimization proce-
dure, they must be chosen to satisfy constraints of the form (2.12). Note that some of the
eigenvalues may be complex, and it is assumed that they will take one of the following
two forms:
• if n is even, then
λi,k = αi,k + jβi,k, i = 1, ..., n, (2.13)
• if n is odd, then
λi,k = αi,k + jβi,k, i = 1, ..., n− 1,
λn,k = αn,k.
This type of parametrization of the eigenvalues is chosen because, when n is odd, at
least one real eigenvalue must exist. Since we are dealing with real systems, it is clear
that the eigenvalues are either real or they must occur in complex conjugate pairs. To
generalize this to any value of n, let us first define the even integer
N =
n, if n is even,
n− 1, if n is odd.(2.14)
Since the coefficients of the characteristic equations are real values, the following con-
straint should also be imposed:
gk =
β1,k + β2,k
α1,kβ2,k + α2,kβ1,k
β3,k + β4,k
α3,kβ4,k + α4,kβ3,k
...
βN−1,k + βN,k
αN−1,kβN,k + αN,kβN−1,k
= 0. (2.15)
Chapter 2. Simultaneous Stabilization 11
Another consideration is that the solution to problem P2 may be unbounded. In a
particular simultaneous stabilization problem, it may be possible to shift the real part of
the greatest eigenvalue and thus the objective function I all the way to negative infinity,
and therefore a minimum will not exist. This difficulty can be easily overcome, however,
by placing simple bounds on the magnitudes of the elements in the feedback gain vector
f of the form
−µ ≤ fi ≤ µ, i = 1, ..., n , (2.16)
where µ > 0 is chosen to be real and finite. With these constraints, f can not be
made arbitrarily large, and therefore the objective function is always bounded. If, for a
particular value of µ there is no solution to problem P2, µ can be increased until a solution
is found, or until it is clear that no solution exists to the simultaneous stabilization
problem. This has an added advantage in limiting the magnitude of the control law
through a single parameter.
The performance of the controller is another issue to be considered, since it is desirable
to find a simultaneously stabilizing controller with acceptable performance. Even in a
stabilized system, a very large imaginary part βi,k compared to the real part αi,k will
result in rapid oscillation and poor performance. One solution is to limit the magnitude
of the imaginary part with respect to that of the real part. A new parameter η ≥ 0 is
now introduced and the constraints
βi,k ≥ 0, i = 1, 3, ..., N − 1, k = 1, ...,m
βi,k ≤ 0, i = 2, 4, ..., N, k = 1, ...,m
αi,k + ηβi,k ≤ 0, i = 1, 3, ..., N − 1, k = 1, ...,m
are imposed. Now if the solution to the simultaneous stabilization problem does not
yield good performance, η can be increased in order to improve the performance of the
controller.
Finally, the general optimization problem which gives a solution to the simultaneous
stabilization problem is stated as follows:
Problem P3. Given parameters µ > 0 and η ≥ 0, Choose f ∈ Rn, γ ∈ R, αi,k ∈ R, and
Chapter 2. Simultaneous Stabilization 12
βi,k ∈ R to minimize the objective function
I = γ, (2.17)
subject to the constraints
−µ ≤ fi ≤ µ, i = 1, ..., n,
αi,k ≤ γ, i = 1, ..., n, k = 1, ...,m,
Gkf + hk = ck, k = 1, ...,m,
gk = 0, k = 1, ...,m,
βi,k ≥ 0, i = 1, 3, ..., N − 1, k = 1, ...,m,
βi,k ≤ 0, i = 2, 4, ..., N, k = 1, ...,m,
αi,k + ηβi,k ≤ 0, i = 1, 3, ..., N − 1, k = 1, ...,m.
If after optimization I is negative, the f that makes it negative is the solution to the
simultaneous stabilization problem, and if I is greater than or equal to zero, no solution
can be found within the given boundary for the elements of the control law f . It was also
seen that when applied to a number of case studies, the optimization process results in
a single best solution regardless of the chosen initial point. Hence, one can assume that
the obtained simultaneously stabilizing controller is the global optimum.
2.4 Optimal Stabilization via Linear State Feedback
Control
In this section, another optimization methodology is introduced as was originally pre-
sented in [23], which focuses on designing a control law with optimal transient response.
Again, consider the collection of m linear time-invariant systems:
x (t) = Akxk (t) +Bkuk (t) ; k = 1, 2, ...,m (2.18)
where x (t) ∈ Rn is the state and u (t) ∈ R is the control, and Ak ∈ Rn×n, Bk ∈ Rn are
independent of time. We assume that [Ak, Bk] is controllable for each k = 1, 2, ...,m. Let
Chapter 2. Simultaneous Stabilization 13
us define the objective function for the kth system as follows:
Jk =1
2
∫ ∞0
(xTkQxk + uTkRuk
)dt, (2.19)
where Q ∈ Rn×n is symmetric and positive definite, and R > 0. This objective function
will ensure that a good transient response is obtained without using an unnecessary
amount of control. We consider linear state feedback controllers of the form:
uk = −fTxk, (2.20)
where f is known as the feedback gain vector. We require f to be the same for all the
systems, and −µ ≤ fi ≤ µ for all i = 1, ..., n where µ is a given positive real constant.
Substituting (2.20) into (2.18) and (2.19), one obtains
xk (t) = Ak (f)xk (t) , k = 1, 2, ...,m (2.21)
and
Jk (f) =1
2
∫ ∞0
[xTk(Q+ fRfT
)xk]dt, k = 1, 2, ...,m, (2.22)
where
Ak (f) =(Ak −Bkf
T), k = 1, 2, ...,m. (2.23)
Let F be a subset of Rn such that if f ∈ F , then the matrix Ak (f) is stable for each
k. F is hence called the set of simultaneously stabilizing feedback gain vectors (i.e. if
there is not such a simultaneously stabilizing controller, F is a null subset and the SS
problem has no solution). For brevity, a vector in F will also be called a feasible vector,
since it is a feasible potential solution to the problem of simultaneous stabilization. Let
Fµ be the subset of F consisting of all the vectors that satisfy:
−µ ≤ fi ≤ µ, i = 1, ..., n. (2.24)
The problem is now formally stated as follows:
Problem 1 Find a feasible vector f such that the cost function S (f) =∑mk=1 Jk (f) is
minimized over Fµ.
Chapter 2. Simultaneous Stabilization 14
Note that a solution to this optimization problem will give us a controller which will
simultaneously stabilize all the systems with a limited control law gain, and further-
more due to the choice of the objective function, a desirable transient behaviour will be
obtained without using unnecessary control effort.
If the closed-loop system Ak − BkfT is stable, it is known (Choi and Sirisena, 1974
[9]) that
JK (f) =1
2
∫ ∞0
[xTkQxk +
(−fTxk
)TR(−fTxk
)]dt
=1
2
∫ ∞0
[xTk(Q+ fRfT
)xk]dt =
1
2tr (Pk) ,
where matrix Pk is the positive definite solution of the matrix Lyapunov equation
X(Ak −Bkf
T)
+(Ak −Bkf
T)TX +Q+ fRfT = 0. (2.25)
Since f ∈ F , the matrix Ak − BkfT is stable for each k = 1, 2, ...,m. Thus, for each
k = 1, 2, ...,m, the matrix Lyapunov equation has a unique positive definite solution. This
leads to an easy way of calculating Jk (f). Also, for a given matrix Ak (f) = Ak −BkfT ,
its characteristic polynomial is given by
det(λI − Ak (f)
)= λn + a1λ
n−1 + ...+ an. (2.26)
Then, Ak (f) is stable if and only if the following Hurwitz conditions are satisfied:
an > 0, an−2 > 0, ...
H1 > 0, H3 > 0, ... ,(2.27)
where Hi denotes the ith leading principal minor of the n× n Hurwitz matrix
H =
a1 a3 a5 · · · a2n−1
1 a2 a4 · · · a2n−2
0 a1 a3 · · · a2n−3
0 1 a2 · · · a2n−4
......
.... . .
...
0 0 0 · · · an
(ar = 0, r > n) . (2.28)
Chapter 2. Simultaneous Stabilization 15
Note that there are several alternative versions of Hurwitz’s necessary and sufficient
conditions.
It is concluded that f is a feasible vector, f ∈ F , if and only if the above Hurwitz
stability constraints are satisfied for each k = 1, 2, ...,m. Moreover f ∈ Fµ if, in addition,
constraints −µ ≤ fi ≤ µ for i = 1, 2, ..., n are satisfied as well. Hurwitz’s necessary and
sufficient conditions for matrix stability can also be rewritten as a set of inequalities.
Hence, f ∈ F if and only if
gki (f) < 0, (2.29)
where gki (f) represents elements and leading principal minors of the Hurwitz matrix H
for matrix Ak −BkfT . In the light of this, Problem 1 can now be rewritten as:
Problem 2 Given parameter µ > 0, minimize S (f) =∑mk=1 Jk (f) with respect to f ,
subject to the constraints
−µ ≤ fi ≤ µ i = 1, 2, ..., n,
gkj (f) ≤ 0 k = 1, 2, ...,m, j = 1, 2, ...,M,
where M is the number of Hurwitz stability constraints.
2.5 Bi-Level Decomposition-Based Strategy
In the approach presented in [27], a bi-level design optimization architecture is adopted
in which design of each individual plant is taking place at the bottom level, and the top
level optimization aims for single control convergence of those individual controllers.
2.5.1 Decomposition formulation
An analogy between the simultaneous stabilization problem and the general multidisci-
plinary design optimization (MDO) problem formulation can be established by realizing
that each plant-stabilizing process is equivalent to a discipline in the general MDO pro-
cess, as shown in Fig. 2.1, in which the common design variable among all disciplines is
the simultaneous control gain K.
Chapter 2. Simultaneous Stabilization 16
Figure 2.1: Multidisciplinary optimization and simultaneous stabilization problem [27]
A bi-level optimization strategy that enables decoupling and decomposition is used to
solve the simultaneous stabilization problem. At the system level (SL) the optimization
problem is stated as:
minzSL,ySLf(zSL, ySL)
subject to G∗j [zSL, z∗j , ySL, y
∗j (x∗j , y∗k, z∗j )] = 0
j, k = 1, ...,m, k 6= j
where f(zSL, ySL) represents the system-level objective function and Gjs are the compat-
ibility constraints, one for each subsystem. The lower subsystem-level (SSL) objective
function is formulated such that it minimizes the discrepancy between the given system
level variables and the subsystem variables that meet the local disciplinary constraints.
Thus, the jth subsystem optimization is stated as:
minzj ,yj ,xjGj[zSL, zj, ySL, yj(xj, yk, zj)]
=∑
(zSL − zj)2 +∑
(ySL − yj)2
subject to gj[xj, zj, yj(xj, yk, zj)] ≤ 0
Chapter 2. Simultaneous Stabilization 17
2.5.2 Decomposed equivalent of necessary and sufficient condi-
tions
In simultaneous stabilization process, each plant-stabilization effort will be decomposed
at the subsystem levels, and the system level ensures stabilization with a unique control
gain K. using above formulation, a decomposed equivalent of Eq. (2.4) is:
SL|KSL,γSL min I = γSL (2.30)
subject to G∗j = 0 j = 1, ...,m
SSLj|Kj ,γj min Gj = (Kj −KSL)2 + (γj − γSL)2 (2.31)
subject to Re(λi,j) ≤ γj i = 1, ..., n
A solution to the problem will exist if and only if the system level converges with
an optimal control gain K∗, which minimizes the system-level objective function I and
makes it negative. Furthermore, at convergence, the subsystem compatibility constraints
(G∗j = 0) are met, which means each plant is stabilized by an optimal gain.
2.6 Parameter Optimization Approach
In the approach presented in [12], a parameter optimization methodology is used for
controller design. Since it is difficult to optimize a problem over multiple specifications
directly, a series of cost function are defined such that an improvement in a cost function
results in an improvement in the related specification.
In the parameter optimization framework, the controller elements are the variables
of optimization and hence it is very similar to a multiple objective parameter synthesis
(MOPS) method. However, instead of optimizing over the specifications directly (e.g.,
settling time, rise time, overshoot), cost functions are defined. This method then converts
a multiple-objective problem into a single-objective problem, by use of the weighted-sums
method.
Chapter 2. Simultaneous Stabilization 18
2.6.1 Cost Function Definitions
In what follows, a number of cost function definitions associated with different types of
specifications are obtained. Later, a parameter optimization is performed to minimize a
weighted sum of these cost functions.
There are a number of different types of specifications that must be satisfied in the
design of a control system, some of which are listed in the following:
1. Tracking
To satisfy the tracking specifications, the designer may wish to minimize both the
transient and steady-state errors for a step command response. Hence, the following
cost functions are defined:
δJt =r∑i=1
∫ ∞0
δeidTδeiddt = tr
{BT A−TL0A
−1B}, (2.32)
Jtss =r∑i=1
eidssTeidss = tr
{(D − CA−1B
)T (D − CA−1B
)}, (2.33)
where L0 ≥ 0 satisfies the Lyapunov equation ATL0 + L0A+ CT C = 0.
2. Controller Limitations
It is also desirable to minimize the control response to a step command. To satisfy
the limitation of the control law, the designer may wish to minimize both the
transient and steady-state control effort, as well as the control rate. Thus, the
following cost functions are defined:
δJu =r∑i=1
∫ ∞0
δuiTδuidt = tr
{BTclA−Tcl LuA
−1cl Bcl
}, (2.34)
Juss =r∑i=1
uissTuiss = tr
{(Hcl −GclA
−1cl Bcl
)T (Hcl −GclA
−1cl Bcl
)}, (2.35)
and
Ju =r∑i=1
∫ ∞0
uiT
uidt = tr{BTclLuBcl
}, (2.36)
where Lu ≥ 0 satisfies the Lyapunov equation ATclLu + LuAcl +GTclGcl = 0.
3. Disturbance Rejection
The transient and steady-state response to disturbance can be calculated and the
corresponding cost functions can be defined.
Chapter 2. Simultaneous Stabilization 19
4. Gain Limitations
To ensure that the controller gains do not become too large, we may penalize them
in the cost function by adding the following term:
Jgain = tr{KTK
}. (2.37)
5. Pole Location Constraint
The simplest constraint of this kind is to limit the closed-loop eigenvalues to a
region inside the left half plane (e.g., the closed loop poles are desired to lie to the
left of −δ, δ ≥ 0). The following LQ cost function ensures this specification:
JPL = E[∫ ∞
0xTQxdt
]= tr {PPLM0} , (2.38)
where x(t) is the unforced response of x = (Acl + δI)x, M0 = E[x(0)x(0)T ] > 0,
and PPL is the solution of (Acl + δI)TPPL + PPL(Acl + δI) +Q = 0.
6. Robustness and Fault Tolerance
One way to consider the robustness is to define a finite set of plants which reflects
the allowable deviations of the actual plant. For robust stability and robust per-
formance of this set of plants, LQ cost functions, as well as tracking, controller
limitations, and disturbance rejection cost functions can be added, respectively. In
the case of fault tolerance, the finite set of plants refers to the closed-loop systems
corresponding to allowable component failures.
