adaptive swtching control applied to multwariable … · 2020. 4. 6. · part icular ciass of...

ADAPTIVE SWTCHING CONTROL APPLIED TO MULTWARIABLE

SYSTEMS

Michael Chang

A thesis submitted in conformity with the requirernents for the degree of Doctor of Philosophy

Graduate Department of Electrical and Computer Engineering University of Toronto

@ Copyright by Michael Chang, 1997

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In loving rnernory 01 rny father

and

tu our farnily

Therefore. the art of employing troops is that when the enemy occripies hi&

ground. do not confront him: with his back resting on hills. do not oppose him.

When he pretends to flee. do not piirsue. Do not attack his élite troops. Do

not gobbIe preferred baits. Do not thwart an enemy returning homewards. To

a surrounded enemy yoii must leave a way of escape. Do not press an enemy at

bay.

Adaptive Switching ControI Appiied to Muitivariable Systems

A thesis submitted in conformity wit h the requirements for the degree of Doctor of Philosophy

Graduate Depart ment of Electrical and Computer Engineering University of Toronto

@ Copyright by Michael Chang. i994

Abstract

In this thesis. a family of adaptive control problems is examined and solved using robust

self-tuning switching controllers. The motivation for using t his type of controller is t hat.

often in practise. no suitable mathematical mode1 of the system to be controled is available:

conventional methods of adaptive controller design generally require specific a priori plant

information (e.g. it may have to be known if the plant is minimum phase). and thus cannot

be implemented if such a knowledge is not known.

In contrast. this thesis shall generdy assume that very little a priori plant information is

known - the main assumption being that the plant çan be modelled by a finite dimensional

linear t ime invariant (LTI) system. More specifically. for the âdapt ive cont rol problem

of a family of not necessarily strictiy proper mult i-input mult i-output (XIIMO) plants.

a switching mechanisni which requires less a priori system information than previously

considered is proposed. Utilizing t his framework. vaxious new self-timing controlIers t hen are

presented. which solve the adaptive stabilization problem and the robust servomechanism

problem for potent iaIly unknown MIhI O systems.

The proposed controllers appear to be quite attractive in their overall improved tuning

transient response when compared wit h earlier results. Real- time experimentd results of one

part icular cIass of switching controllers when applied to a rnultivariable hydrauiic apparat us

are presented. and illustrate the feasibility of applying such adaptive controllers to industrial

process control problems.

1 would like to acknowledge the love. encouragement. and support t hat 1 have constantly

received from my parents. for without them. this current endeavoiu would not have been

possible. With the passing of my father. it is to him and our family that I dedicate this

work.

For academic. financial. and emotional support and guidance. many thanks go to my

supervisor. Professor Edward J. Davison. It has been an honour and a pleasiire to work

with him throughout my graduate years. and the knowledge and experiences thst 1 have

gained are imrneasurable.

Financial funding during my graduate stiidies also has been generously provided for by

the Satura1 Sciences and Engineering Research Council of Canada (YSERC) and the Uni-

versity of Toronto through XSERC Post Graduate Scholarships and University of Toronto

Open Fellowships respectively. For this. too, 1 am extremely grateful.

1 am also t hankful for the generous help in BT@2e provided by Christian Meder. and am

indebted to d l the members and s ta f f of the Systems Control Group for making my stay here

so fruitful and memorable. -1s well. my gratitude &O goes out to Mrs. Linda Espeut. our

Group Secretary. and hlrs. Sarah Cherian. our Graduate -Idmissions ancf Progrrtms Officer.

for their expert and thoughtful advice concerning the intricate and sometimes obfuscacory

rules and regdations of the University.

Last. but not least. 1 would iike to t hank al1 of my teachers - past. present. and future.

Contents

1 Introduction I

1.1 Notation 5' - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Adaptive Switching ControI of LTI MIMO Systems 8

. . . . . . . . . . . . . 2.1 Switching Control for Generd Controller Structures 8

. . . . . . . . . . . . . . . . . . 2.1.1 Preliminary Definitions and Results 9

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Main Results 16

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 SimulationResults 24

. . . . . . . . . . . . . . . . . . . . . 2.2.1 -1 Family of Three SIS0 Plants 24

. . . . . . . . . . . . . . . . . . . . . 2.2.2 A Family of Ten hlIMO Plants 28

. . . . . . . . . . . . . . . . . . . . . 7.2.3 X Fanlily of Five MIMO Plants 32

. . . . . . . . . . . . . . . . . . . 22.4 .A Family of UnstabIe SIS0 Plants 34

3 Adaptive Stabilization of LTI MIMO Systems 39

3.1 Adaptive Stabilization of First Order LTI SIS0 Systems . . . . . . . . . . . 39

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 bhinResults 43

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Simuiation Results -44

. . . . . . . . . . . . . . . . . 3.2 Adaptive S tabilization of LTI MIMO Systenis 44

. . . . . . . . . . . 3.2.1 Using a Known Value of the Compensator Order 50

. . . . . . . . . . 3.2.2 Using no Known Value of the Compensator Order 51

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Siniulation Results 53

4 The Self-Tuning Robust Servomechanism 65

. . . . . . . . . . . . . . . . . . . 4.1 Self-Tuning Proportional-Integral Control 65

. . . . . . . . . . . . . . . . . . . 4.1.1 Using an Estimate of the DC Gain 66

. . . . . . . . . . . . . . . . . . . 41.2 Using no Estimate of the DC Gain 71

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 SimulationResults 72

. . . . . . . . . . . . . 1.2 Self-Tuning Proport ional-Integral-Derivat ive Control 75

. . . . . . . . . . . . . . . . . . . 4.2.1 Using an Estimate of the DC Gain 76

. . . . . . . . . . . . . . . . . . . 4.2.2 UsingnoEst imateof theDCGain 79

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 SimulationResults 53

5 The Self-Tuning Servomechanism with Control Input Constraints 94

. . . . . . . . . . . . 5.1 Constrained Self-Tuning Proportional-Integral Control 94

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Simulation Results 101

6 Adaptive Tracking of LTI MIMO Systems 108

. . . . . . . . . . . . . . . . . . . . . . 6.1 Preliminary Definitions and Results LO8

. . . . . . . . . . . . . . . 6.2 Using a Known Value of the Compensator Order 112

. . . . . . . . . . . . . . 6.3 Using no Known Value of the Compensator Order 113

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 . 4 Simulation Results 115

7 Experimental Results 125

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Experimental Apparatus 125

. . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Linearized Model of MARTS 127

. . . . . . . . . . . . . . . . . . . . 7.3 Conventional Controller Design Results 131

. . . . . . . . . . . . . . . . . . . . . . 7.1 Switching Controller Output Results 194

. . . . . . . . . . . . . . . . . . . . . 7.4.1 Using a Known Estimate of 7 136

. . . . . . . . . . . . . . . . . . . . 7.4.2 Using no Known Estimate of 7 138

. . . . . . . . . . . . . . . . . . . . . . . . . . . 7 . 4 3 Using Controller C1 142

8 Conclusions 146

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Summary of Results 146

. . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Future Research Directions 147

vii

A Proofs of Main Results 151

A . 1 Adaptive Switching ControI of LTl MIMO Systenis . . . . . . . . . . . . . . 151

A . l . l Theorem2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

A 2 Theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

A.2 Adaptive Stabilization of LTI Systems . . . . . . . . . . . . . . . . . . . . . 161

2 Theorem3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

A.2.1 Theorem3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

A.3 The Self-Tuning Robust Servoniechanism . . . . . . . . . . . . . . . . . . . 165

3 . 1 Theorem4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

A.3.2 Theorem4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

A.3.3 Theorem 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

B Miscellaneous Data 171

B.1 Controller Parameters for a Family of Five Plants . . . . . . . . . . . . . . . 171

B.2 Partial Decentralized Control of a Multi-Zone Building . . . . . . . . . . . . 174

B.3 A Four Input-Four Output Furnace >Iode1 . . . . . . . . . . . . . . . . . . . 175

B.4 Matrices used for a Binary Distillation Tower . . . . . . . . . . . . . . . . . 180

C Additional Experimental Results

Bibliography

List of Figures

1.1 Simulated results of y ( t ) with (1.2) applied to (1.1). T

1 . . . . . . . . . . . . .

1.2 Simulatedresultswith(1.3)appliedto(1.1). . . . . . . . . . . . . . . . . . 7

2.1 .4 schematic setup of Controller F1 . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Simulated results with Controller FI applied to P3 when f (k ) = jl (k) . . . . 26

2.3 Simulated results with Controller FI applied to e3 when f (k) = f i ( k ) . . . . 26

2.4 Simulated results of P3 with supervisory controller [73] applied . . . . . . . . 27

2.5 New simulated results with Controller F 1 applied to Pi when f ( k ) = f (k) . 29

2.6 New simulated results with Controller F1 applied to P3 when f ( k ) = f2(k) . 29

2.7 Simulated results with Controller FI applied to Pio . . . . . . . . . . . . . . 32

2.8 New simulated results with Controller F1 applied to Pio . . . . . . . . . . . . 33

2.9 Switching tirne instants with ControIler F 1 applied to Plo . . . . . . . . . . . 33

2.10 Reference signals and switching time instants with ControlIer F1 appIied to

P, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.11 Simulated response with Controller F1 applied to f i . . . . . . . . . . . . . . 35

2-13 (q = -0.5) Simulated response with Controller F1 applied to (2.9). . . . . . 37

3.13 (q = 0.125) SimuIated response with Controller F 1 applied to (2.9). . . . . . 35

2.14 (q = 0.5) Simulated response with Controller F1 applied to (2.9). . . . . . . 38

3.1 Simulatedresultsofy(t) withCoritrollerSIappliedto(3.1) . . . . . . . . . . 45

3.2 (w( t ) # O) Simulated results of y ( t ) with Controller S1 applied to (3.1). . . 45

3.3 Simulated results with Controller S2 applied to (3.5) using (3.6). . . . . . . 57

3.4 ( w ( t ) # 0) Simulated results with Controller S2 applied to (3.5) using (3.6). 57

3.5 Simulated results with Controller S2 applied to (3.5) usiug (3.7). . . . . . . 58

3.6 Simulated results with Controuer S2 applied to (3.8). . . . . . . . . . . . . . 60

. . . . . . 3.7 (.w ( t ) # O) Simulated results with Controller S2 applied to (3 .8) . 60

. . . . . . . . . 3.5 Simulated results witb Controller SS applied to Pi (s) (3.10). 62

. . . . . . . . . 3.9 Sirnulated results witli Controller 52 applied to P J s ) (3.10). 62

. . . . . . . . . 3.10 Simulated results with Controller S2 applied to Pj(s) (3.10). 63

. . . . . 3.11 Schematic set-up of the R, design synthesis used for P I ( . s ) (3.10). 63

. . . . . . 3.12 New simulated results with Controller S2 applied to Pi(.s) (3.10). 64

4.1 Simulated results with Controller P2 [14] applied to (4.5). . . . . . . . . . . 74

4.2 Simulated resiilts with Controller PI1 applied to (-4.5). . . . . . . . . . . . . 74

-4.3 Simulated results witti Controller 2 (641 applied to (4.5). . . . . . . . . . . . Y5

. . . . . . . . . . . 4.4 Simulated resiilts with Controller PID 1 applied to (4.5). 85

4.5 Simulated results with Controller PIDI' applied to (4.5). . . . . . . . . . . . Y7

. . . . . . . 4.6 ( p = O ) Simulated results with Controller PIDl applied to (4.5). 87

4.7 Simulated results with Controller PIDl applied to (4.11). . . . . . . . . . . 89

8 Simulated results with Lued integral cuntroller (4.13) ilpplied to (4.11). . . . 89

1.9 (DR # O ) Simulated resiilts w i th (filtered) Controller PIDl applicd to (4.14). 91

. . . . . . . . . . 4.10 Simulated results witti Controller PIDI applicd to (4.15). 93

4.11 ( ( p . E - ) = (0 . O ) ) Simulated resiilts with Controller PIDL applied to (4.15). . 93

5 . L Simulated resiilts with Coiitroller 2 [67] applied to (5.5). . . . . . . . . . . 105

5.3 Simulated results with Controller C l applied to (5.5). . . . . . . . . . . . . 103

5.3 Simulated results with Controller 2 [67] applied to a distillation tower . . . 106

5.4 Simulated results with Controller C l appliecf to a distillation tower . . . . . 106

5.5 Sirnulateci results with Controller 2 [W] applied tu (4.11). . . . . . . . . . . 107

5.6 Simulated results with Controller C l applied to (4.11). . . . . . . . . . . . 107

. . 6.1 Simulated results of y(t) with Controller T l applied to (6.6) iising (6.7). 116

6.2 Simulated results of y(t) with Controller T l applied to (6.8) using (6.9). . 118

6.3 Simulated results of y(t) with Controller T l applied to (6.10) using (6.12). 120

6.4 Simulated results of g ( t ) with Controller T l applied to (3.8) using (6.14). . 134

6.5 Simulated results of y ( t ) with Controller T l applied to (3.8) using (6.15). . 124

. . . 7.1 Schematic diagram of one possible MARTS arrangement (not to scale) 12G

7.2 Cross sectionai view of the intercounecteci columns of -MARTS (not to scale) . 128

. . . . . . . . . . . . . . 7.3 Experimental results with 0 = 30' and (7.5) applied

7.4 Experimental results with 0 = 30°? and with the outputs reversed at t = 1000

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . seconds

7.5 Experimental results with t9 = 0°: and with the outputs reversed at t = 1000

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . seconds

7.6 (N = 3) Experimental results with 0 = 40' and Controller PID 1 applied . . .

. . . 7.7 (N = 5) Experimental results with t9 = 40' and Controller PID 1 applied

. . . . . . . 7 Experimental results with 8 = 40" and Controller PID1' applied

7.9 (Reversed) Experimental results with 0 = 40' and Controller PIDl' applied .

. . . . . . . 7.10 Experimental results with 0 = 20' and Controller PIDI' applied

. . . . . . . . . . . . . . 7.11 Experimental results with Controller PID1' applied

. . . . . . . . . 7.12 Experimental results with 0 = 30" and Controller C1 applied

. . . . . . . 7.13 Experimental results with 0 = 30" and Controller 2 [67] applied

7.14 (Reversed) Experimental results with 6 = 30a and Controller C1 applied . .

7.15 (Reversed) Experimental results with 0 = 30" and Controller 2 [67] applied .

8.1 ( x ( 0 ) = 1) Simulated results with Controller 52 applied to (8.1). . . . . . .

8.2 (x(0) = 0.001) Simulated results with Controller S2 applied to (8.1). . . . .

. . . . . . . . . . . . . . C.1 Experimental results with Controller PID1' applied

. . . . . . . . . . . . . . C.2 Experimental results with Controller PID 1' applied





. . . . . . . . C.7 (Reversed) Experimental results with Controller PID1' applied

. . . . . . . . C.8 (Reversed) Experimental results with Controller PID1' applied

C.9 (Reversed) Experimental results with Controller PIDI' applied . . . . . . . .

C.10 (Reversed) Experimental results with Controller PIDI' applied . . . .

. . . . . . . . . . . . . . C.11 Experimental results with Controller PID1' applied

C.12 Experimental results with Controller PID1' applied . . . . . . . . . . . . . .

List of Tables

. . . . . . 4.1 Summary of the cyclic switching behaviour used for Figure 4.5. 84

7.1 Summary of the major components of !vIXRTS . . . . . . . . . . . . . . . . . 127

C . 1 Summary of the parameters used for Figures C . 1-C . 12 . . . . . . . . . . . . . 153

xii

Chapter 1

Introduction

In the convent iond design of controllers for rnultivariable systems, t Iie general approach

often adopted is to find a suitabie nominal model for the plant, which is often a difficult task,

and then to design a controller based upon this nominal model. In this case, if the nominal

model captures suflicient dynamicd aspects of the plant, effective control of the system

may be possible to attain. However, if large unexpected structural changes subsequently

occur in tlie system, severe limitations iri practical performance will, in general, arise since

conventional control schemes irsually do not have the ability to coritrol systenis wliicli are

subject to unplanned extreme changes.

Currently, one met hod to deal effectively wit h the specific problem of parametric plant

uncertainty is adaptive control; typically, the controllers erriployed in such schcrries are non-

Iinear and tinie-varying, and consist of a compensator augmentcd with a tuning rneciiariisrn

whicli adjusts the conipensator gains to match a prespecified desired plant tnodel. Although

recent attempts to enhance the robustness propert ies [42], [G]; iriiprove tlie closed loop tun-

ing transient response [go], [81], [10]? weaken the restrictive sufficient a priori assuniptious

[69], [74], [70], (991, and eliminate [77] tiie unwanted "bursting" pheno~nena 131 have been

successful for this class of controllers, these results generally are limited in scope due to the

nature of the a priori assumptions still required.

During the past several years, however, switching control in both theory and practice

has been applied successfully to multivariable systems to accomplish a wide variety of tasks

[131, [1011, [W, (591, [61], [64], [62] , (651, (661, [72], (791, [12], (151, [SOI, [16], (191, [SI], (101,

[23], [17], (731, [97]; while many controllers of this type wliich use very little a prion' plant

information have ais0 traditionally enjoyed the extra benefit of being very robust to large

plant urcertainties, one particuiar disadvantage of some of these schemes conimonly has

been an unpleasant closed loop susceptibility to substantial output transient responses. A

brief review of some of the important contributions made in this area can be found in [57]

and [58].

In this thesis, the primary focus therefore will be on designing simple d u s t adaptive

switching control aigorithms which attempt to use as little a priori system information as

possible, and which attempt to provide a reasonable closed loop transient resporise. For

instance, by now allowing for the possibility of cyclic switching to occur in the adaptive

control problem for a finite family of MIMO plants considered by Miller and Davison [65],

decreased closed loop tuning transients and a prion' system assumptions, as well as reduced

switching controller complexity, usually can be attained. Similarly. by utilizing this frame-

work to solve the adaptive stabilization probleni and the robust servomec~ianisni problem

for possibly unknown MIMO systems, corresponding desired t ransierit iniprovenients for

these problems can also gerlerally be achieved by selectirig nori-pathological coritroller and

tuning parameters.

In order to determine the feasibiIity of appIying such controllers to an industrial systelri,

real-tirne appIication studies using one such class of cont rollers wit h almos t uo a prion' plant

information, when applied to an experimentd multivariable liydraulic systerri. are carried

out; these studies show the feasibility of using the obtained controllers in ari iridustrial-

type setting and, in addition, show that the controllers applied display desirable iiitegrity

feat ures.

Before beginning, however, the following preliminary inforniation will be required.

1.1 Notation

The following mat hemat ical notation will be used irr a fairly corisistent rrianrier t hroiighout

t his t hesis.

Let R, IR+, N and @ denote respectively the set of real, positive real, natural: and

complex numbers; Rn (Cn) will be the n-dimensional real (cornplex) vector space. IRmXrL

(Cmxn) the set of m x n real (complex) matrices, C- (C') the set of complex riumbers with

strictly negative (positive) real parts, and @O the set of complex nunibers lying strictiy o n

1.1. Notation 3

the imaginaq axis. For any x? y € N,

x mod y := L - floor (i) Y

where floor(*) rounds the expression * down to the nearest integer.

With x = [xi x2 . . . xnIT E Rn. denote its pnorm as

and its oo-norm as

For any arbitrary A E C n x n . let X ( A ) and eig(A) denote the eigendues of -4. Matriu

A f RnXn will be said to be stable if X(A) c @- and unstable otherwise.

For the geaerd case when A E Xmx ". -+Ir will denote its matrix transpose. rank(;l) its

rank, and. if A also lias full row rank. di = - A ~ ( A A ~ ) - ' its pseudo-inverse. In addition. the

corresponding induced norms of A will be denoted as l1 - -Ll lp with the x-norm calculated as

and the Froberiius norm given as

where aij denotes the (i' j ) element of A.

As a final point, let C" (W") denote the set of Rn-valued functions defined on Bi which

are infinitely differentiable. A function f : Rç u {O) -r Rn will be said to lie in L, (f E L,)

if

1.2. Some Motivation 4

exists.

1.2 Some Motivation

As mentioned earlier, during the past several years, there bas been a considerable amount

of interest and effort made towards developing controller design methods which require

as little a priori plant information as possible [54], (341, (641, [62] , [6517 [72]' [23]: [17].

The motivation for this interest s t e m £rom the fact that it is generally difficirlt and often

impossible to obtain an accurate model representation of an actual industrial plant. As

well, while conventional adaptive controllers currently have the ability to deal effectively

with the problem of pararnetric plant uncertainty? important a priori plant information still

is required; for example, in conventional model reference adaptive control for a single-input

single-output (SISO) system, the four classical assumptions typically made are t hat [68]

[82]:

(i) the plant is minimum phase;

(ii) an upper boiind on the plaiit order exists and is k~iown;

(iii) the relative degree is known; and

(iv) the sign of the Liigh frequency gain is kiiown.

Altliough recent developnients have been able to reniove condition (iv) (691. [74], aiid to

weaken conditions (ii) [70] and (iii) [71], [98], given above, specific plant inforrnatiorl (e-g.

any plant zeros which lie in the open right half plane must be known to lie i r i a finite set

[63]) still is needed. For an excellent historical overview of some of the major advancements

in this area, see [75] and [76].

In this thesis, the primary focus will be on siniple prerouted adaptive switching control

algorithmsl whicli attempt to use as little a priori system information as possible; as well,

due to potent ial impleuientation constraints, t his work will also bc concerned with ro bust

- -. . . - - - -. - -- -

'A switching algorithm is said to bc prerouted if the potcntid sequence of applied controllers is dctermined off-line, and fixed prior to the application of the switching mechanism.

1.2. Some Motivation 5

adaptive schemes (cf. [go], for exampleo and the results contained t herein) which are tolerant

to bounded immeasurable noise disturbances? and which attempt to provide a reasonable

closed loop t ransient response.

As an illustration of the type of improvernent that may be obtained by using the proposed

controllers, consider the problem in [85] of finding a stabilizing controller (in the sense that

z ( t ) + O as t + cc and [x E L,' where ~ ( t ) and ~ ( t ) are? respectively. the state of the

plant and controller) of the form

for the one-dimensional SIS0 plant

with botb b # O and a > O unknown. In ttiis instance. iising the blktensson-type [54]

cont roller (341

i ( t ) = $(t). ( 12a)

h(il) = d G . (1.2b)

and u(t) = y(t) - h(q(t))0-25 - (sin(JT;i;;iljT) + 1) -cos(h(q(t))) ( 1 .k)

with a := 1? b := -1, x ( 0 ) := 1: and q(0) := 1, the undesirably large transient response (with

a peak overshoot p a t e r than 31 1000 iri magnitude) presented in Figure 1.1 is obtained.

Similarly, using the adaptive Nussbaum-type stabilizer [85], [74? pg. 5491

with

1.3. Thesis Outline 6

one can see t hat an arbitrarily small persistent measurernent dist urbance rnay destabilize

the system if ?12(t) is nonintegrable due to variations in e ( t ) . This effect can be seen in

Figure 1.2? where the controller given by (1.3) is applied to (1.1) with a := 1: b := 1.

x ( 0 ) := 1, q(0) := 0, and ~ ( t ) := 0.25sin(100t).

In contrast, using one of the new controllers proposed later in this thesis (Controler SI),

the more desirable responses shown in Figures 3.1 and 3.2 are obtained. whicb correspond

to the respective identical examples given for Figures 1. L and 1.2.

1.3 Thesis Outline

The remainder of this thesis is organized as fdlows.

