Development of Multivariable Control Systems Rev 200

DEVELOPMENT OF MULTIVARIABLE

CONTROL SYSTEMS

MAUNG MAUNG LATT

SCHOOL OF ELECTRICAL AND ELECTRONIC ENGINEERING

A DISSERTATION SUBMITTED IN PARTIAL FULFILMENT OF

THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF SCIENCE IN COMPUTER CONTROL & AUTOMATION

2008

i

Acknowledgements

Firstly, I would like to express the deepest appreciation to my supervisor, Assoc. Prof.

CAI Wenjian, for giving me the opportunity to work in a very interesting area, and

his continuous guidance, encouragement, and invaluable comments during this work.

Without his help, this dissertation would not have been possible.

In addition, I would like to express my gratitude to Nanyang Technological University

(NTU) Library for the technical books and electronic resources, and the technical staff

of the Process Instrumentation Laboratory, Mr. Yock and his colleagues for their

assistance in hardware and software requirements.

Lastly, but most importantly, I am forever indebted to my parents and my friends for

their understanding, endless patience, and continuous encouragement during this

dissertation.

ii

Summary

This dissertation presents three decentralized control systems for the three distillation

systems in the literature. The loop pairing and controller design methods are:

1. RGA based Biggest Log Modulus Tuning (BLT) method

2. dRI based Skogestad Internal Model Control (SIMC) method

3. ERGA based PID control method

Firstly, we discuss the basic theory of a distillation column. In this dissertation, we

assume that we have the transfer function models for the processes to be controlled.

Then we present the usefulness of decentralized control system and, explain loop

pairing methods and the controller design methods in detail. We create programming

codes and block diagrams to simulate the decentralized control systems using

Matlab/Simulink application software. We also discuss performance analysis for each

control method using IAE, ISE, and ITAE.

Finally, the simulation results from this small work can result in the invaluable

conclusion and recommendation for the decentralized control system.

iii

Table of Contents

Acknowledgements ..................................................................................................... i

Summary .................................................................................................................... ii

Table of Contents ...................................................................................................... iii

List of Figures ............................................................................................................ v

List of Tables ........................................................................................................... vii

1 Introduction ............................................................................................................. 1

1.1 Motivation ........................................................................................................ 1

1.2 Objectives ......................................................................................................... 3

1.3 Major Contribution of the Thesis ...................................................................... 3

2 Distillation Column ................................................................................................. 4

2.1 Introduction ...................................................................................................... 4

2.2 System Identification ........................................................................................ 6

2.3 Distillation Models ........................................................................................... 8

3 Decentralized Control ............................................................................................ 10

3.1 Introduction .................................................................................................... 10

3.2 Control Structure ............................................................................................ 10

3.3 Advantages of Decentralized Control .............................................................. 11

3.4 Disadvantages of Decentralized Control .......................................................... 12

3.5 Interaction Measure and Loop Pairing ............................................................. 12

3.6 Some Important Factors in MIMO System Analysis........................................ 14

4 Controller Design Methods .................................................................................... 17

4.1 Introduction .................................................................................................... 17

4.2 BLT Tuning Method based on RGA Paring Criteria ........................................ 19

4.2.1 Interaction Measure and Loop Pairing ...................................................... 19

4.2.2 Decentralized Controller Design............................................................... 22

4.3 SIMC Tuning Method based on dRI Loop Pairing Criteria .............................. 26


4.3.2 Algorithm for Loop Pairing ...................................................................... 35

4.3.3 Dynamic Relative Interaction ................................................................... 37

4.3.4 Decentralized Controller Design............................................................... 42

4.3.5 Algorithm for Decentralized Controller Design ........................................ 50

iv

4.4 GPM Tuning Method based on ERGA Pairing Criteria ................................... 51


4.4.2 Decentralized Control System Design ...................................................... 57

5 Simulation and Performance Analysis.................................................................... 66

5.1 Introduction .................................................................................................... 66

5.2 Simulation and Discussion for Vinante and Luyben System (1972) ................. 66

5.2.1 BLT ......................................................................................................... 67

5.2.2 SIMC ....................................................................................................... 68

5.2.3 ERGA ...................................................................................................... 69

5.3 Simulation and Discussion for Ogunnaike and Ray System (1979).................. 72

5.3.1 BLT ......................................................................................................... 75

5.3.2 SIMC ....................................................................................................... 76

5.3.3 ERGA ...................................................................................................... 79

5.4 Simulation and Discussion for Alatiqi case 1 System (1985) ........................... 80

5.4.1 BLT ......................................................................................................... 84

5.4.2 SIMC ....................................................................................................... 87

5.4.3 ERGA ...................................................................................................... 89

5.5 Performance Analysis ..................................................................................... 91

6 Conclusion and Recommendation .......................................................................... 97

6.1 Conclusion ...................................................................................................... 97

6.2 Recommendation for further research ............................................................. 98

Bibliography........................................................................................................... 100

A Simulink Block Diagram for 2x2 Decentralized Control System ..................... 107

B Simulink Block Diagram for 3x3 Decentralized Control System ..................... 109

C Simulink Block Diagram for 4x4 Decentralized Control System ..................... 110

D Matlab Program for 2x2 Decentralized Control System ................................... 111

E Matlab Program for 3x3 Decentralized Control System ................................... 123

F Matlab Program for 4x4 Decentralized Control System ................................... 137

v

List of Figures

Figure 2.1 Diagram of typical industrial distillation column ........................................ 4

Figure 2.2 Photo of typical industrial distillation column ............................................ 5

Figure 3.1 Typical block diagram of decentralized control structure ......................... 11

Figure 4.1 all loops are open ..................................................................................... 28

Figure 4.2 only loop i jy u is open ........................................................................... 29

Figure 4.3 only loop i jy u is closed ........................................................................ 30

Figure 4.4 all loops are closed .................................................................................. 32

Figure 4.5 Flow chart of variable pairing selection procedure ................................... 36

Figure 4.6 Structure of loop i iy u by structural decomposition................................ 38

Figure 4.7 Closed loop system with control loop i iy u presented explicitly ............. 40

Figure 4.8 Procedure for the independent design of a decentralized PID controller ... 49

Figure 4.9 Frequency response curves and the energy of ( )ijg j (bandwidth,b ij ) .... 53

Figure 4.10 Frequency response curve and effective energy of ( )ijg j (ultimate

frequency,u ij ) ......................................................................................................... 54

Figure 4.11 Interaction mode with 1, 1ii ii ....................................................... 60


Figure 4.13 f Interaction mode with 1, 1ii ii ..................................................... 62


Figure 5.1 Closed loop step responses of y1 for VL System (1972)........................... 71

Figure 5.2 Closed loop step responses of y2 for VL System (1972)........................... 71

Figure 5.3 Step responses for g33 and g33new for OR System (1979) ...................... 74

Figure 5.4 Closed loop step responses of y1 for OR system (1979) ........................... 77



Figure 5.7 Step responses for g12 and g12new for A1 System (1985) ....................... 85

Figure 5.8 Step responses for g42 and g42new for A1 System (1985) ....................... 85

Figure 5.9 Step responses of y1 for A1 System (1985) .............................................. 91

Figure 5.10 Step responses of y2 for A1 System (1985) ............................................ 92

vi



Figure 6.1 Screenshot of SISO Design Tool .............................................................. 98

Figure 6.2 Screenshot of Decentralized Control ........................................................ 99

Figure A.1 Simulink block diagram for 2x2 decentralized control system ............... 107

Figure A.2 subsystem1 ........................................................................................... 108

Figure A.3 Subsystem2 ........................................................................................... 108

Figure B.1 Simulink block diagram for 3x3 decentralized control system ............... 109

Figure C.1 Simulink block diagram for 4x4 decentralized control system ............... 110

vii

List of Tables

Table 4.1 Four Interaction scenarios for loop i jy u ................................................ 28

Table 4.2 Typical Gain and Phase Margin Values ..................................................... 65

Table 4.3 Decentralized Controller Design ............................................................... 65

Table 5.1 The ultimate values and ZN tuning parameters .......................................... 67

Table 5.2 Tuning parameter ...................................................................................... 68

Table 5.3 Decentralized Controllers for VL System (1972) ....................................... 70


Table 5.5 Tuning parameters .................................................................................... 75

Table 5.6 Decentralized Controllers for OR System (1979) ...................................... 80


Table 5.8 Tuning parameters .................................................................................... 86

Table 5.9 Decentralized Controllers for A1 System (1985) ....................................... 90

Table 5.10 Performance Analysis for VL System (1972) .......................................... 95

Table 5.11 Performance Analysis for OR System (1979) .......................................... 95

Table 5.12 Performance Analysis for A1 System (1985) ........................................... 96

1

1 Introduction

1.1 Motivation

Distillation accounts for approximately 95% of the separation systems used by the

refining and chemical industries around the world. (Hurowitz1, et al., 2003)

Distillation has a major impact upon the product quality, energy usage, and plant

throughput of these industries. Within the USA, there are an estimated 40,000

columns, which consume 3% of the total US energy usage. (Ramchandran, et al.,

1995) Moreover, Oil price has increased since 2003. On January 2, 2008, a single

trade was made at $100 per barrel, but the price did not stay above $100 until late

February. Oil price broke through the $110 per barrel mark on March 12, 2008 and on

May 9, 2008, the oil price exceeded $125 per barrel for the first time.(Wikipedia,

2008) Therefore, improved distillation control technique can have a significant impact

on reducing energy consumption, improving product quality, and protecting

environmental resources. Distillation control is a challenging problem because of the

nonlinearity, the presence of several couplings for dual composition control, no

stationary behavior, and many disturbances. It has been considered a well-known

process from control engineering point of view.

Various control techniques for controlling distillation columns have been reported in

the literature. We can divide the control techniques from various points of view. They

are:

1. Centralized and Decentralized Control

2. Classical and Advanced Control

3. Linear and Non-linear Control

4. Feedback and Feedforward Control

The popular control techniques are listed below.

1. Biggest Log Modulus Tuning (BLT) (Luyben, 1986)(Yu, et al., 1986)(Monica,

et al., 1988)

2

2. Internal Model Control (IMC) (Economou, et al., 1986)

3. Proportional, Integral, Derivative (PID) Control

4. Model Predictive Control (MPC) (Bezzo, et al., 2005)

5. Dynamic Matrix Control (DMC)

6. Linear Quadratic Gaussian with Loop Transfer Recovery (LQG/LTR) (Zhou,

et al., 1995)

7. Linear Quadratic Gaussian (LQG)

8. Model Algorithmic Control (MAC)

9. Nonlinear Model Predictive Control (NMPC)

10. Fuzzy Logic (Stem, et al., 1995)

11. Neuro Control (Yu, et al., 1999)

12. H-infinity (H∞) (Makaremi, et al., 2006)(Gu, et al., 2005)

13. H-tow (H2)

14. Mu-Synthesis Method (µ-Synthesis Method) (Gu, et al., 2005)

15. Generalized Predictive Control (GPC)

16. Direct Nyquist Array/ Inverse Nyquist Array (DNA/INA)

17. Cascade Control

18. Statistical Process Control (SPC)

19. Dominant Pole Placement (DPP)

20. Ratio Control

There is probably no such thing of a perfect control techniques. Everything has its own

advantages and disadvantages.

Full-scale multivariable control techniques such as state space based modern control

and, more recently, model predictive control (MPC) have found increasing

applications in the process industries. Nevertheless, decentralized control remains

popular and often predominant, used either alone or as a lower level control on top of

various model based control schemes. The PID controller has been the most popular in

the industry for over fifty years. More than 90% of all control loops are PID. (Astrom,

et al., 2006) The number of publications on multi-loop PID control cited by

ScienceDirect is 72 in 2006. Decentralized control has received great interest from

industrial and academic field in literature because it has some advantages over the

3

others. It is, therefore, interesting to develop an appropriate approach to tune the

parameters of the PI and PID controllers for the control of distillation columns.

1.2 Objectives

Choosing between the control techniques depends on understanding the process and its

interactions. When designing a new process, it is important to exercise some decision-

making methods. The choice of control technique should not be decided until the

project has been qualified and reasonable objectives have been established.

Overlooking this basic principle has caused companies to waste a lot of time and

money.

To be a reliable reference for industrial control engineers and research control

engineers by using the simulation results and conclusions getting from this small work

in selecting the most suitable control technique to design a decentralized control

system for a distillation process is the main objective of this dissertation.

1.3 Major Contribution of the Thesis

This dissertation is organized into six chapters to meet the requirements of this

dissertation. An introductory chapter presents the factors what motivate to do this

dissertation and the purposes of doing this dissertation, and briefly description of each

chapter in this dissertation. Chapter 2 presents the introduction of distillation column,

the importance of system identification and different types of distillation process

models to be used in this dissertation. Chapter 3 focuses on decentralized control

including basic multivariable system analysis. The following chapter presents the

theoretical backgrounds of BLT, SIMC, and ERGA. Emphasis is placed on Chapter 5,

which deals with the simulation and performance analysis of each control method.

This is a key chapter that provides the comparison results from all of the methods by

using the high-level simulation package, Matlab/Simulink. The invaluable conclusions

and, the advantages and disadvantages of different control methods are presented in

the last chapter. A number of appendices include simulation diagrams and

programming codes to help in the reading of the main body of the dissertation.

4

2 Distillation Column

2.1 Introduction

Distillation is defined as a process in which a liquid or vapor mixture of two or more

substances is separated into its component fractions of desired purity, by the

application and removal of heat. (Tham, 2006)

Figure 2.1 Diagram of typical industrial distillation column

Distillation is based on the fact, which the vapor of a boiling mixture will be richer in

the components that have lower boiling points. Therefore, when this vapor is cooled

and condensed, the condensate will contain components that are more volatile. At the

same time, the original mixture will contain more of the less volatile material.

Distillation columns are designed to achieve this separation efficiently. Although

many people have a fair idea what distillation means, the important aspects that seem

to be missed from the manufacturing point of view are as follows:

1. Distillation is the most common separation technique.

2. It consumes enormous amounts of energy both in terms of cooling and heating

requirements.

3. It can contribute to more than 50% of plant operating costs.

5

The best way to reduce operating costs of existing units is to improve their efficiency

and operation via process optimization and control. To achieve this improvement, a

thorough understanding of distillation principles and how distillation systems are

designed is essential.

There are many types of distillation columns, each designed to perform specific types

of separations, and each design differs in terms of complexity.

Figure 2.2 Photo of typical industrial distillation column

http://upload.wikimedia.org/wikipedia/commons/c/cc/Colonne_distillazione.jpg

6

One way of classifying distillation column type is to look at how they are operated.

Thus, we have batch and continuous columns. In batch operation, the feed to the

column is introduced batch-wise. That is, the column is charged with a batch and then

the distillation process is carried out. When the desired task is achieved, a next batch

of feed is introduced. In contrast, continuous columns process a continuous feed

stream. No interruptions occur unless there is a problem with the column or

surrounding process units. They are capable of handling high throughputs and are the

more common of the two types. We shall concentrate only on this class of columns.

Continuous columns can be further classified according to

1. The nature of the feed that they are processing

1. binary column - feed contains only two components

2. multi-component column - feed contains more than two components

2. The number of product streams they have

1. multi-product column - column has more than two product streams

3. Where the extra feed exits when it is used to help with the separation

1. extractive distillation - where the extra feed appears in the bottom product

stream

2. azeotropic distillation - where the extra feed appears at the top product

stream

4. The type of column internals

1. tray column - where trays of various designs are used to hold up the liquid

to provide better contact between vapor and liquid, hence better separation

2. packed column - where instead of trays, packings are used to enhance

contact between vapor and liquid

2.2 System Identification

System identification is the field of mathematical modeling of processes from test or

experimental data. In technical terms, system identification is defined by Zadeh (1962)

as the determination based on input and output, of a system within a specified class of

systems, to which the system under test is equivalent in terms of a criterion. (Zhu,

2001)

7

For an industry control related project, it normally involves the following steps

:(Cai, 2007)

1. Benefit Analysis and Functional Design

2. Pre-testing

3. Identification Test and Model Identification

4. Controller Tuning and Simulation

5. Controller Commissioning

6 . Controller Maintenance

From above procedures, we can see that dynamic models play a central role in process

control. It is essential to know and understand a system before handling it. A system is

known through modeling and identification and understood by analysis. They are

prerequisites to the practice of automatic control.

Several approaches and techniques are available in the literature for deriving the

desired process model. Standard modeling approaches include two main streams:

1. First-principle (white-box approach)

2. Identification of the parameters (black-box approach)

The system identification procedure has the following basic steps:

1. Identification tests or experiments

2. Model order/structure selection

3. Parameter estimation

4. Model validation

Unfortunately, there is no room to describe in full detail of the system identification of

the processes to be used. This dissertation presents decentralized controller design

methods for MIMO systems and all methods are model-based. In this dissertation, we

assume that we have a MIMO transfer function model for the process to be controlled.

8

2.3 Distillation Models

Three distillation systems from the literature will be studied. All the processes vary

from two-input two-output systems up to four-input four-output systems. These three

distillation systems are:

1. Vinante and Luyben System (1972)

The VL system is a 24 tray column separating methanol and water. The controlled

variables are the temperature at tray 17 and the temperature at tray 4 . From the

transfer function matrix, it is easy to tell that the VL column system does not

exhibit open-loop column diagonal dominance.

2. Ogunnaike and Ray System (1979)

2.6 3.5

6.5 3 1.2

9.2 9.4

0.66 0.61 0.0049

6.7 1 8.64 1 9.06 1

1.11 2.36 0.01( )

3.25 1 5 1 7.09 1

34.68 46.2 0.87(11.61 1)

8.15 1 10.9 1 (3.89 1)(18.8 1)

s s s

s s s

s s s

e e e

s s s

e e eG s

s s s

e e s e

s s s s

This distillation column is a 19 plate, 12 inch diameter copper column having

variable feed and side stream draw-off locations. Temperatures are measured

on each tray as well as in the overhead, reflux, feed, and product lines.

Compositions may be determined through various on-line sensors

(densitometry, refractometry, etc.). A model relating outputs to inputs has been

determined through pulse testing of the column. In this model, the outputs are

overhead ethanol mole fraction, side stream ethanol mole fraction and tray #19

temperature, °C (corresponding to bottoms composition). The inputs are reflux

0.3

1.8 0.35

2.2 1.3

7 1 7 1( )

2.8 4.3

9.5 1 9.2 1

s s

s s

e e

s sG s

e e

s s

9

flow rate, gpm (m3/s), side stream product flow rate, gpm (m

3/s) and reboiler

stream pressure, psig (kPa).

3. Alatiqi case 1 System (1985)

Alatiqi presented the transfer function for a 4 4 multivariable complex

distillation column with side stream stripper for separating a ternary mixture

into three products. There are four controlled variables: purities of the three

product streams and a temperature difference to minimize energy consumption.

There are four manipulated variables: reflux R, heat input to the reboiler QR,

heat input to the stripper reboiler QS and flow rate of feed to the stripper LS.

The 4 4 matrix of open loop transfer functions relating controlled and

manipulated variables is shown above.

2.5 0.05 0.2 20

2 2

5 1.01 7.5 2

2 2 2

22 13

2

2.22 2.94(7.9 1) 0.017 0.64

(36 1)(25 1) (23.7 1) (31.6 1)(7 1) (29 1)

2.33 3.46 0.51 1.68

(35 1) 32 1 (32 1) (28 1)( )

1.06 3.511

(17 1) (

G

s s s s

s s s s

s s

e s e e e

s s s s s s

e e e e

s s s ss

e e

s

1.01 0.5

2

2.5 0.01 2.8 1.15

4.41 5.38

12 1) 16.2 1 17 1

5.73 4.32(25 1) 1.25 4.78

(8 1)(50 1) (50 1)(5 1) (43.6 1)(9 1) (48 1)(5 1)

s s

s s s s

e e

s s s

e s e e e

s s s s s s s s

10

3 Decentralized Control

3.1 Introduction

The historically first approach to multivariable control in industry was dividing the

sensors and actuators into n subsets and designing control loops using one of the

sensor sets and one of the actuator sets, which is usually denoted as input/output

pairing. In this way, a complex control problem is divided into n supposedly simpler

ones, with no information exchange between those controllers. This strategy is called

multi-loop, decentralized, or diagonal control.

The decentralized control is also the simplest approach to multivariable control. This

works well if the plant ( )G s is close to diagonal because the plant to be controlled is

essentially a collection of independent sub-plants, and each element in ( )G s may be

designed independently. However, if off-diagonal elements in ( )G s are large, then the

performance with decentralized control may be poor because no attempt is made to

counteract the interactions.

For decentralized controller design, we have to consider the effects of interaction on

multivariable system behavior, assuming that the process has a controllable input-

output selection. The goal for decentralized controller design is to find out how the

responses of a control system are influenced by interaction. The design of multiple

single-loop controllers for multivariable systems processes in two stages:

1. Judicious choice of loop pairing (control configuration selection)

2. Controller tuning for individual loop

3.2 Control Structure

Consider an open loop stable multivariable system with n inputs and n outputs as

shown in Figure 3.1, where ir , i = 1, 2, . . . , n , are the reference inputs; iu , i = 1,2,. . .

, n , are the manipulated variables; iy , i = 1,2,. . . , n , are the controlled variables, ( )G s

11

and ( )cG s are process transfer function matrix and decentralized controller matrix with

compatible dimensions, expressed by

Figure 3.1 Typical block diagram of decentralized control structure

11 12 1

21 22 2

1 2

( ) ( ) ( )

( ) ( ) ( )( )

( ) ( ) ... ( )

n

n

n n nn

g s g s g s

g s g s g sG s

g s g s g s

(3.1)

and

1

2

( ) 0 0

0 ( ) 0( )

0 0 ( )

c

c

c

cn

g s

g sG s

g s

(3.2)

respectively. We will use this control structure throughout this dissertation.