2.6.2 Multiple Objective Design
Now the controller is designed by solving the single-objective optimization problem.
minKJ = wδtδJt + wtssJtss + wδuδJu + wussJuss
+wuJu + wδdδJd + wdssJdss + wPLJPL
+wgainJgain +W, (2.39)
where w ≥ 0 is the specific weight for each cost function, W is the weighted sum of any
additional cost function added for robustness consideration, and K is the control law.
Chapter 2. Simultaneous Stabilization 20
The variable parameters are hence the weights, the damping ratios, and the un-damped
natural frequencies in the desired system. By finding a local minimum to (2.39), an
optimal solution is found with respect to the cost functions. This strategy is further
explained in [12].
2.7 Proposed SS Optimization Methodology
Based on the literature review, the following methodology for solving the problem of
simultaneous stabilization is proposed and further used for the work presented. The
motivation was to decrease the number of design variables and constraints in the opti-
mization problem. The optimization problem could later be solved using the MATLAB
function fminimax which finds a constrained minimum of the largest element of an ar-
ray of non-linear multivariable functions, starting at an initial estimate. In other words,
it minimizes the worst-case (largest) value of a set of multivariable objective functions.
This is generally referred to as the minimax problem. fminimax internally reformulates
the minimax problem into an equivalent Nonlinear Linear Programming problem by ap-
pending additional (reformulation) constraints of the form Fi (x) ≤ γ to the constraints,
and then minimizing γ over the design variable x. It then uses respective functions to
calculate the value of the objective functions and that of the constraints, and finds the
minimum using a sequential quadratic programming (SQP) method.
Again, consider the same collection of m linear time-invariant systems:
x (t) = Akx (t) +Bku (t) , k = 1, 2, ...,m (2.40)
where x (t) ∈ Rn is the state and u (t) ∈ R is the control, and Ak ∈ Rn×n, Bk ∈ Rn
are state and input matrices and independent of time. Also, consider the linear state
feedback controller of the form:
uk = −fTxk (2.41)
where f is known as the feedback gain vector. To limit the elements of the controller, f is
required to be the same for all the systems, and −µ ≤ fi ≤ µ, i = 1, 2, ..., n, where µ is
Chapter 2. Simultaneous Stabilization 21
a given positive real constant. We may now state the simultaneous stabilization problem
as follows:
Problem Given parameter µ > 0 and η ≥ 0, minimize I = max Re (λi,k) with respect
to f , subject to the constraints
−µ ≤ fi ≤ µ, i = 1, 2, ..., n
αi,k + η |βi,k| ≤ 0, k = 1, 2, ...,m, i = 1, 2, ..., n.
Note that a solution to this optimization problem will give us a controller that will
simultaneously stabilize all the systems without using unnecessary control effort. The
application of this design procedure is further shown in the robustness analysis of a nu-
merical example in section 3.4, as well as the linear and nonlinear case studies introduced
in Chapters 5 and 6.
Chapter 3
Robustness Analysis
In the previous chapter, the problem of simultaneous stabilization was introduced and
necessary and sufficient condition for the existance of such a controller was presented,
along with several techniques introduced in the control theory literature to design such a
controller. However, a less explored aspect is the robustness of such a control law. Since
uncertainties and parameter variations can certainly occur in real life aerospace applica-
tions, the designed controller should be able to provide stability in case of uncertainties
such as gusts, sudden change of flight conditions across the flight envelope, aircraft con-
figuration, and aerodynamic changes due to damages and failures. For instance, gusts
or winds influence aircraft dynamics. Low frequency wind has an effect on tracking per-
formance whereas high frequency wind affects the flight stability of an aircraft. That
being said, this thesis aims at investigating the robustness of the simultaneous stabi-
lization design methods for systems represented by a group of linear state-space models
with structured uncertainties. In the following, several robustness analysis approaches,
namely the extreme point solution, the Kharitonov’s theorem, and the stability robust-
ness bounds, are introduced to provide the physical bounds of allowable perturbations.
Furthermore, their application is shown using a numerical example.
22
Chapter 3. Robustness Analysis 23
3.1 Kharitonov’s Theorem
In the simultaneous stabilization problem, the stability of the uncertain plants in the
imaginary space bounded by the given set of simultaneously stabilized plants is not
guaranteed. Hence, a method for analyzing the robustness of the design is discussed,
an application of which can be found in [26]. This approach, namely the Kharitonov’s
theorem, is presented as follows:
Definition 1 A Polynomial p(s, q) = {∑ni=0 δi(q)s
i|q ∈ Q} is an interval polynomial if
p(s, q) has an independent uncertainty structure, each coefficient depends continuously on
the uncertainty vector q, and the uncertainty bounding set Q is an n-dimensional box.
Associated with the interval polynomial are four fixed Kharitonov polynomial defined
from the upper and lower bounds of the interval polynomial coefficient as:
K1(s) = δ−0 + δ−1 s+ δ+2 s
2 + δ+3 s
3 + δ−4 s4 + δ−5 s
5 + ..., (3.1)
K2(s) = δ+0 + δ+
1 s+ δ−2 s2 + δ−3 s
3 + δ+4 s
4 + δ+5 s
5 + ..., (3.2)
K3(s) = δ+0 + δ−1 s+ δ−2 s
2 + δ+3 s
3 + δ+4 s
4 + δ−5 s5 + ..., (3.3)
K3(s) = δ−0 + δ+1 s+ δ+
2 s2 + δ−3 s
3 + δ−4 s4 + δ+
5 s5 + .... (3.4)
Theorem 1 An interval polynomial with invariant degree is robustly stable (Hurwitz for
every point in the uncertainty space) if and only if its four Kharitonov polynomials are
stable.
Proof Several proofs of Kharitonov’s Theorem are available in the literatures (see [16],[31],
and [25]).
The Kharitonov polynomials also provide the basis for a graphical check of robust
stability, using Zero Exclusion Condition.
Definition 2 Associated with the Kharitonov polynomials is a Kharitonov rectangle
whose four vertices are obtained by evaluating the four Kharitonov polynomials at s =
jw0. The size and position of this rectangle change with w, while its sides always remain
parallel to the real and imaginary axis.
If an interval polynomial with an invariant degree of n is robustly stable, the Kharitonov
polynomials will move in a counter-clockwise direction through n quadrants of the com-
Chapter 3. Robustness Analysis 24
plex plane without touching or passing through the origin (Zero Exclusion Condition).
Theorem 2 An interval polynomial having invariant degree is robustly stable if and
only if the origin of the complex plane is excluded from the Kharitonov rectangle at all
nonnegative frequencies, i.e. 0 /∈ p(jw, q),∀w ≥ 0.
The Kharitonov’s theorem allows for significant reduction in the number of polyno-
mials to be checked for stability in order to assure robust stability of the design. For
the controller design purposes, an interval polynomial is defined from the closed-loop
characteristic equation.
3.2 Extreme Point Solution
In this approach, robustness of the designed controller can be investigated by checking
robust stability of a polytope of matrices. This method, as presented in [38] and [37],
provides a solution to the problem in the form of extreme points, using the fact that robust
stability problem can be converted to the robust nonsingularity problem involving the
Kronecker Lyapunov Matrix. Consider the following linear state-space description:
x = A(q)x(t) q ∈ Q, (3.5)
where x(t) ∈ Rn and q is an s vector of uncertain parameters varying in the prescribed
compact set Q. Let qi be given upper and lower bounds such that qi ≤ qi ≤ qi for
i = 1, ..., s. Now the matrix A(q) can be written as
A(q) = A0 +s∑i=1
qiAi (3.6)
where A0 is the nominal matrix and Ai are constant specified matrices, reflecting the
structure of the uncertainty. The set of possible matrices A(q) = [A(q) : q ∈ Q] forms a
polytope of matrices in Rn×n. Denoting qi as the ith extreme point of the set Q, generated
by each element in q taking its minimum or maximum value, and the extreme matrix
A(qi) as Ai, the above matrix family can be written as
A =
{A =
l=2s∑i=1
αiAi, αi ≥ 0,
∑αi = 1
}. (3.7)
It is assumed that all the vertex matrices Ai are asymptotically stable.
Chapter 3. Robustness Analysis 25
3.2.1 Stability as a Nonsingularity Problem via the ‘Kronecker
Lyapunov Matrix’
As said before, the robust stability problem can be converted to a robust nonsingularity
problem involving the Kronecker Lyapunov matrix denoted by L in the ‘Dagger’ space,
defined as follows:
L = A† = A⊗ In + In ⊗ A, (3.8)
where In is the identity matrix and L is a square matrix of dimension m = 12n(n + 1).
By defining Li as Li = (Ai)†, any member of the matrix polytope can simply be written
as a member of the ‘dagger polytope L’ as:
L =
{L =
l=2s∑i=1
αiLi, αi ≥ 0,
∑αi = 1
}. (3.9)
It is explained in [38] that the polytopic matrix family ‘A’ with all its vertex matrices
being asymptotically stable, is robustly stable if and only if the matrix family ‘L’ is
robustly ‘real axis nonsingular’ and thus is robustly asymptotically stable. Here, ‘real
axis nonsingularity’ means the same sign nature of the real eigenvalues and the real parts
of the complex pair eigenvalues of the ‘Kronecker Lyapunov’ matrix family.
Now let λj be the jth eigenvalue (j = 1, 2, ...,m) of the ‘dagger space’ matrix L with
r real eigenvalues, and let D(L) denote the ‘Weighted Determinant’ of matrix L as:
D(L) = (−1)mλ1λ2...λm. (3.10)
By defining the concepts of ‘Weighted Real Axis Determinant’ and ‘Real Axis Non-
singularity Scalar’ as below, it is clear that they are constrained to be positive and real
for a real axis nonsingular matrix.
Γ(L) =([(−1)rλ1λ2...λr]
1r
) ([(−1)m−rλr+1λr+2λm
] 1r
)= ∆(L)σ(L). (3.11)
Thus, the polytopic matrix family ‘A’ is robustly stable if and only if each member
of the matrix family ‘L’ possesses a positive real Γ.
Chapter 3. Robustness Analysis 26
3.2.2 Necessary and Sufficient Vertex Solution for Robust Sta-
bility
The interior matrices belonging to the matrix polytope L can be expressed as a con-
vex combination of not just the vertex matrices Li, but of a set of special matrices
labelled as ‘Virtual Center Anchored Virtual Rays’, namely Lvc,k(ρc, i1, i2, .., ik, j) =
[Lj + ρcLvc,k(i1, i2, ..ik)]. For Example:
L = γ(α1L1 + α2L
2 + α3L3) = γ1L
1 + γ2L2 + γ3L
3 =
α1[L1 + ρ1(L1 + L2 + L3)] + α2[L2 + ρ2(L1 + L2 + L3)] + α3[L3 + ρ3(L1 + L2 + L3)].
Now define a set of ‘Real Axis Nonsingularity’ matrices as follows:
Lrn,k(i1, i2, ..ik, j) = −[(Lvc,k(i1, i2, ..ik))
−1 Lj], (3.12)
where Lvc,k(i1, i2, ..ik) denotes the ‘virtual center’ matrix formed with k matrices taken
at a time (i1 = 1, 2, ..l; ..., ik = 1, 2, ..l) as:
Lvc,k(i1, i2, ..ik) = (L1 + L2 + ...+ Lk). (3.13)
A necessary and sufficient ‘extreme point’ condition for checking the robust stability
of a polytope of matrices, as further explained in [38], is presented in the following.
Theorem All the matrices belonging to the polytopic matrix family ‘A’ are asymptot-
ically stable if and only if for all (i1 = 1, 2, ..l; i2 = 1, 2, ..l; ..., il = 1, 2, ..l) the ‘Real
Axis Nonsingularity Matrices’ Lrn,k(i1, i2, ..ik, j) with k taking on values from 2 thru l
are ‘Real Axis Nonsingular’ (i.e., possess positive real Γ) and thus are ‘Asymptotically
Stable’.
3.3 Stability Robustness Bounds
As discussed before, the problem of maintaining the stability of a nominally stable system
subjected to perturbations has been of considerable interest to researchers. In this section,
the focus is on the analysis of system stability with parameter uncertainty in state-
space models. In particular, we are interested in obtaining some bounds on the system
Chapter 3. Robustness Analysis 27
uncertainties that guarantee the stability of the perturbed system, assuming that the
nominal system is already stable.
Stability analysis in the area of time domain stability conditions has been available for
some time. However, explicit bounds on the perturbation of a linear system in order to
maintain stability is fairly a new topic which has first been reported only by Patel, Toda,
and Sridar [22], and Patel and Toda [24]. These bounds are given for ‘’highly structured
perturbations‘’ as well as for ‘’weakly structured perturbations‘’. For a given model
structure, highly structured perturbations are those for which only a magnitude bound
on individual matrix elements is known. Weakly structured perturbations are those for
which only a spectral norm bound for the error is known. Mathematical approaches
presented here will provide a bound for highly structured perturbations.
In what follows, the problem of robust stability analysis of linear systems in state-
space models is considered. It is known that every possible state matrix or the polytope
of the matrices can be written as the sum of a nominal stable plant plus some uncertainty
matrices. Considering a system with structured uncertainty and using a Lyapunov matrix
equation solution, an upper bound on allowable structured perturbations can be found
which maintains the stability of the nominal system. Different methods in driving this
upper bound is proposed in the literatures with different levels of conservatism. In [24],
the following state space representation of a perturbed dynamic system is given:
x = Ax(t) + Ex(t) = (A+ E)x(t), (3.14)
where x is the n-dimensional state vector, A is an n × n time invariant, asymptotically
stable nominal matrix, and E is an n × n error matrix. Moreover, the entries of E are
such that
|Eij| ≤ ε, (3.15)
where ε is the magnitude of the maximum deviation allowed. It was shown in [24] that
the perturbed system is stable if
ε <1
nσmax [P ]= µεP , (3.16)
where σmax [P ] represents the largest singular value of P , and P is the solution of the
Chapter 3. Robustness Analysis 28
Lyapunov matrix equation:
ATP + PA+ 2In = 0, (3.17)
where In is an n× n identity matrix.
Patel and Toda [24] derived the stability robustness bounds with the assumption that
every element of the system matrix is perturbed independently of the other. Yedavalli
[36] on the other hand, obtained a less conservative robustness bound by assuming more
structure on the perturbation, as presented in the following. It was shown in [36] that
the system matrix A+ E of Eq. (3.14) is stable if:
|Eij|max = ε <1
σmax [|P |Un]s= µεY , (3.18)
where |P | denotes a matrix formed by taking the absolute value of every element of P ,
[P ]s denotes the symmetric part of P , and Un is an n × n matrix whose entries are
unity, i.e., Unij = 1 for all i, j = 1, ..., n if the corresponding element in A is subject to
perturbation. Moreover, P satisfies the Lyapunov equation given in Eq. (3.17).
That being said, it has been assumed in both approaches presented in the above that
the perturbations in the various elements of the system matrix are independent of one
another, which will introduce additional conservatism in the perturbation bounds. In
order to take this fact into account, another approach for finding the stability robustness
bounds for systems with structured uncertainty was proposed in [40]. This method is
hence less conservative than the previously presented approaches. Assume the error
matrix E has the following form:
E =r∑i=1
kiEi, (3.19)
where Ei are constant matrices, and ki are uncertain parameters assumed to vary in
intervals around zero, i.e., ki ∈ [−εi, εi]. It is seen that this type of formulation allows
for perturbations in different elements of the system to be dependent one another. The
proposed theorem in [40] is as follows.