In Chapter 2, the adaptive coritrol problem for a finite fanlily of MIXI0 LTI plants is

considered from the point of view of either stabilization or servomechanism control. and

new t heoreticd results for a generalized class of switching controllers are obtained. Chapter

3 then re-examines the adaptive stabilization problem for the case when the MIMO LTI

plant is almost entirely unknown (Le. it is assumed only that the plant is stabilizable and

detectable). In Chapter 4. the case of potentially unknown open loop stable 5IIMO plants

is considered. and self-tuning PI and PID controllers wliich solve the robust servoniech-

anism problem for constant reference and coustant dist urbauce inputs are proposed: the

corresponding PI case with control input constraints is next exaniined in Chapter 3. Tfiese

servomechanism controlIer results are generalized in Chapter 6 for the case of possibly un-

known MIMO plants. which may be open loop unstable. by applying an alternative niethod

for resolving the adaptive servomechanism problem. Real-time experimental results of one

class of coritrollers when applied to a multivariable hydraulic system then are preserited in

Ctiapter 7, showing the successful impleuientatiou of the PI and PID controllers developed

in Chapters 4 and 5.

1.3. Thesis Outline 7

Output response using a Mmensson-typc conmllrr. I

Figure 1.1: Simiilated results of y ( t ) with (1.2) applied to (1 .1 ) .

Output response usinor a Nussbriurn-rypt conuollrr. 4.5 r

. - - / - - - 1 - -

Figure 1.2: ( ~ ( t ) := 0.25 sin(100t)) Simulated results with (1.3) applied to (1.1) with s (solid) and q (daslied).

Chapter 2

Adaptive Switching Control of LTI

MIMO Systems

In this chapter. we assume that the plant to be controIled can be niodelled by a finite dimen-

sioiial LTI system which is contained in a specificd finite family of plant morlels. For each

of these plant rnodels. it is assumed that there is an associateci servomediariisin controller

(which has been separately designeci): and it is ciesired to obtain a switdiirig controller

which has the property that it will select the correct stabilizing controller froni ttiis faniiiy

of controllers using as little struct iiral information as possible. Moreover. a swi tdling con-

trol mechanism which is robust in nature to a11 boirnded piecewise contiriuoiis references

and disturbances, and whicb reqiiires less a prton' systern information t haxi prcvioiisly as-

siimed in [G], is presented. As well. plant detectability and the construction of a I~ouiiding

function f are shown to be sufficient to ensure that switching evcntrially stops. Simulation

results using this new controller are dso presented. and compareci witli the corresponding

output responses obtained using the schemes given in [65i and [73].

2.1 Switching Control for General Controller Structures

In this section, switching control for a finite set of s plants, subject to the general control

law

2.1. Switching Control For General Controller Structures 9

2.1.1 Preliminary Definit ions and Results

Let each element

belonging to the finite setL of possible plants to be controlled

be of the finite dimensional form

where x E IRn1 is the state. u IRm is the control input. g E Z r is the plant output to be

regulated, w E Rq is the disturbance, and e E Pr is the difference betweeri the specified

reference input g,,f and the output y. Ili tlie disciissious wtiicli will follow. we do not

necessarily assume that 7 ~ i , Ai: Bi? Cz, Di, Ei7 or Fi are known. and we do not restrict

X(Ai) C @-, i E {1,2,. . . , s ) .

Observe that upon applying Controller Ki to coritrol plant mode1 Pi ? the resiilting closed

loop system is

'since PI is a 6-tuple, a çlight abuse of notation is used to define P in an effort to maintain notational simplicity.

2.1. Switchine: Control for Generd Controller Structures 10

w here

fi := (1 - D,L~) - ' ( 2 . 3 ~ )

BiMi i B ~ L ~ Ï ~ D ~ M ~ Ei + B ~ L ~ Ï ~ F * and Bi := (2.3d)

Ji + Hi fi Di Mi H, Ïi F*

Preliminary definitions and results which are needed before proceeding are given as

folIows.

Definition 2.1: Consider matrices (C. A: B) E BrXn x Rn X n x RnXm. Then (C. A) is said

to be detectable if there exists a matrix K E Rnxr such that X(A + KC) C C-. and (A. B)

is said to be stabilizable if there exists a matrix L E Rmxn such that X ( A + B L ) C C-.

Definition 2.2 : (131. pg. 171) The transmission rems of (C. A. B. D ) E X r x n x RnXn x

Rn"" x WrXm are defined to be the set of complex numbers X which satis@ the following

inequali ty :

Definition 2.3: A plant Pi (2.2) is said to be m i n i m u m phase if al1 OF its transmission

zeros lie in @-; otherwise, phn t Pi is said to be non-minimum phase.

Definition 2.4: ( [ 2 6 ] ) Consider the triple (C. .4B) E ZarXn x Rn"" x Px ": t hen the set

of centralized féced modes of (C. A. B ) , denoted by A(C. -4, B) , is defined as follows:

Definition 2.5: ([24]) Consider (C, A, B' D) E WrX" x P nXn x W n X m x RrXm; then the

set of decentralized &ed modes of (C. A, B. D) with respect to K denoted by A(C. A B, D),

is defined as follows:

2.1. Switchin~: Control for General Controller Striictures 11

where K E IK is also chosen such that ( I - DK)-[ exists.

Remark 2.1: Given D E Rrxm: then for almost d l [30] L E B m X r . (1 - DL) E RrXr is

invertible (i.e. given a fixed matrk D. then ( I - DL) is invertible for generic L) . 0

Remark 2.2: Fkom Lemma 2.3.3 of 137: pg. 591, if 3 E gnXn and 11.T-11 < 1. tlien (1 - 3)

is nonsingular (Le. (1 - F)-l exists). 0

Definition 2.6: ([64]) A function f : N - 33- is said to be a s t m n g botrnding f irnction

(f E SBF) if it is strictly increasing and if

Definition 2.7: A function f : N + Ri is saki to be a modified strong bounding function

(f E MSBF) if it is strictly increasing and if? for al1 constants (cc) . ci , 0) 3- x Rt x Zt'.

Proposition 2.1: There exists a bISBF (e.g f ( 1 ) = i e x p ( i 2 ) ) .

Proof: The proof follows upon first observing that

for i > 2. In addition, since

i exp (i' ) i exp(i2) i- I 1

co + cl (i - 1) + c? cxp(i2) co + ci(i - 1) + c z C jexp(j2)

2.1. Switching Ccrntrol for General Controuer Structures 12

for (CO. ci. c 2 ) E P+ x !Ri x Wt: and since

the resuh immediately follows. D

Proposition 2.2: .Assume that -4, given in (2.3) is stable for a given cimice of (G,. Hi.

Ki, Li): then

X(Ai) êo for almost dI (Gi. Hi. Kt. L,).

Proof: The proof follows upon observing that -Xi can alternatively be written as

using Definition 2.4 and the fact that Ai is stable for a given choice of controllcr parameters

(G,. Hi. Kz. L,). it therefore foilows that -4, bas no fked modes lying in Co for al1 feedback

matrices

satidying the structure

where * is an arbitrary rnatrix having appropriate dimensions. Hence, it immediately follows

[30] that for alrnost ad controiler parasieters (Gia Hz: Ki. Li): X(Ai) CO. O

2.1. Switching Control for General Controller Structures 13

Remark 2.3: When Di = O for the proof of Proposition 2.2, one can alternatively express

Lemrna 2.1: Consider the system given in (2.2) with u( t ) given in (2.1) applied at t i z e

t = O. Assume that in (2.3): the matrk

is stable. and t hat yref(t) and w ( t ) are bounded piecewise cont inuous signais liaving respec-

tive L, norms of ljre and 6. Then there exist constants (Ci! CÏ' G' c-4) E P- x 22- x R' x R'

independent of z(0) = z0 := [ Z ( O ) ~ T ~ ( o ) ~ ] ~ such tiiat

for al1 t E [O,=).

Proof: Since Ai is stable, there exist coristants (A. C l ) E Bi x Rf sucli tliat / ~e - ' l ~ l l 5

for t 2 O; using the additional fact t hat

w here

t herefore


Likewise, since

it therefore follows that

w here

for al1 t E [O' co). (7

111 order to corisider the situation when gr, ( t ) and w ( t ) are boiinded coiit inuous signals.

and when Di = O for al1 i E ( 2 . 2 , . . . . s)' label Controller F1 as

t 1 := 0, and where, for each k 2 2 such that t k - # m, the switching tirne t k is defined by

1 i) t > t a - [ , and if this riiinimum exists

2.1. Switching Control for GeneraJ Controiier Structures 15

with f E MSBF. In addition, let Assumption F1 be the following:

i) Ilrl(0)ll < J(Ur

ii) l Ie(0) l~ < f (l)?

iii) for each plant Pi and each corresponding applied Controller K,, i E { 1.3. . . . . s ) ?

the closed loop system is stable (and controller parameters (G,, H i , .J,. K,. L,, -LIi)

provide acceptable error regulat ion/dis t urbance rejection when the plant Pi is subject

to bounded piecewise constant reference and disturbance inputs):

iv) for each plant Pi, (Ci, Ai) is detectable: and

v) for each i: j E {1.2.. . . .SI. (1 - DiLJ) is invertible (see Remark 2.1).

The switching mechanisrn described by Controller FI is schematically shown in Figure 2.1.

LU f

Plant

Figure 3.1: A schematic setup of Controller FI.

1'

Switch

In Controller F 1, norm bounds on q ( t ) and e ( t ) are used in an attempt

-

to detect closed

Cont roller

&

loop instability which might be caused if Controller Ki is applied to plant P,. i # j . If this

upper bound is met at a.ny time during the tuning process. then a controller switch occurs,

4

' ~ h i s condition is required so thac switching time tr, is well defined for Controller FI; given e ( 0 ) . it can be met easily by appropriatelx xaling f ( i ) .

2.1. Switchine Control for General Controller Structures 16

and is reset to zero irnmediately following this switch. This reset action is performed

since al1 candidate feedback controllers need not necessarily be of the same order, and

since past experimental investigations [14] have indicated that reduced tuning transient

responses generally can be attained via such a sclieme. However, for the case wtien al1

candidate controllers have the same order, i.e. when gi = gj for al1 i7 j E { 1: 2' . . . s } . rl(t:)

need not necessarily be reset to zero after each switch; one cari choose to continue to form

q( t ) iising the set of piecewise LTI systems given by (GiT Hi, J * ) witfi ~ ( t ; ) = q( t k ) .

2.1.2 Main Results

Continuous Signais

For the situation when Di = O for al1 i E {1'2,. . . .s), and wtieri yrel(t) and w ( t ) are

bounded continuous signals, the following result can be obtained:

Theorem 2.1: Corisider a plant P E P with Di = O, i E {1,2.. . . , s ) . and witli Coritroller

FI applied at tirne t = 0; tlien for every j E MSBF. for every bounded contiriiious reference

and disturbance signal, and for every initial condition z (0 ) := [ x ( o ) ~ v ( ~ ) T ] T for which

Assumption F1 holds, the closed loop systexxi has the properties tfiat:

i) t h e exist a finite tirne t,, 2 O and coristant matrices (Gss, H,,, J,,, Kss7 L,,, iCI,,)

such that (G( t ) , H ( t ) , J ( t ) , K ( t ) : L ( t ) h.l(t)) = (Gss, Hss: Jss: Kss, L,,, Alss) for al1 t >_

tss;

ii) the controller states q E Lm and the plant states x E L,; and

iii) if the reference and disturbance inputs are constant signais. tlien for alriiost al1 con-

trolier parameters (Gi, Hi, Ki Li), asymptotic error regulation occurs, i.e. e ( t ) + O as

t -+ XI.

Piecewise Continuous Signals

For the situation when Di # O for some i E { 1,2, . . . , s} and/or wlien yrel(t) or w(t) are

bounded piecewise continuous signals, the switching criterion given for t irne tr; in Controlier

F1 may not bc well-defined. ln order to circumvent such poteritiai problenis, Coritroller F1


can be simply modified by filtering the error signal r ( t ) , and defining e ( t ) as

Hence, label Assumption F2 and Controller F2 to be? respectively. Xssumption F1

with e ( t ) replaced by e ( t ) , and Controller F1 with e ( t ) replaced by ef ( t ) in the definition

of switching time t k .

Lemma 2.2: Consider the closed loop system fornied by augmenting (2.3) together with

(2.5). Assume that X(Ài) C @- and that y re f ( t ) and w(t ) are boiinded piecewise continuous

signals. Then there exist constants (Ci : C2) E W' x W+ independent of i ( 0 ) := [x(OIT r l ( ~ ) T

e such that

for al1 t E [ O , o o ) .

The following result can now be obtairied.

Theorem 2.2 : Consider a plant P E P with Controller F2 applied at time t = 0: then

for every f E MSBF and X E B+. for every bounded piecewise continuous reference and

disturbance signal, and for every initial condition Z(0) := [ Z ( O ) ~ ,"(O)= el (0)*IT for which

Assumption F2 liolds, the closed loop systeni lias the properties tliat:

i 1

ii)

iii)

there exist a finite time t,, > O and constant matrices (Cs, , H,,! Jss7 Kss? L,,. :LIss)

such that ( G ( t ) , H ( t ) : J ( t ) , K ( t ) , L ( t ) , h . l ( t ) ) = (Gss? Hss, Jss. K,,. L,,! 3.I,,) for al1 t 3

tss;

the controller states 17 E L,, the plant states x E Lm, and the filtercd error signal

e l E Ç,; and

if the reference and distubarice inputs are constant signals, theri for alniost al1 con-

troller parameters (Gi, Hi, Ki, Li), asyrnptotic error regulation occiirs, Le. e ( t ) + O as

t + 00.

Controller F2 provides a great deal of generality aud versatility sirice:


atl finite dimensional MII'vIO LTI plant models Pi are assunied to have the general

form given by (2.2), with Ai not necessarily stable, Di not necessarily equal to zero.

and Pi not necessarily minimum phase;

the plant models Pi need not be controllable and/or observabie:

rn the set of al1 admissible controllers need only satisfy the structure presented in (2. l) ,

and thus the candidate controllers need not have the same dimension;

rn the class of piecewise cont inuous reference and dis turbance sigrials allowable for the

servomechanism controller design [27] of Ki (and the irnplementation of Controller F2)

is quite large provided only that gr,/ E L, and w E Lm (e-g. the class of sinusoidal

references and disturbances is allowed);

a priori bounds on either y r e l ( t ) or w(t) are neither rieeded nor estiniated for the

proposed controller;

a rio extensive a pr ior i on-line calculations are iieeded in order to inipleme~it Controller

F2 (cf. the schenies given in [34] and [65]);

the controller switching meclianism is very simple to iniplement in real-time. and is

therefore attractive from a practical point of view;

the switching meclianism does not depend directly on any explicit knowledge of the

matrices associated with plant Pi or candidate Controller K,;

no on-line estimation period is needed; and

rn the switching mechanism is robust and will riot suffer frorri chattering in the stcady-

state (cf. [39], for instance) for al1 bounded piecewise continuous reference arid distur-

bance inputs.

As well, Theorems 2.1 and 2.2 will clearly d s o hold even if the finite number of candidate

controllers is greater than or equal to the nuniber of possible plants3.

In addition, in Theorem 2.2, tlie requirement that ( t ) and w(t) be bounded piecewise

continuous functions and the restriction that switching carmot occur infinitely fast guarau-

tees the existence and uriiqueness of a solution [41] to the set of differential equations given

3 ~ n fact, Theorems 2.1 and 2.2 will hold for an infinite number of plants (cg. see Section 2.2.4) so long as there exist a finite number of candidate controllers which satisfy Xssumptions F1 and F2 rcspectively.

2.1. Switchine: Control for General Controller Structures 19

by (2.3) and (2.5). Furthermore, without any loss of properties i) to iii) given in Theorem

2.2: filtered error signal e (t) couid also have been defined as

where II E EUr, X ( r l f ) C f -. and Cl and B are botli invertible.

In fact. for the general situation when plant P, is described by

properties i) and ii) of Theorem 3.2 will also hold for al1 boiinded pieccwise coritinuoiis

noise s ipals (pi: p) E R'i~~l x W z wi th ( N I - Q i ) E Rnt'<qbl x I P r " ' ~ ~ ~ . This follows since

the closed loop systeni with Controller K, applied rnay be expresseci as

w here

Moreover! if II[~: pr]Tll -t O, property iii) of Theoreni 2.2 will once again be recavered.

Since corresponding comments sirnilar to those given liere cari dso be made for Tlicorerns

2.1. S witchine; Control for General Controller Structures 20

3.1, 3.2, 3.3, 4.2, 4.3, 4.4, 4.5, 6.1, and 6.2 (provided that the output signals are filtered

accordingly) by using an identical argument, the details of these additional extensions will

be omit ted for brevity.

Remark 2.4: Let the switching time t k be defined as

if this minimum exists

I O 0 ot herwise,

and define Assumption A to be Assumption F1 with the following additional condition:

vi) H, is left invertible for al1 j E {l , 2, . . . , s ) (i.e. H j has full column rank. and hence,

there exists H: := (HTH~)-'HT such that HJH, = 1).

Then properties i) to iii) of Theorem 2.1 will also hold for al1 bounded piecewise coiitinuous

reference and disturbance signals with Dj not necessarily equal to zero for all j . This follows

since, for any plant Pm E P, the pair

is detectable. Furt herniore, under Assumpt ion A, if swit chhg based iipon filtered error

signal e (t) is still desired. then Theorem 2.2 will also hold for any (CI , Al B I ) E Wrf "f x

Wnf Xnf x Wnf x r with A(AI) c @-. 0

Remark 2.5: Define

where X E W'. Assume that

i) I I Y ~ ( O ) I I < f ( U a d

ii) lluj(0)ll < f (1)

2.1. Switching Control for Generd Controller Structures 21

both additionally hold. Then Theorem 2.2 wilf also hold using the switching criterion

defined by

i ) t > t k A l r and if this minimum exists t k :=

ii) I[S(t)lJ = f (Cc - I )

I ot herwise.

u- here

- ( t ) := [ T j ( q T yf(t)T]T.

Given the general nature of Controller F2. oue may aho want to bouud iridividually

~ ( t ) ruid e l ( t ) by differenr bounding functions. This can be done as showu in the following

definition and existence resul ts.

Definition 2.8: Functious f i : N -t Ri and f2 : N -t R- are said to be choosubie modified

stmng boundingfvnctions ((fi! f2) E CMSBF) if. for t E {l. 2}. fk is strictly increasing and

if, for al1 constants (CO, ci? C?. c3) E Ri x Ri x Ri x Wr.

Proposition 2.3 : There exist functions J 1 and f- such tbat (f [, f.?) E CMSBF (e.g.

ji ( 2 ) = cri exp(i2) and f2(i) = ,& exp(i2) where (a? 0) E xi x Et+).

2.1. Switching Control for Cenerai Controller Structures 22

Remark 2.6: Consider the switching criterion for Coritroller F2 given by

if this uiiuiniuni exists

where ( f i , f2) E ClviSBF. Then with

md witb conditions iii) to v) of A4ssiimption F3 assunled to be triie. Theoreni 2.2 will also

hold true in this situation. 0

Remark 2.7: The modifications given in Remaxk 2.6 can also be made to the switching

mechanisms given in Theorerus 3.2. 3.3. 4.2. 4.3. 4.4, 4.5. 6.1. and 6.2. 0

Remark 2.8: For simplicity, consider the case when Di = O. i f { 1.2. . . . . ..; ): tlien the ro-

bust servomechanism problem can be solved (if possible) iising the servocorr~peiisator design

method given in [27]. For example. maintaiiiing the structure. notation. aiid assiiniptioiis

given in [27]: with

e -- .- l/ - IJrej.

a full order Luenberger observer of the form

2 = (.Ai + C,Ci)i? + Biu - Ciy

c m be constructed to yield


where := 5 - z. Since the closed loop system rnay be expressed as

w here

IL = PO,i + Pt<.

one can therefore choose matrices Po, and Pi siich that

is stable. Hence. the final controller may now be expressed in the forni

One can tlierefore apply switdiing Controller F3 to repulate i~11d reject adaptivcly tiiis

particular class of bounded reference and disturbance signals. 0

Remark 2.9: For simulation purposes, with plant P liaving system matrices given by

(A, B. C. O, E. F ) ,

and with Controller Ki applied, the closed loop system may be expressed as

A Bpi B ~ o , [i l = [ -C;C B-G Biri C* Ai + CiCi O + BIPol ] [ j] + [;* -CtF U' ] [ J ï f ] 7

2.2. Simulation Results 24

where the same notation as given in Remark 2.8 is maintained.

2.2 Simulation Results

In this section, the output response obtained by applyiug Controller F1 to a faniily of

strictly proper plants will be considered. In the first example, results £rom the simulation

will be compared with other output responses formed using the schemes given in [65] and

[73]. For the second and third example. the iinstable batch reactor mode1 [23! will be used.

This section is concluded wit h results obtained by using the fmi ly of plants given in [31].

2.2.1 A Family of Three SISO Plants

Consider the following family of three SISO LTI plant models taken froni [65]:

Model Pl :

with open loop eigenvalues of -4.5 + 1.5j:

Model Pz:

witb open loop eigenvalues of 0.031 and -24.031; and

Model P3:


wit h open Ioop eigenvalues of -3.535 and - 10.565. As one can v e r i k using

Controller KI : 7 j = e? u = -2.7577:

Controller ICz : 7 j = e. u = -2q + 79: and

Controller K3: 7j = e. u = 25q - ~ I J

conditions iii) and iv) of .4ssumption F1 are both satisfied. and al1 controller-plant mis-

matches result in a closed loop unstable system4.

In Figures 2.2 and 2.3: the output response of the closed loop system with Controller F1

applied to plant P3 is given for the case when f (k) = f i (k) and j(k) = f2(k) respectiveiy.

w here

for each figure.

~ ( 0 ) := 0: w (t) := 2. and yref(t) is a square wave (beginning a t time t = O) having zero DC

offset, a peak magnitude of 10. and a period of 20 seconds. Furthermore. in both instances.

switches occur due to bounds on q ( t 2 ) and 71(t3) being met or exceeded.

As can be seen. in each case, the transient response rnight be considered to be quite

reasonable taking into account the fact t hat t here are t hree differeut poteut i d cont roller

candidates which m u t be considered. In coutrast, the transient peaks obtained using

Controllers 1 and 2 given in [65] are substantially larger for the same set of controller

candidates5. This occurrence is related in large part to the fact that Controller FI has a

potential cyclic switching action which may occur. whereas the switching rnechanisnis given

in [65] will, a t rnost, try each possible controller only once.

"-4 controller-plant mismatch is said to occur if Controller K, is applied to piant P,. where i f j . o or Controllers 1 and 2, the peak magnitude of the output trançients are approximately 2600 and 110

respec tively.


Themtid sontinuous time switching conuol output rcsults. 1

, K v switchine rime instants. ,00 Conuul s i e d u venus fime.

,

0: O 1 4 6 O 50 100

Tirne I seconds Tirne isecondsl

Figure 2.2: Simulated results wit h Controller F 1 (having t hree candidate controllers) applied to plant P3 for the case when f (k) = fi(k).

Theoretical conirnuous rime swiichine control ciurpur mufis. IO - 1

-70' O 10 20 30 40 j0 60 70 Y0 90 100

Tirne (seconds)

, K v swrichine time instanu. Cantml s i e d u vcrsus time.

0; 1 1 O 10 10 30 O 50 100

Time (seconds 1 Time (seconds)

Figure 2.3: Simulated results with Controller F1 (having tliree candidate controllers) applied to p l a t l3 for the case when f (k) = f . - ( k ) .


20 l Sirnulaed output response of P-3.