3.3 Advantages of Decentralized Control

Decentralized PID control has the following advantages:

1. It has flexibility and fault tolerance as independent loops can be turned on and

off, due to operator decision or faults, without excessive degradation on the

performance of the rest of loops.

2. Independent design of SISO regulators is allowed with easier online tuning.

iu

iy

ir

+

- ( )G s

( )c

G s

12

3 . The cost of implementation of a decentralized control system is

significantly lower than that of a centralized controller.

4 . It can save on modeling effort.

5 . They are often easier to understand by operators.

6 . They reduce the need for control links.

7 . Their tuning parameters have a direct effect.

8 . They tend to be less sensitive to uncertainty, for example, in the input

channels.

9 . Computation load in decentralized control is lesser than that in centralized

control.

10. It is simplicity for implementation and manual tuning.

11. The controller structure is simple and easy to handle loop failure.

12. We can use the standard equipment.

13. Decentralized controllers involve far fewer parameters, resulting in a

significant reduction in the time and cost of tuning.

3.4 Disadvantages of Decentralized Control

1. It may not work in strongly coupled systems.

2. The optimal solution is very difficult mathematically.

3. A number of pairing option is !n for a n n plant and thus increase

exponentially with the size of the plant.

4. The optimal controller is in general of infinite order and may be non-unique.

5. It has a lack of flexibility for interaction adjustment and few powerful tools for

its design compared with general multivariable control.

3.5 Interaction Measure and Loop Pairing

In this section, we will present various interaction measure and loop pairing methods

from literature because it is one of the two main steps to design decentralized

controllers for multivariable processes.

The main difference between a scalar SISO system and a MIMO system is the

presence of directions in the latter. Compared with single-input single-output (SISO)

counterparts, MIMO systems are more difficult to control due to the existence of

13

interactions between input and output variables. Tuning the controller parameters of

one loop can affect the performance of the others. For interactive plants and finite

bandwidth controllers, there is a performance loss with decentralized control because

of the interactions caused by non-zero off-diagonal elements in plant transfer function.

The interaction may also cause stability problems. To apply multi-loop control

successfully, a methodology to assess the degree of interaction between the loops is

needed.

The number of combinations is !n so some criteria should be used in deciding loop

pairing. In fact, poor control performance can result from the improper choice of

manipulated/controlled variable pairings. Loop pairing is a vitally important aspect of

multivariable processes such as distillation process. A key element in decentralized

control is therefore to select good pairings of inputs and outputs, such that the effect of

the interactions is minimized.

Several interaction measure and loop-pairing methods have been proposed in the

literature starting from Bristol to present. The popular loop pairing methods are shown

below.

1. Relative Gain Array (RGA) (Bristol, 1966)

2. Niederlinski Index (NI) (Niederlinski, 1971)

3. Relative Interaction (RI) (Bristol, 1967)

4. Dynamic Relative Interaction (dRI) (He, et al., 2006)

5. Generalized Relative Dynamic Gain (GRDG) (Gagnon, et al., 1999)

6. Decomposed Relative Gain Array (DRGA) (Avoy, et al., 2003)

7. Effective Relative Gain Array (ERGA) (Xiong, et al., 2005)

8. IMC Interaction Measure (Economou, et al., 1986)

9. Rijnsdorp’s Interaction Measure (Rijnsdorp, 1965)

10. Nyquist Array (NA) (Grosdidier, et al., 1986)

11. Generalized Interaction (GI) (He, et al., 2004)

12. Interaction Measure (Grosdidier, et al., 1986)

13. Gramian based Interaction Measure (Conley, et al., 2000)

14

3.6 Some Important Factors in MIMO System Analysis

G(s) denotes a square plant with elements ijg . We then have that the controller ( )K s

is diagonal ( { }idiag K ), and we also introduce

11

22

( ) 0 0

0 ( ) 0( ) { }

0 0 ... ( )

ii

nn

g s

g sG s diag g

g s

as the matrix consisting of the diagonal elements of G. The magnitude of the off-

diagonal elements in G (the interactions) relative to its diagonal elements are given by

the matrix

1( ) .E G G G (3.3)

It can be shown that

( ) ( )( )I GK I ET I GK (3.4)

so multi-loop behavior is equal to interaction analysis times nominal individual loops.

To achieve stability of the overall system with all loops closed, we must require that

the interactions do not cause instability. As the nominal loops will be stable, the

overall loop will be stable if ( )I ET does not have RHP zeros. If the plant is

diagonal-dominant, then E is small.

For decentralized control, it is desirable that the system can be tuned and operated one

loop at a time. Assume therefore that G is stable and each individual loop is stable by

itself. Then from the factorization, sensitivity function ( ( ))S S I ET s and the

generalized Nyquist theorem, it follows that the overall system is stable if and only if

det( ( ))I ET s does not encircle the origin as s traverses the Nyquist D-contour. We

then have that the overall system is stable if

( ( )) 1,ET j (3.5)

This sufficient condition for overall stability can be used to obtain a number of even

weaker stability conditions.

15

For the least conservative approach by using the structured singular value, the entire

system is closed-loop stable if

( ) max 1/ ( ) iT t E (3.6)

where ( )E is called the structured singular value interaction measure, and is

computed with respect to the diagonal structure ofT .

Another approach is to use Gershgorin’s theorem, we then derive the following

sufficient condition for overall stability in terms of the rows ofG :

( ) 1

( ) ,( )( )

ii

i

iijj i

g jt j i

g j

(3.7)

These conditions refer to row sums. Substituting ijg for

jig results in a different

condition based on column sums. This is a sufficient condition for closed loop

stability. If ,ijg j i are small, i.e., diagonal dominant, the above bound is a large

number, indicating that good control can be achieved at that frequency. The frequency

for which ( ) 0.5i can be considered as an orientative designed bandwidth limit for

multi-loop control.

When considering decentralized control of a plant, we should first check that the plant

is controllable with any controller.

A desirable property of the decentralized control system is that it has integrity, that is ,

the closed loop system should remain stable as subsystem controller are brought in and

out of service of when inputs saturate. The decentralized controller ( )C s can be

decomposed into ( ) ( ) /C s N s K s , where ( )N s is the transfer function matrix of the

dynamic compensator, which is diagonal and stable and does not contain integral

action, and diag , 1,2,...,iK k i n diag , 1,2,...,iK k i n . In the design of

decentralized control system, it is desirable to choose input/output-pairings such that

the system possesses the property of DCLI. It is defined that a stable plant is said to be

DCLI, if it can be stabilized by a stable decentralized controller, which contains

integral action and if it remains stable after failure occurs in one or more of the

16

feedback loops. Given an n n stable process ( )G s , the closed-loop system of

decentralized feedback structure possesses DCLI only if

( ) 0, 1,... ; 1,...m iiG m n i m (3.8)

or

0, 1,...,mNI G m n (3.9)

where mG an arbitrary m m principal submatrix of G . Either RGA or NI can be used

as a necessary condition to examine the DCLI of decentralized control systems.

17

4 Controller Design Methods

4.1 Introduction

Many design methods based on decentralized control have been reported in literature.

The popular controller design methods are listed below.

1. The biggest log modulus (BLT) tuning approach proposed by Luyben and

co-workers (Luyben, 1986)(Yu, et al., 1986)(Monica, et al., 1988)

2. IMC-PID Controller Based on dRI Analysis proposed by Mao-Jun He,

Wen-Jian Cai and Bing-Fang Wu (He, et al., 2006)

3. Effective transfer function method in terms of ERGA interaction

measurement method proposed by Qiang Xiong and Wen-Jian Cai(Q.

Xiong, 2006)

4. Analytical Design by extending the generalized IMC–PID method for

single input/single output (SISO) systems proposed by M. Lee, K. Lee, C.

Kim and J. Lee (Lee, et al., 2004)

5. Simple tuning method derived from controller synthesis method proposed

by I-Lung Chien, Hsiao-Ping Huang and Jen-Chien Yang(Chien, et al.,

1999)

6. Design by method of inequalities proposed by R. Balachandran and M.

Chidambaram (Balachandran, et al., 1997)

7. Ziegler–Nichols with detuning factor approach proposed by McAvoy

(McAvoy, 1985)

8. Sequential loop tuning approach proposed by Shen and Yu (Shen, et al.,

1994)

9. Relay based auto-tuning approach proposed by Loh et al. (Loh, et al., 1993)

10. Describing function matrix approach proposed by Loh and Vasnani (Loh,

et al., 1994)

11. Nyquist stability analysis based tuning approach proposed by Dan Chen

and Dale E. Seborg (Chen, et al., 2003)

12. Design of Multi-loop PID Controllers Based on the Generalized IMC-PID

Method with Mp Criterion proposed by Truong Nguyen Luan Vu, Jietae

Lee, and Moonyong Lee(Vu, et al., 2007)

18

13. Dominant Pole Placement method proposed by Yu Zhang, Qing-Guo Wang

and K. J. Astrom (Zhang, et al., 2005)

14. Internal Model Control (IMC) based on Dynamic Interaction Measure by

Relative Error Matrix proposed by Jietae Lee and Thomas F. Edgar (Lee, et

al., 2004)

15. SSV-IM interaction measure based Decentralized control proposed by

Pirere Grosdidier and Manfred Morari (Grosdidier, et al., 1986)

16. Multiloop IMC Design proposed by Constantin G. Economou and Manfred

Morari (Economou, et al., 1986)

17. Continuation method proposed by Jietae Lee and Thomas F. Edgar (Lee, et

al., 2006)

18. Mp Criterion Based Multiloop PID Controllers Tuning method proposed

by Dong-Yup Lee, Moonyong Lee, Yongho Lee and Sunwon Park (Lee, et

al., 2003)

19. Iterative continuous cycling (ICC) method proposed by Jietae Lee, Wonhui

Cho and Thomas F. Edgar (Lee, et al., 1998)

20. DynPLS (Dynamic Partial Least Squares) model-based multiloop PID

controller design proposed by Junghui Chen, Yi-Chun Cheng and Yuezhi

Yea (Chen, et al., 2005)

21. Multi-loop version of the modified Ziegler-Nichols method proposed by

Qing-Guo Wang, Tong-Heng Lee and Yu Zhang (Wang, et al., 1998)

22. Independent Design method proposed by Sigurd Skogestad and Manfred

Morari (Skogestad, et al., 1989)

23. Multiloop PID Controller Tuning approach based on Gain and Phase

Margin Specifications (Ho, et al., 1997)

24. Direct method based on Effective Open-loop Process (EOP) proposed by

Hsiao-Ping Huang, Jyh-Cheng Jeng, Chih-Hung Chiang and Wen Pan

(Huang, et al., 2003)

25. Singular Value Decomposition (SVD) based method proposed by

Goshaidas Ray, A.N. Prasad and T.K. Bhattacharyya (Raya, et al., 1999)

We select the first three of the controller design methods mentioned above for this

dissertation. We select BLT tuning method because it is the first approach to

19

decentralized or multi-loop multivariable control. It is also a benchmark method for

decentralized control. ERGA is interesting because it considers four combination

modes of gain and phase changes for a particular loop when other loops are closed.

This equivalent transfer function can effectively approximate the dynamic interactions

among loops. Consequently, the decentralized controllers can be independently

designed by employing the single loop tuning techniques. The method is simple,

straightforward, easy to understand and implemented by field engineers. dRI analysis

is also reasonable and it combines with the famous SIMC-PID tuning method.

(Luyben, 1986) (Q. Xiong, 2006) (He, et al., 2006)

4.2 BLT Tuning Method based on RGA Paring Criteria

A common way is to first design individual controller for each control loop by

ignoring all interactions, and then detune each loop by a detuning factor. Luyben

proposed the Biggest Log-modulus Tuning (BLT) method for multiloop PI/PID

controllers. In the BLT method, the well-known Ziegler-Nichols rule is modified with

the inclusion of the detuning factors, which determines the tradeoff between stability

and performance of the system.


shown in Figure 3.1. The transfer functions and the controllers are typically in the

form of

1

2 3 4

( 1)

( 1)( 1)( 1)

ijD s

ij ij

ij

ij ij ij

K s eg

s s s

(4.1)

and

( ) Ici P D

Kg s K K s

s (4.2)

respectively.

4.2.1 Interaction Measure and Loop Pairing

In this section, we will use RGA, NI, and MIC for interaction measure and loop

pairing. We need to mention the basic theory of these methods.

20

The relative gain array (RGA) was introduced by (Bristol, 1966). Many of its

properties were stated by Bristol, but they were not proven rigorously until the work

by (Grosdidier, et al., 1985). Some additional properties are given in (Hovd, et al.,

1992). The Relative Gain Array of a complex non-singular n n matrix G denoted by

RGA (G) or ( )G is a complex n n matrix defined by

1( ) ( )TG G G (4.3)

where is the hadamard product and 1( )TG

is the transpose of the inverse ofG .

With Matlab, we write

RGA=G.*inv(G’);

The RGA has a number of interesting algebraic properties, of which the most

important are:

1. It is independent of input and output scaling.

2. Its rows and columns sum to one.

3. The sum-norm of the RGA, sum

, is very close to the minimized condition

number. This means that plants with large RGA-elements are always ill-

conditioned but the reverse may not hold.

4. A relative change in an element of G equal to the negative inverse of its

corresponding RGA-element yields singularity.

5. The RGA is the identity matrix if G is upper or lower triangular. From this

property, it follows that the RGA provides a measure of two-way interaction.

The immediate result on pairing selection is the Niederlinski criterion. (Niederlinski,

1971) Given a stable multivariable system n nG and a diagonal controller with integral

action /K s , the closed loop will be unstable if (0) (0)L G K verifies the sign

condition:

1

sign(det( )) sign( )n

iiiL L

(4.4)

The Niederlinski criterion can exclude many candidate pairings, assuming that the sign

of the integral gain in iik is the same as that of the DC gain of iig as expected in a

21

decentralized design. In this case, iiL has positive gain so the criterion reduces to

det( (0)) 0L as a sufficient condition for instability. On the other hand, under closed-

loop conditions in all n loops, the system will be unstable for all possible values of

controller parameters if the Niederlinski Index (NI) defined below is negative, i.e.:

1

(0)

(0)n

ii

i

GNI

g

(4.5)

With Matlab, we write

n=length(G);

K=dcgain(G);

NI=det(K);

for i=1:n

NI=NI/K(i,i);

end

Manipulated and controlled variables in a decentralized control system should be

paired in the following way:

1. The paired RGA elements are closest to 1.0.

2. NI is positive.

3. All paired RGA elements are positive.

4. Large RGA elements should be avoided.

In this criterion, both RGA and NI offer important insights into the issue of control

structure selection. RGA is used to measure interactions, while NI is used as a

sufficient condition to screen out the closed-loop unstable pairings. The main

advantage of RGA based method is that the interpretation of interaction depends on

only the steady state gains. This information is easily obtained from simple

identification experiments or steady state design models. The weakness of the method,

however, is the same fact that it only uses the steady state gains without considering

the phase conditions when determining loop pairing.

22

Algorithm for loop pairing

1. Eliminate pairings with negative RGAs. Pairing on negative RGA’s is

undesirable because it produces either an unstable system or a system that

lacks integrity.

2. Eliminate pairings with negative Niederlinski Indexes. The Niederlinski Index

(NI) is defined as shown in Equation (4.5). If all the SISO controllers contain

integral action and have positive loop gains, a negative value of the

Niederlinski Index shows that the system will be closed-loop unstable with this

variable pairing for any controller tuning.(Niederlinski, 1971)

All the remaining pairing possibilities must be examined. For each case, controllers

are tuned by using the BLT procedure. Then the closed-loop load-rejection

performances are compared in order to find the pairing, which does the best job of

keeping controlled variables constant in the face of load disturbances.

4.2.2 Decentralized Controller Design

BLT-1: The first step in BLT-1 tuning method is to calculate the Ziegler-Nichols

setting for each individual loop. The ultimate gain uK and ultimate frequency u of

each diagonal transfer function are calculated in the classical SISO way. To do this

numerically, a value of the frequency is guessed. The phase angle is calculated, and

frequency is varied until the phase angle is equal to 180 . This frequency is u . The

ultimate gain is the reciprocal of the real part of ( )iig s at u . There are several ways to

find the ultimate data settings, including the method of sustained oscillations, the relay

experiment, or even straightforward Nyquist plotting using software like MATLAB.

Next, a factor F is assumed. Typical values vary from 2 to 5. The gains of all

feedback controllers CK are calculated by dividing the Ziegler-Nichols gain iZNK by

the factor ,F

iZN

Ci

KK

F (4.6)

23

where,

2.2

i

i

u

ZN

KK (4.7)

The reset times ( I ) of all controllers are calculated by multiplying the Ziegler-

Nichols reset times ( ZN ) by the same factor ,F

iIi ZNF (4.8)

where,

2

1.2iZN

ui

(4.9)

The F factor can be considered a detuning factor, which is applied to all loops. The

larger the value of F, the more stable the system will be but the more sluggish will be

the set point and load responses. The method outlined below yields a value of F ,

which gives a reasonable compromise between stability and performance in

multivariable systems.

The closed-loop system can be expressed by

( )cy Gu GG r y (4.10)

Solving for y gives,

1

c cy I GG GG r

(4.11)

Since the inverse of a matrix has the determinant of the matrix in the denominator, the

closed-loop characteristic equation of the multivariable system is the scalar equation

det( ) 0cI GG (4.12)

The number of right half plane zeros of the closed-loop characteristic equation can be

found by plotting the left side of Equation (4.12) as a function of frequency and

looking at the encirclements of the origin. In order to make this multivariable plot look

just like the SISO scalar Nyquist plot, we subtract one from Equation (4.12) and look

at the encirclements of the (-1,0) point. It is convenient to define a new function

( ) 1 det( )( )cW s I GG s (4.13)

24

W is plotted as a function of frequency. The closer W approaches the (-1,0) point,

the closer the multivariable system is to closed-loop instability. The quantity

( )

1 ( )

W s

W s (4.14)

will be similar to the closed-loop servo-transfer function for a SISO loop

( ) ( )

1 ( ) ( )

c

c

g s g s

g s g s (4.15)

Therefore, based on intuition and empirical grounds, a multivariable closed-loop log

modulus cmL are defined as

20 log1

cm

WL

W

(4.16)

The proposed tuning method is based on varying the factor F until the biggest log

modulus max( )cmL is equal to some reasonable number. Thus, the proposed tuning

method is called the biggest log modulus tuning (BLT).

Now we must determine what is a reasonable value to use for max( )cmL for different

orders of the systems. For a SISO system where N = 1, we know from long experience

that a value of 2 dB for max( )cmL gives reasonable time domain responses for set

point and load disturbances. The best tuning criterion for this method is

max( ) 2cmL N (4.17)

For 1N , this reduces to the classical +2dB criterion. For a 4 4 system, a max( )cmL

of 8 should be used. These empirical findings suggest that the higher the order of the

system, the more under damped the closed-loop system must be to achieve reasonable

responses.

This tuning method should be viewed as giving preliminary controller settings, which

can be used as a benchmark for comparative studies. Note that this procedure

guarantees that the system is stable with all controllers on automatic and that each

individual loop will be stable if all others are on manual. Thus, a portion of the

integrity question is automatically answered.

25

The method weighs each loop equally; i.e., each loop is detuned by the same factor F.

If it is more important to keep tighter control of some variable than the others, the

method can be easily modified by using different weighting factors for different

controlled variables. The less-important loop could be detuned more than the more-

important loop.

We will summarize BLT-1 tuning procedures mentioned above as follow.

1. Compute the Ziegler-Nichols tuning parameters of the diagonal elements of the

process transfer function matrix ( )iig s as though the diagonal elements

represented SISO systems.

2. Choose a detuning factor F . F should be greater than one.

3. Compute CiK , and Ii for each loop by

iZN

Ci

KK

F (4.18)

iIi ZNF (4.19)

where iZNK and

iZN are the Ziegler-Nichols PI values.

4. Calculate the function W over the appropriate frequency range (i.e., near the

( 1,0) point), where

1 det ( ) ( )cW I G j G j (4.20)

5. Compute the function ( )cL j where

( ) 20 log1

c

WL j

W

(4.21)

6. Adjust F until the peak in the cL log modulus curve cmL is equal to 2N ,i.e.,

max 2cL N , where cmL is the biggest log modulus in the cL curve and N is the

number of SISO loops in the multivariable system.

It is important when applying this procedure to confirm the form of the Nyquist plot of

the function W , because the log modulus function merely denotes nearness to the

26

critical point. An unstable system could be tuned to a maximum log modulus of 2N

because it encircles the ( 1,0) point but is not near it.

BLT-2: After tuning by BLT-1, derivative action can be incorporated through the

steps that follow. We will consider derivative action for SOPTD processes in this

dissertation.

1. Choose a second detuning factor DF . It should be greater than one.

2. Compute Di , where

Di ZNDi

DF

(4.22)

and Di ZN is the Ziegler-Nichols value for Di and is equal to

0.125Di ZN uP .- (4.23)

3. Repeat steps 4 and 5 of the BLT-1 procedure. Change DF until max

cL

is

minimized, maintaining 1.DF The trivial case may result where max

cL is

minimized for DF , i.e., no derivative action.

4. Reduce DF , and repeat steps 3, 4, and 5 of the BLT-1 procedure until

max 2 .cL N

5. Repeat steps 4 and 5 above until no further reduction in F is possible. By this

process, we are shifting the phase angle of the closed-loop characteristic

equation Nyquist contour by tuning the derivative terms so that they have the

greatest effect near the closed-loop resonant frequency.