Theorem Consider the linear system in Eq. (3.14) with A being nominal and stable,
and E of the form of Eq. (3.19). Let P be the solution to the Lyapunov equation (3.17).
Define Pi as:
Pi = (ETi P + PEi)/2 = [PEi]s , i = 1, 2, ...r. (3.20)
Chapter 3. Robustness Analysis 29
The perturbed system (A+ E) is stable if
|Kj| < 1/σmax
(r∑i=1
|Pi|), j = 1, 2, ...r. (3.21)
3.4 Robustness Analysis of a Numerical Example
In this section, a numerical example as introduced in [19] has been chosen in order to
show the application of the previously introduced robustness investigation techniques.
It is the problem of stabilizing an F4E fighter aircraft as was done by Petersen (1987)
and Wu et al. (1990), where there are four three-dimensional systems to be stabilized,
and each system corresponds to the aircraft’s travelling at a different altitude and speed.
The numerical problem of stabilizing an F4-E fighter jet aircraft is described by the
state-space model asx1,k
x2,k
x3,k
=
a11 a12 a13
a21 a22 a23
0 0 −30
x1,k
x2,k
x3,k
+
b1
0
30
uk. (3.22)
where x1,k represents the normal acceleration a, x2,k represents the pitch rate q, and
x3,k represents the elevator angle δe. This state-space model includes the dynamics of
the actuator with the input being the stick position and the output being the elevator
deflection. The values of the parameters of the above model are given in Table 3.1, and
each system states the dynamics of the aircraft at a different altitude and speed. Using
the proposed optimization strategy in section 2.7, a simultaneously stabilizing controller
has been found. Later, three different approaches for investigating the robustness of the
designed controller, namely the extreme point solution, the stability robustness bounds,
and the Kharitonov’s theorem, have been applied to investigate the robustness of this
simultaneously stabilizing controller. The following simultaneously stabilizing controller
is hence constructed as specified in section 2.7, which ensures the stable operation of the
aircrfat at four different flight conditions.
f =
0.13241
2
−0.27757
. (3.23)
Chapter 3. Robustness Analysis 30
Table 3.1: F4-E flight operating conditions
Operating Point 1 2 3 4
Altitude, ft 5000 35000 5000 35000
Mach number 0.5 0.9 0.85 1.5
a11 -0.9896 -0.6607 -1.702 -0.5162
a12 17.41 18.11 50.72 26.96
a13 96.15 84.34 263.5 178.9
a21 0.2648 0.08201 0.2201 -0.6896
a22 -0.8512 -0.6587 -1.418 -1.225
a23 -11.89 -10.81 -31.99 -30.38
b1 -97.78 -272.2 -85.09 -175.6
In the following sections, the robustness of the controller constructed for the numerical
example is initially investigated by applying the extreme point solution method. Then,
Kharitonov’s theorem is applied to the same example and again robustness of the design is
investigated. Finally, the stability robustness bounds are found to give an understanding
of the robustness of the design.
3.4.1 Extreme Point Solution Application
In the following, the extreme point solution approach is used for analyzing the robustness
of the controller designed for the numerical example. For the robustness investigation,
the first plant out of the set of 4 plants for which the simultaneously stabilizing controller
has been designed, was considered and the closed-loop system was obtained. The state
matrix of the stable closed-loop system has then been chosen as the nominal stable plant,
assuming that every element of the state and input matrices is subject to perturbations.
Uncertainty vector and constant specified matrices Ai were hence formulated and max-
imum and minimum bounds on the elements of the uncertainty vector has also been
found according to the boundary of perturbations for the elements of the state and input
matrices. The machinery behind this kind of formulation is briefly presented here. The
Chapter 3. Robustness Analysis 31
closed-loop system in state-space model has the form of:
x (t) = (A−BfT )x (t) ,
where A is the state matrix, B is the input matrix, and f = [f1 f2 f3] is the controller.
Hence:
A−BfT =
a11 a12 a13
a21 a22 a23
a31 a32 a33
−b1
b2
b3
[f1 f2 f3
]
=
a11 − b1f1 a12 − b1f2 a13 − b1f3
a21 − b2f1 a22 − b2f2 a23 − b2f3
a31 − b3f1 a32 − b3f2 a33 − b3f3
= A0,
where ai,j ≤ ai,j ≤ ai,j for i, j = 1, 2, 3, and bi ≤ bi ≤ bi for i = 1, 2, 3. Therefore the
perturbed closed loop system will have the form of
Acl = A0 + ∆A = A0 +
∆a11 − (∆b1)f1 ∆a12 − (∆b1)f2 ∆a13 − (∆b1)f3
∆a21 − (∆b2)f1 ∆a22 − (∆b2)f2 ∆a23 − (∆b2)f3
∆a31 − (∆b3)f1 ∆a32 − (∆b3)f2 ∆a33 − (∆b3)f3
,
where ai,j − ai,j = ∆ai,j ≤ ∆ai,j ≤ ∆ai,j = ai,j − ai,j for i, j = 1, 2, 3, and bi − bi =
∆bi ≤ ∆bi ≤ ∆bi = bi − bi for i = 1, 2, 3. Now the uncertainty vector q and the constant
specified matrices can be defined as:
q =
q1
q2
...
q9
=
∆a11 − (∆b1)f1
∆a12 − (∆b1)f2
...
∆a33 − (∆b3)f3
,
and
A1 =
1 0 0
0 0 0
0 0 0
, A2 =
0 1 0
0 0 0
0 0 0
, · · · A9 =
0 0 0
0 0 0
0 0 1
,
Chapter 3. Robustness Analysis 32
where the lower and upper bounds on the uncertainty vector elements, qs, can be found
as ∆ai,j − (∆bi)fj = qs ≤ qs ≤ qs = ∆ai,j − (∆bi)fj, for i, j = 1, .., 3, and s = 1, .., 9.
It can be seen that using this formulation, the uncertainty matrix has 9 elements and
hence there are nine constant specified matrices. Applying the extreme point solution
method, all the 29 different vertices, Ai, can be found by substituting 29 different combi-
nations of lower and upper values of the uncertainty vector elements in the closed-loop
state matrix. By transforming these vertices into the dagger space and constructing
Li matrices, and then generating real axis nonsingularity matrices, we can check their
asymptotic stability. If all of the real axis nonsingularity matrices as well as vertex ma-
trices are stable, it can be concluded that all the matrices belonging to the polytopic
matrix family of A, which is the set of all the possible perturbed closed-loop systems,
are asymptotically stable as well.
In the analysis process, using the first plant as the nominal system, a code was
developed to generate the uncertainties, their corresponding constant specified matrices,
and the real axis nonsingularity matrices in the dagger space. Arbitrary bounds for
the elements of the state and input matrices A1, B1 were chosen and in each case the
performance of the closed-loop system subjected to perturbations was investigated. For
some cases of the arbitrarily chosen boundaries, there are a number of vertices that are
not asymptotically stable. Thus, some of the matrices in the perturbed matrix family
A are not stable as well. In other words, the nominal stable plant is not guaranteed
to maintain its stability when it is subjected to perturbation in its elements within the
selected boundary. On the other hand, it is possible to find a boundary for perturbations
in the system elements that allows the nominal plant to maintain its stability when
subjected to disturbances in that specific system element.
In this investigation, the initial effort was to find a boundary for the state and input
matrix elements that would contain those of the four primary plants, while maintaining
the stability of every possible perturbed closed-loop system within that range. However,
it was not a realistic expectation since the four initial plants are located far away from
each other across the flight envelope.
In Table 3.2, a region of perturbation in the state and input matrix elements has been
Chapter 3. Robustness Analysis 33
chosen arbitrarily, for which all of the real axis nonsingularity matrices are assymptoti-
cally stable. Thus, any perturbed closed-loop matrix lying inside the region specified by
those boundaries, is also stable. Hence, it is guaranteed that by imposing these bound-
aries on the system uncertainties, the perturbed closed-loop system remains stable. The
simulation results in Fig. 3.1 show the response to initial condition of the first system
when it is arbitrarily perturbed within the allowable boundaries of change given in Table
3.2. It is obvious that since every perturbed closed-loop system is proved to be stable
within the given range, the state response to initial condition should be converging to
zero as well. This is clearly shown in Fig. 3.1, where the response to initial condition is
in fact converging to zero.
Table 3.2: Allowed perturbation resulting in a stable matrix family
∆a11 ∆a12 ∆a13 ∆a21 ∆a22 ∆a23 ∆a31 ∆a32 ∆a33 ∆b1 ∆b2 ∆b3
min 0.1 0.5 0.5 0.2 0.2 1 0.2 0.2 1 0.7 0.2 1
max -0.1 -0.5 -0.5 -0.2 -0.2 -1 -0.2 -0.2 -1 -0.7 -0.2 -1
Table 3.3 shows another case of arbitrarily chosen boundaries which will result in
some of the vertices being unstable. Therefore, it is not guaranteed that all of the
perturbed closed-loop systems will maintain stability when disturbed within the chosen
ranges shown in Table 3.3. This is equivalent to saying that the designed controller is not
robust for the perturbations in the chosen boundary around the first nominal plant. This
can be regarded as an example of a perturbed nominal system for which stability within
the chosen boundaries is not guaranteed. It is hence concluded that the simultaneously
stabilizing controller is not robust if the first nominal system is subjected to perturbation
within the given boundaries shown in Table 3.3.
Table 3.3: Allowed perturbation resulting in an unstable matrix family
∆a11 ∆a12 ∆a13 ∆a21 ∆a22 ∆a23 ∆a31 ∆a32 ∆a33 ∆b1 ∆b2 ∆b3
min 0.5 0.5 0.5 0.2 0.2 1 0.2 0.2 1 0.7 0.2 1
max -0.5 -0.5 -0.5 -0.2 -0.2 -1 -0.2 -0.2 -1 -0.7 -0.2 -1
Chapter 3. Robustness Analysis 34
Figure 3.1: response to initial condition of the perturbed nominal plant
3.4.2 Kharitonov’s Theorem Application
In the following, the Kharitonov’s Theorem has been used for investigating the robust-
ness of the simultaneously stabilizing controller designed for the numerical example. For
this investigation, the first plant out of the set of 4 plants was considered again and the
closed loop system was generated. The characteristic equation of this closed loop system
is found by solving the equation det(A − sI) = 0. Assuming that every element of the
state and input matrices can be perturbed, the coefficients of this characteristic equa-
tion is subject to changes accordingly. By varying the closed-loop system elements in a
respective boundary and finding the perturbed closed-loop systems, the corresponding
characteristic equation and its perturbed coefficients can easily be obtained. The last
step is to find the minimum and maximum values for each of these coefficients and gen-
erate the four Kharitonov’s polynomials, as introduced before. If the four Kharitonov’s
polynomials are stable, the family of the perturbed closed-loop systems is also stable.
Chapter 3. Robustness Analysis 35
To investigate the effect of uncertainties on the system’s stability, the first element
of the state matrix (i.e. a11) was subjected to perturbations such as −1.2 ≤ a11 ≤ −0.6.
For different values of a11, the closed-loop system A1 − B1fT was generated and using
the ss2tf command in MATLAB, the characteristic equation and its coefficients were
obtained. Later, four Kharitonov’s polynomials were constructed and checked for sta-
bility. Table 3.4 presents the four Kharitonov’s polynomials obtained for the numerical
example, and Fig. 3.2 shows the response to initial condition of the closed-loop sys-
tem with an arbitrary value of a11 in the given allowable boundary. The stability of
Kharitonov’s polynomials was investigated and confirmed, and it was concluded that the
perturbed closed-loop system maintains stability for the given arbitrary disturbance in
the a11 element. Therefore, the response to initial condition of the system is expected to
be converging to zero, as plotted in Fig. 3.2.
Table 3.4: Kharitonov’s polynomials coefficients
Kh = a3s3 + a2s
2 + a1s+ a0
a3 a2 a1 a0
Kh1 1 8.1 766.5 1149.7
Kh2 1 7.5 768.8 1575.4
Kh3 1 7.5 766.5 1575.4
Kh4 1 8.1 768.8 1149.7
3.4.3 Stability Robustness Bound Application
In this section, the approaches proposed in [24] and [36] are applied to the numerical
example in order to find an upper bound on allowable perturbations, while maintaining
the stability of the disturbed nominal system. Again, The first closed-loop system has
been chosen as the nominal stable plant and the error matrix due to the introduced
perturbations is found. Following the results of [24], the Lyapunov matrix equation is
solved for the nominal plant A1 − B1fT , and the solution P is found. By finding the
maximum singular value of matrix P and using Eq. (3.16), an upper bound on the entries
Chapter 3. Robustness Analysis 36
Figure 3.2: response to initial condition of the perturbed nominal plant
of the error matrix (or uncertainty vector elements) is obtained. In another approach, the
method presented in [36] uses the same Lyapunov matrix P , but results in a different and
less conservative upper bound (Eq. 3.18). These bounds represent the range of allowable
perturbation in the state and input matrix elements, which guarantees the stability of
the perturbed nominal plant.
The results obtained using the two methods presented in [24] and [36], are shown
in Table 3.5. These results have been obtained following the same previous formulation
for robustness investigation, with the first closed-loop system being the nominal plant
A0, and its perturbation ∆A being the error matrix. It has been assumed that the
perturbations in the elements of the state and input matrices are independent, due to
the assumptions in [24]. However, if the formulation accounts for the perturbations of
these elements to be dependent on each other, it is expected that a less conservative
robustness bound should be obtained.
It is hence concluded that the two techniques, namely the Extreme Point Solution
and the Kharitonov’s Theorem, work with an introduced range of perturbation applied to
the elements of a stable system, and determine whether or not the perturbed system will
Chapter 3. Robustness Analysis 37
Table 3.5: Stability robustness bounds
Patel and Toda [24] Yedavalli [36]
µεP µεY
0.0418 0.0540
remain stable under uncertainties. The extreme point solution method, however, obtains
an upper bound on the allowable perturbation for which the stability of the disturbed
system is guaranteed. Finally as it is seen in Table 3.5, the method provided in [36] gives
a less conservative result compared to the one proposed in [24].
Chapter 4
Robust Simultaneous Stabilization
Problem
In the previous chapters, the concept of Simultaneous stabilization problem and several
methods to solve the problem were presented, along with techniques and tools that can be
used to measure the robustness of the designed controller and the level of the closed-loop
system’s vulnerability to disturbances. However, it is now clear that such a controller
has to be able to not only simultaneously stabilize a set of plants, but also maintain
that stability when encountered with perturbations and uncertainties. It is therefore
important to address the robustness of the simultaneous stabilization, now known as
the Robust Simultaneous Stabilization (RSS) problem. Hence, the new objective is to
provide stability for all of the systems simultaneously, while maximizing the stability
robustness bounds.
In this chapter, the robustness tools introduced before are implemented in the design
process such that the final design is an improved controller in terms of robustness. First,
such a design approach presented in [29] is introduced which aims at improving the
robustness. Then the concept of multi-objective modelling is briefly addressed as shown in
[28], which is later used when a new design approach for the problem of RSS is presented.