1

-IO il 10 10 j0 10 50 60 70 Y0 'H) 100

Time (seconds)

J r K v swiichtne tirne instmis.

- -

"0 0.05 0.1 0.15 0.2 0.3 0.3 0.35 0.4 0.45 0.5

Time iscconds)

Figure 2.4: Simulated results of plant Pi with supervisory controller [73] applied.

For further cornparison" Figure 2.4 shows the output response of plant Pi obtained

when using the SISO supervisory control switctiing mechanism given ixi [73]. Here. <Irurll

time rr, is set to be equal to 0.1 seconds. with

and Controiier K i is applied initictlly at time t = O. Simila to the simulations shown in

Figures 2.2 and 2.3,

w ( t ) := " M .

q ( 0 ) := o.

and ~ ( 0 ) := [1 21T.

Remark 2.10: The switching mechanisms for the controllers given in [65] and [73] gener-

ally necessitate more system information or a priori computation t han required for proposed

Controllers F1 and F2. For instance, explicit use of the family of SISO transfer functions

6 ~ h e author acknowledges the gracious help and assistance of Wen-CIiung Chang and A. S. Morse for providing the simulation code uscd to generate Fibwrc 2.4.

2.2. Simulation Resdts 28

is needed in [73], while knowledge of each candidate

(A. B. C. O. E. F )

system matrix is requùed in [65]. As such. it is conjectured that the switching mechanisms

given by Controllers F1 and F2 will be more robust with respect to unmodelled plant

perturbations and immeasurable noise disturbances than those mechanisms given in [65]

and [73]. O

Remark 2.11 : In Figure 2.4? dwell time rg c m be chosen to be 0.1 seconds since r~

can be made arbitrarily srnall without sacrficing performance in the absence of un,modelled

dynarnics and meusurement emors 1731. O

In Figures 2.5 and 2.6. the system output response obtained (corresponding to Fi,wes

2.2 and 2.3 respectively) for the case when P2 and K2 are replaced by

(wit h open loop eigenvalues of - 1 and - 15 & 2 j) and

Cont roller K2: r j=e . u = I ~ + 3 e

is shown. Using this new controller (K2): one can verify that controller-plant cnisrnatçh

&- P3 results in a stable closed loop system. This fact thcrefore accourits for the results

shown in Figure 2.6. where K2 is chosen as the final steady state controller.'

2.2.2 A Farnily of Ten MIMO Plants

In this example. we illustrate the implementation of Controller F1 when applied to the

foilowing (unstable batch reactor) MIMO plant taken froni [25]:

--

?III both Figures 2.5 and 2.6, al1 switches are due to the bound on r l ( t ) being met or exceeded.


3: T h e o w d continuous rime switching contrai output rtrults.

I

I

l

-10 ' 1

O 10 10 30 40 50 60 70 Y0 90 100

Time (seconds i

K v switchine tirne insunrs. UIO Cunml signai u vmus timc l

Time (seconds) Timr (seconds 1

Figure 2.5: (New P2 and Kz) Simulated results with Controller F 1 (having t hree candidate controllers) applied to plant e3 for the case when j ( k ) = fi (k).

10 Thmieucd contrnuous tirne switching conuoi uutput resulrs.

Figure 2.6: (New P2 and K2) Simulated results with Controller F 1 (having three candidate controllers) applied to plant i3 for the case when f (k) = _Jr(k).


with open loop eigenvalueç of 1.99, 0.0635, -5.0566, -8.6659. Let the corresponding con-

troller ( K I O ) be

which fias been designed to stabilize and regulate mode1 Plo subject to constant references

and constant disturbances [XI.

For simplicity, assume in the simulations that

and set x ( 0 ) := [l 2 3 4lT? witli al1 0 t h iiiitial coriditions defiiied to be eqiial to zero at

time t = O. III addition, let yre l ( t ) (with Y:e j ( t ) = -y;I>el(t) and T J ! ~ ~ ( ~ ) := 10 For t E [O. 1 0 ) )

be a square wave having zero DC offset, a peak magnitude of 10, and a period of 20 secoiids.

The other potential controller candidates, obtaiiied by using conventiorial cotitroller design

methods, are listed below:


Controller K4:

Controller Ks :

Controller Xe:

Controller KT:

Controller Ka:

Controller Kg:

7j = e. ,u = K (77 + e). K :=

.Tj = e . u = K ( q + e ) . K:=

Using the above controllers. one can verifi that conditions iii) and iv) of .4sstimption F I

are both satisfied8. and that al1 controller-plant (Pie) niismatclies result in a closed loop

unstable system ezcep t for Controller Kn.

In Figure 3.7'. simulation results with Controller FI applied to plant PLU aiid

f (k) := 2Ok exp (k) are given. In this instance. two switches occur due to the bounds on q ( t ) and e ( t ) wtiich are

met or exceeded a t the respective times of 0.245 and 0.31 seconds. uid switcliing stops once

Controller ICs is applied. Similar to the output response shown in Figure 2.6. the above

results again emphasize and illustrate the fact that the final steady-state controller gains

may not necessarily correspond to the controller puameters associatecl witli actrid plant

Pm. It is to be noted. however, that if the closed loop systern consisting of plant P; with

contmlier parameters (G,, Hj, J j K,. L,. il.IJ) is stable if and only if i = J 1 t hen the final

controuer gains applied wiii almost aiways be ensured to be (Gi7 H,, Ji Ki ! Li: Mi).

'For brevity, the system matrices of the othcr nine plants will not be given.

2.2- Simulation Resdts 32

10, Thmrcticd contrnuous rime swirchine conuoIler resultr.

1

Tirne (seconds

Figure 2.7: Simulated results with Controller F1 (having ten candidate controllers) spplied to plant PLo with y1 (solid) and y? (dashed).

In contrast to these results. Figures 2.8 and 2.9 illustrate the output response of plant

PLo using the identical situation given for Figure 2.7. but with ControIler K3 now defined

as follows:

-1.37'76 -0.0131 ] y + [ 1.0000 0.0000 ] Controller K3: 7j = e . u = 'I -

-0.0131 -1.3753 0.0000 1 .O000

In this instance, one can verify that Assumption F1 is satisfied. and that the closed loop

system will indeed be stable if and only if Controller XII, is applied.

2.2.3 A Family of Five MIMO Plants

In this example, Controller F1 will again be applied to (2.8)" As before. assume that

'1x1 this example, howevcr, the unstable batch reactor mil1 be labelled as plant Pg.

2 -2. Simulation Resul ts 33

200 r Thasorericd caniinuous timr swttchine conuolirr resuits.

i

?O fheorerid continuous rime switchin~ conmller rcsults. I I

Figure 2.8: Simulated results with Controller F1 (having ten candidate controllers) applied to plant PLo with y1 (solid) and y3 (dashed).

Ttme (seconds)

Figure 2.9: Switching time instants with Controller F l applied to plant Pio. (Controllers which are applied due to a previous bound on ~ ( t ) or e ( t ) being met or exceeded are marked by a 'N' or an 'E' respectively.)


and define

f (k) :=

Let al1 other initial conditions be equal to zero at tirne t = 0, and let g r e l ( t ) be a periodic

triangular wave of the form shown in Figure 2.10. For completeness, the parameters of all

candidate controllers are listed in Section B. 1.

As one can verify, Assumption F1 is satisfied aud al1 controller-plant (Ps) mismatches

result in a closed loop unstable system. In Figures 2.10 and 2.1 1. the output response as

well as the switching time instants of the closed Ioop system are given: here. al1 switclies are

due to bounds on e ( t ) being met or exceeded. Simi1a.r to the predicted t heoreticd results.

Controller ICs is also selected correctly (after approximately 0.6 seconds).

2.2.4 A Farnily of Unstable SIS0 Plants

In this last exampIe, consider the respective set of unstabIe plants aiid controllers given by

[34, pg. 1102]

2.2. Simulation Resuits 35

Time r seconds i

6 ! K v switchine time insrnu.

l

Figure 2.10: Reference signals & ( t ) (solid) and y&(t ) (dash-dotted) (top plot): and switching time instants with Controller F1 applied to plant Pi (bottom plot).

Throretical continuou iimr swrrchinr contmller rcsulis. 10

-10 ' I

5 10 15 10 15 30 35 40 45 50

Time isecondsi

Figure 2.11: Simulated output response with Controlier FI (having five candidate controllers) applied to plant Ps with 91 (solid) and y2 (dashed).


and

where q E [-0.5.0.510 and

as one can verify, (2.9) is controllable and observable for dl values of q E R. alid the closed

loop system can be expressed as

where X(A) = {-1, -2. -4). The control objective here will be to stabilize the closed loop

sys t em.

With

[q(O) 4 0 ) q ( O ) ] := [l 2 O]!

(it is known froni [34] t hat at least one Controller Ki, i E { 1.2. . . . . 5 1, will stabilize the

system given by (2.9) for a fixed value of q, q E [-0.5.0.51) the output resiilts showri in

Figures 2.12, 2.13, and 2.14 are obtained1° for the indicated values of q. Although the final

loin these simulations, r l ( t k ) is not reset to zero inimcdiatcly follom-ing any controller switch.


Theoreficl continuou iimr switching conrrollrr resulls. 1

7 Swirching tirne instants of controller K v. - I 1

Timc (seconds)

Figure 2.12: (q = -0.5) Simulated output response with Controller F1 (havirig five candidate controllers) applied to (2.9) with xi (solid) and x2 (dashed).

steady-stote controllers do not necessarily correspond in al1 instances to those given in [34]_

the output transient responses are still coniparableii in nature to those showri in Fi y r e s

3, 4, and 5 of [34, pg. 11021".

"In Figure 2.14, the output rcsponse iç improvcd noticeably over the transients shown in Figure 5 of 13.11. This occurrence is due, in part, to the fact that wben using the controller of [34], there exists a positive time intervd immediately follotving each controller switch during which no further switching ruay occur.

''These latter results are, however, obtained by using a much more cornputationally intensive on-line switching mechanism; in addition, a considerable amount of a prion calculations rriust also be done in order to implement the scheme given by 1341.


4 Swiichine t h e insrrints of controller K v.

l

Figure 2.13: (q = 0.125) Simulated output response with Controller F1 (having five candidate controllers) applied to (2.9) with X I (solid) and s* (dashed).

30 Theoreticai continuous tirne switchinp: conmller mlts.

I 1

-20 ' 1

O 5 10 15 70 3 30

Timr (seconds)

5 Swiichinp lime mlants of conuoIlrr K-v.

1

Timr (seconds)

Figure 2.14: (q = 0.5) Simulated output response with Controller F1 (having five candidate controllers) appIied to (2.9) with xi (solid) and x2 (dashed).

Chapter 3

Adaptive Stabilization of LTI

MIMO Systems

Using the methods and techniques developed in Chapter 2. we now propose a new stabi-

k i n g adaptive controller for the class of first order strict ly proper SISO LTI systerns. and

for the general class of finite dimensional strictly proper !vZI?vIO LTI systerns considered in

[54], [61], and [62]. Xs in Chapter 2. the controller is potentially cyclic in nature. and the

emphasis will be to provide a robust switching niechanisrn which is insensitive to bounded

piecewise continuous disturbances w ( t ) and which attempts to provide an acceptable tran-

sient response; the switching mechanism proposed here. tiowever. is sinipler in nature t han

that given in [61]. and it does not require a prelirninary identification period as given in

[621

3.1 Adaptive Stabilization of First Order LTI SISO Systems

In this section, we again utilize the switching mechanisni given in Section 2.1 to stabilize

adaptively (in the sense that z ( t ) -t O as t + zc and [r uIT E C, with w( t ) = 0. and [x

,uIT E Lm with w( t ) # O and UJ E L,) the following fint order SISO system:

(3. la)

(3. l b)

3.1. Adaptive Stabilization of Fkst Order LTI SIS0 Systemç 40

where x E R is the state. u E IR is the control input, 9 E W is the plant output, w E R4

is a bounded piecewise continuous disturbance, (a, b, c) E R x R x R? b # O: c # 0, and

(eT. f T, E R4 x Rq; by maintaining the assumption that 6 # O arid c # 0' the system given

by (3.1) is both stabilizable and detectable for ali a E R.

In the past, one class of non-linear (one-dimensionai) smoot h adaptive stabilizing con-

trollers for t his type of system has been considered [85]: and a particular stabilizing noise

sensitive controller (which potent ially gives a uery large transient response, as O bserved in

Figure 1.1) is given in the form [34]

Howe . .

!ver, due to the nature of the problem considered in t his section, and for brevity, the

simulations presented here will exclude any cornparison wit h ot her convent ional adaptive

control methods (381: (81, (781, [95] which are &O knowo to be able to solve this problem.

Definition 3.1: A furiction f : Ri + Ri is said to be a SI bounding junction (f E SLBF)

if it is strictly increasing and if. for al1 constants (ca' cl. Q: m 1 , rns: T , W ) E Ei x Bi x Rf x

Rf x R' x IRi x IR+,

asi-kx.

Proposi t ion 3.1: There exists a SlBF (e.g. f ( i ) = i' exp(i3)).

Maintaining the notation used earlier, let Control ler S1 be given as follows:

where k E {l, 2 ,3 , . . . },

S := {(eo,r) : €0 > OJ > 1)'

3.1. Ada~t ive Stabilization of First Order LTI SIS0 Systems 41

K ( t ) = , € 1 . . i = ((k - 1) mod 2) + L. t E ( t t . tk ; i ]?

t1 := 0, and where. for each k 2 2 such that tk- 1 # X. the switching time tk is defined by

i) t > t k - l : and if t his minirnuni exis ts t k :=

ii) I y ( t ) ( = f (k - 1)

l m O t herwise

with f E SlBF. Label Assumption SI to be

Here, the restrictiori that 1 y(0)l < f (1) is required in order to ensure once agairi that

the switching time tk is well defined for Controller S1. As well. Controller SI works by

monitoring plant output y ( t ) in an a t tempt to detect instability. Following each controller

switch, and similar to the results presented in [LOl]. the sign of K is clianged. and gain E is

increased with the goal of using high gain output feedback to stabilize (3.1).

Remark 3.1: Consider the SISO system (3.1): where (a, 6. c) E X x R x R. b $ 0 , c # 0.

and K := { K i : K2): then for almost al1 (E& T) E S.

Lemma 3.1 : Consider the first order SISO plant (3.1) with Controller S1 applied at

tirne t = 0, and assume that the controller [lever stops switching; let w ( t ) be a piecewise

continuous signal with w E Cm. and let sign(bKjc) = -I for one j E {1'2}. Then with k

3.1. Ada~t ive Stabilization of First Order LTI SIS0 Systems 42

sufliciently large such that ((k - 1) mod 2) + 1 = j and

the following properties hold for al1 t E (tr. t i+l] (with 1 2 k : ((1 - 1) mod 2) i L = j ) :

wliere (ci, C . e: 6, c6) E R+ x RT x Ri x Pi x B' x 8' are constants independent of

I and i r ( t r ) .

Proof: The proof follows upon first observing that

for d l 2 k' ( ( 1 - 1) mod 2 ) + 1 = j : hence. there exists a constant X E R' such that

for 1 > k. ( ( 1 - 1) mod 2) + 1 = j : t 3 0.

On defining

it therefore now foilows that

and thus that

3.1. Ada~ t ive Stabilization of First Order LTI SIS0 Systerns 43

The result therefore follows upon defining

3.1.1 Main Results

Theorem 3.1 : Consider the system given by (3.1) with Controller S 1 applieti at time

t = 0: then for every f E SlBF and ( E ~ : r) E S. for every bounded continuoiis disturbance

signal. and for every initial condition ~ ( 0 ) for which Assiimption S1 iioids. the closeci loop

system lias the properties that:

i) c h r e exist a finite time t,, 1 0. a finite constant E,, > O. aiid a constant f<,, such

that ~ ( t ) =E,, and K ( t ) = Kss for al1 t 2 t,,;

ii) the plant state r~ E Cs; and

iii) if the disturbance inputs ur(t) = 0. tlien for almost al1 ( ru . T ) E S. r(t) i O as t + s.

Remark 3.2: A s in Tlieorem 2.2. Controller S 1 will also work for boiinclcd piecewise

continuous disturbance signals upon filtering y ( t ) as

where Assumption S l and Controller S1 now are defined to be. respectively. Assurnption

S1 with y ( t ) replaced by yy / ( t ) ' and Controller S1 with y ( t ) replaced by y l ( t ) in the given

definition of switching time t k . 0

3.2. Adaptive Stabilization of LTI bîTbIO Systems 44

3.1.2 Simulation Results

To demonstrate the potential t r a i e n t improvement that might be attained by iisi~ig The-

orem 3.1. consider the system (3.1) with

In Figure 3.1. the output response obtctined iising Controller S1 is shown. Similady. iising

identicai controller parameters with

(c .u .6 .e . f) := (I.1.1.0.1).

~ ( 0 ) := 1.

and ~ ( t ) := 0.25 sin(100t).

the output illustrated in Figue 3.2 is obtained. In both instances. these resiilts compare

favourably when contrasted with the respective outputs shown in Figues 1.1 and 1.2. where

a peak overshoot greater than 311000 in magnitude. and closed Ioop i~istability. respectively

result.

3.2 Adaptive Stabilization of L T I MIMO Systems

In t his section, the gencrd problem of adapt ively stabilizing the finite dimensiorial strictly

proper (stabilizable and detectable) PVIIMO LTI system given by

is examined, where z E Rn is the state, E W m is the control input. !j E Rr is the plant

output, and ul f Rq is the disturbance. The candidate feedback controllers which will be

3.2. Adaptive Stabilization of LTI MIMO Systems 45

d r Theoretid coniinuous tirne switchrng conuollrt results.

3 1 Switchine rime instants oi conuolla K=K-V.

1

Tirnç i seconds I

Figiire 3.1: ( w ( t ) := O ) Simulated results of g ( t ) witli Controller S1 applied to (3.1) .

15, Tharericd continuous time swnchine conuollrr results. 7 l

-3

O 1 4 6 Y 10 12 14 16 18 10

Time c seconds)

3 r Switchine time Insranri of conuolla K=K v. 1

I

Timr: (seconds )

Figure 3.2: (w(t ) := 0.25sin(100t)) Simulated results of y ( t ) witfi Controllcr S 1 applied to (3.1).

3.2. Adaptive Stabilization of LTf MIMO Systems 46

considered are of the form

Similar to Section 2.1? in the discussions which will follow, we do not necessarily assume

that n, A, B, C, E, or F are known. and we do not restrict X(A) C E-.

Preliminary definitions and results which are needed before proceeding are given as

follows.

Definition 3.2: A function f : W -+ 22' is said to be a S2 bounding fmct ion (f S2BF)

if it is strictly increasing and if. for al1 constants (CO. ci. mi. m7. T. 15) E BT x R- x Rf x

Wf x Ri x R'.

CO + cl x(n, i- m 2 i J ) ( f ( j ) i W)

Proposition 3.2: Tliere exists a S2BF (e.g. f ( 2 ) = 2 e x p ( i g )).

Definition 3.3: ([92. pg. 441) X set D is dense in W ifB = W. where v is the closureofD.

(i) { h ( i ) : i E W} is dense in ~ ( " ~ g ~ ) ~ ( ~ ~ g ~ ) for a hxed d u e of gi E NU (O}: and

(ii) for h ( i ) :=

3.2. Adaptive Stabilization of LTI P v ~ l O Systems 47

for constants ( T ~ 3) E IR+ x IR+.

Proposition 3.3: Given gi E N U {O). there exists a h E CTF.

Proof: Consider the situation where each element of h( i ) siiccessively examines. in a nestecl 1

fashion, each intervai [-n. ni, n E N. and tries points apart. Tlien since Il h( 1) I I = (r+q*).

and since each element of h(i) will increase in magnitude at most by one following czny given

switch. t herefore

the result foilows. [7

.As an example of how one might construct h( i ) . consider the situation wtien. for instance.

gi = 1. i E N. and rn = r = L: thcn h ( i ) c m be defined as follorvs:


For the current problcm under consideration, define

Since the closed loop system niay tlierefore be expressed as

the stabilization of (3.2) can be secn to be equivalent to the probleni of firidirig a feedback

niatrix & E ~ ( ~ ~ g ~ l ~ ( ~ ~ g ~ ) such tliat x(A + B K ~ ~ ) C ê-. In fact: with 1 E NU {O): - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Cl := {(C, A, B) : there exists an I't li order LTI stabilizirig comperisator

(3.3) which stabilizes (3.2) ),

and

then C l C and (C, A, B) E C if and otily if ( A , 8) is stabilizable and (C, A) is

detectable. As well, usirig t his particular frmework, t lie followiiig result is ob tained.

Lemma 3.2: Consider the plant (3.2), and assume that there exists a LTI co~itroller (3.3)

of known order gi E N U {O) that stabilizes the closed loop system (3.4) (i-c. it is known

3.2. Ada~tive Stahilization of LTI MIMO Systems 49

that with ü = Kicl A(A + B K i 8 ) c ê- for some value of K~). TThen witli

applied at time t = 0, there exist constants (y,Ci,Cz,G,C4) E IR+ x Wf x WC x IRç x W+

independent of 2(0) such that

Proof: By assurnption, there exists a niatrk ki E R ( ~ + ~ ~ ) ~ ( ~ + " ) such tliat A(.<+ B R,S) c

Ce. Hence, by the continuity of eigenvalues [96], there exists a constant y E IR+ such tint

for al1 1 1 ~ ~ ~ 1 1 5 7. In addition, there therefore also exist constants (a: X) E W+ x IRt such

that

ü := (K* + AI?,)^,

À := À + B K ~ C + B A K ~ C ,

and Ë := Ë + BK*P + BAKiF,

and note that


With

t herefore

arid Ilij(t)ll 5 llCll - ( ~ I I ~ ( O ) I ~ + $@lu) + !IFII . 5:

hencc, the resuit follows upon defining

c 1

(2

C3

and

3.2.1 Using a Known Value of the Compensator Order

For the case when it is known t hat comperisator (3.3) will stabilize plant (:3.2) with gi = p.

p E N U {O)' using sonie appropriate choice of controller matrices Gi. H,. K,. L,. label

Controller S2 as

where k E {l. 2 . 3 , . . . }, h E CTF.


t l := 0, and where, for each k- 2 2 such that t k - l # OC, the switching time t k is defined by


ot herwise

with J E S2BF. In additiori, let Assumption S2 be

Theorem 3.2: Consider the system given by (3.2) with Controller S2 applied at time

t = 0; then for every f E SSBF and h E CTF, for every bounded continuoiis disturbance

signal, and for every initial condition % ( O ) := [x (0 )= q ( ~ ) T ] T for which Assurnption S2 Iiolds.

the closed loop system has the properties that:

ii) the controller states rj E Loo and tlie plant states z E Lm; and

iii) if tlie disturbance inputs w( t ) = 0. tlien for alniost al1 coiitroller pararileters (G. H :

K, L ) , s( t ) -t O as t + W .

3.2.2 Using no Known Value of the Compensator Order

For the case when the order gi of a stabilizing compensator is unknown, but a lower boirnd

a E N U {O) and an upper bound y E N u {O} is known such that a 5 gi $ 7 with

a < y, ControlIer S2 can be modified such that, starting with O := a, one can search the

w ( ~ ~ ~ ) ~ ( ~ ~ O) parameter space near zero wit h a certain degree of fiiieness, arid t lien iricrease

(if necessary) the order of o by one and repeat the process with an iiicreased degree of

fineness. This idea is formalized in the following definitiori.

