gumussoy suat

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OPTIMAL H-INFINITY CONTROLLER DESIGN AND

STRONG STABILIZATION FOR TIME-DELAY AND

MIMO SYSTEMS

DISSERTATION

Presented in Partial Fulfillment of the Requirements for

the Degree Doctor of Philosophy in the

Graduate School of The Ohio State University

By

Suat Gumussoy, B.S.E.E., M.S.* * * * *

The Ohio State University

2004

Ph.D. Examination Committee:Professor Hitay Ozbay, Adviser

Professor Andrea Serrani

Professor Hooshang Hemami

Approved by

Adviser

Department of ElectricalEngineering


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ABSTRACT

In this thesis, the problems of optimal H controller design and strong stabi-

lization for time-delay systems are studied. First, the optimal H controller design

problem is considered for time-delay plants with finitely many unstable zeros and

infinitely many unstable poles. It is shown that this problem is the dual version of

the same problem for the plants with finitely many unstable poles and infinitely many

unstable zeros, that is solved by the so-called Skew-Toeplitz approach. The optimal

H controller is obtained by a simple data transformation. Next, the solution of the

optimal H controller design problem is given for plants with finitely many unstable

poles or unstable zeros by using duality and the Skew-Toeplitz approach. Necessary

and sufficient conditions on time-delay systems are determined for applicability of

the Skew-Toeplitz method to find optimal H controllers. Internal unstable pole-

zero cancellations are eliminated and finite impulse response structure of the optimal

H controller is obtained. The problem of strong stabilization is studied for time

delay and MIMO finite dimensional systems. An indirect approach to design a stable

controller achieving a desiredH performance level for time delay systems is given.

This approach is based on stabilization of H controller by another H controller

in the feedback loop. In another approach, when the optimal controller is unstable

(with infinitely or finitely many unstable poles), two methods are given based on a

search algorithm to find a stable suboptimal controller. In this approach, the main

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idea is to search for a free parameter which comes from the parameterization of sub-

optimal H controller, such that it results in a stable H controller. Finally, the

strong stabilization problem and stable H controller design for finite dimensional

multi-input multi-output linear time invariant systems are studied. It is shown that

if a certain linear matrix inequality condition has a solution then a stable controller,

whose order is the same as the order of the generalized plant, can be constructed.

This result is applied to design stable H controller with the order twice of the order

of the generalized plant.

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To my mom and dad, Dilek and Kamil,

and my brother, Murat for ...

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ACKNOWLEDGMENTS

I would like to thank my advisor, Professor Hitay Ozbay, for his understand-

ing, support and help throughout this academic experience. He was the advisor

just I wished. I would also like to thank Professor Andrea Serrani and Professor

Hooshang Hemami for serving in my examination committee. I am thankful to Pro-

fessor Vadim I. Utkin for his valuable discussions.

I would also like to acknowledge the financial support from the National Science

Foundation, AFRL/VA and AFOSR.

My special thanks are for my parents, Kamil and Dilek Gumussoy, and my brother,

Murat Gumussoy who helped and assisted me at every stage in my life.

Many thanks to my office-mates, Pierre F.Quet and Xin Yuan for nice chats and

valuable discussions. They were just there, when I need to speak Hish.

I am thankful to my early-Ph.D.friends, Tankut Acarman, Veysel Gazi, Mehmet

Onder Efe, Umit Ogras, Oguz Dagc, Peng Yan and late-Ph.D friends, Cosku Kas-

nakoglu, Alvaro E. Gil, Nicanor Quijano, Jorge Finke. Since it is difficult to mention

all the names, I am also thankful to CRL graduate students and faculty.

I want to thank my non-departmental friends Cezmi Unal, Yelda Serinagaoglu,

Gokhan Korkmaz, Sahika Vatan, Tansu Demirbilek, Yucel and Derya Demirer for

spending their time with me.

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Although a graduate study in abroad causes to lost many friendship in your home

country, I am thankful to Kursad Ergut, Hatice Dinc, Ulas Bars Sarsoy, Ilke and

Unsal Soysal for everything they did in spite of distances.

While I were studying all night with no sleep, I were always happy to see my

friend Douglas Ellis, janitor, with smile on his face at 6:00a.m.

My last thanks are for my love, Ipek. Everything will be too much difficult without

you. When I started to doubt that there exist the one, you came...

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VITA

May 16, 1977 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Born - Aksaray, Turkey

September, 1999 . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.S. in Electrical and Electronics Engi-neeringMiddle East Technical University

Ankara, TurkeySeptember, 1999 .... . . . . . . . . . . . . . . . . . . . . . . . .B.S. in Mathematics

Middle East Technical UniversityAnkara, Turkey

August, 2001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M.S. in Electrical and Computer Engi-neeringThe Ohio State UniversityColumbus, Ohio

September, 2000 - Present . . . . . . . . . . . . . . . . . . Graduate Research AssociateThe Ohio State UniversityColumbus, Ohio

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PUBLICATIONS

Research Publications

Suat Gumussoy and Hitay Ozbay, On the Mixed Sensitivity Minimization for Sys-tems with Infinitely Many Unstable Modes, to appear in Systems and Control

Letters, 2004 (available on publishers web site since June 10, 2004).

Suat Gumussoy and Hitay Ozbay, Remarks on Strong Stabilization and Stable H-infinity Controller Design, to appear in Proceedings of 43rd IEEE Conference onDecision and Control, The Bahamas, December 2004.

Suat Gumussoy and Hitay Ozbay, On Stable H-infinity Controllers for Time Delay

Systems, Proceedings of the conference on Mathematical Theory of Network andSystems, July 2004.

Murat Saglam, Sami Ezercan, Suat Gumussoy and Hitay Ozbay, Controller tuning

for active queue management using a parameter space method, Proceedings of theconference on Mathematical Theory of Network and Systems, July 2004.

Suat Gumussoy and Hitay Ozbay, Control of Systems with Infinitely Many Unstable

Modes and Strong Stabilizing Controllers Achieving a Desired Sensitivity, Proceed-ings of the conference on Mathematical Theory of Network and Systems, August 2002.

FIELDS OF STUDY

Major Field: Electrical Engineering

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TABLE OF CONTENTS

Page

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

Chapters:

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

1.1 Literature Review on Optimal H Controller Design for Time-DelaySystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Literature Review on Stable H Controller Design . . . . . . . . . 41.3 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2. On the Mixed Sensitivity Minimization for Systems with Infinitely Many

Unstable Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . 14

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3. H Controller Design for Neutral and Retarded Delay Systems . . . . . 163.1 Preliminary Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.1 Neutral and Retarded Time Delay Systems with Finitely

Many Unstable Zeros . . . . . . . . . . . . . . . . . . . . . . 213.1.2 FIR Structure of Neutral and Retarded Time Delay Systems 24

3.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2.1 Factorization of the Plants . . . . . . . . . . . . . . . . . . . 27

3.2.2 Optimal H Controller Design . . . . . . . . . . . . . . . . 293.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3.1 IF Plant Example . . . . . . . . . . . . . . . . . . . . . . . 333.3.2 FF Plant Example . . . . . . . . . . . . . . . . . . . . . . . 36

3.4 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4. Stable H Controllers for Delay Systems: Suboptimal Sensitivity . . . . 404.1 Optimal Sensitivity Problem for Delay Systems . . . . . . . . . . . 414.2 Sensitivity Deviation Problem . . . . . . . . . . . . . . . . . . . . . 43


5. On Stable H Controllers for Time-Delay Systems . . . . . . . . . . . . 495.1 Structure ofH Controllers . . . . . . . . . . . . . . . . . . . . . . 505.2 Stable suboptimal H controllers, when the optimal controller has

infinitely many unstable poles . . . . . . . . . . . . . . . . . . . . . 52

5.3 Stable suboptimal H controllers, when the optimal controller hasfinitely many unstable poles . . . . . . . . . . . . . . . . . . . . . . 56

5.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.4.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.4.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 64


6. Remarks on Strong Stabilization and Stable H Controller Design . . . 706.1 Strong stabilization of MIMO systems . . . . . . . . . . . . . . . . 71

6.2 Stable H controller design for MIMO systems . . . . . . . . . . . 746.3 Numerical examples and comparisons . . . . . . . . . . . . . . . . . 75

6.3.1 Strong stabilization . . . . . . . . . . . . . . . . . . . . . . 75

6.3.2 Stable H controllers . . . . . . . . . . . . . . . . . . . . . 766.4 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . 79

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7. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Appendices:

A. Skew-Toeplitz Approach for Mixed Sensitivity Problem . . . . . . . . . . 85

B. IF and FI Plant Calculations . . . . . . . . . . . . . . . . . . . . . . . . 88

B.1 Example: IF Plant Case . . . . . . . . . . . . . . . . . . . . . . . . 88

B.2 Example: FF Plant Case . . . . . . . . . . . . . . . . . . . . . . . . 89

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

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LIST OF FIGURES

Figure Page

3.1 Optimal H Controller for IF plants . . . . . . . . . . . . . . . . . . 30

3.2 Optimal H Controller for FI plants . . . . . . . . . . . . . . . . . . 31

3.3 fT(t) for IF plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4 fopt(t) for IF plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.5 fT(t) for FF plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.6 fopt(t) for FF plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.1 (i) F = 0: the weighted sensitivity is H optimal; (ii) F = 0: thecontroller K is stable. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.1 wmax and max versus u . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.2 Z(s) plot for right half plane . . . . . . . . . . . . . . . . . . . . . . . 64

5.3 min versus n2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.4 Magnitude plot ofU(j ) . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.5 u values resulting stable H controller . . . . . . . . . . . . . . . . 696.1 Standard Feedback System . . . . . . . . . . . . . . . . . . . . . . . . 71

6.2 Comparison for plant G1 . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.3 Comparison for plant G2 . . . . . . . . . . . . . . . . . . . . . . . . . 76

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LIST OF TABLES

Table Page

6.1 Stable H controller design for combustion chamber . . . . . . . . . 77

6.2 Stable H controller design for Example in [1] . . . . . . . . . . . . . 79

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CHAPTER 1

INTRODUCTION

Mathematical modeling of physical systems is typically based on certain simpli-

fying assumptions and physical laws of nature. Hence it always introduces modeling

errors, i.e., the plant model from which a controller is to be designed is just an ap-

proximation of the actual plant. Therefore, it is important to use design techniques

which guarantee stability and performance against such uncertainties. Robust control

is an important field of feedback control theory, that is concerned with stability and

performance of systems with uncertainties.