More specifically, a combination of optimal stabilization and parameter optimization
methods (introduced in Chapter 2) has contributed in the development of this design
approach, with the emphasis of the cost function not on the performance or transient
38
Chapter 4. Robust Simultaneous Stabilization Problem 39
response of the design, but rather on its robustness. The resulted multiple-objective
problem is later converted into a single-objective problem.
4.1 An Extended Decomposition-Based Strategy for
the RSS Problem
A decomposition approach for the solution of the simultaneous stabilization and simul-
taneous optimal control problem has been proposed in [27], as described in Chapter 2.
The decomposition approach was later extended in [29] to consider stability robustness
bounds. As presented here, This strategy was proposed to solve the problem of simul-
taneously stabilizing a set of plants while maximizing the upper bound on the allowable
structured uncertainty, which preserves the stability of each closed-loop system.
Again, consider the set of m different plants described by the state-space equation
xk (t) = Akxk (t) +Bkuk (t) , k = 1, 2, ...,m (4.1)
yk (t) = Ckxk (t) (4.2)
where xk ∈ Rn and u is a scalar control. The goal of the robust simultaneous stabilization
is to simultaneously stabilize a set of different systems by designing a single feedback
controller which can maintain stability in the presence of bounded uncertainties for each
plant. The structured uncertainty for each plant is represented by an n× n error matrix
Ek and the state-space equations:
xk (t) = (Ak + Ek)xk (t) +Bkuk (t) , (4.3)
yk (t) = Ckxk (t) , k = 1, 2, ...,m (4.4)
The entries of the uncertainty matrix Ek is bounded as
|Ek| ≤ ε (4.5)
Chapter 4. Robust Simultaneous Stabilization Problem 40
where |Ek| denotes the absolute value of the entries of matrix Ek, and ε is the magnitude
of the maximum deviation in matrix entries. Again, a single feedback control law
uk (t) = −fTxk (t) (4.6)
is sought such that each one of the perturbed closed-loop system
xk (t) =(Ak −Bkf
T)xk (t) + Ekxk (t) = (ACLk + Ek)xk (t) (4.7)
remains stable under uncertainties.
The decomposition approach to the simultaneous stabilization problem originally pro-
posed in [27] was based on its analogy to a general multi-disciplinary optimization (MDO)
problem. Now an extended version of that approach, aimed at solving the RSS problem
[29], is proposed:
SL
∣∣∣∣∣∣∣∣min max
fSL,γSL− µ∗eYk
s.t. G∗k = 0, k = 1, 2, ...,m
SSLk
∣∣∣∣∣∣∣∣minfk,γk
Gk = (fk − fSL)2
s.t. Re (λi,k) ≤ γk < 0, i = 1, ..., n
(4.8)
where µ∗eYk is the upper bound on the structured uncertainties maintaining the stability
of the kth closed-loop system, corresponding to the subsytem optimal solution gain of
f ∗k . Such upper uncertainty bound is defined in [36] as:
|Ek|max = ε <1
σmax [|P ∗k |Un]s= µ∗eYk (4.9)
where Un is an n× n matrix whose entries are unity for each element of the closed-loop
system which is subjected to perturbations, and zero otherwise. P ∗k is the solution to the
Lyapunov matrix equation:
P ∗k(Ak −Bkf
∗T)
+(Ak −Bkf
∗T)TP ∗k + 2In = 0 (4.10)
where In is an n× n identity matrix.
Figure 4.1 shows the concept of the solution that simultaneously stabilize a set of m
plants while improving the robustness. The system-level optimization process maximizes
Chapter 4. Robust Simultaneous Stabilization Problem 41
stability robustness bounds while the constraint enforces a unique gain solutionn for all of
the systems. The subsystem-level optimization process minimizes the discrepancy with
the system level gain solution while enforcing internal stabilization of each system under
the feedback control law.
Figure 4.1: Robust Simultaneous Stabilization solution approach
This methodology tries to increase the value of µ∗eYk for the system with the lowest
stability robustness bound at each step, which is obviously the most vulnerable system to
disturbances at that time. This may result in the overall optimization of the robustness
bounds being hurt. The reason is that the value of µ∗eYk for a single vulnerable plant can
be increased at the cost of decreasing that of the other plants substantially. As a result,
some of the plants may see a huge drop in their stability robustness boundaries and if
this outcome is not desirable, another optimization methodology should be presented.
The problem of increasing the robustness boundary of the more vulnerable plants while
improving or maintaing the same level of robustness for the others, can be treated as a
multi-objective modelling problem.
Chapter 4. Robust Simultaneous Stabilization Problem 42
4.2 Multi-Objective Optimization
Many real-world search and optimization problems are naturally posed as non-linear pro-
gramming problems having multiple objectives. A multi-objective optimization problem
(MOOP) deals with more than one objective function and in most practical decision-
making problems, multiple objectives or multiple criteria are evident. Due to the lack
of suitable solution techniques, such problems were artificially converted into a single-
objective problem and solved. However, there exist a number of fundamental differences
between the working principles of single- and multi-objective optimization algorithms.
In a single-objective optimization problem, the task is to find one solution which op-
timizes the sole objective function. A multi-objective optimization is much more than
this simple idea. A MOO problem also has two different search spaces. The objective
functions constitute a multi-dimensional space, in addition to the usual decision variable
space common to all optimization problems.
The difficulty arises because such problems give rise to a set of trade-off optimal
solutions (known as Pareto-optimal solutions), instead of a single optimum solution. All
such trade-off solutions are optimal solutions to a multi-objective optimization problem.
Often, such trade-off solutions provide a front on an objective space plotted with the
objective values. This front is called the Pareto-optimal front and all such trade-off
solutions are called Pareto-optimal solutions.
4.2.1 Formulation & the Concept of Pareto Optimality
Where ‘many-wish’ attributes are involved in the design process, a multi-objective opti-
mization formulation can be applied to the design process where every ’wish attribute’
is modelled by an objective function. This formultion is presented mathematically as:
minimize[f1 (x) , f2 (x) , · · · , fNobjs (x)
]Twith respect to x ∈ Rn
subject to g (x) ≤ g0,
xl ≤ x ≤ xu
Chapter 4. Robust Simultaneous Stabilization Problem 43
where the optimal solution x∗ can be found using already known optimization algo-
rithms. Most multi-objective optimization algorithms use the concept of dominance in
their search. To better understand this term, the concept of ’Pareto Optimality’ must
first be defined.
Definition A design vector x∗ ∈ D is Pareto optimal if and only if there is no vector
x ∈ D with the characteristics
fi (x) ≤ fi (x∗) for all i = 1, 2, ..., Nobjs (4.11)
and
fi (x) < fi (x∗) for at least one i, 1 ≤ i ≤ Nobjs (4.12)
By applying this definition to the decision variable space, there will be a set of solu-
tions, any two of which do not dominate each other. This set also has another property.
For any solution outside of this set, we can always find a solution in this set which will
dominate the former. Thus, this particular set has a property of dominating all other so-
lutions which do not belong to this set. In simple terms, this means that the solutions of
this set are better compared to the rest of solutions. This set, the resulted non-dominated
set for the given set of solutions, is given a special name called the Pareto-optimal set.
More importantly, the Pareto-optimal set always consists of solutions from a particular
edge of the feasible search region.
In design optimization, Pareto optimality represents a rational choice of solutions
and provides more information as the tradeoffs in the design process are now visible.
However, a design supercriterion or preference is necessary to choose the best solution
from a set of Pareto-optimal solutions.
4.2.2 Weighted-Sum Method
In this section, a number of commonly-used classical multi-objective optimization meth-
ods are briefly mentioned. In most cases, multi-objective problems can be converted into
a single-objective problem or a series of constrained optimization problems by modelling
design preferences and objective tradeoffs.
Chapter 4. Robust Simultaneous Stabilization Problem 44
Figure 4.2: Pareto set in a convex objective space
Figure 4.3: Pareto Set in a non-convex objective space
Chapter 4. Robust Simultaneous Stabilization Problem 45
(a) Weighted-sum method in a convex objec-
tive space
(b) Weighted-sum method in a non-convex ob-
jective space
Figure 4.4: Pareto sets in a weighted-sum optimization problem
Generally, if a relative preference factor among the objectives is known for a specific
problem, there is no need to solve a multi-objective optimization problem. A simple
method would be to form a composite objective function as the weighted sum of the ob-
jectives, where a weight for an objective is proportional to the preference factor assigned
to that particular objective. The values of the weights depend on the importance of each
objective in the context of the problem and a scaling factor.
This method of scalarizing an objective vector into a single composite objective func-
tion converts the multi-objective optimization problem into a single-objective optimiza-
tion problem. When such a composite objective function is optimized, in most cases it
is possible to obtain one particular trade-off optimal solution. However, the trade-off so-
lution obtained by using the preference-based strategy is largely sensitive to the relative
preference vector w used in forming the composite function, and a change in this prefer-
ence vector will result in a different trade-off solution. Having said that, this procedure
cannot be used to find Pareto-optimal solutions which lie on the non-convex portion of
the Pareto-optimal front, as shown in Figure 4.4.
In the most common case of weighted-sum method, a composite objective function
Chapter 4. Robust Simultaneous Stabilization Problem 46
can be formed by summing the weighted objectives and the problem is then converted
to a single-objective optimization problem as follows:
minimize f (x, p) =Nobjs∑i=1
wifi (x, p) (4.13)
A few other commonly-used classical multi-objective optimization methods, such as
the Global Criterion Method or the Compromise Programming Method, are also ad-
dressed in [28]. Each one of them could have been used in the process of proposing a
new design approach to the problem of RSS.
4.2.3 Multiple Robustness Optimization
The designed controller has a certain degree of robustness in case of uncertainties, de-
pending on the closed-loop system for which it is beeing used. Assuming a system with
structured uncertainty and a stable nominal plant, an upper bound on the allowable
perturbations can be found, which maintains the stability of the uncertain closed-loop
system. As before, assume the following perturbed state-space equation:
x = ACLx (t) + Ex (t) = (ACL + E)x (t) (4.14)
where x is the n-dimensional state vector, A is an n×n time invariant and asymptotically
stable matrix, and E is the n× n error matrix such that
|Ei,j| ≤ ε
and ε is the magnitude of the maximum allowable deviation.
As it was stated before, the perturbed closed-loop system ACL + E is stable if:
|Ei,j|max = ε <1
σmax [|P |Un]s= µeY (4.15)
where σmax [.] represents the largest singular value of a matrix, |.| is formed by taking
the absolute value of every element of that matrix, and [.]s denotes the symmetric part
of it. Here, P is the solution of the Lyapunov matrix equation:
ATCLP + PACL + 2In = 0 (4.16)
Chapter 4. Robust Simultaneous Stabilization Problem 47
In the approach presented in Chapter 2, the goal is to simultaneously stabilize all the
systems by minimizing the maximum real part of the eigenvalues as much as possible and
making it negative, within a given range for the controller enteries and a performance in-
dex. However, the following solution to the RSS problem tries to simultaneously stabilize
all the systems by imposing it as a constraint, while maximizing the stability robustness
bounds as the new objective function. Therefore, The wish attribute for each system
is to maximize its stability robustness bounds, µeYk . To avoid any confusion in Multi-
Objective Optimization problems, most applications convert a maximization problem
into a minimization problem and treat every problem as a combination of minimizing all
objectives. Now in order to simultaneously increase the robustness bound of each sys-
tem, µeYk , as much as possible, let us define the following wish attribute as an objective
function for the k-th system:
Jk =1
µeYk, k = 1, 2, ...,m (4.17)
where minimizing the above objective function is equall to maximizing the robustness
bound of each closed-loop system. Now the problem of maximizing µeYk is converted to
the problem of minimizing the objective function Jk = 1µeYk
, as the greater the stability
robustness boundary, the lower the new objective function Jk. Since the goal is to
increase the robustness of all of the systems under a single stabilizing controller f , this
problem can be treated as a ”many wish-many attributes” problem. In the previous
section several methods for solving such a problem were discussed. Here, the Wighted-
Sum Method is applied to convert this multi-objective problem to a single-objective
optimization problem. As for the weights, it has been assumed that there is no specific
preferences between the systems in terms of which one is more desired to have a higher
µeYk . Therefore, every objective function Jk corresponding to different systems has been
assigned the same weights of wk = 1.
In what follows, a new optimization methodology for the Robust Simultaneous Sta-
bilization Problem is proposed which yields a better solution in terms of robustness, as
seen in the follwoing investigations. This will ensure that all of the closed-loop systems
remain stable under a unique controller while increasing the robustness bounds in case
Chapter 4. Robust Simultaneous Stabilization Problem 48
of uncertainties. The proposed optimization problem is stated as:
Problem Given parameters µ > 0 and η ≥ 0, choose the controller f ∈ Rn to
minimize the objective function
I =m∑k=1
1
µeYk(4.18)
with respect to f , and subject to the constraints
−µ ≤ fi ≤ µ, i = 1, ..., n (4.19)
αi,k = Re (λi,k) < 0, i = 1, ..., n, k = 1, ...,m (4.20)
αi,k + ηβi,k ≤ 0, i = 1, ..., n, k = 1, ...,m (4.21)
The simultaneous stability of the systems is insured by constrainting the real part of
all of the eigenvalue, Re (λi,k), to be negative at all times. As before, constraints on the
value of the controller entries and performance of the closed loop systems are imposed.
The controller which minimizes the new objective function, will also stabilize all the
plants simultaneously and yields a solution with the desired performance depending on
the value of η, without using unnecessary control effort. The problem formulation shown
in Eq. (4.18) is believed to be convex since Eqs. (4.15) through (4.17) are also convex
in nature. Hence, the weighted-sum method can be used to find the set of pareto-
optimal solutions to this problem. This methodology also enforces a way to increase
or decrease the robustness boundary of each system depending on its importance, by
assigning different weights to their respective objective functions in the multi-objective
optimization problem.
The application of the multiple robustness optimization and bi-level optimization
solutions to the RSS problem, is shown using two linear and nonlinear case studies
introduced in Chapters 5 and 6.
Chapter 5
Linear Simulation: An F4-C Flight
Control Case Study
In this chapter, a simultaneously stabilizing feedbcak control law is designed for an F4C
fighter aircraft operating at 4 different flight conditions using the method described earlier
in section 2.7. Later, its robustness is investigated using methods described in [40], [38],
and [31]. The robustness is then relaxed and a new controller is obtained using the
multiple objective optimization methodology described in Chapter 4. The solution to
the RSS problem is further compared to the previous SS problem solution in terms of
robustness. Moreover, this design is again compared to the solution obtained using a
bi-level optimization methodology introduced in section 4.1 as was presented in [29].