Definition 3.5: Given tliat a E M U {O) is a lower bound alid that 7 E N U {O} is an

upper bourid on gi such that cr 5 gi 6 y with a < y' a function h : N + ~ ( " ' + 9 ~ ) ~ ( ' + * ) is


a modified controller tuning function (h E CTF') if, with (gi+[ - gi) E { O . I}. gl := a, and

h(i) := Ki, the following properties hold:

(iii) for h(i) := 1 1 R(rrl+$ 1 x (.+!hi,

for constants (ri. F ~ ) E Wç x IR+.

Proposition 3.4: There exists a h E CTP.

Proof: Consider the situation where each element of h( i ) successively examines, in a nested 1

fashion, each interval [-n'n], n E N, and tries points - apart, and where? upon completing 2"

each nested search accordingly, gi is increased (if necessary) by one and the nested search

is restarted; then

and lience the result immediately follows.

Remark 3.3: In an attenipt to clarify the terse statemerits given in the proof of Proposition

3.4, consider the situation when (a, y, m, r ) = (O, I, 1 , f ): then h( i ) can be defined as follows:


h(5) = -1.0

For the case when it is known that compensator (3.3) will stabilize plant (3.2) with

gi = p using some CY 5 p 5 y with a, y E N U {O), a < y, and using sonie appropriate

choice of controller parameters Gi? &, K,, Li, label Controller S2' to be Controller S2.

but with h ECTF'.

Theorem 3.3: Consider the system given by (3.2) with Controller 52' applied a t time

t = 0; then for every f E S2BF and h E CTF', for every bounded continuous disturbarice

sigiial! and for every initial condition *(O) := [z(OIT i I ( ~ ) T ] T for which Assumption S2 holds.

the closed loop system has the properties that:

ii) the controller states E Lm and the plant states z E Lm: and

iii) if the disturbance inputs w ( t ) = O? ttien for almost al1 coritroller paranieters (G. H:

K, L), z( t ) + O as t + m.

Here, the relative corriputational simplicity of Controllers S2 and S2' compare favourably

when contrasted, for instance, with the controllers giveu in [74], [ol]. [62]. arid [57]. 111

essence, by properly constructing f to have certain known a priori propertics: the adaptive

stabilization problem for stabilizable and detectable MIMO LTI systenis can be solved by

monitoring norrn bounds on ~ ( t ) and y ( t ) .

Rernark 3.4: As ooted in (621, if it is known a priori that there exists a gain Ki in a known

set S c ~ [ p ( ~ + g ~ ) ~ ( ' + g ' ) such that the closed loop system is stable, then one can restrict the

search of the ~ ( ~ + g ~ ) ~ ( ' + " ) parameter space to S. For example: if

3.2. Adaptive Stabilization of LTI hIIM0 Systems 54

then one can define

( EL: if ( ( i - 1) m o d p ) t L = L

E2: if (( i - 1) mod p) + 1 = 2 h(2) :=

K if ((i - 1) mod p) + 1 = p

as a controller tuning function. 0

Hence. if one can restrict the search of the Btm'S1)X(r-*) puameter space to S for

stabilizing controller parameters IC,? where

then Theorem 3.2 reduces to Theorem 2.1 with yref ( t ) := O. In this instance. f (k) need

only satisfy the property that

as i + x for al1 constants (col cl. c- ) E R' x 32- x 2'.

Remark 3.5: If w ( t ) is a bounded piecewise continuoiis signal. tlien corresponding coni-

ments equivalent to the ones given in Remark 3.2 are d s o applicable to SLieorerns 3.2 and

3.3 provided that

are additionally satisfisd for (r3: T . ~ ) E Ri x RT. (This additional restriction can always be

met by using the construction rncthods given in the proof of Proposition 3.3 and Remark

3.3.) O

3.2. Adaptive S tabilization of LTI Pc/IIPc,IO Systems 55


Example 1: SISO Unstable Nonminimum Phase Plant

Consider the following (controllable and observable) unstable rionminimum phase SISO

plant taken Ekom [62. pg. 6041:

1 u + Ew.

wit h poles given by -2 and 0.5 kj. and zeros given by - L.5 and 0.5. Assume that the plant

is unknown. but that it is known that there exists a zero'th-order stabilizing compensator

for the system (i.e. it is known that for some d u e of L, E W. the closed loop system will

be stable with u = L i g ) . In addition. define h( i ) as

1 so that each successive i n t e d [-n. n]. n E M. is exarnined. and points , - apart are chosen.

Let

Lk, l L k . 1 1 5 and f(k) :=

20(k - 15)' exp((k - 15)"). k > 15.

3.2. Adaptive Stabilization of LTI %DM0 Svstems 56

and observe t hat the closed loop system is stable if and only if

In Figures 3.3 and 3.4, output results obtained using Controller S2 are shown for .w ( t ) :=

O and w( t ) := sin(2t) respectively. In both instances, t hese results are comparable to those

given in [61] and [62] , and. in accordance with Theorem 3.2. L ( t ) remains constant after a

finite number of switches (L,, = 2) .

For cornparison, using Controller S2 and the same initial conditions and parameters as

given for Figure 3.3. but with h( i ) defined as

the response shown in Figure 3.5 is obtained. In this iristance. L,, = 3. ami. unlike the

results presented in Figure 1 and the highly oscillatory response shown in Figure 2 of [6210

the state transient response suffers froru only an approxiniate four-fold iricrertse in peak

magnitude when compared with Fi-pre 3.3 '.

Example 2: Two Input-One Output Unstable Minimum Phase Plant

As anot her example. consider the (stabilizable and observable) unstable minimum phase

MIS0 plant [62. pg. 605)

'in [62], Figure 2 suffers from an approximace seven-fold increase in peak maguitude when conipared with Figure 1.


15 Theoretical conrtnuous tirne switching conmller renilts. 1

rot A

Tirne (seconds)

6 Switchine rime instants of conuoller L=L v.

01 I

O 1 2 3 4 5 6 7 Y

Time (seconds

Figure 3.3: ( w ( t ) := O) Simulated results with Controller S2 applied to (3.5) using (3.6) with xi (dotted), xz (dashed), x3 (dash-dotted), and y (solid).

10 [ Thmrertcal continuous tirne switchine conuoller w!ts.

Time (seconds)

6 i Switching rime m s m s of conuolIer L=L v.

Time (seconds)

Figure 3.4: (w(t) := sin(2t)) Simulated results with Controller S2 applied to (3 .5 ) using (3.6) with zi (dotted), x2 (dashed), x3 (dash-dotted), and y (solid).

3.2. Adaotive Stabilization of LTI MIMO Systemv 58

30, T h r o ~ r i d continuous time swirchtne conmller rcsulis. 1 1

Figure 3.5: ( ~ ( t ) := O ) Simulated results with Controller S2 applied to (3.5) using (3.7) with LT 1 (dotted), z- (dashed), s3 (dash-dotted). and y (solid).

assume once again that the plant is i~nknourn. but that it is known that ttiere exists a

zero'th-order stabilizing compensator (Le. i t is known that for some value of L, E P'. the

closed loop system will be stable with u = L,y). For

the closed loop system will be stable if and only if (1 l , 1-1 ) Iies in the region defined by

-11L-412i - 2 > O.

and I l l + 1 > 0.

3.2. Ada~t ive S tabilization of LTI bIIhIO Systerns 59

and let

6 + 2k. l ~ k - < l O and f(k) :=

10(k - LO)' exp((k - 10)"). k > 10.

In Figures 3.6 and 3.7. the output response is shown with w ( t ) := O and u(t ) := sin(2t)

respect ively: once again. in accordance with Theorem 3.2. L ( t ) remains coustant after a

finite number of switches ( L , , = [l -llT) even with w ( t ) # O.

Example 3: Simultaneous Stabilization Problem with Three Plants

Alternatively? consider the simultaneous stabilization problem [94]. [100]: [11] of the foilow-

ing family of three unstable non-minimum phase SIS0 plant models taken Erom [36' pg.

1 los]:

S - 7 5 - 3 .S - 6 Pr ( s ) := - pi&) := P3 (s) :=

s - 4.6' zs - 2.6: 4.8s - 24.6'

let the corresponding

3.2. Ada~tive Stabilization of LTI MIMO Systems 60

Thmretid continuous m e switchinn conuolla results. 'O 1

Time (seconds )

Switchinp rime insfan~ of conuoller L=LLv. t

! 0 ;

O 0 2 0.4 0.6 0.3 I 1.1

Time (srcon&)

Figure 3.6: ( w ( t ) := O) Simulated results with Controller S2 applied to (3.8) using (3.9) with xi (dotted), z2 (dashed). xn (dash-dotted), and y (solid).

Throreiical continuous time switchine conuoller ~ . ~ ~ l t s . IO r 1

!

Tirnr (seconds)

3 Switchine timr instuits of controller L=L v.

"O 0.1 0.7 0.3 0.4 O3 0.6 0.7 0.Y 0.9 1

Time (seconds)

Figure 3.7: (w(t) := sin(2t)) Simulated resdts with Controller S2 applied to (3.8) using (3.9) with X I (dotted), xz (dashed), x3 (dash-dotted) and y (solid).

3.2. Adaptive Stabilization of LTI ~ ~ 1 0 Systems 61

state space representation for each plant be given as

and define their respective

controller parameters to be

One can veri@ that each respective controller-plant pair yields chsed Ioop poles at i- 1. - 1).

{-1: - l}, and {-0.5' -0.5). Furthermore. as shown in (361 and [LI. pg. 811: there does not

exist a Lxed finite dimensional LTI controller which can simdtaneously stabilize plants Pi

fi: and Pj.

Now using Controller S2 and the cornmentu given in Remasks 2.6 and 3.4. with

and f2(k) :=

the output response for plants Pl, Pz, and f i is shown in Figures 3.8' 3.9. and 3.10 respec-

tively. Xlthough the transient response in Figure 3.10 is significantly larger when cornpared

to the results presented in Figures 3.8 a d 3.9, this fact may be attributed to the x t u a l


8 Tharet~cJ conrinuou urne swirching convol output resultr.

1 O' - W . -

O 5 IO 15 10 15 30

fime (seconds)

Timr (seconds 1

Figure 3.8: Simulated results with Controller 52 applied to plant Pi (s) (5.10) witli x (dashed) and y (solid).

75 Theoreiicd continuous time switching conuol ouipul raulrs.

n 1

1

Timc (seconds)

3 i K v switchine time insmts.

1

I

1

i 0 1

O 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time (seconds)

Figure 3.9: Simulated results with Controller S2 applied to plant &(s) (3.10) with x (dashed) and I/ (solid).

3.2. Ada~t ive Stabilization of LTI hfi%IO Svstems 63

1500, Thmerical çonunuous tirne switchine convol ouipui results. i

Time i seconds

J K v switchine lime insmts.

O' O 0.1 0.2 0.3 0.4 0.5 0.6 O.? 0 . Y Y I

Timr t seconds I

Figure 3.10: Simulated results with Controuer S2 applied to pIant P3(s) (3-10) with ;c

(dashed) and y (soiid).

Figure 3.11: Schematic set-up of the X, design synthesis used for plant P7 ( Y ) (3.10).

3.2. Adaptive S tabilization of LTI PViIbIO Systems 64

controller designed for plant PÎ (s). Indeed, using Figure 3.11 and the R, design technique

[33] wit h

and €2 := 1 x 10'.

one can obtain

and the noticeably improved closed loop transient response shown in Figure 3.12.

m. Thzareucal conunuous orne swiichme conuol output results.

Tirne (seconds)

Figure 3.12: Simulated resuits with Controller S2 (cf. Figure 3.10) applied to plant Pi(s) (3.10) with x (dashed) and y (solid).

Chapter 4

The Self-Tuning Robust

Servomechanism

In this chapter, the robust self-tuning controllers presented in [64], [lj]. and [16] for cases

involving both a known and an unknown estirnate of the steady-state DC gain matriu 7

will be reconsidered: as well, two riew self-tunirig proportional-integral-derivative (PID)

controllers for similar corresponding cases will also be deveroped. In an attenipt to irnprove

the output transient response, the proposed controllers now switch based upon norrn bounds

on q( t ) and e ( t ) . The results presented here ttierefore indicate that a more general controller

structure than originally given by Miller and Davison in [64] can be implemented on an open

loop stable system using a switching criterion with very Little a prioll system information.

4.1 Self-Tuning Proportional-Integral Control

Consider the finite dimensional LTI systeni given by

k = ,42+ Bu + Ew, y = Cx+ Du + Fw? e := Y r e j - y

(4.la)

(4.1 b)

(4. lc)

where x E Rn is the state, u E W m is the control input, y E Rr is the plant output to be

regulated, w E WQ is the disturbance, and e E Wr is the differcnce betweeri the specified

reference input y,,/ and the output g. Assume that m > r, that A is stable, that rL, A, B,

4.1. Self-Tuning Proportional-Integral Control 66

C' D. E. or F are not necessarily known. and restrict fief and w to be bounded piecewise

constant signals; let f E MSBF. and define 7 := D - C.4-L B. For the case wlien the

estimate of 7, T' has full row rank, let K := p. (An estirnate (n of 7 can be obtained

via rn steady state experiments outlined in [21].)

In t his section, the self- tuning ro bust servomechanism controllers considered in [64]: [ls], and [16] for cases involving both a known and an unknown estimate of the steady-state DC

gain matrix 7 will be re-examined. As in these earlier results. assunie throiiglmiit chat 7

has fui l row rank in order to form a tractable problem. and niaintain the definitions (e.g.

f E MSBF) and notation given in previous cbapters.

Definition 4.1: X function g : N + Et' is a tuning function (9 E TF) if lim g(k) = O. If A--=

fa g E TF and if there exist finite constants €0 > O and T > L so that g ( k ) = - for k E N.

rk

then define g E TF' to be a modified tuning fiinction.

4.1.1 Using an Estimate of the DC Gain

When an estimate of 7. given by T. is available. define the set of admissible controller

parameters as

Label Assumrition PI1 to be

iii) -TT+ is stable: and

€0 iv) (1 t p - D K ) . i E (1.2.3.. . . ). is invertible for Lxed p 2 O (see Remark 4.1): rL

then with a = ( j . g, p) E a. define Controller PI1 as


t l := 0 , and where. for each k 2 2 such that t k - L # X. the switching time t k is defined by

i) t > t k - ~ . and if this minimum exists t k :=

ii) 1llq(t)* e ( t ) T ] T ~ ~ = f (k - 1)

I X O t herwise.

Remark 4.1 : Given p > O and D E Rr m. then for almost d l E E 2.- and for almost d l

matrices K E Rm"', (1 + p d K ) is invertible. 0

The following results will also be needed.

Theorem 4.1 : ([48. pg. 571) Consider the singularly perturbed system given by

where ( X ~ ~ Q . E ) E RnL x Rn- x 2-. If -4;; exists. then as c i O. ni eigenvalites of (4 .2)

tend to

1 while the remaining ny eigenvalues of (4.2) tend to infinity. with the rate of -. dong the

E l

asymptotes defined by - X (A2 - ) . E

Corollary 4.1 : ([48, pg. 581) Consider the singularly perturbed system given by (4.2).

If ~2 exists. and if A. := ilii - A ~ ~ A ~ ~ A ~ ~ and AT2 axe asymptoticdly stable matrices.

then there exists an E* E Ri such that for dl E E (O, E * ) the system (4.2) is asymptotically

stable.

4.1. Self-Tuning Proport ional-Integral Control 68

Lemma 4.1: Consider the closed loop system

wit h

Assume that 1 exists (see Remark 4.1). and that -4 and - T K are both stable. Tlisu tliere

exist constants (a. P . E * ) E Xt x 32- x R- with the property that for every initial condition

and for every pair of bounded piecewise continuous reference and disturbance signals.

for é E (O.€'). t >_ 0.

Proof: To prove Lemma 4.1. observe from (4.3) that

eig(-<(p. c ) ) = e . eig(A(p. E ) )

w here

Rom the comments given in [48. pg. 481 and Corollary 4.1. there then exists an E' E R-

such that eig(A(p. E)) C C- for all E E (O. E'). Hence, by the continuity of eigenvdues, for

foced p 2 O. there exist constants (a. a) E R' x W' such that

for d l E E ( O . E * ) , t 2 0.


Define

Since

it therefore follows upon taking norms that

for al1 E E (O? E * ) , t 2 0.

Lemma 4.2: Consider the matrix

where i E {l? 2 , 3 , . . . ) and

€0 and À := A - p - ~ ~ f ~ . 7'

Assume that both A and -TK are stable: then for almost al1 T ) E S,

Proof: Since this proof closely follows the proof of Leninia 4 given in (641, only the major

necessary modifications will be provided. To proof Lemma 4.2, observe that

4.1. Self-Tuning Proport ionai-Integral Control 70

(wbich is assumed to exist (see Remark 4.1)) can alternatively be rewritten as

where Ï is a matrix whose elements are polynornials in a and the elements of K. and where

!2 is a polynomid in E and the elements of K with the property that

det [ SI - .A + ~ B K Ï C -BKÏ 1 L =- fin-r det +A

EÏC + E ~ D K

where

and since -i is a rnatrk whose elements are polynomial; of c and the elements of K. the

proof of Lemma 1.2 cao now continue using the identical method given in [W. pg. 5211. 0

Lemmas 4.1 and 4.2 ensble us to obtain the t'ollowing.

Theorem 4.2: Consider the stable plant (4.1) with D = O and with Controller PI1 applied

at time t = O: t hen for every 5 E n. for every bounded constant reference and disturbance

signal, and for every initial condition r (0 ) := [x(o)= q ( ~ ) T ] T for which Xssumption PI1

holds. the closed loop system has the properties that:

i) there exist a finite time t,, 2 O and a finite constant es, > O such that e ( t ) = es, for

al1 t 2 t,,:

ii) the controller states E Cs and the plant states z E C,: and

iii) if the reference and disturbance inputs are constant signals and g E TF', then for

almost all ( Q , T ) E S, e ( t ) + O as t -P oc.

4.1.2 Using no Estimate of the DC Gain

For the situation when no estimate of 7 is available? define the set of admissible controller

parameters as

- fi .- .- {( f , g , pl LI) : / E MSBF. g E TF'. p 2 O? U E Etmxrn and is nonsingular}

and let S' := {(Q, T , U) : €0 > 0, T > 1: Lr E IRmXm and is nonsingular};

with b = (f, g, p, U) E 6 and Assumption PIl' defined to be

i) il~(0)Il c f (1);

ii) Ile(0)ll < f (1): and

iii) (1 + ~ I - D K ) , i E (1: 2 , 3 , . . . } ' is invertible 7'

label Controller PI1' as

rt

for fixed p ': O (see Rernark 4.

where k E {1.2,3,. . . },

t := 0, and where, for eacli k 2 2 such that t k - L # cc: the switcllirlg time ta is defiried by

min t 3

i) t > t k - 1 , and if this minimum exists t k :=

ii) 11[~(t)~ e ( t ) T I T ~ ~ = m - 1)

30 otherwise

with

K ( t ) = UWi, z { 2 s i = ( ( k - 1 ) r n o d s ) + l , t E ( tkl tk+i] .

4.1. Self-Timing Proportional-Intemal Control

An explicit method for constructing K, := UW, for the case wlien m 2 r is given in [55]

which ensures t hat - W W , is stable for at least one j E (1 , 2, . . . . s ). s E N.

Using the same method as shown in the outline of the proof of Lemma 4.2, and following

exactly the proof given in [64, pp. 521-5223, the following result can also be obtained.

Lemma 4.3: Consider the niatrix A(p7 B , K) defined in (4.4) wliere A is stable and p 2 O

is fixed; the11 with K := { K I : j E (1,2, . . . . s)} (an explicit met hod for calculating K, is

given in [55]), for almost al1 ( r o , T. U) E Sf7

Theorem 4.3: Consider the stable plant (4.1) with D = O and with Controller P I l f applied -

at time t = 0; tlien for every b E fio for every bounded constant refererice and disturbance

signal, and for every initial condition z(0) := ( ~ ( 0 ) ~ qt(0)*IT for which Assuniptiori PI1'

holds, the closed loop system has the properties that:

i) there exist a finite time t,, 2 O, a finite constant es, > 0. and a matrix K3, such that

~ ( t ) = es, and K( t ) = Kss for al1 t 2 t,,;

ii) the controller states 71 E Cm and the plant states z E Lm; and

iii) if the reference and disturbance inputs are constant sigrials and g E TF1' then for

almost ail (eO, r ,U) E S ' , e ( t ) + O as t + m.


To illustrate th2 effect of this new switching mechanisni on a system's potential closed loop

transient response, consider the following MIMO plant taken Erom [64, pg. 5171:

4.1. Self-Tuning Proportional-Integrd Control 73

which has a DC gain given by

and a nominal transfer function rnatrix (Le. with E = O ) given by

Assume that the following estimate of T is known:

let

and set dl controller-plant initial conditions to be equal to zero at time t = O. For t his

example. do not reset controller States q ( t ) to be equal to zero immediately followiug any

controller switch.

In Figures 4.1 and 4.2. the output resporise of the closed Ioop system is shown with

Controllers P2 1141 and PI1 applied respectively. As can be seen' in Figure 4.2. a l controller

switches are due to norm bounds on e ( t ) being met or exceeded: however, although the

actual switching time instants shown in Figures 4.1 aud 4.2 are relatively close to each

other, a substantially improved transient response occurs in Figure 4.2. This result can be

attributed to the fact that Coatroller PI1 now uses an additional norm bound ( J l e ( t ) JI) in

rui attempt to detect instability.

4.1. Self-Tuning Proport ional-Intepal Control

Theoreticai conunuous tirne swiiching contmllrr output responsr.

Tmi: (seconds)

5 . Switchine ttrnc instants of the wplid controlla.

1

!

Figure 4.1: Simulated results of y1 (solid) and y2 (dashed) with Controller P2 [I-l] applied to (4.5).

10 Themucd çontmuous urne switchine controller autput mponse. i

Figure 4.2: Simulated results of g1 (solid) and y2 (dashed) with Controller PI1 applied to (4.5).

4.2. Seff-Tuning Proportional-Integral-Derivative Control 75

4.2 Self-Tuning Proport ional-Integral-Derivat ive Control

In t his section, pract icd self- tuning proportional- integral-derivat ive ( PID ) controllers of the

form [6]

will be considered when applied to the system given in (4.1) for situations involving both a

known and an unknown estimate of the steady-state DC gain matrix 7. A s in Section 4.1.

assume throughout that T has fùll row rank in order to form a tractabie problem.

Note that for constant parameters p. c. €1, Q. :V. and K. the closed loop system formed

by augmenting (4.1) toget ber wit h (4.7) c m be expressed as

where (see Remark 4.2)

4.2. Self-Tuning Proportional-Integral-Derivative Control 76

Furthermore, here? we consider "derivative" terms of the form

- P

and not of the form

since, as noted in [6, pg. 71, the reference signal yrel(t) is normally piecewzse constant in

nature. As well, the proposed first order filter (d(s)/y(s)) is used in order to limit t lie noise

sensitivity produced by the derivat ive action in any pract ical situation.

4.2.1 Using an Estimate of the DC Gain

For the self-tuning PID controller using a known estimate of 7, given by j. define the set

of admissible controller parameters as

Q P ~ D := {(f, 9.914~ p. 1V) : f E MSBF, g E TF'. gl E TF', E TF'. p 2 O. N > O } ,

iii) -'TF is stable; and

iv) (1 + p % ~ ~ + % N D K), i E { I l 2 ,3 , . . . }. is invertible for (eu, ri co2. 3) E S x S Tl T$

and for fixed p >_ O, N > O (see Remark 4.2).