One of the most important tools of robust control is H control. Since the

H norm can interpreted as the worst-case gain of the system, it is an appropriate

criterion to pose the disturbance minimization problems in this setting. There are

three major types of plant uncertainties, i.e., additive, multiplicative, coprime factor

uncertainties. Considering these types of uncertainties of the plant, different types of

H

controllers are designed to achieve the closed loop stability (Robust Stabilization

Problem) and pre-specified performance level (Robust Performance Problem) [2].

Stable controller design for a given plant is called strong stabilization. It is de-

sirable to have a stable controller for practical and theoretical reasons, ([3, 4]). For

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example it is well known that the unstable poles of the controller degrades the track-

ing performance and/or disturbance rejection capabilities of the closed loop system

([5, 6, 7]). Moreover, unstable controllers are highly sensitive and their response to

sensor-faults and plant uncertainties/nonlinearities are unpredictable ([8]). It is also

well-known, [3], that the strong stabilization problem has a close connection with

simultaneous stabilization of two or more plants by a single controller. Also, stable

controllers can be tested off-line to check some faults in the implementation and to

compare with the theoretical design specifications. Therefore, it is desirable to use

strongly stabilizing controllers in the feedback loop whenever possible.

In this thesis, a fundamental tool in robust control, H control theory, will be used

to design optimal H controllers for time-delay systems and stable H controllers for

time-delay and multi-input multi-output (MIMO) systems. Unless otherwise speci-

fied, the discussion will be primarily be confined to single-input single-output (SISO)

systems. For the rest of this Chapter, a brief overview of the literature on optimal

H

controller and stable H

controller design will be given and the outline of the

dissertation can be found in the last section of this Chapter.

1.1 Literature Review on Optimal H Controller Design forTime-Delay Systems

It is well known that H controllers for linear time invariant systems with finitely

many unstable modes can be determined by various methods, see e.g. [9, 10, 11,

12, 13, 14, 15]. In particular, H control problem for time-delay systems is studied

by many researchers. When the plant is a dead-time system in the form ehsP0(s),

where P0 is a rational transfer function, optimal H controller design problem is

solved in [16, 17, 18, 19, 20]. A general class of infinite dimensional H control

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problems are solved by operator theoretic methods. In this thesis we will refer to one

of these methods known as the Skew-Teoplitz approach. In [2], a mixed sensitivity

minimization problem is converted into two-block, and then to an one-block H

problem after a series of factorizations and transformations, and the resulting Nehari

problem is solved using the Commutant Lifting Theorem. In [15] the overall algorithm

is significantly simplified. The optimal H controller can be obtained by this method

for SISO dead-time systems.

State-space methods for H control of dead-time systems are given in [21, 22].

In these papers, the infinite dimensional problem is reduced to finite dimensional

problems and the corresponding problems are solved. Another solution to same prob-

lem is given in [23] by using the J-spectral factorization approach. Moreover, in the

same work, a special structure of the optimal and suboptimal H controllers are

shown (i.e., infinite dimensional part of the controller can be represented in finite

impulse response filter (FIR)). Optimal H controller design for MIMO plants with

input/output delays is solved in [24]. In this paper, multiple input/output delays

are decomposed into dead-time systems (so-called adobe plant problems), then opti-

mal controller is constructed from these systems. The FIR structure of the infinite

dimensional part of the optimal H controller is also shown for MIMO plants with

input/output delays.

Although the Skew-Toeplitz approach [2] solves the mixed sensitivity problem for

general infinite dimensional systems, connections of these systems with time-delay

systems are little known. In this thesis, the link between Skew-Toeplitz approach and

time-delay systems will be established. Moreover, general time-delay systems will be

considered including dead-time systems.

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1.2 Literature Review on Stable H Controller Design

A necessary and sufficient condition for the existence of a stable controller is the

parity interlacing property (p.i.p.), [25]. A plant satisfies the p.i.p. if the number of

poles of the plant (counted according to their McMillan degrees) between any pair

of real right half plane blocking zeros is even. There are several design methods for

constructing strongly stabilizing controllers, see e.g. [3, 25]. In general, the procedures

involve construction of a unit in H satisfying certain interpolation conditions. A

parameterization of all stabilizing controllers for SISO case is obtained in [3]. For

stable controller design with rational controllers see [26, 27]. It is known that if the

plant is arbitrarily close to violating p.i.p., then the order of the strongly stabilizing

controller can be unbounded [28]. Certain bounds on the norm and the degree of

unit interpolants are given in [29] by using Nevanlinna-Pick interpolation theory. A

conservative bound on controller order of strongly stabilizing controller is obtained in

[30]. Stable controller design for MIMO systems is considered in [7]. The necessary

and sufficient condition for strong stabilization, parity interlacing property, is shown

in [31] for SISO delay systems. A design method to find strongly stabilizing controller

for SISO systems with time delays is given [32] in which the stable controller is

constructed by using the unit satisfying some interpolation conditions.

The H performance cost minimization with stable controller has also been stud-

ied in the literature. For finite dimensional SISO plants, optimal stable

H con-

troller for weighted sensitivity minimization is given in [33]. Using the interpolation

conditions of sensitivity function and Nevanlinna-Pick approach, optimal stable H

controller is obtained. An explicit formula for this optimal controller is given in [34]

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where certain parameters of the controller are found from the solution of a set of non-

linear equations. The SISO discrete-time version of the weighted sensitivity problem

is solved in [35] by using a convex integer programming approach. For the mixed

sensitivity problem, certain conditions on the weighting functions and the plant are

determined in [36] for stability of the optimal controller. In [37] a sufficient condi-

tion is obtained for the synthesis of SISO finite dimensional suboptimal stable H

controllers, by converting the problem into a Nevanlinna-Pick interpolation problem.

For the same problem with finite dimensional MIMO systems, a design method

has been developed in [38]. The problem of weighted sensitivity minimization with

stable controllers is reduced into finite dimensional optimization problem of finding

a set of integers and some admissible matrices. In [39], the stability of the controller

is guaranteed if nonnegative definite solutions exist to the design equations; further-

more, under this condition, the controller order is less than or equal to that of the

plant. Similarly, in [40], a sufficient condition is obtained for the existence of a stable

H

controller. This condition is expressed in terms of an Algebraic Riccati Equation

(ARE) derived from the state-space realization of the central controller. The dimen-

sion of the stable H controller determined in [40] is less than or equal to 2n, where

n is the dimension of the generalized plant. In [41] strong stabilization problem is

considered for MIMO finite dimensional linear time invariant systems. It is shown

that if a two block H problem is solvable, then a strongly stabilizing controller can

be derived. The controller is of the same order as the plant. Moreover, under the

given sufficient conditions finite dimensional characterizations of strongly stabilizing

controllers are obtained. Another sufficient condition is presented for the existence

of a stable H controller in [42] using chain-scattering approach where the controller

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design algorithm requires solution of an ARE. An improved H control design al-

gorithm with strong stability condition is presented in [8]. By choosing a weighting

function, the conservatism in two-block problem for stabilizing controller design is

reduced. However, the weight function causes a substantial increase in the order of

the controller.

The stable H controller design for infinite dimensional systems is still an open

research problem. In this thesis, this problem will be addressed and stable H

controller design methodologies will be given for time-delay systems as well as MIMO

systems.

1.3 Dissertation Outline

In Chapter 2, optimal H controller design for time-delay systems with finitely

many zeros and infinitely many unstable poles are considered. It is assumed that the

unstable zeros and unstable poles of the plant are captured in inner terms and the

stable part of the plant is represented as an outer term. It is shown that this problem

is the dual case of the usual Skew-Toeplitz approach [2, 15], and the optimal H

controller is given for this plant. In Chapter 3, using this result and the Skew-Toeplitz

method, the optimal H controller design is extended for the time-delay plants with

finitely many zeros or poles. Necessary and sufficient conditions are obtained for

the applicability of the Skew-Toeplitz approach on generalized time delay systems.

Moreover, the controller is constructed such that there are no internal unstable pole-

zero cancellations and the FIR structure of the H controllers are shown for the same

class of plants.

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An indirect approach to design stable controller achieving a desired H perfor-

mance level for dead time systems is given in Chapter 4. This approach is based on

stabilization of H controller by another H controller in the feedback loop. Sta-

bilization is achieved and the sensitivity deviation is minimized. Then in Chapter

5, two other design methods are given for the same problem. Moreover in Chapter

6, stable H controller design for finite dimensional linear invariant MIMO plants is

considered. The design method in [41] is generalized using linear matrix inequalities.

