A Novel Parametrized Controller Reduction Technique based on … · A Novel Parametrized Controller...

A Novel Parametrized Controller Reduction

Technique

based on Different Closed-Loop Configurations

by

Pantazis Constantine Houlis

A thesis submitted to the School of Electrical, Electronic

and Computer Engineering in partial fulfilment of the

requirements for the degree of Doctor of Philosophy

Faculty of Engineering, Computing and Mathematics

University of Western Australia

December 2008

Author’s Publications

International Journal Paper to Appear

P. C. Houlis and V. Sreeram, A Parametrized Controller Reduction Technique

via a New Frequency Weighted Model Reduction Formulation, IEEE Transac-

tions on Automatic Control, Vol. 54, No 2, February 2009.

International Journal Paper Submitted

P. C. Houlis and V. Sreeram, A Generalized Controller Reduction Technique,

submitted to the European Journal of Control, January 2009.

P. C. Houlis and V. Sreeram, Interconnections between Different Classical and

Modern Control Systems Block Diagrams, submitted to Journal of Dynamic

Systems, Measurment, and Control, January 2009.

Refereed International Conference Papers

P.C. Houlis and V. Sreeram, Controller reduction via a new double-sided fre-

quency weighted model reduction formulation, Joint 20th IEEE International

Symposium on Intelligent Control (ISIC’05) and 13th Mediterranean Confer-

ence on Control and Automation (MED’05), pp. 537-542 , Limassol, Cyprus,

June 2005.

P.C. Houlis and V. Sreeram, A Parametrized Controller Reduction Technique,

Proceedings of the 45th IEEE Conference on Decision and Control, pp. 3430-

3435, San Diego, USA, December 2006.

ii

P. C. Houlis and V. Sreeram, Connections between Classical and Modern Control

Block Diagrams, 6th Asian Control Conference, Bali, Indonesia, July 2006.

P. C. Houlis and V. Sreeram, An Interconnection between Combined Classi-

cal Block Diagrams and Linear Fractional Transformation Block Diagrams, 9th

International Conference on Control, Automation, Robotics and Vision, Singa-

pore, December 2006.

iii

Acknowledgements

I would like to thank my parents Constantine and Kleoniki for their continuous

support throughout the years, my family, as well as my supervisor Prof. Victor

Sreeram for his guidance and his inspiring positive attitude. I would also like to

thank some really good friends with whom I shared many good moments during all

those years, namely, Chang Su Lee, Martin Masek, Simon Kwok, Abdul Ghafoor,

Weiqun Zheng, Shafishuhaza Sahlan, Yves Hwang, and Ivan Neubronner. Finally, a

big thank you to all the School’s helpful staff, especially Doris Dennis, Clive Dennis,

and Rob Mattaboni. This Thesis is dedicated to all of them.

iv

Abstract

This Thesis is concerned with the approximation of high order controllers or the

controller reduction problem.

We firstly consider approximating high-order controllers by low order controllers

based on the closed-loop system approximation. By approximating the closed-loop

system transfer function, we derive a new parametrized double-sided frequency weighted

model reduction problem. The formulas for the input and output weights are derived

using three closed-loop system configurations: (i) by placing a controller in cascade

with the plant, (ii) by placing a controller in the feedback path, and (iii) by using the

linear fractional transformation (LFT) representation. One of the weights will be a

function of a free parameter which can be varied in the resultant frequency weighted

model reduction problem. We show that by using standard frequency weighted model

reduction techniques, the approximation error can be easily reduced by varying the

free parameter to give more accurate low order controllers. A method for choosing

the free parameter to get optimal results is being suggested. A number of practi-

cal examples are used to show the effectiveness of the proposed controller reduction

method.

We have then considered the relationships between the closed-loop system config-

urations which can be expressed using a classical control block diagram or a modern

control block diagram (LFT). Formulas are derived to convert a closed-loop system

represented by a classical control block diagram to a closed-loop system represented

by a modern control block diagram and vice versa.

v

Contents

Acknowledgements ii

Acknowledgements iv

Abstract v

List of Tables x

List of Figures xi

1 Introduction 1

1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Preliminaries 9

2.1 State Space Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Controllability and Observability Gramians . . . . . . . . . . . . . . . 11

2.3 Hankel Singular Values . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.5 Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . 14

vi

2.6 The Linear Fractional Transformations . . . . . . . . . . . . . . . . . 14

2.7 Model Reduction Problem . . . . . . . . . . . . . . . . . . . . . . . . 16

2.8 Balanced Truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.9 Frequency Weighted Model Reduction Problem . . . . . . . . . . . . 21

2.9.1 H∞ Controller Reduction . . . . . . . . . . . . . . . . . . . . 21

2.9.1.1 Stability Consideration . . . . . . . . . . . . . . . . . 21

2.9.1.2 Closed-loop Transfer Function Consideration . . . . 22

2.9.2 Frequency Weighted Model Reduction . . . . . . . . . . . . . 23

2.9.3 Enns’ Technique . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.9.4 Wang’s Technique . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.10 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.10.1 Linear Quadratic Regulator . . . . . . . . . . . . . . . . . . . 30

2.10.2 The Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . 32

2.10.3 H∞ Controller Design . . . . . . . . . . . . . . . . . . . . . . 33

3 The Double Controller Technique 43

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2 The Double Controller Technique for Cascade Systems . . . . . . . . 45

3.2.1 Original Methods . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2.2 The Main Results . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2.3 Relationships between closed-loop configurations . . . . . . . . 48

3.2.4 The Derivation of New Frequency Weights . . . . . . . . . . . 52

3.2.5 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.2.6 The behavior of H(c) . . . . . . . . . . . . . . . . . . . . . . . 61

vii

3.2.7 The Application of the Double Controller Technique . . . . . . 67

3.2.8 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.3 The Double Controller Technique for Feedback Systems . . . . . . . . 70


3.3.2 The Main Results . . . . . . . . . . . . . . . . . . . . . . . . . 73


3.3.4 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 79


3.3.6 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.4 The Double Controller Technique for LFT Systems . . . . . . . . . . 83


3.4.2 The Main Results . . . . . . . . . . . . . . . . . . . . . . . . . 86


3.4.4 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 93


3.4.6 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4 Design Examples for the Double Controller Technique 98

4.1 Single-Input Single Output (SISO) Examples . . . . . . . . . . . . . . 98

4.1.1 Cascade System Example . . . . . . . . . . . . . . . . . . . . 99

4.1.2 Feedback System Example . . . . . . . . . . . . . . . . . . . . 101

4.1.3 LFT System Example . . . . . . . . . . . . . . . . . . . . . . 103

4.2 Multiple-Input Multiple-Output (MIMO) Examples . . . . . . . . . . 104

4.2.1 Cascade System Example . . . . . . . . . . . . . . . . . . . . 104

viii

4.2.2 LFT System Example . . . . . . . . . . . . . . . . . . . . . . 107

5 Conversions between Classical Systems and Modern Systems 112

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.2 Notation and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 113

5.3 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.4 Classical to Modern Conversion . . . . . . . . . . . . . . . . . . . . . 117

5.4.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.4.2 The Cascade System . . . . . . . . . . . . . . . . . . . . . . . 118

5.4.3 Subcase 1: AGK = GKA . . . . . . . . . . . . . . . . . . . . 123

5.4.4 The Feedback System . . . . . . . . . . . . . . . . . . . . . . . 125

5.4.5 The Combined System . . . . . . . . . . . . . . . . . . . . . . 130

5.5 Modern to Classical Conversion . . . . . . . . . . . . . . . . . . . . . 133

5.5.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.5.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

6 Conclusions 147

6.1 Main Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

6.2 Recommendations for future work . . . . . . . . . . . . . . . . . . . . 148

Bibliography 150

Appendixes 159

ix

List of Tables

4.1 Approximation Error Comparison for First and Second Order Controllers100

4.2 Approximation Error Comparison using Enns’ Method for First up to

Fifth Order Controllers (MIMO Example) . . . . . . . . . . . . . . . 108

4.3 Approximation Error Comparison using Wang’s Method for First up

to Fifth Order Controllers (MIMO Example) . . . . . . . . . . . . . . 109

x

List of Figures

1.1 The three ways to find a low order controller [7] . . . . . . . . . . . . 5

2.1 Singular Value Decomposition is applied when the matrix is non-square. 39

2.2 Block diagram of an LFT system (upper LFT). . . . . . . . . . . . . 40

2.3 Closed loop system diagram with a high order controller. . . . . . . . 40

2.4 Closed loop system diagram with a reduced order controller. . . . . . 40

2.5 Input-output frequency weighted error system. . . . . . . . . . . . . . 41

2.6 Input frequency weighted error system. . . . . . . . . . . . . . . . . . 41

2.7 Output frequency weighted error system. . . . . . . . . . . . . . . . . 41

2.8 Input-output augmented systems. . . . . . . . . . . . . . . . . . . . . 42

3.1 A closed-loop system with plant G and controller K . . . . . . . . . . 50

3.2 The new closed-loop system with plant G and controller K . . . . . . 50

3.3 The plant G is itself a closed-loop system with plant G and controller

K − K = K(I − C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.4 The double controller technique uses a more complex way for controller

reduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.5 The upper and lower bounds for h(c), where λ = ‖GK‖∞. . . . . . . 64

xi

3.6 A feedback system with plant G and controller K. . . . . . . . . . . . 72

3.7 A generalization of the feedback system gives us an extra parameter c.

Also, H(c) = I +GK(1− c). . . . . . . . . . . . . . . . . . . . . . . . 73

3.8 A detailed view of the generalized block diagram of the feedback system. 74

3.9 LFT system with plant P (made of the submatrices P11, P12, P21, and

P22) and controller K. . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.10 A generalization of the LFT system gives us an extra parameter c. . . 87

3.11 A detailed view of the generalized block diagram of the LFT system. 87

4.1 Enns’ and Wang’s method results for e(c, 1) for second order controllers.101

4.2 The error function e(c, 1) is bounded by e(1, 1)h−1(c) and e(1, 1)h(c) . 102

4.3 The error function e(c, 1) could be approximated by e(c, 1) . . . . . . 103

4.4 The error function e(c1, c2) for Enns’ Method in a three dimensional

plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.5 The error function e(c1, c2) for Wang’s Method in a three dimensional

plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.6 The error functions e(c, 1) and e(c, 1) for the MIMO example when

applying controller reduction of order 4. . . . . . . . . . . . . . . . . 106

4.7 The error functions e(c, 1) and e(c, 1) for the MIMO example when

applying controller reduction of order 5. . . . . . . . . . . . . . . . . 107

4.8 Enns’ method results for e(c) for second order controllers. . . . . . . . 109

4.9 Wang’s method results for e(c) for second order controllers. . . . . . . 110

4.10 The error function e(c) using Enns method is being approximated by

e(c) for second order controllers. . . . . . . . . . . . . . . . . . . . . . 110

xii

4.11 Enns’ method results for e(c) for second order controllers. . . . . . . . 111

4.12 Wang’s method results for e(c) for fifth order controllers for the MIMO

LFT case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.1 Block diagram of cascade classical controller. . . . . . . . . . . . . . . 144

5.2 Block diagram of feedback classical controller. . . . . . . . . . . . . . 144

5.3 Block diagram of combined classical controller. . . . . . . . . . . . . . 144

5.4 Block diagram of modern controller (upper LFT). . . . . . . . . . . . 145

5.5 Block diagram of modern controller (upper LFT) rotated by 180 degrees.145

5.6 Block diagram of modern controller (lower LFT). . . . . . . . . . . . 145

5.7 The Redheffer star product. . . . . . . . . . . . . . . . . . . . . . . . 146

5.8 The Redheffer star product, when replacing P ′ by P22. . . . . . . . . 146

5.9 The Redheffer star product, when replacing P ′ by GK1. . . . . . . . . 146

xiii

Elementary Notations and Terminologies

Transfer function G(s) =

A B

C D

= C(sI − A)−1B +D

⇔ State-space realization {A,B,C,D}

‖G(s)‖∞ Infinity norm of the system G(s)

P > 0 Positive definite matrix P

Symmetric matrix P with positive eigenvalues

P ≥ 0 Positive semidefinite matrix P

Symmetric matrix P with non-negative eigenvalues

XT or X ′ Transpose of matrix or vector X

X∗ Complex conjugate transpose of matrix or vector X

X−1 Inverse of matrix X

λi(X) Eigenvalues of X

σ(X) Singular values of X

xiv

Chapter 1

Introduction

Model reduction is very important in system analysis and design. There are plenty

applications of model reduction in various areas such as model identification [9,35,56],

model simplification [40,42,43,48], and uncertain model reduction [10–12].

The aim of model reduction is to find a reduced order system which approximates

the input and the output behaviour of the original high order system. Many different

classes of model reduction have been developed, depending on criteria such as reducing

the H∞ norm of the error transfer function (H∞ norm model reduction), reducing

the H2 norm of the error transfer function (H2 norm model reduction), using Hankel

norm reduction, and many more.

Other ways to classify model reduction is as unweighted model reduction, fre-

quency weighted model reduction and controller reduction, uncertain systems model

reduction, and many more.

In this Thesis, we will focus on frequency weighted model reduction and controller

reduction problems for linear continuous systems, and using their H∞ norm error.

1

1.1 Overview

For many decades the model reduction problem has attracted considerable attention.

Because of this, many methods have been proposed. In the beginning, the main

methods of model reduction were parameter matching methods and optimization

methods.

The parameter matching methods retain dominant model parameters, which in-

clude times moments, Markov parameters, and eigenvalues. For instance, the methods

of Pade [55] and Routh [30] belong to this category. Those methods have drawbacks,

such as lack of performance guarantee and no clear choice of the retained parameters.

The optimization methods are based on minimization of the output error between

the high order system and the reduced order system. Those methods include Wilson’s

methods [63, 64] which minimize the integral of the impulse response error squared

between the full order and the reduced order model, Obinata and Inooka’s methods

[51,52] which minimize the equation error. But the early-stage optimization methods

can only be applied on special cases, and the optimization results cannot be predicted.

The developments of the optimal Hankel norm approximation [21,36] and the bal-

anced truncation [45,53], have significantly changed the way we use model reduction.

Those two techniques have almost perfect characteristics. More specifically, all their

reduced order models are stable, their solutions are of closed form, and they have a

priori frequency response error bounds.

Mullis and Roberts [46,47] were the first who introduced balanced representation,

and this was extended to model reduction by Moore [45]. In balanced realization,

each state is equally controllable and observable, and the reduced order models are

2

obtained by truncating the least controllable and least observable states. There are

many modifications to the balanced truncation method [3, 13, 18, 58], some of which

include singular perturbation approximation [18, 41]. The optimal Hankel norm ap-

proximation was based on Nehari’s theory [49]. Several papers about infinite Hankel

matrices and approximation problem were published by Adamjan et al [1,2]. Similar

developments have also been made by others [36, 44, 54]. A characterization of all

optimal Hankel norm approximations, in both time and frequency domain, was given

by Glover [21]. More details of the early model reduction techniques may be found

in [56,57].

In the middle of the 80s decade, the developments of the unweighted model re-

duction were reaching saturation. Because of this, research was now focusing the

frequency weighted model reduction, a relatively new area for which a number of tech-

niques for frequency weighted model reduction has been proposed [15,17,31,38,59,60].

In frequency weighted model reduction, we are interested in reducing the error be-

tween the high order system and the low order model, but only over certain bands

of frequencies of our interest. Those bands of frequencies of our interest is directly

related to an input weight and an output weight.

Many balanced unweighted model reduction methods and optimal Hankel norm

approximation methods, eventually evolved to today’s most known frequency weighted

model reduction methods [6, 17, 29,33,38,39,59,60,67].

The concept of frequency weighted model reduction was first proposed by Enns

[17]. There are two types of methods, single-sided frequency weighted (by using

either an input or an output weight), and double-sided frequency weighted (by using

both input and output weights). Despite the good results in Enns’ examples, more

3

concrete proofs are needed. In the double-sided frequency weighted case, a proof for

asymptotic stability and formulas for the error bounds, are essential to establish the

theory. Another extension of this method gives a frequency response error bound [4,5],

but it is limited to only a certain class of systems and weights, where the number of

states of the frequency weights must be equal to the number of inputs of the original

system. In the last years, many modifications of Enns’ method have been proposed,

some of which guarantee stability of the reduced order model in the double-sided

frequency weighted case [39, 60].

Extensions to the Hankel norm approximation method to accommodate the fre-

quency weighted cases, have also been proposed. For instance, by minimizing the

frequency weighted Hankel norm error (by introducing a frequency weighted rational

function) [38], by using a more general weight [29], or by deriving a L∞ error bound

for the weighted error [6]. In recent years, the frequency weighted model reduction

problem is a blend between different methods. Those methods use frequency weighted

Hankel norm approximation to find the poles of the reduced order model, and convex

optimization is used to find the zeros. By using such methods, in many cases we may

achieve near optimal results in terms of minimizing H∞ of the frequency weighted er-

ror [69]. Note that, convex optimization may also be used together with the balanced

model reduction methods.

The unweighted techniques work generally better than the frequency weighted

ones. And although many such techniques exist already, none of them can guarantee

to give the best results for all cases. This is because of the complexity which is

involved (when compared to the unweighted cases), and each method yields a better

approximation depending on many criteria. Therefore, the model characteristics may

4

Figure 1.1: The three ways to find a low order controller [7]

dictate which method should be used.

The frequency weighted model reduction method is not only limited in solving

the frequency weighted model reduction problem, but it can also solve the controller

reduction problem. When a controller is designed for complex real world applications,

it is usually of high order. And it is always preferable to use a lower order linear con-

troller. We now summarize the three practical ways of finding a low order controller

from the original higher order system as presented in Figure 1.1:

• Reducing the original high order system to a low order model using model

reduction techniques. Then use, standard controller design techniques (LQG or

H∞-infinity design) to obtain a low order controller.

• Directly obtain a low order controller from the given high order system.

• Use the given original high order system to design a high order controller using

5

standard techniques (LQG or H∞-infinity design), and then reduce this high

order controller by approximating the closed-loop performance.

Analyzing the above approaches, we find that the first approach does not work

in practice because the model reduction (approximation) is involved in the first step.

The controller design step which follows, will only magnify the errors introduced in

the approximation step. As far as the second approach is concerned, there is no

commercial software available which will directly give a low order controller from the

original high order plant. The last approach is the most logically motivated, since the

controller design step is performed before the controller reduction (approximation).

The controller reduction is based on approximating the closed-loop behaviour of

the system. Obtaining a low order controller by approximating the closed-loop be-

haviour gives rise to the frequency weighted model reduction problem [7]. There are

two ways in which this problem can be formulated: single-sided frequency weighted

model reduction and double-sided frequency weighted model reduction problems.

When the controller reduction is based on closed-loop stability criteria, a single-

sided frequency weighted model reduction problem is obtained. When a closed-loop

system approximation is the main criteria in controller reduction, then a double-sided

frequency weighted model reduction problem is obtained.

In this Thesis, the controller reduction problem is solved via a new double-sided

frequency weighted model reduction problem formulation. The new formulation is a

function of a user chosen free parameter, unlike the standard formulation [7]. It is

shown that by varying this parameter, more accurate low order controllers can be

obtained.

6

1.1.1 Thesis Outline

In Chapter 2, we present some fundamental concepts which are required for under-

standing the material presented in this Thesis. Here we present some basic state-

space system concepts, norms, model reduction and controller reduction problems,

and standard controller design techniques. In addition, we present some important

model reduction and frequency weighted model reduction methods, which will be used

later in this Thesis.

In Chapter 3, we present the proposed Double Controller Technique, which is a

new controller reduction method. The controller reduction problem is formulated as

a new double-sided frequency weighted model reduction problem. The formulas for

the input and output weights required for the frequency weighted model reduction

are derived. These formulas are a function of a free parameter, unlike the standard

formulas [7]. Here we use standard techniques ( [17, 62]) to solve the new frequency

weighted model reduction problem derived. By varying the free parameter, we can

easily reduce the frequency weighted approximation error (and obtain more accurate

controllers). Three different sets of formulas for the weights are derived, based on

three closed-loop configurations.

In Chapter 4, we demonstrate the proposed technique using practical examples

taken from the literature. Both Single-Input Single-Output (SISO) examples, as

well as Multiple-Input Multiple-Output examples are considered. Is is clear from

those examples, that by choosing the correct parameter for the Double Controller

Technique, we may achieve a significant reduction of the approximation error.

In Chapter 5, we derive relationships between classical control block diagrams

7

and modern control block diagrams (LFT). We consider two cases of classical control

block diagrams, one with a controller in cascade with the plant, and the other with

the controller in the feedback path. Formulas are derived for converting classical

control block diagrams to modern control block diagrams, and vice versa.

Finally, in Chapter 6, we summarize the main contributions and outline future

research directions.

8

Chapter 2

Preliminaries

In this chapter, we briefly summarize some mathematical background required for

understanding the Thesis. We will denote some important concepts and definitions,

which will be used in the theories which will be introduced later.

2.1 State Space Systems

A finite dimensional linear time invariant system may be described by the differential

equations

x = Ax+Bu, x(t0) = x0 (2.1)

y = Cx+Du

where A,B,C,D are constant matrices of appropriate dimensions, x(t) ∈ Rn is

the system state (which has an initial condition x(t0) = x0), u(t) ∈ Rm is the system

input, and y(t) ∈ Rp is the system output. If m = p = 1, then the system is called

9

Single Input Single Output (SISO) system, and in any other case it is called Multiple

Input Multiple Output (MIMO) system.

Let us now define as U(s) and Y (s) the Laplace transforms of u(t) and y(t) using

the initial condition x(0) = 0. Then we will have

Y (s) = G(s)U(s) (2.2)

G(s) = C(sI − A)−1B +D (2.3)

The standard notation to describe the above is

A B

C D

= C(sI − A)−1B +D

Given the input u(t) and the initial condition x(t0), the system’s response for x(t)

and y(t) (when t ≥ t0) is

x(t) = eA(t−t0)x(t0) +

∫ t

t0

eA(t−τ)Bu(τ)dτ (2.4)

y(t) = Cx(t) +Du(t) (2.5)

The input and output are related via the convolution equation:

y(t) = (g ∗ u) :=

∫ ∞−∞

g(t− τ)u(τ)dτ =

∫ t

−∞g(t− τ)u(τ)dτ (2.6)

To follow the standard notations, the Laplace transform variable will always be s,

and the time variable will always be t. For the sake of simplicity, we may omit those

variables in some cases.