With o p r ~ = (f, g,gi , g2, p, N) E ClprD, define Controller PIDl as follows:


where k E {1!2.3 ,... }, K := ?!

t l := 0 , and where, for each k > 2 such that tk- l $ OC, the switching time t k is defined by

I -. ot herwise.

Remark 4.2: Given p > O and D E Etrxrn: then for almost al1 ( E ~ . Q , N ) E IRr x Rt x Rt

and for alrnost d l matrices K E !Etmx Y (1 + pi D K + E ~ L V D K ) is invert ible. 0

Once again, the following two lemmas can be obtained via methods siinilar to those

used for Leminas 4.1 and 4.2. Iu particular. to prove Lemma 4.5' one can again rewrite f as

where Î is a matrix whose elements are polynomials in €2: and the elements of K' and

where fi is a polynomial in € 1 : €2: and the elements of K with the property that

Lemma 4.4: Consider the closed loop system (4.8) where A and -7-K are both stable.

Then there exist constants (a' 8, es) E R+ x R+ x W' with the property that for every initial

condition and for every pair of bounded piecewise continuous reference and disturbance

signals,

IIz(t) I I 5 alIz(o) I I + f l s u ~ ( l l ~ r . f ( ~ ) T ~ O I I + I l w ( ~ ) I l )

for E E (O,E*), € 1 E (O, E * ) , E* E (O, E * ) , t 3 0.


Proof: Consider the situation when ( € 1 , E-) = (0: O ) , and observe t hat

eig(-xPrD) = c baeig(AprD)

w here

Rom Corollary 4.1. there then exists an E f R' such that eig(APID) C C- for ail E E (O. Z).

Hence. by the continuity of eigenvalues. there exist constants (o. 6'. 1) E W- x B- x R I

such that. with 1 := (O? cl) .

it therefore foIlows upon taking norms that

for dl ( E . E ~ , E ' ? ) Z x Z x 2. t 3 O. 6l

Lemma 4.5: Consider the matriv Apro(p. el. € 2 : K. N ) given in (4.8) where -4 is stable

and p 2 O. iV > O are h e d : then if - T K is stable. for almost al1 (q. T. col. ri, ~ 0 ~ .

4.2. Self-Timing Proportional-Integral-Derivative Control 79

These results enable us to obtain the following.

Theorem 4.4: Consider the stable plant (4.1) with D = O and with Controller PIDL

applied at time t = O ; then for every op [ * E f l P I D : for every bounded constant reference

and disturbance signal. and for every initial condition -(O) := [Z(O)' O(0)T a ( 0 ) ~ 1 ~ for

which Xssumption PIDl holds, the closed loop system has the properties t hat:

i) t here exist a finite time t,, 2 O and constants (eSs: E , ~ , ~ , E,.$- j E R- x R' x R' such

that ~ ( t ) = E , ~ . ~ l ( t ) = E ~ ~ ~ . c 2 ( t ) = for al1 t 2 t,,:

ii) the controller states q. a E 15,. and the plant states ~c E C,: and

iii) if the reference and disturbance inputs are constant signais and g E TF'. gl E TF'.

g2 E TF'. then for almost di (Q. r-60,. q . c o 2 . -) E S x S x S. e( t ) -t O as t + x.

4.2.2 Using no Estimate of the DC Gain

For the self-tuning PID controller using no known estimate of T. define the set of admissible

controller parameters as

let Assumption PID 1' be Assumption PID 1 with condition (iii) rernoved.

With o'prD = (f > g, 91 g2:p. 1V. U ) E Q ( P I D : define Controller PIDI' as

4.2. Self-Tuning Proportional-Integral-Derivative ControL 80

where k E (1 2.3. . . . ).

t := 0: and where. for each k >_ 2 such that t k - 1 $ X. the switching time tk is defined by

O t herwise

wit h

K ( t ) = UbViv,, i E {l. 2 . . . . ..s). i = ((k - 1) mod s ) + 1. t E (tt. t k - L ] .

An explicit method for constructing the K, := CrCVj for the case when m > r is given in

[55] which ensures t hat -TKj is stable for a t least one j E { 1.2.. . . . s } . s E N.

Lemma 4.6: Consider the matrix d p l D (P . é. c 1. € 2 . K. LV) given in (4.8) wbere -4 is stable

and p > O. iV > O are k e d : then with K := {h; : j {l. 2.. . . . s)}. for almost al1 (eu r.

eo,? TI: E O ? . - :L I ) E S x S x Sr'

Theorem 4.5: Consider the stable plant (4.1) with D = O and with Controller PIDL'

applied at time t = O: then for every o>plD E O'pro. for every bounded coustant reference

and disturbance signal, and for every initial condition r (0) := [z(O)' rl(0)T a(0)'lT for

wliich Assumption PID1' holds. the closed loop system has the properties that:

i) there exist a finite time t,, 2 0' a matrix Kss, and çonstaots (c,,'a ,,,. a,,?) E Rf x

W+ x R+ such that K ( t ) = Kss? c ( t ) = ers, 8 1 ( t ) = e,,,. e2( t ) = es,, for d l t 2 t,,:

ii) the controiler states 77: a E Coc? and the plant states x E L,; and

4.2. Self-Tunine: Pro~ortional-Integral-Derivative Control 51

iii) if the reference and disturbance inputs are constant signals and g E TF'. g~ E TF',

g2 E TF'. then for almost al1 (cor .eo, . r l : ~ ~ ~ . i l , Li) E S x S x S': e(t) i O as t i p.

Remark 4.3: If LI # O and/or if yref(t) or w ( t ) are bounded piecewise constant reference

and disturbance signals. 'hen Theorems 4.2. 4.3. 4.4. and 4.5 can be rnodified. as in Tlieorem

2.2, by letting X E 8'. by defining

ef := -Xef + Xe.

and by switching based upon norm bouods on (rl(t)T a( t )= e1(t)'lT or [rl(t)T a(tlT ef(tjT

~ ~ ( t ) ~ ] ~ (assuming that el(0) < f ( 1) and uf (0) < f (1)). 0

In Theorems 4.4 and -4.5. relatively little a priori plant information and on-Iinc conipii-

tation is required in order to successfully apply either Controller PID 1 or Controller PID 1'

to a potentially unknown (not necessarily strictly proper) b1IMO open loup stable systcm.

In essence. with n. A. B. C. D. Er. or F potentially unknown. but with 'T having full row

rank and X(A) C C-. al1 proposed controllers will alrnost always provide asymptotic error

regdation and disturbance rejection with y,,f and w bounded constant signals. Furthcr-

more. in an attempt to reduce the transient tuning response. the switching criterion riow is

based partially upon the norm bound of e ( t ) .

These facts therefore compare favourably when contrasteci. for instance, witti the SIS0

results outlined in j1021. [SI. [35]. [o]. [do]. (71. and with the (strictly proper) M M 0

results given in [83]. [88]: [89]. [53]. [AS]. [52]. More specifically. it is assumed that rn = r

(wit h each decentralized control agent meauring only one plant output and manipuhting

only one plant input) and tliat an accurate mode1 representation is available in [83]: tliat

on-line manual tuning of the proportional-integral controllers occurs in [88]: that m = r.

rn 5 TL, and that input-output decoupling (in the sense of 132. pg. 6521) occurs in [89] and

[45]; that rn = r and that diagonal decentralized control occurs in [53]; and that diagonal

dominance occurs in [52]. Moreover' the scheme of [83] admits the possibility of Iiaving to

first design m! c~ntrollers~ and 3 s based on a t heorem and two heurist i d .

Remark 4.4: Consider the situation when

4.2. Self-Tuning Proportional-Integrai-Derivative Control 52

i) (f 1. f3) € CMSBF:

Then, similax to Remark 2.4, Theorems 4.2. 4.3. 4.4. and 4.5 will also hold for al1 botrnded

piecewise constant reference and disturbance signais with D not necesszily equal to z e n

and with the switching time t k given by

, min t 3

i) t > t k -L : and if t his minimum t'xists

ii) Ilq(t)ll = f i ( k - I ) and/or (4.9)

This occurs since. with Lxed admissible controller parameters. Controller PIDL can be

expressed as

[:] = [; - N I " [ [ u ] - [il !/+ [;]fief.

in its most generd form (4.10). As well. in this instance. witii p := O and € 2 := 0. (4.9)

reduces essentially to the origind definition of switctiing time t k given for Controllers 2 and

2' in [64]. 0

Remark 4.5: Theorems 4.4 and 4.5 are both equally valid for the (PI) case when E - := O

(see Theorems 4.2 and 4.3) and for the (ID) case when p := 0. 0

Theorems 2.4 and 4.5 are also valid for self-tuning PID controllers of the form [9. pg.

2221

4.2. Self-Tuning Proportional-Intemal-Derivative Control 83

where (b . N ) E Ri x P'. This follows since the closed loop system may be expressed as

(4.8) where

and

Furthermore. in this particular case. the comments given in Remarks 4.3. 4.4. and 4-21 also

hold true.


Example 1: Two Input-Two Output System

Consider once again the systern given by (4.5). As before. assume that the fbllowing estimate

of 7- is known:

let f (i) be given by (4.6). with

1 O g ( i ) := - 2i ?

( t ) := [-2 - 2IT.

and set ail controller-plant initial conditions to be equal to zero at time t = O.

4.2. Self-Tuning Proport ional-Integrd-Derivat ive Control 84

Using Controller 2 defined in [64], the output respoose shown in Figure 4.3 is obtained.

For coniparison, the results obtained by applying Controller PID 1 wit h

p := 1. and N := 1 are given in Fiogre 4.4; in this case, al1 initial conditions are also set to

be equal to zero a t time t = 0' the states of q ( t ) are not reset to zero after each controller

switch, and the same rnodified strong bounding function f ( 2 ) as defined in (4.6) is used.

As can be seen, in t his instance, the output transient response of Figure 4.4 is not iceabiy

improved over that shown in Figure 4.3. (The one switch which occurs at 0.475 seconds in

Figure 4.4 is due to the bound on [T(t)T a ( t ) T ] T . ) For further cornparison. using similar

initial condit ions and parameter functions/values (unless o t herwise noted) as in Figure 4.4'

the plant output obtained using Controller PID1' and Controller PID1 is sliown in Figures

4.5' and 4.6 respectively. In Figure 4.5,

0.2 - ((k - 1) niod 6):

and the cyclic switching action as suminarized in Table 4.1 is iniplenietited2; for Figure

Table 4.1: Summary of the cyclic switching behaviour used for Figure 4.5.

' ~ e r e , no estimatc of 7 is assumed to be knonn. 2 ~ n this case, fivc cycles through each of the six possible feedback matrices are requircd before switching

stops; once again, the states of q ( t ) are not reset to zero aftcr each controller switch.

Theoreticai conunuom urne switching conuoller output respnse.

i

J r Switchine time instants o f the applird conmlla.

I I 1

Figure 4.3: Simulated results of y1 (solid) and g? (dashed) with Controller 2 given by Miller and Davison [64] applied to (4.5).

O r Theorcrical continuous urne swirchinn conuoller outpur rrsponsr.

I Switchine unw insmts of ~hc mpliai conuoIler.

1

l

01 i O 0.05 0.1 0.15 0.7 0.3 03 0.35 0.4 0.45 0.5

Tirne (seconds)

Figure 4.4: Simulated results of y1 (solid) and yz (daslied) with Controller PIDl applied to (4.5).

4.2. Self-Tuning Proportional-IntegraI-Derivative Control 86

4.6, p is set to be equal to zero (with q( t ) not reset to zero immediateIy following the one

controller switch). Note that in Figure 4.5: switches which are due to the bound on e ( t )

or [r)(t)T a(tlTlT axe denoted by an 'E' or a .X' respectively. while in Figure 4.6. the one

switch is again due to the bound on [q(t)= a ( t ) T ] T .

Remark 4.6: For a two input-two output system. the six possible feedback matrices are

and

Example 2: Four Input-Four Output Building Temperature System

As anot her example. consider the (four input-four output) partial decentralized temperature

control problem of a multi-zone building [29] described by

x = . ~ L c + B u + E ~ T ~ ~ ~ ? (4.1 la)

y = Cx. (4.11b)

For cornpleteness. matrices A, B. C. and E are given in Section 8.2. As one can verify.

X(A) C @- [29: pg. 121 and

has full r a d .


Thco~ical conrinuous urne switchine conuolIer ourput femme. 1

-31 {

O 5 10 15 10 -5 30 35 U) 15 5 0

Time c seconds)

3 , Fdback K=W v switchine insrana.

1

6 r K x Y Et I

Y Y Y

1r Y Y Y E l Y Y Y i :1

2 LE e Y E E l b Y Y Y E d

UL J

O 0 5 I l .j 7 15 3

Timr istronds i

Figure 4 3 : Simulated results of y1 (solid) and y? (dashed) with Controller PID1' applied to (4.5).

u, Thearcticri1 connnuous trmc switchine conual1er ouiput mponsc. 1

3 i Swiichine tune instants of the ripplird conuollcr. I

01 j

O 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.1 0.15 0.5

Timr !seconds)

Figure 4.6: (p = O ) Simulated results of y1 (solicl) and y2 (dashed) with Controller PEDL applied to (4.5).


Assume that an estimate of 7, given by

is available, and let

and K := T-' ;

tiere, y,, ( t ) and 6Tmt(t) represent respectively tlie perturbed reference (tenlperattire) aiid

disturbance inputs about riominal values sel and Tm(. With q( t ) not reset to zero aftcr each controller switcli and Controller PIDl applied

a t time t = 0, and using zero initial coriditions and the parmieter values in (4.12): the

output response shown in Figure 4.7 is obtained. We note that two switches occur at

approximately 0.518 and 2.244 minutes due to tlie bound on [q( t )T ~ z ( t ) ~ ] ~ . In additiou.

since K # ( -CA-'B)- ' and dTmt(t) # 0, one tlierefore can also conclude that the coiitroller

is robust. Alternatively, upon applying the h e d integral controller [29] given by

with identical initial conditions, references, and disturbances, the discernibly more oscilla-

tory response stiown in Figure 4.8 is obtained.


Thmretic;il conunuous urne switcfiing controllri output response. J r

Time (minutes I

Figure 4.7: Simulated results of y1 (solid). y2 (dotted). (dash-dotted). and y4 (dashed) with Controller PID1 applied to (4.1 1).

Fued controller output respanst!.

Figure 4.8: Simulated results of yl (solid), y:! (dotted), y~ (dash-dotted). and y4 (dashed) with fixed integral controller (4.13) applied to (4.11).

4.2. Self-Tuning Proportional-Integal-Derivative Control 90

Example 3: Four Input-Four Output Furnace System

For a final example, consider the situation when an estirnate of 7- is assunieci to be knowri.

and w hen ( filtered) ControIler PID 1 ( using (4.10)) is applied to t Lie following niodel-reduced

(four input-four output) furnace systern taken originally from [gl . pg. 1991'':

with X( ,AR) C (C- and D R f O. Since

one c m therefore verify that

(Le. K := I ) satisfies condition iii) of .4ssurnption PIDL.

In Figure 4.9. the output response of the closed loop system iising Controller PID 1 is

shown wit b

3 ~ o r brevity, the systcrn matrices uscd arc giwn in Section B.3.

4.2. Self-Tuning Proportional-integral-Derivative Control 91

and ail initial conditions of the controller

t = O. Here. one controller switch occurs at

and plant defined to be eqiial to zero at time

0.352 seconds (due to the norm bound on [s( t )T

a(t)=IT being met or exceeded), and. similar to the earlier results. q ( t ) is not reset to zero

immediately foliowing this switch. Moreover: due to the relatively poor estimate of 7 which

is used for t his particular class of reference and dist urbance signals (117 - j l l = 1.33). and

due to the fact that

one can therefore also conclude that Controller PIDZ is indeed robust in nature.

Thmrrtrcal conunuous urne switching conuolla output rrsponse. IO.

Figure 4.9: ( D R # O ) Simulated results of (solid), y* (dotted). y3 (dash-dotted), and y,l (dashed) with (filtered) Controller PID 1 applied to (4.14).

Finally, we include for completeness Figure 4.10. which shows the output response4

'1x1 this case. no further controller snitches occur for t > 0.

4.2. Self-Tunino Pro~ortional-Intemal-Derivat ive Cont rol 92

obtained using the same controIIer, initial conditions, and parameter funct ions/vaIues as

given for Figure 4.9 for the nominal (unreduced) input-output transfer funct ion rnatriv

given by

and Figure 4.11, which shows the corresponding (integral control) output resiilts5 produced

using p := O and €2 := 0.

'NO furthcr controller switches occur for t > O in Figure 4.11.

4.2. Self-Tuning Proportional-Integral-Derivat ive Control 93

Theoretical continuous urne switching contmiIer output response. OS,

Figure 4.10: Simulated results of (solid), y;! (dotted), y3 (dash-dotted), and 94 (dasbed) with Controller PIDl applied to (4.15).

ThmreticaI conunuous timr switching conûotlrr output rtrponsr. 0.5 r

8 10 12

Time (seconds)

Figure 4.11: ((p, E Z ) = (0, O)) Simulated results of y1 (solid), y2 (dotted), y3 (dash-dotted), and y4 (dashed) with Controller PIDl applied to (4.15).

Chapter 5

The Self-Tuning Servomechanism

with Control Input Constraints

Similar to the adaptive tracking problern with control input constraints considered originally

in [60] and (671: in this chapter. the general structure of the previously proposed class

of proportional-integrai (PI) controllers (Controllers PII and PI1') is further modified to

incorporate control signal saturation constraints. As in Chapter 4. the controller presented

here at tempts to improve the tuning output transient response over t hat obtained by using

conventional integral (1) control. Unlike. however. the continuous time resiilts given in

j-161: [93] and the discrete time SIS0 settings considered in [Se]. [Y?]. [2]. [l]. and [-LI: for

example, where certain structural information typically is assumed to be known in advance.

the results presented here are given for cont inuous tirne: finite dimensional mult ivariable

systems and once again require very little a priori system information. Simulation results

as well as initial experimental studies (see Section 7.4.3) obtained when using tbis new

controuer tend to indicate that desirable improvements in the closed loop tuning response

generally can be achieved.

5 .1 Constrained Self-Tuning Proportional-Integral Control

Consider the finite dimensional LTI system given by

(5. la)

(5.1b)

5.1. Constrained Self-Tuning Proportional-1ntegra.l Control 95

where x E Rn is the state, u E Wm is the control input. y E Rr is the plant output to be

regulated, w E Wq is the disturbance, and e E Rr is the daerence between the specified

reference input y,,f and the output y. Assume that rn 2 r , that A is stable? that n. A. B.

Cl D, E, or F are not necessarily known, and restrict gr.! and w to be bounded constant

signals; let T := D - CA-' B. In addition. in order to form a tractable problem? assume

throughout that 7 has full row rank, and that. if available. the estimate of 7. T. has full

row rank.

With

and uyin < u:axT i E {1'2.. . . ? m}. define U := { u E Rm : UT'" 5 Ui 5 ilyG}. aU to be the boundary of U, UO to be the interior of U. and uC to be the center of U. Let

(b , É, A, p) E Bi x R' x Rf x K E W m X r with rank(K) = r. and define

On augmenting (5.1) together with (5.2)? the closed loop systern and equilibrium points can

be expressed respect ively as

5.1. Constrained Self-Tuning Proportional-Integral Control 96

and

where A-' (p' E: A' K) is well defined. and is given by

A-' (p , E: A? K) :=

Before proceeding, the following preliminary definitions and results are required.

Proposition 5.1: ([67, pg. 8781) With y r e j E 8' and w E Wq bounded constant signais.

there eas ts a control input signal u : [O. x) + U so that lim e f t ) + O if and only if t - a

Definition 5.1: ( [67 . pg. 8801) X function ri. : N + { K E X m x r : il KI] = 1. rank(K) = r )

is a K tuning fvnction (k E KTF) if

Remark 5.1: If rn = r = l, then k ( i ) := (-l)'+' is a K tuning function.

Proposition 5.2: Consider a cornplex matrix M E cdX'' defined by

5.1. Constrained SeE-Tuning Proportional-Integrai Control 97

w here

and A* := a - bj.

for d l admissible parameter values.

Proof: Observe that

w here

Hence' the result irnmediately follows upon constructing the Routli table [dg. pp. 164-1681

for (5.4). (7

Remark 5.2: For the case when

5.1. Constrained Self-Tuning Proportional-Integrd Control 98

with (a. X?p) E P' x R' x Rt. it immediately follows that

eig(M) C C-

det (r I - hl) = r2 - ï trace(M) + det(M).

Furthermore. since

(a t X + ,da)' - 4aX = ( A - a)' + 2a2Xp + 2 a ~ ' ~ + a"'$,

and hence

O < (a + X i ,da)' - 4aX < ( a i X + p ~ a ) ' .


Lemma 5.1 : With (A. p ) E R' x B'. and eig(-TK) C 'CA. T K E Wr". çonsider the

matrix M E R' '~ '~ giwn by

for ail admissible paramet er values.

Proof: Let

5.1. Constrained SeIf-Tuning Proport ional-Integral Control 99

be the block diagonal Jordan decomposition [37. pg. 3391 of TK. and let

e ig(7K) = :17, . . . , Ar).

Observe that

w here

hence, since

( i) eig(-TK) c C-: and

(ii) d l eigenvalues of TK must occur in coniplex conjugate pairs

the result follows upon applying Proposition 3.2 and noting Reuiark 5.2.

Theorem 4.1 and Lemma 5.1 euable one to obtain the following result.

Proposition 5.3 : Consider the closed Ioop system given by (5.3) : then with (A. p) E

IR+ x Etf and eig(-TK) C T K E RrXr . there exists a constant E' E P' such thst

for al1 E E (O, C).

Definition 5.2 : ([67]) Bounded constant input signals g , ,~ E Rr and w 6 R'l are said to

be feasible with respect to ( A , B. C, D. E. F) if

5.1. Constrained Self-Tuning Proportional-Integral Control 100

For the case when no estimate of 7 is available, define Controller Cl as

where k E ( l , Z , 3 , . . . ),

t l := 0, and where, for each k 2 2 such that t k - l # X, the switching time t k is defined by

i) t > t k - i , and if this minimum exists t k :=

ii) u( t ) E dU

I= O t herwise

wit h

and g E TF', P E TF, k. E KTF, and (b. A ) E Ri x R'.

Remark 5.3: Equation (5.3) can alteruatively be written as

where Û := pKeb. 0

A direct coiisequence of Reiriark 5.3 and Theoreni 2 of [67] is the following.

Theorem 5.1: Corisider the stable plant (5.1) with rank('T) = r. and with Coritroller C l


applied at time t = 0: then for every (6: A ) E IIPt x Pt. for every g E TF'. 5 E TF. & E KTF.

for eviry bounded constant reference and disturbance signal (yreI. w ) E 3' x R(/ wtiich is

feasible with respect to (-4: B. C. D. E. F). and for evcry initial coiidition x(0) E Rn. the

closed loop system has the properties that:

i) there exist a finite time t,, 2 0. a matrix K,,. and constants (clS. p r s ) E $2- x Ef

such that K( t ) = Kss: ~ ( t ) = es,! p ( t ) = pss for al1 t 3 t,,:

ii) the controller states q! eb E L,. and the plant states z f Lm: and

iii) if the final closed loop systern lias rio cigenvaliics lying or1 .fO. tlien e ( t ) i D as t -+ x.

As in [67], Controller Cl essentially works by detuiiing tlie control systeni via parameter

~ ( t ) each time any input control signal saturates. Since

each controler switch lessens the filter -'speed" attributed to e b ( t ) . As weIlt for property

iii) of Theorem 5.1. one can d s o show that the final closed loop system gcnericnlly will have

no eigenvalues lying in CO.