The links between the Chapters of thesis are as follows: Chapters 2 and 3 are

concerned with optimalH controller design for general time-delay systems. In

Chapters 4, 5 and 6, the strong stabilization problem is considered. More specifically,

Chapters 4 and 5 give stable H controller design methods for time-delay systems,

and Chapter 6 deals with the same problem for finite dimensional MIMO systems.

Finally, in Chapter 7 concluding remarks are given.

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CHAPTER 2

ON THE MIXED SENSITIVITY MINIMIZATION FORSYSTEMS WITH INFINITELY MANY UNSTABLE

MODES

The main purpose of this Chapter is to show that H controllers for systems with

infinitely many unstable modes can be obtained by the Skew-Toeplitz approach [2],

using a simple data transformation. An example of such a plant is a high gain system

with delayed feedback (see Section 2.2). Undamped flexible beam models, [43], may

also be considered as a system with infinitely many unstable modes.

In earlier studies, e.g. [15],

H controllers are computed for weighted sensitivity

minimization involving plants in the form

P(s) =Mn(s)

Md(s)No(s) (2.1)

where Mn(s) is inner and infinite dimensional, Md(s) is inner and finite dimensional,

and No(s) is the outer part of the plant, that is possibly infinite dimensional. In

the weighted sensitivity minimization problem, the optimal controller achieves the

minimum H cost, opt, defined as

opt = inf C stabilizing P

W1(1 + P C)1W2P C(1 + P C)1

, (2.2)

where W1 and W2 are given finite dimensional weights. Note that in the above

formulation, the plant has finitely many unstable modes, because Md(s) is finite

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dimensional, whereas it may have infinitely many zeros in Mn(s). In this section, by

using duality, the mixed sensitivity minimization problem will be solved for plants

with finitely many right half plane zeros and infinitely many unstable modes.

2.1 Main Result

Assume that the plant to be controlled has infinitely many unstable modes, finitely

many right half plane zeros and no direct transmission delay. Then, its transfer func-

tion is in the form P = NM

, where M is inner and infinite dimensional (it has infinitely

many zeros in C+, that are unstable poles ofP), N = NiNo with Ni being inner finite

dimensional, and No is the outer part of the plant, possibly infinite dimensional. For

simplicity of the presentation we further assume that No, N1o H.

To use the controller parameterization of Smith, [44], we first solve for X, Y H

satisfying

N X + M Y = 1 i.e. X(s) =

1 M(s)Y(s)

Ni(s)

N1o (s). (2.3)

Let z1,...,zn be the zeros of Ni(s) in C+, and again for simplicity assume that they

are distinct. Then, there are finitely many interpolation conditions on Y(s) for X(s)

to be stable, i.e.

Y(zi) =1

M(zi)i = 1, . . . , n .

Thus by Lagrange interpolation, we can find a finite dimensional Y

H and infinite

dimensional X H satisfying (2.3), and all controllers stabilizing the feedback

system formed by the plant P and the controller C are parameterized as follows, [44],

C(s) =X(s) + M(s)Q(s)

Y(s) N(s)Q(s) (2.4)

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where Q(s) H and Y(s) N(s)Q(s)) = 0.

Now we use the above parameterization in the sensitivity minimization problem.

First note that,

(1 + P(s)C(s))1 = M(s)(Y(s) N(s)Q(s)),

P(s)C(s)(1 + P(s)C(s))1 = N(s)(X(s) + M(s)Q(s)). (2.5)

Then,

infC stab. P

W1(1 + P C)1

W2P C(1 + P C)1

= inf QH

W1(Y N Q)W2N(X + M Q)

(2.6)

where Y(s) N(s)Q(s) = 0, W1 and W2 are given finite dimensional (rational)weights. From (2.3) equation, we have W1Y W1N QW2N1MYN + W2M N Q

=

W1(Y Ni(NoQ))W2(1 M(Y Ni(NoQ)))

.(2.7)

Thus, the H optimization problem reduces to

opt = inf Q1H and YNiQ1=0

W1(Y NiQ1)W2(1 M(Y NiQ1))

(2.8)

where Q1 = NoQ, and note that W1(s), W2(s), Ni(s), Y(s) are rational functions, and

M(s) is inner infinite dimensional.

The problem defined in (2.8) has the same structure as the problem dealt in

Chapter 5 of the book by Foias, Ozbay and Tannenbaum [2], where Skew-Toeplitz

approach has been used for computing H optimal controllers for infinite dimensional

systems with finitely many right half plane poles. Our case is the dual of the problem

solved in [2, 45], i.e., there are infinitely many poles in C+, but the number of zeros in

C+ is finite. Thus, by mapping the variables as shown below, we can use the results

of [2, 45] to solve our problem:

WFOT1 (s) = W2(s) (2.9)

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WFOT2 (s) = W1(s)

XFOT(s) = Y(s)

YFOT(s) = X(s)

MFOTd = Ni(s)

MFOTn (s) = M(s)

NFOTo (s) = N1o (s),

and the optimal controller, C, for the two block problem (2.6) is the inverse of optimal

controller for the dual problem in [2], i.e.,

CFOTopt

1.

If we only consider the one block problem case, with W2 = 0, then the minimiza-

tion of

W1(Y NiQ1)

is simply a finite dimensional problem. On the other hand, minimizing

W2(1 M(Y NiQ1))

is an infinite dimensional problem.

2.2 Example

In this section, we illustrate the computation ofH controllers for systems withinfinitely many right half plane poles. The example is a plant containing an internal

delayed feedback:

P(s) =R(s)

1 + ehsR(s)

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where R(s) = k

sas+b

with k > 1, a > b > 0 and h > 0. Note that the denominator

term (1 + ehsR(s)) has infinitely many zeros n jn, where n o = ln(k)h > 0,

and n (2n + 1), as n . Clearly, P(s) has only one right half plane zero at

s = a.

The plant can be written as,

P(s) =Ni(s)

M(s)No(s) (2.10)

where

Ni(s) = s as + a

No(s) =1

1 + (sb)k(s+a)

ehs

M(s) =(s + b) + k(s a)ehs(s b)ehs + k(s + a)

It is clear that No is invertible in H, because sb

k(s+a) < 1. By the same argument,

M is stable. To see that M is inner, we write it as

M(

s) =

m(s) + f(s)

1 + m(s)f(s)with m(s) =

sas+a

ehs, and f(s) = s+b

k(s+a). Note that m(s) is inner, m(s)f(s) is

stable, and M(s)M(s) = 1. Thus M is inner, and it has infinitely many zeros in

the right half plane.

The optimal H controller can be designed for weighted sensitivity minimization

problem in (2.2) where P is defined in (2.10) and weight functions are chosen as

W1(s) = , > 0 and W2(s) = 1+s+s , > 0, > 0, < 1. As explained before,

this problem can be solved by the method in [2] (see Appendix A for details) after

necessary assignments are done, WFOT1 (s) =1+s+s

, WFOT2 (s) = , MFOTd =

sas+a

,

MFOTn (s) =(s + b) + k(s a)ehs(s b)ehs + k(s + a)

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NFOTo (s) =(s b)ehs + k(s + a)

k(s + a).

We will briefly outline the procedure to find the optimal H controller.

1) Define the functions,

F(s) =

s

a + bs

, =

1 222 2 for > 0

where a =

1 + 22 22, and b =

(1 22)2 + 2.

2) Calculate the minimum singular value of the matrix,

M =

1 j M F(j) jM F(j)1 a M F(a) aM F(a)

M F(j) jM F(j) 1 jM F(a) aM F(a) 1 a

for all values of (max{,

1+22}, 1

) and M F(s) = M(s)F(s). The

optimal gamma value, opt, is the largest gamma which makes the matrix M

singular.

3) Find the eigenvector l = [l10, l11, l20, l21]T such that Moptl = 0.

4) The optimal H controller can be written as,

Copt(s) =kf + K2,FIR(s)

K1(s)

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where kf is constant, K1(s) is finite dimensional, and K2,FIR(s) is a filter whose

impulse response is of finite duration

K1(s) =

k(l21s + l20)

opt(+ s) ,

kf =

kboptl11 optl21

2opt 2

,

K2,FIR(s) = A(s) + B(s)ehs,

kf + A(s) =k(s + a)c(s) + (s + b)d(s)

((1 2opt2) + (2opt 2)s2)(s a),

B(s) =(s b)c(s) + k(s a)d(s)

((1 2opt2) + (2opt 2)s2)(s a)

where c(s) = (aopt + bopts)(l11s + l10) and d(s) = opt( s)(l21s + l20).

As a numerical example, if we choose the plant as P(s) =2( s3s+1)

1+2( s3s+1)e0.5sand

the weight functions as W1(s) = 0.5, W2(s) =1+0.1s0.4+s

, then the optimal H cost is

opt = 0.5584, and the corresponding controller is

Copt(s) =

0.558s + 0.223

2s + 3.725 (1.477 + K2,FIR(s))

where K2,FIR(s) is equal to

(2.081s2 6.302s 0.826) (0.615s3 0.768s2 5.269s + 1.587)e0.5s(0.302s3 0.905s2 + 0.950s 2.850)

and its impulse response is of finite duration, k2,FIR(t) = L1(K2,FIR(s)) can be

written as

0.27e3t + 7.16 cos(1.77t) + 0.36 sin(1.77t) 2.037(t 0.5) 0 t 0.5

0 t > 0.5.