10

Out of all possible realizations (A,B,C,D) of G, those whose state dimension

is the smallest possible will be called minimal realizations. This is equivalent to

having (A,B) controllable and (C,A) observable. Also, if (A,B,C,D) is a minimal

realization, then the poles of G are defined as eigenvalues of A, which describes the

behaviour of the system. In general, a system G is stable if Re(λi(A)) < 0, where λi

are the eigenvalues of A.

2.2 Controllability and Observability Gramians

Given a stable realization (A,B,C,D), the Controllability Gramian P of G is defined

as

P =

∫ ∞0

eAtBB′eA′t ≥ 0 (2.7)

It can be verified (by direct substitution) that the Controllability Gramian satisfies

the Lyapunov equation

AP + PA′ +BB′ = 0 (2.8)

Similarly, the Observability Gramian Q of G is defined as

Q =

∫ ∞0

eA′tC ′CeAt ≥ 0 (2.9)

It can be verified (by direct substitution) that the Observability Gramian satisfies

the Lyapunov equation

11

A′Q+QA+ C ′C = 0 (2.10)

We have that (A,B) is controllable if and only if P > 0, and (C,A) is observable if

and only if Q > 0. A very interesting property is that the eigenvalues of the product

PQ are invariant under invertible state transformations.

2.3 Hankel Singular Values

We now consider a stable transfer function G with realization (A,B,C,D). Then the

Hankel Singular Values of G are defined as

σi(G) =√λi(PQ) (2.11)

and they are in decreasing order, i.e. σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0.

Let Σ = diag(σ1, σ2, . . . , σn), and Σ2 = Λ. This new realization with controllabil-

ity and observability Gramians P = Q = Σ will be referred to as balanced realization

(also known as internally balanced realization) [45].

2.4 Norms

The knowledge of the maximum energy gain from bounded inputs and bounded out-

puts is very important. A common and efficient way to represent the maximum energy

gain is by using the H∞ norm.

12

Definition 1 The space L∞ (also known as Banach space), is the space of matrix

valued functions which are essentially bounded on the imaginary axis. The L∞ norm

is defined as

‖G‖∞ = ess supω∈Rσ(G(jω)). (2.12)

In state-space theory, L∞ corresponds to the proper transfer functions which have no

poles on the jω axis.

Definition 2 We define as H∞, the closed subspace of functions in L∞ which are

analytic and bounded in the open half plane. Then, the H∞ norm is defined as

‖G‖∞ = ess supRe(s)>0σ(G(s)) = ess supω∈Rσ(G(jω)). (2.13)

Definition 3 We define as H−∞, the closed subspace of functions in L∞ which are

analytic and bounded in the open left half plane. The norm here is defined as:

‖G‖∞ = ess supRe(s)<0σ(G(s)) = ess supω∈Rσ(G(jω)). (2.14)

In state-space theory, H∞ corresponds to the transfer functions which have no

poles on the imaginary axis or in the open right plane, and H−∞ corresponds to the

transfer functions which have no poles on the imaginary axis or in the open left plane.

By defining a transfer function G, such that G ∈ H∞, it implies that G is stable.

A prefix R is used to denote functions which are real and rational. In this Thesis

we will always use real and rational functions unless stated otherwise. So if a transfer

function G is real and rational, then G ∈ RH∞ if and only if G is proper and

stable [19].

13

2.5 Singular Value Decomposition

Many times in linear algebra, we are required to find the eigenvalues and eigenvectors

of a square matrix. But what happens if the matrix is an m×n matrix, where m 6= n?

A logical solution to this, is to temporary transform the matrix into square form.

Truncation of the matrix is not advisable as this would result in loss of potentially

valuable information. An efficient alternative is use the Singular Value Decomposi-

tion. This is done by multiplying the given matrix with its transpose, and the result

will be a square symmetric matrix. Then, once we find the eigenvalues of the result-

ing square matrix, we may derive the singular values of the initial matrix by simply

calculating their square root. A nice way to present this well known concept is shown

in Figure 2.1.

2.6 The Linear Fractional Transformations

In complex variable function theory, a mapping F : C 7→ C such that

F (s) =a+ bs

c+ ds(2.15)

where a, b, c, d ∈ C, is called a linear fractional transformation or LFT. Moreover,

in the case c 6= 0, we have

F (s) = α + βs(1− γs)−1 (2.16)

where α, β, γ ∈ C. By generalizing Equation 2.16 for matrices, we obtain the

linear fractional transformations related to modern control theory.

14

Definition 4 A linear fractional transformation system (as seen in Figure 2.2) will

be denoted as LFT.

More specifically, there are two equivalent types of LFTs, the lower LFT Fl(P,K),

and the upper LFT Fu(P,K) (which is shown in Figure 2.2). In this Thesis we will

focus on the upper LFT (all the procedures are equivalent when using a lower LFT).

Let us define a LFT system with plant P

P =

P11 P12

P21 P22

(2.17)

where P11, P12, P21 and P22 are matrices with appropriate dimensions with respect

to the input values u, u0 and the output values y, y0.

Then, the block diagram in Figure 2.2 represents the two equations

y0

y

= P

u0

u

=

P11 P12

P21 P22

u0

u

(2.18)

u0 = Ky0 (2.19)

Let us assume that we have some LFT system, with P a given linear time-invariant

plant, and a stabilizing high order controller K as shown in Figure 2.2.

We may now write the definition of Fu(P,K) in terms of the submatrices of P (as

shown in [69]) and Fu(P,K):

Fl(P,K) = P11 + P12K(I − P22K)−1P21 (2.20)

15

Fu(P,K) = P22 + P21K(I − P11K)−1P12 (2.21)

2.7 Model Reduction Problem

The derivation of a reasonable mathematical model is fundamental to the analysis and

design of a dynamic system. In practice, one can obtain a fairly complex large scale

model of the system. However it is difficult to obtain a good understanding of the

behavior of the system. The analysis and design of such a system is easier if a lower

order model (that provides a good approximation) is derived. The process of deriving

a low order model from a high order model is known as model reduction. In general,

the aim of model reduction is to find a reduced order model which approximates the

input-output behavior of the original system.

The McMillan degree of a transfer-function (matrix) is the total number of poles in

the diagonal elements of the matrix in its diagonalized (McMillan) form. This number

determines the order of any minimal state-space realization of the transfer-function

matrix.

Given a stable model G(s) ∈ RH∞ of McMillan degree n, the open loop approx-

imation problem is to calculate G(s) ∈ RH∞ of McMillan degree k (k < n) such

that

minG(s)∈RH∞‖G(s)− G(s)‖∞

A lower bound for this approximation problem also exists:

16

σk+1(G) ≤ infG(s)∈RH∞‖G(s)− G(s)‖∞

There is no reduction technique which can give a better approximation than this

bound. Many reduction techniques also satisfy an upper bound on the approximation

error which may be calculated a priori. The tightness between the upper bound for

a particular reduction technique and the general lower bound, gives us an indication

of how well the technique may perform. Generally speaking, good approximation

techniques are those which use simple calculations, are efficient, and derive a low

order model whose response accurately matches the response of the corresponding

model.

2.8 Balanced Truncation

For a given transfer matrix, there are infinitely many state space realizations. But

only some of those have been shown to be useful in control engineering. Some of those

useful cases are:

• The internally balanced realization, which gives an indication of the dominance

of the system states in the input/output behavior.

• The balanced realization, which is an asymptotically stable and minimal real-

ization, where the controllability and the observability Gramians are equal and

diagonal.

Let us define the original full order stable system as G(s) = C(sI − A)−1B + D

where {A,B,C,D} is its nth order minimal realization. Let P > 0 and Q > 0 be

17

the controllability and the observability Gramians satisfying the following Lyapunov

equations:

AP + PAT +BBT = 0 (2.22)

ATQ+QA+ CTC = 0 (2.23)

Also, let T be the transformation obtained by simultaneously diagonalizing the

Gramians P and Q

T TQT = T−1PT−T = Σ =

Σ1 0

0 Σ2

where Σ1 = diag{σ1, σ2, . . . , σr}, Σ2 = diag{σr+1, . . . , σn}, σi ≥ σi+1, i = 1, 2, . . . , n−

1, σr > σr+1 and σi are the Hankel singular values. We now transform and partition

the original system:

A = T−1AT =

A11 A12

A21 A22

, B = T−1B =

B1

B2

,C = CT =

[C1 C2

], D = D (2.24)

where A11 ∈ Rr×r (r < n). The reduced order model is given by Gr(s) = C1(sI −

A11)−1B1 +D.

Remark 1 We now summarize some of the most important properties of the balanced

realization and model reduction:

18

1. A given realization {A,B,C,D} can be transformed by a state transformation

to a balanced realization {A, B, C, D} (also known as internally balanced real-

ization [45]) if and only if it is asymptotically stable and minimal.

2. The balanced realization is unique up to the ordering of the Hankel singular

values σi and an orthogonal transformation1 that commutes with Σ.

3. Any subsystem realization {Akk, Bk, Ck, D} for k = 1, 2 obtained via partition-

ing and truncating the realization{A, B, C, D

}is internally balanced and stable

if Σ1 and Σ2 have no diagonal entries in common [53]. Furthermore, if all the

diagonal elements of Σ are distinct, then every possible subsystem is asymptot-

ically stable.

4. ‖G(s)−Gr(s)‖∞ ≤ 2n∑

i=r+1

σi and ‖G(s)−Gn−1(s)‖∞ = 2σn.

Remark 2 For any stable system G(s) with a non minimal realization

{A,B,C,D}, there exists a nonsingular transformation T , such that the realization

{T−1AT, T−1B,CT,D} has the controllability Gramian diag(Σ1,Σ2, 0, 0) and the ob-

servability Gramian diag(Σ1, 0,Σ3, 0) where Σ1, Σ2, Σ3 are diagonal and positive

definite [21, 69]. The realization corresponding to Σ1 is the balanced realization.

Remark 3 The stability condition (that is, Σ1 and Σ2 have no diagonal entries in

common) does not require that the balanced realization is ordered in any way. The

only assumption is that the partitioning of Σ does not split the states associated with

a multiple σi. Moreover, this stability condition is only sufficient [69].

1SΣ = ΣS where S is a diagonal matrix with the diagonal elements ±1.

19

Remark 4 The balanced truncation method [45] produces a very good approximation

of the original system, but it does not yield an optimal approximation in the infinity

norm sense [69].

Remark 5 Other closely related realizations are the input and the output normal

realizations. A realization is said to be input normal (output normal, respectively)

if its controllability Gramian is an identity matrix and its observability Gramian is

diagonal (its controllability Gramian is diagonal and its observability Gramian is the

identity matrix, respectively) [32]. Note that if {A, B, C, D} is a balanced realization,

then {Σ−1/2AΣ1/2, Σ−1/2B, CΣ1/2, D} is its input normal and

{Σ1/2AΣ−1/2, Σ1/2B, CΣ−1/2, D} is its output normal realization.

Remark 6 Note that, the above description of balanced realization/truncation is

given for continuous time systems. The discrete time system case description of

balanced realization/truncation follows similarly with the exception of the Lyapunov

equations. Given the original full order stable system Gd(z) = Cd(zI −Ad)−1Bd +Dd

where {Ad, Bd, Cd, Dd} is its nth order minimal realization. The controllability and

the observability Gramians Pd > 0 and Qd > 0 respectively satisfying the following

Lyapunov equations:

AdPdATd − Pd +BdB

Td = 0 (2.25)

ATdQdAd −Qd + CTd Cd = 0 (2.26)

The important properties (like stability of the reduced order model and error bounds)

of the continuous time system balanced realization/truncation also holds for the dis-

20

crete time system balanced realization/truncation with the exception that, in the dis-

crete time system balanced truncation case, the reduced order model is not guaranteed

to be balanced [41].

2.9 Frequency Weighted Model Reduction Prob-

lem

2.9.1 H∞ Controller Reduction

A generalization of the classical feedback system is the H∞ feedback system. Let

P (s) =

P11(s) P12(s)

P21(s) P22(s)

be the transfer function matrix of a linear time-invariant

generalized plant, where K(s) and Kr(s) are the full order and the reduced order

stabilizing controllers respectively as shown in Figures 2.3 and 2.4.

Let the closed-loop transfer function with the full order controller K(s) and re-

duced order controller Kr(s) respectively be expressed in linear fractional transfor-

mation form as [69]:

Tzw(s) = P11(s) + P12(s)K(s)(I − P22(s)K(s))−1P21(s)

Tzw(s) = P11(s) + P12(s)Kr(s)(I − P22(s)Kr(s))−1P21(s)

2.9.1.1 Stability Consideration

Suppose K(s) and Kr(s) have same number of right half plane poles, then the closed

loop system Tzw(s) is stable if either of the following sufficient conditions is satisfied

∥∥(I − P22(s)K(s))−1P22(s)(K(s)−Kr(s))∥∥∞ < 1

21

or

∥∥(K(s)−Kr(s))(I − P22(s)K(s))−1P22(s)∥∥∞ < 1

2.9.1.2 Closed-loop Transfer Function Consideration

Tzw − Tzw(s) = P12(s)K(s)(I − P22(s)K(s))−1P21(s) (2.27)

−P12(s)Kr(s)(I − P22(s)Kr(s))−1P21(s)

≈ P12(s)(I −K(s)P22(s))−1 (K(s)−Kr(s)) (I − P22(s)K(s))−1P21(s)

Equation 2.27 defines an approximation problem. In this problem, we must find

the reduced order controller Kr(s) such that the full order controller K(s) and the

reduced order controller Kr(s) have the same number of poles in the open right half

plane, and the index ‖P12(s)(I −K(s)P22(s))−1 (K(s)−Kr(s)) (I − P22(s)K(s))−1P21(s)‖∞

is minimized.

Minimizing ‖P12(s)(I −K(s)P22(s))−1 (K(s)−Kr(s)) (I − P22(s)K(s))−1P21(s)‖∞

does not guarantee an optimal solution. We seek a stabilizing reduced order controller

Kr(s) such that

∥∥P12(s)(I −K(s)P22(s))−1 (K(s)−Kr(s)) (I − P22(s)K(s))−1P21(s)

∥∥∞ < γ

where γ is a positive constant.

Note that, in a special case when P (s) =

0 P12(s)

P21(s) 0

=

0 W (s)

V (s) 0

,

then Tzw(s)− Tzw(s) = P12(s) (K(s)−Kr(s))P21(s).

22

2.9.2 Frequency Weighted Model Reduction

The above controller reduction problems can be summarized as a frequency weighted

model reduction problem. Given the original full order stable system G(s) = C(sI −

A)−1B + D, the stable input weighting system V (s) = CV (sI − AV )−1BV + DV

and the stable output weighting system W (s) = CW (sI − AW )−1BW + DW , where

{A,B,C,D}, {AV , BV , CV , DV } and {AW , BW , CW , DW} are nth, pth and qth order

minimal realizations respectively, the objective is to find a lower order stable system

Gr(s) = Cr(sI−Ar)−1Br+Dr where {Ar, Br, Cr, Dr} is an rth order (r < n) minimal

realization, such that

‖W (s) (G(s)−Gr(s))V (s)‖∞

is made as small as possible. This is known as the two sided frequency weighted model

reduction problem (see Figure 2.5). If one of the weights is an identity, the problem

is known as one sided frequency weighted model reduction, where the objective is to

find a stable lower order model Gr(s), such that ‖(G(s)−Gr(s))V (s)‖∞ (in case of

input weighting) and ‖W (s) (G(s)−Gr(s))‖∞ (in case of output weighting) is made

as small as possible. Figures 2.6 and 2.7 represent the input and output frequency

weighted error systems respectively. Enns’ [17] was the first to formulate this problem

by introducing frequency weightings to the balanced truncation [45] scheme.

2.9.3 Enns’ Technique

Given the original full order stable system G(s) = C(sI − A)−1B + D, the stable

input weighting system V (s) = CV (sI −AV )−1BV +DV and stable output weighting

23

system W (s) = CW (sI − AW )−1BW + DW , the augmented systems (see Figure 2.8

are given by

G(s)V (s) = Ci(sI − Ai)−1Bi +Di

W (s)G(s) = Co(sI − Ao)−1Bo +Do

where

{Ai, Bi, Ci, Di} =

A BCV

0 AV

, BDV

BV

, [ C DCV

], DDV

{Ao, Bo, Co, Do} =

AW BWC

0 A

, BWD

B

, [ CW DWC

], DWD

Let

Pi =

PE P12

P T12 PV

, Qo =

QW QT12

Q12 QE

satisfy the following Lyapunov equations:

AiPi + PiATi +BiB

Ti = 0 (2.28)

AToQo +QoAo + CTo Co = 0 (2.29)

Remark 7 Note that the realizations {Ai, Bi, Ci, Di} and {Ao, Bo, Co, Do} may not

necessarily be minimal especially in certain frequency weighted model reduction appli-

cations like controller order reduction.

24

Expanding (1,1) and (2,2) blocks respectively of the equations 2.28 and 2.29 yield the

following:

APE + PEAT +X = 0 (2.30)

ATQE +QEA+ Y = 0 (2.31)

where

X = BCV PT12 + P12C

TVB

T +BDVDTVB

T (2.32)

Y = CTBTWQ

T12 +Q12BWC + CTDT

WDWC (2.33)

Simultaneously diagonalizing the weighted Gramians PE and QE, we get

T TQET = T−1PET−T = diag{σ1, σ2, . . . , σn} (2.34)

where σi ≥ σi+1, i = 1, 2, . . . , n − 1 and σr > σr+1. Transforming and partitioning

the original system, we get

A = T−1AT =

A11 A12

A21 A22

, B = T−1B =

B1

B2

,C = CT =

[C1 C2

], D = D

whereA11 ∈ Rr×r. The reduced order model is given byGr(s) = C1(sI−A11)−1B1+D.

Remark 8 For input weighting only, the symmetric positive matrices PE and Q are

simultaneously diagonalized in 2.34. Similarly, for output weighting only, the sym-

metric positive matrices P and QE are simultaneously diagonalized in 2.34, where P

and Q satisfy following Lyapunov equations:

AP + PAT +BBT = 0

ATQ+QA+ CTC = 0

25

Remark 9 The realization{A, B, C,D

}, obtained via applying the weighted balanc-

ing transformation T on the original system {A,B,C,D}, may not be balanced in a

strict sense (i.e., the unweighted controllability and observability Gramians of the re-

alization{A, B, C,D

}may not be diagonal and equal). Consequently, any realization

{Akk, Bk, Ck, D} for k = 1, 2, obtained via partitioning and truncating the realization{A, B, C,D

}, may not be balanced in contrast to the unweighted internally balanced

realization.

Remark 10 When the frequency weighted Hankel singular values are distinct, the

frequency weighted balanced realization{A, B, C,D

}is unique within a change of

sign of a state variables [17]. As a result, the reduced order model and the frequency

weighted approximation error for a given order are invariant under a similarity trans-

formation of the original system and the weighting functions. Note that, the reduced

order model is not invariant under similarity transformation applied on the augmented

realizations. In the following sections, we see that applying a similarity transforma-

tion on augmented realization yield a new frequency weighted model reduction meth-

ods [39, 59].

Remark 11 Since the reduced order models are obtained directly from truncating the

realization{A, B, C,D

}, they do not necessarily retain the frequency weighted Hankel

singular values of the original system.

Remark 12 Truncating the least dominant Hankel singular values does not guarantee

a lower weighted approximation error. Some numerical examples are produced in [20]

to support this fact.

26

Theorem 1 [20] Let G(s) be asymptotically stable original system. Let V (s) and

W (s) are minimum phase input and output weighting functions, respectively. Let

Gr(s) is stable reduced order system obtained using Enns method. There exist no

upper bound of the type

‖W (s) (G(s)−Gr(s))V (s)‖∞ ≤ f(σr+1, · · · , σn, C, A,B)n∑

i=r+1

σi

where f(.) depending only on its arguments.

Remark 13 Note that, the arguments of the function f(.) in above theorem do not

include weighting functions W (s) and V (s), the inclusion of weighting function may

provide extra freedom.

Following is a special case of Enns one sided frequency weighted model reduction

scheme.

Theorem 2 [67] Let G(s) be square and minimum phase transfer matrix. The

reduced order system Gr(s) obtained using Enns one sided frequency weighted model

reduction scheme is minimum phase and satisfies

∥∥G−1(s) (G(s)−Gr(s))∥∥∞ ≤

n∏i=r+1

(1 + 2σi

√1 + σ2

i + 2σ2i )− 1

∥∥G−1r (s) (G(s)−Gr(s))

∥∥∞ ≤

n∏i=r+1

(1 + 2σi

√1 + σ2

i + 2σ2i )− 1

Remark 14 The error bound also holds for possibly a non square transfer matrix,

when G(s) is full row rank for output weighting case, or dually G(s) is full column

rank for input weighting case.

27

Remark 15 An equivalence between Enns’ one sided frequency weighted model re-

duction scheme and balanced stochastic truncation [14] is established in [67], where

it is shown that for square and minimum phase transfer matrix G(s), the balanced

stochastic realization can be obtained by solving a pair of Lyapunov equations, instead

of one Lyapunov equation and one algebraic Riccati equation.

Remark 16 One of the important properties of balanced stochastic truncation is that

right half plane zeros of the original system are preserved in the reduced order system

[22]. Kim et al [34] extended balanced stochastic truncation technique to carry one

side weighting with the aim to reduce the index ‖G−1(s) (G(s)−Gr(s))V (s)‖∞. An

assertion in [34] that the number of right half plane zeros are preserved in the reduced

order model in the weighted balanced stochastic truncation case. We note that the

number of right half plane zeros are not guaranteed to be preserved in the reduced

order model.

Remark 17 Since the symmetric matrices X and Y in equations 2.30 and 2.31 may

not be positive semidefinite, the models obtained by Enns technique may not be stable

for two-sided weighting case. Examples obtaining unstable reduced order model are

produced in [60]. Note that, the positive semidefinite condition on matrix X and

Y for guaranteeing stability of reduced order model is only sufficient. Inspired from

the stability of reduced order model obtained using one sided frequency weighted model

reduction by Enns method, we note following sufficient condition for existence of stable

reduced order model in the case of two sided frequency weighted model reduction by

Enns method:

1. X ≥ 0 and{A,X

12

}is controllable, or

28

2. Y ≥ 0 and{Y

12 , A

}is observable.