Remark 5.4: ([ü7. pg. 8811) If a good estimate of 7. T. is availablc by. for instance,

conducting rn steady state experiments [21] on the open loop system. then one can set

K ( t ) := ? assuming that -7-fl is stable. Except in the SISO case. Iiowcvcr. now the set

of admissible gr,/ E Wr and w E g4 does not include tlie set of al1 feasible pairs: however. if

(%, j , w ) are feasible. tlien it follows that (yreI. w ) are admissible provided tliat III< - ( 1

5.2 Simulation Results

Example 1: SISO Nonminimum Phase System

Consider the SISO nonminimum phase stable plant [67! pg. 8831


whose nominal transfer function is aven by

where

Assume that (umfn. umm) = ( -5 .5) . and set

( y . w ) := (2.1) .

and x ( 0 ) O.

In Figures 3.1 and 5.2. the output responses respectively obtaincd iipcin applying Con-

trollers 2 [67] and C l to ( 5 . 5 ) are sliown for the above paranieter fiiiictiotis and values. 4

Observe that in both instances. the final closed loop systenis are stabie (E,, = - and '> .5 - il

e,, = ;iTT respectively). reference tracking aiid disturbance rejection occiirs. ;md the control - input constraint pIaced on ~ ( t ) is satisfied for d l time. Furtherrnore. despite the relatively

smaller value of E,, in Figure 5.2 when compared with that of Figure 5.1. tlie transient

responses of both figures are roughly comparable in nature.

ExampIe 2: Three Input-Three Output Distillation Column

-4s another example. consider a MIM0 minimum phase binary distillation tower with pres-

sure variation [20]. whose mode1 is stable and is obtained by a Iiueariïation of the system

about a standard operating point. Let tlie t h e e control s i p a l inputs be tlic rcboiler s t e m

temperature ui ( O F ) , the condenser coolant temperature uz ( O F ) . and the refiux ratio un;

the three outputs to be rebgdated are the bottom product composition y1 (mole Baction of


IO, Thromticai continuous rime switching controller outpur mponse. 1 ,

Timr (seconds)

IO r Conuol stem1 u versus rime. (

Figure 5.1: Simulated results with Controller 2 1671 applied to ( 5 . 5 ) .

IO i Theore!ic;il continuous urne swttchine rontroller ciuipur rcsponse.

A

Tirnc t seconds)

'0; Conml simal u v m rime. Conml siend u versus rime.

'O'

Figure 5.2: Simulated results with Controller CI applied to (5.3).


the more volatile component in liquid phase). the top product composition (mole fraction

of the more volatile component in Liquid phase). and the pressure 93 (atmosphere) [67, pg.

8843. With primary disturbance input w (mole Eraction of the more volatile component in

Iiquid phase) âssociated with changes in the input feed composition stream of the system.

restrict

and assume. after using the methods outlined in [21]. that the following estimate of 7 to

two significant digits is available:

Now with system matrices (A. B, C. D. E. F) given in Section B.4.

K( t ) := ? (see Remark 5.4).

(6. A ) := (2. LOO).

and x(0) O.

the results shown in Figures 5.3 and 5.4 are respectively obtained upon applying Controller

2 1671 and Controller C l to the distillation tower. In these figures. y ( t ) and ui ( t ) . y ( t )

and u2( t ) , and y3 ( t ) and un(t) are represented as solid, dashed. and dCuh-dot ted lines.

Observe that in this instance, the transient response shown in Figure 5.4 is discernibly

improved over that shown in Figure 5.3. In addition. in both of these cases. the final 0.5 0.5

closed loop systems are once again stable ( c , , = - and c,, = - respe~tiveiy)~ reference 27 35

tracking and disturbance rejection occurs. and the control input constraint imposed on u(t)


is successfully maintained for al1 time.

Example 3: Four Input-Four Output lblulti-Zone Building Control

As a final example, consider the ILIIMO minimum phase part iaI decentralized teruperat ure

control problem of a multi-zone building whose stable mode1 was given earlier in (4.11). Let

g ( k ) :=

p f q :=

( b J ) :=

Y,&) :=

&Tm, (t) :=

and x(0) =

and restrict

Upon using these values and on applying Controllers 2 (671 and C l respectively to (4.11).

the output results shown in Fioves 5.5 and 5.6 are obtained, where y l ( t ) and ur(L). y2(t)

and u2(t), y3(t) and u3(t) , and y4 ( t ) and u l ( t ) are represented as solid. dashed. da&-dot tedt 10

and dotted Iines. Again, as eupected, the final closed loop systems are stable (E,, = in d

bot h instarices) , reference tracking and dist urbance rejection occurs, and the control input

constraint imposed on u(t ) is successfuily maintained for al1 time.

5.2. Simulation Resdts 106

Theoretical ccntinuous time switchine controlier output nrponse. 0.7 1 t

I l . i

= O J t - . , 1

C - . .

JJ - _ _ - - , . .---- . , . --- . . . . . - - - . .

1 - -- - . . - 1 . .

. . I I

l

-0.7 1 O 0.5 I 1.5 2 2.5 3 3.5 J 4.5 5

Time (seconds l 1 lo4

Coniml siend u vernis tirne. 2 , Cunuol s i e d u versus rime. 2 [ I t

, . . , . . . , . . . . . .. . .-. , , , ' < . .,. .. . .--

ii . , - ..-..--..-.---.-.-A . . . . . 1 ,, . . -. : .. : . -. . -- -1" l

1

-7 -0 500 iooo O 5

Timr (seconds) Tirne (seconds) s IOJ

Figure 5.3: Simulated resdts with Controller 2 [67] applied to a distillation tower.

O. 1 l-t~eoretical çonunuous rime switchinr conuoller output rcspunsc.

Conrml siend u vmus tirne. 2 Contml siend u vcnus rime. 11 I l

Figure 5.4: Simuiated resdts with Controuer C l applied to ü distillation tower.


Theoreticd cononuous u n e swilching connolla output response- . . . 7 I

O 10 20 30 JO 50 Ml

Time (minutes)

Figure 5.5: Simulated results with Controller 2 [67] applied to (4.1 1 ) .

Conuol signal u venus rime. 2 - Convol s i e d u versus time. 2

J : Thcoreiicai conunuous lime switchine conuoller output responsr. _ . - . _ _ 1

-J O IO 20 30 JO 50 60

Timc (minutes)

1

Conuol sienal u venus time. 2,

. - _ - _ . . . 3

-7 -2 -0 5 O 70 JO 60

Time (minutes) Time (mrnuces)

- _ - . 1

Canuol si na1 u venus rime. 2i

- . .

Timr (minutes) Time (minutes)

Figure 5.6: Simulated results with Controller C l applied to (4.1 1 ) .

Chapter 6

Adaptive Tracking of LTI MIMO

Systems

Using the metliods and results presented in Chapter 3: we consider now the finite dimen-

sional strictly proper MIMO adaptive tracking problem for the general class of reference and

disturbance signals wliose behaviour can be described by a Iinear cornbinatiori of botinded

piecewise continuous sinusoidal and polynomial functions. Ouce again. by monitoriiig plant

output y ( t ) and/or error signal e ( t ) the emphasis will be to provide a robust controller

which is insensitive to bounded piecewise continuou disturbances w ( t ) and whicli attempts

to provide au acceptable transient resporise. Initial simulation studies usirig t hese new

controllers tend to indicate that such desirable improvernents in the tuning response usu-

ally can be achieved rvtien compared with. for instance. the outputs obtained usirig the

computationally more intensive algorithms presented in 1591 and (661.

6.1 Preliminary Definitions and Results

Similar to Section 3.2, consider the finite dimensional strictly proper (stabilizable and de-

tectable) MIMO LTI system given by

x = -42 +Bu+ Ew,

y = Cz+ Fw,

e = Y r e j - Y

( 6 . l a )

(6.1 b)

(6. l c )

6.1. Preliminary Definitions and Resuits 109

where x E Rn is the state? u E Rm is the control input, y E Rr is the plant output to be

regulated, w E Rq is the disturbance, and e E Pr is the difference between the specified

reference input y,,~ and the output y. In addition? assume that n. -4. B. C. E. or F are

no t necessarily known. P

Let a(s ) := C aisL with p E M and np := 1. and let the roots of a ( s ) lie in C+ U CO: as r=O

weU, restrict

The control objective in this chapter will be to stabilize the system given by (6.l j such that

asymptotic reference tracking and disturbance rejection occurs for the class of yref(t) and

w(t ) defined by (6.2). Thus, in order to form a tractable problem [27]. assume too t hat

(i) m 2 r: and

(ii) the transmission zeros of (C. -4. B' O ) do not coincide with the zeros of a(s ) .

To form the servocompensator. consider the situation where matrices .i E RPXP and

B E RP are chosen so that det(s I - A) = ~ ( s ) and (.A. B) is controllable. Usirig [-71. defirie

.A8 := block diagonal (A.. . . . A) E RPrxPr .

B. := block diagorial (B.. . . . fi) E i lprxr .

and let the servocompensator < be given by

Combining plant (6.1) together with servocompensator (6.3): the augmented system can

therefore be written as

6.1. Preliminarv Definitions and Results 110

w here

Lemma 6.1: ([-SI) Given the plant (fi 1). the servocompensator (6 .3)- and the augnented

sys tem

then (A. B) is stabilizable and (G. -4) is detectable.

Due to the structure of the constructed servocompensator E , uny LTI controller wliich

stabilizes (6.4) will provide asymptotic error re ylation and dis turbance rejection [27].

Therefore, using the results given in Lemma 6.1. let the adaptive stabilizing controller

for (6.4) be of the form

6.1. Preliminary Definitions and Results 111

In addition, note that upon applying controller (6.5) to system (6.4). there always exists a

gi 5 n such that the final closed loop system is stable.

(i) {h(i) : i E W} is dense in for 6xed values ofgi E NU{O}. p E .lu{O};

and

(ii) for h(i) := 1 :: J p ~ . r - ~ r - g , ) ,

for constants ( r l : TV) E $3- x Sf.

Proposition 6.1: Given gi E N U (O). p E N u {O). there exists a h E GCTF.

Definition 6.2: Given that o! E NU { O } is a lower bound and that -y E .I u {O} is an

upper bourid on gi such that CI 5 gi < y with a < 7. a function h : M i ~ ( ' ~ - " ) ~ ~ - ~ ~ ~ g ~ )

is a modified yenerul controiier tuning Junction (h E GCTF') if. with (g,+ - gi) E {O. l}.

gi := cr? h(i) := Ri, and p E NU {O}. the following properties hold:

(iii) for h( i ) := 14 , - ~ ( - - g , ~ ~ ~ r + p r - g , ) .

for constants (rl, n) E R+ x R+.

Proposition 6.2: There exists a h E GCTF'.

Using the adaptive stabilization resuits of Chapter 3, the following results are obtaiued.

6.2. Using a Known Value of the Compensator Order 112

6.2 Using a Known Value of the Compensator Order

For the case whcn it is known that compensator (6.5) will stabilize the augrnented system

(6.4) with gi = 1 , 1 E N U {O}, using some appropriate clioice of controller matrices Gi, Hi,

Ki, Li, define Controller Tl as

where k E (1: 2: 3 , . . . }: h E GCTF.

t l := 0, and where, for each k 2 2 such that tk-1 # ccl the switching time tr; is defined by

I min t 3

if ttiis rniiiiniurri exists

ot herwise

with f E S2BF. Label Assumption T l to be

i ) IW) 1[ < f (1); and

Theorem 6. l: Consider the systern given by (6. l), and assume t h tliere exists a solut ion

to the robust servomechariism probiern for the class of reference and disturbarice signals

defined by (6.2); then with the corresponding servocompensator (6.3) implernented, and

with Controller T l applied at time t = 0, for every f E SSBF and h E GCTF, for every

bounded reference and disturbance signal of the form given by (6.2), and for every initial

condition [x(o)* ~ ( 0 ) ~ q ( ~ ) T ] T for which Assumption Tl holds, the closed loop systeni lias

the properties that:

6.3. Using no Known Value of the Compensator Order 113

ii) the controller states <, 77 L,, and the plant states x E L,; and

iii) for almost al1 controller parameters (G1 H, K, L), the final eigenvalues of the closed

loop system will lie in C.

6.3 Using no Known Value of the Compensator Order

For the case when it is known that compensator (6.5) will stabilize the augrnented system

(6.4) with gi = 1 using some a 5 1 5 y with a, y E N U {O): cr < y' and using some

appropriate choice of controller parameters Gi, H*, Ki , Li: label Controller Tl' to be

Controller Tl, but with h E GCTF'.

Theorem 6.2: Consider the system given by (6.1) , and assume that t here exists a solution

to the robust servomeclianism problem for the class of reference and disturbancc signals

defined by (6.2): t hen wit h the corresponding servocomperisator (6.3) implemented. and

with Controller Tl' applied a t time t = 0. for every f E S2BF and h E GCTF'. for every

bounded reference and disturbance signal of the form given by (6.2): and for every initial

condition [x(0)= ~ ( 0 ) ~ q ( ~ ) T ] T for which Assumption Tl Iiolds. the closed loop system has

the properties that:

ii) the controller states c , I) E Lm? and the plant states x E L,: and

iii) for almost al1 controller parameters (G, H , K, L). the final eigenvalues of the closed

loop system will lie in @-.

If A is stable and if a ( s ) = s, then the controllers proposed in Chapter 4 of this t hesis

can be used to achieve asymptotic reference tracking and disturbance rejection for t his

particular class of signals. As well, upon filtering Y ( t ) as

6.3. Usine: no Known Value of the Compensator Order 114

and upon making corresponding changes in t fie assumpt ion and controller defini t ions (e.g.

see Theorem 2.2 and Rernark 3.5), Theorems 6.1 and 6.2 will also work for those refer-

ence and disturbance signals whose behaviour can be described by a linear combination of

bounded piecewise continuous sinusoidal and polynomial functions. Furtherniore, the com-

ments given in Remark 3.4 are equally valid for for Theorems 6.1 and 6.2. In this instance,

by using more system information, and tience, by reducing the parameter space scarch for

a stabilizirig h ( i ) , one would therefore also expect an improved output transient response.

Remark 6.1: ([21]) If A is stable in (6.1) and the roots of a ( s ) lie in @O. tlieo there exists

a gain K E IRmxpr so that Â + B[O K]C is stable. Hence? if plant (6.1) Iias an l'th-order

s tabih ing compensator, then the augmerited system given by (6.4) does as well. 0

One can dso combine plant (6.1) together with servoconipensator (6.3) to form the

augrnented system

A* z* B* E* fi*

where (A*, B*) is stabilizable and (Cu. A') is detectable. Hence. Controllers T l and Tl'

will also work iising the switching mechanism defined by

niin t 3

i) t > t k - ~ , and t,, :=

ii) llE(t)ll = f (k - 1)

00


O t herwise,

where

6.4. Simulation ResuIts L 15

assuming that lle(0)11 < f(1) and 11<(0)11 < f (1) also hold at time t = 0.

6.4 Simulation Resuits

Example 1: SISO Unstable Minimum Phase Plant

Consider the (controllable and observable) unstable minimum phase SISO plant [66. pg. 571

with open loop eigenvalues of -2 and 1. and assume that there exists a solution to the

robust servomechanism problem for constant references and constant disturbances. Let

.y,&) := 10' w ( t ) := 1.

3.5k. 1 5 k 5 1 5 f (k) :=

30(k - 15)' exp((k: - 15)". k > 15.

and set al1 controller-plant initial conditions to be equal to zero at time t = O. Since

rn = r = 1 and a ( s ) = .S. a choice of (A*. B * ) = (0.1) yields an appropriate corresponding

servocompensator ( for the particular class of reference and disturbance signals chosen.

Assume that the given plant (6.6) is unknouin. but that it is known that there exists

a zero'th-order stabilizing compensator for the open loup system. In addition. using the

comments given in Remark 6.1 and the results given in [21]. it sutnces to search for a

stabilizing feedbadc matrix Li in the set P x [- 1.11; hence, define h( i ) as


and note that for

the closed loop system is stable if and only if (Z11?lL2) lies in the regions defined by

1 - l L L > 0:

LI, > 0.

and 1:1 + l i i ( l - i l2) - 2 > O.

30, Theoretical contmuous rime swirchinq contmllcr reniirs. 1

1 - s 10; h t

10. Switchine rrmr instuits of conuollrr k t - v . \

l

Figure 6.1: Sirnulated results of g ( t ) with Coutroller Tl applied to systeni (6.6) using (6.7). (Controllers which are applied due to a previous bound ou < ( t ) or y ( t ) being met or exceeded are marked by a 'X' or -Y7 respectively.)

In Figure 6.1, simulated output results are shown using Controller T l with the given

initial condit ions and parameter functions/values defined earlier. (Controllers which are

applied due to a previous bound on y ( t ) or f ( t ) being met or exceeded are marked by a 'Y?

or 'X' respectively.) As expected. Controller T l eveutually stops switching, and L,, = (-2

1). In addition, i11 this instance, e ( t ) -+ O as t -+ m. and the respouse shown in Figure 6.1

is noticeably improved over that shown in Figure 1 of (661.


Example 2: SIS0 Stable Nonminimum Phase Plant

As a second example, consider the (controllable and observable) stable nonminimum phase

SIS0 plant 1501

(with open loop poles at -1 and zeros at 0.5) whem it is assumed that (6.8) is known only

to be stable' and that a solution to the robust servomeclianism problem exists for coristant

references and constant disturbances. Using the results given in Rcrnark 6.1 and [?Il. define

h( i ) as follows:

With

the closed loop system is stable if and only if


- Upon defining (A*. B*) := (0.1). gref(t) := 1. w ( t ) := sin(2t). F := 1. and

with al1 controller-plant initial conditions set to be equal to zero at time t = O. the output

response shown in Figure 6.2 is obtained using Controller Tl. In this instance. although

the transient magnitude is comparable to that given in Figure 2 of [59]. the response shown

here is much more sluggish in nature: however. as expected. the controller is indeed robust

and eventually stops switching (L,, = [O 0.51) even with a sinusoidal disturbance.

3 Theoretlrril continuou time switchinc rontmller rcnilrs. I

-0 10 20 30 Ul 50 60 70 $0 40 100

Time (seconds)

8 * Switchine urne instants of controilcr L=L-V.

O 5 10 15 10 '5 3 0

Time iswonds)

Figure 6.2: Simulated results of y ( t ) with Controller TI applied to (6.8) usiag (6.9) .

Example 3: SISO Unstable Minimum Phase Plant

As anot her example, consider the (controllable and observable) unstable minimum phase

SISO plant [59]


with poles given by -2 and 1: assume again that the plant (6.10) is i~nknouin. t hat there

exists a solution to the robust servomechanism problem for constant reference and constant

disturbance signals. and t hat t here elusts a zero'th-order stabilizing compensator for the

augmented system (6.4) of the forni

With u given by (6.1 l ) , the closed loop system is stable if and o d y if

E (3. x).

and let (A*. B*) := (0,1), grel(t) := 1.

1 g i Y g o and f(k) :-

15(k - 20)' exp ( (k - 20)"). k > 20:

with d controller-plant initial conditions set to be equal to zero at time t = 0. the output

response given in Figure 6.3 is obtained upon applying Controller T l . As anticipated.

Controller T l eventudy stops switching (L,, = [O 41): and, in this case. the transient

response is also improved over that shown in Figure 3 of [59].


Timr ( sesonds )

Figure 6.3: Simulated results of y ( t ) with Controller TL applied to (6.10) usirig (6.12).

Example 4: MIS0 Unstable Minimum Phase Plant

Finaily, consider once again the (stabilizable and observable) unstable minimum phase

MIS0 p l u t [62. pg. 6051 given in (3.8). Assume that the plant (3.8) is unknoiun. that there

exists a solution to the robust servomechanism problem for coristant reference and constant

dist iirbance sipals. and t hat t here exists a zero't h order s tabilizing cornpeusatoc for the

augrnented system (6.4) of the form

One can verify that wit h u given by (6.13). the closed loop system is stable if and onIy if

and (0.1 +O.llll +112+4122)(-2-11L l L 2 > 0.


Let

(A*' B') := (O. l )?

~ l r e f ( t ) := 1:

w ( t ) := sin@),

E := 0,

F := 1.

and set al1 controller-plant initial conditions to be equal to zero at time t = O. Define h(i)

as


One can veri& t hat for the given listed values of h(.i),

h(2) : h(10). and h(20)

form stable closed hop systems.

In Figure 6.4. output results are shown using Controller TI with the prcvioiisly defined

parameters and funct ions. Xgain. as anticipated. the controller eventudly stops switching?

wit h

and the controller is indeed robust even with a persistent. sinusoidd disturbance. For com-

parison. using h( i ) defined as


the larger output transient response given in Figure 6.5 is obtained. and

6.4- Simulation Results 124

Thromid continuous tirne switchine contmllrr mula. 1

Timr 1 seconds)

Figure 6.4: Simulated results of y(t) with Controller T l applied to ( 3 . 8 ) iising (6.11).

100 Thr<lrrncJ continuaus tirne swrtching i o n m l k r wulrs.

Timr (seconds)

Figure 6.5: Simulated results of y(t) with Coutroller TL applied to (3.8) iising (6.15)-

Chapter 7

Experiment al Result s

In t his chapter. the robust self-tuning PI and PID controllers presented iri Chapters 4 and

5 (for cases involving both a known and an unknown estimate of the steady-state DC gain

m a t r ~ ~ T) will be implemented and examined on a multivariable industrial system cailed

LUIARTS (Multivariable Apparatus for Red Time Control S tudies). located at the Systems

Control Group. Department of Electrica! and Computer Engineering. University of Toronto.

This experimental equipment cousists of a collection of industrial ac t iiators rmd sensors to

produce a highly interacting multivariable system. hforeover. using MARTS. it wil! be

shown how these new controllers possess certain desirable integrity féatures in spite of goss

changes which may occur unexpectedfy to the ,LIARTS configuration.

7.1 Experimental Apparatus

The MARTS facility comists of an intercomect ion of industrial conimercial actuators and

sensors which monitor and control a nonlinex hydraulic dynamic process. Although there

exist many different control configurations for this apparatus. the system which we will

primarily be concerned with is depicted in Figure 7.1. Here, the control objective using this

particular arrangement will be to regulate the level of both column heights, if possible, for

al1 initial conditions and disturbances applied.

In t his system, the by-pass, drainage, and interconnecting vaives are al1 adj ustable

rnanually to enable the selection of desired equilibrium column heights, and to control the

degree of interaction existing between both columns. Xctuator valves for both columns abo

enable one to individudly apply positive constant disturbances wi and .wz. To mesure

7.1. Experimental Apparat us 126

By-pass Valves ControI Valves,

~ c t u a t o r Valve (to produce

disturbances)

Output 91 l Column 1 i Column 2

L (to produce

dis t ur bances)

7 Tub Valves -

Figure 7.1: Schematic diagram of one possible SLARTS arrangement (not to scale).

both column heights (through the use of a curent-pressure transdiicer) and to control the

actuator and control valves (by using a current-pneumatic actuator). a Texas Instruments

(TM W O / 101/bIX-1) real time digital computer using the PDOS operating system with an

addit iond 12 bit analog-digital (-A/D) and digital-analog (D / -1) board (not s hown) is used.

Valid inputs u to the control valves are t herefore restrictcd numerically CO the intervai

O (shut) 5 u 5 4095 (fully open)!

while column height measurements h generalIy fali in the range (due to the sensors' cali-

bration) of

1200 (bottom) < h 5 4095 (top).