2.3 Summary of Results

In this Chapter, we have considered H control of a class of systems with in-

finitely many right half plane poles and infinitely many right half plane zeros. We

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have demonstrated that the problem can be solved by using the existing H control

techniques [2] for infinite dimensional systems with finitely many right half plane

poles. An example from delay systems is given to illustrate the computational tech-

nique. This result is used in the next Chapter to design optimal H controller for

general time-delay systems, including dead-time systems.

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CHAPTER 3

H CONTROLLER DESIGN FOR NEUTRAL ANDRETARDED DELAY SYSTEMS

In this Chapter, the mixed sensitivity minimization problem is solved for Neutral

and Retarded Delay Systems. Assume that p(s) is a quasi-polynomial which can be

written as

p(s) = p1(s)eh1s + . . . + pn(s)e

hns

where h1 < h2 < .. . < hn and p1(s), p2(s), . . . , pn(s) are polynomials. If the degree of

the polynomial p1(s) is larger or equal to p2(s), . . . , pn(s), then p(s) is called as neutral

quasi-polynomial and if the degree of the polynomial p1(s) is strictly larger than all

the other polynomials, then p(s) is called as retarded quasi-polynomial. Assume that

q(s) is a stable polynomial and the degree of q(s) is equal to that of p1(s) and define

R(s) =p(s)

q(s),

Ri(s) =

pi(s)

q(s) , i = 1, . . . , n .

Similarly, R(s) is called neutral delay system if p(s) is neutral quasi-polynomial and

R(s) is called retarded delay system if p(s) is retarded quasi-polynomial. The plant

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is assumed to have a transfer function whose numerator and denominator can be

expressed as Neutral or Retarded Delay System, i.e.,

P(

s) =

R(s)

T(s) = ni=1 Ri(s)e

his

mj=1 Tj(s)ejs .In this Chapter, optimal H controllers are designed for neutral/retarded delay

systems. First, a necessary and sufficient condition is derived for neutral and retarded

delay systems to have finitely many right half plane zeros. Optimal controllers are

designed for the time-delay systems with finitely many unstable zeros or poles. Spe-

cial structures of the corresponding controllers for these types of plants are studied.

Moreover, in this Chapter, the largest class of neutral/retarded delay systems (see

equation (3.5)) are determined for which the techniques of [2] and [15] are applicable.

The link between [15] and [23] is established for neutral/retarded plants, i.e., the

finite impulse response structure in the controller exists not only for plants in the

form ehsP0(s), but also for general retarded/neutral delay systems.

In [15], it is assumed that the plant is single input single output (SISO) and admits

the representation as,

P(s) =mn(s)No(s)

md(s)(3.1)

where mn(s) is inner, infinite dimensional and md(s) is inner, finite dimensional and

No(s) is outer, possibly infinite dimensional. The optimal H controller, Copt, stabi-

lizes the feedback system and achieves the minimumH cost, opt:

opt =

W1(1 + PCopt)1W2PCopt(1 + PCopt)1

(3.2)

where W1 and W2 are finite dimensional weights for the mixed sensitivity minimization

problem.

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In Chapter 2, the optimal H controller design is given for infinite dimensional

systems with finitely many unstable zeros by using the duality with the problem (3.2).

The plant has a factorization as,

P(s) =md(s)No(s)

mn(s)(3.3)

where mn is inner, infinite dimensional, md(s) is finite dimensional, inner, and No(s)

is outer, possibly infinite dimensional. For this dual problem, the optimal controller,

Copt, and minimum H cost, opt, are found for the mixed sensitivity minimization

problem which can be defined as,

opt = W1(1 + PCopt)1W2PCopt(1 + PCopt)1 . (3.4)

Optimal controller designs for problems (3.2) and (3.4) are outlined in Appendix A.

In this Chapter, we will design optimal H controller for a class of neutral and

retarded time-delay systems,

P(s) =R(s)

T(s)=

ni=1 Ri(s)e

his

mj=1 Tj(s)e

js.

The assumptions on the plant is given in Section 3.1 as A.1 A.7. We apply the

design methods in [15] and Chapter 2 for this plant. Note that P is the generalized

version of time-delay systems for SISO case. We assume that P has finitely many

right half plane zeros or poles (assumption A.7 in Section 3.1). We give the necessary

and sufficient conditions when this assumption is valid in Section 3.1.1. The special

structure ofH controllers is given in this section. Note that single delay plant with

finite dimensional plant is a special case of the plant P. By showing finite impulse

response filter structure of neutral and retarded time-delay systems, we eliminate

right half plane pole-zero cancellation in the controller. The resulting controller in

this structure is numerically stable and easy for implementation.

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3.1 Preliminary Work

We consider the plant, P, which has a transfer function,

P(s) = R(s)T(s)

= ni=1 Ri(s)ehismj=1 Tj(s)e

js. (3.5)

satisfying the following assumptions,

A.1 Ri and Tj are finite dimensional, stable, proper transfer functions,

A.2 Time delays are nonnegative rational numbers shown as hi and j with hi < hi+1

for i = 1, . . . , n

1 and j < j+1 for j = 1, . . . , m

1,

A.3 h1 1,

A.4 Relative degree of R1 is smaller than or equal to that of Ri, j = 2, . . . , n and

the relative degree of T1 is smaller than or equal to that of Ti, i = 2, . . . , m,

A.5 Relative degree of T1 is smaller than or equal to that of R1,

A.6 P has no imaginary axis zeros or poles,

A.7 R or T has finitely many zeros in the right half plane.

These assumptions are not restrictive. It is always possible to obtain a representa-

tion such that A.1 A.2 are satisfied. The first assumption results in stable transfer

functions in the controller which is desired for implementation purposes. The assump-

tion A.3 guarantees that the plant is causal. The numerator and denominator of the

plant is neutral or retarded time delay system by the assumption A.4. The plant is

proper transfer function if the assumption A.5 is valid. Note that the assumptions

A.1A.5 are necessary by definition of neutral and retarded time delay systems. The

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condition A.6 is a technical assumption which can be removed, but it is assumed for

simplicity in derivation. The last condition comes from the limitation of standard the

skew-toeplitz approach [2] and its dual approach given in Chapter 2.

For example, consider the plant whose dynamical behavior is described in time

domain by the following set of equations:

x1(t) = x1(t 0.2) x2(t) + u(t) + 2u(t 0.4),

x2(t) = 5x1(t 0.5) 3u(t) + 2u(t 0.4),

y(t) = x1(t). (3.6)

Its transfer function is

P(s) =s + 3 + 2(s 1)e0.4ss2 + se0.2s + 5e0.5s

, (3.7)

and it can be re-written in form of (3.5) as,

P =R

T=

R1eh1s + R2e

h2s

T1 + T2e2s + T3e3s(3.8)

where

R1 =s+3

(s+1)2, R2 =

2(s1)(s+1)2

,

T1 =s2

(s+1)2, T2 =

s(s+1)2

, T3 =5

(s+1)2,

are stable proper finite dimensional transfer functions and the delays are

h1 = 0, h2 = 0.4,

1 = 0, 2 = 0.2, 3 = 0.5.It is clear that A.1 A.3 are satisfied. The relative degree of R1 is equal to that of

R2 which is 1 and the relative degree of T1, which is 0, is smaller than that of T2 and

T3, which are 1 and 2, respectively. Also, the relative degree of T1 is smaller than

that of R1. Therefore assumptions A.4 A.5 hold. The plant P has no imaginary

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axis zeros. By a simple computation, T has finitely many right half plane zeros at

0.4672 1.8890j, whereas R has infinitely many unstable zeros converging to the

asymptote 1.7329 (5k + 2.5)j as k . In conclusion, this system satisfies all

the assumptions, A.1 A.7.

We define the conjugate of R(s) =n

i=1 Ri(s)ehis as R(s) := ehnsR(s)MC(s)

where MC is inner, finite dimensional whose poles are poles of R. For example, we

can calculate the conjugate of R(s) in the previous example,

R(s) =s + 3

(s + 1)2+

2(s 1)(s + 1)2

e0.4s

where h2 = 0.4 and MC(s) = s1

s+12. The conjugate of R(s) can be written as,

R(s) = eh2sR(s)MC(s),

=

2

(s + 1)+

(s 3)(s + 1)2

e0.4s

.

We will design an optimal H controller for the plant in (3.5) which satisfies the

assumptions A.1

A.7. In the next sections, we will give an equivalent condition

to check whether the assumption A.7 is valid or not and standard Skew-Toeplitz

approach to design H controller for infinite dimensional systems will be outlined.

3.1.1 Neutral and Retarded Time Delay Systems with FinitelyMany Unstable Zeros

We begin with preliminary observations (see also [46] for further details). First,

recall that R1(s), . . . , Rn(s) are stable and proper transfer functions. Therefore,

R(s) =n

i=1 Ri(s)ehis has (infinitely) finitely many right half plane zeros if and

only if

RG(s) = 1 +R2(s)

R1(s)eh1s + . . . +

Rn(s)

R1(s)ehns

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has (infinitely) finitely many unstable zeros respectively.

The following fact will be used later in this Chapter and in Chapter 5.

Claim: RG(s) has no unstable zeros with real part extending to infinity.

Proof: Assume that s = 0 +j . For sufficiently large o, we can write the following

inequalities,

i Ri(0 + j)R1(0 + j)

+i i = 2, . . . , nbecause all Ri(s) terms are finite dimensional. Following holds by the triangle in-

equality,

1 2 eh20 . . . n ehn0 |RG(0 + j)| 1 + +2 eh20 + . . . + +n ehn0

This inequality tells us that RG(s) has no unstable zeros with real part extending to

infinity because lim0 |RG( + j)| = 1. Therefore, R(s) has no unstable zeros

with real part extending to infinity.