2.9.4 Wang’s Technique

The stability problem is solved in Wang et al’s technique [62] by making the matrices

X (2.32) and Y (2.33) positive semidefinite. In this technique, the new controlla-

bility and observability Gramians PWSL and QWSL are obtained as the solutions to

Lyapunov equations respectively:

APWSL + PWSLAT +BWSLB

TWSL = 0

ATQWSL +QWSLA+ CTWSLCWSL = 0

are simultaneously diagonalized

T TQWSLT = T−1PWSLT−T = diag{σ1, σ2, . . . , σn}

where σi ≥ σi+1, i = 1, 2, . . . , n − 1 and σr > σr+1. The matrices BWSL and

CWSL in the above Lyapunov equations are fictitious input and output matrices

BWSL = UWSL|SWSL|1/2 and CWSL = |RWSL|1/2V TWSL, respectively. The terms

UWSL, SWSL, VWSL, and RWSL are obtained from the orthogonal eigen-decomposition

of symmetric matrices X = UWSLSWSLUTWSL and Y = VWSLRWSLV

TWSL, where

SWSL = diag(s1, s2, · · · , sn), RWSL = diag(r1, r2, · · · , rn), |s1| ≥ |s2| ≥ · · · ≥ |sn| ≥ 0

and |r1| ≥ |r2| ≥ · · · ≥ |rn| ≥ 0. Reduced order models are obtained by transforming

and partitioning the original system. Since X ≤ BWSLBTWSL ≥ 0, Y ≤ CT

WSLCWSL ≥

0 and the realization {A,BWSL, CWSL} is minimal, the stability of the reduced order

model in case of double-sided weighting is guaranteed.

29

Remark 18 To establish the relationship between system input matrix B and new fic-

titious input matrix BWSL, the existence of rank [BWSL B] = rank [BWSL] is shown

in some sense [62].

Theorem 3 [62] The following error bounds hold (subject to fulfillment of

rank

[BWSL B

]= rank

[BWSL

]and rank

CWSL

C

= rank

[CWSL

]):

‖W (s)(G(s)−Gr(s))V (s)‖∞ ≤ 2‖W (s)L‖∞‖KV (s)‖∞n∑

i=r+1

σi

where

L = CVWSLdiag(|r1|−12 , |r2|−

12 , · · · , |rni|−

12 , 0, · · · , 0)

K = diag(|s1|−12 , |s2|−

12 , · · · , |sno|−

12 , 0, · · · , 0)UT

WSLB

ni = rank [X] and no = rank [Y ].

2.10 Controller Design

In this Section, we consider some methods for constructing controllers K which pro-

vide internal stability. Those controllers are constructed under certain strict condi-

tions, which is the main reason they are of high order.

We are mainly interested in linear feedback system design and we will briefly

outline the theory behind those methods.

2.10.1 Linear Quadratic Regulator

We assume that we have a finite dimensional linear time-invariant system, with state

space matrices (A,B,C,D). Recall the state space Equations in Section 2.1:

30

x = Ax+Bu, x(t0) = x0 (2.35)

y = Cx+Du (2.36)

and we define the linear quadratic index

J = 12

∫ ∞0

(xT (t)Qx(t) + uT (t)Ru(t))dt (2.37)

where Q is a symmetric non-negative definite (i.e. QT = Q ≥ 0), and R is

symmetric positive definite (i.e. RT = R > 0).

The pair (A,B) is stabilizable if there exists a state feedback u = Fx such that

the system is stable. Also, the pair (C,A) is detectable, if A + LC is stable for

some L. By assuming that the pair (A,B) is stabilizable, the purpose of the Linear

Quadratic Regulator (LQR), is to minimize the index J . To achieve this, we may use

u = −Fx as a feedback condition. It is well known [8,37] that the feedback gain which

minimizes J will satisfy the condition F = R−1BTPc, where Pc is the non-negative

definite solution to the Riccati Equation

PcA+ ATPc − PcBR−1BTPc +Q = 0 (2.38)

And since Q ≥ 0, there exists a p× n real matrix H such that Q = HTH.

As a consequence, if the pair (A,H) is detectable, then the linear quadratic closed-

loop regulator is stable, i.e. Re(λi{A−BF} < 0.

For the sake of simplicity, we may assume R = I. Then Equation 2.38 will

become [41]

31

(I + L(jω))∗(I + L(jω)) = I +W ∗(jω)W (jω), (2.39)

where L(jω) = F (jωI −A)−1B, and W (jω) = H(jωI −A)−1B. From the above,

it is easy to conclude that

σ(I + L(jω)) ≥ 1,∀ω ∈ R (2.40)

i.e. the minimum singular value of the LQ regulator’s input return difference is

bounded below by unity at all frequencies. Therefore, there is a noticeable improve-

ment by using LQ feedback at all frequencies ω.

2.10.2 The Kalman Filter

The system equations (presented in Equation 2.36) can be subject to noise. Then the

equations may become:

x(t) = Ax(t) +Bu(t) + w(t), x(t0) = x0 (2.41)

y(t) = Cx(t) + u(t) (2.42)

where w(t) and u(t) are zero-mean Gaussian noise and are independent, with

covariances Wδ(t − τ) and V δ(t − τ) respectively. By assuming that V > 0, W ≥

0, (A,W12 ) is controllable, and (A,C) is detectable, we are able to design a state

estimator to obtain an estimate x(t) of the state x(t). This would help us to apply

the state feedback control law. The estimator equation is

32

˙x(t) = Ax(t) +Bu(t) + L(y(t)− y(t)), x(t0) = x0 (2.43)

y(t) = Cx(t) (2.44)

where L = PfCTV −1 is the estimator gain, and Pf is non-negative definite and

satisfies yet another Riccati equation:

APf + PfAT − PfCTV −1BPf +W = 0 (2.45)

It is well known that by using the above assumptions, the estimator gain L will

provide a stable estimator in Equation 2.44, and at the same time will minimize the

performance index

J = lim tr(E){(x(t)− x(t))(x(t)− x(t))T} (2.46)

By using the estimates of the state in the LQ feedback control law (i.e. imple-

menting u(t) = −F (t)), we will obtain a dynamic compensator transfer function

K(s) = F (sI − A+BF + LC)−1L (2.47)

Although this compensator provides a stable closed-loop system, it is not neces-

sarily open-loop stable [65].

2.10.3 H∞ Controller Design

Infinity norm (H∞) methods are commonly used in control theory to design stable

controllers with robust performance. Firstly, the control problem is analyzed as a

33

mathematical optimization problem and then we find the controller which satisfies

the conditions of the problem. The techniques using the H∞ norm have an advan-

tage over classical control techniques in that they are readily applicable to problems

involving multivariable systems with cross-coupling between channels. But there also

disadvantages, as the require a high level of mathematical understanding in order to

apply them successfully and then obtain a reasonably good model of the system to be

controlled. The formulation of the mathematical problem is very important, because

most of the times, the conditions are very sensitive to small changes.

H∞ is defined as the space of matrix-valued functions that are analytic and

bounded in the open right-half of the complex plane defined by Re(s) > 0. The

H∞ norm is then the maximum singular value of the function over that space.

Moreover, simultaneously optimizing robust performance and robust stabilization

is difficult. One method that comes close to achieving this is by using H∞ loop-

shaping, which allows us to apply classical loop-shaping concepts to the multivariable

frequency response to get good robust performance, and then optimizes the response

near the system bandwidth to achieve good robust stabilization.

We now consider the system defined by the block diagram in Figure 2.3, where

bothG andK are real, rational and proper. We requireK to provide internal stability.

A controller is said to be admissible if it is real-rational, proper, and stabilizing.

By assuming that the realization of G is described by

G(s) =

A B1 B2

C1 D11 D12

C2 D21 0

=

A B

C D

34

we may make the following assumptions:

(1) (A,B2) is stabilizable and (C2, A) is detectable.

(2) D12 =

0

I

, and D21 =

[0 I

].

(3)

A− jωI B2

C1 D12

has full column rank for all ω.

(4)

A− jωI B1

C2 D21

has full row rank for all ω.

The first assumption is necessary for the existence of a stabilizing controller. The

second assumption normalizes the sensor noise weighting and makes it non-singular.

The third and fourth assumptions are made for technical reasons.

We now define:

35

R = D13∗D13 −

γ2I 0

0 0

, where D13 =

[D11 D12

](2.48)

R = D31∗D31 −

γ2I 0

0 0

, where D31 =

D11

D21

(2.49)

H∞ =

A 0

−C1∗C1 −A∗

− B

−C1∗D13

R−1

[D13

∗C1 B∗]

(2.50)

J∞ =

A 0

−B1B1∗ −A

− C∗

−B1D31∗

R−1

[D31B1

∗ C

](2.51)

F =

F1∞

F2∞

= −R−1

[D13

∗C1 +B∗X∞

](2.52)

L =

[L1∞ L2∞

]= −

[B1D31

∗ + Y∞C∗

]R−1 (2.53)

where X∞ and Y∞ are the solutions of the Riccati equations X∞ = Ric(H∞) and

Y∞ = Ric(J∞) [69]. The partition of D, F1∞, and L1∞ is done as

F ′

L′ D

=

F ∗11∞ F ∗12∞ F ∗2∞

L∗11∞ D1111 D1112 0

L∗12∞ D1121 D1122 I

L∗2∞ 0 I 0

Theorem 4 [69] Let us assume that the assumptions (1),(2),(3), and (4), are all

satisfied by G. Then:

1. There exists an admissible controller K(s) such that such that ‖Fl(G,K)‖∞ < γ

(i.e. ‖Tzw‖∞ < γ ) if and only if

36

• γ > max(σ[D1111, D1112], σ[D1111∗, D1121

∗], (where σ denotes the largest sin-

gular value),

• H∞ ∈ dom(Ric) with X∞ = Ric(H∞) ≥ 0,

• J∞ ∈ dom(Ric) with Y∞ = Ric(J∞) ≥ 0,

• ρ(X∞Y∞) < γ2 (where ρ(X∞Y∞) is the spectral radius of X∞Y∞).

2. Given that all the above conditions are satisfied, then all rational internally

stabilizing controllers K(s) satisfying ‖Fl(G,K)‖∞ < γ are given by

K = Fl(M∞, Q), for arbitrary Q ∈ RH∞ such that ‖Q‖∞ < γ (2.54)

where

M∞ =

A B1 B2

C1 D11 D12

C2 D21 0

, (2.55)

D11 = −D1121D1111∗(γ2I −D1111D1111

∗)−1D1112 −D1122,

D12, D21 are any matrices satisfying

D12D12∗

= I −D1121(γ2I −D1111D1111

∗)−1D1121,

D21

∗D21 = I −D1112(γ

2I −D1111∗D1111)

−1D1112,

and

B2 = Z∞(B2 + L12∞)D12,

C2 = −D21(C2 + F12∞),

B1 = −Z∞L2∞ + B2D12−1D11,

C1 = F2∞ + D11D21−1C2,

A = A+BF + B1D21−1C2,

37

where Z∞ = (I − γ−2Y∞X∞)−1.

A proof of the above Theorem and a more detailed description of the H∞ formu-

lation, may be found in [69].

Remark 19 Although the assumptions are made for real transfer functions, the re-

sults here are also true for complex cases.

38

Figure 2.1: Singular Value Decomposition is applied when the matrix is non-square.

39

Figure 2.2: Block diagram of an LFT system (upper LFT).

Figure 2.3: Closed loop system diagram with a high order controller.

Figure 2.4: Closed loop system diagram with a reduced order controller.

40

Figure 2.5: Input-output frequency weighted error system.

Figure 2.6: Input frequency weighted error system.

Figure 2.7: Output frequency weighted error system.

41

Figure 2.8: Input-output augmented systems.

42

Chapter 3

The Double Controller Technique

3.1 Introduction

Controller reduction problems are usually solved via frequency weighted model reduc-

tion problem [7, 17, 24, 50, 62, 69]. The frequency weighted model reduction problem

can be classified into single-sided or double-sided frequency weighted problems. The

single-sided frequency weighted model reduction problem is based on stability mar-

gin considerations. The reduced-order controller should satisfy the same conditions as

are listed in the above references. The double-sided problem is based on closed-loop

system approximation and attempts to minimize an index of the form:

e = ‖V1(K −Kr)V2‖∞

where V1 = (I +GK)−1G and V2 = (I +GK)−1

and G, K, and Kr are the plant, the original controller and the reduced controller

respectively.

There are a few methods for the solution of the frequency weighted model reduc-

43

tion problem [17,62]. However, approximation errors obtained using these techniques

are large. In general, the techniques are not as good as the techniques available for

the unweighted case [45].

In this Thesis, a new frequency weighted model reduction formulation is proposed

using standard techniques [17,25,61,62,66–68]. The formulation is based on deriving a

new set of weights for double-sided frequency weighted model reduction problem. Out

of the two new weights derived, one of the weights can be made to be a function of free

parameters. By varying those free parameters in the resulting double-sided frequency

weighted model reduction problem, the frequency weighted error can be significantly

reduced, subsequently obtaining more accurate low-order controllers. Various results

are reported in [26–28].

Three closed-loop configurations are considered for deriving the new double-sided

frequency weighted model reduction problems.

case (i) closed loop configuration with the controller in the forward path (see Figure

3.1),

case (ii) closed-loop configuration with the controller in the feedback path instead of the

forward path (see Figure 3.6) and,

case (iii) plant and controller in an LFT configuration (see Figure 2.2).

Using the above configurations, three new frequency weighted model reduction

problem formulations are derived. The advantage of these formulations is that one

of the weights becomes a function of free parameters which can be varied to reduce

the approximation errors obtained by using standard frequency model reduction tech-

niques [17, 62].

44

The methods presented here are applicable to both SISO (Single Input and Single

Output) and MIMO (Multiple Input Multiple Output) systems. The notation used

throughout the Thesis for transfer functions (plants, controllers, and any combination

of them) will represent both continuous and discrete cases. For example, a plant G

will represent both G(s) (continuous case) and G(z) (discrete case), unless stated

otherwise. Furthermore, we introduce an efficient searching methods for finding the

optimal parameters.

We will use the following definitions:

Definition 5 A closed-loop control system with a controller in the cascade loop (as

seen in Figure 3.1) will be called a cascade system.

Definition 6 A closed-loop control system with a controller in the feedback loop (as

seen in Figure 3.6) will be called a feedback system.

Definition 7 A closed-loop control system using an LFT configuration (as seen in

Figure 3.9) will be called an LFT system.

3.2 The Double Controller Technique for Cascade

Systems

3.2.1 Original Methods

Consider the closed-loop system shown in Figure 3.1, with the plant G and the con-

troller K. The transfer function of the closed-loop system is given by

W = GK(I +GK)−1 (3.1)

45

In the closed-loop system configuration shown in Figure 1, if the original controller

K is replaced by a reduced-order controller, Kr, then the closed-loop system transfer

function is given by

Wr = GKr(I +GKr)−1 (3.2)

Remark 20 In the controller reduction problem, the objective is to find a reduced-

order controller, Kr such that the closed-loop systems are approximately equal. Be-

cause of the order simplification, it is not possible to have (in general) Wr = W .

Therefore, a more realistic approach is to minimize the index ‖W −Wr‖∞, so that

the closed-loop systems W and Wr can become approximately equal, i.e., Wr ≈ W .

Assuming that the second order terms are negligible in K − Kr, we write the

following [7]

W −Wr = (I +GK)−1G [K −Kr] (I +GK)−1 (3.3)

A more detailed expansion of the above Equation exists in the Appendix.

Remark 21 In the above expression, we are neglecting the second and higher order

terms of K−Kr. By obtaining a reduced controller Kr, the difference K−Kr is already

very small when compared to either K or Kr. Moreover, a term of the form (K−Kr)2

will become extremely small, and although it is making the approximation formula

more complicated, it has no effect on the actual approximation. The same justification

holds for orders higher than two. More specifically, it can be easily shown (using simple

calculations) that the difference of the two parts of the above approximation is equal

to

(I +GK)−1G[G(K −Kr)(I +GK)−1

]2(3.4)

46

which in turn means that since the term G(K −Kr)(I + GK)−1 is already supposed

to be very small, the approximation in Equation 3.3 is acceptable.

Therefore, the controller reduction problem can be reduced to a double-sided

frequency weighted model reduction problem, which aims to minimize an index of

the form:

e = ‖V1(K−Kr)V2‖∞, (3.5)

where V1 = (I+GK)−1G and V2 = (I+GK)−1 (3.6)

Since the method proposed in this Thesis is applicable to both continuous and dis-

crete systems, the notation used throughout this Thesis for transfer functions (plants,

controllers, and any combination of them) will represent both continuous and discrete

cases. For example, a plant G will represent both G(s) (continuous case) and G(z)

(discrete case), unless stated otherwise.

3.2.2 The Main Results

Consider the closed-loop block diagram shown in Figure 3.1. This system has a plant

G and controller K and a closed-loop transfer function, W . We will first show that

this system can be expressed in another closed-loop configuration (see Figure 3.3)

having the same closed-loop system transfer function, W . The closed-loop system

configuration (Figure 3.2) in turn can be expressed in a third closed-loop configuration

as shown in Figure 3.3. The new configuration uses the original plant G and two

controllers KC and K(I − C) instead of one, where C = [cij] is a non-singular

constant matrix. Note that both closed-loop configurations in Figures 3.1 and 3.3

47

have the same input and output and hence the same closed-loop system transfer

function. Furthermore, Figure 3.3 is structurally equivalent to Figure 3.3, since we

can obtain Figure 3.3 from Figure 3.2 by replacing K and G in Figure 3.2 with KC

and a closed-loop system consisting of G in the forward path and K(I − C) in the

feedback loop (as shown by dashed lines in Figure 3.3) respectively.

Recall that for deriving frequency weights in standard techniques [7,17,25,61,66–

68], we use the closed-loop system configuration shown in Figure 3.1. For deriving

the new set of frequency weights we use the closed-loop configuration in Figure 3.3.

Definition 8 The configuration in Figure 3.3 will be called the Double Controller

Form of the closed-loop system W . Moreover, the use of the Double Controller Form

to manipulate the frequency weights by changing the matrix C of free parameters, will

be called the Double Controller Technique.

The advantage of using the Double Controller Form is that one of the weights

will be a function of the matrix C. By varying the parameters of the matrix, we

can significantly reduce the approximation error when using any frequency weighted

model reduction technique.

3.2.3 Relationships between closed-loop configurations

In this subsection we derive the relationships between the closed-loop configurations

shown in the block diagrams, Figures 3.1 - 3.3. In particular we will derive the

relationships between the new plant G and the new controller K in terms of the old

plant G, the old controller K and a matrix C of free parameters.

48

Let W = GK(I+GK)−1, W = GK(I+GK)−1 be closed-loop systems with plants

and controllers G, K and G, K respectively. Let us also define H(C) = I+GK(I−C).

By assuming that (I + GK)−1 exists, it can be shown that (I + GK(I − C))−1 also

exists for given C, except for a finite number of values for s (continuous case) or z

(discrete case). The proof (which is a simple extended version of the proof in [26],

but for matrices) is omitted due to space restrictions. We will disregard those finite

number of values, as we have infinite choices for s or z. For C = I we have H(I) = I.

We also define H = I +GK and H = I + GK.

Lemma 1 Assume we have the closed-loop systems W = GKH−1 and W = GKH−1

as defined above. If C is a non-singular real matrix such that K = KC and G =

H−1(C)G, then we have

W = WC. (3.7)

Proof 1 First we have

H−1 = (I + GK)−1 = (I +H−1(C)GKC)−1

= (H(C) +GKC)−1H(C)

= (I +GK(I − C) +GKC)−1H(C)

= (I +GK)−1H(C) = H−1H(C) (3.8)

Since the inverses of H−1(C) and H exist, we have that H−1 also exists:

H−1 = H−1H(C) (3.9)

49

Figure 3.1: A closed-loop system with plant G and controller K

Figure 3.2: The new closed-loop system with plant G and controller K

Note that for any closed-loop system W, the commutative property GKH−1 =

H−1GK holds [50]. Then we use this property and the definitions of G, K and H−1

to obtain W :

W = GKH−1 = H−1GK = H−1H(C)H−1(C)GKC

= H−1GKC = GKH−1C = WC (3.10)

Therefore, for C nonsingular, we could replace the closed-loop system W in Figure

3.1 by the closed-loop system WC−1 in Figure 3.2.

Remark 22 By using the definitions of G and K we have

G = (I +G(K − K))−1G. (3.11)

50

Figure 3.3: The plant G is itself a closed-loop system with plant G and controller

K − K = K(I − C)

Figure 3.4: The double controller technique uses a more complex way for controller

reduction.

And as shown by the dashed lines in Figure 3.3, G can itself be regarded as a

closed-loop feedback system with plant G and controller K − K = K(I − C).

The real gain here is the revelation of a new matrix C of free parameters. By

looking at the figures, it is clear that Figure 3.3 is a generalization of Figure 3.1, and by

setting C = I both block diagrams will become identical. The input and output always

remain the same regardless of C, which means that we may manipulate the parameter

C without affecting the system, but we may improve the non-linear procedure for

51

calculating a reduced-order controller. In our technique, instead of directly calculating

the reduced system from the original system, we use three steps:

1. We multiply the system W by the constant matrix C (linear procedure).

2. We reduce the system by using the standard controller reduction technique (non-

linear procedure). For large C, there is more room for improvement between WC

and WrC, than between W and Wr.

3. Finally, after finding Wr′C, we multiply it by C−1 (linear procedure).

In Figure 3.4, we can see how the new technique works. The dashed lines show

the linear manipulations, while the full lines show the non-linear ones. We will show

that the constant matrix C plays a major role in decreasing the approximation error

in controller reduction.

3.2.4 The Derivation of New Frequency Weights

The main aim of controller reduction is to obtain a low-order controller by approxi-

mating the closed-loop behavior of the system.

In this subsection, we derive the new set of frequency weights using the closed-loop

configuration shown in Figure 3.3. This is achieved by approximating the difference

between the closed-loop systems W and Wr, where Wr = WrC, and Wr, Wr are the

closed-loop systems with the lower order controllers Kr(C), Kr respectively. The

reduced controller Kr(C) is obtained from a system W with plant G = (I +GK(I −

C))−1G and controller K = KC. Therefore it is dependent on C. Since the procedure

for obtaining a reduced controller is non-linear, for C1 6= C2 we should also have

Kr(C1) 6= Kr(C2).

52

Let us now consider the closed-loop system W = WC with the assumptions used

in Lemma 1.

We can express the difference WC − WrC by the difference W − Wr. More

specifically:

(W −Wr)C = W − Wr (3.12)

From Equation 3.3 we have

W −Wr = H−1G(K −Kr)H−1 and

W − Wr = H−1G(K − Kr(C))H−1.

Therefore, Equation 3.12 may be rewritten as

H−1G(K−Kr)H−1C = H−1G(K−Kr(C))H−1. (3.13)

Remark 23 Without loss of generality, and since the controller Kr(C)) is to be con-

structed with regards to our own specifications, we may strengthen the quality of all

the above approximations by assuming that the left and right terms of Approximation

3.13 are extremely close, if not equal.