Physically, this later interval corresponds to a height range of about 1.3 meters. or. equiv-

alently, to approximately 0.4 mm. per digital division of resolution.

In Table 7.1, a listing of some of the va.rious major components of the LIARTS appa-

7.2. Linearized Model of MARTS 127

ratus is given. Note that as the interconnecting valve angle increases from O" to 45'. the

ry

Control valve transducsr 1 Foxboro E69P current-to-pneumatic valve posit ioner 1

By-pass vdve Interconnec t ion valve Pump Level sensor Control valve

Table 7.1: Summary of the major cornponents of MARTS.

Part Drainage valve

1/2 inch globe valve 1/2 inch bal1 valve Iwaki centrifuga1 pump (1/ 12 HP) Taylor 3400T Series pressure differential transmit ter Foxboro V4A 1/2 inch linear valve

plant becomes increasingly difficult to controII and that due to the valve transducers used,

bot h coutrol valves utilize a mechanical feedback Linkage to give an equilibriurn valve po-

sition that is proportional to t lie input current signalL. Additional informat ion concerning

the equipment used, the system startup procedure, as well as ottier possible experimental

configurations can be found in [22].

Since niariy commonly used controller synthesis design techniques need an accurate

representation of the actual plant, and since conventional adapt ive cont rollers still typically

require specific plant information (e.g, the order of a stabilizing controller. an upper boiind

on the order of the plant, the relative degree of the plant) in order to c a r a n t e e acceptable

controlIer performance, the variable structure of MXRTS preseuts an ideal situation for

one to implement and examine the robust self-tuuing PI and PID cotitrollers proposed iri

Chapters 4 and 5.

Descript ion 1/2 inch globe valve

7.2 Linearized Model of MARTS

In order to denionstrate the general difficuity and uncertainty ili obtainirig an accurate

mode1 representation of an unknown system. consider the interconnectioii structure of

MARTS shown in Figure 7.2.

Let i E {l ,2) , and define

ui := input to controi valve z,

'This occurs as opposed to using pneumôtic feedback to give an equilibrium output pressure that is proportional to the input current signal.

7.2. Linearized Mode1 of MARTS 128

m -4 1 .-h - - -

li-iterconnection

e

-A2 Column 2

Figure 7.2: Cross sectional view of the interconnected columns of !vIARTS (not to scale).

measured output (height ) of column i.

cross sectional area. j E { 1.2.3} .

acceleration due to gravity.

liquid Lieight in column i.

input flow rate to colunin i.

output flow rate from columri i.

interconnection vaIve angIe (O0 5 19 5 90').

coefficient of discharge.

coefficient of discharge with respect to 8.

O (i.e. 8 = 0° e interconnection valve is shut).

Here, the time delay occurring between the input signal ui ( t ) and the output flow rate Q i ( t )

is ignored, and it is assumed t hat any actuator valve nonlinearit ies have been eliminated

(by using, for instance, nonlinear compensation gains).

Defining

7.2. Linearized Mode1 of MARTS 129

where hi, and ui: are. respectively. tlie steady state lieigtit of aiid input to coliinin i and

control valve i! the following linearized niodel of MXRTS can be obtained for the case wfien

hls > h2s [l4]: [511:

w here

In (7.1), is the perturbed liquid level of column i (i E {1.2}). and (hli is the perturbed

input liquid flow rate froni control valve i entering coIumn i.

Remark 7.1 : .A similar derivation for the case when hl, < h2, yields t Lie sarrie basic

format for (7.1): with

being tlie only slight modification needed. In addition. for ttic case when hl, S i~? , arid

B # O". one can also show [ I I ] that tlie systern beliaves as a single colttmn apparatils obeying

the equations

where X := 6hi + Sh2: and where the (redistic) assumption is made tliat 0 := Pi % h.

Hence, (7.1) may be seeu to be equally valid under this partictilar condition. 0

Using the experimental apparat us with the interconnection valve angle B set at 30" : (Bi !

7.2. Linearized Mode1 of b1ARTS 130

,&? ~ ( 0 ) ) were found experimentally to be (0.0237.0.0260.0.0135). which implies that the

CVIARTS system is described approximately as

In this case, one can also verify that -0.0248 and -0.0519 are the stable eigenvalues of A.

Alternatively, using singular vdue analysis and the model identification and reduction

algorithm given in [50], a discrete îime model given by

( k ) = H.c(k) .

corresponding to a sampiing time period T of 2 seconds. w a

for 8 = 30". where:

experimentally identifid [Id]

with the stable eigenvalues of .? given by 0.98 11 and 0.9609.

The above results show the generai difficulty in obtaining consistent mathematical mod-

els for an unknown system: although both models (C. -4.8) in (7.1) and ( H . 3. G) in (7.3)

approximately describe the beliaviour of the system, the models are. in fact. not consistent

with each other (e.g. with T = 2, ~ ( e ~ - ' ) = {0.901,0.952)).

Remark 7.2: Due to the apphed nonlinear compensation gains (to remove any actuator

valve nonlinearities) and the mechanical feedback nature of the control valves, any adrnis-

sible value of dui calculated by a control algorithm can be used directly to control the

actuator valves [44, pp. 2542591. 0

In t his chapter, unless otherwise stated, the models listed above will nat be used in any

mamer to obtain the experimental results which will be presented in Section 7.4.

7.3. Conventional Controller Design Results 131

7.3 Conventional Controller Design Results

In this section, experimental results obtained by using the high performance controller

design met hods given in [28j and [25] (for constant reference and disturbance inputs). which

both require that an accurate mat hematical mode1 of the system be available, are presented.

In particular, upon using the MARTS mode1 given in (7.2) for 8 = 30°, and the performance

index [28]

wit h é E the optimal controller which minimizes (7.4) is given by

where x := [bhi 6h2IT, e := - r), and y, ,~ is the desired reference tracking signal:

hence, with E = 1. the optimal controller obtained for the MARTS system is

where

When controller (7.5) is digitally implerrrented on the MAARTS system usirig a sarnpling

time period T of 0.4 seconds, a refereuce input signal of

1 (3500,2000), 1500 5 t < 2000 seconds,

and an interconnection valve angle 0 of 30": the experimental results prese~ited in Figure

7.3 are obtairied. As can be seen, in this instance, the controller (7.5) provides excellent

7.3. Conventional Controller Design Results 132

performance, and the desired control objective is achieved. (The sensitivity exhibited by

column height ~ ( t ) when g:ej(t) = 2000 can be attributed to the (relatively small) value

of E used in (7.4), and hence, to au undesirably Large closed loop system banrlwidth.)

Expenmental ( o p ù m l conuallrr) resuits for qsi lon=l. 3600 i

1 34001

Figure 7.3: Experimental proportional-integral results of y1 (dotted) and y2 (dashed) with T = 0.4 seconds, 0 = 30°, and conventional controller (7.5) applied to the MARTS system.

However, when an unexpected event occurs usiiig the conventional controller (7.5), catas-

trophic and unacceptable results may, and alniost certainly will. occur, as demonstrated in

the sample experimental output response shown in Figure 7.4. Here? tlie followiiig goss

change in the MARTS configuration was niade at t = 1000 seconds:

With the controller (7.5) (wliich is designed for the case when (O , 6) = (30°, 1)) im-

plemented on the MARTS system, the plant's configuration was suddenly clianged a t

t = 1000 seconds by reversing output leads y , ( t ) and y2(t) witli the reference input

signal given by

(3500,2500), O 5 t < 1000 (3;. j (t) 1 t ~ : ~ j ( t ) ) :=

(2500,3500), t 1 1000 seconds

applied.

7.3. Conventiond Controller Desim Results 133

Figure 7.4: Experimental proportional-integral results of y1 (dotted) aud y2 (dashed) with T = 0.4 seconds, 6 = 30°, and with the outputs reversed at t = 1000 seconds, showing failure of convent ional controller (7.5).

Figure 7.5: Experimental proportional-integrd results of y1 (dotted) and g2 (dashed) with T = 0.4 seconds, O = O", and with the outputs reversed a t t = 1000 seconds, showing failure of convent ional controller (7.5).

7.4. Switching Controller Output Results 134

As can be seen, in this situation, the closed loop system fails to track (7.6) for such a

severe codguration change: indeed, even with 8 = O". Figure 7.5 shows that a sirnilar

problem still occurs on applying controller (7.5) with T = 0.1 seconds to the MARTS

system. Unfortunately. such failures of this type r'if controller ;ire not unexpected. since

drastic changes in the plant have occurred at t = 1000 seconds.

It will now be shown that one class of the self-tuning robust servornechanism controllers

proposed previously does not have this limitation in the sense that these specified controllers

can readily adjust to severe plant contiguration changes'.

7.4 Switching Controller Output Results

As evidenced in Section 7.3. conventional controller (7.5) is unable to carry out its specified

mandate when gross structurai changes occur to the MARTS configuration. Hence. consider

the following definition of intelligent coutrol adopted from [23] and [17].

Definition 7.1 : An intelligent controller for a plant is a controller wliicli successfully

carries out its mandate for the plant's nominal operating conditions. as well as for any

unezpected (i.e. ,unplanned) events which may occur.

Mt hough Definition 7.1 is loose in the sense t hat the wording mi& not bc considered to

be well defined. the flavour of the definition is clear. For example. if one drs igns a controller

so that it successfuily controls the nominal plant -,4f" as well as when the plant is in some

failure mode .X'? Say. then the controller has the property that it displays certain integrity

leatures (which is very desirable). but the controller is not intelligent: the controller is

intelligent only if it successfully controls the plant. either in nominal mode or failure

mode .X'. in spite of the fact that failure mode 'K' was not unticipated in the design of the

controuer.

In this section, the self-tuning proportional-integral (PI) and proportional-integal-

derivat ive (PID) controllers presented earlier in Chapters 1 and 5 will be implemented

and exmined on the MARTS apparatus. Using almost no a priori system informat iono in

Sections 7.4.2: 7.4.3, and Appendix C, Controllers PID1' (Theorem 4.5) and Cl (Theorem

' ~ u e to the nature of the controllers considered, plant configuration chimges whicli maintain the assumptions that eig(=l) C @- and rank(7) = r only will be considcred.


5.1) will be shown to possess intelligent-like properties. Furthermore. where applicable, the

generalized switching crit erion defined by

i) t > tk-l$ and if t bis minimum exists

ii) I l [q~( t )~ a ( t ) T ] T ~ ~ = fI(k - 1) and/or

ot henvise

will be used in the experiments.

For brevity, the details concerning various important practical issues (e.g. saturation

constraints and integrator "windup" of both ~ ( t ) and a ( t ) (but not ef(t))) will not be

repeated here, but can instead be found in [14]. As well. similar to [l-l], a zero order hold

is also used to obtain

and

as the discrete time equivalerits to the continuous time systen~s giveri by

and

eI = -Xej + Xe, X E W+

respect ively.


7.4.1 Using a Known Estimate of 7

Since a n estimate T of the DC gain matrix 7 can be obtained by performing two steady-

state experiments on the KARTS apparatus. with 8 = 30". consider the following ernpirical

measurement of T [14]:

observe that

and that. from Remark 4.6?

Hence. in an attempt to maintain some consistency in norm values with those candidate

feedback matrices to be used later in Section 7.4.2. define

W i t h Controller PID 1 a p plied using

T := 0.75 seconds.


Figure 7.6: ( N = 3) Experimental proport ional-integral-derivat ive results of y1 (solid) and g2 (dashed) with 6, = 40' and Controller PIDl applied to the MAARTS systern.

Figure 7.7: (N = 5) Experirnental proportionai-integrai-derimtive results of y1 (solid) and y:, (dashed) with 0 = 40° and Controller PID l applied to the MARTS systern.

7.4. Switching Controuer Output Results 138

(:3000.2500). 1200 5 t < 1500

(2500.2000). t 3 1500 seconds.

p := 20. (7.8d)

and X := 10. (7.8e)

the output responses shown in Figures 7.6 and 7.7 are obtained for the case when LV := 3

and N := 5 respectively. In both examples. q ( t ) is not reset to be eqiial to zero immediately

following any controller switch, and. as expected. the controller eventrially stops switching

(a total of three switches occurs in each figure). In addition, altliough the intercon~iection

valve angle 0 is now set to be equal to -10". tracking of the given rcference Iieights occiirs.

and the controller is indeed robust.

7.4.2 Using no Known Estimate of 7

In Figure 7.8. the output response obtained using Controller PIDI' and (7.8) witli

P(k) := 3 5 . (((k - 1) mod 6) + I), (7.90

and no a priori estimate of T is shown; in this instance, utilizing the cyclic switching

action summarized in Table 4.1 (with a ( t ) and e f ( t ) both additionally reset to be equal

7.4. Switching Controffer Output Results 139

to zero immediately following any controller switch), a total of 18 switches occurs within

approximately one minute. and tracking of ( 7 . 8 ~ ) once again is achieved. As in Figures 7.6

and 7.7. the sluggish behaviour which is apparent in yi(t) during the initial transition in

height from 3000 to 2500 can be attributed primarily to valve saturation constrainis.

For cornparison. in Figure 7.9, experimental results are presented using Controller PIDI'

and the identical system setup as given for Figure 7.8. but wit h output l a d s y l ( t ) and ?j2( t )

init ially reversed.

(h:,I ( t ) hLl ( t ) ) := (3000.2500). t 3 O.

and with positive constant disturbance w2 applied to column 2 a t time t = O. In tbis case.

there is a total of 23 switches. and tracking of (7.10) occurs after approximately 800 seconds.

Furthermore. from (7.7): these results are entirely consistent in the fact that for Figure 7.8.

wfiile for Figure 7.9:

As a final example, in Figure 7.10. output results for Controller PID1' are shown iising

(7.9) with

T := 0.75 seconds,

8 := 20':

( ( t ) ( t ) ) := (2500.2250). t 2 0.

p := 20.

X := 107

and wit h both output leads reversed after 1200 seconds. Once again. as anticipated, tracking

of the given reference heights occurs, and k,, = 24. Xlternatively. using (7.9) and (7.11)

7.4. Swi tching Controller Output Results 140

Figure 7.5: Experimental proportional-integral-derivative results of 91 (solid) and y? (dashed) with 13 = 40" and Controller PIDl' appIied tu the MXRTS systeni.

1 LOO T

Figure 7.9: Experirnental proportional-integrai-derivative results of hl (dashed) and hl (solid) with 8 = 40' and Coritroller PID 1' applied to the reversed MARTS systeni.

7.4. Switching ControlIer Output Results 141

l 1 1

-*-- -1 ml-

Time (seconds)

Figure 7.10: Experimental proportional-integral-derivat ive rcsrilts of y 1 (solid) and y2

(dashed) with 6 = 30° and Controller PIDL' applied to the MARTS system.

Figure 7.11: Experimental proportional-integraI-derivative results of 91 (solid) and (dashed) with Controller PIDI' applied to the MARTS system.


with no output lead reversa1 and

40'. O 5 t < 750 Q ( t ) :=

O". t 2 750 seconds.

the resuits shown in Figure 7.11 are obtained and k,, = 19.

Additional experimental results obtained ilpon applying Controller PID 1' to 41ARTS

are summarized in Appendix C.

7.4.3 Using ControlIer Cl

For a final set of experimental results, let

èb = -€Xeb + cX(b?~,. - y ) . (6. e . A) E B' x BT x R1.

On applying Controller C 1 (in order to eliminate any potent ial reset windup issues caused

by control signal saturation constraints) with no a prion estirnate of 7. and on using the

cyclic switching action for K ( t ) siinimarized in Table 4.1. together witli

g2(k) :=

8 :=

(6:X) :=

.- and ( ~ ~ e ~ ( t ) ~ ~ ~ t = ~ ( t ) ) * -

0.5 seconds.

10 $(k- 1 ) mod 6)+ 1 '

2 2 ( ( k - L) mod i-o)+i ' 30".


the output results shown in Figures 7.12 and 7.13 are respectively obtained for &(k) :=

5 - g2(k) and P(k) := O. In both instacces. switching eventually ceases (k , , = 85 aad

kss = 61). and tracking of yref(t) occurs after approximately 500 seconds.

Alternatively. for the case when (7.12) is applied with Controikr CL using

(6.X) := (1.50):

( y ( ) ( t ) ) := (25OO.%OO).

and with output leads y ( t ) and yu ( t ) suddenly reversed at t = 1500 seconds. the respective

results shown in Figures 7.14 and 7.15 are obtained for p ( k ) := 0.05 . ga(k) cznd @(k) := 0.

Once again. switching eventually ceases (k,, = 66 and k,, = 72). and tracking of the given

reference inputs still occurs despite the unanticipated drastic plant changes w hich take

place at t = 1500 seconds. Moreover: in both of the situations presented here. Controller

C l does indeed successfuily result in a noticeably improved tuning transient response when

compared with that obtained by using Controller 2 given in [67].


Figure 7.12: Experimental proportional-integral results of y1 (solid) and y:, (dashed) with 9 = 30" and Controller C l applied to the MARTS system.

Timr (seconds 1

Figure 7.13: Experimental integral results of y1 (solid) and y:! (daslied) with 8 = 30" and Controller 2 [67] applied to the MARTS system.

7.4. Switching Controiier Output Results 145

Figure 7.14: Experimental proportional-integrd results of 91 (solid) and yr (dashed) with 8 = 30" and Controller C l applied to the MXRTS system.

Fiogre 7.15: Experimental integrai results of y1 (solid) and y2 (dashed) with O = 30° and Controuer 2 [67] applied to the MARTS system.

Chapter 8

Conclusions

In this chapter. the main contributions of this work are Iisted. and possible future resemch

directions are discussed.

8.1 Summary of Results

In t his t hesis. a new class of switching controllers using a variant of the switching mechanism

originally presented by Miller and Davison in [64! has been constructed to solve a number

of problems in which as little a pr ior i plant information as possible is assunied to be known.

For exampie, the following classes of probleuis have be investigated and solved:

a general adaptive switching probIeni for a faniiIy of not rlecessariIy strictly proper

MIMO plants (Chapter 2):

0 an adap t ive stabilizat ion problem for possibly unknown MIMO systems (Chapter 3) :

0 a self-t uning proport ional-integral-derivative (PID) robust servomechanism problem

for constant reference and constant disturbance inputs for stable plants (Chapter 4);

a a self-t uning proport ional-integral (PI) robust se~omechanism probleru wit h control

input constraints for constant reference and constant disturbance inputs for stable

plants (Chapter 5); and

an adap tive servomechanism problem for potent ially unknown MIMO plants (Chapter

6)-

8.2. Future Research Directions 147

As well, al1 controllers developed here are robust with respect to bounded immeasurable

additive noise dis t urbances applied to control signal .u( t ) and/or plant output y ( t ) .

Since one of the main objectives of this work has been to reduce the potentiai closed

loop tuning response through tlie use of as little a priori plant informatio~i as possibie,

the controllers presented here t herefore are significant in the sense t hat, wit h the exception

of Controller Cl , switching now is based partially upon direct norm bounds on the system

output error. Moreover, initial simulation results obtained when using t hese new controllers

tend to indicate that a desirable improvement in the system transient response generally

can be achieved by selecting non-pathological controller and tuning parameters.

An experirnental real-time application study of one such class of switching controllers.

using almost no a prion' plant information when applied to a multivariable hydraulic systeni

(MARTS), indicates that the proposed controllers are feasible to iniplement in ari iridustrial

environment: in fact, i t has been shown that such controllers operate satishctorily in the

presence of extreme structural changes occurring unexpectedly in the plant.

8.2 Future Research Directions

While many of the results presented here are positive. and tlierefore give assurailce as to the

development of a full theory of general adaptive switchirig coritrollen. riuIIieroiis outstaiidiiig

points still reuiain open for future exaniination. For exaniple, althougli switctiiiig now is

based partially upon direct norm bounds ori tlie system output error. an unacccptably

large output transient response may still occur if there exists a large nurnber of candidate

feedback controllers. This situation can be seen in Figures 2.8 and 3.11: and lience. further

investigations into the reduction of initial tuning t rwien t s are warranted.

However, as one example of the potential generality afforded by the faniily of switching

controllers proposed in this thesis. consider the time-varying plai t [56. pg. 111

which is outside the class of systerns considered in this work. Assunie that the plant is

unknown, biit that it is kriown that there exists a zero'th order stabilizirig corriperisator for

8.2. Future Research Direct ions 148

the system (Le. it is known that for some value of Li E R. the cIosed loop system w i l be

stable with u = L a ) . As one can veri&, since the closed loop system may be expressed as

(3 + Li) t (L, - 1) sin(2t) ~ ( t ) = exp +

4

for a fixed value of Li E R: ( A + BL,C) is exponentially stable (in the sense made precise

in [47. pp. 167-1681} if and o d y if Li < -3.

Upon applying Controller S2 and on defining h( i ) as

wit h

the sample output responses given in Figures 8.1 and 8.2 are obtained for the case when

x ( 0 ) = 1 aud ~ ( 0 ) = 0.001 respectively. Since successtiil controI occurs? it therefore is

conjectured that subsequent extensions of these coritrollers to various cluses of linear time-

varying systems may also be conceivable.

Indeed, using the many weU known techniques that have been applied so successfully to

conventional mode1 reference adaptive control problems. it may likewise be advantageous to

replace bounding function /(k) by a dynaniic one. j? where j would have certain resetting

properties (during time periods when no further switches occur) and could possibly be

governed by the general nonlinear different ial equat ion

8.2. Future Research Direct ions 149

Time c seconds 1

15 1 Controller K u versus urne.

1

" O I 3 J 5 6 7

Timr (seconds)

Figure 8.1: (x (0 ) = 1) Sirnulated results with Controller S2 applied to (8.1) using (8.2) with x (dashed) and y (solid).

3 0 0 7 Thmreiicai conmuous urne switchine conuoller results.

1

15 Controlla K v vmus umc 1

Timr (seconds)

Figure 8.2: ( x ( 0 ) = 0.001) Sirnulated results with Controller S2 applied to (8.1) using (8.2) with x (dashed) and y (solid).

8.2. Future Research Directions 150

F u t hermore. due to the simplistic switching mechanisrns proposed in t his t hesis. addit ional

improvements in the current transient tuning response (through. for instance. the use of a

more complex switching structure and/or increased a priori system informat ion) also are

surmised to be viable in aay future work.

Appendix A

Proofs of Main Results

In this appendiu. remaining detailed proofs of the main results presented in CIlapters 2. 3,

and 4 are given.

A.1 Adaptive Switching Control of LTI MIMO Systems

A L 1 Theorem 2.1

Proof: The proof is by contradiction. and essentially works by constructing a Luenberger

observer to estimate the unknown plant state z( t ) . üsing norm bounds on controller state

7 7 ( t ) as well as u priori properties of the bounding function f . a contradiction can t hen be

shown to occur.

The proof proceeds in the following four phases:

Phase I obtains a bound on the observer error:

me Phase 2 obtains a bound on the estimated plant states:

Phase 3 obtâins a bound on the augmented p l a t and controlIer states: and

Phase 4 uses a priori properties of bounding function f and Lemma 2.1 to show that

a contradiction occurs.

Phase 1: To prove property i): assume that there exist a controlIer parameter f E MSBF.

a continuous reference yrej having norm a continuous disturbance w having norni 6,

and an initial condition z (0 ) = [ x ( o ) ~ D ( ~ ) T ] T for which Assumption F1 liolds' but property

A.1. Adaptive Switching Control of LTI MIMO System 152

i) does not: it follows that ti is defined for al1 i E N. Furthermore,

for some constant m E {1 ,2 , . . . , s}.