4.3 SIMC Tuning Method based on dRI Loop Pairing Criteria

In this section, a simple yet effective decentralized PID controller design methodology

will be presented based on dynamic interaction analysis and Skogestad Internal Model

Control (SIMC) principle. Based on structure decomposition, the dynamic relative

interaction (dRI) is defined and represented by the process model and controller

explicitly. An initial decentralized controller is designed first by using the diagonal

elements and then implemented to estimate the dynamic relative interaction to

27

individual control loop from all others. With the obtained dRI, the multiplicate model

factor (MMF) is derived, and then simplified to a pure time delay function at the

neighborhood of each control loop critical frequency to obtain the equivalent transfer

function for the particular control loop. Consequently, appropriate controller

parameters for individual control loop are determined by applying the SIMC-PID

tuning rules for the equivalent transfer function.


shown in Figure 3.1. ijG denotes the matrix G with its thi row and thj column

removed. , ,r u and y are vectors of references and manipulated and controlled

variables, while , ,i ir u and iy are the ith references and manipulated and controlled

variables, and , ,i ir u and iy denote the reference, input, and output vectors with

variables , ,i ir u and iy removed. The process transfer functions and the controllers1

are typically of the form as

'

( ) , ' .( 1)( 1)

ijs

ij

ij ij ij

ij ij

k eg s

s

(4.24)

and in parallel form as

1

( ) (1 )i Pi Di

Ii

C s k ss

(4.25)

or in series form as

1

( ) (1 )( 1).i Pi Di

Ii

C s k ss

(4.26)

respectively. And to investigate the interactions between an arbitrary loop i jy u and

all other loops of the multivariable system, the process from u to y can be explicitly

expressed by

ij j

i ij j i

i ij ij j

j j

y g u g u

y g u G u

(4.27)

1 ( ) ( ).cC s G s

28

where ij

ig and ij

jg denote the ith row vector and the thj column vector of matrix G

with element ijg removed.


For an arbitrary loop i jy u

and the remaining subsystem block, there are four

interaction scenarios corresponding to the combination of their open and closed states,

which are listed in Table 4.1 and indicated in Figure 4.1 to Figure 4.4.

Case loop i jy u All other loops Figure

1 open loop open loop Figure 4.1

2 open loop closed loop Figure 4.2

3 closed loop open loop Figure 4.3

4 closed loop closed loop Figure 4.3

Table 4.1 Four Interaction scenarios for loop i jy u

Case 1: All loops are open, as shown in Figure 4.1. Obviously, no interaction exists

among loops; thus, the process gain of an arbitrary pairing i jy u is the element

ijg in

transfer function matrixG .

Figure 4.1 all loops are open

29

Case 2: Loop i jy u

is open and all other loops are closed, as shown in Figure 4.2.

This is the very condition for deriving the RGA, and the direction of interactions is

from the subsystem block of closed loops to loopi jy u . RI can be obtained by matrix

operation as

11( )ij ij ij

ij i j

ij

g G gg

(4.28)

Figure 4.2 only loop i jy u is open

Case 3: Loop i jy u is closed and all other loops are open, as shown in Figure 4.3. In

this case, the interaction is from closed loop i jy u to all other open loops, and the RI

can be derived from the 2 2 subsystem that includes loop i jy u and an arbitrary

loop k ly u of subsystem ( and )ijG k i l j . This describes the 2 2 subsystem as

.ij ili j

kj klk l

g gy u

g gy u

Hence, we obtain the RI of loop k ly u from loop i jy u as

,CLi j

ijkl y u klij kl

kl

kl kl

g g g

g g

(4.29)

where

30

il kjij

kl

ij

g gg

g (4.30)

is the incremental process gain of loop k ly u

when loop i jy u is closed. By

extending Equation (4.29) and Equation (4.30) to all loops of the subsystem ijG , we

have

,CL

( )i j

ij ij ij ij ij ij

y uG G G G G (4.31)

where indicates element-by-element division, and

1ij ij ij

j i

ij

G g gg

(4.32)

is the incremental process gain matrix of subsystem ijG when loop i jy u is closed.

Thus, ij is called the RIA to the open subsystem ijG when loop

i jy u is closed.

Figure 4.3 only loop i jy u is closed

Case 4: All loops are closed, as shown in Figure 4.4. In this case, the interaction

between the closed loop i jy u and all other closed loops is bidirectional. Because the

interaction to loop i jy u from all other closed loops is the same as what has been

studied in case 2, the interactions in this direction are not considered here. We now

focus on the interaction from the isolated loop i jy u to all other closed loops.

Suppose all loops of subsystem ijG are closed and the following:

31

1. If loop i jy u is open, then the gain of an arbitrary loop k ly u is affected by

the RGA value of ijG ; therefore, according to the definition of RGA, it is

ˆ klkl ij

kl

gg

(4.33)

where arc indicates the changed process gain in the closed subsystem ijG .

When Equation (4.33) is extended to all loops of subsystem ijG , the closed-

loop transfer function matrix is obtained as

ˆ ij ij ijG G (4.34)

2. If loop i jy u is closed, then the subsystem that includes loops

i jy u and

k ly u in the closed subsystem ˆ ijG is given as

.ˆ

ij ili j

kj klk l

g gy u

g gy u

Thus, the RI from loop i jy u to loop k ly u is obtained as

,CL

ˆ ˆ

ˆ ˆ

i j

ijkl y u klij kl

kl

kl kl

g g g

g g

(4.35)

where ij

klg is the increment of closed-loop process gain ˆklg after loop

i jy u is

closed and it is the same as that shown in Equation (4.32). Extending Equation (4.35)

to all loops of the subsystem ˆ ijG , we obtain

,CL

ˆ ˆ ˆ ˆ( )i j

ij ij ij ij ij ij

y uG G G G G (4.36)

where ijG is the incremental matrix of subsystem ˆ ijG after loop i jy u is closed and

it is the same as that shown in Equation (4.32). Thus, ij is called the RIA to the

closed subsystem ˆ ijG , which reflects the interactions to ˆ ijG when loop i jy u is

closed.

As an arbitrary element ofG , ijg may be zero. In such a case, the values of ij

kl and

ij

klg in Equation (4.29) and (4.30) are indefinite. Fortunately, those variables are only

used during the course of derivation (where 0 may be used to replace the zero

elements).

32

For practical calculation, ij of the thij element ijg can be directly obtained by

substituting Equation (4.3) with (4.34) into (4.36) to result in

( )ij ij ij TG G (4.37)

To explore the inherent relationship between RGA and RI and on the basis of the

definition of the RGA, rewrite Equation (4.35) as

ˆ ˆ

ij ijij ij ijkl kl klkl kl klij

kl kl kl

g g g

g g g

(4.38)

Figure 4.4 all loops are closed

Extending Equation (4.38) to all loops of subsystem ˆ ijG , we obtain another important

matrix form for ij

ij ij ij (4.39)

Furthermore, the relationship between ij and

ij for an arbitrary nonzero element of

system G is given by the following theorem.

For an arbitrary nonzero element ijg ofG , the corresponding ij is the sum of all

elements inij , i.e.

1, 1,

n nij ij

ij kl

k k i l l j

(4.40)

33

where

is the summation of all matrix elements. Using Equation (4.28), Equation

(4.31) and Equation (4.32), we have

Then from Equation (4.37), we obtain the result of Equation (4.40).

Because ij reveals the information on loop-by-loop interaction, we define it as DRIA.

We can summarize as follows:

1. The pairing structure with a small value of ij

kl , i.e., 0ij

kl , should be

preferred.

2. A large value of ij

kl implies that the interaction from the closed loop i jy u an

arbitrary loop k ly u in the closed subsystem ˆ ijG is large. Therefore, the

corresponding pairing structure should be avoided.

3. Since the RI from loop i jy u to loop k ly u in the closed subsystem ˆ ijG is

the product of ij

kl and ij

kl , and the ideal situation is 1ij

kl and 0ij

kl , ij

kl

should be 0ij

kl for minimal interactions.

4. Compared with Case 3, the RI from loop i jy u to an arbitrary loop k ly u of

the subsystem ijG in Case 4 is changed by a factor of ij

kl , whereas this RGA

is determined by the other loops of subsystem ijG ; thus, the element ij

kl of

DRIA reflects more information on the interaction effect to i jy u from all of

the other loops working together. Apparently, the best pairing structure should

be 1ij

kl , which is consistent with the conventional pairing rule.

11( )

1( )

( ) ( )

ij ij ij

ij i j

ij

ij ij ij T

j i

ij

ij ij T

g G gg

g g Gg

G G

34

5. ij given in a matrix form provides loop-by-loop information of the RIs

between i jy u and all other loops as well as their distributions; therefore, it is

more precise than ij in measuring the loop interactions.

The analysis above suggests that the RGA and RI may not be able to reflect the

interactions among loops accurately, while through DRIA analysis, loop interactions

in the matrix form can be revealed categorically.

Based on the DRIA, we can now define loop-by-loop interaction energy. The GI ij is

the interaction energy (2-norm) of matrix ij , and correspondingly

ij is the thij

element of the GI array (GIA)

2, , 1,2,...,ij

ij ij i j (4.41)

where 2

denotes the 2-norm of the matrix defined as

max2( ).ij ij

Equation (4.41) represents the interaction energy to all closed loops of the subsystem

ijG from loopi jy u . Moreover, it also reflects the interaction energy to loop

i jy u

from all remaining 1n closed loops in G . Therefore, GI reflects the intensity of the

interactions among all loops. In analogy to RGA and NI, we here provide some

important properties of the GI:

1. The GI only depends on the steady-state gain of the multivariable system.

2. The GI is not affected by any permutation ofG .

3. The GI is scaling-independent.

4. If the transfer function matrix G is diagonal or triangular, the corresponding

GI is equal to zero.

Manipulated and controlled variables in a decentralized control system should be

paired in such a way that

1. All paired RGA elements are positive.

35

2. NI is positive.

3. The pairings have the smallest ij value.

For two or more pairing structures that have passed all three variable pairing steps but

with similar GI values, the pairing structure that has th9e smallest product of all GIs

are preferred because the interactions are transferable through interactive loops, i.e.,

selecting

1

minn

i

i

(4.42)

where i is the GI corresponding to the ith output.

4.3.2 Algorithm for Loop Pairing

Based on this loop-pairing criterion, an algorithm to select the best control structure

can be summarized as follows:

1. For a given transfer function ( )G s , obtain steady-state gain matrix (0)G .

2. Calculate RGA and NI.

3. Eliminate pairs having negative RGA and NI values.

4. Calculate ijG andij .

5. Calculate ij and form .

6. Select the pairing that has the smallest value of ij in .

7. Select the best one if two or more pairings have similar GI values.

8. End

The procedure for the variable-pairing selection is demonstrated by the flowchart as

shown in Figure 4.5.

For the system that has the pure integral element, GIA and DRIA can be calculated by

using a method similar to that proposed by Arkun and Downs (Arkun, et al., 1990).

For 2×2 system, we obtain the GI value of element 11g as

22 11

11 11 22 11 22 .

36

Figure 4.5 Flow chart of variable pairing selection procedure

37

Furthermore, from properties of RGA and RI, we can obtain an equation as

11 12 11 121 1

which indicates 11 and 12 must have the same sign, and a smaller 11 means less

interaction. Therefore, selecting the pairings that have the smallest GIA elements is

equivalent to the RGA-based loop-pairing criterion to select the RGA value closest to

unity. Even though the procedure to derive the new loop-pairing criterion is tedious,

the calculation for the control structure configuration can be achieved automatically

and can easily be programmed into a computer by using Matlab/Simulink application

software.

4.3.3 Dynamic Relative Interaction

To examine the transmittance of interactions between an individual control loop and

the others, the decentralized control system can be structurally decomposed into n

individual SISO control loops with the coupling among all loops explicitly exposed

and embedded in each loop.

Figure 4.6 shows the structure of an arbitrary control loop i iy u after the structural

decomposition. In

Figure 4.6, the interaction to an individual control loop i iy u from the other 1n

control loops is represented by the RI, 1( )ii n s

, and the equivalent transfer function of

an individual control loop i iy u denoted by ˆ ( )iig s , can be obtained in terms of

, 1( )ii n s by

, 1

ˆ ( ) ( ) ( )ii ii ii ng s g s s (4.43)

with

, 1 , 1( ) 1 ( )ii n ii ns s (4.44)

where , 1( )ii n s is defined as multiplicate model factor (MMF) to indicate the model

change of an individual control loop i iy u after the other 1n control loops are

closed. Once the RI , 1( )ii n s is available, the corresponding MMF , 1( )ii n s and the

equivalent process transfer function ˆ ( )iig s can be obtained, which can be directly used

to design independently the PID controller for each individual control loop i iy u .

38

We use RI as a basic interaction measure to investigate the interactions among control

loops and derive equivalent transfer function for each loop. Sometimes, we will omit

the Laplace operator s for simplicity unless otherwise

Figure 4.6 Structure of loop i iy u by structural decomposition

specified. The RI (Bristol, 1967) for control loop i jy u is defined as the ratio of two

elements: the increment of the process gain after all other control loops are closed and

the apparent gain in the same loop when all other control loops are open, that is

, 1

( / ) ( / ) 11

( / )

i j all other loops close except for loop yi uj i j all loops open

ij n

i i all loops open ij

y u y u

u y

Since the RI cannot offer effective measure on the reverse effect of individual control

loop and loop-by-loop interactions, He and Cai decomposed the RI as the elements

summation of decomposed relative interaction array (DRIA) to give important insights

into the cause-effects of loop interactions (He, et al., 2004). However, the obtained

results are limited to the steady state, which are less useful for controller design than

the dynamic representations. Hence, it is necessary to derive the dynamic interaction

among control loops represented explicitly by the process models and controllers.

Because the dynamic interactions among control loops are controller dependence,

appropriate controllers have to be designed and implemented into the control system

for investigation of the dynamic interactions. For an arbitrarily decentralized PID

control system of a multivariable process, we can redraw Figure 3.1 as Figure 4.7 for

the convenience of analyzing the interactions between an arbitrary control loop i iy u

and the others, where iy is a vector indicating the effects of iu on other outputs while

39

iy indicates the reverse effect of iy by all of the other closed control loops. , ,i i ir u y

and ( )iC s indicate , ,r u y and ( )C s with their thi elements, , ,i i ir u y and ( )ic s ,

removed, respectively.

Because the dRIs are input independent, without loss of generality, the references of

the other 1n control loops are set as constants, i.e.

0 or ( ) 0 1,..., ; ,kk

drr s k n k i

dt

in the analysis of the dynamic interaction between control loop i iy u and the other

controlled closed-loops. Then, we have

i ii i

iy G u y (4.45)

i i iu C y (4.46)

and

ii

i iy g ui (4.47)

ii i

i iy g u (4.48)

where iiG is the transfer function matrix G with its thi row and the thi column

removed, and *

ii

ig and *

ii

ig indicate the thi row and the thi column of G with the thii

element iig removed, respectively. Combining Equation (4.45) and Equation (4.46),

we can write

1( )i ii

iu G y (4.49)

where

1.G G C (4.50)

Furthermore, iy in Equation (4.48) can be represented by the summation of the

following row

1 1 , 1, , ; .i i ik k in ny g u g u g u k n k i

(4.51)

40

Figure 4.7 Closed loop system with control loop i iy u presented explicitly

where A

is the summation of all elements in a matrix .A Using Equation (4.47) –

Equation (4.51), we obtain

* *

* *

* *

( )

( )

( ) .

ii ii ii

i i i i

ii ii ii T

i i i

ii iiii Ti i

ii i

ii

y g G g u

g g G u

g gG g u

g

Consequently, those steady state relationships provided in [42] can be extended as

follows. Define

, 1

1 ii ii

ii n i i

ii

G g gg

as the incremental process gain matrix of subsystem iiG when control loop i iy u is

closed, then the dynamic DRIA (dDRIA) of individual control loop i iy u in an n n

system can be described as

, 1 , 1 ( )ii T

ii n ii nG G

(4.52)

and the dRI , 1ii n , is the summation of all elements of , 1ii n , i.e.

, 1 , 1 , 1 ( )ii T

ii n ii n ii nG G

(4.53)

where the Hadamard product and ( )ii TG is the transpose of the inverse of matrix

41

.iiG From Equation (4.50), G can be factorized as

11 1 12 1

21 22 2 2

1 2

1/

1/

1/

n

n

n n nn n

g c g g

g g c gG

g g g c

G P

where

11 11

11 1

22 22

22 2

11 1

11 1

11 1 nn nn

nn n

g c

g c

g c

g cP

g c

g c

G P

(4.54)

Hence, the dDRIA and dRI can be obtained respectively by,

, 1 , 1 ( )ii ii T

ii n ii nG G P

(4.55)

and

, 1 , 1 ( )ii ii T

ii n ii nG G P

(4.56)

In dDRIA and dRI of Equation (4.55) and Equation (4.56), there exists an additional

matrix P , which explicitly reveals interactions to an arbitrary loop by all the other

1n closed loops under band-limited control conditions. Thus, for the given

decentralized PID controllers, the dynamic interaction among control loops at an

arbitrary frequency can be easily investigated through the matrix P , moreover, the

calculation remains simple even for high dimensional processes.

For decentralized PI or PID control, we have

1( 0) 0C j

and

( 0) ( 0)G j G j

which implies that the RI and the dRI are equivalent at steady state. However, because

42

the dRI measure interactions under practical unperfected control conditions at some

specified frequency points, it is more accurate in estimating the dynamic loop

interactions and more effective in designing decentralized controllers.

The significance of above development is as follows:

1. The interaction to an individual control loop from the other loops is derived in

matrix form, and the relationship between RI and DRIA is extended to the

whole frequency domain from the steady state;

2. Equation (4.52) and Equation (4.53) indicate that the dRI is a combination of

the interactions to individual control loops from the others; there may exist

cause-effect cancelation among them such that selecting of loop pairings by

dRI may be inaccurate (He, et al., 2004).

3. Because interactions among control loops are controllers dependent, the

dDRIA represents how the decentralized controllers interact with each other,

while the dRI provides the overall effect to an individual control loop from the

others.

4. The dRI can be obtained easily once process transfer function elements and

controllers are available, which is not limited by the system dimension.

4.3.4 Decentralized Controller Design

This method involves three main steps in designing decentralized controllers for

multivariable processes:

1. the design of individual controllers by ignoring the loop interactions;

2. the determination of the equivalent transfer function for each individual loop

through use of the dRI;

3. the fine-tuning of each controller parameter based on the equivalent

transfer function.

Because the interactions among control loops are ignored in the first step, all

available SISO PID controller design techniques can be applied to the design of

the initial individual controllers for the diagonal elements. Thus, according to the

43

expected control performance, one can select the most suitable tuning rules, such

as Ziegler and Nichols tuning, IMC tuning, and some other optimal design

methods. O bviously, different tuning rules provide different controller settings

and, correspondingly, different estimations of loop interactions. Even these

different estimations will affect the decentralized controllers designed in the third

step; they have no influence on the design mechanism of this method.

The IMC design procedure is brief studied as follows. The process model g is

factorized into an all-pass portion g and minimum phase portion g , that is

.g g g

The all-pass portion g includes all the open right half plane zeros and delays of

g and has the form

( 1), Re{ } 0,s

i

i

g e s

where 0 is the time delay and 1

i

is the right-half-plane zero in the process

model.

Then the IMC controller and the complementary sensitivity function are derived

respectively as

1 ,cg g f

and

,T g f

where f is the IMC filter and has the form

1,

( 1)r

C

fs

where the filter order r is selected large enough to make cg proper, and the

adjustable filter parameter C provides the tradeoff between performance and

robustness. The key advantage of the IMC design procedure is that all controller

parameters are related in a unique, straightforward manner to the model

parameters. There is only one adjustable parameter C which has intuitive appeal

44

because it determines the speed of response of the system. Furthermore, C is

approximately proportional to the closed-loop bandwidth, which must always be

smaller than the bandwidth over which the process model is valid.

Even though more precise higher order process models can be obtained by either

physical model construction (following the mass and energy balance principles)

or the classical parameter identification methods, from a p ractical point of view,

the lower order process model is more convenient for controller design.

For the SO PDT SISO system given by Equation(4.24), the SIMC-PID settings in

the form of Equation (4.26) was suggested as

1 ii

Pi

ii Ci ii

kk

(4.57)

min{ ,4( )}Ii ii Ci ii (4.58)

'

Di ii (4.59)

where, Ci is the desired closed-loop time constant. It is recommended to select

the value of Ci as ii for a tradeoff between:

1. the fast speed of the response and good disturbance rejection (favored by a

small value of Ci );

2. the stability, robustness, and small input variation (favored by a large value of

Ci );

Step 1: Let diagG G . The initial controllers C can be designed for G through the

SIMC-PID tuning rules, such that

11 1

11 1

0 0

0 0

0 0 nn n

g c

g cGC

g c

45

with the open loop transfer function for each individual control loop is

1

.( )

iis

ii i

Ci ii

g c es

(4.60)

Using first order Taylor series expansion for (4.60), the closed-loop transfer

function, open-loop gain crossover frequency ci and matrix P , can be obtained

respectively by

1

1iisi

i Ci

ye

r s

1

,ci

Ci ii

(4.61)

and

11

22

1

2

( 1) 1 1

1 ( 1) 1

1 1 ( 1) nn

s

C

s

C

s

Cn

s e

s eP

s e

(4.62)

Because each controller is designed around the critical frequency of its transfer

function, the dRI of individual control loop i iy u can be estimated at the critical

frequency crij

, 1 , 1( ) ( ) ( ( ) ( )) .ii ii T

ii n cri ii n cri cri crij G j G j P j

(4.63)

For interactive multivariable processG , it is desirable to have the same open-loop

transfer function as multivariable control system GC . Because controller C is

designed for the diagonal elements of G without considering the couplings

between control loops, the designed controller should be detuned by

, 1 , 11

i ii

ii n ii n

c cc

(4.64)

to result in approximately the same closed-loop control performance. However,

it is impractical to use , 1ii n directly to fine-tune the controller because it has

different values at different frequencies. To solve this problem, one possible way

is to use some appropriate transfer functions to identify those MMFs. Because at

46

the neighborhood of the critical point the transmission interaction can be

considered as a linear function, it is reasonable to represent the MMF by a low-

order transfer function involving the first two items of its Taylor series, which can

be further simplified for controller design by a pure time-delay transfer function

as

, 1

, 1 , 1 ,i n s

ii n i nk e

(4.65)

where,

, 1 , 1 , 1( ) 1 ( )i n ii n cri ii n crik j j (4.66)

and

, 1 , 1

, 1

arg( ( )) arg(1 ( )),

ii n cri ii n cri

i n

cri cri

j j

(4.67)

with cri indicating the critical frequency of the thi control loop i iy u .