Kr(C) = Kr(C)C−1 (3.14)

Now define Kr(C) = Kr(C)C−1. For C = I, it is clear that H(I) = I, G = G and

K = K, which implies Kr = Kr(I). We also define I(C) = HCH−1H(C)C−1.

Theorem 5 Using the assumptions in Lemma 1, we have

H−1G(K−Kr(I))H−1 = H−1G(K−Kr(C))H−1I(C) (3.15)

53

Proof 2 To prove the above Theorem, we must bring the RHS of Equation 3.13 into

a more explicit form.

Recall that K = KC and Kr(C) = Kr(C)C−1. By using Equation 3.9 it can be

shown that

H−1G(K−Kr(C))H−1 = H−1G(K −Kr(C))CH−1H(C). (3.16)

Then we have:

H−1G(K−Kr(C))H−1 = H−1G(K −Kr(C))CH−1H(C)

= H−1G(K −Kr(C))ICH−1H(C)

= H−1G(K −Kr(C))(H−1H)CH−1H(C)

= H−1G(K −Kr(C))H−1(HCH−1H(C))

= H−1G(K −Kr(C))H−1(HCH−1H(C))I

= H−1G(K −Kr(C))H−1(HCH−1H(C))(C−1C)

= H−1G(K −Kr(C))H−1(HCH−1H(C)C−1)C

= H−1G(K −Kr(C))H−1I(C)C.

Thus, by substituting the last part of the above Equation into Equation 3.13, and

then canceling out the constant matrix C, we obtain the required result.

Remark 24 The new weights follow immediately from the RHS of Equation 3.15:

V1 = H−1G = (I +GK)−1G and V2(C) = H−1I(C) (3.17)

54

For C = I, we have V2(C) = V2 and we get the standard weights defined in

Equation 3.6.

Remark 25 Comparing the two set of weights in Equations 3.6 and 3.17, observe

that the output weights, V1 are exactly the same while the input weights are different.

The new input weight V2(C), is now a function of a matrix C of free parameters,

whereas the input weight V2 in the standard techniques is a fixed transfer function

[7,17]. The proposed method is based on varying this free parameter to achieve lower

approximation errors and hence better low-order controllers.

Remark 26 Since GK is a transfer function matrix, it can only commute with the

constant matrix cI (where c is a scalar parameter) or, in very rare cases (provided

the transfer function matrix GK has a certain structure), with a constant matrix cC ′,

where C ′ 6= I is dependent on the structure of GK.

Lemma 2 If in Theorem 5 we have C = cI, where c is a real number, then C

commutes with GK, that is CGK = GKC, and Equation 3.15 will be simplified into

the form

H−1G(K−Kr(I))H−1 = H−1G(K−Kr(C))H−1H(C) (3.18)

Proof 3 Since C = cI, it commutes with all the matrix terms GK, H, H−1, H(C)

and H−1(C). Therefore, regarding I(C), the term C will be canceled out by the

term C−1, and consequently, H will be canceled out by H−1. Therefore, I(C) will be

simplified to H(C).

55

Remark 27 According to the above Lemma, if C commutes with GK, the new weights

will now be:

V1 = H−1G = (I +GK)−1G (3.19)

V2(C) = H−1H(C) = (I +GK)−1[I +GK(I − C)] (3.20)

Remark 28 We now consider the simplest of all commutative matrices, that is C =

cI. The case of non-commutative matrices C is under investigation (i.e., using I(C)

instead of H(C), which restricts C to only one parameter). Specifically, we seek to

generalize this theory along the following lines:

1. Let C = diag[c1, c2, ..., cn], which requires more complex calculations, but would

provide improved error reduction. The entries, ci, can be viewed as weighting

parameters in the optimization of the error.

2. Let C be a full matrix.

3. For a known plant G and controller K, perform a search over the general form

matrices C that commute with GK.

We seek to generalize the theory for future usage. Although it is beyond the scope of

this Thesis, under certain circumstances, we may partially use (wherever the inverse

of C is not required) commutative matrices which have a number of diagonal entries

equal to a non-zero c and all the other entries equal to zero. Another usage might be

for a known plant G and a controller K, the search of the general form of matrices C

which commute with GK. All these cases are under investigation. But since we are

56

also using MIMO cases, it should be a good practice to include all the paths from the

generalization of an arbitrary matrix C toward the simplification of a matrix cI. The

case where we may use non-commutative matrices C is also under investigation (i.e.

using I(C) instead of H(C) which restricts the matrix C to only one parameter), and

the possibilities are too many to list here. In the case where C = diag[c1, c2, ..., cn,

we are required to use more complex calculations, but there is certainly potential for

an even better error reduction. For example, certain parameters ci could be given a

higher or lower weight, depending on our specifications for the reduced controller or the

optimization of the error. The same could be said for a full matrix C = [cij], where

we have even more freedom of choice, but the complexity of the problem increases

exponentially. In this Thesis, we will use the simplest of all commutative matrices,

that is C = cI.

By defining C = cI where c is a constant, the matrix clearly commutes with GK,

and we can safely use the weights defined in Equations 3.19 and 3.20, that is, we may

use H(C) instead of I(C). The matrix I(C) (which is a part of the right weight V2(C)

as shown in Equation 3.17) is too complex to use, as it will involve any matrix C,

and a much more expanded error analysis.

Furthermore, we will have H(C) = I + GK(1 − c). Since C now depends on the

scalar parameter c, in this Thesis the notations H(c), Kr(c) and Kr(c) will be used

instead of H(C), Kr(C) and Kr(C).

57

3.2.5 Error Analysis

Given G, K and c, it is standard procedure to derive a lower order controller Kr =

Kr(c) by minimizing ‖H−1G(K −Kr)H−1H(c)‖∞ using any of the standard double-

sided frequency weighted model reduction techniques [69], [17]. Let us define E(c1, c2)

and e(c1, c2) as

E(c1, c2) = H−1G(K −Kr(c1))H−1H(c2) (3.21)

e(c1, c2) = ‖E(c1, c2)‖∞. (3.22)

In other words, the term e(c1, c2) symbolizes the approximation error that uses

Kr(c1) as the reduced controller, and H−1G and H−1H(c2) as left and right weights.

Note that, the controller Kr(c1) is defined as being found by minimizing the error

e(c1, c1), not e(c1, c2). Moreover, we have H(1) = I, and e(1, 1) corresponds to the

approximation error obtained using standard weights (Equation 3.6) in double-sided

frequency weighted model reduction techniques [69], [17]. Without loss of generality,

and for the benefit of obtaining a better approximation error e(c1, c2)), we are allowed

to replace Kr(1) by a controller Kr(c1), and H−1 = H−1H(1) by H−1H(c2), since

they all correspond to approximate systems.

Remark 29 The controller Kr(c1) depends on c1 and H(c1), but the way Kr(c1)

is influenced by H(c1) (in making the approximation error smaller) is more complex,

and it mostly depends on the mechanism of the frequency weighted balanced truncation

method that is used at the time (for example Enns’s [17] or Wang et al’s [62] method).

58

If we right multiply E(c, 1) by H(c), then from Equation 3.15 we will get

Lemma 3

E(c, c) = E(1, 1). (3.23)

(3.24)

E(1, c) = E(1, 1)H(c). (3.25)

E(c, 1) = E(1, 1)H−1(c). (3.26)

Proof 4 Equation 3.23 is directly derived from Equation 3.18 by using the definitions

of E(c, c) and E(1, 1).

Equation 3.25 is also directly derived from the definitions of E(1, c) and E(1, 1),

by using the formula in Equation 3.21.

Regarding Equation 3.26, we use the formula in Equation 3.21 and the definitions

of E(c, 1), E(c, c) to get E(c, 1)H(c) = E(c, c). Then, by using Equation 3.23 we

obtain E(c, 1)H(c) = E(1, 1), which finally gives E(c, 1) = E(1, 1)H−1(c).

We will also have

E(c, 1)H(c) = E(1, 1)

E(c, 1)[(1− c)I + cH−1] = E(1, 1)H−1. (3.27)

59

Lemma 4 Let h(c) = ‖H(c)‖∞ and h(c) = ‖H−1(c)‖∞. Then we have

e(c, c) = e(1, 1). (3.28)

(3.29)

e(1, 1)h−1(c) ≤ e(1, c) ≤ e(1, 1)h(c). (3.30)

e(1, 1)h−1(c) ≤ e(c, 1) ≤ e(1, 1)h(c). (3.31)

e(c)h−1(c) ≤ eH(c) ≤ e(c)h(c). (3.32)

Proof 5 Equation 3.28 is a direct consequence of Equation 3.23.

Regarding Inequality 3.31, we will first prove the left part of the above expression

and then the right part. From Equation 3.26 we have:

E(c, 1) = E(1, 1)H−1(c)

‖E(c, 1)‖∞ = ‖E(1, 1)H−1(c)‖∞

‖E(c, 1)‖∞ ≤ ‖E(1, 1)‖∞‖H−1(c)‖∞

e(c, 1) ≤ e(1, 1)h(c)

60

Similarly, we have

E(c, 1) = E(1, 1)H−1(c)

E(c, 1)H(c) = E(1, 1)

‖E(c, 1)H(c)‖∞ = ‖E(1, 1)‖∞

‖E(c, 1)‖∞‖H(c)‖∞ ≥ ‖E(1, 1)‖∞

e(c, 1)h(c) ≥ e(1, 1)

e(c, 1) ≥ e(1, 1)h−1(c)

3.2.6 The behavior of H(c)

It is clear from the above that the term H(c) and its two forms of infinity norm

h(c) and h−1(c) are of great importance, and so it is essential to understand their

properties.

Remark 30 Recall Equation 3.27. We can see very clearly that by increasing the

value of the parameter c, the term [I(1− c) + cH−1] will become large. And since the

RHS of this Equation is a constant with respect to c, the size of the term E(c, 1) will

decrease, and the derived controller Kr(c) will give a smaller approximation error.

Theoretically speaking, by looking at Equation 3.27, the approximation error

e(c, 1) tends to zero. This is not the case in real world applications as the error

can never become zero, but we will still achieve a significant error reduction as large

values of c will force the approximation error to converge to a better minimum.

By looking at Equation 3.27, the approximation error e(c, 1) becomes smaller as

c increases. Because of the difference between the high order and low-order transfer

61

functions, the approximation error e(c, 1) can never become zero, but we will still

achieve a significant error reduction as large values of c will force the approximation

error to converge to a better minimum. And as c goes to plus or minus infinity, this

minimum becomes a constant value, which may not be the optimal for all c, but it is

always less than the original error when c = 1. By using some rough assumptions, we

may approximate a value for c, that will give us an approximation error which will

be almost equal to the constant number that e(c, 1) is converging when c tends to

±∞. This means, that if we choose greater values for c, there will be no significant

difference.

Let us choose some number γ, such that we want e(c, 1) = γe(1, 1), where 0 <

γ < 1. From Equation 3.27 we have

‖E(c, 1)[(1− c)I + cH−1]‖∞ = ‖E(1, 1)H−1‖∞

‖E(c, 1)‖∞‖(1− c)I + cH−1‖∞ ≥ ‖E(1, 1)H−1‖∞ (3.33)

If we choose a large c, then it can be assumed without loss of generality that

[I(1− c) + cH−1] = [I(−c) + cH−1] = [c(H−1 − I)]. Then Equation 3.33 becomes

‖E(c, 1)‖∞‖c(H−1 − I)‖∞ ≥ ‖E(1, 1)H−1‖∞

|c|‖E(c, 1)‖∞‖(H−1 − I)‖∞ ≥ ‖E(1, 1)H−1‖∞

|c| ≥ ‖E(1, 1)H−1‖∞[‖E(c, 1)‖∞‖(H−1 − I)‖∞]−1 (3.34)

By assuming that the original transfer function W = GKH−1 has no poles on the

jω-axis, we can easily show that the terms I−GKH−1, H−1, H−1−I, and E(1, 1)H−1

62

have no poles on the jω-axis. Thus, we may safely use their corresponding infinity

norms to calculate c. And if we substitute e(c, 1) by γe(1, 1) we finally have

|c| ≥ ‖E(1, 1)H−1‖∞[γe(1, 1)‖(H−1 − I)‖∞]−1 (3.35)

For γ = 13, we may choose a big enough c that gives the required approximation

error. As we have stated, in practice, we may not expect to derive an approximation

error e(c, 1) as small as γ times (in this case 13) of the original error e(1, 1), but we will

derive a big improvement of this error, equal to the value of e(c, 1) for c converging

to ±∞. A satisfactory assumption would always require a number γ which is “small

enough”. The reason is, that a derived low-order controller Kr(c) can never be equal

to the original controller K, and there is a limit (depending on the order of the low-

order controller) of how small their difference (and subsequently the difference of their

corresponding systems) can be.

The term H−1(c) can be written in a more explicit way as

1

1− c(I

1

1− c+GK)−1

The above expression, if seen as a quantity, tends to zero as c tends to infinity.

This affects both h−1(c) and h(c) which will also become very small. As a result, the

error e(c, 1) in Equation 3.31 would also become very small when compared to e(1, 1),

as c tends to infinity.

Firstly, we investigate the properties of h(c). Recall that h(c) = ‖I+GK(1−c)‖∞

and define λ = ‖GK‖∞. By using standard norm properties, we have

|‖A‖∞ − ‖B‖∞| ≤ ‖A+B‖∞

63

Figure 3.5: The upper and lower bounds for h(c), where λ = ‖GK‖∞.

for any matrices A,B, we have

|‖I‖∞ − ‖GK(1− c)‖∞| ≤ h(c) ≤ ‖I‖∞ + ‖GK(1− c)‖∞

|1− ‖GK‖∞|1− c|| ≤ h(c) ≤ 1 + ‖GK‖∞|1− c|

|1− λ|1− c|| ≤ h(c) ≤ 1 + λ|1− c| (3.36)

And since λ is a fixed positive number (for given G and K), we know that the

boundaries for h(c) as shown in the bright shaded area in Figure 3.5.

Secondly, we investigate the properties of h−1(c), which is closely related to h(c).

We will use the sub-multiplicative property of the infinity norm to help us find the

properties and boundaries of h−1(c). The sub-multiplicative property states that

‖AB‖∞ ≤ ‖A‖∞‖B‖∞ for any functions A and B, and is one of the properties of the

64

infinity norm. So by assuming that the inverse of H(c) exists, we have

H(c)H−1(c) = I

‖H(c)H−1(c)‖∞ = ‖I‖∞

‖H(c)‖∞‖H−1(c)‖∞ ≥ ‖I‖∞

h(c)h(c) ≥ 1

h(c) ≥ h−1(c) (3.37)

Therefore, we only need to find a lower bound for h−1(c). Recall that H(c) = I +

GK(1− c). It cab be easily shown that

H(c)H−1(c) = I

(I +GK(1− c))H−1(c) = I

H−1(c) = I −GK(1− c)H−1(c) (3.38)

By taking the infinity norm of both sides, we have

‖H−1(c)‖∞ = ‖I −GK(1− c)H−1(c)‖∞

h(c) ≤ ‖I‖∞ + ‖GK(1− c)H−1(c)‖∞

h(c) ≤ ‖I‖∞ + ‖GK(1− c)‖∞‖H−1(c)‖∞

h(c) ≤ 1 + λ|1− c|h(c) (3.39)

h(c)(1− λ|1− c|) ≤ 1 (3.40)

where λ = ‖GK‖∞. For λ−1λ< c < λ+1

λall components in Inequality 3.40 are positive

and by solving for h−1(c) we get

65

h−1(c) > 1− λ|1− c| (3.41)

So, from Inequality 3.41, for c ≤ λ−1λ

or c ≥ λ+1λ

, h−1(c) will only have zero as a

lower bound.

Thus, the upper and lower bounds for both h(c) and h−1(c) are represented by

the entire shaded area in Figure 3.5. By observing the dependence of e(c, 1) on both

h−1(c) and h(c) we can see that by finding a parameter c that makes h(c) large enough,

we may be able to construct a controller with an improved approximation error.

Remark 31 From Figure 3.5, we may observe that the values of the error func-

tions h(c) and h−1(c) become approximately linear when c lies inside the intervals

(−∞, λ−2λ

) and (λ+2λ,∞), where λ = ‖GK‖∞. But for λ−2

λ< c < λ+2

λ, there seem to

be many changes to the way both h(c) and h−1(c) behave. So preferably, if we wish

to improve the approximation error, we should choose an interval for c, that includes

the points λ−2λ, λ+2

λ. Accuracy will depend on the number of points chosen to represent

this interval in a discrete way.

Remark 32 The result e(c, c) ≈ e(1, 1) in Equation 3.31 could be misinterpreted

as that e(c, c) is always constant for different values of c, when this is certainly not

the case. This is more clear for e(c, 1) in Equation 3.28. The quantities e(c, c) and

e(c, 1) can become small, always subject to the order difference between the high order

and low-order controllers. Both will have a common lower bound that could not be

optimized any more. On the other hand, both h(c) and h−1(c) can become larger than

1 for c greater than α+2α

or smaller than α−2α

, and as c tends to infinity, their limits

will also go to infinity.

66

Remark 33 The use of all the inequalities above, may seem to raise some questions

regarding the accuracy of the method. But there are new results which show that the

inequality approximations are acceptable.

3.2.7 The Application of the Double Controller Technique

The proposed double controller technique solves the frequency weighted model reduc-

tion problem e(c1, c2) = ‖V1(K−Kr(c1))V2H(c2)‖∞ to find a new improved controller

Kr(c1), by replacing the old weights with new ones (V1 and V2 are the standard weights

(Equation 3.6)). To solve the frequency weighted model reduction problem, we can

use any of the standard techniques, e.g. Enns’ technique [17] and Wang et al’s tech-

nique [62]. Since the weight V2H(c1) is a function of c1, we get different low-order

controllers Kr(c1) for different c1. Note that, c1 = 1 corresponds to the lower order

controller Kr(1) obtained by using the standard weights (Equation 3.6). A logical

approximation for e(c, 1) which lies between e(1, 1)h−1(c) and e(1, 1)h(c), would be

e(c, 1) ≈ e(c, 1) = e(1, 1)h−1(c) + h(c)

2, (3.42)

that is, by taking the mean of the lower and upper bounds of e(c, 1), which is

very easy to calculate. This sum is able to reveal important changes of the behavior

of e(c, 1). The local minimum points of the smooth curve represented by e(c, 1)

are candidate values for which c gives optimal controllers. Note that, this method

depends on the quality of the approximation given in Equation 3.42. Thus, we may

not get the best controller (found after an inefficient exhaustive search), but we will

get a controller whose approximation error is close to the optimum and has better

67

approximation error to the one using the original method with the standard weights.

Summarizing, there are two ways to improve the approximation error.

1. Calculate e(c, 1), and use the values cmin that correspond to its local minimum

points to construct an optimal controller.

2. Construct an optimal controller by using a large value c (as shown in Equation

3.35).

Based on the approximation error used, we propose an algorithm (of one param-

eter) for finding the optimum controller Kr(c) by using weights V1, V2H(c).

3.2.8 The Algorithm

The algorithms presented here are easy to program and use, as there are no restrictions

for the usage of controller reduction methods, when using the extra parameter c.

1. Given plant G and controller K, define the weights V1 = (I + GK)−1G and

V2(c) = (I +GK)−1(I +GK(1− c)).

2. Calculate e(c, 1) and find the values cmin that correspond to its local minimum

points. Alternatively, a large value c may be used instead.

3. Solve the frequency weighted model reduction problem ‖V1(K − Kr(c))V2(c)‖

for those c to compute Kr(c) by using standard techniques (Enns [17] and Wang

et al [62]).

Remark 34 If we apply the double controller technique twice on the same closed-loop

system, this time by using a new parameter matrix c, then it becomes identical to the

68

original technique for c replaced by cc. Therefore, reapplying the technique has no real

effect.

Remark 35 The searching procedure for finding Kr(cmin) may seem expensive, but it

is limited to a small portion of the real numbers (namely the interval (σ1, σ2)). Fur-

thermore, there are strong hints that for large absolute values of c, the approximation

error is always less when compared to the original approximation error for c = 1,

while the optimal values for c seem to be near the inflection point of the function

e(c, 1). Both cases are may be investigated as a future project.

Remark 36 Summarizing, the RHS of Equations 3.3 and 3.18 give rise to two dif-

ferent double sided frequency weighted model reduction problems as follows:

(P1) Find a reduced order controller Kr such that

J1 = ‖(I +GK)−1G [K −Kr] (I +GK)−1‖∞

is minimum.

(P2) Find a reduced order controller Kr such that

J2 = ‖(I +GK)−1G [K −Kr(C)] (I +GK)−1H(C)‖∞

is minimum.

The solution of the problems ((P1) and (P2)), gives low order controllers Kr and

Kr(C) respectively. The Kr obtained using the first method (P1) is unique, while

Kr(C) obtained by the second method (P2) is non-unique, and is a function of a user

chosen free parameter C. By varying this parameter (by using a searching technique),

69

it is possible to find more accurate low order controllers. Furthermore, when C = I,

both methods yield the same Kr. This is because the block diagrams in Figures 3.1

and 3.2 are equivalent. Problem (P2) corresponds to Figure 3.1, and problem (P2)

corresponds to Figure 3.2.

3.3 The Double Controller Technique for Feedback

Systems


In this section we present a double sided frequency weighted model reduction problem

formulation. This is based on extensions of Equation 3.3 and Lemma 2 to the closed-

loop configurations with the controller in the feedback loop (as shown in Figures 3.6

and 3.7 instead of the closed-loop configuration with the controller in series with the

plant (as shown in Figures 3.1 and 3.3). The equivalence between the closed-loop

configurations shown in Figures 3.6 and 3.8 is first established, which is then used to

derive the new set of input and output weights for the new double sided frequency

weighted model reduction formulation.

The advantage of this formulation is that we have a better searching method than

the searching method available for the formulation given in Lemma 2.

Consider the feedback system shown in Figure 3.6, with plant G and controller

K. The transfer function of the closed-loop system is given by

W = (I +GK)−1G (3.43)

In the closed-loop system configuration shown in Figure 3.6, if the original con-

70

troller K is replaced by a reduced-order controller Kr, then the closed-loop system

transfer function is given by

Wr = (I +GKr)−1G (3.44)

Lemma 5 Given a feedback system with plant G, controller K, and reduced order

controller Kr (assuming that the second order terms are negligible in [K −Kr]), we

will have

W −Wr = (I +GK)−1G [K −Kr] (I +GK)−1G (3.45)

Proof 6 Please see the Appendix for the proof.



the form:

e = ‖V1(K −Kr)V2‖∞, where V1 = V2 = (I +GK)−1G (3.46)

The above result is an intermediate step, which although not used directly, will

help to understand the generalized procedure (Lemma 7).