5.1 Introduction of the Test Case
This test case considers the problem of simultaneous stabilization of the longitudinal
dynamics Short-Period mode of an F4C fighter aircraft at 4 distinct operating points, as
was originally proposed by Ackermann [2]. For this test case a simultaneously stabiliz-
ing controller is designed and then using several robustness analysis tools provided, its
robustness is later evaluated in the following section. The system is described in linear
time-invariant state-space form for all flight conditions as:
x(t) = Ax(t) +Bu, y(t) = Cx(t), (5.1)
49
Chapter 5. Linear Simulation: An F4-C Flight Control Case Study 50
where
A =
a11 a12 b1
a21 a22 b2
0 0 −20
, B =
0
0
20
, C = I (5.2)
The original Short-Period state vector (angle of attack α, and pitch rate q) is also ex-
tended to include the elevator actuator dynamics, represented by a first-order low-pass
filter with a time constant Te of 0.05 s. Thus:
x =[α q δe
]T(5.3)
Four distinct flight conditions have been chosen for this study, with variation in both
altitude and speed, as shown in Table 5.1.
Table 5.1: F4-C flight operating conditions
Operating Point 1 2 3 4
Altitude, ft Sea Level Sea Level 35000 45000
Mach number 0.206 1.1 0.6 2.15
a11 -0.4535 -2.112 -0.2978 -0.484
a12 0.9792 1.0 0.9866 0.9997
a21 -0.3693 -772.08 -1.803 -42.75
a22 -0.4615 -3.126 -0.4436 -0.3718
b1 -0.0290 -0.2098 -0.0411 -0.0419
b2 -1.459 -63.48 -4.989 -17.72
Using the optimization approach introduced in section 2.7, and with assumed values
of µ = 2 (limiting the feedback gain vector entries) and η = 0.2 (limiting the damping
ratio by introducing a performance factor), a simultaneously stabilizing controller is ob-
tained, as presented in Table 5.2. Note the negative value of the maximum real part of
the eigenvalues, showing that a controller has been found that minimizes the maximum
real part of the eigenvalues and makes it negative, thus assuring that simultaneous stabi-
lization has been achieved. Table 5.3 also shows the closed loop system eigenvalues and
Chapter 5. Linear Simulation: An F4-C Flight Control Case Study 51
all pole locations for all plants are in the negative real half of the complex plane, stating
the fact that all of the closed loop systems reached stability under the same design.
u (t) = −fTx (t) (5.4)
Table 5.2: Simultaneous Stabilization solution
Proposed SS optimization methodology (sec 2.7)
f -2.0000
-1.8404
0.1423
I = max Re(λ) -1.7721
Table 5.3: Closed-loop system eigenvalues
System Eigenvalues
A1 −B1fT -20.2173, and -1.7721 ± 1.2421i
A2 −B2fT -6.2202, and -10.9322 ± 54.6610i
A3 −B3fT -1.7721, and -10.9079 ± 6.8181i
A4 −B4fT -2.9612, and -10.3706 ± 23.3970i
For further illustration, the initial response simulation of the obtained closed loop
systems is presented here. Figure 5.1 shows the time history of the states subject to
the initial condition X = [ 1 1 1]T at time t = 0 for all four flight conditions. It can
be seen from the simulation that when the feedback gain is applied, the states of the
closed-loop system decay to the steady state and thus the closed loop systems are stable.
5.2 Robustness Investigation
It was shown that the controller is able to simultaneously stabilize all four different
flight conditions. However, any introduced perturbation in speed, altitude, airplane’s
Chapter 5. Linear Simulation: An F4-C Flight Control Case Study 52
Figure 5.1: Response to initial condition
configuration or aerodynamic shape, or other flight conditions and characteristics due
to damage, changes the closed loop system and therefor it is desired to check if the
perturbed closed loop system is still stable. Failures such as Loss of effectiveness of a
control surface or aerodynamic shape change due to loss of some parts can be modeled as
a change in aerodynamic derivatives, and it can be assumed that the derivatives change
suddenly. To make the investigation easier it is assumed that flight conditions (speed and
altitude), and all other stability derivatives and flight characteristics remain the same at
each operating point, except for one which preferably has a strong and obvious effect on
the aircraft’s performance.
5.2.1 Perturbations Due to CLαuncertainties
It is seen by looking at the state space representation of the short-period mode given
below that CLα has an obvious effect on the longitudinal dynamics of the aircraft. This
aerodynamic parameter directly affects the lift on the aircraft and the resulting moment
and hence it changes the longitudinal dynamics short-period behavior more significantly
Chapter 5. Linear Simulation: An F4-C Flight Control Case Study 53
than other derivatives. α
q
=
VT − Zα 0
−Mα 1
−1 Zα VT + Zq
Mα Mq
α
q
+
VT − Zα 0
−Mα 1
−1 Zδe
Mδe
δe(5.5)
α
q
=
a11 a12
a21 a22
α
q
+
b1
b2
δe (5.6)
Where the matrix entries can be calculated as:
a11 = ZαVT−Zα
a12 = VT+ZqVT−Zα
a21 = Mα + MαZαVT−Zα
a22 = Mq + Mα(VT+Zq)VT−Zα
b1 = ZδeVT−Zα
b2 = MαZδeVT−Zα
+Mδe
(5.7)
and the dimensional stability derivatives are introduced in terms of dimensionless deriva-
tives as
Zα = − qSm
(CD + CLα)
Mα = qScJyCmα
Zα = − qSc2mVT
CLα
Mα = qScJy
c2VT
Cmα
Zq = − qSc2mVT
CLq
Mq = qScJy
c2VT
Cmq
Zδe = − qSmCLδe
Mδe = qScJYCmδe
(5.8)
The dimensionless stability derivatives at each flight condition along with the geo-
metric characteristics of the aircraft are given in Table 5.4 as was originally presented in
[6], [11], and [8].
Now to investigate the robustness of the controller designed for this case study, assume
that the only source of perturbation is changes in the aerodynamic derivative CLα and
all of the other derivatives keep their initial value. Therefore, the new perturbed closed
loop system can be represented as:
Chapter 5. Linear Simulation: An F4-C Flight Control Case Study 54
A =(A−BfT
)+
−∆CLαqS/mVT 0 0
−∆CLαq2S2c2Cmα/2mJY V
2T 0 0
0 0 0
(5.9)
Effect of ∆CLα on the robust stability bounds can now be investigated.
Stability Robustness Bounds
Assuming that the only source of perturbation comes from changes in CLα results in:
ACL = A0 + E =(A−BfT
)+ E , (5.10)
where
E =
−∆CLαqS/mVT 0 0
−∆CLαq2S2c2Cmα/2mJY V
2T 0 0
0 0 0
.
Therefore r = 1, k1 = ∆CLα , and
E1 =
−qS/mVT 0 0
−q2S2c2Cmα/2mJY V2T 0 0
0 0 0
(5.11)
Using the strategy presented in Ref. [40], the magnitude of the maximum allowable
deviation in CLα is obtained for each flight condition and is presented in Table 5.5.
It is thus guaranteed that if CLα is disturbed within these ranges at each operating
point, the perturbed closed-loop system will remain stable.
The Extreme Point Solution
The extreme point solution strategy [38] can also be used to verify the robust stability
bounds. Assuming that at each operating point only CLα is allowed to perturb in the
maximum deviation range given in Table 5.5, the matrix familyA for each flight condition
can be defined as:
A (q) = A0 +S∑i=1
qiAi =(A−BfT
)+ E (5.12)
Chapter 5. Linear Simulation: An F4-C Flight Control Case Study 55
Tab
le5.
4:F
4-C
char
acte
rist
ics
atdiff
eren
tflig
ht
condit
ions
Fligh
tP
rop
erty
Con
dit
ion
Alt
itude,
ftM
CLα
qS
mVT
cJy
Cmα
1Sea
Lev
el0.
206
2.8
62.6
530
3892
423
016
.04
1221
86-0
.95
2Sea
Lev
el1.
13.
317
9253
038
925
1228
16.0
412
2193
-1.2
335
000
0.6
2.8
126
530
3892
558
416
.04
1221
93-1
.3
445
000
2.15
2.3
1004
530
3892
520
8116
.04
1221
930.
25
Chapter 5. Linear Simulation: An F4-C Flight Control Case Study 56
Table 5.5: Maximum allowable deviation in ∆CLα
Operating Point 1 2 3 4
|∆CLα |max < µεY = 3.5065 0.0474 4.9685 0.9716
E =
−∆CLαqS/mVT 0 0
−∆CLαq2S2c2Cmα/2mJY V
2T 0 0
0 0 0
where s = 1, l = 2s = 2, ∆CLα ≤ q1 = ∆CLα ≤ ∆CLα , and
A1 =
−qS/mVT 0 0
−q2S2c2Cmα/2mJY V2T 0 0
0 0 0
.
For the four different flight conditions, generate two vertex matrices A1 = A0 +(∆CLα .A1
)and A2 = A0 +
(∆CLα .A1
), and their respective vertices in dagger space,
L1 and L2. For each flight condition, ‘Real Axis Nonsinularity’ matrices can now be
defined:
KN1 = −[(L1 + L2
)−1.L1
],
KN2 = −[(L1 + L2
)−1.L2
]. (5.13)
The above matrices have been checked to be asymptotically stable for all four oper-
ating points and thus all of the matrix families A are stable. This verifies the results
obtained in the previous analysis, that if the aerodynamic derivative CLα is bounded
to perturb in the range given in Table 5.5 the resulting perturbed closed-loop system
remains stable.
The Kharitonovs Theorem
Assume CLα is allowed to perturb in the maximum deviation range provided in Table
5.5 for each operating point, and for each value of CLα the closed loop system and
Chapter 5. Linear Simulation: An F4-C Flight Control Case Study 57
its characteristic equation is found. The maximum and minimum of the characteristic
polynomial coefficients is later found and used to generate Kharitonov’s polynomials for
each flight condition, as presented in Table 5.6.
Table 5.6: Kharitonov’s polynomials coefficients
Operating Point s3 s2 s1 s0
1
K1(s) 1 24.1794 66.6442 69.0873
K2(s) 1 23.3426 86.0316 120.2817
K3(s) 1 23.3426 66.6442 120.2817
K4(s) 1 24.1794 86.0316 69.0873
2
K1(s) 1 28.1143 3242.6 19255
K2(s) 1 28.0537 3244.1 19400
K3(s) 1 28.0537 3242.6 19400
K4(s) 1 28.1143 3244.1 19255
3
K1(s) 1 24.0574 193.2548 203.7089
K2(s) 1 23.1174 215.0013 382.7542
K3(s) 1 23.1174 193.2548 382.7542
K4(s) 1 24.0574 215.0013 203.7089
4
K1(s) 1 23.9073 711.5970 1803.4
K2(s) 1 23.4963 721.1680 2075.5
K3(s) 1 23.4963 711.5970 2075.5
K4(s) 1 23.9073 721.1680 1803.4
Each set of Kharitonov’s polynomials was then checked for stability and found to be
stable, that is, all the roots of the polynomials have strictly negative real parts and are
in the left half plane. Therefor, it is guaranteed that if CLα changes in the range pro-
vided in Table 5.5, all four interval polynomials corresponding to four distinct perturbed
flight conditions are stable and thus the pitch-tracking controller has a certain degree of
robustness with respect to changes in CLα .
More over, the controller’s robustness is stablished graphically as all Kharitonov rect-
angles are plotted for different values of w, and it is seen that the Zero Exclusion Con-
Chapter 5. Linear Simulation: An F4-C Flight Control Case Study 58
dition is met as no rectangle touches the point z = 0, as shown in Figure 5.2. It is also
clearly seen in Figure 5.2 that the value sets for each flight condition move from the first
quadrant to the second and eventually to the third (the interval polynomial has a degree
of 3) without touching or passing through the origin, thus providing a graphical confir-
mation of the robust stability of the perturbed closed-loop system for the four different
flight conditions.
(a) Kharitonov rectangles, operating point 1 (b) Kharitonov rectangles, operating point 2
(c) Kharitonov rectangles, operating point 3 (d) Kharitonov rectangles, operating point 4
Figure 5.2: Kharitonov’s robust stability graphical check for different flight conditions
Chapter 5. Linear Simulation: An F4-C Flight Control Case Study 59
5.2.2 Robustness Optimization
Figure 5.3 shows the maximum real part of the closed-loop system eigenvalues for all four
flight conditions, when CLα is allowed to change in the maximum allowable deviation
range obtained before in Table 5.5. Note that the maximum real part of the eigenvalues
remains negative in all cases which means that the closed-loop system remains stable
despite large perturbations in CLα . These methods provide sufficient yet conservative
robust stability conditions and as long as the derivative changes in the given range, the
closed-loop system’s stability is guaranteed. However, because the stability robustness
bound theory provides a sufficient condition and not a necessary one, it is not possible
to draw any conclusion regarding the stability of a disturbed plant with CLα perturbing
outside of the obtained boundaries in Table 5.5. In other words, the perturbed system
may or may not maintain its stability, depending on its closed-loop pole placement. As a
result of this conservatism and as it is seen in Figure 5.3, it is still possible for the system
to remain stable even when moving out of the obtained robust stability bounds.
Figure 5.3: Maximum Eigenvalue of the closed-loop system vs. CLα
Note the general trend in Figure 5.3. The stability of the perturbed close-loop system
is increased (i.e. the maximum of the real parts of the eigenvalues gets more negative)
Chapter 5. Linear Simulation: An F4-C Flight Control Case Study 60
as CLα increases and thus provides more lift and control.
As said before, the robustness analysis provides a sufficient condition and not a nec-
essary one, thus no assumption can be made about the stability of a perturbed system
with the uncertainties outside of the given bounds. Therefore, and due to the fact that
these results are also highly conservative, it can be desirable to increase the robustness
bounds for each system. This was done using the methodologies presented in Chapter
4. The new requirement to increase the robustness introduces a new problem, previously
presented as the Robust Simultaneous Stabilization Problem (RSS) which is solved using
a multi-objective optimization method. For this test case the following wish attribute
has been formulated as the new objective function, converting a maximization prob-
lem to a minimization one, using the wighted sum method to solve the multi-objective
optimization problem of increasing the robustness bounds.
I =4∑j=1
1
|∆CLαj |max=
(1
|∆CLα1|max+
1
|∆CLα2|max+
1
|∆CLα3 |max+
1
|∆CLα4|max
)(5.14)
Note that in this investigation, the robustness of the design is determined with respect
to the value of the maximum allowable deviation in the stability derivative CLα . Finally,
a multiple robustness optimization problem as formulated in section 4.2.3 is solved and
the results are presented in Table 5.7.
Table 5.7: Solutions to the SS and RSS problems
SS Problem RSS Problem: RSS Problem:
Multiple Robustness Optimization Bi-level Optimization
f [−2 − 1.8404 0.1423]T [0.3123 − 2 1.5933]T [1.4481 − 2 1.6869]T
The new control law identified as the solution to the ”RSS problem: Multiple Ro-
bustness Optimization” is now compared to the previously obtained solution to the SS
problem, in terms of robustness of the design. Table 5.8 shows this comparison. For
each distinct flight condition, the stability robustness bounds or |∆CLα|max is obtained
for both the SS and RSS solutions. The 4th column also shows how much the maximum
Chapter 5. Linear Simulation: An F4-C Flight Control Case Study 61
allowable deviation range has been increased when the optimization goal is to expand the
robustness bounds. It is seen that for the first, second, and the forth operating condi-
tions, the maximum allowable deviation, |∆CLα |max, has increased significantly as shown
in Table 5.8, whereas there has been a slight decrease in the value for the third operating
point. This, however, is acceptable if not desirable since it is only a %1 decrease and the
robustness of the other systmes were significantly increased at this cost.