O bviously, the positive/ negative , 1i n

indicates that the interactions from other

control loops bring a phase lag/ lead of individual control loops.

Step 2: From Equation (4.24) and Equation (4.26), the equivalent transfer function

for loop i iy u can be expressed by

'

ˆ ( ) ,( 1)( 1)

i iif ski iiii

ii ii

f kg s e

s s

(4.68)

where,

, 1max 1, ,ki i nf k (4.69)

and

, 1

max 1,1 .i n

i

ii

f

(4.70)

The open-loop transfer function for each individual control loop given by

Equation (4.60) is determined by the SIMC-PID tuning settings. If other tuning

47

rules are applied, some different forms may be derived, and ii ig c always has the

form

( )s

ii i

keg c h s

s

where, ( )h s is proper and rational with ( 0) 0h j .

Because the region between ( 1)ci ii ig c and 180(arg( ) 180 )ii ig c is most

critical for individual control loop design, the crossover frequency of iig c can be

adopted as the critical frequency point for determining the dRI to the particular

loop i iy u to obtain îig .

In Equation (4.69) and Equation(4.70), the factors kif and if are selected to be

not smaller than 1 , such that the equivalent open loop gain and the time delay of

îig are not smaller than those of iig . The reason for such a selection is to make

the resultant controller settings more conservative than those of îig so that loop

failure tolerance property can be preserved.

Step 3: Using tuning rules of Equation (4.57) to Equation (4.59), the controller

parameters for the individual equivalent transfer function of Equation (4.68) can be

determined as

1 ii

Pi

ki ii Ci i ii

kf k f

(4.71)

min{ ,4( )}Ii ii Ci i iif (4.72)

'

Di ii (4.73)

Furthermore, by selecting the desired closed-loop time constant Ci to be the same

value as the time delay i iif , we can obtain a set of simple rules for designing the

parameters of a decentralized PID controller as,

48

1 1 1

2

iiPi

ki i ii ii

kf f k

(4.74)

min{ ,( ,8 )}Ii ii ii i iif (4.75)

'

Di ii (4.76)

Once the closed-loop properties of the diagonal elements and the critical frequencies

of all individual control loops are determined, the dRI and MMF can be obtained

through calculation of the matrix P and dDRIA. Then, the parameters of the PID

controllers can be calculated based on the equivalent transfer function of each

control

49

Figure 4.8 Procedure for the independent design of a decentralized PID controller

Pair inputs and outputs

?i n

End

Design C for G

Construct P

Set i=1

Calculate the critical frequency

cri

Calculate , 1i nk

and , 1i n

for

MMF , 1ii n

Calculate dRI, 1ii n

Calculate detuning factors kif

and if

Detuning Controller Parameters

50

loop and the SIMC-PID tuning rules. In such a design procedure, the overall control

system stability is assured if the loop pairing is structurally stable. This can be

explained as follows:

1. Each ic is designed without considering loop interaction; it is more aggressive

to control loop i iy u than the final control setting ic . Generally, we have

1 2( ) ( ) as , 1max 1, 1ii n , where ( )A is the maximum singular

value of matrix A , 1

1 (1 )GC GC and 1

2 (1 )GC GC .

2. Let 1( )M G G G , and following the definition of the structured singular

value (SSV) :

1

( )min ( ) det( ) 0 for structured m

MI k M

we have 1 2( ) ( )M M which implies there exist smaller interactions

among control loops when the detuned controller C is applied.

3. Because the bigger dRI , 1ii n

is used to determine the detuning factors and kif

and if are not smaller than 1, the resultant controller ic will be more

conservative, with a smaller gain and crossover frequency compared with ic .

4. Conservative control action in each loop presents smaller interaction to other

loops. Because the design based on îig for loop i iy u will end up with a more

conservative ic the stability margin for each individual loop will be further

increased compared with that using the true , 1ii n

4.3.5 Algorithm for Decentralized Controller Design

As a summary of the above results, an algorithm for the design of a decentralized PID

controller for general multivariable processes is given as below.

1. For multivariable process G , pair the inputs and outputs based on the method

given above.

2. Design an individual controller for the diagonal elements of the process

transfer function matrix.

51

3. Construct matrix P .

4. Let 1i , go to step 5.

5. Calculate critical frequency cri , dRI and , 1( )ii n crij

.

6. Determine , 1i nk

and , 1i n

for MMF, , 1ii n

.

7. Obtain the factors kif and if .

8. Fine-tune the controller parameters.

9. If , 1i n i i , go to step 5. Otherwise, go to step 10;

10. End.

The procedure for the independent design of a decentralized PID controller by using

the proposed method is illustrated in Figure 4.8.

4.4 GPM Tuning Method based on ERGA Pairing Criteria

In this section, a novel equivalent transfer function in terms of effective relative gain

array (ERGA) for multivariable processes will be presented. By considering four

combination modes of gain and phase changes for a particular loop when other loops

are closed, this equivalent transfer function can effectively approximate the dynamic

interactions among loops. Consequently, the decentralized controllers can be

independently designed by employing the single loop tuning techniques. The method

is simple, straightforward, easy to understand and implement by field engineers.


shown in Figure 3.1.


In decentralized control system design, individual loop is tuned around its critical

frequency region of the transfer function. Thus, this frequency region should be

focused upon when considering the effect of interactions. Therefore, two factors in the

open loop transfer functions will affect the loop pairing decision:

52

1. Steady state gain: the steady state gain of the transfer function reflects

the effect of manipulated variable to controlled variable .

2. Response speed: response speed is accountable for the sensitivity of the

controlled variable to manipulated variable and, consequently, the

ability to reject the interactions from other loops.

Let

0( ) ( )ij ij ijg j k g j

where, ijk and

0 ( )ijg j are steady state gain and normalized transfer function of

( )ijg j , i.e., 0 (0) 1ijg respectively. In order to use both steady state gain and

response speed information for interaction measure and loop pairing, we now define

the effective gain for a particular transfer function as

, 0

0( )

c ij

ij ij ije k g j d

where, ,c ij for is the critical frequency of the transfer function

and is the absolute value of . In order to calculateije , the critical frequency can be

defined in two ways:

1. , ,c ij b ij , where

,b ij for , 1,2,...,i j n is the bandwidth of the transfer

function 0 ( )ijg j and determined by the frequency where the magnitude plot of

frequency response reduced to 0.707 time, as shown in Figure 4.9, i.e.,

,( ) 0.707 (0) .ij b ij ijg j g

2. , ,c ij u ij , where ,u ij for , 1,2,...,i j n is the ultimate frequency of the

transfer function 0 ( )ijg j and determined by the frequency where the phase

plot of frequency response across , as shown in Figure 4.10, i.e.,

,( ) .ij u ijg j

(0)ijg

ju iy

iyju

ije

, 1,2,i j n ( )ijg j

53

For transfer function matrices with some elements without phase crossover

frequencies, such as first order or second order without time delay, it is necessary to

use corresponding bandwidths as critical frequencies to calculate ije . However, it is

worth to point out that the phase crossover frequency information, i.e., ultimate

frequency ,u ij is recommended if applicable for calculation of

ije , since it is closely

linked to system dynamic performance and control system design. Without loss of

generality, we will use ,u ij as the bases for the following development.

For the frequency response of ( )ijg j as shown in Figure 4.10, ije is the area covered

by ( )ijg j up to ,u ij . Since 0 ( )ijg j represents the magnitude of the transfer

function at various frequencies, ije is considered to be the energy transmission ratio

from the manipulated variable ju to the controlled variable iy .

Figure 4.9 Frequency response curves and the energy of ( )ijg j (bandwidth ,b ij )

Express the energy transmission ratio array as

11 12 1

21 22 2

1 2

...

...

... ... ... ...

...

n

n

n n nn

e e e

e e eE

e e e

54

Figure 4.10 Frequency response curve and effective energy of ( )ijg j (ultimate frequency,u ij )

To simplify the calculation, we approximate the integration of ije by a rectangle area,

i.e.

, , 1,2,..., .ij ij u ije k i j n

Then, the effective energy transmission ratio array is given as:

(0) ,E G

where the operator is the Hadamard product, and

11 12 1

21 22 2

1 2

...

...(0)

... ... ... ...

...

n

n

n n nn

k k k

k k kG

k k k

and

,11 ,12 ,1

,21 ,22 ,2

, 1 , 1 ,

...

...

... ... ... ...

...

u u u n

u u u n

u n u n u nn

55

are the steady state gain array and the critical frequency array respectively. Since ije is

an indication of energy transmission ratio for loopi jy u , the bigger the

ije value is,

the more dominant of the loop will be.

Similar to the definition of relative gain, the effective relative gainij , between output

variable iy and input variable ju is defined as the ratio of two effective energy

transmission ratios:

ˆ

ij

ij

ij

e

e (4.77)

where ije is the effective energy transmission ratio between output variable iy and

input variable ju when all other loops are closed. When the effective relative gains are

calculated for all the input/output combinations of a multivariable process, it results in

an array, ERGA, which can be calculated by

11 12 1

21 22 2

1 2

...

...

... ... ... ...

...

n

n

n n nn

TE E

The introduction of energy transmission ratio is to mathematically represent the

effectiveness of a control loop which is affected by two key factors, i.e., the steady

state gain of the transfer function reflecting the effect of the manipulated variable ju

to the controlled variable iy , and the response speed reflecting the sensitivity of the

controlled variable iy to the manipulated variable ju and, consequently, the ability to

reject the interactions from other loops. Since ERGA is a relative measure, using the

multiplication of the two parameters to approximate the energy transmission ratio in

ij can simplify the calculation while captures the key elements in a multivariable

control system.

The ERGA is used to determine the best variable paring. In the following sections, we

56

will employ this interaction measure to develop effective transfer functions (ETFs)

under decentralized control structure.

Since both RGA and ERGA use relative gains, the properties of RGA can be directly

extended to ERGA:

1. The value of is a measure of the effective interaction expected in the

loop if its output is paired with .

2. The elements of the ERGA across any row, or down any column, sum up to 1,

i.e.

.

3. Let ije represent the loop effective gain when all the other loops are closed,

whereas represents the normal, open loop effective gain, then:

1ˆ

ij

ij ije e

.

4. When is negative, with other loops open, produces a change in in

response to a change in totally opposite in direction to that when the other

loops are closed.

As 0NI , based on steady state information provides a sufficient condition for the

loop configuration to be unstable, it can used to eliminate those structure unstable

pairing options. Similar to RGA and NI based pairing rules, the ERGA, and NI based

loop pairing rules requires that manipulated and controlled variables in a decentralized

control system be paired:

1. Corresponding ERGA elements are closest to 1.0.

2. The NI is positive.

3. All paired ERGA elements are positive.

4. Large ERGA elements should be avoided.

Here, both ERGA and NI play important roles for control structure selection. ERGA is

used to measure interactions, while NI is used as a sufficient condition to rule out the

ijthi

iy ju

1 1

1n n

ij ij

i j

i

ij

ij iy

ju

57

closed loop unstable pairings. Since we are using effective gains instead of simply

steady state gains, transient response can be effectively reflected. Therefore,

comparing REGA method with RGA and DRGA, we may expect that:

1. In additional to steady state gains, only time constant and time delay are

required in ERGA method, it is far easier to calculate than those DRGA

methods.

2. ERGA combines both steady state gains and equivalent error in measuring the

loop interactions, it should provide better pairing results than that of RGA

based pairing and comparable with DRGA ones.

3. Since ERGA only uses information of open loop process transfer functions, it

is controller type independent.

4.4.2 Decentralized Control System Design

This method involves two main steps in designing decentralized controllers for

multivariable processes:

1. the determination of the equivalent transfer function for each individual loop

through use of the ERGA

2. the tuning of each controller parameter based on the equivalent transfer

function

Step 1: Suppose that the best loop configuration has been determined and the best pair

is diagonally placed in the transfer function matrix as shown in Figure 3.1. Similar to

the open loop gain, we let the effective energy transmission ratio ije when all other

loops are closed be ,ˆˆ ˆ (0) , 1,2,..., ,ij ij u ije g i j n where ˆ (0)ijg and ,

û ij are the steady

state gain and ultimate frequency between output variable iy and input variable iu

when all other loops are closed, respectively. Then, from Equation (4.77)

,

,

(0)ˆˆ (0) .

ij u ij

ij u ij

ij

gg

(4.78)

By the definition of RGA, we have

58

(0)

ˆ (0) .ij

ij

ij

gg

(4.79)

where ij is the relative gain. Substitute Equation (4.79) into Equation (4.78) and

rearrange to result

,

,ˆ

ij u ij

ij

ij ij

(4.80)

where ij represents the critical frequency change of loop i j when other loops are

closed, defined as relative critical frequency. When the relative frequencies are

calculated for all the input/output combinations of a multivariable process, it results in

an array, i.e., relative frequency array (RFA).

Since control loop transfer functions when other loops closed will have similar

frequency properties with when other loops open if it is well paired, we can let the

ETFs have the same structures as the corresponding open loop transfer functions but

with different parameters

ˆ

ˆ ˆ( ) (0) ( ) ,iid sr

ii ii iig s g g s e (4.81)

where ( )r

iig s is defined by

0( ) ( ) iid sr

ii iig s g s e

and îid is the time delay of the ETF.

As the change in ultimate frequency of a control loop is generally affected by changes

in both time constant and time delay when other loops are closed, and they are

exchangeable by linear approximation, it is reasonable to change only time delay to

reflect the phase changes. In Equation (4.81), ˆ (0)ijg can be determined by using

Equation (4.79), while by the definition of the ultimate frequency,

, , , ,ˆ ˆ ˆ( ) ( ) .r r

ii u ii ii u ii ii u ii ii u iid g j d g j

îid can be easily determined by

, ,

, ,

ˆ ˆ( ) ( )ˆ .ˆ

r r

ii u ii ii u ii

ii ij

u ii u ii

g j g jd

(4.82)

59

Notice that

1. ( )r

iig s is usually low order transfer functions, their contribution to the phase

change at low frequency range are small and can be equivalently represented

by the additional time delay term.

2. In many decentralized control system designs, such as gain and phase margin

method, an individual loop is tuned around the critical frequency region of

each control loop. Accurate estimation of overall variation is required around

the critical frequency, not who contribute to the change.

By letting , ,ˆ( ) ( )r r

ii u ii ii u iig j g j , we can make further simplification to Equation

(4.82) as

, ,ˆ ˆ .ii u ii ii u iid d

which results by considering Equation (4.80)

,

,

ˆ .ˆ

u ii

ii ii ii ii

u ii

d d d

(4.83)

This is the practical formula, which will be used to derive the ETFs. Even though

Equation (4.83) is less accurate than Equation (4.82), several simulation results have

showed that the control system performances are comparable by the two

approximations, but Equation (4.83) is much more straightforward and easier

explainable and understandable than Equation (4.82).

Since it is necessary that the controlled system possesses integrity property, i.e., the

overall control system remained to be stable regardless put in and/or taken out of other

control loops, ˆ (0)iig and îid in ETF must take different values for different

combination of ii and ii . For the four different combinations of ii and ii , ˆ ( )iig s

may take different modes shown in Figure 4.11 - Figure 4.14, and are discussed below:

Case 1: 1, 1ii ii

In this case, 1

( 1) 0ii

and ( 1) 0.ii According to Equation (4.79) and Equation

(4.83), we have ˆ (0) (0)ii iig g , and ˆ .ii iid d

60

1. ˆ (0) (0)ii iig g , this means that the magnitude of the frequency response when

the other loops closed is not less than that of when the other loops open. Since

the retaliatory effect from the other loops magnifies the main effect of iu on iy ,

we need to reduce the controller gain to assure system stability. In this case, the

gain is by Equation (4.79)

(0)ˆ (0) .ii

ii

ii

gg

Figure 4.11 Interaction mode with 1, 1ii ii

2. îi iid d , this means that the time delay when the other loops closed is not

bigger than that of when other loops open. The reduced time delay will

increase the phase margin. However, by considering the control system

integrity, the time delay needs to be kept as before, i.e.,

ˆ .ii iid d

Case 2: 1, 1ii ii

In this case, 1

( 1) 0ii



1. ˆ (0) (0)ii iig g , same as in Case 1,

61

(0)ˆ (0) .ii

ii

ii

gg

2. ˆ .ii iid d , this means that the time delay when the other loops closed is bigger

than that of when the other loops open. The enlarged time delay will reduce the

phase margin. In this case, the time delay is determined by Equation (4.83)

ˆ .ii ii iid d


Case 3: 1, 1ii ii

In this case, 1

( 1) 0ii



1. ˆ (0) (0)ii iig g , this means that the magnitude of the frequency response when

the other loops closed is smaller than that of when the other loops open. Even

if the retaliatory effect from other loops acts in opposition to the main effect of

iu on iy , we cannot enlarge the controller gain for better performance due to the

system integrity consideration. Hence, the gain should be unchanged, i.e.,

ˆ (0) (0).ii iig g

2. ˆ .ii iid d , same as in Case 1

ˆ .ii iid d

62

Figure 4.13 f Interaction mode with 1, 1ii ii

Case 4: 1, 1ii ii

In this case, 1

( 1) 0ii

and ( 1) 0.ii . According to Equation (4.79) and Equation


1. ˆ (0) (0)ii iig g , same as in Case 3,

ˆ (0) (0).ii iig g

2. ˆ .ii iid d , same as in Case 2,

ˆ .ii ii iid d


63

A unique problem for decentralized control of MIMO processes is the zero crossing.

Stable or unstable zeros might be introduced into a particular control loop when other

loops are closed. If an unstable zero is introduced, it will result phase shift to the left in

the frequency domain. In order to guarantee the entire system stability, the controllers

are normally conservatively designed by conventional detuning approaches. By

introducing the relative critical frequency, ii , to indicate phase changes after the other

loops closed, the effects of unstable zeros can be accurately estimated in each control

loop. Consequently, the resultant control systems will be much less conservative.

Mathematically, the equivalent transfer function should incorporate the controllers of

all other loops. To solve such a complex problem, recursive solution is required by

first assigning initial controllers, then finding the equivalent loop transfer functions

and designing controllers again. This process is continuous until a stable solution is

obtained. To simplify the problem, both detuning and independent methods proposed

so far assume that all other closed loops are under perfect control when designing the

controller for a particular loop and consider only the gain change. In this method, the

changes are considered for both gain and frequency. Especially, Equation (4.79)

focuses on the gain impact while Equation (4.83) contributes to time delay portion,

i.e., frequency impact. As will be shown later, it is far more accurate than those

existing methods.

Step 2: Without loss of generality, we assume that each main loop, i.e., diagonal

element in the transfer function matrix is represented by a second order plus dead time

(SOPDT) model, which can be used to describe most of the industrial processes:

0,

2

2, 1,

( ) .1

iiii d s

ii

ii ii

bg s e

a s a s

Similarly, ETF is represented as

ˆ

2

2, 1,

ˆ (0)ˆ ( ) .

1iid sii

ii

ii ii

gg s e

a s a s

The decentralized controllers can then be independently designed by single loop

approaches based on the corresponding ETFs. Here we employ the gain and phase

margins (GPM) approach. This is primary because the frequency response method

64

provides good performance in the face of uncertainty in both plant model and

disturbances.

The PID controller of each loop is supposed of the following standard form:

,

, ,( ) .i i

ci p i d i

kg s k k s

s

The controller can be rewritten as

2 2, , ,

( )d i p i i i

ci

k s k s k As Bs Cg s k

s s

where , ,/ , /k i p iA k k B k k and

, / .i iC k k

By selecting 2 1,A a B a and 1C , the open loop transfer function becomes

ˆˆ (0)ˆ( ) ( ) .iid sii

ci ii

gg s g s k e

s

Denoting the gain and phase margin specifications as ,m iA and

,m i , and their

crossover frequencies as ,g i and

,p i respectively, we have

, , ,

, , , ,

, , ,

, , , ,

ârg ( ) ( )

ˆ( ) ( ) 1

ˆ( ) ( ) 1

ârg ( ) ( ) .

c i g i ii g i

m i c i g i ii g i

c i p i ii p i

m i c i p i ii p i

g j g j

A g j g j

g j g j

g j g j

By substitution and simplification to above equations, we obtain

,

,

,

,

, ,

ˆ2

ˆ (0)

ˆ (0)

ˆ2

g i ii

g i

m i

ii

ii p i

m i p i ii

d

Akg

kg

d

which results

,

,

,

1(1 )

2

.ˆ ˆ2 (0)

m i

m i

m i ii ii

A

kA d g

65

By this formulation, the gain and phase margins are interrelated to each other, some

possible gain and phase margin selections are given in Table 4.2.

,m i / 4 / 3 3 / 8 2 / 5

,m iA 2 3 4 5

Table 4.2 Typical Gain and Phase Margin Values

The PID parameters are given by (Wang, et al., 2002)

, 1,

,

,, 2,

1ˆ ˆ2 (0)

p i ii

i i

m i ii iid i ii

k a

kA d g

k a

(4.84)

Applying Equation (4.84) for each case discussed in section above, we can easily

obtain both ETFs and the PID parameters, which are summarized in

Table 4.3.