Remark 37 Each of the two weights is equal to the feedback system’s transfer func-

tion (I +GK)−1G.

Remark 38 The closed loop configuration in Figure 3.6 can be expressed in another,

more generalized closed-loop configuration shown in Figure 3.7. A more detailed view

71

Figure 3.6: A feedback system with plant G and controller K.

of the new configuration of Figure 3.7 is shown in Figure 3.8. The new configuration

uses the original plant G and two controllers C2K and K(I−CH(C)) instead of one,

where C is a constant matrix. Observe that both closed-loop configurations in Figures

3.6 and 3.7 (and consequently in Figure 3.8) give the same input and output. Hence,

they give the same closed-loop system transfer function.

Remark 39 From now on, for the sake of simplicity, we will use a scalar constant

C = c.

Definition 9 The configuration in Figures 3.7 and 3.8 will be called the Double Con-

troller Form of the feedback system W . Moreover, the usage of the Double Controller

Form to manipulate the frequency weights by changing the parameter c, will be called

the Double Controller Technique.

Remark 40 The advantage of using the Double Controller Form is that one of the

72

Figure 3.7: A generalization of the feedback system gives us an extra parameter c.

Also, H(c) = I +GK(1− c).

weights will be a function of the parameter c. We will show that by increasing the

absolute value of this parameter, we can significantly reduce the approximation error

when using any standard frequency weighted model reduction technique.



shown in Figures 3.6 and 3.8. In particular, we will derive the relationships between

the new plant G and the new controller K (as shown in Figure 3.7) in terms of the

old plant G, the old controller K and a parameter c.

Remark 41 By observing Figure 3.6, the outputs number of G are equal to the inputs

number of K, and the outputs number of K are equal to the inputs number of G. Thus,

we may assume that G and K have dimensions m × n and n ×m respectively, and

73

Figure 3.8: A detailed view of the generalized block diagram of the feedback system.

this also ensures the existence of GK and KG.

Let W = (I +GK)−1G, W = (I + GK)−1G be two closed-loop transfer functions

with plants and controllers G, K and G, K respectively.

Definition 10

H(c) = I +GK(1− c)

J(c) = I + (I − cH(c))GK.

H ′ = I +KG

H = I + GK

Remark 42 By assuming that (I +GK)−1 exists, it can be shown that (I +GK(1−

c))−1 and (I + (I − cH(c))GK)−1 also exist for a given c, except for a finite number

of values for s (continuous case) or z (discrete case) [27]. We will disregard those

finite number of values, as we have infinite choices for s or z.

74

Assume now that we have the closed-loop systems W = H−1G and W = H−1G

as defined above. Then we have:

Lemma 6 If we are given K = c2K and G = (I + (I − cH(c))GK)−1H(c)c−1G =

J−1(c)H(c)c−1G (where c 6= 0), then

W = H(c)c−1W. (3.47)

Proof 7 Note that, the commutative property GKH−1 = H−1GK holds [50]. As a

consequence, and after some simple calculations, the property H−1(c)H = HH−1(c)

also holds. Now we have

W = (I + GK)−1G

= (I + J−1(c)H(c)Gc−1c2K)−1J−1(c)H(c)c−1G

= (I + J−1(c)H(c)cGK)−1J−1(c)H(c)c−1G

= (J(c) + J(c)J−1(c)cH(c)GK)−1H(c)c−1G

= (J(c) + cH(c)GK)−1H(c)c−1G

= (I + (I − cH(c))GK + cH(c)GK)−1H(c)c−1G

= (I +GK)−1H(c)c−1G = H−1H(c)c−1G

= H(c)c−1H−1G = H(c)c−1W (3.48)

Therefore, for c 6= 0, we have shown that we can replace the closed-loop system

W in Figure 3.6 by the closed-loop system H−1(c)cW in Figure 3.7. The real gain

here is the revelation of a new parameter c.

75

Remark 43 As shown by the dashed lines in Figure 3.8, G can itself be regarded as

consisting of a feedback system with plant G and controller K(I − cH(c)).

Remark 44 By looking at the figures, it is clear that Figure 3.3 is a generalization

of Figure 3.6. This fact can be verified by setting c = 1. Then we will have H(1) =

H ′(1) = J(1) = I and K(I − cH(c)) = K(I − IH(1)) = K(I − I) = 0, which when

substituted, will make both block diagrams identical.

Remark 45 The input and output always remain the same regardless of c, which

means that we may manipulate the parameter c without affecting the system, to get a

more accurate low order controller.

We will show that the parameter c plays a major role in decreasing the approxi-

mation error in controller reduction.


The main aim of controller reduction is to obtain a low order controller by approx-

imating the closed-loop behavior of the system. This is achieved by approximating

the difference between closed-loop systems (W and Wr) which leads to the frequency

weighted model reduction problem as shown in Equation 3.45.


configuration shown in Figures 3.7 and 3.8. This is achieved by approximating the

difference between the closed-loop systems W and Wr.

Let us assume that the lower order controllers Kr(c) and Kr, correspond to the

systems Wr and Wr respectively.

76

Definition 11 We will denote by Kr(c) the reduced order controller obtained from a

system W = H−1(c)cW (which has identical input and output with W ).

Remark 46 The system W has a plant G = J−1(c)H(c)c−1G and a controller K =

c2K. We may observe that the system H−1(c)cW has identical input and output with

W . Although the input and output of the systems W and H−1(c)cW are identical,

the internal structure of the later system is dependent on c. And since the procedure

for obtaining a reduced order controller is non-linear, for c1 6= c2 we should have

Kr(c1) 6= Kr(c2).

Remark 47 We already know that W = H(c)c−1W , or in more analytical terms,

H−1G = H(c)c−1H−1G. Therefore, to relate the differences W −Wr and W − Wr, it

is required to have:

W − Wr = H(c)c−1(W −Wr) (3.49)

or equivalently,

W −Wr = H−1(c)c(W − Wr) (3.50)

By generalizing Equation 3.45 by using W − Wr instead of W −Wr, we get

W − Wr = (I + GK)−1G[K − Kr

](I + GK)−1G (3.51)

Thus, by substituting G and K by J−1(c)H(c)c−1G and c2K, the following formula

may be obtained for W − Wr:

77

Lemma 7

W − Wr = H−1G[K − Kr(c)c

−2]H−1GH ′(c)c (3.52)

Proof 8 By using Equation 3.51 the difference W − Wr will be

W − Wr =

H−1G[K − Kr(c)

]H−1G =

H(c)c−1H−1G[c2K − Kr(c)

]H(c)c−1H−1G =

H(c)H−1G[K − Kr(c)c

−2]H(c)H−1G (3.53)

From Equation 3.50, to relate the difference W − Wr to the difference W −Wr,

we must pre-multiply Equation 3.53 by H−1(c)c. Equation 3.53 then becomes

W − Wr = cH−1G[K − Kr(c)c

−2]H(c)H−1G (3.54)

Also, we have H(c)H−1G = H−1H(c)G = H−1GH ′(c), where H ′(c) = I+KG((1−

c). Therefore, Equation 3.54 finally becomes

W − Wr = H−1G[K − Kr(c)c

−2]H−1GH ′(c)c (3.55)

Remark 48 Instead of attempting to find reduced order controllers using the standard

weights V1 = H−1G and V2 = H−1G, we may use the weights V1 = H−1G and

V2 = H−1GH ′(c)c.

78

Definition 12 We define Kr(c) = Kr(c)c−2.

It is obvious that for c = 1, we will have V1 = V1 and V2 = V2, and Kr(1) = Kr.

The advantage here, is that c does not have to be equal to 1, but it can be varied, to

obtain a more accurate low order controller.


It is standard procedure to derive a lower order controller Kr = Kr(c) by minimizing

‖H−1G [K −Kr(c)]H−1GH ′(c)c‖∞

using any of the standard double-sided frequency weighted model reduction tech-

niques [17, 62].

Definition 13

E(c) = H−1G [K −Kr(c)]H−1G (3.56)

e(c) = ‖E(c)‖∞ (3.57)

EH(c) = H−1G [K −Kr(c)]H−1GH ′(c)c (3.58)

eH(c) = ‖EH(c)‖∞ (3.59)

The term e(c) symbolizes the approximation error which includes the original

weights and has Kr(c) as the reduced order controller, while the term eH(c) symbolizes

the approximation error which has H−1G and H−1GH ′(c)c as left and right weights

and Kr(c) as the reduced order controller.

79

Again, for c = 1, we have H ′(1) = I, and e(1) = eH(1) which is the original

approximation error obtained by the standard double sided frequency weighted model

reduction techniques [17,62].

Remark 49 Since the inputs and outputs of the two systems in Figures 3.1 and 3.3

are identical, their approximation errors e(1) and eH(c) would be equal.

We will now isolate the part of EH(c), which contains the weights used in E(1).

In other words, we are interested in calculating e(c).

Lemma 8 Let e(c), E(1), and H ′ be as defined above. Then we will have

e(c) ≤ ‖E(1)H ′−1

(c)‖∞|c−1| (3.60)

Proof 9 From Equation 3.49 we have:

W − Wr = H(c)c−1(W −Wr)⇔

H(c)H−1G [K −Kr(c)]H(c)H−1G = H(c)c−1H−1G [K −Kr]H−1G⇔

H−1G [K −Kr(c)]H(c)H−1G = c−1H−1G [K −Kr]H−1G⇔

H−1G [K −Kr(c)]H−1GH ′(c) = c−1H−1G [K −Kr]H

−1G⇔

E(c)H ′(c) = c−1E(1)⇔ E(c) = E(1)H ′−1

(c)c−1

Hence, from Equation 3.58, and by using the norm properties, we will have

e(c) = ‖E(1)H ′−1

(c)c−1‖∞ ⇒

e(c) ≤ ‖E(1)H ′−1

(c)‖∞|c−1| (3.61)

80

Remark 50 If the absolute value of c increases, the RHS of Equation 3.60 will be-

come smaller. To compensate for this, the LHS of Equation 3.60 is forced to produce

a reduced order controller Kr(c) which is closer to K than the original reduced order

controller Kr(1) (both with respect to the original frequency weights). It is a fact that

the difference between two controllers of different order can never be zero, but optimal

results can be obtained by using the proposed double controller technique.



tion problem e(c) = ‖V1(K−Kr(c))V2(c)‖∞ to find a new improved controller Kr(c),

by replacing the old frequency weight V2 with a new frequency weight V2(c) which

depends on c. The weight V1 remains the same.

To solve the frequency weighted model reduction problem, we can use any of the

standard techniques, e.g. Enns’ method [17] and Wang et al’s method [62]. Since

the weight V2(c) is a function of c, we get different low order controllers Kr(c) for

different values of c. And as the absolute value of c increases, we obtain a better

approximation of the original high order controller, and those low order controllers

will perform better. Note that c1 = 1 corresponds to the lower order controller Kr(1)

obtained by using the standard weights in Equation 3.46.

Remark 51 By observing Equation 3.60, as a general rule, the error becomes stable

as c goes to ±∞. Therefore, we only need to choose a c which is big enough, forcing

the approximation error e(c) to belong to those stabilized values.

Using an extremely large number c may be a good a choice, but it is not a necessary

81

one. From various experiments, we have deduced that for |c| ≥ 70, we may cover

all the cases of controllers which are of non-linear order. Note that, for first order

controllers, there is not much room for improvement, because the order is too low.

For non-linear orders (second or more), the double controller technique will always

give very improved results.

Remark 52 There is a very important difference between using the double controller

technique for the feedback systems, and using it for the cascade systems. The extra

parameter |c−1| in Equation 3.61 only exists in the feedback systems case. Because of

this, a large c produces a better approximation error without any other requirements

(e.g. finding the function which approximates the behaviour of H(c)). This can be

better seen when comparing the following two approximation error conditions:

e(c) ≤ ‖E(1)H ′−1

(c)‖∞|c−1| (3.62)

e(1)h−1(c) ≤ e(c) ≤ e(1)h(c) (3.63)

In Inequality 3.62, the condition corresponds to the feedback case, and it is clear

that whenever c increases, e(c) becomes smaller. This is a straightforward condition.

But in Inequality 3.63 (which corresponds to the cascade case), we can see that things

are not very simple. The approximation error e(c) is bounded by two quantities, each

of which has a constant part (i.e. e(1)), and a function which depends on H(c).

Hence, a further analysis and search was needed here. And this is the reason why the

double controller method for the feedback systems case, is superior. (Note that, we

used a slightly different notation for the cascade case, i.e. e(c, 1) instead of e(c), but

they symbolize the same notion).

82

3.3.6 The Algorithm

1. Given plant G and controller K, define the weights W1 = (I + GK)−1G and

W2(c) = (I +GK)−1G(I +KG(1− c))c.

2. Solve the frequency weighted model reduction problem ‖W1(K −Kr(c))W2(c)‖

for a large enough value of |c| (|c| ≥ 70 is recommended).

3. The derived controller for |c| ≥ 70, is the one that should be used (instead of

the one when c = 1).

Remark 53 It can be clearly seen that the algorithm above, provides an efficient

searching technique, without using any brute force search. This can only occur in

feedback systems, and even without the requirement to find an approximation function,

as demonstrated for the cascade systems case [26,27]. The main reason for obtaining

a lower approximation error is because a new parameter is added to the basic method

which calculates a reduced order controller.

3.4 The Double Controller Technique for LFT Sys-

tems


In this section we present a double sided frequency weighted model reduction problem

formulation. This is based on extensions of Equation 3.3 and Lemma 2 to the more

generalized closed-loop configurations described by the LFT systems (as shown in

Figures 3.9 and 3.10, instead of the closed-loop configuration with the controller in

83

series with the plant (as shown in Figures 3.1 and 3.3). The equivalence between the

closed-loop configurations shown in Figures 3.9 and 3.10 is first established, which is

then used to derive the new set of input and output weights for the new double sided

frequency weighted model reduction formulation.

The advantage of this formulation is that we cover all cases of closed-loop systems

described by the modern LFT configuration.

Consider the LFT system shown in Figure 2.2, with plant P and controller K.

The transfer function of the closed-loop system is given by

W = Fu(P,K) = P22 + P21K(I − P11K)−1P12

In the closed-loop system configuration shown in Figure 2.2, if the original con-

troller K is replaced by a reduced-order controller Kr, then the closed-loop system

transfer function is given by

Wr = Fu(P,Kr) = P22 + P21Kr(I − P11Kr)−1P12 (3.64)

The next Lemma is well known and it is added to ensure completeness for ex-

plaining the method we use.

Lemma 9 [69] Given LFT system with plant P , controller K, and reduced order

controller Kr (assuming that the second order terms are negligible in [K −Kr]), we

will have

W −Wr = P21(I −KP11)−1 [K −Kr] (I − P11K)−1P12 (3.65)

Proof 10 Please see the Appendix for the proof.

84



the form:

e = ‖V1(K −Kr)V2‖∞,

where

V1 = P21(I −KP11)−1, V2 = (I − P11K)−1P12. (3.66)

The above result is an intermediate step, which although not used directly, will

help to understand the generalized procedure (Lemma 11).

Remark 54 The closed loop configuration in Figure 3.9 can be expressed in another,

more generalized closed-loop configuration shown in Figure 3.10. A more detailed view

of the new configuration of Figure 3.10 is shown in Figure 3.11. The new configuration

uses a modification P of the original plant P , and has controller =cK instead of K.

Observe that both closed-loop configurations in Figures 3.9 and 3.10 (and consequently

in Figure 3.11) give the same input and output for any non zero value of the parameter

c. Hence, they have the same closed-loop system transfer function.

Remark 55 Instead of a scalar parameter c, we could have used a matrix parameter

C. But for the sake of simplicity, we will continue to use a scalar parameter C = c

throughout this Thesis.

Furthermore, Figure 3.2 is structurally equivalent to Figure 3.3, since we can

obtain Figure 3.3 from Figure 3.2 by replacing K and G in Figure 3.2 with KC and a

closed-loop system consisting of G in the forward path and K(I −C) in the feedback

loop (as shown [7,17]

85

Figure 3.9: LFT system with plant P (made of the submatrices P11, P12, P21, and

P22) and controller K.

Definition 14 The configuration in Figures 3.10 and 3.11 will be called the Double

Controller Form of the LFT system W . Moreover, the usage of the Double Controller

Form to manipulate the frequency weights by changing the parameter c, will be called

the Double Controller Technique.

Remark 56 The advantage of using the Double Controller Form is that one of the

weights will be a function of the parameter c. We will show that by choosing an

appropriate value for this parameter, we can significantly reduce the approximation

error when using any standard frequency weighted model reduction technique.



shown in Figures 3.9 and 3.11. In particular, we will derive the relationships between

the new plant P and the new controller K (as shown in Figure 3.10) in terms of the

old plant P , the old controller K and a free parameter c.

86

Figure 3.10: A generalization of the LFT system gives us an extra parameter c.

Figure 3.11: A detailed view of the generalized block diagram of the LFT system.

Let W = Fu(P,K), W = Fu(P , K) be two closed-loop transfer functions with

plants and controllers P , K and P , K respectively.

87

Definition 15

H(c) = I + (c− 1)P11K

H = I − P11K

H = I − P11K

H ′(c) = I + (c− 1)KP11

H ′ = I −KP11

H ′ = I − KP11

Remark 57 Note that, the definition of H(c) for the LFT systems, is the same to

the ones for the cascade and feedback cases by setting P11 = −G.

Remark 58 By assuming that (I−P11K)−1 (and consequently (I−KP11)−1) exists,

it can be shown that (I + (1− c)P11K)−1 (and consequently (I + (1− c)KP11)−1) also

exists for a given c, except for a finite number of values for s (continuous case) or

z (discrete case) [27]. We will disregard those finite number of values, as we have

infinite choices for s or z.

Assume now that we have the closed-loop systems W = Fu(P,K) and W =

Fu(P , K) as defined above. Then we have:

Lemma 10 If we are given K = cK and

P =

P11 P12

P21 P22

, and P =

H−1(c)P11 H−1(c)P12

c−1P21 P22

(3.67)

88

such that c 6= 0, then

Fu(P , K) = W = W = Fu(P,K). (3.68)

Proof 11 From the LFT definition for W , we have

W = P22 + P21K(I − P11K)−1P12

= P22 + c−1P21cK(I −H−1(c)P11cK)−1H−1(c)P12

= P22 + P21K(I −H−1(c)P11cK)−1H−1(c)P12

= P22 + P21K(H(c)−H(c)H−1(c)P11cK)−1P12

= P22 + P21K(H(c)− P11cK)−1P12

= P22 + P21K(I + (c− 1)P11cK − P11cK)−1P12

= P22 + P21K(I − P11K)−1P12 = W. (3.69)

Therefore, for c 6= 0, we have shown that we can replace the closed-loop system

W in Figure 3.9 by the closed-loop system W in Figure 3.10. The real gain here is

the revelation of a new parameter c.

Remark 59 By looking at the figures, it is clear that Figure 3.11 is a generalization

of Figure 3.9. This fact can be verified by setting c = 1. Then we will have H(1) =

H ′(1) = I, which when substituted, will make both block diagrams identical.

Remark 60 The input and output always remain the same regardless of c, which

means that we may manipulate the parameter c without affecting the system, to get a

more accurate low order controller, but we may improve the non-linear procedure for

calculating a reduced order controller.

89

We will show that the parameter c plays a major role in decreasing the approxi-

mation error in controller reduction.


The main aim of controller reduction is to obtain a low order controller by approx-

imating the closed-loop behavior of the system. This is achieved by approximating

the difference between closed-loop systems (W and Wr) which leads to the frequency

weighted model reduction problem as shown in Equation 3.65.


configuration shown in Figures 3.10 and 3.11. This is achieved by approximating the

difference between the closed-loop systems W and Wr.

Let us assume that the lower order controllers Kr(c) and Kr, correspond to the

systems Wr and Wr respectively.

Definition 16 We will denote by Kr(c) the reduced order controller obtained from a

system W = W (which has identical input and output with W ).

Remark 61 We may observe that the system W with plant P and controller K = cK

has identical input and output with W . Although the input and output of the systems

W and W are identical, the internal structure of the later system is dependent on

c. And since the procedure for obtaining a reduced order controller is non-linear, for

c1 6= c2 we should have Kr(c1) 6= Kr(c2).

Remark 62 We already know that W = W , or in more analytical terms, Fu(P , K) =

Fu(P,K). Therefore, to relate the differences W −Wr and W − Wr, it is required to

90

have:

W − Wr = W −Wr (3.70)

Thus, the following formula may be obtained for W − Wr:

Lemma 11

W − Wr = P21H′(c)(I −KP11)

−1 [K −Kr(c)] (I − P11K)−1P12 (3.71)

Proof 12 By using Equation 3.65 the difference W − Wr will be

91

W − Wr =

P21(I − KP11)−1[K − Kr(c)

](I − P11K)−1P12 =

P21c−1(I − cKH−1(c)P11)

−1 [cK − cKr(c)]

(I −H−1(c)P11cK)−1H−1(c)P12 =

P21(I − cKH−1(c)P11)−1 [K −Kr(c)]

(I −H−1(c)P11cK)−1H−1(c)P12 =

P21(I − cKH−1(c)P11)−1 [K −Kr(c)]

(H(c)−H(c)H−1(c)P11cK)−1P12 =

P21(I − cKH−1(c)P11)−1 [K −Kr(c)]

(H(c)− P11cK)−1P12 =

P21(I − cKH−1(c)P11)−1 [K −Kr(c)]H

−1P12 =

P21(I −H ′−1(c)cKP11)−1 [K −Kr(c)]H

−1P12 =

P21(I −H ′−1(c)cKP11)−1 [K −Kr(c)]H

−1P12 =

P21(H′(c)− cKP11)

−1H ′(c) [K −Kr(c)]H−1P12 =

P21(I −KP11)−1H ′(c) [K −Kr(c)]H

−1P12 =

P21H′(c)(I −KP11)

−1 [K −Kr(c)]H−1P12

P21H′(c)H ′−1 [K −Kr(c)]H

−1P12 (3.72)

Note that, we used above the property H ′(c)H ′−1 = H ′−1H ′(c), where H ′(c) =

I +KP11((1− c).

Remark 63 Instead of attempting to find reduced order controllers using the standard

92

weights V1 = P21(I − KP11)−1 and V2 = (I − P11K)−1P12, we may use the weights

V1 = P21H′(c)(I −KP11)

−1 and V2 = (I − P11K)−1P12.

Definition 17 We define Kr(c) = Kr(c)c−2.

It is obvious that for c = 1, we will have V1 = V1 and V2 = V2, and Kr(1) = Kr.

The advantage here, is that c does not have to be equal to 1, but it can be varied, to

obtain a more accurate low order controller.