Table 5.8: Effect of relaxing the robustness on |∆CLα |maxunrelaxed robustness relaxed robustness
SS Problem RSS Problem: Multiple Robustness Optimization
Flight Condition |∆CLα|max |∆CLα|max % of increase in robustness
1 3.5065 4.4500 % 27
2 0.0474 0.0599 % 26
3 4.9685 4.8953 % -1
4 0.9716 1.0488 % 7
Now to give a sense of how desirable these results are, they are compared to the
solution of the ”RSS problem: Bi-level Optimization” which was presented in section 4.1.
Therefore, the following objective function was constructed where the goal is to maximize
the robustness boundary of the system with the most vulnerability to perturbations, i.e.
maximizing the minimum of the robustness bounds at each time step.
I = max −∣∣∣∆CLαj ∣∣∣max = max (−|∆CLα1|max,−|∆CLα2|max,−|∆CLα3 |max,−|∆CLα4|max)
(5.15)
The previously proposed extended decomposition based problem is solved as formu-
lated in Eq. 4.8 and the designed controller is shown in Table 5.7. The robustness of
this design is investigated as well and compared to that of the previous RSS solution,
as shown in Table 5.9. Having the smallest robustness bounds, both the second and the
forth operating points experienced an increase in their |∆CLα|max values. Whereas that
of the first and third operating points have seen a significant drop. Having said that, it
is seen that maximizing the minimum value of ∆CLα has been acheived, but only at the
Chapter 5. Linear Simulation: An F4-C Flight Control Case Study 62
cost of loosing the robustness of the other systems to a great extent. It can be said that
the multiple objective optimization problem (MOOP) methodology introduced before,
delivers a better result as it provides almost the same level of increase to ∆CLα for the
2nd and 4th systems while increasing that of the other two operating points as well.
Table 5.9: Results from different RSS optimization methodologies and objective functions
RSS Problem: RSS Problem:
Multiple Robustness Optimization Bi-level Optimization
Flight |∆CLα|max robustness increased by |∆CLα|max robustness increased by
Condition
1 4.4500 % 27 1.4883 % -57
2 0.0599 % 26 0.0606 % 27
3 4.8953 % -1 0.6901 % -86
4 1.0488 % 7 1.0795 % 11
For a more graphical comparison between these two approaches, Figure 5.4 shows the
values of the maximum real part of the eigenvalues for the 4 different closed-loop systems
versus the respective maximum allowable deviations in CLα as presented in Table 5.9. In
all cases, the maximum of the real part of the eigenvalues are negative, indicating that all
systems under different controllers have poles in the left half plane and thus simultaneous
stabilization was achieved. The set of pictures also clearly shows that the solution to the
Multi-objective optimization problem does a much better job overall in terms of increasing
the maximum allowable range of perturbations in CLα , when compared to the solution of
the bi-level decomposition problem (where robustness of the most critical case is increased
at the cost of reducing the robustness of some of the other plants significantly). This
behaviour can be explained due to the nature of the weighted-sum method optimization
technique, which aims for a more balancing effect when minimizing a multi-objective
problem. Note that the bi-level strategy yields a more conservative result in terms of
the allowable perturbation in CLα compared to that of the multi-objective optimization
problem, although conservatism is a characteristic of the stability robustness bounds
Chapter 5. Linear Simulation: An F4-C Flight Control Case Study 63
technique introduced in Chapter 3.
Figure 5.4: Maximum Eigenvalue of the closed-loop system vs. CLα
Results presented in Table 5.10 show all four closed-loop system’s poles under three
different control laws given in Table 5.7. As suggested by the pole locations, all of
the controllers provide simultaneous stability since the real part of the eigenvalues were
negative for all of the operating points.
Figures 5.5, 5.6, and 5.7 show the initial response of the unperturbed systems (i.e.
∆CLα = 0) to the initial condition x = [1 0 0]T for the SS problem solution, Multiple
Robustness Optimization solution, and Decomposition-Based Strategy solution, respec-
tively. In all three plots, states are decaying to the steady state and the systems are
stable. However, states are converging much faster to the steady states under the S.S.
control law than under the two other problem solutions. In other words, the other two
problem solutions are slightly overdamped and sluggish in reaching the steady states.
This is specially true for the decomposition-based bi-level optimization problem which is
sluggish in decaying the initial condition and converging to the steady states. This can
be expected since the primary goal of the RSS problem is not to optimize the pole place-
ment or performance, but rather to optimize the robustness of the design. Moreover, the
Chapter 5. Linear Simulation: An F4-C Flight Control Case Study 64
Tab
le5.
10:
Aco
mpar
ison
ofth
ecl
osed
-loop
eige
nva
lues
for
diff
eren
top
tim
izat
ion
met
hodol
ogie
s
Clo
sed-l
oop
Eig
enva
lues
Syst
emSS
Pro
ble
mR
SS
Pro
ble
m:
RSS
Pro
ble
m:
(Fligh
tC
ondit
ion)
Mult
iple
Rob
ust
nes
sO
pti
miz
atio
nB
i-le
vel
opti
miz
atio
n
A1−B
1fT
-20.
2173
,an
d-1
.772
1±
1.24
21i
-50.
7045
,-0
.640
7,an
d-1
.435
8-5
2.62
00,
-0.1
510,
and
-1.8
820
A2−B
2fT
-6.2
202,
and
-10.
9322±
54.6
610i
-12.
8762
,an
d-2
2.11
39±
50.2
965i
-12.
9720
,an
d-2
3.00
20±
49.6
732i
A3−B
3fT
-1.7
721,
and
-10.
9079±
6.81
81i
-47.
6306
,-0
.598
0,an
d-4
.378
8-4
9.64
96,
-0.0
680,
and
-4.7
618
A4−B
4fT
-2.9
612,
and
-10.
3706±
23.3
970i
-3.9
594,
and
-24.
3812±
2.88
62i
-3.2
880,
-20.
1510
,an
d-3
1.15
49
Chapter 5. Linear Simulation: An F4-C Flight Control Case Study 65
decomposition-based formulation does not consider the performance of the controller as
a design criteria. This issue can be addressed later by imposing additional constraints
on the bi-level optimization problem which will improve the damping characteristics of
the design.
Figure 5.5: Response to initial condition: SS problem
Chapter 5. Linear Simulation: An F4-C Flight Control Case Study 66
Figure 5.6: Response to initial condition: Multiple Robustness Optimization solution
Figure 5.7: Response to initial condition: Decomposition-Based Strategy solution
Chapter 6
Non-Linear Simulation: A CRJ-200
Flight Control Case Study
In this chapter, a simultaneously stabilizing pitch tracking controller for a CRJ-200 re-
gional jet aircraft, operating at 4 different flight conditions, is designed with the elevator
deflection being the control input. The design of this control law is based on the linear
model of the plane’s longitudinal dynamics, using the design process proposed before in
section 2.7. Later, the effectiveness of this controller under the introduced normal op-
erating conditions as well as gust encountered flight conditions is investigated, with the
help of a state of the art CRJ-200 flight training device and simulation platform. The
robustness of the designed controller will also be further investigated using the method
described in section 3.3. Moreover, The robustness of the designed controller is relaxed
and a new control law is obtained using the optimization methodology proposed in Chap-
ter 4, which is later compared to previously obtained results. This new control law design
is further compared to a controller obtained using the method presented in [29], which is
also designed to have a more relaxed robustness using a bi-level optimization methodology
with a different objective function.
The state-of-art Flight Training Device (FTD) which can be used for pilot training,
presents a more realistic airplane model, as shown in Figure 6.1. It features a generic
regional twin jet aircraft cockpit, hydraulic sticks, cockpit panels and aircraft compo-
nents, flight systems, navigation and communication systems, and other electrical and
67
Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study68
(a) (b)
Figure 6.1: The Flight Training Device (FTD) Facility
mechanical systems, as well as a visual system providing 150◦ (horizontal) x 35◦ (verti-
cal) field of view and an instructor operating station running a model of a Bombardier
CRJ-200. When using the FTD as a test bed for interactive pitch tracking control design
and simulation, the designed controller takes over as soon as the Simulink simulation is
started, overwriting the built-in autopilot.
6.1 Modelling of the CRJ-200
In order to derive the aircraft’s equations of motion, some principal reference frames used
in the flight dynamics should be listed in the following.
Earth–Fixed Reference Frame, FE: In many applications, any reference frame fixed
to the earth can be used as an inertial frame. The Earth–fixed reference frame is
an earth-surface frame with it’s origin located at any point of the earth surface
near the vehicle. The xE–axis points north, the zE–axis is directed vertically down
towards the center of the earth, and the yE–axis completes the right-handed system,
pointing towards the east.
Body–Fixed Reference Frame, FB: The origin of this frame is the center of mass.
The xB–axis lies in the plane of symmetry and is directed towards the aircraft’s
Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study69
nose. The yB–axis is perpendicular to the plane of symmetry and is positive to-
wards the starboard side. The zB–axis completes the right–handed system, directed
downward.
Air–Trajectory Reference Frame or Wind Axes, FW : This frame has its origin fixed
to the plane at the center of mass. The xW is directed along the velocity vector
V of the vehicle relative to the atmosphere, and the zW axis lies in the plane of
symmetry. The yW–axis completes the right–handed system.
Stability Axes, FS: The stability axis system can be derived from the body–fixed sys-
tem by rotating the coordinate frame by −α0 around the yB–axis. Here, α0 is the
angle of attack during a reference flight condition. This system is commonly used
for flight control purposes. In particular, throughout this document the stability
coordinate system is used unless otherwise specified. For simplicity of notation, the
index S has often been omitted.
6.1.1 Nonlinear Model
In general, the mathematical models describing the motion of an aircraft are subject
to few assumptions. For example, the earth is a sphere rotating on an axis fixed in
inertial space with g being a radial vector, the atmosphere is at rest relative to the earth,
and the vehicle is a rigid body with no elastic mode taken into account. Moreover,
for a rigid vehicle having a plane of symmetry and with the flat–earth approximation
(the simplification of treating the earth as a flat stationary plane in inertial space), the
equations of motion describing the plane’s unsteady behaviour are reduced to nine scalar
differential equations, including six Euler equations of motion and three equations for
calculating the Euler angles φ, θ and ψ. These reduced equations obtained by neglecting
the spherical rotation of the earth are collected below:
X = mg sin θ +m (u+ q w − r v), (6.1)
Y = −mg cos θ sinφ+m (v + r u− pw), (6.2)
Z = −mg cos θ cosφ+m (w + p v − q u), (6.3)
Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study70
L = Ix p− Ixz r + q r (Iz − Iy)− Ixz p q, (6.4)
M = Iy q + r p (Ix − Iz) + Ixz(p2 − r2), (6.5)
N = Iz r − Ixz p+ p q (Iy − Ix) + Ixz q r, (6.6)
φ = p+ (q sinφ+ r cosφ) tan θ, (6.7)
θ = q cosφ− r sinφ, (6.8)
ψ = (q sinφ+ r cosφ) sec θ, (6.9)
where the state vectore is defined as x = [u, v, w, p, q, r, φ, θ, ψ]T , and the control input
influencing the air reaction forces and moments is u = [δe, δp, δa, δr]T , representing eleva-
tor deflection, throttle setting, aileron, and rudder deflection respectively. The nonlinear
equations of motions are derived in detail and can be found in [15] as well as many other
references.
6.1.2 Linear Model
In most cases, a nonlinear system can be locally described by a linearized model about a
reference condition. Here, the reference steady-state is the symmetric steady rectilinear
flight over a flat earth with no angular velocity, meaning that v0 = p0 = q0 = r0 =
φ0 = ψ0 = 0. The steady-state values are denoted by subscript 0 and deviation from the
steady-state is represented by the prefix ∆, that is: x = x0 + ∆x. For vehicles with a
plane of symmetry, two uncoupled sets of longitudinal and lateral equations are found.
When using the small–disturbance theory, it is convenient to use wind axes for the lift-
force and drag-force equations, and the stability axes for the presentation of longitudinal
equations. Therefore, and because of the choice of the coordinates, w0 = 0 and u0 is
equal to the reference flight velocity. The forces and moments acting on the body is also
splitted into a steady-state reference value and a deviation from the reference condition:
X = X0 + ∆X, (6.10)
Y = Y0 + ∆Y, (6.11)
Z = Z0 + ∆Z, (6.12)
L = L0 + ∆L, (6.13)
Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study71
M = M0 + ∆M, (6.14)
N = N0 + ∆N. (6.15)
For a symmetric rectilinear flight, it is obvious that X0 = mg sin θ0, Z0 = −mg cos θ0,
and Y0 = L0 = M0 = N0 = 0. To arrive at an approximation for the air reactions forces
and moments, the stability derivatives can now be used:
∆X = Xu∆u+Xw∆w +Xq∆q +Xu∆u+Xw∆w +Xq∆q +
Xδe∆δe+Xδp∆δp, (6.16)
∆Y = Yv∆v + Yp∆p+ Yr∆r + Yv∆v + Yp∆p+ Yr∆r +
Yδa∆δa+ Yδr∆δr, (6.17)
∆Z = Zu∆u+ Zw∆w + Zq∆q + Zu∆u+ Zw∆w + Zq∆q +
Zδe∆δe+ Zδp∆δp, (6.18)
∆L = Lv∆v + Lp∆p+ Lr∆r + Lv∆v + Lp∆p+ Lr∆r +
Lδa∆δa+ Lδr∆δr, (6.19)
∆M = Mu∆u+Mw∆w +Mq∆q +Mu∆u+Mw∆w +Mq∆q +
Mδe∆δe+Mδp∆δp, (6.20)
∆N = Nv∆v +Np∆p+Nr∆r +Nv∆v +Np∆p+Nr∆r +
Nδa∆δa+Nδr∆δr. (6.21)
Since the stability derivatives are found to be negligible with respect to time derivatives
of the state variables, they can be ommitted with the exception of Zw and Mw. The
longitudinal dynamics obtained by linearizing the non-linear equations of motion and
substituting the air reaction forces and moments accordingly, is as follows:
Along =
Xum
Xwm
0 −gcosθ0
Zum−Zw
Zwm−Zw
Zq+mu0
m−Zw−mgsinθ0m−Zw
1Iy
[Mu + MwZu
m−Zw
]1Iy
[Mw + MwZw
m−Zw
]1Iy
[Mq + Mw(Zq+mu0)
m−Zw
]−Mwmgsinθ0
Iy(m−Zw)
0 0 1 0
,
(6.22)
Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study72
Blong =
Xδem
Xδpm
Zδem−Zw
Zδpm−Zw
Mδe
Iy+ MwZδe
Iy(m−Zw)
Mδp
Iy+
MwZδpIy(m−Zw)
0 0
, (6.23)
and
xlong = [u w q θ]T , (6.24)
ulong = [δe δp]T . (6.25)
The stability derivatives of the CRJ-200 in different flight conditions can be calculated
using the Aircraft Conceptual Design Toolbox for Matlab, written by Ruben Perez, Uni-
versity of Toronto. Furthermore, since the FTD is not equipped with an automatic
throttle control, it is reasonable to set δp = 0. Finally, a first order lag filter was found
to be adequate for modeling the behaviour of the actuators, as shown in [4]. The time
constants used are:
Televator = 1.6 s
Taileron = 2.75 s
Trudder = 0.1 s
6.2 Introduction of the Test Case
The aircraft’s longitudinal dynamics is represented by the following system for each flight
condition:
x = Ajx+Bjδe, (6.26)
δe = − 1
Teδe+
1
Teδec. (6.27)
where the elevator dynamics is given by a first order differential equation in the time
domain. Therefore for this case study, the longitudinal control augmented system (CAS)
Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study73
of the aircraft, including both short period and phugoid modes and elevator actuator
dynamics, is represented here:
∆u
w
q
∆θ
δe
︸ ︷︷ ︸
x
=
Aj Bj
0 − 1Te
︸ ︷︷ ︸
A
∆u
w
q
∆θ
δe
︸ ︷︷ ︸
x
+
0
1Te
︸ ︷︷ ︸
B
δec︸︷︷︸u
, (6.28)
where δec is the elevator command input, the actuator system has a time constant Te of
1.6 s and the plant state coefficient matrices and control coefficient vectors are given as:
Aj =
a11j a12j 0 a14j
a21j a22j a23j a24j
a31j a32j a33j a34j
0 0 1 0
, Bj =
0
b2j
b3j
0
Cj = I. (6.29)
The control input is δec, and the state vector is defined as x = [u w q θ δe]T .