Since (Cm, A,) is detectable, this irnplies that there exists a matrix fi such that X(A, + MC,) c @-. Hence, one can [theoretically) construct a (full order) Luenberger observer

of the form

with I ( t i ) an arbitrary constant vector in Pnrn and X(A, + MC,) C @-. Also? with

t herefore

e = 2 - 5

= ( A , +-GfC,rt)é

and

for t E ( t l , t , ] , p E N, p 2 2. Upon recalling that the stability of niatrix (A, + iGC,)

irnplies tha t there exist constants (6, -X) E Pi x Ri so that ~ l e ( " ~ n ~ " ~ ~ ) ' l l 5 ireit for

t 2 0, it follows (after taking norms) t hat for t E (t ,, t,],

Phase 2: Similarly, since

A.1. Adaptive Switching Control of LTI MIMO Systems 153

t herefore

upon defining

and on noting that

for t E ( t l : t,]? it also follows that

Phase 3: Now iising the fact that

and thus that

A.1. Adaptive Switchinn ControI of LTI hIIb10 Systems 154

for t E ( t i . t p ] : it t herefore follows that

for t E ( t l . t,] and for finite constants CO > O. ci > 1. In addition. because

t here fore

for t E ( t l : t,].

Phase 4: Since Lemma 2.1 holds for al1 siich tiiat ( ( j - 1) mod s) = nr - 1. ;iiid sinçe

f E WSBF, tliere exists a finite j _ ( ( j - 1) mod s) = rn - 1. sricii that

(ii) c31'(3) + C < f ( j )

are both satisfied for t E ( t ; : t ;_ ,]: if we now set t = t ;_ , : tiieii t lie iiieqtialit ics

contradict our definition of t;+,: tience property i) is true.

From property i) and the bound given in (AI) , property ii) follows: also. from i): there

exist matrices Gss, Hss, Jss, K,,, L,,, M,,, and a t,, 2 O such that (G(t), H ( t ) , . i ( t ) ,

K ( t ) , L ( t ) ! iW( t ) ) = ( G s s , Hss, J,,. Kss, L,,: rLI,,) for al1 t > t,,; it therefore follows from

Proposition 2.2 that for almost al1 (Gi, Hi, Kt, Li)? Ai will Iiave no eigenvalues in CO ; Lience.

property iii) follows since for almost all (Gi, Hi, Ki$ Li), tiie excited modes of t lie final closed

loop system will be stable. 0

Proof: To prove property i): assume that there exist coritroller parameters E MSBF

and X E R+. a piecewise continuous reference g , , ~ Iiaving norm a piecewise continuous

disturbance w liaving norm W. and an initial coridition Z(0) = [ X ( O ) ~ r l ( ~ ) T el(0)']* for

which Xssumption F2 holds? but property i) does iiot: it follows that t , is <hfined for ail

i E N. Furthermore,

P € P P=P*

for some constant rn E { 1 .2 . . . . . s ) : as such. for t E (tp-l? t,]: p E N. p 3 2. it also follows

where x ( t G l ) = x(tP- 1).

and

Since (Cm, A,) is detectable. consider the partial state estimation problein [18. pg. 3611

A. 1. Adaptive Switching Control of LTI MIMO Systerns 156

given by

for t E (t,- ,, t,]. Upon setting

on noting that

and upon defining


A. 1. Adaptive Switching Control of LTI bIIMO Systems 157

w here

In addition, by rewriting (-4.2) as

and on defining

t herefore

Now since (Cm, 4,) is detectable. let matrix &! I>e chosen sucli that A(-& + ,@cm) c

@-. Observe tliat (Agl, A;'~) is detectable for al1 p My p > 2. and consider

A.1. Adaptive Switchin~ Control of LTI PvITiLIO Systems 158

for t E ( t p - 1. t,] with 2( t i ) an arbitrary constant vector in Rnm and A(.-$', i (;p-lji<l) =

A(-& + MC,) c C-. Also. with

t herefore

and

for t E ( t l ' t,]. Upon recalling that the stability of matriu (-4, + -F~C,) iniplies that there

exist constants (6 . - A ) E Rf x R- so that / le(-4m'"Cm)t~~ < 6eXt for t 2 0. it bllows (after

taking norms) that for t E ( t i . t p ] .

Similady. upon defining

A. 1. Adaptive Switching Control of LTI MIMO Systems 159

for t E (t,-l? $1, and on noting that

t herefore

however, since

for t E ( t 1, t,], and witli

it also follows tfiat

Now using the fact that

A.2. Adapt ive Stabilization of LTI Systems 161

From property i) and the bound given in (A.3). property ii) follows: &o. from i): there

exist matrices G,,, H,,, J,,? K,,? L,,, iMSbgI,,: and a t,, 2 O such that (G( t ) . H ( t ) . . J ( t ) .

K ( t ) , L ( t ) , iU( t ) ) = (G,,? Hssl*Jss. K,,. L , , . M 9 , ) for dl t > t,,; it therefore follows from

Proposition 2.2 that for almost al1 (Gi, Hi: K t , L,): Ai will have no eignvalues in Co: hence.

property iii) follows since for almost al1 (Gi: Hi , K,. Li), the excited modes of the final closed

loop system will be stable. U

A.2 Adapt ive Stabilizat ion of LTI Syst ems

Proof: To prove property i ) ? assume t hat there exist controller parameters f f S LBF and

( E ~ , T) E S: a continuous disturbance w having norm G. and an initial condition z(0) for

wbich Assumption S 1 holds. but property i) does not: it follows that t , is defined for d l

i @Ne

Since

a d c + 0: therefore

for dl t 2 O. As weil, using the fact that

for t E ( t I , t,], p E N, p 2 2. it therefore follows that

A.2. -4dapt ive Stabilization of LTT Systems 163

for t E (tlT t,] and for constants (co.cl) E B' x X'.

Consider now a j' sufncient ly large so t hat Lemma 3.1 holds for al1 q 2 j . t E ( t,. t,[,

with ( ( q - 1) mod 2) = ( ( j ' - 1) rnod 2). Since f E SlBF. there exists a finite > ;. ((3 - 1) rnod 2) = ((j - I ) rnod 2). such that Lemma 3.1 holds and

I Y ( ~ ; + J < f (3

contradicts our definition of t j T I ; hence property i) is mue.

Rom property i) and the bound given in

exist an E,, E R'. K,, E {l. -1}. and a t,, ..-t

al1 t 3 t,,. Since K,, E K and E,, = for 50

for almost di T ) E S. ( a + ~E, ,K , , c ) # O:

(A.4). property ii) follows: also. from i) . there

2 O such that ~ ( t ) = E,, and K ( t ) = K.%, for

some i E W. it follows from Remark 3.1 that

hence property iii) foIlows since for almost al1

(€0 . T ) E S. the excited modes of the final closed loop system will be stable.

A.2.2 Theorem 3.2

Proof: To prove property i). assume that there exist cnntroller parameters f E S3BF and

h E CTF, a continuous disturbance cu having norm W. and an initial condition = ( O ) = [r (O)=

~ I ( o ) ~ ] ~ for which .Assumption S2 holds. but property i) does not: it foliows that t , is defined

for al1 i E !Y.

Since (C. A) is detectable. this implies that there exists a niatr~u such that A(-4 i

MC) C @-. Hence. one can (theoretically) construct a (full order) Luenberger observer of

the form

ii = (A+iîdC)?+ Bu - M y + ( E + M F ) w

with Z( t i ) an a r b i t r q constant vector in Rn and X(A t MC) C C-. Also. with

A.2. Adaptive Stabilization of LTI Systems 163

t herefore

and

for t E ( t l : t ,] , p E PI, p 2 2. Upon recalling that the stability of matriw ( A + MC) implies

that t here exist constants (a. -X ) E 8' x P' so t hat le('\+."")^ 11 < - de*' for t 2 O. it follows

(after taking norms) that for t E ( t t,].

t herefore

upon defining

a~id on noting that

Ilv(t)ll I f b - 1)

and Ilu(t)ll <- .<-'f @ - 1) + f @ - 1 )

A.2. Adap tive Stabilizat ion of LTI Systems 164

for constants (q'n) E IR+ x Ri and for t E ( t l : t p ] , it also foUows that

0 11 5 a .. ll*(t i ) l l + 2 (mi (TP-' f @ - L) + <-If (p - 1)) + m J ( p - 1) + mniü) . 1x1

Now using the fact that

and thus that

for t E ( t : t,]: it t herefore Follows t bat

For t E ( t l . t,] and for finite constants ca > O. ci > 1. Q > 0. r > O. In addition. because

t herefore

(A..)

for t E ( t l , t ,] .

Since h E CTF, and since it is known that gi = q will stabilize (3.2). where q E N U { O ) :

for some ci E ~ ( ~ ' g ~ ) ~ ( ' ~ ~ ~ ) . consider now a j' sufficiently large so that Lemma 3.2 holds

for t E (t;. t;,,]: with f E S2BF and h E CTF, there therefore exists a finite J' 2 j' such

t hat Lemma 3.2 holds and

A.3. The Self-Tuning Robust Servomechanism 165

is satisfied for t E ( t ; , t ; + l ] ; if we now set t = t;, , , tlien tlie inequality

contradicts our definition of t;, : Lience property i) is true.

From property i) and the bound given in (A.5). property ii) follows: also. from i),

there exist matrices G,,, Hss, Kss. L,,, and a t,, > O such that (G( t ) . H ( t ) . K ( t ) ? L ( t ) ) =

(G,,. Hss7 K,,. L,,) for al1 t 2 t,,; it therefore follows frorn Proposition 2.2 tliat for alrnost

al1 (Gi, Hi , l(,' Li) rIi will have no eigenvalues in @O: tience. property iii) follows sirice for

almost al1 matrices (G,, Hi, KI, L,). the excited modes of the filial closed loop systern will

be stable. 0

A.3 The Self-Tuning Robust Servomechanism

A.3.1 Theorem 4.2

Proof: To prove property i). assume tliat there exist a controller paranieter 5 E fi. a

constant reference gr,== Iiaving norrn ire!. a constant distiirbaiicc io Iiaviiig norni iü. niid

a n initial condition z (0 ) = [z(o)* l l ( ~ ) T ] T for wliich Assumptioli PI1 Iiolds. 1)ut property i)

does not: it follows tliat t , is defiued for al1 i E P!.

Since X ( A ) c C-. ttiere exist constants ((5. -X) E 2- x 8' so tliat lJe''tJl 5 GeAt for

t > O. In addition. since

i = .+lx t Bu i Ew.

upon defining

A.3. The SeIf-Tunine: Ro bust Servomechanism 166

and on noting that

for t E ( t l , t i ] , i 3 2. it also follows that

for t E ( t l , ti] and for finite constants co > 0. ci > 1. In addition. because

t herefore

for t E ( t i , t i ] .

Since Lemma 4.1 holds for a sufftciently large 3. and since f E MSBF and g E TF'. there

exists a finite 3 2 3 such that

(i) d ( j ) + P(,j,,, + G) < f (j): and

are both satisfied for t E (tj , t;+,]; if we now set t = t;,,, then the inequalities

contradict our definition of t;,,; Lence property i) is true.

Rom property i) and the bound given in (A.6) , property ii) follows; also. from i), there

A-3. The Self-Tunine: Robust Servomeciianism 167

€0 exist an E,, E !Et+ and a t,, 2 O such that c ( t ) = es, for ail t > t,,. Since es, = - for

some i E W, it follows from Lernma 4.2 that for almost al1 (Q, T) E S. A(p. c,,) will have

no eigenvalues in @O; hence, property iii) follows since for alrnost al1 (eo7 T) E S. the excited

modes of the final closed Ioop system will be stable. 0

A.3.2 Theorem 4.4

Proof: Although the proof of Theorem 4.4 is very similar in nature to that provided for

Theorem 4.2, it will nevertheless be given in complete detail. To prove property i). assume

that there exist a controller parameter a p l ~ E ClPIDI a constant reference y,,! having norm

grel: a constant disturbance w having norm 6, and an initial condition z(0) = [z(OIT q(0)T

a ( ~ ) ~ ] ~ for which Assurnption PIDl holds? but property i) does not: it follows tliat t i is

defined for a11 i E N.

Since X(A) C C-. there exist constants (& ' -A) E R' x P' so that I ~ ~ " ' I I 5 ~ e " for

t > O. In addition, since

t herefore

upon defining

and on noting that

A.3. The Self-Tuning Robust Servomechanism 168

for t E ( t l ? t i ] , i 2 2, it also follows that

for t E ( t i , ti] and for finite constants CO > 0: ci > 1. In addition. because

t herefore

for t E ( t l , t i ] .

Since Lemma 4.4 holds for a sufficiently large 3, and since / E MSBF and ( g . g1 , g2) E

TF' x TF' x TF', there exists a finite 3 2 j' such ttiat

(i) al?(;) + + fi) < f ( j ) ; and

( i ) [ - -Ï;DK 9 2 ( j ) ~ 2 Ï j ~ ~ ] I I - ~ ( ~ ) + I I I - P ~ ~ ( ~ ) ~ ~ D K I I - G ~ . ~ + I ~ ~ ~ F I I . @ < f ( 3 )

are both satisfied for t E ( t j , t; , ,]; if we now set t = tj+,: then the inequalities

contradict our definition of t ; + [ ; tience property i) is true.

From property i) and the bound given in (A.7), property ii) follows; also, from i) , there

exist ( e s , , e , , , ,c , , , ) E W+ x Et+ x WC and a t , , 2 O such that e ( t ) = es,, c i ( t ) = c,,,, €0 € 0 2 and t p ( t ) = c,,, for al1 t > t,,. Since es , = -, es , , = 3 aud ess2 = - . for some rL ri ' 7.;

i E N, it follows from Lemma 4.5 that for almost al1 ( t g , T , €0, , ~ 1 ? eo2, r2) E S x S x Si

À P I D ( P 7 es,, t,,, , es,, , K, N) will have no eigenvalues in CO; Iierice, property iii) follows since

for almost al1 (ea, T , EO,, 71, C O - , T ~ ) E S x S x S, the excited riiodes of the filial closed loop

A.3. The Self-Tuning Robust Servomechariism 169

system will be stable.

A.3.3 Theorem 4.5

Proof: The proof of Theorem 4.3 is very similar in nature to the proof given for Theorem

4.4 upon making appropriate changes to accommodate the Jact that there are now .s possible

feedbnck mutrices: hence. only the major niodifications to the latter wiI1 be given. To prove

property i ) , one can form (in a manner anaiogous to the proof giveri for Theorem 4.4) the

inequality

where finite constants CO > O, ci > 1. and where t E ( t l : tijl i E M. i >_ 2. If we now

set q E {l. 2 . . . . . s } such that -TK,, is stable. and find a siifficiently large so that

Lemma 1.4 liolds for al1 i 2 3 witli ((i - 1) tiiod s ) = q - 1. ttieri since f E SISBF and

(9, gi.y2) E TF' x TF' x TF'. tliere exists a finite j 2 ;. witti ( ( j - 1) n i d s ) = q - 1. sticli

t hat

(i) d ( j ) + + 6 ) < f ( j ) : and

are both satisfied for t E (t;. t;,, 1; if we now set t = tj,,, tlien the iricquülitics

contradict o u definition of tj,,; hence property i ) is true.

Rom property i) and the bound given in (A.8): property ii) follows: &o. from i)? there

exist a matrix Kss, constants (es,, E . ~ ~ ~ . esS2) E B- x Bi x Rf . and a t,, 3 O siich that

K ( t ) = K.ss, ~ ( t ) = es,, c l ( t ) = essi, and ~ ( t ) = es,, for d l t 2 t s s . sincc Kss E K, and €0 €0 1 €O2

cSs = - Ti cssl = -, csSl = - for some i E N. it follows fiom Lemrna 4.6 that for alrnost al1 7: r;

A.3. The SeIf-Tuning Robuçt Servomechanisrn 1 70

(sa, T' , T I ' €a2. T2, U) E S x S x S'. A p l D ( p o Easi c,,, ? K,,. N ) will have no eigenvalues

in cor hence, property iii) foiiows since for alrnost al1 (€0 . T. €0, - ri? col: 72. Lr) E S x S x Sf7

the excited modes of the final closed Ioop system will be stable. Cl

Appendix B

Miscellaneous Data

System matrices used for various sin~ulation examples are given in this appendix.

B.1 Controller Parameters for a Family of Five Plants

ln this section, the controller parameters for the structure

used for the simulatiori example provided in Section 2.3.3 (Figure '2.11) are listed below:

B.1. Controller Parameters for a Family of Five Plants 172

B. 1. Controller Parameters for a Family of Five Plants 173

The true plant is given by ( A , B? C).

A =

B.2. Partial Decentraiized Control of a hduiti-Zone Building 174

B.2 Partial Decentralized Control of a Multi-Zone Building

The following system matrices were used to obtain Figure 4.7:

A = Columns 1 through 7

B.2. Partial Decentralized Control of a Multi-Zone BiiiIdin~

Columns 22 through 28

B.2. Partial Decentralized Control of a Multi-Zone Building

Columns 29 through 32

Columns 1 through 7 O O

Columns 8 through 14 O O O

Columns 15 through 21 O O O

B.3. A Four Input-Four Output Furnace Model 175

Columns 22 through 28 O O O 0.0070 0.0050 0.0050 0.0060

Columns 29 through 32 0.0060 O. 0060 O. 0045 O. 0050

A Four Input-Four Output Furnace Mode1

The following matrices were used in the simulation to obtain Figure 4.9:

A-R = -0.2003 -0.0044 O. 0002 O. 0008 o. O000 o. O000 0.0000

B-R = -0.5606 -0.4696 -0.5579 O. 76% -0.1268 O. 3292 O. O658

C-R = -0.1154 -0.1498 -0.1628 -0.1303

D-R = 0.0042 -0.0002 -0.0059 0.0015

E-R = o. 1000 o. 1000 0.1000 O. 1000 O. 1000

B.3. A Four Input-Four Output Furnace Mode1 179

These matrices were obtained upon using the mode1 reduction methods given in [26] on

the original furnace mode1 given in [!IL. pg. 1991 and !26jL. The system matrices

obtain Figures 4.10 and 4.11 are Iisted below.

A = Columns 1 through 7 -0.2323 O. 0023 0.0067 O. 02% O. 0055 -0.0224 -0.0572 -0.00 18

Column 8 O. 0045 -0.0123 O. 0057 0.0073 -0.0055 o. 0020 -0.0003 -0.2038

B = -0.3339 -0.1606 O. 1476 O. 1986 -0.1571 O. 0764 -0.0196 -0.0376

C = Columns 1 through 7 -0.4177 -0.3583 -0.3344 -0.2647 -0.4481 -0.1575 -0.3225 O. O 137 O. 3904 -0.2685 0.2041 -0.2276

used to

'ln essence, the system given by (4.14) wm formuiated for didaçtic purposes with DR # 0.

B.4. Matrices used for a Binary Distillation Tower

B.4 Matrices used for a Binary Distillation Tower

The folIowing matrices were used to obtain Figures 5.3 and 5.4:

A = Columns 1 through 6 -1.4000e-02 4.3000e-03 9.5000e-03 -1.3800e-02

O 9.5000e-03 O O O O O O O O O O O O O O

2.5500e-02 O

Columns 7 through 11 O O O O O O O O

B.4. Matrices med for a Binarv Distillation Tower 181

Appendix C

Addit ional Experiment al Result s

In t his appendix, additional experimentd results. obtained upon applying Controller PID 1'

(for the case when no estimate of T is available) to the SIXRTS apparatus. are presented for

the class of piecewise constant reference and disturbance inputs. A listing of the controller

parameters and the individual system setup used for each figure is siimmarized in Table

C.1, where the reference Lieights referred to are the foilowing:

(3000.2500), O 5 t < 600

Y Y ( (2500.2000). 600 5 t < 1200

(3000.2500), t 2 1200 seconds.

(3000.2300). O 5 t < 2250

(2500.2000). t 3 2250 seconds.

1 (3250,3000). O 5 t c 2250

h . h ) := (3000.2500). 2250 5 t < 3000

(3250.3000). t >_ 3000 seconds.

( t ) h t ) := (800.2500)' t 3 0.

(hfe ( t ) : hp, ( t ) ) := (XIOO. 2500). t 2 0.

In addit ion, for al1 instances, uriless ot herwise stated,

C. Addit ional Experimental Results 183

and T := 0.75 seconds.

Table (2.1: Summary of the parameters used for Figures C.1-C.12. wliere o ( k ) :=

floor (y) a d ( k ) := 5 - (((k - 1) mod 6 ) + 1)-

1015" O 5 -4 LVl 'i 10/5" O 15 4 ?j Y (CA) - 10/3O O 3 6 \VI Y/Y (CA) -

Additional notes and comrnents:

10/3" 10/3" 10/3" 10/3"

(a) Iri Figures C.11 and C.12. output leads y i ( t ) and p ( t ) are reversed at t = 2250

seconds.

20 20 20 20

3 20 3 3

10/3" 20 10/3" 1 3

20 5 20 10

10/5" 10/za

6 5 6 5

20 20

5 6 4 4

ttVl tVt

1

11,

?3 S 'r' Y 1- Y

Y/?- (C.3) / X/E' X/Y

/ N/X / (C.2) N/Y (C.3) -

(CA) (CA)

Y I ( 4 (a)

- -

S/N j (c-5) (a) [

C. Additional Experimental Results 184

Fiorne C. 1: Experimental integral-derivat ive results of y 1 (solid) and y- (dashed) wit h Controlier PID1' applied to the MXRTS system.

fime (seconds)

Figure C.2: Experimental integral-derib*at ive results of y 1 (solid) and y2 (dashed) wit h Controlier PIDI' applied to the MARTS system.

C. Addit ional Emerimental Results 185

Figure C.3: Experimental integral-derivative results of y1 (solid) and fi (dashed) wi th Controller PID1' applied to the MXRTS system.

Expenmrnul (ID i rcsults usine T 4 . 7 5 ~ ~ . 3100 : I

Figure C.4: Experimental integral-derimtive results of y1 (solid) and 12 (dashed) with Controller PID1' applied to the MARTS system.


fime i seconds )

Figure C.5: Experimental proportional-integral-derivative results of y1 (solid) and y:! (dashed) with Controller PID1' applied to the MARTS system.

1

1600' I O 7 0 UXI 600 $00 lûûû 1700 1100 1600 1800

Timr (seconds)

Figure C.6: Experimental proportioual-integral-derivative results of y1 (solid) and y:! (dashed) with Controiler PID1' applied to the MARTS system.


Timt: (seconds)

Figure (2.7: Experimental proportional-integral-derivat ive results of hl (dashed) and h2 (solid) wit h Controller PID 1' applied to the reversed MARTS system.

1 5 0 0 ~ 1 O 500 Io00 1500 Zoo0 300 3000 3500 U)c]O

Time (seconds)

Figure C.8: Experimental proportional-integral-derivat ive results of h 1 ( d a . hed) and hs (solid) wit h Controller PID 1' applied to the reversed MARTS system.

C. Addit ional Emeriment al Results 188

Loo-

1000 -

Figure C.9: Experimental proport iond-integral-derivative results of h i (dashed) and hz (solid) wit h Controuer PID 1' applied to the reversed MARTS system.

Time (srconds)

Figure C. 10: Experimental proport ional-integral-derivat ive results of h (dashed) and h2 (solid) wit h Controller PID1' appiied to the reversed MARTS systeui.


Time (seconds)

Figure C. 11: Experimentd proportional-integral-derivat ive results of 91 (solid) and 92 (dashed) with Controller PID1' applied to the MARTS system.

Figure C. 12: Experimentd proportionai-integrai-derivat ive results of $1 (solid) and y2 (dashed) with Controuer PIDI' appiied to the MARTS system.

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