Lemma 3.1.1. Assume that R(s) =

ni=1 Ri(s)e

his is a neutral or retarded time

delay system with no imaginary axis zeros and poles, then the system R has finitely

many right half plane zeros if and only if all the roots of the polynomial, 1+ 2rh2h1 +

+ nrhnh1 has magnitude greater than 1 where

i = lim

Ri(j)R11 (j) i = 2, . . . , n ,

hi =hi

N, N, hi Z+, i = 1, . . . , n .

If the system R satisfies this condition, it is called as F-system where F is an abbre-

viation for finitely many unstable zeros.

Proof. Since the system is neutral or retarded time delay system, it may have infinitely

many right half plane zeros extending to infinity in imaginary part with fixed positive

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real part [46]. R has infinitely many right half plane zeros if and only if for fixed

o > 0, lim R(o +j) has infinitely many unstable zeros. Define r = e(o+j)/N,

then

lim

R(o + j)

R1(o + j )= rh1 + 2r

h2 + + nrhn. (3.9)

Let r0 is the root of (3.9) which satisfies |ro| > 1. Then the system R do not have

infinitely many right half plane zeros, since

|ro| = eo/N,

o =

Nln

|ro

|< 0

which is a contradiction. Therefore, if all the roots of the polynomial (3.9) has

magnitude less than 1, then the system, R, does not have infinitely many unstable

zeros. Assume that there exist a root, ro, of (3.9) with magnitude smaller than 1.

Then, there are infinitely many right half plane zeros ofR converging to the asymptote

ro,k = ln |ro|N jN(ro + 2k) as k where k Z and ro is the phase of the

complex number ro. The lemma is also valid when delays are real numbers. If the

delays are rational numbers, the resulting function is polynomial which is easy to

check its roots. When the delays are real numbers, it will be difficult to find the

roots of the function. Therefore, the delays are assumed as rational numbers for

convenience.

We define R(s) as I-system (where I is an abbreviation for infinitely many unstable

zeros) if R is F-system. In order to find whether R is an I-system, we can check the

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magnitude of all the roots of the polynomial, 1 + 2rh2h1 + + nrhnh1 where

i = lim

Ri(j)R11 (j) i = 2, . . . , n ,

hi =

hi

N, N, hi Z+, i = 1, . . . , n .

If the magnitude of all the roots is smaller than 1, then R is an I-system.

In this section, the plant in (3.5) has finitely many unstable zeros or poles. This

is guaranteed by assumption A.7. By Lemma 3.1.1, it is easier to check that the

assumption A.7 is valid by finding the magnitude of all roots of the given polynomial.

We can conclude that the assumption A.7 is satisfied if and only if either R or T is a

F-system (or both).

3.1.2 FIR Structure of Neutral and Retarded Time DelaySystems

In this section, we will show special structure of neutral and retarded time delay

systems. This is a key lemma to be used in the next section for the proofs of main

results.

Lemma 3.1.2. LetR be as in Lemma 3.1.1 and MR be a finite dimensional system.

Define S+z be the set of common C+ zeros of R and MR. Then RMR , can always be

decomposed as,

R(s)

MR(s)= HR(s) + FR(s) (3.10)

where HR is an infinite dimensional system which does not have any poles in S+z and

FR is a finite impulse filter.

Proof. For clarity of the presentation we give the proof for the case where the entries

z1, z2, . . . , z nz of S+z are distinct. For the general case main idea is the same, except

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that some of the expressions below may become more complicated (notation needs to

be modified to include the multiplicity of each common zero, and the partial fraction

expansion and inverse Laplace transformation needs to consider these multiplicities).

So, we assume R(zk) = MR(zk) = 0, k = 1, . . . , nz. We can rewrite the expressionR

MRas

R

MR=

ni=1

Ri

MRehis.

We can decompose each term by partial fraction, RiMR

= Hi + Fi where the poles ofFi

are elements ofS+z and define the terms HR and FR as

HR(s) =n

i=1

Hi(s)ehis,

FR(s) =n

i=1

Fi(s)ehis.

Note that each term, Fk is strictly proper and FR(zk) is finite k = 1, . . . , nz. The

lemma ends if we can show that FR is a FIR filter. Inverse Laplace transform of FRcan be written as

fR(t) =nz

k=1

ni=1

Res{Fi(s)}

s=zkezk(thi)uhi(t)

where Res(.) is the residue of the function and uhi(t) is the unit step function delayed

by hi. The impulse response is equivalent to

fR(t) =nz

k=1

ezkt

n

i=1

Res{Fi(s)}

s=zkehizk

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for t > hn. It is clear from partial fraction that Res{Fi(s)}

s=zk= Ri(zk) which

results fR(t) is FIR filter,

fR(t) =nz

k=1 ezktn

i=1 Ri(zk) ehizk ,=

nzk=1

ezktR(zk) 0 for t > hn,

since zk is the zero of R, i.e., R(zk) = 0. Therefore, we can conclude that FR is a

FIR filter with support [0, hn].

Note that this decomposition eliminates right half plane pole-zero cancellation in

RMR and brings into form which is easy for numerical implementation. Lemma 3.1.2

explains the FIR structure of H controllers. Since these controllers satisfy inter-

polation conditions, right half plane pole-zero cancellations occur. By Lemma 3.1.2,

we can see that it is always possible to form an FIR structure in H controllers for

time-delay systems.

Assume that P is in the form of P(s) =

nk=1 Pk(s)e

hks where Pk are stable,

proper, finite dimensional transfer functions. Also, P0 is a bi-proper, finite dimen-

sional system. By partial fraction, we can write PkP0

as

Pk(s)

P0(s)= Pk,p(s) + Pk,0 k = 1, . . . , n

where the Pk,0 is a proper transfer function whose poles are same as the zeros of P0.

Then, the decomposition operator, , can be defined as,

(P(s), P0(s)) = HP(s) + FP(s)

where HP =n

k=1 Pk,p(s)ehks and FP =

nk=1 Pk,0(s)e

hks are infinite dimensional

systems. Note that if the zeros of P0 are also unstable zeros of P, then FP is an FIR

filter shown in Lemma 3.1.2.

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3.2 Main Results

In this section, we will construct the optimal H controller for the plant, P, with

a transfer function in the form of (3.5) satisfying assumptions A.1 A.7. The plant,P = R

T, is assumed to be one of the following:

i) R is I-system and T is F-system (IF plant),

ii) R is F-system and T is I-system (FI plant),

iii) R is F-system and T is F-system (FF plant).

For each case, we will find an optimal H controller and obtain a structure such that

there is no internal pole-zero cancellations in the controller.

3.2.1 Factorization of the Plants

In order to apply skew-toeplitz approach [2], we need to factorize the plants as in

(3.1) or (3.3).

IF Plant Factorization

Assume that the plant in (3.5) satisfies the assumptions A.1 A.7, R is I-system

and T is F-system, then we can factorize the plant as

mn = e(h11)sMR(s)

{eh1sR(s)}R(s)

,

md = MT(s),

No =

R(s)MR(s)

{e1sT(s)}MT(s)

(3.11)

where MR is an inner function whose zeros are the right half plane zeros of R(s).

Since R is an I-system, the conjugate of R has finitely many right half plane zeros,

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therefore MR is well defined. Similarly, zeros of MT are right half plane zeros of T.

Note that mn and md are inner functions, finite and infinite dimensional respectively.

No is an outer term. Although there are right half plane pole-zero cancellations in

md and No, this problem will be solved in Section (3.2.2) by using the technique in

Section 3.1.2.

FI Plant Factorization

The plant satisfies the assumptions A.1 A.7, R is F-system and T is I-system.

Since the design method in Chapter 2 requires the plant and its inverse to be causal

proper transfer functions, we restrict P in (3.5) to satisfy the following additional

assumptions:

B.3 h1 = 1 = 0,

B.5 Relative degree ofT1 is equal to that of R1.

Then the plant P can be factorized as in (3.3),

mn = MT(s)T(s)

T(s),

md = MR(s),

No =

R(s)MR(s)

T(s)MT(s)

. (3.12)

The zeros of inner function MR are right half plane zeros of R. The unstable zeros

of

T(s) are the same as the zeros of inner function MT. Similar to previous section,

conjugate ofT has finitely many right half plane zeros since T is a I-system. The right

half plane pole-zero cancellations in mn and No will be eliminated in Section 3.2.2.

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FF Plant Factorization

For FF plants, assumptions for IF plants are valid (i.e., assumptions A.1 A.7),

but R and T are F-systems. The plant P has a factorization as in (3.1),

mn = e(h11)sMR(s),

md = MT(s),

No =

{eh1sR(s)}MR(s)

{e1sT(s)}MT(s)

(3.13)

where MR and MT are inner functions whose zeros are right half plane zeros of R and

T respectively. Note that when h1 = 1, mn is finite dimensional term. Then exact

internal pole-zero cancellations are possible for this case (except the ones in No).

3.2.2 Optimal H Controller Design

By using the methods in [15] and Chapter 2, given the appropriately chosen weight

functions W1 and W2, the optimal H cost, opt can be found. After opt is calculated,

it is easy to obtainE

opt,F

opt andL

. We will give the structure of optimal H

controllers for each type of plant.

Controller Structure of IF Plants

By using the method in [15] and Chapter 2, we can obtain opt, Eopt, Fopt, L.