It is standard procedure to derive a lower order controller Kr(c) by minimizing

‖P21H′(c)(I −KP11)

−1 [K −Kr(c)] (I − P11K)−1P12‖∞

using any of the standard double-sided frequency weighted model reduction tech-

niques [17, 62].

Definition 18

E(c) = P21H′−1 [K −Kr(c)]H

−1P12 (3.73)

e(c) = ‖E(c)‖∞ (3.74)

EH(c) = P21H′(c)H ′−1 [K −Kr(c)]H

−1P12 (3.75)

eH(c) = ‖EH(c)‖∞ (3.76)

The term e(c) symbolizes the approximation error which includes the original

weights and has Kr(c) as the reduced order controller, while the term eH(c) symbolizes

93

the approximation error which has P21H′(c)(I−KP11)

−1 and (I−P11K)−1P12 as left

and right weights and Kr(c) as the reduced order controller.

Again, for c = 1, we have H ′(1) = I, and e(1) = eH(1) which is the original

approximation error obtained by the standard double sided frequency weighted model

reduction techniques [17,62].

Remark 64 Since the inputs and outputs of the two systems in Figures 3.9 and 3.11

are identical, their approximation errors e(1) and eH(c) would be equal.

We will now isolate the part of EH(c), which contains the weights used in E(1).

In other words, we are interested in calculating e(c).

Lemma 12 Let h(c) = ‖H ′(c)‖∞ and h(c) = ‖H ′−1(c)‖∞. Then we have

e(c) = e(1). (3.77)

e(1)h−1(c) ≤ e(c) ≤ e(1)h(c). (3.78)

Proof 13 From Equation 3.68 we have

EH(c) = E(1) (3.79)

By taking the norms we obtain Equation 3.77.

Regarding Inequality 3.78, we will first prove the left part of the above expression

and then the right part. From Equation 3.79 we have:

94

EH(c) = E(1)⇒

E(c)H ′(c) = E(1)⇒

E(c) = E(1)H ′−1(c)⇒

‖E(c)‖∞ = ‖E(1)H ′−1(c)‖∞ ⇒

‖E(c)‖∞ ≤ ‖E(1)‖∞‖H ′−1(c)‖∞ ⇒

e(c) ≤ e(1)h(c)

Similarly, we have

EH(c) = E(1)⇒

E(c)H ′(c) = E(1)⇒

‖E(c)H ′(c)‖∞ = ‖E(1)‖∞ ⇒

‖E(c)‖∞‖H ′(c)‖∞ ≥ ‖E(1)‖∞ ⇒

e(c)h(c) ≥ e(1)⇒

e(c) ≥ e(1)h−1(c)

It is clear from the above that the term H ′(c) and its two forms of infinity norm

h(c) and h−1(c) are of great importance, and so it is essential to understand their

properties. Both h(c) and h−1(c) are known function of c, and by varying the param-

eter c, we could derive a graph which will reveal the potential optimal values for c.

In fact, the error function e(c) behaves in a similar way to the function e(c) [26, 27],

where

95

e(c) =h(c) + h−1(c)

2e(1). (3.80)



tion problem e(c) = ‖V1(c)(K−Kr(c))V2‖∞ to find a new improved controller Kr(c),

by replacing the old frequency weight V1 with a new frequency weight V1(c) which

depends on c. The weight V2 remains the same.

To solve the frequency weighted model reduction problem, we can use any of the

standard techniques, e.g. Enns’ method [17] and Wang et al’s method [62]. Since

the weight V1(c) is a function of c, we get different low order controllers Kr(c) for

different values of c. In general, as the absolute value of c increases, we obtain a

better approximation of the original high order controller. But if we want to obtain

an optimal controller, we need to take into consideration the behavior of h(c) and

h−1(c).

Remark 65 By observing Equation 3.78, the error seems to become stable as c goes

to ±∞. Therefore, we only need to choose a c which is big enough, forcing the

approximation error e(c) to belong to those stabilized values. Unfortunately though,

this is not the general rule, and as stated above, it is more appropriate to use the

approximation of the functions h(c) and h−1(c) which dictate the bounds for e(c).

Note that, for first order controllers, there is not much room for improvement,

because the order is too low. For non-linear orders (second or more), the double

controller technique will always give very improved results.

96

3.4.6 The Algorithm

1. Given an LFT system with plant P and controller K, define the weights W1(c) =

P21H′(c)(I −KP11)

−1 and W2 = (I − P11K)−1P12.

2. Create a graph for the function e(c) = h(c)+h−1(c)2

e(1), where h(c) = ‖I +

KP11((1 − c)‖∞ and h(c) = ‖(I + KP11(1 − c))−1‖∞, and choose the c which

gives the smallest value (both functions h(c) and h(c) behave in a similar way).

3. Solve the frequency weighted model reduction problem ‖W1(c)(K −Kr(c))W2‖

for this value of c.

4. The derived controller is the one that should be used (instead of the one when

c = 1).

Remark 66 Again, it can be clearly seen that all the algorithm above, also provides

an efficient searching technique, without using any brute force search. This is a very

important result, especially since we describe a very generalized system type such as

the LFT system. The procedure is similar to the cascade system algorithm [26,27], but

this time it covers a much broader range of systems, including those with controllers

in series and in the feedback path.

97

Chapter 4

Design Examples for the Double

Controller Technique

In this chapter, we apply the Double Controller Technique to benchmark examples

from practical applications.

4.1 Single-Input Single Output (SISO) Examples

To demonstrate the flexibility of the Double Controller Technique, we will present

examples with a system having the controller in the feedback path, and a system

with an LFT configuration. For completeness reasons, and to demonstrate certain

similarities, an example for the cascade system configuration [26] is also included.

98

4.1.1 Cascade System Example

We denote by ε(c), the actual error (i.e. the infinity norm of the direct difference

between the original and reduced systems), such that the reduced controller was

obtained by using the parameter c. That is,

ε(c) = ‖GK(I +GK)−1 −GKr(c)(I +GKr(c))−1‖∞ (4.1)

Consider the example presented in Kim et al [33], where

Gc(s) =(s+ 0.8)(s+ 2)

(s+ 1.5)(s2 + 1.4s+ 1),

Kc(s) =10.3544(s+ 1.86183)(s+ 0.745649)

(s+ 19.8229)(s+ 2.00134)(s+ 0.800627).

Consider the example presented in [16] and Kim et al [33].

Then we will obtain σ1 = −25 and σ2 = 25, so we should search for a parameter

c inside the interval (−25, 25).

Let us define as e(cmin, 1), e(∞, 1), and ε(copt), the corresponding approximation

errors for c = cm giving local minimum to the function e(c, 1), for c converging to

±∞, and the actual best approximation error at c = copt. In table 4.1, we list all the

results from this example.

Table 4.1 gives the approximation errors for each case.

Figure 4.1 shows the change of the approximation error e(c, 1) with respect to

different values of c. We can notice a very big improvement by using the double

controller technique. For large values of c (where the error becomes stabilized), the

error e(∞, 1) may not be optimal, but there is still significant improvement compared

99

Enns

Order e(1, 1) e(cmin, 1) e(∞, 1) copt ε(copt)

1 .01658 .01657 .01657 5.5184 .01510

2 .00104 .00056 .00067 21.3712 .00055

Wang


1 .02497 .02363 .02363 2.3077 .01710

2 .00118 .00056 .00066 -6.9231 .00054

Table 4.1: Approximation Error Comparison for First and Second Order Controllers

to the original error. Also, by finding local minimum points of the function e(c, 1)

we can get very close to obtaining an optimal approximation error e(cmin, 1). All

these results can be seen in Table 4.1, and may be compared to the value of the

lowest possible approximation error ε(copt). In Figure 4.3, demonstrates that e(c, 1)

can be approximated by e(c, 1), whose local minimum points reveal the values (or

neighborhood) of the minimum approximation error.

The error function e(c1, c2) for the above example may be seen while both param-

eters change in a three dimensional plot. Figures 4.4 and 4.5 represent the plots for

Enns’ method and Wang’s method respectively.

100

Figure 4.1: Enns’ and Wang’s method results for e(c, 1) for second order controllers.

Notice that ε(c) is almost always less than e(c, 1). Because of this, it is no surprise

that the minimum value of ε(c) is also less than e(c, 1) (as shown in Table 4.1).

Note that by using the double controller technique on Wang’s method for second

order controllers, there is a reduction of almost 60% of the original error. The rest

of the examples had also achieved significant error reduction. All results these can

verified by using the corresponding controller to calculate the real error between the

original and reduced systems.

4.1.2 Feedback System Example

Let us consider again the plant G of the example presented in [33], which has a con-

troller in the feedback loop. The MATLAB Control Design Tools is used to design an

101

Figure 4.2: The error function e(c, 1) is bounded by e(1, 1)h−1(c) and e(1, 1)h(c)

optimal LQG regulator (controller) Kf for a simple regulation loop. More specifically,

we will have a feedback system with plant Gf and controller Kf defined as:

Gf (s) =(s+ 0.8)(s+ 2)

(s+ 1.5)(s2 + 1.4s+ 1),

Kf (s) =−283.4s2 − 723.9s− 373.5

s3 + 105.6s2 + 289.5s+ 164.5

102

Figure 4.3: The error function e(c, 1) could be approximated by e(c, 1)

4.1.3 LFT System Example

The cascade system example may be translated into an LFT system with plant Pc

and the same controller Kc, such that

Pc =

−Gc I

Gc 0

(4.2)

This feedback system example may also be translated into an LFT system with

plant Pf and the same controller Kf , such that

Pf =

−Gf Gf

−Gf Gf

(4.3)

103

Figure 4.4: The error function e(c1, c2) for Enns’ Method in a three dimensional plot.

The results by applying the Double Controller Technique are the same, something

which validates that the LFT systems are a very useful generalization of the Cascade

and Feedback systems.

4.2 Multiple-Input Multiple-Output (MIMO) Ex-

amples

4.2.1 Cascade System Example

We now consider an example from the H∞ and H2 Optimization Toolbox in SLICOT

[23], defined with a plant G and a designed controller K (of sixth order, four inputs,

and four outputs), whose state space matrices (AG, BG, CG, DG, and AK , BK , CK , DK

104

Figure 4.5: The error function e(c1, c2) for Wang’s Method in a three dimensional

plot.

respectively) are given below

AG =

−1 0 4 5 −3 −2−2 4 −7 −2 0 3−6 9 −5 0 2 −1−8 4 7 −1 −3 02 5 8 −9 1 −43 −5 8 0 2 −6

, BG =

−3 −42 0−5 −74 −6−3 91 −2

,

CG =(

1 −1 2 −4 0 −3−3 0 5 −1 1 1

), DG = ( 1 −2

0 4 ) ,

AK =

−2.8043 14.7367 4.6658 8.1596 0.0848 2.52904.6609 3.2756 −3.5754 −2.8941 0.2393 8.2920−15.3127 23.5592 −7.1229 2.7599 5.9775 −2.0285−22.0691 16.4758 12.5523 −16.3602 4.4300 −3.316830.6789 −3.9026 −1.3868 26.2357 −8.8267 10.4860−5.7429 0.0577 10.8216 −11.2275 1.5074 −10.7244

,

CK =( −0.2480 −0.1713 −0.0880 0.1534 0.5016 −0.0730

2.8810 −0.3658 1.3007 0.3945 1.2244 2.5690

),

BK =

−0.1581 −0.0793−0.9237 −0.57180.7984 0.66270.1145 0.1496−0.6743 −0.23760.0196 −0.7598

, DK = ( 0.0554 0.1334−0.3195 0.0333 ) .

Figures 4.6 and 4.7 reveal that in the fourth order and fifth order reduced con-

trollers cases (for Enns’ Method), e(c, 1)’s local minimums directly point out the

105

Figure 4.6: The error functions e(c, 1) and e(c, 1) for the MIMO example when ap-

plying controller reduction of order 4.

possible c’s which give optimal results. Sometimes though, it is e(∞, 1) which gives

a better approximation error. This usually happens in the cases where the order re-

duction is done by an odd number, which could result in replacing a pair of complex

poles by a real one, and yield results which are not as accurate as the ones when the

order reduction is done by an even number [41]. However, there are also cases where

e(∞, 1) becomes too large. In general, we notice again a very big improvement of

error reduction by using the double controller technique.

All the results for controllers with a reduced order from 1 to 5, can be seen in

Tables 4.2 and 4.3. We may comment that in most cases, copt is directly found by

the cmin’s for each different order.

106

Figure 4.7: The error functions e(c, 1) and e(c, 1) for the MIMO example when ap-

plying controller reduction of order 5.

4.2.2 LFT System Example

We now consider a more general example for the LFT system. We will demonstrate

how the technique works in the generalized LFT framework for the MIMO case. So

we consider the MIMO LFT example from the H∞ and H2 Optimization Toolbox

in SLICOT. Note that, this is a extension of the example shown in [28] (with more

inputs and outputs), defined with a plant P and a designed controller K (of sixth

order, four inputs, and four outputs).

whose state space matrices (AP , BP , CP , DP , and AK , BK , CK , DK respectively)

are given below

107

Enns


1 3.6608 2.7510 3.6909 -0.6122 2.7510

2 3.1186 2.2697 2.2993 1.0204 2.2697

3 5.3129 2.8736 1.5485 1.0204 1.5485

4 0.9745 0.7643 1.3900 2.2449 0.7643

5 0.2271 0.0846 0.3172 0.2041 0.0846

Table 4.2: Approximation Error Comparison using Enns’ Method for First up to Fifth

Order Controllers (MIMO Example)

AP =

−1 0 4 5 −3 −2−2 4 −7 −2 0 3−6 9 −5 0 2 −1−8 4 7 −1 −3 02 5 8 −9 1 −43 −5 8 0 2 −6

, BP =

−3 −4 −2 1 02 0 1 −5 2−5 −7 0 7 −24 −6 1 1 −2−3 9 −8 0 51 −2 3 −6 −2

,

CP =

(1 −1 2 −4 0 −3−3 0 5 −1 1 1−7 5 0 −8 2 −29 −3 4 0 3 70 1 −2 1 −6 −2

), DP =

(1 −2 −3 0 00 4 0 1 05 −3 −4 0 10 1 0 1 −30 0 1 7 1

),

AK =

−2.8043 14.7367 4.6658 8.1596 0.0848 2.52904.6609 3.2756 −3.5754 −2.8941 0.2393 8.2920−15.3127 23.5592 −7.1229 2.7599 5.9775 −2.0285−22.0691 16.4758 12.5523 −16.3602 4.4300 −3.316830.6789 −3.9026 −1.3868 26.2357 −8.8267 10.4860−5.7429 0.0577 10.8216 −11.2275 1.5074 −10.7244

, BK =

−0.1581 −0.0793−0.9237 −0.57180.7984 0.66270.1145 0.1496−0.6743 −0.23760.0196 −0.7598

,

CK =( −0.2480 −0.1713 −0.0880 0.1534 0.5016 −0.0730

2.8810 −0.3658 1.3007 0.3945 1.2244 2.5690

), DK = ( 0.0554 0.1334

−0.3195 0.0333 ) .

108

Wang


1 4.6013 3.1764 3.8629 -1.0204 3.1764

2 2.5725 2.3780 2.7582 -0.6122 2.3780

3 1.9823 1.0293 7.1506 -0.2041 1.0293

4 1.0475 0.8282 1.0028 0.2041 0.8282

5 0.3209 0.2431 0.3477 -0.2041 0.2431

Table 4.3: Approximation Error Comparison using Wang’s Method for First up to

Fifth Order Controllers (MIMO Example)

Figure 4.8: Enns’ method results for e(c) for second order controllers.

109

Figure 4.9: Wang’s method results for e(c) for second order controllers.

Figure 4.10: The error function e(c) using Enns method is being approximated by

e(c) for second order controllers.

110

Figure 4.11: Enns’ method results for e(c) for second order controllers.

Figure 4.12: Wang’s method results for e(c) for fifth order controllers for the MIMO

LFT case.

111

Chapter 5

Conversions between Classical

Systems and Modern Systems

In this chapter, we will present a link between simple forms of classical control sys-

tems and linear fractional transformations, whilst preserving the controller. A more

generalized form of classical control systems is also linked with linear fractional trans-

formations. It is shown that under certain conditions, those classical control system

block diagrams (and systems) may always be represented by a family of linear frac-

tional transformation block diagrams (and systems), and proofs where the inverses

of those representations are true are also provided. There have been many times

where examples of such links have been demonstrated, but here we present a more

generalized framework. This framework was also used to connect different one of the

examples for the Double Controller Technique.

112

5.1 Introduction

In other words, given a closed-loop system described by block diagrams (Figures 5.1,

5.2, or 5.3) our objective is to obtain an equivalent LFT representation (Figure 5.5)

having the same closed-loop transfer function, and vice versa.

We will also describe the restricted cases where the opposite may be considered.

As we will see later on, internal stability is preserved throughout this procedure by

simply examining the form of the corresponding space state representations.

5.2 Notation and Preliminaries

Since the proposed method is applicable to both continuous and discrete systems,

the notation used throughout this Thesis for transfer functions (plants, controllers,

and any combination of them) will represent both continuous and discrete cases. For

example, a plant G will represent both G(s) (continuous case) and G(z) (discrete

case), unless stated otherwise.

Definition 19 A system with a controller in the forward path (shown in Figure 5.1)

will be called a cascade system and its corresponding block diagram as a cascade block

diagram.

Definition 20 A system with a controller in the feedback path (shown in Figure 5.2)

will be called a feedback system and its corresponding block diagram as a feedback block

diagram.

Definition 21 A system with two controllers, one in the forward path and one in

113

the feedback path (shown in Figure 5.3), will be called a combined system and its

corresponding block diagram as a combined block diagram.

Definition 22 The cascade, feedback, and combined systems will be called classical

systems, and their corresponding block diagrams as classical block diagrams.

Definition 23 A system expressed by a linear fractional transformation (shown in

Figure 5.5) will be called a modern system and its corresponding block diagram as a

modern block diagram.

It is well known that the cascade system and feedback system transfer functions

are GK(I+GK)−1 (Figure 5.1) and (I+GK)−1G (Figure 5.2) respectively. Moreover,

the transfer function for the combined system in Figure 5.3 is (I +GK1K2)−1GK1.

In complex variable function theory, a mapping F : C 7→ C such that

F (s) =a+ bs

c+ ds(5.1)

where a, b, c, d ∈ C, is called a linear fractional transformation or LFT. Moreover,

in the case c 6= 0, we have

F (s) = α + βs(1− γs)−1 (5.2)

where α, β, γ ∈ C. By generalizing Equation 5.2 for matrices, we obtain the linear

fractional transformations related to modern control theory.

More specifically, there are two types of linear fractional transformations, the lower

LFT Fl(P,K) (shown in Figure 5.6) and the upper LFT Fu(P,K) (shown in Figure

5.4). In this Thesis we will focus on the upper LFT (the procedure is equivalent

114

when using a lower LFT). For consistency regarding the directions of the input and

the output, the upper LFT may be expressed by the block diagram in Figure 5.5,

which is exactly the same as in Figure 5.4 , but rotated by 180 degrees.

Let us define a plant P of a modern system (expressed in LFT form) as

P =

P11 P12

P21 P22

(5.3)

where P11, P12, P21 and P22 are matrices with appropriate dimensions with respect

to the input values u, u0 and the output values y, y0.

Then, the block diagram in Figure 5.5 represents the two equations

y0

y

= P

u0

u

=

P11 P12

P21 P22

u0

u

(5.4)

u0 = Ky0 (5.5)

while the block diagram in Figure 5.4 represents the equations

y

y0

= P

u

u0

=

P11 P12

P21 P22

u

u0

(5.6)

115

u0 = Ky0 (5.7)

We may now write the definition of Fu(P,K) in terms of the submatrices of P [69]:

Fl(P,K) = P11 + P12K(I − P22K)−1P21 (5.8)

Fu(P,K) = P22 + P21K(I − P11K)−1P12 (5.9)

5.3 Problem Formulation

while a modern system (Figure 5.5) is represented by a linear fractional transformation

with a matrix plant P [69]. We need to be careful while attempting to unify the

classical and modern notions as we must consider that the output, input, as well

as the choice and behaviour of the corresponding controller are identical for any K.

Moreover:

• the controller K which is used, must be exactly the same when going from a

modern to a classical system (and vice versa).

• the plant G (corresponding to the classical systems) and the plant P (corre-

sponding to the modern system), must be independent of K.

Using the two conditions above, we will try to retrieve all the types of classical-

to-modern (and modern-to-classical) systems.

116

At this stage, we will focus on the lower LFT form (solving the problem with an

upper LFT is an equivalent procedure with the only difference that P22, P21, P11, P12

are re-arranged).

We will always assume that the inverses of I+GK and I−P11K are well defined,

and that whenever G is a matrix, it will be of full rank. This condition is required

for the proofs, but it is not necessary when verifying the results.

If we are dealing with the Multiple Input Multiple Output (MIMO) case, the

matrices that correspond to G and K are not necessarily square. More specifically,

we may assume without loss of generality that the matrices G and K have dimensions

m × n and n ×m respectively. We will always assume that the dimensions between

all the matrices are compatible.

Moreover, during the proofs of theorems that will be presented, it will be required

that a matrix commutes with either GK or K. In those cases we will have m = k or

n = k respectively.

In general, we will always assume that during matrix operations, all matrices

are of compatible dimensions. It must be clear by now that G does not necessarily

commute with K.

5.4 Classical to Modern Conversion

5.4.1 Main Results

Given a closed-loop system described by classical block diagrams (Figures 5.1, 5.2,

and 5.3) our objective is to obtain an equivalent LFT representation (Figure 5.5)

117

having the same closed-loop transfer function, and vice versa.

Regarding the cascade and feedback systems, we make the following logical as-

sumptions of this derivation:

• The controller K is exactly the same in both representations.

• The plant G (in the classical block diagrams) and the plant P (in the LFT

representation) are independent of K.

• The plant G is of full rank and the inverses of I + GK and I − P11K always

exist.

• For compatibility regarding matrix operations, we may assume (without loss of

generality) that the plant G and the controller K have dimensions m × n and

n×m respectively.

It must be clear by now that in this case, G does not necessarily commute with

K, and we will always assume that during matrix operations, all matrices are of

compatible dimensions.

Remark 67 The assumptions regarding the combined system will be slightly different

and will be mentioned later.

5.4.2 The Cascade System

In this section we will present the conditions under which a cascade system (Figure

5.1) may be represented by a linear fractional transformation (Figure 5.5), and vice

versa. We have assumed that G and K have dimensions m×n and n×m respectively.