Mode ˆ ( )iig s ,P iik ,i iik

,d iik

1, 1ii ii 2

2, 1,

(0) /( )

1iid sii ii

ii

ii ii

gg s e

a s a s

1,

2 (0)

ii ii

m ii ii

a

A d g

2 (0)

ii

m ii iiA d g

2,

2 (0)

ii ii

m ii ii

a

A d g

1, 1ii ii 2

2, 1,

(0) /( )

1ii iid sii ii

ii

ii ii

gg s e

a s a s

1,

2 (0)

ii ii

m ii ii ii

a

A d g

2 (0)

ii

m ii ii iiA d g

2,

2 (0)

ii ii

m ii ii ii

a

A d g

1, 1ii ii 2

2, 1,

(0)( )

1iid sii

ii

ii ii

gg s e

a s a s

1,

2 (0)

ii

m ii ii

a

A d g

2 (0)m ii iiA d g

2,

2 (0)

ii

m ii ii

a

A d g

1, 1ii ii 2

2, 1,

(0)( )

1ii iid sii

ii

ii ii

gg s e

a s a s

1,

2 (0)

ii

m ii ii ii

a

A d g

2 (0)m ii ii iiA d g

2,

2 (0)

ii

m ii ii ii

a

A d g

Table 4.3 Decentralized Controller Design

66

5 Simulation and Performance Analysis

5.1 Introduction

We have three typical control structures, which are shown in Appendix A, Appendix

B, and Appendix C respectively. We also have three typical programs for 2 2 , 3 3

and 4 4 systems. They are shown in Appendix D, Appendix E, and Appendix F

respectively. We do parameter estimation for some transfer functions. Performance

analysis is done based on IAE, ISE and ITAE for all systems. One click solution is

also provided to execute the programs, which means that sim command calls and

executes the Simulink block diagram and, the responses for the individual loops will

be come out sequentially with a single click on Run command from Debug menu.

5.2 Simulation and Discussion for Vinante and Luyben System (1972)

Firstly, we start with 2x2 Vinante and Luyben System (1972) whose transfer function

matrix is given as

0.3

1.8 0.35

2.2 1.3

7 1 7 1( ) .

2.8 4.3

9.5 1 9.2 1

s s

s s

e e

s sG s

e e

s s

The dimension of the plant is 2 2 and so the pairing option is 2!, which is equal to 2.

The DC gains and NI values for all possible pairings for this process are

1

1.3 2.2

4.3 2.8k

and 2

2.2 1.3,

2.8 4.3k

and

Pairing 1 2

NI -1.5989

0.6152

67

respectively. After using NI screening tool, the RGA values for possible pairing is

Pairing 1

11

1.6254

5.2.1 BLT

The first step for loop pairing in this method is to eliminate undesirable pairings by

the screening tools NI and RGA. NI values for the best pairing is 0.6152. The first

pairing option is cancelled because NI value for that pairing is negative. The DC gain

and RGA value for the best pairing are

2.2 1.3

2.8 4.3k

and

1.6254 0.6254

0.6254 1.6254

respectively. The ultimate values and Ziegler-Nichols tuning parameters of the

diagonal elements of this process transfer function matrix are shown in Table 5.1.

Parameters Loop 1 Loop 2

uK -5.2912 9.7507

u 1.6568 4.5561

ZNK -2.4051 4.4322

ZN 3.1603 1.1492

Table 5.1 The ultimate values and ZN tuning parameters

68

We assume that F is equal to 2.3 and after executing the while loop in Appendix D

the value of F becomes 2.29. At the same time, controller parameters can be

calculated by using tuning factor F . See Table 5.3 for controller values. The following

table shows the assumption and final values for F .

Parameter Assumption Final value

F 2.3 2.29

Table 5.2 Tuning parameter

We do not try to find D in this case. However, we will find it in the later two cases by

using dF derivative factor.

5.2.2 SIMC

The first step for loop pairing in this method is to eliminate undesirable pairings by

the screening tools NI, RGA and GIA. The DC gain and RGA values for the best

pairing are the same to those in BLT. Then GIA can be calculated as

0.3848 1.0000

1.0000 0.3848

Selecting 2Ci ii , the SIMC-PID controller for diagonal system G is obtained as

1.061 0.15150

2.038 0.48520

s

sC

s

s

resulting in the critical frequencies for control loops 1 1y u and 2 2y u as

1 20.5 rad/s and 1.4286 rad/scr cr , respectively. Then we have P matrix and dRI

at the critical frequencies

0.7268 0.9572 1.0000.

1.0000 0.7268 0.9572

jP

j

69

and

11,1

22,1

( 0.5) 0.3900 - 0.3206

( 1.4286) 0.0033 + 0.0476

j j

j j

respectively. Subsequently MMF can be calculated as

0.6800

11,1

( 0.0498)

22,1

1.4265

1.0044

s

s

e

e

and 1 11.4265 and 1.6800kf f and, 2 21.0044 and 1.0000kf f are used to fine-

tune the PI/PID controller setting for each control loop. Then the equivalent transfer

functions of both control loops are constructed as

1.68

11

0.35

22

3.138ˆ

7 1

4.319ˆ .

9.2 1

s

s

eg

s

eg

s

Consequently, the decentralized PI controller parameters are determined by SIMC and

shown in Table 5.3. Figure 5.1 and Figure 5.2 shows the responses for each control

loop.

5.2.3 ERGA

In this section, the loop pairing is based on the criterion of ERGA plus Niederlinski

index (NI). The gain and phase margin for all loops are specified to be 3 and / 3 ,

respectively. These margins are generally regarded as making the best compromise

between performance and robustness in process control.

The DC gain, Critical Frequency Array, RGA, ERGA and RFA are

2.2 1.3

2.8 4.3k

,

0.1425 0.1425

0.1050 0.1084

1.6254 -0.6254,

-0.6254 1.6254

1.5939 -0.5939

-0.5939 1.5939

and

70

0.9806 0.9497

0.9497 0.9806

respectively.

Both RGA and ERGA pairing rules indicate diagonal pairing (NI =0.6152>0). Since

1ii and 1ii for 1,2i , we do not need to detune and ETFs for two loops are

11

2.2

7 1ˆ ( )

se

sg s

and

0.35

22

4.3

9.2 1ˆ ( )

se

sg s

respectively.

The controller parameters for all methods are shown in Table 5.3.

Method Loop CK I D

BLT

1 -1.0502 7.2372 0

2 1.9354 2.6318 0

SIMC

1 -0.7048 7.0000 0

2 1.9449 4.2000 0

ERGA

1 -1.6660 7.0000 0

2 3.2007 9.2000 0

Table 5.3 Decentralized Controllers for VL System (1972)

Figure 5.1 and Figure 5.2 show the responses for all control methods with unit set

points change in 1r at t=0 and 2r at t=50.

71

Figure 5.1 Closed loop step responses of y1 for VL System (1972)

Figure 5.2 Closed loop step responses of y2 for VL System (1972)

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

y1

Time

Closed loop step responses of y1 for VL System (1972)

BLT

SIMC

ERGA

0 10 20 30 40 50 60 70 80 90 100-0.2

0

0.2

0.4

0.6

0.8

1

1.2

y2

Time

Closed loop step responses of y2 for VL System (1972)

BLT

SIMC

ERGA

72

5.3 Simulation and Discussion for Ogunnaike and Ray System (1979)

Secondly, we continue to present 3x3 Ogunnaike and Ray System (1979) whose

transfer function matrix is given as

2.6 3.5

6.5 3 1.2

9.2 9.4

0.66 0.61 0.0049

6.7 1 8.64 1 9.06 1

1.11 2.36 0.01( ) .

3.25 1 5 1 7.09 1

34.68 46.2 0.87(11.61 1)

8.15 1 10.9 1 (3.89 1)(18.8 1)

s s s

s s s

s s s

e e e

s s s

e e eG s

s s s

e e s e

s s s s

The dimension of the plant is 3 3 and so the pairing option is 3! which is equal to 6.

The DC gains and NI values for all possible pairings for this process are

1

-0.0049 -0.6100 0.6600

-0.0100 -2.3600 1.1100 ,

0.8700 46.2000 -34.6800

k

2

-0.0049 0.6600 -0.6100

-0.0100 1.1100 -2.3600 ,

0.8700 -34.6800 46.2000

k

3

-0.6100 -0.0049 0.6600

-2.3600 -0.0100 1.1100 ,

46.2000 0.8700 -34.6800

k

4

-0.6100 0.6600 -0.0049

-2.3600 1.1100 -0.0100 ,

46.2000 -34.6800 0.8700

k

5

0.6600 -0.6100 -0.0049

1.1100 -2.3600 -0.0100 and

-34.6800 46.2000 0.8700

k

6

0.6600 -0.0049 -0.6100

1.1100 -0.0100 -2.3600 .

-34.6800 0.8700 46.2000

k

and

Pairing 1 2 3 4 5 6

Value -1.3039

2.0810 2.4718 -0.8877 0.3859 -1.7149

respectively. After using NI screening tool, the RGA values for possible pairings are

73

Pairing 1 2 3

11

-0.2864

-0.7220 2.0084

22 -0.6460 -0.1786 1.8246

33 -0.1026 -0.3624 1.4650

Since SIMC and ETF only use the standard transfer function models such as FOPTD

and SOPTD, we need to do parameter estimation for 33( )g s in the system mentioned

above. We can use the following methods to estimate parameters. They are:

1. pem

2. modred

3. ident

4. log

5. two points

6. area

7. relay feedback and so on.

We use pem and modred to estimate the parameters in this dissertation. The following

is Matlab program code to find K, T and L.

%*******************************************************************

%Parameter Estimation for g33

g33=tf([0.87*11.61 0.87],[3.89*18.8 (3.89+18.8) 1]...

,'InputDelay',1);

sys=g33;

[sys,elim_val] = balreal(sys); % Compute balanced realization

elim = (elim_val<1e-8);

rsys = modred(sys,2,'MatchDC'); % Remove negligible states

g33=tf(rsys);

t=0:1:10000;

[y t]=step(g33,t);

data33=iddata(y,t); % The IDDATA object

m33 =pem(data33,'p1d','kp',0.87, 'kp','fix','td' ,...

1,'td','fix' ,'tp1' ,{'max',10},'tp1' ,{'min',...

8.5});% fixing the parameter kp to the value 0.87

g33new=tf(m33.kp.value,[m33.tp1.value 1],...

'InputDelay' ,m33.td.value);

74

disp('The new transfer function for g33 is ')

g33new;

figure;

[y t x]=step(g33);

[ynew tnew xnew]=step(g33new);

plot(t,y,'b-',tnew,ynew,'r:','LineWidth',1.5);

ylabel('Amplitude');

xlabel('Time');

grid on;

title('Step responses for g33 and g33new for OR System (1979)');

legend('g33','g33new',4);

%*******************************************************************

The new transfer function model in the form of FOPTD is

33

0.87

10 1

s

new

eg

s

(4.85)

and the following diagram shows the comparison of step responses for both transfer

functions.

Figure 5.3 Step responses for g33 and g33new for OR System (1979)

0 10 20 30 40 50 60 700

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Am

plit

ude

Time

Step responses for g33 and g33new for OR System

g33

g33new

75

5.3.1 BLT

The DC gain and RGA values for the best pairing are

0.6600 0.6100 0.0049

1.1100 2.3600 0.0100

34.6800 46.2000 0.8700

k

and

2.0084 -0.7220 -0.2864

-0.6460 1.8246 -0.1786

-0.3624 -0.1026 1.4650



Parameters Loop 1 Loop 2 Loop 3

uK 7.1317 -1.3935 12.4487

u 0.6865 0.6266 1.7014

ZNK 3.2417 -0.6334 5.6585

ZN 7.6272 8.3564 3.0774


We assume that F and dF are equal to 2.2 and 1 respectively and after executing the


F 2.2 2.19

dF 1 2.18

Table 5.5 Tuning parameters

76

while loop in Appendix E the values of F and dF become 2.19 and 2.18 respectively.

At the same time, controller parameters can be calculated by using tuning factor F and

dF . See Table 5.6 for controller values. Table 5.5 shows the assumption and final

values for F and dF . In this case, we also find D using dF derivative factor.

5.3.2 SIMC

The DC gain and RGA values for the best pairing are the same to those in BLT. Then

GIA can be calculated as

0.7096 1.7802 393.4131

2.2007 0.7536 473.8140

3.1877 7.3665 0.7701

Selecting 0.5Ci ii , the SIMC-PID controller for diagonal system G is obtained as

2.603 0.38850 0

0.4708 0.094160 0

7.663 1.2770 0

s

s

Cs

s

s

resulting in the critical frequencies for control loops 1 1y u , 2 2y u and 3 3y u as

1 2 30.2564 rad/s, 0.2222 rad/s and 0.6667 rad/scr cr cr , respectively. Then we

have P matrix and dRI at the critical frequencies

0.5798 + 0.8803 1 1

1 0.5798 + 0.8803 1 .

1 1 0.5798 + 0.8803

j

P j

j

and

11,2( 0.2564) -9.9431 - 13.7285j j

22,2( 0.2222) 0.2948 + 3.4206j j

33,2( 0.6667) -0.0729 - 0.0639j j

77

Figure 5.4 Closed loop step responses of y1 for OR system (1979)


0 200 400 600 800 1000 12000

0.2

0.4

0.6

0.8

1

1.2

1.4

y1

Time

Closed loop step responses of y1 for OR System (1979)

BLT

SIMC

ERGA

0 200 400 600 800 1000 12000

0.2

0.4

0.6

0.8

1

1.2

1.4

y2

Time


BLT

SIMC

ERGA

78



8.3779

11,2

( 5.4402)

22,2

0.1032

33,2

16.3844

3.6575

0.9293

s

s

s

e

e

e

and,

1 116.3844 and 4.2223kf f ,

2 23.6575 and 1.0000kf f and

3 31.0000 and 1.1032kf f

are used to fine-tune the PI/PID controller setting for all control loops. Then the

equivalent transfer functions of all control loops are constructed as

0 200 400 600 800 1000 1200-14

-12

-10

-8

-6

-4

-2

0

2

y3

Time


BLT

SIMC

ERGA

79

11

11

3

22

1.1

33

10.81ˆ

6.7 1

8.632ˆ

5 1

0.87ˆ

10 1

s

s

s

eg

s

eg

s

eg

s


shown in Table 5.6. Figure 5.4 to Figure 5.6 show the responses for each control loop.

5.3.3 ERGA

The gain and phase margin for all loops are specified to be 7 and 3 / 7 respectively.


0.6600 0.6100 0.0049

1.1100 2.3600 0.0100

34.6800 46.2000 0.8700

k

,

0.6865 0.5123 1.6380

0.3521 0.6266 1.3927

0.2250 0.2108 1.6319

2.0084 -0.7220 -0.2864

-0.6460 1.8246 -0.1786 ,

-0.3624 -0.1026 1.4650

1.2928 -0.2026 -0.0902

-0.2076 1.2475 -0.0399

-0.0852 -0.0449 1.1301

and

0.6437 0.2806 0.3149

0.3213 0.6837 0.2237

0.2352 0.4377 0.7714

respectively.


1ii and 1ii for 1,2,3i , we do not need to detune and ETFs for two loops are

2.6

11

0.66

6.7 1ˆ ( )

se

sg s

3

22

2.36

5 1ˆ ( )

se

sg s

and

80

33

0.87

10 1ˆ ( )

se

sg s

respectively. The controller parameters for all methods are shown in Table 5.3. The

PI/PID controller parameters for all methods are listed in Table 5.6. The closed-loop

responses, where the unit set-points change in 1r at t=0, 2r at t=400 and 3r at t=800

are shown in Figure 5.4 to Figure 5.6.

Method Loop PK I D

BLT

1 1.4802 16.7038 1.1441

2 -0.2892 18.3007 1.2535

3 2.5838 6.7397 0.4616

SIMC

1 0.1334 6.7000 0

2 -0.0938 5.0000 0

3 6.9234 6.0000 0

ERGA

1 0.8762 6.7000 0

2 -0.1585 5.0000 0

3 2.5793 10.0000 0

Table 5.6 Decentralized Controllers for OR System (1979)

5.4 Simulation and Discussion for Alatiqi case 1 System (1985)

Finally, we end with 4x4 Alatiqi case 1 System (1985) whose transfer function matrix

is given as

81

The dimension of the plant is 4 4 and so the number of pairing options is 4!which is

equal to 24. The first six of all DC gains for all possible pairings for this process are

1

-0.6400 0.0170 -2.9400 2.2200

1.6800 -0.5100 3.4600 -2.3300,

-5.3800 4.4100 3.5110 -1.0600

4.7800 -1.2500 4.3200 -5.7300

k

2

-0.6400 0.0170 2.2200 -2.9400

1.6800 -0.5100 -2.3300 3.4600,

-5.3800 4.4100 -1.0600 3.5110

4.7800 -1.2500 -5.7300 4.3200

k

3

-0.6400 -2.9400 0.0170 2.2200

1.6800 3.4600 -0.5100 -2.3300,

-5.3800 3.5110 4.4100 -1.0600

4.7800 4.3200 -1.2500 -5.7300

k

4

-0.6400 -2.9400 2.2200 0.0170

1.6800 3.4600 -2.3300 -0.5100,

-5.3800 3.5110 -1.0600 4.4100

4.7800 4.3200 -5.7300 -1.2500

k

5

-0.6400 2.2200 -2.9400 0.0170

1.6800 -2.3300 3.4600 -0.5100,

-5.3800 -1.0600 3.5110 4.4100

4.7800 -5.7300 4.3200 -1.2500

k

2.5 0.05 0.2 20

2 2

5 1.01 7.5 2

2 2 2

22 13

2

2.22 2.94(7.9 1) 0.017 0.64

(36 1)(25 1) (23.7 1) (31.6 1)(7 1) (29 1)

2.33 3.46 0.51 1.68

(35 1) 32 1 (32 1) (28 1)( )

1.06 3.511

(17 1) (

G

s s s s

s s s s

s s

e s e e e

s s s s s s

e e e e

s s s ss

e e

s

1.01 0.5

2

2.5 0.01 2.8 1.15

4.41 5.38

12 1) 16.2 1 17 1

5.73 4.32(25 1) 1.25 4.78

(8 1)(50 1) (50 1)(5 1) (43.6 1)(9 1) (48 1)(5 1)

s s

s s s s

e e

s s s

e s e e e

s s s s s s s s

82

6

-0.6400 2.2200 0.0170 -2.9400

1.6800 -2.3300 -0.5100 3.4600,

-5.3800 -1.0600 4.4100 3.5110

4.7800 -5.7300 -1.2500 4.3200

k

and so on and NI values for all possible pairings for this process are

Pairing 1 2 3 4 5 6

NI -2.9420

12.9251 -0.3452 -6.5842 2.9519 0.6800

Pairing 7 8 9 10 11 12

NI 33.6224

-147.7151 10.6540 64.8207 -29.0609 -20.9847

Pairing 13 14 15 16 17 18

NI -0.4179

-2.5429 0.1548 2.9519 0.4194 -0.1338

Pairing 19 20 21 22 23 24

NI 1.0167

0.7341 0.1193 -0.3740 -1.1802 -0.2719

respectively. After using NI screening tool, the RGA values for possible pairings are

83

Pairing 1 2 3 4 5 6

11

-1.5016

-1.5016 -1.5016

0.0502 0.0502 0.0502

22 -0.7028 -3.3628 -3.3628 2.2971 2.7685 2. .7685

33 -0.1203 0.1305 2.2635 0.1305 -1.2736 -0.1203

44 -0.8224 -0.6109 -0.8224 0.9552 0.9552 1.4781

Pairing 7 8 9 10 11 12

11

-1.0766 -1.0766 -1.0766 3.5280 3.5280 3.5280

22 2.2971 2.2971 -3.3628 -0.7028 -0.7028 2.7685

33 2.2635 -0.1203 -1.2736 0.1305 -1.2736 2.2635

44 0.9552 -0.6109 -0.6109 1.4781 -0.8224 1.4781

The elements 12 ( )g s and 42 ( )g s in this process are not standard models. They have

zeros at 0.1266s and 0.0400s respectively. The following are Matlab

programming codes for parameter estimation of 12 ( )g s . Figure 5.7 and Figure 5.8

show the step responses of their original and new transfer functions.

%*******************************************************************


g12=tf([(-2.94*7.9) -2.94],[(23.7*23.7) (23.7+23.7) 1],...

'InputDelay',0.05);

84

sys=g12;




g12=tf(rsys);

t=0:1:1000;

[y t]=step(g12,t);


m12 =pem(data12,'p1d','kp',-2.94, 'kp','fix','td' ,0.05,...

'td' ,'fix' ,'tp1',{'max',40}, 'tp1',{'min',...

30});% fixing the parameter kp to the value -2.94

g12new=tf(m12.kp.value,[m12.tp1.value 1],'InputDelay',m12.td.value);


g12new;

figure;

[y t x]=step(g12);




xlabel('Time');

grid on;

title('Step responses for g12 and g12new for A1 System (1985)')


%*******************************************************************

The new transfer function models for 12g and 42g in the form of FOPTD are

0.05

12

2.94

40 1

s

new

eg

s

(4.86)

and

0.01

42

4.32

28 1

s

new

eg

s

(4.87)

respectively.