The optimal controller can be written as,

Copt =

Kopt(s)e1sT(s)

MT(s) R(s)MR(s)

+ e1sR(s)Fopt(s)L(s)(3.14)

where Kopt(s) = Eopt(s)Fopt(s)MT(s)L(s). In order to obtain the structure of

controller:

1. Do the necessary cancellations in Kopt,

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(Hopt 1) + Fopt

HT + FT- - -d

6

+

Figure 3.1: Optimal H Controller for IF plants

2. Partition Kopt as Kopt(s) = opt(s)T(s), where opt is a bi-proper transfer

function. The zeros of opt are right half plane zeros of EoptMT,

3. Obtain (HT, FT), (HR,1,FR1) and (H2, FR2) by using the partitioning operator

HT + FT = (e1sT T, MT),

HR1 + FR1 = (R, MRopt),

HR2 +F

R2 = (e1sRFoptL, opt),

as explained in Section 3.1.2.

Then, the optimal controller has the form

Copt =HT + FT

Hopt + Foptwhere HT, Hopt = HR1 + HR2 are neutral/retarded time-delay systems and

FT,

Fopt = FR1 + FR2 are FIR filters. The controller has no right half plane pole-zero

cancellations and its structure is shown in Figure 3.1.

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(HR 1) + FR

Hopt + Fopt- - -d

6

+

Figure 3.2: Optimal H Controller for FI plants

Controller Structure of FI Plants

After the data transformation is done as shown in (2.9) in Chapter 2, we can

find opt, Eopt, Fopt, L as in IF plant case. We can write the inverse of the optimal

controller similar to (3.14):

C1opt =Kopt(s)

R(s)

MR(s)

T(s)

MT(s)+ T(s)Fopt(s)L(s)

(3.15)

where Kopt

(s) = Eopt

(s)Fopt

(s)MR

(s)L(s). We can obtain the optimal controller

following exactly same steps for IF plant case (note that R and T are interchanged,

and h1 = 1 = 0 from assumptions):

Copt =Hopt + Fopt

HR + FR .

The optimal controller is the dual of the controller for the IF plants, as expected.

There is no internal right half plane pole-zero cancellations and its structure is shownin Figure 3.2.

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Controller Structure of FF Plants

Structure of FF plants is similar to that of IF plants. We can calculate opt, Eopt,

Fopt, L by the method in [15], Chapter 2 and then write the optimal controller as:

Copt =Kopt(s)

{e1T(s)}MT(s)

{eh1sR(s)}MR(s)

+ e1sR(s)Fopt(s)L(s)(3.16)

where Kopt(s) = Eopt(s)Fopt(s)MT(s)L(s). We can find controller structure follow-

ing similar steps as in IF plants case:

1. Do the cancellations in Kopt,

2. Factorize Kopt and find opt, T,

3. Obtain (HT, FT), (HR1, FR1)and (HR2, FR2) by

HT + FT = (e1sT T, MT),

HR1 + FR1 = (eh1sR, MRopt),

HR2 + FR2 = (e1sRFoptL, opt).

The controller structure can be written as:

Copt =HT + FT

Hopt + Foptwhere Hopt = HR1 + HR2 and Fopt = FR1 + FR2. Note that the resulting structure

is same as IF plant case. It is possible to cancel the zeros of opt with denominator

when h1 = 1 = 0 which is considered in Example 3.3.2.

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3.3 Examples

We will show the optimal H controller design for IF and FF plants. Since FI

plant case is the dual of IF plant case, example of this section is omitted. The plants

in the examples are SISO plants. Sensitivity and complementary sensitivity weight

functions are chosen as,

W1(s) = 2

s + 1

10s + 1

,

W2(s) = 0.2(s + 1.1)

which are the same weight functions taken in [2].

3.3.1 IF Plant Example

We will design optimal H controller for the plant whose time domain represen-

tation is given by (3.6) in Section 3.1. The transfer function for this plant is as in

(3.7). The plant can be written in the form of (3.5) as demonstrated in (3.8). Note

that R is a I-system, since its corresponding polynomial (see Lemma 3.1.1) is 1 + 2 r

and the magnitude of root is smaller than 1. It is clear that T is a F-system, since

its corresponding polynomial is a constant, 1. Therefore it has no roots and satisfies

the condition trivially. Therefore, the plant is a IF plant with h1 = 1 = 0. In order

to find optimal controller, the plant is factorized as in (3.1),

mn = MR(s)R(s)

R(s)

,

md = MT(s),

No =

R(s)MR(s)

T(s)MT(s)

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where

MR =s 0.2470s + 0.2470

,

MT =

s2

0.9344s + 3.7866

s2 + 0.9344s + 3.7866 ,

R =2(s + 1) + (s 3)e0.4s

(s + 1)2.

For the given plant and weight functions, the optimal H cost (3.2) is opt = 0.7203

and the corresponding optimal controller is [2] (see A.3 at Appendix A),

Copt = Eopt(s)md(s)N1o (s)Fopt(s)L(s)

1 + mn(s)Fopt(s)L(s)

where

Eopt =3.4812 + 47.8803s2

0.5188(1 100s2) ,

Fopt =0.5188(1 10s)

1.384s2 + 2.842s + 1.381,

L =0.589s2 0.3564s + 0.16160.589s2 + 0.3564s + 0.1616

.

The optimal controller,

Copt, has internal unstable pole-zero cancellations (i.e., thezeros of Eopt and md are cancelled by the denominator of the controller). In order

to eliminate this, we will follow the procedure given in Section 3.2.2:

steps 1-2 After cancellation, Kopt = EoptFoptMTL can be factorized as

Kopt = opt(s)T(s),

where

opt =3.4812 + 47.8803s2

(1.384s2 + 2.842s + 1.381)MT(s),

T =L(s)

(1 + 10s).

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Impulse response of FT(s)

time

Figure 3.3: fT(t) for IF plant

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Impulse response of F(s)

time

Figure 3.4: fopt(t) for IF plant

step 3 We can find (HT, FT), (HR,1,FR1) and (HR2, FR2) by the partitioning opera-

tor, . The expressions can be found in Appendix B (see equations (B.1-B.6)).

Then, the optimal controller can be found from,

Copt = HT + FTHopt + Fopt

where Hopt = HR1 + HR2 and Fopt = FR1 + FR2 . The impulse responses ofFopt and

FT are shown in Figures 3.3 and 3.4.

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3.3.2 FF Plant Example

We will find the optimal H controller for the plant whose time-domain repre-

sentation is

x1(t) = x1(t) + 3x2(t 0.2),

x2(t) = x2(t) + 2x3(t 0.5),

x3(t) = x3(t) + 2x1(t 0.1) + u(t),

y(t) = x1(t) + x2(t) + x3(t). (3.17)

The corresponding transfer function for this plant is

P(s) =(s 1)2 + 2(s 1)e0.5s + 6e0.7s

(s 1)3 12e0.8s

and it can be rewritten as

P(s) =R(s)

T(s)=

(s1)2

(s+1)3+ 2(s1)

(s+1)3e0.5s + 6

(s+1)3e0.7s

(s1)3

(s+1)3 12

(s+1)3e0.8s

.

It is clear that both R and T are F-system. Since h1 = 1 = 0, exact cancellation inthe controller is possible, otherwise optimal controller design is the same as previous

example. We can factorize the plant as,

mn = MR(s),

md = MT(s),

No =

R(s)MR(s)

T(s)MT(s)

(3.18)

where

MR =s2 2.757s + 4.455s2 + 2.757s + 4.455

,

MT =s3 4.09s2 + 8.185s 9.128s3 + 4.09s2 + 8.185s + 9.128

.

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Given the weight functions of the previous example, the optimal H cost is computed

as opt = 0.7031. The optimal controller is as in (A.3) where

Eopt =3.5056 + 45.4406s2

0.4944(1 100s2),

Fopt =0.4944(1 10s)

1.3482s2 + 2.7681s + 1.3446,

L =0.1364s3 + 0.3852s2 + 0.5654s + 0.1154

0.1364s3 0.3852s2 + 0.5654s 0.1154 .

Since mn is finite dimensional, 1 + mn(s)Fopt(s)L(s) is finite dimensional. Therefore,

the exact cancellation between opt and 1 + mn(s)Fopt(s)L(s) is possible. We will

follow the procedure of Section 3.2.2 with a slight modification:

steps 1-2 After the cancellation, Kopt = EoptFoptMTL can be factorized as

Kopt = opt(s)T(s),

where

opt =3.5056 + 45.4406s2

1.3482s2

+ 2.7681s + 1.3446

MT(s),

T =L(s)

(1 + 10s).

Then, the controller can be written as,

Copt =

T TMT

R

MR

1+MRFoptL

opt

step 3 We perform the cancellations in K = 1+MRFoptLopt as shown in Appendix B(see equations (B.7-B.8)),

step 4 We can find the controller parameters (HT, FT) from

HT + FT = (T T, MT)

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.4

0.3

0.2

0.1

0

0.1

0.2

0.3

0.4

0.5

Impulse response of FT(s)

time

Figure 3.5: fT(t) for FF plant

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.4

0.3

0.2

0.1

0

0.1

0.2

0.3

Impulse response of F(s)

time

Figure 3.6: fopt(t) for FF plant

and (Hopt,Fopt) from

Hopt + Fopt = (RK, MR).

See Appendix B for these partitions (equations (B.9-B.12)). In Figure 3.5 and

3.6, impulse responses ofFT and Fopt are given.

Note that the structure of controller is same, however exact cancellation simplifies

the controller expression.