118

From Equations 2.18 and 5.5, we conclude that all the submatrices P22, P21, P11, and

P12, will have dimensions m×m, m× n, m× n, and m×m respectively.

We now present a theorem that establishes the conditions under which the gen-

eralized version of a cascade system is equivalent to a family of linear fractional

transformations.

Theorem 6 A cascade system with a plant G (which is a matrix of full rank) and a

controller K, with closed loop transfer function (I +GK)−1GK (Figure 5.1), may be

represented by a linear fractional transformation (Figure 5.5) with controller K and

plant Pc (which is independent of K), if and only if

Pc =

−G αI

α−1G 0

(5.10)

where α is any non-zero scalar.

Proof 14 We need to represent a cascade system (Figure 5.1) in LFT form (Figure

5.5), that is

(I +GK)−1GK = P22 + P21K(I − P11K)−1P12 (5.11)

To ensure that the LFT form can represent the cascade system, we assume without

loss of generality that P12 is of full rank. From Equation 5.11 if any of P21 or P12

was not of full rank, that would mean that P22 will depend on K, a contradiction.

119

We assume that at least one of P21 or P12 is not of full rank. Then, if we multiply

Equation 5.11 by V ∈ ker{P21K(I − P11K)−1P12} (such that V 6= 0), we will get

[(I + GK)−1GK − P22]V = 0. Since G (and consequently (I + GK)−1G) is of full

rank, P22 is not independent of K, a contradiction. Therefore, both P21 and P12 are

assumed to be of full rank.

Post-multiplying Equation 5.11 by P12−1, yields the following RHS

P22P12−1 + P21K(I − P11K)−1 =

(P22P12−1(I − P11K) + P21K)(I − P11K)−1 =

(P22P12−1 − P22P12

−1P11K + P21K)(I − P11K)−1 (5.12)

And since GK(I +GK)−1 = (I +GK)−1GK, Equation 5.11 will become

(I +GK)−1GKP12−1 = (P22P12

−1 − P22P12−1P11K + P21K)(I − P11K)−1(5.13)

By multiplying each part of Equation 5.13 by (I + GK) on the left, and by (I −

P11K) on the right, we get

GKP12−1(I − P11K) = (I +GK)(P22P12

−1 − P22P12−1P11K + P21K) (5.14)

By rearranging those terms we end up with a more analytic form:

−GKP12−1P11K +GKP22P11KP12

−1−GKP21K+

(P22P11KP12−1− P21K +GKP12

−1−GKP22P12−1)+

(−P22P12−1) = 0 (5.15)

120

The third line of Equation 5.15 contains the only element that is not multiplied

by positive powers of K. Hence, since P12 is of full rank, and both P12, P22 are

independent of K, we have P22 = 0, and Equation 5.15 may be rewritten as

−GKP12−1P11K −GKP21K + (−P21K +GKP12

−1) = 0 (5.16)

Again, we need to bring the above expression into a more compatible form. We

recall that P12 is of full rank and is independent of K. Without loss of generality we

define P12 = A, and Equation 5.16 will become

GKA−1P11K +GKP21K + P21K −GKA−1 = 0 (5.17)

By replacing K by −K, Equation 5.17 becomes

GKA−1P11K +GKP21K − P21K +GKA−1 = 0 (5.18)

If we add and subtract Equations 5.17 and 5.18, we will obtain respectively:

GK(A−1P11 + P21)K = 0 (5.19)

P21K −GKA−1 = 0 (5.20)

And since G is assumed to be of full rank, while K is a free parameter, we get

A−1P11 = −P21 (5.21)

P21K = GKA−1 (5.22)

121

Denote as K−1 the pseudo inverse of K. Then Equation 5.22 may be written as

P21 = GKA−1K−1 (5.23)

The square matrix GK, the controller K, and the pseudo inverse K−1 of the con-

troller matrix K do not necessarily commute with A. This means that the submatrix

P21 is not independent of K, unless A was of the form αI, where α is a scalar and I

is the identity matrix.

Then Equation 5.22 becomes

P21K = GKA−1

= GK(αI)−1

= GKα−1

= α−1GK (5.24)

And since the above Equation must be true for any controllers K we have that

P21 = α−1G (5.25)

P11 = −G. (5.26)

We have two subcases:

• A commutes with GK,

• A commutes with K.

122

It is possible to have both statements true at the same time (for example in the

SISO case, or if A = I and provided we have compatible matrix dimensions).

5.4.3 Subcase 1: AGK = GKA

Equation 5.22 will become

(P21 − A−1G)K = 0

P21 − A−1G = 0

P21 = A−1G (5.27)

Finally, Equation 5.20 will become

A−1(P11 +G)K = 0

P11 +G = 0

P11 = −G (5.28)

We also have:

123

−GKA−1P11K−GKP21K− P21K +GKA−1 = 0

−GKA−1P11K−GKP21K− P21K + A−1GK = 0

(−GKA−1P11 −GKP21 − P21 + A−1G)K = 0

−GKA−1P11 −GKP21 − P21 + A−1G = 0

−GK(A−1P11 + P21)− P21 + A−1G = 0

(5.29)

The last two terms of the above Equation are independent of K, so we have

−P21 + A−1G = 0

P21 = A−1G (5.30)

Moreover, from the first two terms of Equation 5.29 we have

GK(A−1P11 + P21) = 0

A−1P11 + P21 = 0

A−1P11 + A−1G = 0

P11 +G = 0

P11 = −G (5.31)

So we finally obtain:

124

P11 = −G (5.32)

P12 = αI (5.33)

P21 = α−1G (5.34)

P22 = 0 (5.35)

where α is any non-zero scalar. Therefore, the matrix P regarding the modern

system that emulates a cascade system will be of the form

Pc =

−G αI

α−1G 0

(5.36)

To prove the inverse of this Theorem, we just need to use the same steps backwards.

5.4.4 The Feedback System

In this section we will present the conditions under which a feedback system (Figure


versa. We have assumed that G and K have dimensions m×n and n×m respectively.

From Equations 2.18 and 5.5, we conclude that all the submatrices P22, P21, P11, and

P12, will have dimensions m× n.


eralized version of a feedback system is equivalent to a family of linear fractional

transformations.

125

Theorem 7 A feedback system with a plant G (which is a matrix of full rank) and

a controller K, with closed loop transfer function (I +GK)−1G (Figure 5.2), may be

represented by a linear fractional transformation (Figure 5.5) with controller K and

plant Pf (which is independent of K), if and only if

Pf =

−G βG

−β−1G G

(5.37)

where β is any non-zero scalar.

Proof 15 We need to represent a feedback system (Figure 5.2) in LFT form (Figure

5.5), that is

(I +GK)−1G = P22 + P21K(I − P11K)−1P12 (5.38)

To ensure that the LFT form can represent the feedback system, we assume without

loss of generality that P12 is of full rank. From Equation 5.38, if any of P21 or P12 was

not of full rank, that would mean that P22 will depend on K, a contradiction. Also,

if P12 was singular, that would mean that P22 will depend on K, a contradiction.

We assume that at least one of P21 or P12 is not of full rank. Then, if we multiply

Equation 5.38 by V ∈ ker{P21K(I − P11K)−1P12} (such that V 6= 0), we will get

[(I + GK)−1G − P22]V = 0. Since G (and consequently (I + GK)−1G) is of full

rank, P22 is not independent of K, a contradiction. Therefore, both P21 and P12 are

assumed to be of full rank.

126

By having both P12 and G of full rank, we may assume that P12 = BG, where B

is an invertible m×m matrix. We may also assume that P22 = JG, for some m×m

matrix J . Since G is of full rank, Equation 5.38 will become

(I +GK)−1G = JG+ P21K(I − P11K)−1BG

(I +GK)−1 = J + P21K(I − P11K)−1B (5.39)

If we post-multiply Equation 5.39 by B−1, the RHS becomes

JB−1 + P21K(I − P11K)−1 =

(JB−1(I − P11K) + P21K)(I − P11K)−1 =

(JB−1 − JB−1P11K + P21K)(I − P11K)−1 (5.40)

Then Equation 5.39 will become

(I +GK)−1B−1 = (JB−1 − JB−1P11K + P21K)(I − P11K)−1 (5.41)

By multiplying each part of Equation 5.41 by (I + GK) on the left, and by (I −

P11K) on the right, we get

B−1(I − P11K) = (I +GK)(JB−1 − JB−1P11K + P21K) (5.42)

By rearranging those terms we have:

127

GKJB−1P11K−GKP21K+

(JB−1P11K− P21K−B−1P11K−GKJB−1)+

(I − J)B−1 = 0 (5.43)

The third line of Equation 5.43 contains the only term which is not multiplied

by positive powers of K. Hence, since B−1 is of full rank, and both B−1, I − J are

independent of K, we have J = I, and consequently, P22 = G. Then Equation 5.43

may be rewritten as

GK(P21 −B−1P11)K + P21K +GKB−1 = 0 (5.44)

By replacing K by −K, Equation 5.44 becomes

GK(P21 −B−1P11)K − P21K −GKB−1 = 0 (5.45)

If we add and subtract Equations 5.44 and 5.45, we will obtain respectively:

GK(P21 −B−1P11)K = 0 (5.46)

P21K +GKB−1 = 0 (5.47)

And since G is assumed to be of full rank, while K is a free parameter, we get

P21 = B−1P11 (5.48)

P21K = −GKB−1 (5.49)

128

Denote as K−1 the pseudo inverse of K. Then Equation 5.49 may be written as

P21 = −GKB−1K−1 (5.50)

The square matrix GK, the controller K, and the pseudo inverse K−1 of the con-

troller matrix K do not necessarily commute with B. This means that the submatrix

P21 is not independent of K, unless B was of the form βI, where β is a scalar and I

is the identity matrix.

Then Equation 5.49 will become

P21K = −GKB−1 = −GK(βI)−1 = −GKβ−1 = −β−1GK (5.51)

And since Equation 5.51 must be true for any controller K we have that

P21 = −β−1G (5.52)

P11 = −G. (5.53)

So we finally obtain:

P11 = −G (5.54)

P12 = βG (5.55)

P21 = −β−1G (5.56)

P22 = G (5.57)

129

where β is any non-zero scalar. Therefore, the matrix P regarding the modern

system that emulates a cascade system will be of the form

Pf =

−G βG

−β−1G G

(5.58)

To prove the inverse of this Theorem, we just need to use the same steps backwards.

Remark 68 The final results can be easily verified, by simply using the plant Pc to

obtain the transfer function GK(I +GK)−1 that corresponds to the classical cascade

system, or by using the plant Pf to obtain the transfer function (I + GK)−1G that

corresponds to the classical feedback system. More specifically:

Fu(Pc,K) =GK(I +GK)−1

Fu(Pf ,K) = (I +GK)−1G

During this verification, both G and K may even be singular, without affecting the

end result.

5.4.5 The Combined System

In this section we will present the conditions under which a combined system (Figure


versa. The system will have m inputs and n outputs.

The plant G of the combined system, will have dimensions m× n. The controller

K1 (as in Figure 5.3) may only have n×n as compatible dimensions. Without loss of

130

generality, we may assume that K1 is a nonsingular matrix. Similarly, the controller

K2 (as shown in Figure 5.3) can only be an n ×m matrix. If any of K1 or K2 has

different dimensions than those mentioned, the number of inputs or outputs will be

affected and the system will be changed.

For the modern system, we may assume without loss of generality that P12 is a

nonsingular n× n, square matrix. From Equations 5.4 and 5.5, we conclude that the

other submatrices P21, P11, and P22 will have dimensions m× n′, n× n′, and m× n

respectively.


eralized version of a combined system is equivalent to a family of linear fractional

transformations.

Theorem 8 Assume that the matrices G ,K1, K2, P11, P12, P21, P22, and K have

the same dimensions as defined above.

A combined system with a plant G (which is a matrix of full rank), a controller K1

in the forward path, a controller K2 in the feedback path, and with a closed loop trans-

fer function (I+GK1K2)−1GK1 (Figure 5.3) may be represented by a linear fractional

transformation (Figure 5.5) with controller K and plant P (which is independent of

K,K1, and K2), if and only if

P =

P11 P12

P21 P22

(5.59)

for any P11, P21, P22, and any nonsingular P12, and

131

K = L1−1L2 (5.60)

such that

L1 = (GK1P12−1P11 +GK1K2P21 + P21 −GK1K2P22P12

−1P11 − P22P12−1P11(5.61)

L2 = (GK1 −GK1K2P22 − P22)P12−1 (5.62)

and L1 is nonsingular.

Proof 16 First we assume that we have a combined system with a plant G (which

is a matrix of full rank), a controller K1 in the forward path, a controller K2 in the

feedback path, with a closed loop transfer function (I + GK1K2)−1GK1 (Figure 5.3).

We seek to represent it as a linear fractional transformation. Then from Equation

5.9 we will have:

(I +GK1K2)−1GK1 = P22 + P21K(I − P11K)−1P12

(I +GK1K2)−1GK1 = P22 + P21K(P−1

12 − P−112 P11K)−1

(I +GK1K2)−1GK1 = (P22P

−112 − P22P

−112 P11K + P21K)(P−1

12 − P−112 P11K)−1(5.63)

If we multiply Equation 5.63 by (I+GK1K2) on the left and by (P22P−112 −P22P12)

on the right, we will get

GK1(P−112 − P−1

12 P11K) = (I +GK1K2)(P22P−112 − P22P

−112 P11K + P21K) (5.64)

Solving Equation 5.64 for K, will give

132

K = L1−1L2 (5.65)

where

L1 = (GK1P12−1P11 +GK1K2P21 + P21 −GK1K2P22P12

−1P11 − P22P12−1P11)(5.66)

L2 = (GK1 −GK1K2P22 − P22)P12−1 (5.67)

Of course, for Equation 5.65 to make sense, L1 must be nonsingular.

5.5 Modern to Classical Conversion

We now present a theorem that establishes the conditions under which a linear frac-

tional transformation may be represented by a generalized version of a combined

system.

Theorem 9 Assume that the matrices G, K1, K2, P11, P12, P21, P22, and K have

the same dimensions as defined above.

A linear fractional transformation (Figure 5.5) with plant P and controller K

may be represented by a combined system with a plant G (which is a matrix of full

rank), a controller K1 in the forward path, a controller K2 in the feedback path, and a

transfer function (I+GK1K2)−1GK1 (Figure 5.3), if and only if K2 is any controller

of compatible dimensions, and

GK1 = M1M2−1 (5.68)

133

such that

M1 = P22P12−1 − P22P12

−1P11K + P21K (5.69)

M2 = P12−1 − P12

−1P11K −K2P22P12−1 +K2P22P12

−1P11K −K2P21K(5.70)

and M2 is nonsingular.

Proof 17 Assume that we have a linear fractional transformation with a plant P as

defined in Equation 5.59. We seek to represent it as a combined system with a plant

G (which is a matrix of full rank), a controller K1 in the forward path, a controller

K2 in the feedback path, and a transfer function (I +GK1K2)−1GK1 (Figure 5.3).

By setting the two terms (I + GK1K2)−1GK1 and P22 + P21K(I − P11K)−1P12

equal to each other, we will have the same results as in Equations 5.63 and 5.64.

This time, we solve Equation 5.64 for GK1, and this will give

GK1 = M1M2−1 (5.71)

where

M1 = P22P12−1 − P22P12

−1P11K + P21K (5.72)

M2 = P12−1 − P12

−1P11K −K2P22P12−1 +K2P22P12

−1P11K −K2P21K(5.73)

Again, for Equation 5.71 to be valid, the matrix M2 must be nonsingular.

Remark 69 Note that, if we are dealing with a MIMO system that has the number

of inputs equal to the number of outputs, the plant G of the combined system will be a

134

square matrix. Without loss of generality we may assume that G is nonsingular, and

multiplying on the left the two parts of Equation 5.71 by G−1 will give

K1 = G−1M1M2−1 (5.74)

where M1 and M2 are defined in the previous Theorem.

5.5.1 Analysis

Let us assume that the minimal realizations of a plant G and a controller K are

respectively

G =

AG BG

CG DG

and K =

AK BK

CK DK

We have shown that a cascade system with plant G and controller K may be

represented by the linear fractional transformation Fu(Pc, K) with plant Pc (as shown

in Equation 5.10) and the same controller K.

Without loss of generality (since the transfer function remains unchanged for any

non-zero values for α), we set α = 1. Then we will have

Pc =

−G I

G 0

(5.75)

By knowing the state space representation of G and by using Equation 5.75, the

state space representation of Pc will be

135

Pc =

AG 0 0 0 BG 0

0 0 0 0 0 0

0 0 AG 0 BG 0

0 0 0 0 0 0

−CG 0 0 0 −DG I

0 0 CG 0 DG 0

(5.76)

The above configuration is not minimal, but it can be shown (please see the

Appendix section “Representing Transfer Functions by a Minimal Realization”) that

it can be expressed by the following minimal realization:

Pc =

AG BG 0

−CG −DG I

CG DG 0

(5.77)

Similarly, a feedback system with plant G and controller K may be represented

by the linear fractional transformation Fu(Pf , K) with plant Pf (as shown in Equa-

tion 5.37) and the same controller K. Without loss of generality (since the transfer

function remains unchanged for any non-zero values for β) we may set β = 1. Then

we will have

136

Pf =

−G G

−G G

(5.78)

and the corresponding realization would be

Pf =

AG 0 0 0 BG 0

0 AG 0 0 0 BG

0 0 AG 0 BG 0

0 0 0 AG 0 BG

−CG CG 0 0 −DG DG

0 0 −CG CG −DG DG

(5.79)

The above configuration is not minimal, but it can be shown (please see the

Appendix section “Representing Transfer Functions by a Minimal Realization”) that

it can be expressed by the following minimal realization:

Pf =

AG −BG BG

CG −DG DG

CG −DG DG

(5.80)

137

Since G is assumed to be internally stable, then so are both Pc and Pf (as shown

in Equations 5.77 and 5.80) whose state space representations share the same matrix

AG. Clearly, the inverse is also true.

Finally, regarding a combined system, let us assume that the minimal realizations

of a plant G and the controllers K1 and K2 are respectively

G =

AG BG

CG DG

, K1 =

A1 B1

C1 D1

, and K2 =

A2 B2

C2 D2

We also construct GK1 using the terms A,B,C,D for convenience:

GK1 =

A1 0

BGC1 AG

B1

BGD1

DGC1 CGDGD1

=

A B

C D

(5.81)

Then the transfer function for the closed loop combined system may be expressed

with the following state space representation:

(I +GK1K2)−1GK1 =

A−BD2R1−1C −BR2

−1C2 BR2−1

B2R1−1C A2 −B2DR2

−1C2 B2DR2−1

R1−1C −R1

−1DC2 DR2−1

(5.82)

such that R1 = I +DD2, R2 = I +D2D.

138

It is well known that cascade and feedback systems, can be viewed as special cases

of the Redheffer star product (pages 266-267 [69]). Such a star product (Figure 5.7)

is defined as a matrix containing certain LFT forms:

S(K ′, P ′) =

Fl(K ′, P ′11) K ′12(I − P ′11K

′22)−1P ′12

P ′21(I −K ′22P′11)−1K ′21 Fu(P ′, K ′22)

The definition of the star product is dependent on the partitioning of the matrices,

and we will assume that we are using partitions which are well defined. By using

an observation, we can give a more specific result of how our previously discussed

combined system is connected to a Redheffer star product. Let us define P , K, and

K ′:

P =

P11 P12

P21 P22

=

A′ B′1 B′2

C ′1 D′11 D′12

C ′2 D′21 D′22

=

C ′1(sI − A)−1B′1 C ′1(sI − A)−1B′2

C ′2(sI − A)−1B′1 C ′2(sI − A)−1B′2

,

K =

AK BK

CK DK

, and K ′ =

0 I

I K

where P and K are the plant and the controller of the modern system respectively.

Then we will have

139

P22 = C ′2(sI − A)−1B′2 =

A′ B′2

C ′2 D′22

(5.83)

By using K ′ from Equation 5.83, and setting P ′ = P22, it can be shown that (page

499 [69]):

S(K ′, P ′) =

A′ +B′2DKR−1C ′2 B′2R

−1CK B′2R−1

BKR−1C ′2 AK +BKD

′22R

−1CK BKD′22R

−1

R−1C ′2 R−1D′22CK D′22R−1

(5.84)

such that R = I −D′22DK , R = I −DKD′22. Figure 5.8 shows the corresponding

block diagram.

For Equations 5.82 and 5.84 to be equal, we will need A′, B′2, C′2, D

′22, AK , BK ,

CK , DK , R, R to be equal to A, B, C, D, A2, B2, −C2, −D2, R1, R2 respectively.

This is true if

P22 =

A′ B′2

C ′2 D′22

=

A B

C D

= GK1, and

K =

AK BK

CK DK

=

A2 B2

−C2 −D2

= −K2

140

Note that, the terms R, R are equal to the corresponding terms R1, R2, as a direct

consequence by having D′22, DK equal to D, −D2 respectively.

Therefore, the star product will be equal to the closed loop transfer function

(I +GK1K2)−1GK1 if the following two conditions are satisfied:

1. P22 = GK1, and

2. K = −K2.

Then the systems described in Figures 5.8, 5.9, and consequently Figure 5.3, will

have the same closed loop transfer functions. The other three submatrices P11, P12, P21

of P , may be chosen arbitrarily, as they do not affect the final result.

Remark 70 When using Theorems 8 and 9, the linear fractional transformation with

plant P and the combined system with G, are supposed to have stable closed loop

systems. Therefore, although we get a parametrization for P or G (depending on

which way we perform the transformation), it will make sense if we choose a plant P

or G which is stable.

We will now demonstrate the construction of a simple way of representing a com-

bined system with a linear fractional transformation and vice versa.

In the case of representing a combined system as a linear fractional transformation,

we may observe from Theorem 8 that the only restriction required for the plant P is

to have a nonsingular submatrix P12, as well as having the expression L1 nonsingular.

By setting

141

P =

0 I

I G

(5.85)

that is, by setting P11 = 0, P12 = I, P21 = I, and P22 = G (which is one of the

best choices regarding compatibility in dimensions), Equation 5.60 will become

K = (I +GK1K2)−1G(K1 −K1K2G− I) (5.86)

If G, K1, and K2 are stable by definition, then both P and K will also be stable.

Conversely, let us have a plant P and its corresponding controller K. By setting

K2 = 0, Equation 5.71 will become

GK1 = (P22P−112 − P22P

−112 P11K + P21K)(I − P11K)−1P12 (5.87)

Clearly, both Equations 5.86 and 5.87 do not require any extra conditions, except

the standard desired conditions for the closed loop function’s existence.