Four different flight conditions have been chosen with variation in both altitude and
speed, as shown in Table 6.1. The values of matrices parameters are also given in Table
6.1.
In order to design a controller to track a pitch angle command as illustrated in Figure
6.2, let us define Cz = [0 0 0 1 0] and extend the state vector x by the integral state
xI , where xI = e and e is the tracking error. The extended system now has the form of:
x
xI
︸ ︷︷ ︸
˙x
=
A 0
−Cz 0
︸ ︷︷ ︸
A
x
xI
︸ ︷︷ ︸
x
+
B
0
︸ ︷︷ ︸
B
u+
0
1
︸ ︷︷ ︸W
θc, (6.30)
where
u = −[Kx KI ]x = −Kx. (6.31)
Using the proposed optimization methodology introduced in section 2.7, with the
controller enteries being limited by a factor of µ = 2 and a performance index of η = 1,
Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study74
Table 6.1: CRJ-200 flight operating conditions
Flight Condition
Property 1 2 3 4
Total Weight 40,000 40,000 40,000 40,000
xcg, ft 0.16c 0.16c 0.16c 0.16c
Ix, slug.ft2 55,717 55,717 55,717 55,717
Iy, slug.ft2 369,830 369,830 369,830 369,830
Iz, slug.ft2 411,017 411,017 411,017 411,017
Ixz, slug.ft2 17,789 17,789 17,789 17,789
Mach number 0.74 0.55 0.48 0.42
Altitude, ft 33,000 33,000 19,000 19,000
angle of attack, deg 0.7 0.7 0.7 0.7
Flight path angle, deg 0 0 0 0
a11 -0.0084 -0.0002 -0.0001 -0.00001
a12 0.0066 0.0146 0.0171 0.0206
a14 -32.198 -32.198 -32.198 -32.198
a21 -0.1221 -0.1375 -0.1425 -0.1584
a22 -0.8338 -0.5586 -0.8251 -0.7095
a23 718.744 533.551 489.778 428.541
a24 -0.3935 -0.3927 -0.3921 -0.3919
a31 0.0001 0.0001 0.0001 0.0001
a32 -0.0135 -0.0095 -0.0142 -0.0123
a33 -0.6610 -0.4397 -0.6490 -0.5564
a34 0.0001 0.0001 0.0001 0.0001
b2 -52.179 -26.656 -36.754 -36.884
b3 -7.0588 -3.6180 -4.9911 -5.0157
Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study75
Figure 6.2: Pitch angle tracker
a simultaneously stabilizing controller is obtained as:
K = [0.0006 0.0010 − 0.2932 − 1.5976 1.0112 0.4683] . (6.32)
Table 6.2 presents the aircraft open-loop system eigenvalues and it can be seen that
except for the first flight condition, the longitudinal system is phugoid unstable.
Now, the closed loop system is given by
˙x =(A− BK
)x+ Wθc. (6.33)
Table 6.3 shows the pitch tracking closed-loop system eigenvalues, and it is seen that
all pole locations for all of the plants are in the negative real axis, and hence the closed
loop system is stable.
Table 6.2: Open-loop system eigenvalues
System Eigenvalues
A1 -0.7478 ± 3.1074i, and -0.0037 ± 0.0745i
A2 -0.4999 ± 2.2539i, and 0.0007 ± 0.0912i
A3 -0.7380 ± 2.6316i, and 0.0009 ± 0.0955i
A4 -0.6343 ± 2.2887i, and 0.0013 ± 0.1073i
6.3 Results: Ordinary and Gust-Encountered Flight
The effectiveness and performance of the pitch-tracking controller given in (6.32) is now
investigated under ordinary operating conditions for which it was designed, as well as
Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study76
Table 6.3: Closed-loop system eigenvalues
System Eigenvalues
A1 −B1K -0.6051 ± 3.2348i, -0.2483 ± 0.2879i, -1.0447, and -0.0088
A2 −B2K -0.3355 ± 2.3586i, -0.1643 ± 0.2340i, -1.2519, and -0.0039
A3 −B3K -0.5537 ± 2.7338i, -0.2149 ± 0.2819i, -1.1908, and -0.0032
A4 −B4K -0.4436 ± 2.3843i, -0.1857 ± 0.2653i, -1.2598, and -0.0047
gust encountered flight conditions. This was done with the help of our integrated system
design and flight simulation platform, as discussed in [4] and [21], which is currently
being used for control system design, simulation, and testing. It consists of a state of the
art flight training device research simulator that is interconnected either with a real time
system simulator or with Matlab/Simulink in a computer terminal. This testbed repre-
sents a more complete and accurate aircraft model and hence the obtained simulation
results are much more accurate and precise than those obtained by linear or nonlinear
offline simulations in Simulink. For flying under ordinary operating conditions, the pitch
tracking control performance for a step input of 5 deg pitch angle at four different flight
conditions, is obtained from the testbed and is shown in Figure 6.3. It can be seen that
the designed pitch tracking controller provides adequate tracking at all flight conditions.
A sudden change in flight conditions, aircraft configuration or any other flight char-
acteristics due to damage results in nonlinear behavior, and although previously intro-
duced robustness investigation approaches are practical, due to the nonlinear nature of
the testbed a mathematical investigation of robust stability bounds is difficult. It was
found that these methods result in a very conservative region of stability which is also a
sufficient condition and not a necessary one. Hence, no conclusion can be made about the
stability outside of this region. Therefore to better illustrate the robustness of the con-
troller at each flight condition, the implemented ‘’Moderate Turbulence‘’ option within
the flight simulation platform was activated, which introduces sinusoidal gust to the sim-
ulation. The pitch-tracking control performance at four different flight conditions under
gust encounter is shown in Figure 6.4. Although highly oscillatory, and although the
controller is not able to damp out the external disturbances because it was not initially
Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study77
desigend with disturbance rejection criteria in mind, the controller is still able to track
the step pitch command. This oscillation is due to the fact that the gust alleviation
characteristics was not taken into account at the time of designing the controller.
6.4 Robustness Investigation & Optimization
As it was shown, the controller is able to simultaneously stabilize all four different flight
conditions. However, any introduced perturbation in speed, altitude, airplane’s config-
uration or aerodynamic shape, or other changes in flight conditions and characteristics
due to damage, disturbes the closed loop system and therefore it is desired to check if the
perturbed closed loop system is still stable. This was also shown in the simulation and
as presented before, the controller was still able to stabilize the aircraft when encoun-
tered with the incorporated medium gust settings on the flight test bed. However, to
get more accurate results in terms of how much perturbation the state matrix’s elements
can toplerate while maintaining the stability of the closed-loop systems, a theoretical
investigation is needed with the help of the method presented in [36].
As discussed before, the state space representation of the aircraft’s longitudinal dynamics
is:
xlong =
Xum
Xwm
0 −gcosθ0
Zum−Zw
Zwm−Zw
Zq+mu0
m−Zw−mgsinθ0m−Zw
1Iy
[Mu + MwZu
m−Zw
]1Iy
[Mw + MwZw
m−Zw
]1Iy
[Mq + Mw(Zq+mu0)
m−Zw
]−Mwmgsinθ0
Iy(m−Zw)
0 0 1 0
xlong
+
Xδem
Zδem−Zw
Mδe
Iy+ MwZδe
Iy(m−Zw)
0
δe,
where stability derivatives are defined as
Xu = qSwCxuV
, Cxu = − (CDu + 2CD) ,
Xw = qSwCxαV
, Cxα = (CL − CDα) ,
Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study78
(a) Pitch angle tracking for a 5-deg pitch angle step com-
mand
(b) Elevator deflection for a 5-deg pitch angle step com-
mand
(c) Pitch rate for a 5-deg pitch angle step command (d) Forward velocity changes for a 5-deg pitch angle step
command
(e) Upward velocity changes for a 5-deg pitch angle step
command
Figure 6.3: Time history of the states at different flight conditions subject to a 5-deg
step input
Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study79
(a) Pitch angle tracking for a 5-deg pitch angle step com-
mand
(b) Elevator deflection for a 5-deg pitch angle step com-
mand
(c) Pitch rate for a 5-deg pitch angle step command (d) Forward velocity changes for a 5-deg pitch angle step
command
(e) Upward velocity changes for a 5-deg pitch angle step
command
Figure 6.4: Time history of the states at different flight conditions subject to a 5-deg
step input, when encountered with gust
Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study80
Zu = qSwCzuV
, Czu = − (CLu + 2CL) ,
Zw = qSwCzαV
, Czα = − (CLα + CD) ,
Zq =qSwMACwCzq
2V, Czq = −CLq ,
Zw =qSwMACwCzα
2V 2 , Czα = −CLα ,
Mu = qSwMACwCmuV
, Cmu = (Cmu + 2Cm),
Mw = qSwMACwCmαV
, Cmα = Cmα ,
Mq =qSwMAC2
wCmq2V
, Cmq = Cmq ,
Mw =qSwMAC2
wCmα2V 2 , Cmα = Cmα ,
Xδe = qSwCxδe , Cxδe = −CDδe ,
Zδe = qSwCzδe , Czδe = −CLδe ,
Mδe = qSwMACwCmδe , Cmδe = Cmδe .
Assuming that the perturbation is due to sudden changes in altitude or flight mach
number, every matrix element which is a function of these variables will also change as
a result. Therefore, the perturbed closed loop system is formulated as:
ACL = A0 + E =(A− BK
)+ E ,
E =
∆a11 ∆a12 0 0 0 0
∆a21 ∆a22 ∆a23 ∆a24 ∆b2 0
∆a31 ∆a32 ∆a33 ∆a34 ∆b3 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
.
Hence:
U =
1 1 0 0 0 0
1 1 1 1 1 0
1 1 1 1 1 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
. (6.34)
Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study81
Applying the methodology in [36] to this test case, the magnitude of the maximum
allowable deviation in aij is obtained for each flight condition, which is presented in
Table 6.4.
Table 6.4: Maximum allowable deviation in ai,j
Operating Point 1 2 3 4
|∆ai,j|max < µεY = 9.0093e-006 8.7701e-006 1.5187e-005 1.5141e-005
As it is seen from this table, these results are far too conservative. Moreover, this
method provides sufficient yet conservative robust stability conditions and as long as the
state matrix elements change in the given range, the closed-loop system’s stability is
guaranteed. However, because the stability robustness bound theory provides a sufficient
condition and not a necessary one, it is not possible to conclude anything regarding
the stability of the disturbed state matrix with it’s elements perturbing outside of the
obtained boundaries in Table 6.4, and depending on the closed-loop pole placement, the
perturbed system may or may not maintain its stability. As a result, it is still possible
for a perturbed system to remain stable even when disturbance is out of the obtained
robust stability bounds, and no assumption can be made regarding the stability of such a
perturbed plant. Since these results are highly conservative, it can be desirable to increase
the robustness bounds for each system. Hence, in order to deal with the conservatism
and relax the robustness bounds, the optimization problem must be looked at from a
different perspective. The new objective to increase the robustness bounds, introduces
a new problem previously discussed as the Robust Simultaneous Stabilization Problem
(RSS), which is solved using a multi-objective optimization methodology. Note that in
this investigation, the robustness of the design is addressed in terms of the values of
the maximum allowable deviation in the enteries ai,j of the state matrix. For this case
study, the following wish attribute is formulated as the new objective function, converting
a maximization problem to a minimization one and using the wighted sum method to
Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study82
solve the multi-objective optimization problem of increasing the robustness bounds.
I =4∑j=1
1
µεY j=
1
µεY 1
+1
µεY 2
+1
µεY 3
+1
µεY 4
. (6.35)
Finally, By solving the above multiple robustness optimization problem proposed in sec-
tion 4.2.3, a new simultaneously stabilizing controller with a generally improved robust-
ness boundary is found:
K = [0.0001 0.0001 − 1.0000 − 0.1480 − 0.9999 0] . (6.36)
The robustness of this new controller identified as the solution to the ‘’RSS problem:
Multiple Robustness Optimization‘’ is now compared to the robustness of the previously
obtained solution to the SS problem. Table 6.5 shows this comparison. For each flight
condition, the stability robustness bounds, µεY j, is obtained for both the SS and RSS
control law solutions. The 4th column also shows how much the maximum allowable
deviation range has been increased when the optimization objective is to expand the
robustness bounds. As it is seen, for all of the four operating conditions, the maximum
allowable deviation in ai,j has increased significantly overall. The first plant correspond-
ing with the aircraft flying at the first operating condition, saw a significant improvement
in it’s stability robustness bounds by %88. The second, third, and fourth systems also ex-
perienced great improvement in their stability robustness values, by %14, %48, and %23
respectively. The results can again be explained due to the nature of the weighted-sum
method optimization technique, which introduces a more balancing effect when minimiz-
ing a multi-objective problem.
Table 6.6 presents closed-loop eigenvalues for all four closed-loop systems under the
two feedback control laws obtained by solving the SS and RSS problems. It is suggested
by the negative real part of the eigenvalues, indicating that all systems corresponding to
different flight conditions have poles in the left half plane, that both SS and RSS control
law solutions provide simultaneous stability.
The effectiveness and performance of the RSS pitch-tracking control law obtained
in (6.36) is further investigated under four flight conditions for which it was designed.