In this Chapter, we determined a necessary and sufficient condition for general

time-delay systems for applicability of the Skew-Toeplitz approach [2]. Note that

the neutral and retarded delay systems are the generalized versions of the time-

delay systems. The optimal H controllers for these types of generalized time delay

systems can be obtained by using the Skew-Toeplitz method. Moreover, the internal

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unstable pole-zero cancellation problem is eliminated by using special structure of the

controller.

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CHAPTER 4

STABLE H CONTROLLERS FOR DELAY SYSTEMS:SUBOPTIMAL SENSITIVITY

In this Chapter, an indirect method for strongly stabilizing H controller designfor systems with time delays is given. It is known that for a certain class of time

delay systems the optimal H controllers designed for sensitivity minimization lead

to controllers with infinitely many unstable modes, [47, 48]. An indirect way to

obtain a strongly stabilizing controller, in this case, is to internally stabilize the

optimal sensitivity minimizing H controller, while keeping the sensitivity deviation

from the optimum within a desired bound. The proposed scheme is illustrated in

Figure 4.1: the objective is to have a stable feedback system, and to minimize the

weighted sensitivity function W S := W(1 + P K)1, with a stable K. We will assume

that for given W and P, the optimal H controller Copt is determined. Then, F will

be designed to yield a stable K, such that the feedback system remains stable, and

W S is relatively close to the optimal weighted sensitivity W Sopt := W(1+P Copt)1.

When the plant P contains a time delay, and the sensitivity weight W is bi-proper,

the indirect approach outlined above requires internal stabilization of C (which con-

tains infinitely many unstable modes) by F. In Chapter 2, a solution to the two-block

H control problem involving a plant with infinitely many poles in the open right

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r

z

y

+ +

C P

FK

W

Figure 4.1: (i) F = 0: the weighted sensitivity is H optimal; (ii) F = 0: thecontroller K is stable.

half plane is given. Then, the results of that Chapter will be used to derive sufficient

conditions for solvability of the stable H controller design problem considered here

for systems with time delays. Namely, in this Chapter we investigate the indirect

method of obtaining a strongly stabilizing controller for systems with time delays,

subject to a bound on the deviation of the sensitivity from its optimal value. First

we examine the optimal sensitivity problem for stable delay systems and illustrate

that the corresponding optimal controller has the structure of the plant introduced

in Chapter 2.

4.1 Optimal Sensitivity Problem for Delay Systems

Consider the feedback system shown in Figure 4.1, where P(s) = ehsNp(s) and

W(s) = 1+ss+

. We assume that Np,N1

p

H. By using the method developed in [2,

15], we calculate the optimal controller Copt(s) minimizing the weighted sensitivity

W(1 + P C)1 over all stabilizing controllers, as follows (See Appendix A for details).

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The smallest satisfying the phase equation given below is the optimal (smallest

achievable) sensitivity level:

h + tan

1 + tan

1

=

(4.1)

where =

122

22, and < < 1

. Once opt is computed as above, the corre-

sponding optimal controller is

Copt(s) =(1 2opt2) + (2opt 2)s2

opt(+ s)(1 + s)

N1p (s)

1 + opt

s1+s

ehs

. (4.2)

Also, define the optimal sensitivity function as Sopt(s) = (1 + P(s)Copt(s))1, then,

Sopt(j) =1 + opt(j)

1+j ejh

1 +

1jopt(+j)

ejh

. (4.3)

In [47], it was mentioned that H-optimal controllers may have infinitely many right

half plane poles. Here we will give a proof based on elementary Nyquist theory:

if S1opt(j ) encircles the origin infinitely many times, we can say that Copt(s) has

infinitely many right hand poles, because P(s) does not have any right half plane

poles. For s = j as , we have

S1opt(j ) 1

optejh

1 opt

ejh

and |S1opt(j)| opt . Since < opt < 1, we can say that |S1opt(j)| has constant

magnitude between 0 and 1 for sufficiently large . For k =2k

h, as k the

phase of S1opt(jk) tends to

. In other words, S1opt(j ) intersects negative part of

the real axis near k =2k

h, as k . Similarly, S1opt(j) intersects positive part of

the real axis near k =(2k+1)

has k . Thus S1opt(j) encircles the origin infinitely

many times, which means that Copt(s) has infinitely many poles in C+.

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Remark. Let m1(j) =

j+j

ejh , m2(j) =

1j1+j

ejh and g(j ) =

opt

+j1+j

. Then,

W(j)Sopt

(j) = opt g

1(j) + m1(j)

1 + g1(j)m2(j) = opt1 + g(j)m1(j)g(j ) + m2(j ) and hence |W(j )Sopt(j )| = opt as expected.

4.2 Sensitivity Deviation Problem

Recall that the H optimal performance level was defined as

0 := opt = inf C stab. P

W(1 + P C)1

where W(s) = 1+ss+

, with > 0, > 0, < 1, and P(s) = Np(s)Mp(s), with

Np, N1

p H, and Mp is inner and infinite dimensional, e.g. Mp(s) = ehs. We

have obtained the optimal controller for the sensitivity minimization problem in (4.2).

Claim: The optimal H controller is in the form

Copt(s) =N1p (s)Nc(s)

Dc(s)(4.4)

where Dc is inner infinite dimensional and Nc, N1c H.

It is easy to verify this claim by comparing (4.2) with (4.4): we see that

Nc(s) =2optW

2(s)m2(s) m1(s)1 + 1optW(s)m2(s)

,

=1

2opt(+ s)2

(1 2opt2) + (2opt 2)s2

1 + 1opt

1s+s

ehs

(4.5)

Dc(s) =1optW(s) + m1(s)

1 + 1optW(s)m2(s)= 1optW(s)

1 + m1(s)optW1(s)1 + m2(s)

1optW(s)

,

= 1optW(s)(1 + P(s)Copt(s))1,

=

s+s

ehs + 10

1+s+s

1 + 10

1s+s

ehs

(4.6)

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where m1(s) =

s+s

ehs, m2(s) =

1s1+s

ehs. Note that Nc(s) has no right half

poles or zeros (it has only two imaginary axis poles that are cancelled by the zeros

at the same locations). Therefore Nc, N1c

H. Also, it is easy to check that Dc is

inner and infinite dimensional.

Note that,

Dc = 10 W S0 =

10 W(1 + P Copt)

1 =

10 W

Dc

Dc + MpNc

.

Our goal is to have a stable controller K, by an appropriate selection of F:

K(s) =

Copt(s)

1 + F(s)Copt(s) .

At the same time we would like to have the resulting sensitivity function,

S(s) = (1 + P(s)K(s))1 =

1 + Mp(s)Np(s)

N1p (s)Nc(s)Dc(s)

1 + F(s)N1p (s)Nc(s)Dc(s)

1, (4.7)

to be close to the optimal sensitivity, Sopt = (1 + P Copt)1. By the parameterization

of the set of all stabilizing controllers for Copt [44], F can be written as,

F(s) =X(s) + Dc(s)Q(s)

Y(s) N1p (s)Nc(s)Q(s)

with N1p (s)Nc(s)X(s) + Dc(s)Y(s) = 1 which can be solved as Y = 0 and X =

N1c Np where Q H, Q(s) = 0. Then, in terms of the design parameter Q, the

functions F(s), K(s) and S(s) can be re-written as,

F(s) = N1c (s)Np(s) + Dc(s)Q(s)

N1p (s)Nc(s)Q(s) = (Q1

(s) + C1

opt(s)) (4.8)

K(s) =Copt(s)

1 Copt(s)(Q1(s) + C1opt(s))= Q(s) (4.9)

S(s) = (1 + Mp(s)Np(s)(Q(s)))1. (4.10)

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Also, the sensitivity function S(s) should be stable. We can define the relative devia-

tion of the sensitivity as WS0SS

, then minimizing this deviation over Q H,Q(s) = 0 is equivalent to

1,opt = inf QH

WS0 SS

,

= inf QH

W(MpNc)(1 + DcNpN1c Q)Dc + MpNc

. (4.11)

Note that, |Dc(j) + Mp(j)Nc(j)| = |10 W(j)| as shown before. Then,

1,opt = inf

QH

0Nc(1 + Dc

Q) (4.12)

where Q = NpN1c Q. For stability of the feedback system formed by the resultingcontroller K and the original plant P, we also want the sensitivity function S =

(1 MpNpQ)1 to be stable. Once the optimal Q is determined from (4.12), a

sufficient condition for stability of S (and hence the original feedback system) can be

determined as

|Np(j)| < |Q(j )|1 (4.13)

Note that the problem defined in (4.12) is equivalent to a sensitivity minimization

with an infinite dimensional weight 0Nc for a stable infinite dimensional plant

Dc. For the case where both the plant and the weight are infinite dimensional,

sensitivity minimization problem is difficult to solve. So, we propose to approximate

the weight by a finite dimensional upper bound function: find a stable rational weight

W1 such that |0Nc(j )| |W1(j )|. We suggest an envelope which is in the form,

W1(s) = 0 Ks + 1s + 1

,

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where

K = 1 + 1opt

1 = (opt + )(1 opt)1

1

and 1 is determined in some optimal fashion.