5.4.1 BLT

The DC gain and RGA values for the best pairing are

2.2200 -2.9400 0.0170 -0.6400

-2.3300 3.4600 -0.5100 1.6800

-1.0600 3.5110 4.4100 -5.3800

-5.7300 4.3200 -1.2500 4.7800

k

and

85

Figure 5.7 Step responses for g12 and g12new for A1 System (1985)

Figure 5.8 Step responses for g42 and g42new for A1 System (1985)

0 50 100 150 200 250-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

Am

plit

ude

Time

Step responses for g12 and g12new for A1 System (1985)

g12

g12new

0 20 40 60 80 100 120 140 160 1800

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Am

plit

ude

Time

Step responses for g42 and g42new for A1 System (1985)

g42

g42new

86

3.5280 -1.0766 0.0502 -1.5016

-3.3628 2.7685 -0.7028 2.2971

-0.1203 0.1305 2.2635 -1.2736

0.9552 -0.8224 -0.6109 1.4781



Parameters Loop 1 Loop 2 Loop 3 Loop 4

uK 11.3027 14.5683 5.8583 10.0365

u 0.1634 1.5749 1.5936 0.4250

ZNK 5.1376 6.6219 2.6629 4.5620

ZN 32.0454 3.3247 3.2857 12.3190


We assume that F and dF are equal to 2.5 and 1 respectively, and after executing the

while loop in Appendix F the values of F and dF become 2.49 and 1 respectively. At

the same time, controller parameters can be calculated by using tuning factors F and

dF . See Table 5.9 for controller values. The following table shows the assumption and

final values for F and dF .


F 2.5 1

dF 2.49 1

Table 5.8 Tuning parameters

In this case, we also find D using dF derivative factor.

87

5.4.2 SIMC

The DC gain and RGA values for the best pairing are the same to those in BLT. Then

GIA can be calculated as

2.0965 6.6211 512.7859 11.3807

2.9807 3.5689 26.4433 5.2120

11.6651 7.5100 0.5561 0.8550

9.0448 7.5043 20.4058 5.8869

Selecting 2Ci ii , the SIMC-PID controller for diagonal system G is obtained as

2

2

0 0 0s

3.052 s + 0.25180 0 0

1.212 s

54.05 2.162 0.07207

14.55 2.911 0.2109

+ 0.10 0 0

0 0 0

sC

s

s

s s

s s

resulting in the critical frequencies for all control loops as

1 0.1333 rad/s,cr

2 0.3300 rad/s,cr

3 0.3300 rad/scr

and

4 0.2899 rad/scr

respectively. Then we have P matrix and dRI at the critical frequencies

0.7268 + 0.9572 1 1 1

1 0.7268 + 0.9572 1 1.

1 1 0.7268 + 0.9572 1

1 1 1 0.7268 + 0.9572

j

jP

j

j

88

and

11,3( 0.1333) 0.2244 + 0.5093j j

22,3( 0.3300) -0.0084 + 0.0083j j

33,3( 0.3300) 0.0023 - 0.0022j j

44,3( 0.2899) -0.0256 + 0.2630j j


( 2.9566)

11,3

( 0.0255)

22,3

0.0066

33,3

( 0.9094)

44,3

1.3261

0.9916

1.0023

1.0093

s

s

s

s

e

e

e

e

and,

1 11.3261 and 1.0000kf f ,

2 21.0000 and 1.0000kf f

3 31.0023 and 1.0065kf f and

4 41.0093 and 1.0000kf f

are used to fine-tune the PI/PID controller setting for all control loops. Then the

equivalent transfer functions of all control loops are constructed as

2.5

11

1.01

2

2

22

1.02

33

1.15

44

2.944ˆ

3.46ˆ

32 s + 1

4.42ˆ

16.2 s + 1

4.82

900 61 1

2ˆ

40 5

4

3 1

s

s

s

s

eg

eg

eg

s s

s s

eg

89


shown in Table 5.9. Figure 5.9 to Figure 5.12 show the responses for each control

loop.

5.4.3 ERGA

The gain and phase margin for all loops are specified to be 5 and 2 / 5 respectively.


2.2200 -2.9400 0.0170 -0.6400

-2.3300 3.4600 -0.5100 1.6800

-1.0600 3.5110 4.4100 -5.3800

-5.7300 4.3200 -1.2500 4.7800

k

,

0.0209 0.0249 0.0302 0.0221

0.0183 0.0312 0.0200 0.0229,

0.0377 0.0534 0.0616 0.0587

0.0195 0.0356 0.0219 0.0206

3.5280 1.0766 0.0502 1.5016

3.3628 2.7685 0.7028 2.2971,

0.1203 0.1305 2.2635 1.2736

0.9552 0.8224 0.6109 1.4781

1.8820 0.1371 0.0487 1.0678

1.4541 2.1913 0.0159 0.2469

0.0265 0.0174 1.9648 0.9209

0.5987 1.3110 1.0294 2.7417

and

0.5334 0.1273 0.9710 0.7111

0.4324 0.7915 0.0226 0.1075

0.2203 0.1330 0.8680 0.7231

0.6267 1.5942 1.6850 1.8549

respectively.


1ii and 1ii for 1,2,3i , we do not need to detune for those loops but for 1ii

and 1ii for 4i we need to detune and, ETFs are

90

2.

2

5

11

2.22

900 61 1ˆ ( )

se

s sg s

1.01

22

3.46

32 s + 1ˆ ( )

se

g s

1.01

33

4.41

16.2 s + 1ˆ ( )

se

g s

and

2.13

244240 5

4

3

7

1

. 8ˆ ( )

se

s sg s

respectively. The controller parameters for all methods are shown in Table 5.9.

Method Loop CK I D

BLT

1 2.0632 79.7944 4.8068

2 2.6594 8.2786 0.4987

3 1.0694 8.1816 0.4929

4 1.8321 30.6749 1.8479

SIMC

1 1.6305 30.0000 25.0000

2 3.0523 12.1200 0

3 1.2070 12.1462 0

4 2.8838 13.8000 5.0000

ERGA

1 3.4529 61.0000 14.7541

2 2.8768 32.0000 0

3 1.1426 16.2000 0

4 1.6330 53.0000 4.5283

Table 5.9 Decentralized Controllers for A1 System (1985)

91

Figure 5.9 Step responses of y1 for A1 System (1985)

Figure 5.9 to Figure 5.11 shows the responses for all control methods with unit set

points change in 1r at t=0, 2r at t=400, 3r at t=800 and 4r at t=1200.

5.5 Performance Analysis

The ability of the closed-loop system to remain stable despite changes in process

parameters is called robustness. The study of the controller tuning methods illustrated

above cannot be completed without a robustness test. The performance of the control

system should be checked when the values of the process parameters deviate from the

estimated nominal values.

Doyle and Stein (1981) proposed two useful measures of robustness.

1DSO= ( )I GB (4.88)

1DSI= ( )I BG (4.89)

0 200 400 600 800 1000 1200 1400 16000

0.2

0.4

0.6

0.8

1

1.2

1.4

y1

Time

Closed loop step responses of y1 for A1 System (1985)

BLT

SIMC

ERGA

92



0 200 400 600 800 1000 1200 1400 1600-0.2

0

0.2

0.4

0.6

0.8

1

1.2

y2

Time


BLT

SIMC

ERGA

0 200 400 600 800 1000 1200 1400 1600-5

-4

-3

-2

-1

0

1

2

3

4

5

y3

Time


BLT

SIMC

ERGA

93


The final variable pairing and controller tuning should be checked for robustness by

plotting DSO and DSI as functions of frequency. Singular values below 0.3 0.2

indicate a lack of stability robustness. Further controller detuning may be required.

The structured singular value (denoted Mu, mu, SSV or ) is a function which

provides a generalization of the singular value and the spectral radius . We will

use to get necessary and sufficient conditions for robust stability and for robust

performance. It is defined as mathematically,

1( ) min ( ) det( ) 0 for structured M I M (4.90)

Clearly, ( )M depends not only on M but also on the allowed structure for . This is

sometimes shown explicitly by using the notation ( )M .

0 200 400 600 800 1000 1200 1400 1600-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

y4

Time


BLT

SIMC

ERGA

94

To evaluate the output control performance, we consider a unit step setpoint change

for each loop, and the integrated absolute error (IAE), the integrated square error (ISE)

and the integrated time absolute error (ITAE) of i i ie y r are used to evaluate the

control performance such that

0

iIAE e dt

(4.91)

2

0iISE e dt

(4.92)

and

0

i iITAE e t dt

(4.93)

respectively. The followings are Matlab codes to find IAE, ISE and ITAE for OR

system.

%*******************************************************************

% Performance Index *

%*******************************************************************

%IAE

for i=1:n

BLT_IAE(i)=trapz(blt_tout,abs(blt_error(:,i)));

SIMC_IAE(i)=trapz(simc_tout,abs(simc_error(:,i)));

ERGA_IAE(i)=trapz(erga_tout,abs(erga_error(:,i)));

end

disp('BLT_IAE1, SIMC_IAE1 and ERGA_IAE1 are ')

sprintf('\t%f\t%f\t\t%f',BLT_IAE(1),SIMC_IAE(1),ERGA_IAE(1))



%*******************************************************************

%ISE

for i=1:n

BLT_ISE(i)=trapz(blt_tout,(blt_error(:,i)).^2);

SIMC_ISE(i)=trapz(simc_tout,(simc_error(:,i)).^2);

ERGA_ISE(i)=trapz(erga_tout,(erga_error(:,i)).^2);

end

disp('BLT_ISE1, SIMC_ISE1 and ERGA_ISE1 are ')

sprintf('\t%f\t%f\t\t%f',BLT_ISE(1),SIMC_ISE(1),ERGA_ISE(1))



%*******************************************************************

%ITAE

for i=1:n

BLT_ITAE(i)=trapz(blt_tout,abs(blt_error(:,i)).*blt_tout);

SIMC_ITAE(i)=trapz(simc_tout,abs(simc_error(:,i)).*simc_tout);

ERGA_ITAE(i)=trapz(erga_tout,abs(erga_error(:,i)).*erga_tout);

end

disp('BLT_ITAE1, SIMC_ITAE1 and ERGA_ITAE1 are ')

sprintf('\t%f\t%f\t\t%f',BLT_ITAE(1),SIMC_ITAE(1),ERGA_ITAE(1))



%*******************************************************************

These codes will generate the contents of Table 5.10 to Table 5.12.

95

Methods Controller IAE ISE ITAE

BLT

Loop 1 6.700269 3.041136 113.736678

Loop 2 2.142164 0.890127 73.807713

SIMC

Loop 1 11.055941 4.879077 215.310918

Loop 2 2.104946 0.810928 64.872263

ERGA

Loop 1 4.141611 2.054463 65.335278

Loop 2 2.541929 0.827885 71.617611

Table 5.10 Performance Analysis for VL System (1972)


BLT

Loop 1 40.669637 11.913219 5207.841148

Loop 2 122.673663 47.662144 29961.832043

Loop 3 213.878712 1092.095839 29028.759469

SIMC

Loop 1 154.647127 71.142149 26730.952140

Loop 2 90.409888 24.020309 24398.624396

Loop 3 65.994708 19.914998 16407.995639

ERGA

Loop 1 31.766417 11.258064 3374.608357

Loop 2 70.727090 31.085399 14662.244667

Loop 3 350.391320 2003.041302 48476.558050

Table 5.11 Performance Analysis for OR System (1979)

96


BLT

Loop 1 127.399980 40.959631 52132.564747

Loop 2 9.684995 2.088477 4766.423058

Loop 3 135.942491 131.136045 65605.190341

Loop 4 130.146470 78.475350 61284.463291

SIMC

Loop 1 91.620452 36.804210 33937.547835

Loop 2 7.910612 2.182851 4095.297639

Loop 3 90.242356 95.401712 43643.197559

Loop 4 41.880308 12.395012 21448.368157

ERGA

Loop 1 59.343257 19.869040 20765.241418

Loop 2 15.798839 2.746638 8191.105350

Loop 3 102.073710 77.894160 44768.557853

Loop 4 146.755354 50.885016 57869.975745

Table 5.12 Performance Analysis for A1 System (1985)

97

6 Conclusion and Recommendation

6.1 Conclusion

The results show that the BLT settings give stable and reasonable responses. In all

cases, the BLT method converged very quickly to settings that worked. The

computation horsepower required to implement the basic method is quite nominal. The

only calculation required is the evaluation of the determinant of a N N matrix. No

matrix inversion or eigenvalue calculations are required. These calculations are well

within the capability of a personal computer.

The simplicity and effectiveness of the model is based on the energy transmission ratio

of each individual transfer function which provides necessary information of gain and

frequency changes when all other loops are closed. Consequently, the decentralized

controllers can be obtained by simply using single loop design approaches. Simulation

results for variety of industrial 2 2 , 3 3 and 4 4 processes show that ERGA

method provides overall better performance compared to other reported design

approaches. The advantage of ERGA method is even more significant when applied to

higher dimensional processes with complicated interaction modes. The method can be

easily integrated into an auto-tuning control structure combined with some on line

parameter identification module and implemented for industrial control systems.

dRI based Skogestad Internal Model Control (SIMC) method is a simple and effective

design method for decentralized PID controller design. On the basis of structure

decomposition, the dynamic relative interaction was defined and represented by the

process model and controller explicitly. An initial decentralized controller was

designed first by using the diagonal elements and then implemented to estimate the

dRI to individual control loop from all others.

According to the simulation results in chapter 5, we conclude that the three

decentralized control methods presented in this dissertation can be easily applied to

high dimensional processes with complicated interaction modes such as distillation

column.

98

6.2 Recommendation for further research

The SISO Design Tool is a graphical user interface (GUI) that allows you to analyze

and tune SISO feedback control systems. Using the SISO Design Tool, you can

graphically tune the gains and dynamics of the compensator (C) and prefilter (F) using

a mix of root locus and loop shaping techniques.

Figure 6.1 Screenshot of SISO Design Tool

There are many decentralized control methods in literature. They are shown in Chapter

4. We should create MIMO Design Tool for all pairing and decentralized control

methods. Better theory and design have to be established for multivariable PID control

to reach the same maturity and popularity as the single variable case. Figure 6.2 shows

the simple idea to compare these methods with each other.

99

Figure 6.2 Screenshot of Decentralized Control

100

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107

A Simulink Block Diagram for 2x2 Decentralized Control

System

Figure A.1 Simulink block diagram for 2x2 decentralized control system

108

Figure A.2 subsystem1

Figure A.3 Subsystem2

109

B Simulink Block Diagram for 3x3 Decentralized Control

System

Figure B.1 Simulink block diagram for 3x3 decentralized control system

110

C Simulink Block Diagram for 4x4 Decentralized Control

System

Figure C.1 Simulink block diagram for 4x4 decentralized control system

111

D Matlab Program for 2x2 Decentralized Control System

clear;

clc;

%*******************************************************************

%Transfer Function

disp('Vinante and Luyben System (1972)')

g11=tf(-2.2,[7 1],'InputDelay',1);

g12=tf(1.3,[7 1],'InputDelay',0.3);

g21=tf(-2.8,[9.5 1],'InputDelay',1.8);

g22=tf(4.3,[9.2 1],'InputDelay',0.35);

g=[g11 g12;g21 g22];

%*******************************************************************

% BLT *

%*******************************************************************

%Loop pairing

n=length(g);%Dimension of the plant

pair_opt=factorial(n);%No: of pairing options

index=perms([1:1:n]);%All possible permutations

[nrow ncol]=size(index);

k=dcgain(g);

for i=1:pair_opt

temp1=[];

temp2=[];

for j=1:ncol

temp2=[temp2 g(:,index(i,j))];

temp1=[temp1 k(:,index(i,j))];

end

G(:,:,i)=temp2;%All possible permutations for G

K(:,:,i)=temp1;%All possible permutations for K

end

%*******************************************************************

%Step 1&2: Calculate NI and RGA, and eliminate paris having negative

% NI and RGA values.

for i=1:pair_opt

temp3=det(K(:,:,i));

for j=1:n

temp3=temp3/K(j,j,i);

end

NI(:,:,i)=temp3;%All possible permutations for NI

end

temp4=G;

temp5=K;

for i=pair_opt:-1:1

if NI(:,:,i)<0

temp4(:,:,i)=[];%Delete ith pairing of G

G=temp4;

temp5(:,:,i)=[];%Delete ith pairing of K

K=temp5;

end

end

final=size(K,3);

for i=1:final

RGA(:,:,i)=K(:,:,i).*inv(K(:,:,i)');%Calculate RGA

end

temp6=G;

temp7=K;

temp8=RGA;

112

final=size(RGA,3);

for i=final:-1:1

if RGA(1,1,i)<0 | RGA(2,2,i)<0



temp8(:,:,i)=[];%Delete ith pairing of RGA

G=temp6;

K=temp7;

RGA=temp8;

end

end

final=size(K,2);

final1=size(K,3);

for i=1:final1

for j=1:final

K_dia(j,j,i)=diag(K(j,j,i));

end

end

final=size(K,3);

for i=1:final

E(:,:,i)=(K(:,:,i)-K_dia(:,:,i))*(K_dia(:,:,i))^-1;

E_tf=ss(E(:,:,i));

[ssv_value(i),non]=ssv(E_tf.a,E_tf.b,E_tf.c...

,E_tf.d,0);

end

[val_min,nmin]=min(ssv_value);%Minimum Structured Singular value

G=G(:,:,nmin);

K=K(:,:,nmin);

RGA=RGA(:,:,nmin);

%*******************************************************************

%Controller

%Step 1: Compute the Ziegler-Nichols tuning parameters of the

% diagonal elements of the process transfer function matrix.

for a=1:n

if sign(K(a,a))==1

S(a)=allmargin(G(a,a));

else

S(a)=allmargin(-G(a,a));

S(a).GainMargin=-S(a).GainMargin;

end

Ku(a)=S(a).GainMargin(1,1);

Wu(a)=S(a).GMFrequency(1,1);

end

for b=1:n

KcZN(b,b)=S(b).GainMargin(1,1)/2.2;

TauiZN(b,b)=2*pi/(1.2*S(b).GMFrequency(1,1));

end

p=sym('p');

[n,m]=size(G);

Gs=zeros(n,m);

Gs=sym(Gs);

for i=1:n

for j=1:m

[num,den]=tfdata(G(i,j),'v');

l=length(den);

order=l;

for k=1:l

A(order,1)=p^(k-1);

order=order-1;

end

n=num*A;

113

d=den*A;

Gs(i,j)=n/d;

clear A;

end

end

Gs=simple(Gs);

delays=exp(-totaldelay(G)*p);

Gs=Gs.*delays;%Transform transrer function into symbolic matrix.

%*******************************************************************

%Step 2: Choose a detuning factor.

F=2.3;%Initial value

df=0.01;%Increament

%*******************************************************************

%Step 3,4,5&6: Compute F, Kc and Taui.

n=length(G);

wrange=logspace(-1,0.8,200);

j=sqrt(-1);%Imaginary unit

s=j*wrange;

m=1;%Initialization

flagm=-1;%Initialization

flagp=-1;%Initialization

while m<10

Kc=KcZN/F;

Taui=TauiZN*F;

nwtot=length(wrange);

for w=1:nwtot

Gs_value=subs(Gs,s(w));

Gcs(1,1)=Kc(1,1)+Kc(1,1)/(Taui(1,1)*s(w));

Gcs(2,2)=Kc(2,2)+Kc(2,2)/(Taui(2,2)*s(w));

wnyquist(w)=-1+det(eye(size(Gs_value))+Gs_value*Gcs);

Lc(w)=wnyquist(w)/(1+wnyquist(w));%Calculate Lc function

dbcl(w)=20*log10(abs(Lc(w)));

end

[dbmax,nmax]=max(dbcl);%Pick off peak closed-loop log modulus

wmax=wrange(nmax);

m=m+1;

if dbmax>2*n%Test if +4 dB and regusee F factor

if flagp>0

df=df/2;

end

flagm=1;

F=F+df;

else

if flagm<0

df=df/2;

end

flagp=1;

F=F-df;

if F<1

F=1;

end

end

end%End of while loop to find correct F factor

F%Final value of F

Kp=Kc;

Ki=Kc./Taui;

n=length(G);%Dimension of the plant

for i=1:n

Gc(i,i)=tf([Kp(i,i) Ki(i,i)],[1 0]);

end

disp('**********The controller one for BLT **********')

114

Gc11=Gc(1,1)

disp('**********The controller two for BLT **********')

Gc22=Gc(2,2)

%*******************************************************************

tStart=0;

tFinal=100;

timespan=[tStart tFinal];

sim('blt2x2',timespan);%Simulation

blt_tout=tout;%From Simulink

blt_simout;%From Simulink

blt_error;%From Simulink

%*******************************************************************

% SIMC *

%*******************************************************************

%Loop pairing

%Step 1: Obtain steady state gain matrix.





k=dcgain(g);

for i=1:pair_opt

temp1=[];

temp2=[];

for j=1:ncol



end



end

%*******************************************************************



for i=1:pair_opt


for j=1:n


end


end

temp4=G;

temp5=K;

for i=pair_opt:-1:1

if NI(:,:,i)<0


G=temp4;


K=temp5;

end

end

final=size(K,3);

for i=1:final


end

temp6=G;

temp7=K;

temp8=RGA;

final=size(RGA,3);

for i=final:-1:1

if RGA(1,1,i)<0 | RGA(2,2,i)<0

115




G=temp6;

K=temp7;

RGA=temp8;

end

end

%*******************************************************************

%Step 4: Calculate delijG and ijPsi.

final=size(K,3);

for t=1:final

l=1;

temp9=K(:,:,t);

for i=1:n

for j=1:n

temp9(i,:)=[];

temp9(:,j)=[];

ijG(:,:,l,t)=temp9;

temp9=K(:,:,t);

l=l+1;

end

temp9=K(:,:,t);

end

end

final=size(K,3);

for t=1:final

l=1;

temp10=K(:,:,t);

for i=1:n

for j=1:n

row=temp10(i,:);

row(:,i)=[];

column=temp10(:,i);

column(i,:)=[];

delijG(:,:,l,t)=-(column*row)/temp10(i,j);

l=l+1;

end

end

end

final=size(ijG,4);

for t=1:final

temp11=ijG(:,:,:,t);

temp12=delijG(:,:,:,t);

final1=size(ijG,3);

for i=1:final1

ijPsi(:,:,i,t)=temp12(:,:,i).*inv(temp11(:,:,i)');

end

end

%*******************************************************************

%Step 5: Calculate Omega.

final3=size(ijG,4);

for t=1:final3

final4=size(ijG,3);

for j=1:final4

w(:,:,j,t)=norm(ijPsi(:,:,j,t),2);

end

end

final5=size(ijG,4);

for t=1:final5

Omega(:,:,t)=[w(:,:,1,t) w(:,:,2,t);w(:,:,3,t) w(:,:,4,t)];

116

dia(:,t)=diag(Omega(:,:,t));

pro(t)=prod(dia(:,t));

end

%*******************************************************************

%Step 6: Find the best pairing.