As explained before, the simultaion was performed using the integrated system design
Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study83
Table 6.5: Effect of relaxing the robustness on |∆ai,j|maxunrelaxed robustness relaxed robustness
SS Problem RSS Problem: Multiple Robustness Optimization
Flight Condition |∆ai,j|max |∆ai,j|max % of increase in robustness
1 9.0093e-6 1.6951e-5 % 88
2 8.7701e-6 1.0051e-5 % 14
3 1.5187e-5 2.2439e-5 % 48
4 1.5141e-5 1.8692e-5 % 23
and flight simulation platform (FTD). Figure 6.5 shows the pitch tracking controller
performance for a step input of 5 degree pitch angle at four different flight conditions,
as obtained from the testbed. It is seen that the aircraft flying under the RSS problem
solution is stable and able to track the pitch angle command, however, the pitch angle
is converging much faster to the pitch angle command under the S.S. control law than
under the RSS control solution. After a long enough simulation time, which is not
satisfactory, the states are just starting to converge to their steady value. Overall, the RSS
solution is overdamped and sluggish in reaching the input command. This, however, is
expected, since the main objective of the RSS problem formulation is not to optimize the
performance of the controller, but rather to maximize the robustness of the design at every
flight condition. Clearly, this consideration takes away from the tracking performance
of the controller. Since the RSS problem formulation does not take the performance
into account as a design criteria, the issue can later be addressed by imposing additional
constraints which will improve the damping and tracking characteristics of the design.
Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study84
Tab
le6.
6:A
com
par
ison
ofth
ecl
osed
loop
eige
nva
lues
for
diff
eren
top
tim
izat
ion
met
hodol
ogie
s
Clo
sed-l
oop
Eig
enva
lues
Syst
emSS
Pro
ble
mR
SS
Pro
ble
m:
(Fligh
tC
ondit
ion)
Mult
iple
Rob
ust
nes
sO
pti
miz
atio
n
A1−B
1fT
-0.6
051±
3.23
48i,
-0.2
483±
0.28
79i,
-1.0
447,
and
-0.0
088
-0.6
191±
3.72
70i,
-0.1
227±
0.14
35i,
-0.0
196,
and
-0.0
001
A2−B
2fT
-0.3
355±
2.35
86i,
-0.1
643±
0.23
40i,
-1.2
519,
and
-0.0
039
-0.4
085±
2.69
31i,
-0.0
850±
0.14
10i,
-0.0
115,
and
-0.0
001
A3−B
3fT
-0.5
537±
2.73
38i,
-0.2
149±
0.28
19i,
-1.1
908,
and
-0.0
032
-0.6
116±
3.14
22i,
-0.1
193±
0.15
19i,
-0.0
126,
and
-0.0
001
A4−B
4fT
-0.4
436±
2.38
43i,
-0.1
857±
0.26
53i,
-1.2
598,
and
-0.0
047
-0.5
225±
2.73
14i,
-0.1
043±
0.15
57i,
-0.0
123,
and
-0.0
001
Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study85
(a) Pitch angle tracking for a 5-deg pitch angle step com-
mand
(b) Elevator deflection for a 5-deg pitch angle step com-
mand
(c) Pitch rate for a 5-deg pitch angle step command (d) Forward velocity changes for a 5-deg pitch angle step
command
(e) Upward velocity changes for a 5-deg pitch angle step
command
Figure 6.5: Time history of the states at different flight conditions subject to a 5-deg
step input, under the RSS problem solution
Chapter 7
Conclusion and Future
Developments
This chapter summarizes the conclusions derived from the different optimization method-
ologies, investigations and simulations presented in this thesis. A discussion on several
possible improvements for future research is presented as well.
7.1 Conclusions
Simultaneous Stabilization addresses the stability of a number of distinct plants under a
single feedback controller. Such a controller stabilizes the closed loop system and provides
simplicity and reliability. However, the less explored aspect is the robustness of the
controller designed for simultaneous stabilization. Since uncertainties such as parameter
variations can occur in flight, the designed controller should be able to provide some
degree of stability in case of perturbations such as gust, structural damage and failure,
etc. For instance, low frequency wind has an effect on tracking performance whereas
high frequency wind affects flight stability. It is therefore important to address the
robustness of the simultaneous stabilization or the Robust Simultaneous Stabilization
(RSS) problem, and to provide simultaneous stability for all systems while maximizing
their stability robustness bounds.
The objective of this thesis was to develop an optimization methodology which can
86
Chapter 7. Conclusion and Future Developments 87
deliver such a feedback control law. The optimization methodology proposed initially for
designing a simultaneously stabilizing controller was based on the previous research on
the topic, borrowing the same concept of a necessary and sufficient condition and con-
straints. The difference, however, was in the number of design variables and the imposed
constraints. The main addition to this initial optimization algorithm was to account for
the stability robustness bounds in the design process, hence finding a solution to the
RSS problem. The new developed optimization solution includes a different objective
function formulated to address systems’ robustness, using the concept of stability ro-
bustness bounds and multi-objective optimization. The optimization process is further
expanded to include the simultaneous stabilization condition as a constraint along with
the imposed constraints on the pole placement and the performance.
Two flight control case studies were considered for robustness optimization. The
investigation was performed through numerical and flight simulations, using a research
flight training device (FTD). The first case was the simultaneous stabilization of the
short-period mode of an F4-C fighter jet longitudinal dynamics, and the second case was
the simultaneously stabilizing pitch-tracking control for a CRJ-200 regional jet aircraft.
Several robustness analysis approaches were introduced to provide and test the physical
bounds of allowable perturbations. The proposed optimization solution to the SS problem
was applied to both case studies and simultaneously stabilizing controllers were designed.
These controllers were then compared to the Multiple-Robustness optimization solution
to the RSS problem, in terms of robustness bounds. To give a better comparison, both
control law solutions were also compared to the Bi-level optimization solution to the RSS
problem, in terms of stability robustness bounds.
In the first case study, the SS problem solution had a degree of robustness to varia-
tions in the aerodynamic derivative CLα , which was also verified by two other different
robustness investigation strategies. When expanding the robustness bounds through the
Multiple-Robustness optimization solution, the first, second, and the forth operating
conditions experienced a significant increase in their corresponding maximum allowable
deviation, |∆CLα|max, by %27, %26, and %7 respectively. There was a slight decrease of
%1 in the maximum allowable deviation for the third operating point, and the robustness
Chapter 7. Conclusion and Future Developments 88
of the other three systems was significantly increased at this cost. When the robustness
was to improve through the Bi-level optimization solution to the RSS problem, the sec-
ond and the forth operating points experienced an increase in their maximum allowable
deviation by %27 and %11 respectively; whereas that of the first and third operating
points saw a significant drop of %-57 and %-86. The multiple-objective optimization
problem (MOOP) methodology delivered a better result in terms of increasing the sta-
bility robustness bounds.
In the second case study, a simultaneously stabilizing pitch-tracking controller was
designed and the simulation results were obtained using a state of the art flight simula-
tion platform. Later, the robust behaviour of the controller was shown by activating the
incorporated “Moderate Turbulence” function. When expanding the robustness bounds
through the Multiple-Robustness optimization solution, the maximum allowable devia-
tion in ai,j saw significant improvement. The first plant corresponding with the aircraft
flying at the first operating condition saw an improvement in its stability robustness
bounds by %88. The second, third, and fourth systems also experienced great improve-
ment in the stability robustness values by %14, %48, and %23, respectively.
7.2 Future Developments
Several improvements can be made to the design optimization strategy. It could be
interesting to investigate the controller’s stability robustness with respect to variations
in other prominent stability and aerodynamic derivatives such as Cmq , CLδe , etc., as well
as altitude and Mach number across the flight envelope. This would be more realistic
and of practical interest.
The behaviour of the closed loop systems when encountered with gust can also be a
topic of further investigation. It would be interesting to investigate the effect of different
gust models on the stability of the system at different operating points and the robustness
bounds.
One interesting addition to the optimization methodology would be to optimize for
a combination of robustness bounds and performance, in order to improve the transient
Chapter 7. Conclusion and Future Developments 89
response of the design. The desired transient behaviour of the system could be included
in the objective function or it could be imposed as a constraint.
The proposed multiple-robustness optimization solution to the RSS problem could in-
clude other classical multi-objective optimization methods, which are more complicated
than the weighted-sum method in terms of formulation. A change in the objective func-
tion to represent more complex trade-offs is usually accompanied by a different optimum
solution to the problem. Performance criteria could also be implemented in the Bi-level
optimization solution to the RSS problem. This would result in improved stability while
maintaining the performance of the design.
Bibliography
[1] J. Ackerman. Parameter space design of robust control systems. IEEE Transactions
on Automatic Control, 25:1058–1072, 1980.
[2] J. Ackermann. Longitudinal control of fighter aircraft f4c. In in a Collection of
Plant Models and Design Specifications for Robust Control. DFVLR Press, Oberp-
faffenhoffen, Germany, 1982.
[3] J. Ackermann. Robustness against sensor failures. Automatica, 20:211–215, 1984.
[4] Holger Berndt. Integrated flight simulation and control of a regional jet aircraft.
Master’s thesis, University of Toronto, 2005.
[5] V. Blondel. Simultaneous stabilization of linear systems, volume 191 of lecture notes
in control and information sciences. Springer-Verlag, London, 1994.
[6] et al Bonine, W. J. Model f/rf-4b-c aerodynamic derivatives. Technical report, MAC
Report 9842, 10 Feb. 1964.
[7] Balakrishnan V. Feron E. Boyd, S. and L. El Ghaoui. Control system analysis and
synthesis via linear matrix inequalities. In American Control Conference, pages
2147–2154, June 1993.
[8] B. C. Bridges. Calculated longitudinal stability and performance characteristics
of the f-4b/c/d/j and rf-4b/c aircraft plus the an/asa-32h automatic flight control
system. Technical report, MAC Report F934, 19 Apr. 1963.
90
Bibliography 91
[9] S.S. Choi and H. R. Sirisena. Computation of optimal output feedback gains for
linear multivariable systems. IEEE transactions on Automatic Control, 3:257–258,
1974.
[10] J. H. chow. A pole-placement design approach for systems with multiple operating
conditions. IEEE Transactions on Automatic Control, 35:278–288, 1990.
[11] W. N. Crawford and G. Nadler. Static and dynamic control system characteristics
for the f-4 aircraft. Technical report, MAC Report F218, 16 Dec. 1966.
[12] Anton H. J. de Ruiter and Hugh H. T. Liu. A parameter optimization approach to
multiple-objective controller design. IEEE Transactions on Control Systems Tech-
nology, 16(2):330–339, March 2008.
[13] A. Debowski and Kurylowicz. Simultaneous stabilization of linear single-
input/single-output plants. International Journal of Control, 44:1257–1264, 1986.
[14] Yang W. Dorato, P. and C. Abdallah. Application of quantifier elimination theory to
robust multi-objective feedback design. Technical report, Technical report EECE 95-
007, University of New Mexico Department of Electrical and Computer Engineering,
Sep. 1995.
[15] Reid L. D. Etkin, B. Dynamics of Flight: Stability and Control. John Wiley & Sons,
Inc., New York, 1996.
[16] Y. K. Foo and Y. C. Soh. Strong kharitonov theorems for low-order polynomials.
IEEE Transactions on Automatic Control, 37(11):1816–1820, November 1992.
[17] S. N. Franklin and J. Ackermann. Robust flight control - a design example. AIAA
Journal of Guidance and Control, 4:597, 1981.
[18] B. K. Ghosh and C. I. Byrnes. Simultaneous stabilization and simultaneous pole-
placement by nonsitching dynamic compensation. IEEE Transactions on Automatic
Control, 28:735–741, 1983.
Bibliography 92
[19] G. Howitt and R. Luus. Simultaneous stabilization of linear single input systems
by linear state feedback control. International Journal of Control, 54(4):1015–1030,
1991.
[20] Chow J. H. Kale, M. A. and K. D. Minte. A controller parametrization and pole-
placement design for simultaneous stabilization. In Proceeding of the 1990 American
Control Conference, pp. 116-121, 1990.
[21] H. Liu and H. Berndt. Interactive design and simulation platform for flight vehicle
systems development. Journal of Aerospace Computing, Information, and Commu-
nication, 3:550–561, Nov. 2006.
[22] R. V. Patel M. Toda and B. Sridhar. Robustness of linear quadratic state feedback
designs in the presence of system uncertainty. IEEE Transactions on Automatic
Control, 22:945–949, December 1997.
[23] Sreeram V. Teo K.L. Paskota, M. and A.I. Mees. Optimal simultaneous stabilization
of linear single-input systems via linear state feedback control. International Journal
of Control, 60:483–498, 1994.
[24] R. V. Patel and M. Toda. Quantitative measures of robustness for multivariable
systems. In Joint Automat. Contr. Conf., 1980.
[25] Vijay V. Patel and Kanti B. Datta. Classification of unit in h∞ and an alterna-
tive proof of kharitonov’s theorem. IEEE Transactions on Circuits and Systems,
44(5):454–458, May 1997.
[26] R. E. Perez and H. H. T. Liu. Robustness analysis of decomposition-based simulta-
neous stabilization with optimal control approach.
[27] R. E. Perez and H. H. T. Liu. Decomposition-based simultaneous stabilization with
optimal control. Journal of Guidance, Control, and Dynamics, 31(3):647–655, 2008.
[28] Ruben E. Perez. Lecture Notes for ”Advanced Design of Engineering Systems”.
Royal Military College of Canada.
Bibliography 93
[29] Ruben E. Perez and Hugh H.T. Liu. Robustness of simultaneous stabilization in
flight control. AIAA Guidance, Navigation, and Control Conference, Honolulu,
Hawaii, Aug 2008.
[30] I. R. Petersen. A procedure for simultaneously stabilizing a collection of single-input
linear systems using nonlinear state feedback control. Automatica, 23:33–40, 1987.
[31] J. J. Anagnost R. J. Minnichelli and C. A. Desoer. An elementary proof of
kharitonov’s stability theorem with extensions. IEEE Transactions on Automatic
Control, 34(9):995–998, september 1989.
[32] R. Saeks and J. Murray. Fractional representation, algebraic geometry, and the
simultaneous stabilization problem. IEEE Transactions on Automatic Control,
27:895–903, 1982.
[33] W. E. Schmitendorf and C. V. Hollot. Simultaneous stabilization via linear state
feedback control. IEEE Transactions on Automatic Control, 34:1001–1005, 1989.
[34] M. Vidyasagar and N. Viswanadham. Algebric design techniques for reliable stabi-
lization. IEEE Transactions on Automatic Control, 27:1085–1095, 1982.
[35] Gao W. B. Wu, D. N. and M. Chen. Algorithm for simultaneous stabilization
of single-input systems via dynamic feedback. International Journal of Control,
51:631–642, 1990.
[36] R. K. Yedavalli. Improved measures of stability robustness for linear state space
models. IEEE Transactions on Automatic Control, 30:577–579, 1985.
[37] R. K. Yedavalli. A necessary and sufficient vertex solution for checking robust sta-
bility of interval parameter matrices. In AIAA Guidance, Navigation, and Control
Conference and Exhibit, number AIAA 2008-7007, Honolulu, Hawaii, Aug 2008.
[38] R. K. Yedavalli. A necessary and sufficient extreme point solution for checking robust
stability of polytopes of matrices. Proceedings of the American Control Conference,
page 1822, June 2000.
Bibliography 94
[39] Bongiorno J. J. Youla, D. C. and C. N. Lu. Single-loop feedback stabilization of
linear multivariable plants. Automatica, 10:159–173, 1974.
[40] K. Zhou and P. Khargonekar. Stability robustness bounds for linear state-space mod-
els with structured uncertainty. IEEE Transactions on Automatic Control, 32:621–
623, 1987.