5.5.2 Conclusions

As we have seen from the two main Theorems, we conclude that a classical system

may be represented in many ways as a modern system (LFT), because of the freedom

of choice for α or β (in the cascade and feedback case) and the freedom of choice

for P (which is independent of any controllers in the combined case). Similarly, a

modern system may be represented as a cascade or a feedback system (with a plant

142

G and a controller K), provided that the plant P of the modern system (which is

independent of K) can be expressed as one of the matrices Pc (as in Equation 5.10), or

Pf (as in Equation 5.37) respectively. Finally, a modern system may be represented

in many ways as a combined system, because of the freedom of choice for G (which

is independent of any controllers) and K2.

The derived results may be used in many ways, and we mention some of them,

for example, providing a theoretical overview on how classical and modern systems

are interconnected (Equations 5.10 and 5.37, the two Theorems 8 and 9, as well

as the corresponding state space representations). Also, by transforming classical

control system examples or applications into their corresponding modern form, we

are able to analyze and manipulate them within modern control methods or software

packages. Moreover, by observing the form of a modern system (LFT), we may easily

deduce if it is possible to convert it into a classical system (cascade or feedback) whose

configuration is simple and easy to construct.

143

Figure 5.1: Block diagram of cascade classical controller.

Figure 5.2: Block diagram of feedback classical controller.

Figure 5.3: Block diagram of combined classical controller.

144

Figure 5.4: Block diagram of modern controller (upper LFT).

Figure 5.5: Block diagram of modern controller (upper LFT) rotated by 180 degrees.

Figure 5.6: Block diagram of modern controller (lower LFT).

145

Figure 5.7: The Redheffer star product.

Figure 5.8: The Redheffer star product, when replacing P ′ by P22.

Figure 5.9: The Redheffer star product, when replacing P ′ by GK1.

146

Chapter 6

Conclusions

In this chapter, we summarize the main contributions and outline future research

directions.

6.1 Main Contributions

• Formulas for new set of weights required for solving controller reduction problem

via double-sided frequency weighted model reduction techniques are derived. It

is shown that one of the frequency weights in a double-sided frequency weighted

model reduction problem can be expressed as a function of a free matrix pa-

rameter C. It is shown that by varying this matrix parameter, the approxima-

tion error in double-sided frequency weighted model reduction problem can be

greatly reduced, yielding more accurate low order controllers.

• There is a very important difference between using the double controller tech-

nique for the feedback systems, and using it for the cascade and the LFT sys-

147

tems. The extra parameter |c−1| in Equation 3.61 only exists in the feedback

systems case. Because of this, a large c produces a better approximation error

without any other requirements (e.g. finding the function which approximates

the behaviour of H(c)). And this is the reason why the double controller method

for the feedback systems case, is superior.

• It can be clearly seen that the algorithm used for the application of the double

controller technique on LFT systems, is built on an efficient searching technique,

without using any brute force search. It is a very important result, especially

since we describe an extremely generalized system type such as the LFT system.

The procedure could be said to be similar to the one used for the cascade system

[26,27], but this time it covers a much broader range of systems, including those

with controllers in series and in the feedback path.

• A classical system may be represented in many ways as a modern system (LFT),

and vice versa.

6.2 Recommendations for future work

• So far we have tried various examples using frequency weighted balanced tech-

niques [17,62]. It would be interesting to use other frequency weighted balancing

and optimal Hankel norm approximation techniques in our proposed frequency

weighted model reduction formulation, and compare the results.

• In this Thesis, we have applied our technique to only continuous time examples.

It would be worthwhile to check how the proposed technique performs with

148

discrete time examples.

• Finally, for future work, the use of a constant matrix C with more than one

parameters (instead of one) has clearly the potential to give even better ap-

proximation errors, as well as constructing a much more accurate graph for

e(c, 1).

149

Bibliography

[1] V. M. Adamjan, D. Z. Arov, and M. Krein. Analytic properties of schmidt pairs

for a hankel operator and the generlized schur-takagi problem. Math. USSR

Sbornik, 15:31–73, 1971.

[2] V. M. Adamjan, D. Z. Arov, and M. Krein. Infinite hanekl block matrices and

related extension problems. American Mathematical Society Transl., 111:133–

156, 1978.

[3] P. Agathoklis and V. Sreeram. Truncation criteria for model reduction using

balanced realization. Electronic Letters, 24:837–838, 1988.

[4] U. M. Al-Saggaf and G. Franklin. On model reduction. Proceedings of 23th IEEE

Conference on Decision and Control, pages 1064–1069, 1986.

[5] U. M. Al-Saggaf and G. Franklin. Model reduction via balanced realization: an

extension and frequency weighted technique. IEEE Transactions on Automatic

Control, 33:687–692, 1988.

[6] B. D. O. Anderson. Weighted hankel norm approximation: Calculation of

bounds. Systems and Control Letters, 7:247–255, 1986.

150

[7] B. D. O. Anderson and Y. Liu. Controller reduction: Concepts and approaches.

IEEE Transactions on Automatic Control, 34(8):802–812, Aug. 1989.

[8] B. D. O. Anderson and J. B. Moore. Linear Optimal Control. Prentice Hall,

Englewood Cliffs, New Jersey, 1971.

[9] E. J. Ang, V. Sreerma, and W. Q. Liu. Identification/reduction to a balanced

realization via the extended impulse response grammian. IEEE Transactions on

Automatic Control, 40:2153–2158, 1995.

[10] C. Beck. Minimality for uncertain systems and iqcs. In Proceedings of the 33rd

IEEE Conference on Decision and Control, pages 3068–3073, Lake Buena Vista,

Florida, USA, 1994.

[11] C. Beck. Model reduction and Minimality for Uncertain Systems. PhD thesis,

California Institute of Technology, USA, 1996.

[12] C. Beck, J. Doyle, and K. Glover. Model reduction of multi-dimensional and

uncertain systems. IEEE Transactions on Automatic Control, 41:1466–1477,

1996.

[13] A. M. Davidson. Balanced systems and model reduction. Electronic Letters,

22:531–532, 1986.

[14] U. B. Desai and D. Pal. A transformation approach to stochastic model reduc-

tion. IEEE Transactions on Automatic Control, AC-29(12):1097–1100, 1984.

151

[15] M. Diab, W. Q. Liu, and V. Sreeram. A gradient flow approach to frequency

weighted model reduction. Proceedings of the 13th World Congress of Interna-

tional Federation of Automatic Control Conference, pages 381–386, 1996.

[16] D. Enns. Model reduction for control system design. PhD thesis, Department of

Aeronautics and Astronautics, Stanford University, USA, 1984.

[17] D. Enns. Model reduction with balanced realizations: an error bound and a

frequency weighted generalization. In Proceedings of the 23rd IEEE Conference

on Decision and Control, pages 127–132, Las Vegas, USA, Dec. 1984.

[18] K. V. Fernando and H. Nicholson. Singular perturbation model reduction of

balanced systems. IEEE Transactions on Automatic Control, 27:466–468, 1982.

[19] B. A. Francis. A Course in H∞ Control Theory, Lecture Notes in Control and

Information Sciences. Springer Verlag, New York, USA, 1987.

[20] T. V. Gestel, B. D. Moore, B. D. O. Anderson, and P. V. Overschee. On fre-

quency weighted balanced truncation: Hankel singular values and error bounds.

European Journal of Control, 7:584–592, 2001.

[21] K. Glover. All optimal hankel norm approximations of linear multivariable sys-

tems and their L∞ error bounds. International Journal of Control, 39:1115–1193,

1984.

[22] M. Green. Balanced stochastic realization. Linear Algebra and its Applications,

98:211–247, 1988.

152

[23] D. W. Gu, P. H. Petkov, and M. M. Konstantinov. H∞ and H2 optimization

toolbox in slicot. SLICOT Working Note, Sept. 1999.

[24] S. Gugercin and A. C. Antoulas. A survey of model reduction by balanced

truncation and some new results. International Journal of Control, 77(8):748–

766, 2004.

[25] Y. Halevi. Frequency weighted model reduction via optimal projection. IEEE

Transactions on Automatic Control, 37(10):1537–1542, Oct. 1992.

[26] P. Houlis and V. Sreeram. Controller reduction via a new double-sided fre-

quency weighted model reduction formulation. In Joint 20th IEEE International

Symposium on Intelligent Control (ISIC’05) and 13th Mediterranean Conference

on Control and Automation (MED’05), pages 537–542, Limassol, Cyprus, June

2005.

[27] P. Houlis and V. Sreeram. A parametrized controller reduction technique. In

Proceedings of the 45th IEEE Conference on Decision and Control, pages 3430–

3435, San Diego, USA, Dec. 2006.

[28] P. C. Houlis and V. Sreeram. A parametrized controller reduction technique via

a new frequency weighted model reduction formulation. IEEE Transactions on

Automatic Control, 54(2), Feb. 2009.

[29] Y. S. Hung and K. Glover. Optimal hankel-norm approximation of stable systems

with first-order stable weighting functions. Systems and Control Letters, 7:165–

172, 1986.

153

[30] M. F. Hutton and B. Friedland. Routh approximations for reducing order of lin-

ear time-invariant systems. IEEE Transactions on Automatic Control, 20(3):329–

337, 1975.

[31] D. C. Hyland and D. S. Bernstein. The optimal projection equation for model

reduction and the relationship among the methods of wilson, skelton and moore.

IEEE Transactions on Automatic Control, 30:1201–1211, 1985.

[32] P. T. Kabamba. Balanced forms: Canonicity and parameterizations. IEEE

Transactions on Automatic Control, A.C. 30(11):1106–1109, 1985.

[33] S. Kim, B. Anderson, and A. Madievski. Error bound for transfer function order

reduction using frequency weighted balanced truncation. Systems and Control

Letters, 24:183–192, 1995.

[34] S. Kim, B. Anderson, and A. Madievski. Multiplicative approximation of for

transfer function with frequency weighting. Systems and Control Letters, 25:199–

204, 1995.

[35] S. Y. Kung. A new identification and model reduction algorithm via singular

value decomposition. Proceedings of 12th Asilomar Conference on Circuits, Sys-

tems and Computers, pages 705–714, 1986.

[36] S. Y. Kung and D. W. Lin. Optimal hankel-norm model reduction: Multivariable

systems. IEEE Transactions on Automatic Control, 26:832–852, 1981.

[37] H. Kwakanaak and R. Sivan. Linear Optimal Control Systems. Wiley-

Interscience, New York, 1972.

154

[38] A. Latham and B. Anderson. Frequency weighted optimal hankel norm approx-

imation of stable transfer functions. Systems and Control Letters, 5:229–236,

1986.

[39] A. Lin and T. Chiu. Model reduction via frequency weighted balanced realization.

Control Theory and Adv. Tech., 8:341–351, 1992.

[40] Y. Liu and B. D. O. Anderson. Controller reduction via stable factorizarion and

balancing. International Journal of Control, 44(2):507–531, 1986.

[41] Y. Liu and B. D. O. Anderson. Singular perturbation approximation of balanced

systems. International Journal of Control, 50:1379–1405, 1986.

[42] Y. Liu, B. D. O. Anderson, and U. L. Ly. Coprime factorization controller

reduction with bezout identity induced frequency weighting. Automatica, 26:507–

531, 1986.

[43] D. McFarlane, K. Glover, and M. Vidyasagar. Reduced-order controller design

using coprime factor model reduction. IEEE Transactions on Automatic Control,

35:369–373, 1981.

[44] G. Michaletzky. Hankel norm approximation of a rational function using stochas-

tic realizations. Systems and Control Letters, 12:211–216, 1989.

[45] B. C. Moore. Principal component analysis in linear systems: controllability,

observability, and model reduction. IEEE Transactions on Automatic Control,

A.C. 26:17–32, 1981.

155

[46] C. T. Mullis and R. A. Roberts. Roundoff noise in digital filters: Frequency

transformations and invariants. IEEE Transactions on Acoustic, Speech and

Signal Processing, 24:538–550, 1976.

[47] C. T. Mullis and R. A. Roberts. Synthesis of minimum roundoff noise fixed point

digital filters. IEEE Transactions on Circuits and Systems, 23:551–561, 1976.

[48] D. Mustafa and K. Glover. Controler reduction by H∞-balanced truncation.

IEEE Transactions on Automatic Control, 36:668–682, 1991.

[49] Z. Nehari. On bounded bilinear forms. Annals of Mathematics, 15:155–162, 1957.

[50] G. Obinata and B. Anderson. Model Reduction for Control System Design.

Springer Verlag, London, UK, 2001.

[51] G. Obinata and H. Inooka. A method for modelling linear time-invariant systems

by linear systems of low order. IEEE Transactions on Automatic Control, 21:602–

603, 1976.

[52] G. Obinata and H. Inooka. Author’s reply to comments on model reduciton

by minimizing the equation error. IEEE Transactions on Automatic Control,

28:124–125, 1983.

[53] L. Pernebo and L. M. Silverman. Model reduction via balanced state space

representation. IEEE Transactions on Automatic Control, AC-27:382–387, 1982.

[54] M. G. Safonov. L∞ sensitivity optimization and hankel approximation. IEEE

Transactions on Automatic Control, 30:279–280, 1985.

156

[55] Y. Shamash. Stable reduced order models using pade type approximation. IEEE

Transactions on Automatic Control, 19:615–616, 1974.

[56] N. K. Sinha and Kuszta. Modelling and Identification of Dynamic Systems. Van

Nostrand Reinhold, New York, USA, 1983.

[57] R. E. Skelton. Dynamic Systems Control. Wiley, New York, USA, 1988.

[58] V. Sreeram and P. Agathoklis. Model reduction using balanced realization with

improved low frequency behaviour. Systems and Control Letters, 12:33–38, 1989.

[59] V. Sreeram, B. Anderson, and A. Madievski. Frequency weighted balanced re-

duction technique: A generalization and an error bound. Proceedings of the 34th

IEEE Conference on Decision and Control, 4:3576–3581, 1995.

[60] V. Sreeram, B. Anderson, and A. Madievski. New results in frequency weighted

balanced reduction technique. Proceedings of American Control Conference,

pages 4004–4009, 1995.

[61] A. Varga and B. Anderson. Accuracy-enhancing methods for balancing-related

frequency-weighted model and controller reduction. Automatica, 39:919–927,

2003.

[62] G. Wang, V. Sreeram, and W. Liu. A new frequency-weighted balanced trun-

cation method and an error bound. IEEE Transactions on Automatic Control,

44:1734–1737, 1999.

[63] D. A. Wilson. Optimum solution of model reduction problem. Proc. IEE Part

D, 117:1161–1165, 1970.

157

[64] D. A. Wilson. Model reduction for multivariable systems. International Journal

of Control, 20:57–64, 1974.

[65] D. Youla, J. B. Jr, and C. N. Lu. Single-loop feedback stabilization of linear

multivariable dynamical plants. Automatica, 10:159–173, 1974.

[66] K. Zhou. Frequency weighted model reduction with L∞ error bounds. Systems

and Control Letters, 21:115–125, 1993.

[67] K. Zhou. Frequency weighted model L∞ norm and optimal hankel norm model

reduction. IEEE Transactions on Automatic Control, 40(10):1687–1699, Oct.

1995.

[68] K. Zhou. Relative/multiplicative model reduction for unstable and non-

minimum-phase systems. Automatica, 31:1087–1098, 1995.

[69] K. Zhou, J. Doyle, and K. Glover. Robust and Optimal Control. Prentice Hall,

New Jersey, USA, 1996.

158

Appendix

Expansion of Equation 3.3

W −Wr = (I +GK)−1GK − (I +GKr)−1GK =[

(I +GK)−1 − (I +GKr)−1]GK =[

I − (I +GKr)−1(I +GK)

](I +GK)−1GK =

(I +GKr)−1 [(I +GKr)− (I +GK)] (I +GK)−1GK =

(I +GKr)−1 [GKr −GK] (I +GK)−1GK =

(I +GKr)−1G [K −Kr] (I +GK)−1GK

Replacing K by Kr in the first term (I+GKr)−1 is equivalent to neglecting second

order terms of K −Kr whose norm is extremely small and can be neglected. Thus,

we may have

W −Wr = (I +GK)−1G [K −Kr] (I +GK)−1GK

159

Proof of Lemma 5

W −Wr = (I +GK)−1G− (I +GKr)−1G =[

(I +GK)−1 − (I +GKr)−1]G =[

I − (I +GKr)−1(I +GK)

](I +GK)−1G =

(I +GKr)−1 [(I +GKr)− (I +GK)] (I +GK)−1G =

(I +GKr)−1 [GKr −GK] (I +GK)−1G =

(I +GKr)−1G [K −Kr] (I +GK)−1G

Replacing K by Kr in the first term (I+GKr)−1 is equivalent to neglecting second

order terms of K −Kr whose norm is extremely small and can be neglected. Thus,

we may have

W −Wr = (I +GK)−1G [K −Kr] (I +GK)−1G

Proof of Lemma 9

W −Wr =

P21K(I − P11K)−1P12 − P21Kr(I − P11Kr)−1P12 =

P21[K(I − P11K)−1 −Kr(I − P11Kr)−1]P12 =

P21[(I −KP11)−1K −Kr(I − P11Kr)

−1]P12 =

P21(I−KP11)−1[K−(I−KP11)Kr(I−P11Kr)

−1]P12 =

P21(I −KP11)−1[K(I − P11Kr)−

(I −KP11)Kr](I − P11Kr)−1P12 =

P21(I −KP11)−1[K −Kr](I − P11Kr)

−1P12.

160

This result is presented in [69].

The invertibility of H(C)

Recall that H(C) = I+GK(I−C), where C = cI is a constant matrix. By calculating

the determinant of H(C) we will obtain an expression of the form

an(s)cn + an−1(s)cn−1 + · · ·+ a1(s)c+ a0(s)

There are n zeros which cause the matrix to become singular. All those values

for c are not necessarily independent of s. The n′ solutions for c which are functions

of s should be omitted, as we have assumed that c is a constant with respect to s.

Therefore, we only need to exclude the remaining n − n′ constant solutions when

choosing our parameter c to build the matrix H(C) based on a predefined system.

The number n− n′ is finite which means that H(C) will almost always be invertible.

Of course, cases such as GK = δI and C = (δ+1)I may create non-invertible H(c)’s,

but they are never found in practical situations and may be excluded.

Representing Transfer Functions by a Minimal Realization

Let us assume that G has a minimal realization expressed by {AG, BG, CG, DG}. Then

by using matrix properties, the plant Pc (as shown in Equation 5.75) that corresponds

161

to the classical cascade system may be expressed as in Equation 5.76.

Pc(s) =

AG 0 0 0 BG 0

0 0 0 0 0 0

0 0 AG 0 BG 0

0 0 0 0 0 0

−CG 0 0 0 −DG I

0 0 CG 0 DG 0

(1)

The transfer function for Pc may be rewritten as

162

P (s) =

−CG 0 0 0

0 0 CG 0

sI −

AG 0 0 0

0 0 0 0

0 0 AG 0

0 0 0 0

−1

BG 0

0 0

BG 0

0 0

+

−DG I

DG 0

=

−CG 0 0 0

0 0 CG 0

sI − AG 0 0 0

0 sI 0 0

0 0 sI − AG 0

0 0 0 sI

−1

BG 0

0 0

BG 0

0 0

+

−DG I

DG 0

=

−CG 0 0 0

0 0 CG 0

(sI−AG)−1 0 0 0

0 1sI 0 0

0 0 (sI−AG)−1 0

0 0 0 1sI

BG 0

0 0

BG 0

0 0

+

−DG I

DG 0

=

−CG(sI − AG)−1BG 0

CG(sI − AG)−1BG 0

+

−DG I

DG 0

=

−CG

CG

(sI − AG)−1

[BG 0

]+

−DG I

DG 0

=

AG BG 0

−CG −DG I

CG DG 0

=

AP BP

CP DP

163

By assuming that {AG, BG, CG, DG} is a minimal realization for G, we will show

that {AC , BC , CC , DC} is a minimal realization for Pc, where

Ac = AG, Bc =

[BG 0

], Cc =

−CG

CG

, Dc =

−DG I

DG 0

.

The controllability and observability matrices for the realizations {AG, BG, CG, DG}

and {Ac, Bc, Cc, Dc} are

CG =

[BG AGBG . . . AG

n−1BG

]

OG =

[CG CGAG . . . CGAG

n−1

]T

Cc =

[BG 0 AGBG 0 . . . AG

n−1BG 0

]

Oc =

[CG CG CGAG CGAG . . . CGAG

n−1 CGAGn−1

]T

Since CG and OG are full rank matrices and have rank n. We observe that Cc and

Oc are expanded versions of CG and OG, which means that their rank is at least n.

And since the number of rows of Cc is equal to n and the number of columns of Oc is

also equal to n, we conclude that both Cc and Oc are of full rank, which means that

{Ac, Bc, Cc, Dc} is a minimal realization.

Similarly, consider the plant Pf (as shown in Equation 5.78) that corresponds to

164

the classical feedback system which may be expressed as in Equation 5.76.

Pf =

AG 0 0 0 BG 0

0 AG 0 0 0 BG

0 0 AG 0 BG 0

0 0 0 AG 0 BG

−CG CG 0 0 −DG DG

0 0 −CG CG −DG DG

(2)

By using exactly the same methodology which we used for the classical cascade sys-

tem, it is easy to prove that the minimal realization of Pf (as in Equation 5.79)

corresponding to a feedback system will be:

Pf =

AG −BG BG

CG −DG DG

CG −DG DG

. (3)

Non-commutative properties with respect to GK and K

Assume that there exists a matrix D which is not commutative with both GK and K.

We want to show that the expression GKDK−1 is not always the same for different

values of K, and subsequently, is a function of K.

165

Firstly, we assume that D is not commutative with both GK1 and K1, where

G,K1 are without loss of generality two full rank matrices. Secondly, we construct

another matrix K3 = K1K2, where K2 is any (full rank) matrix such that D does not

commute with GK2 and K2.

Since K2 is of full rank and does not commute with D, we have that K2 6= I. So

clearly, K1 and K3 are different matrices. Let us now assume that we have

GK1DK1−1 = GK3DK3

−1

Then the above Equation will become

GK1DK1−1 = GK3DK3

−1

GK1DK1−1 = GK1K2DK1K2

−1

GK1DK1−1 = GK1K2DK2

−1K1−1

GK1D = GK1K2DK2−1

K1D = K1K2DK2−1

D = K2DK2−1

DK2 = K2D

which is a contradiction. Therefore, different matrices K give different matrices

for the expression GKDK−1, which subsequently, is a function of K.

166

Date post:	10-Apr-2020
Category:	Documents
Upload:	others
View:	7 times
Download:	0 times

A Novel Parametrized Controller Reduction Technique based on … · A Novel Parametrized Controller...

Documents