[value,index]=min(pro);

disp('G matrix for the best pairing is ');

G=G(:,:,index);

disp('K matrix for the best pairing is ');

K=K(:,:,index);

disp('RGA matrix for the best pairing is ');

RGA=RGA(:,:,index);

%*******************************************************************

%Controller

for i=1:n%Start of loop for Tau and Tau'

for j=1:n

temp13=pole(G(i,j));

if length(temp13)==2

Tau1(i,j)=-1/temp13(2);

Tau2(i,j)=-1/temp13(1);

else

Tau1(i,j)=-1/temp13(1);

Tau2(i,j)=0;

end

end

end%End of loop for Tau and Tau'

Theta=totaldelay(G);%Time delay matrix

p=sym('p');

[n,m]=size(G);

Gs=zeros(n,m);

Gs=sym(Gs);

for i=1:n

for j=1:m


l=length(den);

order=l;

for k=1:l

A(order,1)=p^(k-1);

order=order-1;

end

n=num*A;

d=den*A;

Gs(i,j)=n/d;

clear A;

end

end

Gs=simple(Gs);


Gs=Gs.*delays;%Transform transrer function into symbolic matrix

%*******************************************************************

%Step 1: Design an individual controller for the diagonal elements.

TauC=2*Theta%Assumption


for i=1:n%Start of loop for diagonal controler

kP(i,i)=(1/K(i,i))*(Tau1(i,i)/(TauC(i,i)+Theta(i,i)));

TauI(i,i)=min(Tau1(i,i),4*(TauC(i,i)+Theta(i,i)));

kI(i,i)=kP(i,i)/TauI(i,i);

TauD(i,i)=Tau2(i,i);

kD(i,i)=kP(i,i)*TauD(i,i);

C(i,i)=tf([kD(i,i) kP(i,i) kI(i,i)],[1 0]);

end%end of loop for diagonal controller

117

p=sym('p');

[n,m]=size(C);

Cs=zeros(n,m);

Cs=sym(Cs);

for i=1:n

for j=1:m

[num,den]=tfdata(C(i,j),'v');

l=length(den);

order=l;

for k=1:l

A(order,1)=p^(k-1);

order=order-1;

end

n=num*A;

d=den*A;

Cs(i,j)=n/d;

clear A;

end

end

Cs=simple(Cs);%Transform transrer function into symbolic matrix

%*******************************************************************

%Step 2: Calculate P.

n=length(G);

for i=1:n

Wc(i,i)=1/(TauC(i,i)+Theta(i,i));%Critical Frequency

end


s=j*Wc;

for i=1:n

for J=1:n

if i==J

P(i,J)=(TauC(i,J)*s(i,J)+1)*exp(Theta(i,J)*s(i,J));

else

P(i,J)=1;

end

end

end

gii=diag(Gs);

for i=1:n

temp=Gs(:,i);

temp(i,:)=[];

iigstari(:,:,i)=temp;

end

for i=1:n

temp=Gs(i,:);

temp(:,i)=[];

iigistar(:,:,i)=temp;

end

temp=Gs;

for i=1:n

for j=1:n

temp(i,:)=[];

temp(:,j)=[];

iiGs(:,:,i)=temp;

iiGs_value(:,:,i)=subs(iiGs(:,:,i),s(i,i));

temp=Gs;

end

end

for i=1:n

delGii(:,:,i)=-(1/Gs(i,i))*iigstari(:,:,i)*iigistar(:,:,i);

temp=subs(delGii(:,:,i),s(i,i));

118

delGii_value(:,:,i)=temp;

end

temp=P;

for i=1:n

for j=1:n

temp(i,:)=[];

temp(:,j)=[];

iiP(:,:,i)=temp;

temp=P;

end

end

%*******************************************************************

%Step 3&4: Calculate dRI and MMF.

for i=1:n

Phi(:,:,i)=delGii_value(:,:,i).*(iiGs_value(:,:,i).*

...iiP(:,:,i))';

end

for i=1:n

Phi_sum(i)=sum(sum(Phi(:,:,i)));

end

for i=1:n

krho(i)=abs(1+Phi_sum(i));

thetarho(i)=-angle(1+Phi_sum(i))/Wc(i,i);

end

%*******************************************************************

%Step 5: Obtain Equivalent Transfer Function.

for i=1:n

fk(i)=max(1,krho(i));

ftheta(i)=max(1,1+(thetarho(i)/Theta(i,i)));

end

for i=1:n

Gcap(i)=tf(fk(i)*K(i,i),[(Tau1(i,i)*Tau2(i,i)) (Tau1(i,i)...

+Tau2(i,i)) 1],'InputDelay',ftheta(i)*Theta(i,i));

end

%*******************************************************************

%Step 6: Fine tuning

for i=1:n

kP(i)=(1/fk(i))*(1/K(i,i))*(Tau1(i,i)/(TauC(i,i)+ftheta(i)...

*Theta(i,i)));

TauI(i)=min(Tau1(i,i),4*(TauC(i,i)+ftheta(i)*Theta(i,i)));

kI(i)=kP(i)/TauI(i);

TauD(i)=Tau2(i);

kD(i)=kP(i)*TauD(i);

gc(i,i)=tf([kD(i,i) kP(i,i) kI(i,i)],[1 0]);

end

disp('**********The controller one for SIMC **********')

gc11=gc(1,1)

disp('**********The controller two for SIMC **********')

gc22=gc(2,2)

%*******************************************************************

tStart=0;

tFinal=100;


sim('simc2x2',timespan)%Simulation

simc_tout=tout;%From Simulink

simc_simout;%From Simulink

simc_error;%From Simulink

%*******************************************************************

% ERGA *

%*******************************************************************

%Loop pairing

119




index=perms([1:1:n]);%%All possible permutations


k=dcgain(g);

for i=1:pair_opt

temp1=[];

temp2=[];

for j=1:ncol



end



end

%*******************************************************************

%Step 2&3: Calculate NI and ERGA, and eliminate paris having

% negative NI and ERGA values.

for i=1:pair_opt


for j=1:n


end


end

temp4=G;

temp5=K;

for i=pair_opt:-1:1

if NI(:,:,i)<0


G=temp4;


K=temp5;

end

end

final=size(K,3);

for t=1:final

for i=1:n

for j=1:n

zeroG(i,j,t)=G(i,j,t)/K(i,j,t);

S=allmargin(zeroG(i,j,t));

Wu(i,j,t)=S.GMFrequency(1,1);%Critical frequency

end

end

end

final=size(K,3);

for i=1:final


end

temp6=K;

temp7=Wu;

final=size(K,3);

for t=1:final

E(:,:,t)=temp6(:,:,t).*temp7(:,:,t);%Transmission Energy Array

end

final=size(K,3);

for i=1:final

ERGA(:,:,i)=E(:,:,i).*inv(E(:,:,i)');%Effect Relative Gain Array

end

temp8=G;

120

temp9=K;

temp10=RGA;

temp11=ERGA;

final=size(ERGA,3);

for i=final:-1:1

if ERGA(1,1,i)<0 | ERGA(2,2,i)<0




temp11(:,:,i)=[];%Delete ith pairing of ERGA

G=temp8;

K=temp9;

RGA=temp10;

ERGA=temp11;

end

end

disp('G matrix for the best pairing is ')

G;

disp('K matrix for the best pairing is ')

K;

disp('RGA matrix for the best pairing is ')

RGA;

disp('ERGA matrix for the best pairing is ')

ERGA;

%*******************************************************************

%Controller

L=totaldelay(G);%Time delay matrix

for i=1:n%Start of loop for a2 and a1

for j=1:n

[nr nc]=size(G(i,j).den{1});

if nc==3

a2(i,j)=G(i,j).den{1}(1);

a1(i,j)=G(i,j).den{1}(2);

elseif nc==2

a2(i,j)=0;

a1(i,j)=G(i,j).den{1}(1);

end

end

end%End of loop for a2 and a1

%*******************************************************************

%Step 1: Calculate Ralative Ferquency Array.

RFA=ERGA./RGA;

%*******************************************************************

%Step 2: Assume Gain Margin.

Am=5%Assumption

%*******************************************************************

%Step 3: Obtain Effective Transfer Function.

for i=1:n

if (RGA(i,i)<1 & RFA(i,i)<=1)

gnew(i,i)=tf((K(i,i)/RGA(i,i)),[a2(i,i) a1(i,i) 1],...

'InputDelay',L(i,i));

kp(i,i)=(pi*a1(i,i))/(2*Am*L(i,i)*K(i,i)/RGA(i,i));

ki(i,i)=pi/(2*Am*L(i,i)*K(i,i)/RGA(i,i));

kd(i,i)=(pi*a2(i,i))/(2*Am*L(i,i)*K(i,i)/RGA(i,i));

elseif (RGA(i,i)<1 & RFA(i,i)>1)


'InputDelay' ,RFA(i,i)*L(i,i));

kp(i,i)=(pi*a1(i,i))/(2*Am*RFA(i,i)*L(i,i)*K(i,i)/RGA(i,i));

ki(i,i)=pi/(2*Am*RFA(i,i)*L(i,i)*K(i,i)/RGA(i,i));

kd(i,i)=(pi*a2(i,i))/(2*Am*RFA(i,i)*L(i,i)*K(i,i)/RGA(i,i));

elseif (RGA(i,i)>=1 & RFA(i,i)<=1)

121

gnew(i,i)=tf(K(i,i),[a2(i,i) a1(i,i) 1],...


kp(i,i)=(pi*a1(i,i))/(2*Am*L(i,i)*K(i,i));

ki(i,i)=pi/(2*Am*L(i,i)*K(i,i));

kd(i,i)=(pi*a2(i,i))/(2*Am*L(i,i)*K(i,i));

elseif (RGA(i,i)>=1 & RFA(i,i)>1)

gnew(i,i)=tf((K(i,i)),[a2(i,i) a1(i,i) 1],...

'InputDelay' ,(RFA(i,i)*L(i,i)));

kp(i,i)=(pi*a1(i,i))/(2*Am*RFA(i,i)*L(i,i)*K(i,i));

ki(i,i)=pi/(2*Am*RFA(i,i)*L(i,i)*K(i,i));

kd(i,i)=(pi*a2(i,i))/(2*Am*RFA(i,i)*L(i,i)*K(i,i));

end

end

for i=1:n

gnew(i,i);

gc(i,i)=tf([kd(i,i) kp(i,i) ki(i,i)],[1 0]);

end

disp('**********The controller one for ERGA**********')

gc11=gc(1,1)

disp('**********The controller two for ERGA**********')

gc22=gc(2,2)

%*******************************************************************

tStart=0;

tFinal=100;


sim('erga2x2',timespan)%Simulation

erga_tout=tout;%From Simulink

erga_simout;%From Simulink

erga_error;%From Simulink

%*******************************************************************

% Simulation *

%*******************************************************************

figure;

plot(blt_tout,blt_simout(:,1), 'b-',...

simc_tout,simc_simout(:,1), 'g:',...

erga_tout,erga_simout(:,1), 'r-.' ,'LineWidth',1.5);

ylabel('y1');

xlabel('Time');

grid on;

title('Closed loop step responses of y1 for VL System (1972)');

legend('BLT','SIMC','ERGA',4);

figure;

plot(blt_tout,blt_simout(:,2),...

simc_tout,simc_simout(:,2),...

erga_tout,erga_simout(:,2),'LineWidth',1.5);

ylabel('y2');

xlabel('Time');

grid on;

title('Closed loop step responses of y2 for VL System (1972)');


%*******************************************************************


%*******************************************************************

%IAE

for i=1:n




end



122



%*******************************************************************

%ISE

for i=1:n




end





%*******************************************************************

%ITAE

for i=1:n




end





%*******************************************************************

123

E Matlab Program for 3x3 Decentralized Control System

clear;

clc;

%*******************************************************************

%Transfer Function

disp('Ogunnaike and Ray System (1979)')

g11=tf(0.66,[6.7 1],'InputDelay',2.6);

g12=tf(-0.61,[8.64 1],'InputDelay',3.5);

g13=tf(-0.0049,[9.06 1],'InputDelay',1);

g21=tf(1.11,[3.25 1],'InputDelay',6.5);

g22=tf(-2.36,[5 1],'InputDelay',3);

g23=tf(-0.01,[7.09 1],'InputDelay',1.2);

g31=tf(-34.68,[8.15 1],'InputDelay',9.2);

g32=tf(46.2,[10.9 1],'InputDelay',9.4);

g33=tf([0.87*11.61 0.87],[3.89*18.8 (3.89+18.8) 1]...

,'InputDelay',1);

g=[g11 g12 g13;g21 g22 g23;g31 g32 g33];

%*******************************************************************

% BLT *

%*******************************************************************

%Loop pairing




[r c]=size(index);

k=dcgain(g);

for i=1:pair_opt

temp1=[];

temp2=[];

for j=1:c



end



end

%%******************************************************************



for i=1:pair_opt


for j=1:n


end


end

temp4=G;

temp5=K;

for i=pair_opt:-1:1

if NI(:,:,i)<0


G=temp4;


K=temp5;

end

end

final=size(K,3);

124

for i=1:final


end

temp6=G;

temp7=K;

temp8=RGA;

final=size(RGA,3);

for i=final:-1:1

if RGA(1,1,i)<0 | RGA(2,2,i)<0 | RGA(3,3,i)<0




G=temp6;

K=temp7;

RGA=temp8;

end

end

final=size(K,2);

final1=size(K,3);

for i=1:final1

for j=1:final


end

end

final=size(K,3);

for i=1:final

E(:,:,i)=(K(:,:,i)-K_dia(:,:,i))*(K_dia(:,:,i))^-1;

E_tf=ss(E(:,:,i));


,E_tf.d,0);

end


G=G(:,:,nmin);

K=K(:,:,nmin);

RGA=RGA(:,:,nmin);

%*******************************************************************

%Controller



for a=1:n

if sign(K(a,a))==1


else



end

end

for b=1:n



TaudZN(b,b)=0.125*2*pi/S(b).GMFrequency(1,1);

end

p=sym('p');

[n,m]=size(G);

Gs=zeros(n,m);

Gs=sym(Gs);

for i=1:n

for j=1:m


l=length(den);

order=l;

125

for k=1:l

A(order,1)=p^(k-1);

order=order-1;

end

n=num*A;

d=den*A;

Gs(i,j)=n/d;

clear A;

end

end

Gs=simple(Gs);



%*******************************************************************



Fd=1;%%Initial value

df=0.01;%Increament

dfd=0.01;%Increament

%*******************************************************************

%Step 3,4,5&6: Compute F, Fd, Kc and Taui.

n=length(G);



s=j*wrange;

m=1;%Initialization



while m<10

Kc=KcZN/F;

Taui=TauiZN*F;

Taud=TaudZN/Fd;


for w=1:nwtot


Gcs(1,1)=Kc(1,1)+Kc(1,1)/(Taui(1,1)*s(w))...

+Kc(1,1)*Taud(1,1)*s(w);

Gcs(2,2)=Kc(2,2)+Kc(2,2)/(Taui(2,2)*s(w))...

+Kc(2,2)*Taud(2,2)*s(w);

Gcs(3,3)=Kc(3,3)+Kc(3,3)/(Taui(3,3)*s(w))...

+Kc(3,3)*Taud(3,3)*s(w);




end


wmax=wrange(nmax);

m=m+1;

if dbmax>2*n%Test if +6 dB and regusee F and Fd factors

if flagp>0

df=df/2;

dfd=dfd/2;

end

flagm=1;

F=F+df;

Fd=Fd+dfd;

else

if flagm<0

df=df/2;

end

flagp=1;

126

F=F-df;

if F<1

F=1;

end

if Fd<1

Fd=1;

end

end

end%End of while loop to find correct F and Fd factor

F%Final value of F

Fd%Final value of Fd

Kp=Kc;

Ki=Kc./Taui;

Kd=Kc.*Taud;


for i=1:n

Gc(i,i)=tf([Kd(i,i) Kp(i,i) Ki(i,i)],[1 0]);

end


Gc11=Gc(1,1)


Gc22=Gc(2,2)

disp('**********The controller three for BLT **********')

Gc33=Gc(3,3)

%*******************************************************************

tStart=0;

tFinal=1200;


sim('blt3x3',timespan)%Simulation




%*******************************************************************

% SIMC *

%*******************************************************************


g33=tf([0.87*11.61 0.87],[3.89*18.8 (3.89+18.8) 1]...

,'InputDelay',1);

sys=g33;




g33=tf(rsys);

t=0:1:10000;

[y t]=step(g33,t);








g33new;

figure;

[y t x]=step(g33);




xlabel('Time');

grid on;

title('Step responses for g33 and g33new for OR System (1979)');

127


%*******************************************************************

g33=g33new;

g=[g11 g12 g13;g21 g22 g23;g31 g32 g33];

%*******************************************************************

%Loop pairing





[r c]=size(index);

k=dcgain(g);

for i=1:pair_opt

temp1=[];

temp2=[];

for j=1:c



end



end

%*******************************************************************



for i=1:pair_opt


for j=1:n


end


end

temp4=K;

temp5=G;

for i=pair_opt:-1:1

if NI(:,:,i)<0


K=temp4;


G=temp5;

end

end

final=size(K,3);

for i=1:final


end

temp6=G;

temp7=K;

temp8=RGA;

final=size(RGA,3);

for i=final:-1:1

if RGA(1,1,i)<0 | RGA(2,2,i)<0




G=temp6;

K=temp7;

RGA=temp8;

end

end

128

%********************************************************************

**


final=size(K,3);

for t=1:final

l=1;

temp9=K(:,:,t);

for i=1:n

for j=1:n

temp9(i,:)=[];

temp9(:,j)=[];

ijG(:,:,l,t)=temp9;

temp9=K(:,:,t);

l=l+1;

end

temp9=K(:,:,t);

end

end

final=size(K,3);

for t=1:final

l=1;

temp10=K(:,:,t);

for i=1:n

for j=1:n

row=temp10(i,:);

row(:,i)=[];

column=temp10(:,i);

column(i,:)=[];


l=l+1;

end

end

end

final=size(ijG,4);

for t=1:final



final1=size(ijG,3);

for i=1:final1


end

end

%*******************************************************************


final3=size(ijG,4);

for t=1:final3

final4=size(ijG,3);

for j=1:final4


end

end

final5=size(ijG,4);

for t=1:final5

Omega(:,:,t)=[w(:,:,1,t) w(:,:,2,t) w(:,:,3,t);

w(:,:,4,t) w(:,:,5,t) w(:,:,6,t);

w(:,:,7,t) w(:,:,8,t) w(:,:,9,t)];



end

%*******************************************************************


129



G=G(:,:,index);


K=K(:,:,index);


RGA=RGA(:,:,index);

%********************************************************************

**

%Controller


for j=1:n

temp=pole(G(i,j));

if length(temp)==2

Tau1(i,j)=-1/temp(2);


else


Tau2(i,j)=0;

end

end


Theta=totaldelay(G);

p=sym('p');

[n,m]=size(G);

Gs=zeros(n,m);

Gs=sym(Gs);

for i=1:n

for j=1:m


l=length(den);

order=l;

for k=1:l

A(order,1)=p^(k-1);

order=order-1;

end

n=num*A;

d=den*A;

Gs(i,j)=n/d;

clear A;

end

end

Gs=simple(Gs);



%*******************************************************************


TauC=0.5*Theta;%Assumption










p=sym('p');

[n,m]=size(C);

Cs=zeros(n,m);

Cs=sym(Cs);

130

for i=1:n

for j=1:m


l=length(den);

order=l;

for k=1:l

A(order,1)=p^(k-1);

order=order-1;

end

n=num*A;

d=den*A;

Cs(i,j)=n/d;

clear A;

end

end


%*******************************************************************


n=length(G);

for i=1:n

Wc(i,i)=1/(TauC(i,i)+Theta(i,i));

end


s=j*Wc;

for i=1:n

for J=1:n

if i==J


else

P(i,J)=1;

end

end

end

gii=diag(Gs);

for i=1:n

temp=Gs(:,i);

temp(i,:)=[];


end

for i=1:n

temp=Gs(i,:);

temp(:,i)=[];


end

temp=Gs;

for i=1:n

for j=1:n

temp(i,:)=[];

temp(:,j)=[];

iiGs(:,:,i)=temp;


temp=Gs;

end

end

for i=1:n




end

temp=P;

for i=1:n

131

temp(i,:)=[];

temp(:,i)=[];

iiP(:,:,i)=temp;

temp=P;

end

%*******************************************************************


for i=1:n

Phi(:,:,i)=delGii_value(:,:,i).*(iiGs_value(:,:,i).*...

iiP(:,:,i))';

end

for i=1:n

Phi_sum(i)=sum(sum(abs(Phi(:,:,i))));

end

for i=1:n



end

%*******************************************************************


for i=1:n



end

for i=1:n

Gcap(i)=tf(fk(i)*K(i,i),[(Tau1(i,i)*Tau2(i,i)) (Tau1(i,i)...