Then, we can solve the one following block problem as in Chapter 2

1,opt 2,opt = inf QH W1(1 + DcQ). (4.14)Note that 2,opt is the smallest value of 2, in the range 0K < 2 < (0K)

11

,

satisfying

= tan1

1

+ h + tan1

10 cos(h) + ( 10 sin(h))(+ 10 cos(h)) +

10 sin(h)

+tan1

1

tan1

10 cos(h) ( 10 sin(h))(+ 10 cos(h)) 10 sin(h)

(4.15)

where =

(oK)22122

21

22(0K)2

. After finding 2,opt, we can write the C2,opt as,

C2,opt(s) = A(s) 11 Dc(s)B(s) (4.16)

where,

A(s) =(20K

221 22,opt21) + (22,opt 20K2)s20K2,opt(1 + s)(1 + s)

B(s) =

2,opt

0K

1 s1 + s

.

In order to calculate Q2,opt(s) corresponding to C2,opt(s), we will use the transforma-tion

Q2,opt(s) = C2,opt(s)1 + P(s)C2,opt(s)

.

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That gives

Q2,opt(s) = A(s)

1

1 Dc(s)B1(s)

=(2

1 2

K

2

1) + (2

K 1)s2

(1 + s)(K(1 + s) Dc(s)(1 s)) (4.17)

where K =2,opt0K

. After finding Q2,opt(s), F(s) can be calculated via (4.8),F(s) = (Q12,opt(s) + C1opt(s))

where Q2,opt(s) and Copt(s) are found in (4.17) and (4.2) respectively.Similarly, the resulting controller K(s) is determined as

K(s) = Q2,opt(s)which is shown in (4.9).


An indirect approach to design stable controller achieving a desired H perfor-

mance level for time delay systems is studied. This approach is based on stabilization

ofH controller by another H controller in the feedback loop. Stabilization of the

controller is achieved and the sensitivity deviation is minimized. However, there are

two main drawbacks of this method. First, the solution of sensitivity deviation brings

conservatism because of finite dimensional approximation of the infinite dimensional

weight. Overall system does not achieve the exact performance level, since the op-

timal H controller is perturbed by deviation. Second, and more importantly, the

stability of overall sensitivity function (feedback system stability) is not guaranteed.

It is possible to keep the feedback system stable as long as the resulting coprime

factor perturbation in the controller is small. However, the perturbation needed to

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stabilize the controller may be relatively large. For this reason we tried an alternative

method, which is the subject of the next Chapter.

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CHAPTER 5

ON STABLE H CONTROLLERS FOR TIME-DELAYSYSTEMS

In this Chapter we focus on the strong stabilization problem for infinite dimen-

sional plants such that the stable controller achieves the pre-specified suboptimal H

performance level. When the optimal controller is unstable (with infinitely or finitely

many unstable poles), two methods are given based on a search algorithm to find

a stable suboptimal controller. However, both methods are conservative. In other

words, there may be a stable suboptimal controller achieving a smaller performance

level, but the designed controller satisfies the desired overall H

norm. The stability

of optimal and suboptimal controller is discussed and necessity conditions are given

for stability ofH controllers.

It is known that a H controller for time-delay systems with finitely many un-

stable poles can be designed by the methods in [17, 16, 15, 45]. In general, weighted

sensitivity problem results in an optimal H controller with infinitely unstable modes

[47, 48].

We assume that the plant is single input single output (SISO) and admits the

representation as in [15],

P(s) =mn(s)No(s)

md(s)(5.1)

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where mn(s) = ehsM(s), h > 0, and M(s), md(s) are finite dimensional, inner,

and No(s) is outer, possibly infinite dimensional. The optimal H controller, Copt,

stabilizes the feedback system and achieves the minimum H cost, opt:

opt = W1(1 + P Copt)1W2P Copt(1 + P Copt)1

, (5.2)

= inf C stabilizing P

W1(1 + P C)1W2P C(1 + P C)1

where W1 and W2 are finite dimensional weights for the mixed sensitivity minimization

problem.

In the next section, the structure of optimal and suboptimalH controllers will

be summarized. The optimal controller with infinitely many unstable poles case is

considered in Section 5.2. The conditions and a design method for stable suboptimal

H controller is given in the same section. Similar work is done in Section 5.3 for

the optimal controller with finitely many unstable poles. Examples related for these

design methods are presented in Section 5.4, and concluding remarks can be found in

Section 5.5.

5.1 Structure ofH Controllers

Assume that the problem (5.2) satisfies (W2No), (W2No)1 H, then optimal

H controller can be written as [2] (for controller calculations, see Appendix A),

Copt(s) = Eopt(s)md(s)N1o (s)Fopt(s)L(s)

1 + mn(s)Fopt(s)L(s)

. (5.3)

Similarly, the suboptimal controller achieving the performance level can be de-

fined as

Csubopt(s) = E(s)md(s)N1o (s)F(s)LU(s)

1 + mn(s)F(s)LU(s). (5.4)

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Note that the unstable zeros of Eopt and md are always cancelled by the denomi-

nator in (5.3). Therefore, Copt is stable if and only if the denominator in (5.3) has no

unstable zeros except the unstable zeros of Eopt and md (multiplicities considered).

Same conclusions are valid for the suboptimal case, Csubopt is stable provided that

the denominator in (5.4) has unstable zeros only at the unstable zeros of E and md

(again, multiplicities considered).

It is clear that the optimal, respectively suboptimal, controllers have infinitely

many unstable poles if and only if there exists o > 0 such that the following inequality

holds

lim

|Fopt(o + j )Lopt(o + j)| > 1, (5.5)

respectively,

lim

|F(o + j)LU(o + j)| > 1. (5.6)

The controller may have infinitely many poles because of the delay term in the de-

nominator. All the other terms are finite dimensional.

Even when the optimal controller has infinitely many unstable poles, a stable

suboptimal controller may be found by proper selection of the free parameter U(s).

In Section 5.2, this case is discussed.

Note that the previous case covers one and two block cases (i.e., W2 = 0 and

W2=0 respectively). When Fopt is strictly proper, then the optimal and suboptimal

controllers may have only finitely many unstable poles. Existence of stable suboptimal

H controllers and their design will be discussed in Section 5.3 for this case.

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5.2 Stable suboptimal H controllers, when the optimal con-troller has infinitely many unstable poles

The following lemma gives the necessary condition for a suboptimal controller to

have finitely many unstable poles.

Lemma 5.2.1. Assume that the optimal controller has infinitely many unstable poles

and U(s) is finite dimensional. Then the suboptimal controller has finitely many

unstable poles if and only if

lim |

F(j)LU(j)

| 1. (5.7)

Proof Assume that the suboptimal controller has infinitely many unstable poles,

then the equation

1 + eh(+j)M( + j)F( + j)LU( + j ) = 0

has infinitely many zeros in the right half plane, i.e. (recall Section 3.1.1) there exists

= o > 0 and for sufficiently large ,

1 + eh(o+j) lim

(F(o + j)LU(o + j )) = 0 (5.8)

will have infinitely many zeros. Since F and LU are finite dimensional,

lim

F(j ) = lim

F( + j )

lim

LU(j ) = lim

LU( + j)

> 0.

By using this fact, we can rewrite (5.8) as,

1 + eh(o+j) lim

(F(j)LU(j )) = 0 (5.9)

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which implies that in order to have infinitely many zeros, the condition in lemma

should be satisfied. Conversely, a similar idea can be used to show that (5.7) implies

finitely many unstable poles.

Note that this lemma is valid not only for only finite dimensional U(s) term, but

also for any U H, U 1 provided that

lim

U(j) = lim

U( + j) = u, > 0. (5.10)

is satisfied where u R. Also, we can find conditions on U which guarantees finitely

many unstable poles by using the lemma.

Assume that U(s) is finite dimensional and bi-proper, and define

f = lim

|F(j)| > 1,

u = lim

U(j ),

k = lim

L2(j)

L1(j).

Lemma 5.2.2. The suboptimal controller has finitely many unstable poles if and only

if the following inequalities hold:

|k| 1f

, |u| 1 f|k|f |k| (5.11)

when (n1 + l) is odd (even) and ku < 0, (ku > 0), and

|k| < 1, f|k| 1f |k| < |u| 0, (ku < 0).

Proof By using Lemma 5.2.1, when (n1 + l) is odd (even) and ku < 0, (ku > 0),

we can re-write (5.7) as

f|k| + |u|

1 + |k||u| 1.

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After algebraic manipulations and using f > 1, we can show that (5.11) satisfies

this condition. Similarly, when (n1 + l) is odd (even) and ku > 0, (ku < 0), (5.7)

is equivalent to

f |k| |u|1 |k||u| 1,

and (5.12) satisfies this condition.

Note that u is a design parameter and the range can be determined, by given f

and k.

Theorem 5.2.1. Assume that the optimal and central suboptimal controller (when

U = 0) has infinitely many unstable poles, if there exists U H, U < 1 suchthat L1U has no C+ zeros and |LU(j)F(j )| 1, [0, ), then the suboptimal

controller is stable.

Proof Assume that there exists U satisfying the conditions of the theorem. By

maximum modulus theorem,

|1 +ehsoM

(s

o)F

(s

o)L

U(s

o)|>

1 eh

|F

(j

)L

U(j

)|>

0,

therefore, there is no unstable zero, so = +j with > 0. Since, all imaginary axis

zeros are cancelled by E, the suboptimal controller has no unstable poles.

The theorem has two disadvantages. First, there is no information for calculation

of an appropriate parameter, U. Second, the inequality brings conservatism and

there may exist stable suboptimal controllers even when the condition is violated. It

is difficult to reveal the first problem, therefore it is better to use first order bi-proper

function for U. For the second problem, define max and max as,

max = max|LU(j)F(j)|=1

,

max = max[0,)

|LU(j)F(j )|.

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It is important to design max and max a

gumussoy suat

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