+Tau2(i,i)) 1],'InputDelay',ftheta(i)*Theta(i,i));

end

%*******************************************************************


for i=1:n

kP(i,i)=(1/fk(i))*(1/K(i,i))*(Tau1(i,i)/(TauC(i,i)+ftheta(i)...

*Theta(i,i)));

TauI(i,i)=min(Tau1(i,i),4*(TauC(i,i)+ftheta(i)*Theta(i,i)));





end


gc11=gc(1,1)


gc22=gc(2,2)

disp('**********The controller three for SIMC **********')

gc33=gc(3,3)

%*******************************************************************

tStart=0;

tFinal=1200;






%*******************************************************************

% ERGA *

%*******************************************************************


g33=tf([0.87*11.61 0.87],[3.89*18.8 (3.89+18.8) 1]...

,'InputDelay',1);

sys=g33;


132



g33=tf(rsys);

t=0:1:10000;

[y t]=step(g33,t);








g33new;

figure;

[y t x]=step(g33);




xlabel('Time');

grid on;

title('Step responses for g33 and g33new for OR System');


%*******************************************************************

g33=g33new;

g=[g11 g12 g13;g21 g22 g23;g31 g32 g33];

%*******************************************************************

%Loop pairing






k=dcgain(g);

for i=1:pair_opt

temp1=[];

temp2=[];

for j=1:ncol



end



end

%*******************************************************************



for i=1:pair_opt


for j=1:n


end


end

temp4=K;

temp5=G;

for i=pair_opt:-1:1

if NI(:,:,i)<0


K=temp4;


G=temp5;

133

end

end

final=size(K,3);

for t=1:final

for i=1:n

for j=1:n

zeroG(i,j,t)=G(i,j,t)/K(i,j,t);

S=allmargin(zeroG(i,j,t));

Wu(i,j,t)=S.GMFrequency(1,1);%Critical frequency

end

end

end

final=size(K,3);

for i=1:final


end

temp6=K;

temp7=Wu;

final=size(K,3);

for t=1:final

E(:,:,t)=temp6(:,:,t).*temp7(:,:,t);%Transmission Energy Array

end

final=size(K,3);

for i=1:final


end

temp8=G;

temp9=K;

temp10=RGA;

temp11=ERGA;

final=size(ERGA,3);

for i=final:-1:1

if ERGA(1,1,i)<0 | ERGA(2,2,i)<0 | ERGA(3,3,i)<0




temp11(:,:,i)=[];%Delete ith pairing of ERGA

G=temp8;

K=temp9;

RGA=temp10;

ERGA=temp11;

end

end


G;


K;


RGA;


ERGA;

%*******************************************************************

%Controller



for j=1:n


if nc==3

a2(i,j)=G(i,j).den{1}(1);

a1(i,j)=G(i,j).den{1}(2);

elseif nc==2

134

a2(i,j)=0;

a1(i,j)=G(i,j).den{1}(1);

end

end


%*******************************************************************

%Step : Calculate Ralative Ferquency Array.

RFA=ERGA./RGA;

%*******************************************************************


Am=8%Assumption

%*******************************************************************


for i=1:n










kp(i,i)=(pi*a1(i,i))/(2*Am*RFA(i,i)*L(i,i)*...

K(i,i)/RGA(i,i));

ki(i,i)=pi/(2*Am*RFA(i,i)*L(i,i)*K(i,i)...

/RGA(i,i));

kd(i,i)=(pi*a2(i,i))/(2*Am*RFA(i,i)*L(i,i)*...

K(i,i)/RGA(i,i));


gnew(i,i)=tf(K(i,i),[a2(i,i) a1(i,i) 1],...








kp(i,i)=(pi*a1(i,i))/(2*Am*RFA(i,i)*L(i,i)...

*K(i,i));


kd(i,i)=(pi*a2(i,i))/(2*Am*RFA(i,i)*L(i,i)...

*K(i,i));

end

end

for i=1:n

gnew(i,i);


end


gc11=gc(1,1)


gc22=gc(2,2)


gc33=gc(3,3)

%*******************************************************************

tStart=0;

tFinal=1200;




135



%*******************************************************************

% Simulation *

%*******************************************************************

figure;

plot(blt_tout,blt_simout(:,1),'b-',...

simc_tout,simc_simout(:,1),'g:',...

erga_tout,erga_simout(:,1),'r-.','LineWidth',1.5);

ylabel('y1');

xlabel('Time');

grid on;

title('Step responses of y1 for OR System (1979)');


figure;




ylabel('y2');

xlabel('Time');

grid on;



figure;




ylabel('y3');

xlabel('Time');

grid on;



%*******************************************************************


%*******************************************************************

%IAE

for i=1:n




end







%*******************************************************************

%ISE

for i=1:n




end







%*******************************************************************

136

%ITAE

for i=1:n




end







%*******************************************************************

137

F Matlab Program for 4x4 Decentralized Control System

clear;

clc;

%*******************************************************************

%Transfer Function

disp('Alatiqi case 1 System (1985)')

g11=tf(2.22,[(36*25) (36+25) 1],'InputDelay',2.5);

g12=tf([(-2.94*7.9) -2.94],[(23.7*23.7) (23.7+23.7) 1],...

'InputDelay',0.05);

g13=tf(0.017,[(31.6*7) (31.6+7) 1],'InputDelay',0.2);

g14=tf(-0.64,[(29*29) (29+29) 1],'InputDelay',20);

g21=tf(-2.33,[(35*35) (35+35) 1],'InputDelay',5);

g22=tf(3.46,[32 1],'InputDelay',1.01);

g23=tf(-0.51,[(32*32) (32+32) 1],'InputDelay',7.5);

g24=tf(1.68,[(28*28) (28+28) 1],'InputDelay',2);

g31=tf(-1.06,[(17*17) (17+17) 1],'InputDelay',22);

g32=tf(3.511,[(12*12) (12+12) 1],'InputDelay',13);

g33=tf(4.41,[16.2 1],'InputDelay',1.01);

g34=tf(-5.38,[17 1],'InputDelay',0.5);

g41=tf(-5.73,[(8*50) (8+50) 1],'InputDelay',2.5);

g42=tf([(4.32*25) 4.32],[(50*5) (50+5) 1],'InputDelay',0.01);

g43=tf(-1.25,[(43.6*9) (43.6+9) 1],'InputDelay',2.8);

g44=tf(4.78,[(48*5) (48+5) 1],'InputDelay',1.15);

g=[g11 g12 g13 g14;g21 g22 g23 g24;g31 g32 g33 g34;g41 g42 g43 g44];

%*******************************************************************

% BLT *

%*******************************************************************

%Loop pairing




[r c]=size(index);

k=dcgain(g);

for i=1:pair_opt

temp1=[];

temp2=[];

for j=1:c



end



end

%*******************************************************************



for i=1:pair_opt


for j=1:n


end


end

temp4=K;

temp5=G;

for i=pair_opt:-1:1

if NI(:,:,i)<0

138


K=temp4;


G=temp5;

end

end

final=size(K,3);

for i=1:final


end

temp6=G;

temp7=K;

temp8=RGA;

final=size(RGA,3);

for i=final:-1:1

if RGA(1,1,i)<0 | RGA(2,2,i)<0 | RGA(3,3,i)<0 | RGA(4,4,i)<0




G=temp6;

K=temp7;

RGA=temp8;

end

end

final=size(K,2);

final1=size(K,3);

for i=1:final1

for j=1:final


end

end

final=size(K,3);

for i=1:final

E(:,:,i)=(K(:,:,i)-K_dia(:,:,i))*(K_dia(:,:,i))^-1;

E_tf=ss(E(:,:,i));


,E_tf.d,0);

end


G=G(:,:,nmin);

K=K(:,:,nmin);

RGA=RGA(:,:,nmin);

%*******************************************************************

%Controller



for a=1:n

if sign(K(a,a))==1


else



end

end

for b=1:n



TaudZN(b,b)=0.125*2*pi/S(b).GMFrequency(1,1);

end

p=sym('p');

[n,m]=size(G);

139

Gs=zeros(n,m);

Gs=sym(Gs);

for i=1:n

for j=1:m


l=length(den);

order=l;

for k=1:l

A(order,1)=p^(k-1);

order=order-1;

end

n=num*A;

d=den*A;

Gs(i,j)=n/d;

clear A;

end

end

Gs=simple(Gs);



%*******************************************************************



Fd=1;%%Initial value

df=0.01;%Increament

dfd=0.01;%Increament

%*******************************************************************

%Step 3,4,5&6: Compute F, Fd, Kc and Taui.

n=length(G);



s=j*wrange;

m=1;%Initialization



while m<10

Kc=KcZN/F;

Taui=TauiZN*F;

Taud=TaudZN/Fd;


for w=1:nwtot


Gcs(1,1)=Kc(1,1)+Kc(1,1)/(Taui(1,1)*s(w))+Kc(1,1)...

*Taud(1,1)*s(w);


*Taud(2,2)*s(w);


*Taud(3,3)*s(w);


*Taud(4,4)*s(w);




end


wmax=wrange(nmax);

m=m+1;

if dbmax>2*n%Test if +8 dB and regusee F and Fd factors

if flagp>0

df=df/2;

dfd=dfd/2;

140

end

flagm=1;

F=F+df;

Fd=Fd+dfd;

else

if flagm<0

df=df/2;

end

flagp=1;

F=F-df;

if F<1

F=1;

end

if Fd<1

Fd=1;

end

end

end%End of while loop to find correct F and Fd factors

F%Final value of F

Fd%Final value of Fd

Kp=Kc;

Ki=Kc./Taui;

Kd=Kc.*Taud;


for i=1:n

Gc(i,i)=tf([Kd(i,i) Kp(i,i) Ki(i,i)],[1 0]);

end


Gc11=Gc(1,1)


Gc22=Gc(2,2)

disp('**********The controller three for BLT **********')

Gc33=Gc(3,3)

%*******************************************************************

tStart=0;

tFinal=1600;


sim('blt4x4',timespan)%Simulation




%*******************************************************************

% SIMC *

%*******************************************************************


g12=tf([(-2.94*7.9) -2.94],[(23.7*23.7) (23.7+23.7) 1],...

'InputDelay',0.05);

sys=g12;




g12=tf(rsys);

t=0:1:1000;

[y t]=step(g12,t);







g12new;

141

figure;

[y t x]=step(g12);




xlabel('Time');

grid on;



%*******************************************************************


g42=tf([(4.32*25) 4.32],[(50*5) (50+5) 1],'InputDelay',0.01);

sys=g42;




g42=tf(rsys);

t=0:1:1000;

[y t]=step(g42,t);


m42 =pem(data12,'p1d','kp',4.32, 'kp','fix','td' ,0.01,...


20});% fixing the parameter kp to the value 4.32



g42new;

figure;

[y t x]=step(g42);




xlabel('Time');

grid on;



%*******************************************************************

g12=g12new;

g42=g42new;


%*******************************************************************

%Loop pairing





[r c]=size(index);

k=dcgain(g);

for i=1:pair_opt

temp1=[];

temp2=[];

for j=1:c



end



end

%*******************************************************************

%Step 2&3: Calculate RGA and NI and eliminate paris having negative

% RGA and NI values.

for i=1:pair_opt

142


for j=1:n


end


end

temp4=K;

temp5=G;

for i=pair_opt:-1:1

if NI(:,:,i)<0


K=temp4;


G=temp5;

end

end

final=size(K,3);

for i=1:final


end

temp6=G;

temp7=K;

temp8=RGA;

final=size(RGA,3);

for i=final:-1:1

if RGA(1,1,i)<0 | RGA(2,2,i)<0




G=temp6;

K=temp7;

RGA=temp8;

end

end

%*******************************************************************


final=size(K,3);

for t=1:final

l=1;

temp9=K(:,:,t);

for i=1:n

for j=1:n

temp9(i,:)=[];

temp9(:,j)=[];

ijG(:,:,l,t)=temp9;

temp9=K(:,:,t);

l=l+1;

end

temp9=K(:,:,t);

end

end

final=size(K,3);

for t=1:final

l=1;

temp10=K(:,:,t);

for i=1:n

for j=1:n

row=temp10(i,:);

row(:,i)=[];

column=temp10(:,i);

column(i,:)=[];

143


l=l+1;

end

end

end

final=size(ijG,4);

for t=1:final



final1=size(ijG,3);

for i=1:final1


end

end

%*******************************************************************


final3=size(ijG,4);

for t=1:final3

final4=size(ijG,3);

for j=1:final4


end

end

final5=size(ijG,4);

for t=1:final5

Omega(:,:,t)=[w(:,:,1,t) w(:,:,2,t) w(:,:,3,t) w(:,:,4,t);

w(:,:,5,t) w(:,:,6,t) w(:,:,7,t) w(:,:,8,t);

w(:,:,9,t) w(:,:,10,t) w(:,:,11,t) w(:,:,12,t);

w(:,:,13,t) w(:,:,14,t) w(:,:,15,t) w(:,:,16,t)];



end

%*******************************************************************




G=G(:,:,index);


K=K(:,:,index);


RGA=RGA(:,:,index);

%*******************************************************************

%Controller


for j=1:n

temp=pole(G(i,j));

if length(temp)==2



else


Tau2(i,j)=0;

end

end


Theta=totaldelay(G);

p=sym('p');

[n,m]=size(G);

Gs=zeros(n,m);

Gs=sym(Gs);

for i=1:n

144

for j=1:m


l=length(den);

order=l;

for k=1:l

A(order,1)=p^(k-1);

order=order-1;

end

n=num*A;

d=den*A;

Gs(i,j)=n/d;

clear A;

end

end

Gs=simple(Gs);



%*******************************************************************


TauC=2*Theta;%Assumption










p=sym('p');

[n,m]=size(C);

Cs=zeros(n,m);

Cs=sym(Cs);

for i=1:n

for j=1:m


l=length(den);

order=l;

for k=1:l

A(order,1)=p^(k-1);

order=order-1;

end

n=num*A;

d=den*A;

Cs(i,j)=n/d;

clear A;

end

end


%*******************************************************************


n=length(G);

for i=1:n

Wc(i,i)=1/(TauC(i,i)+Theta(i,i));

end


s=j*Wc;

for i=1:n

for J=1:n

if i==J


145

else

P(i,J)=1;

end

end

end

gii=diag(Gs);

for i=1:n

temp=Gs(:,i);

temp(i,:)=[];


end

for i=1:n

temp=Gs(i,:);

temp(:,i)=[];


end

temp=Gs;

for i=1:n

for j=1:n

temp(i,:)=[];

temp(:,j)=[];

iiGs(:,:,i)=temp;


temp=Gs;

end

end

for i=1:n




end

temp=P;

for i=1:n

temp(i,:)=[];

temp(:,i)=[];

iiP(:,:,i)=temp;

temp=P;

end

%*******************************************************************


for i=1:n

Phi(:,:,i)=delGii_value(:,:,i).*(iiGs_value(:,:,i).*....

iiP(:,:,i))';

end

for i=1:n

Phi_sum(i)=sum(sum(abs(Phi(:,:,i))));

end

for i=1:n



end

%*******************************************************************


for i=1:n



end

for i=1:n

Gcap(i)=tf(fk(i)*K(i,i),[(Tau1(i,i)...

*Tau2(i,i)) (Tau1(i,i)...

+Tau2(i,i)) 1],'InputDelay',...

146

ftheta(i)*Theta(i,i));

end

%*******************************************************************


for i=1:n

kP(i,i)=(1/fk(i))*(1/K(i,i))*(Tau1(i,i)/...

(TauC(i,i)+ftheta(i)...

*Theta(i,i)));

TauI(i,i)=min(Tau1(i,i),4*(TauC(i,i)+...

ftheta(i)*Theta(i,i)));





end


gc11=gc(1,1)


gc22=gc(2,2)


gc33=gc(3,3)


gc44=gc(4,4)

%*******************************************************************

tStart=0;

tFinal=1600;






%*******************************************************************

% ERGA *

%*******************************************************************


g12=tf([(-2.94*7.9) -2.94],[(23.7*23.7) (23.7+23.7) 1],...

'InputDelay',0.05);

sys=g12;




g12=tf(rsys);

t=0:1:1000;

[y t]=step(g12,t);








g12new;

figure;

[y t x]=step(g12);




xlabel('Time');

grid on;

title('Step responses for g12 and g12new for A1')


147

%*******************************************************************


g42=tf([(4.32*25) 4.32],[(50*5) (50+5) 1],'InputDelay',0.01);

sys=g42;




g42=tf(rsys);

t=0:1:1000;

[y t]=step(g42,t);


m42 =pem(data12,'p1d','kp',4.32, 'kp','fix','td' ,0.01,...


20});% fixing the parameter kp to the value 4.32




g42new;

figure;

[y t x]=step(g42);




xlabel('Time');

grid on;

title('Step responses for g42 and g42new for A1')


%*******************************************************************

g12=g12new;

g42=g42new;


%*******************************************************************

%Loop pairing






k=dcgain(g);

for i=1:pair_opt

temp1=[];

temp2=[];

for j=1:ncol



end



end

%*******************************************************************



for i=1:pair_opt


for j=1:n


end


end

temp4=K;

temp5=G;

148

for i=pair_opt:-1:1

if NI(:,:,i)<0


K=temp4;


G=temp5;

end

end

final=size(K,3);

for t=1:final

for i=1:n

for j=1:n

%zeroG(i,j,t)=G(i,j,t)/K(i,j,t);

%S=allmargin(zeroG(i,j,t));

%Wu(i,j,t)=S.GMFrequency(1,1);%Critical frequency

Wu(i,j,t)=bandwidth(G(i,j,t));

end

end

end

final=size(K,3);

for i=1:final


end

temp6=K;

temp7=Wu;

final=size(K,3);

for t=1:final

E(:,:,t)=temp6(:,:,t).*temp7(:,:,t);

end

for i=1:final


end

final=size(K,3);

temp8=G;

temp9=K;

temp10=RGA;

temp11=ERGA;

final=size(RGA,3);

count=0;

for i=final:-1:1

if RGA(1,1,i)<0 | RGA(2,2,i)<0 | RGA(3,3,i)<0 | RGA(4,4,i)<0

discard(i)=i;

else

temp8=G(:,:,i);

temp9=K(:,:,i);

temp10=RGA(:,:,i);

temp11=ERGA(:,:,i);

keep(i)=i;

count=count+1;

G1(:,:,count)=temp8;

K1(:,:,count)=temp9;

RGA1(:,:,count)=temp10;

ERGA1(:,:,count)=temp11;

end

end

temp12=G1;

temp13=K1;

temp14=RGA1;

temp15=ERGA1;

final=size(ERGA1,3);

count=0;

149

for i=final:-1:1

if ERGA1(1,1,i)<0 | ERGA1(2,2,i)<0 | ...

ERGA1(3,3,i)<0 | ERGA1(4,4,i)<0

discard(i)=i;

else

temp12=G1(:,:,i);

temp13=K1(:,:,i);

temp14=RGA1(:,:,i);

temp15=ERGA1(:,:,i);

keep(i)=i;

count=count+1;

G2(:,:,count)=temp12;

K2(:,:,count)=temp13;

RGA2(:,:,count)=temp14;

ERGA2(:,:,count)=temp15;

end

end

G=G2;

K=K2;

RGA=RGA2;

ERGA=ERGA2;

temp16=[];

final=size(ERGA,3);

for i=1:final

temp16=[temp16 diag(ERGA(:,:,i))];

dia_col=temp16;

end

dia_col_sum=sum(dia_col);

[val_min,nmin]=min(abs(dia_col_sum-n));

G=G(:,:,nmin);

K=K(:,:,nmin);

RGA=RGA(:,:,nmin);

ERGA=ERGA(:,:,nmin);


G;


K;


RGA;


ERGA;

%*******************************************************************

%Step 4: Calculate Ralative Ferquency Array.

RFA=ERGA./RGA;

%*******************************************************************



for j=1:n


if nc==3

a2(i,j)=G(i,j).den{1}(1);

a1(i,j)=G(i,j).den{1}(2);

elseif nc==2

a2(i,j)=0;

a1(i,j)=G(i,j).den{1}(1);

end

end


%*******************************************************************

%Step : Calculate Ralative Ferquency Array.

RFA=ERGA./RGA;

150

%*******************************************************************


Am=5%Assumption

%*******************************************************************


for i=1:n











*K(i,i)/RGA(i,i));

ki(i,i)=pi/(2*Am*RFA(i,i)*L(i,i)*K(i,i)/RGA(i,i));

kd(i,i)=(pi*a2(i,i))/(2*Am*RFA(i,i)*L(i,i)*...

K(i,i)/RGA(i,i));


gnew(i,i)=tf(K(i,i),[a2(i,i) a1(i,i) 1]...

,'InputDelay',L(i,i));








*K(i,i));


kd(i,i)=(pi*a2(i,i))/(2*Am*RFA(i,i)*L(i,i)...

*K(i,i));

end

end

for i=1:n

gnew(i,i);


end


gc11=gc(1,1)


gc22=gc(2,2)


gc33=gc(3,3)


gc44=gc(4,4)

%*******************************************************************

tStart=0;

tFinal=1600;






%*******************************************************************

% Simulation *

%*******************************************************************

figure;


151



ylabel('y1');

xlabel('Time');

grid on;

title('Step responses of y1 for A1 System (1985)');


figure;




ylabel('y2');

xlabel('Time');

grid on;



figure;




ylabel('y3');

xlabel('Time');

grid on;



figure;




ylabel('y4');

xlabel('Time');

grid on;



%*******************************************************************


%*******************************************************************

%IAE

for i=1:n




end









%*******************************************************************

%ISE

for i=1:n




end




152






%*******************************************************************

%ITAE

for i=1:n




end









%*******************************************************************

Documents

Development of Multivariable Control Systems Rev 200