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This is to certify that the
dissertation entitled
A QUANTITATIVE POLE-PLACEMENT APPROACH
FOR ROBUST TRACKING

presented by

Chang - Doo Kee

has been accepted towards fulfillment
of the requirements for

M E

PI] . DI. degree in

 

  
 

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A QUANTITATIVE POLE-PLACEMENT APPROACH

FOR ROBUST TRACKING

BY

Chang - Doo Kee

A.DISSERIAIION

submitted to
Michigan State University
in partial fulfilment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

1987

O

'__'_u
J

¥N) I

If

</5CJ~

ABSTRACT

A QUANTITATIVE POLE- PIACHIENT APPROACH

FOR ROBUST TRACKING

By

Chang- D00 Kee

Uncertainty in plant(p) parameters and in disturbances (d) are
among the principal reasons for using feedback in control systems.
Often control systems are synthesized to satisfy qualitative measures
of performance characterized by such popular choices as overshoot, rise
time, and settling time. Here a framework is considered where the un-

certainties are quantitatively defined in terms of a set of plants

P = {p} and disturbances D = {d}. Acceptable output sets *rd are

defined for the output y in response to command inputs r and distur-

bances d,to be achieved V p e P . This framework is a manifestation
of the need to satisfy the design specifications directly.

The research described in here is based on a novel concept of
tracking called "tracking in the sense of input-output _spheres", first
introduced by Barnard and Jayasuriya [4]. The main approach to the
problem is based on methods of functional analysis motivated by the

topological characterization of the said notion of tracking. Design

criteria formulated within this framework lead to an interesting pole-
placement idea which is quantitative in character. Primary
contributions of this work were motivated by this notion of

" quantitative pole-placenen " .
In this thesis a previous formulation in LEW, 00) is extended to

incorporate external disturbances. It is shown that for sector

bounded nonlinearities computation of norms is somewhat simplified.
L: formulation captures the physical nature of continuous tracking

precisely. The broadness of this function space however, makes it
difficult to obtain simple, yet accurate, criteria for the so called
quantitative pole placement. A somewhat loose form of initializing an
algorithm for this pole placement is developed however. It is shown
that if a Butterworth configuration is chosen as the initial guess for
eigenvalues then the numerical scheme "converges" to the desired solu-

tion.

In order to generate frequency domain interpretations the tracking
problem was next embedded in L:[O, m). This allows a neat geometric

interpretation of the design criteria in terms of the frequency

response characterization of a certain linear operator norm for 8180
. . (1)
systems. These L2 interpretations are somewhat Similar to the H -

criteria recently developed in the literature for linear systems.

Several applications to support the L0° theory and L2 - theory are also

included.

To my parents

iv

ACKNOWLEDGHENT S

I would like to express special thanks to Dr. Suhada Jayasuriya,
my major professor, for his guidance and support throughout my Ph. D.
program, and also for his painstaking review of this manuscript. His
academic exCellence and good personality have provided a constant
source of encouragement.

Also, I wish to extend my sincere thanks to the other members of
my Doctoral Guidance Committee, Drs. Ronald Rosenberg, Clark Radcliffe,
Hassan Khalil, and David Yen, for their valuable comments and
observations concerning this work.

This research was supported partially by the National Science
Foundation under grant No. MEA 8404324 and by the Division of
Engineering Research at Michigan State University. This support is
gratefully appreciated.

My family has always been a source of strength and love for me.
To my father and mother, my sisters and brothers, my father and mother
in-law, and my uncle, Yong Chul Cho, especially to my wife, my
daughter, Wonju, and my son, Won-jin, thank you, I couldn't have done
it without you.

Finally, special thanks go to all my friends for making my
graduate career a successful experience, and for being there when I

needed you.

 

This is to certify that the
dissertation entitled
A QUANTITATIVE POLE ~ PLACEMENT APPROACH

FOR ROBUST TRACKING
presented by

Chang — Doo Kee

WA W1

Chang oo Kee

Date
Docto 0 Philosophy Candidate

has been accepted towards fulfillment
of the requirements for

Doctor of Philosophy Degree in Mechanical Engineering

Michigan State University

by

w 0.2 {a ¥//7J’;Z
Suhada asfifiya "

Date

 

1 Professor
Dis ertation Advisor
Department of Mechanical Engineering

MW 6%
o yR. Lloyd ' /, Date
P ofessor and Chairman
epartment of Mechanical Engineering

Vi

TABLE OF CONTENTS

LIST OF FIGURES

CHAPTER I. INTRDUCTION
1.1 Literature Survey
1,2 General Objectives and Approach

1.3 Organization

CHAPTER II. PROBLEH.FORHULATION FOR.SERVO-TRACKINC
2.1 Servo-Tracking
2.2 System Formulation
2.2.1 Plant
2.2.2 Nominal Plant
2.2.3 Observer and the Controller
2.2.4 Operator Representation
2.3 Design Criteria

2.4 A Special Case

CHAPTER III. SERVO-TRACKING IN A.LDR'E TYPE SYSTEM
3.1 Statement of Problem
3.2 Problem Formulation
3.3 Main Results
3.4 Special Case
3.5 Design Procedure

3.6 An Illustrative Example

vii

ix

10

13

13

14

14

17

21

29

34

35

38

4O

44

58

59

CHAPTER IV AN APPROACH FOR SELECTION OF EIGENVALUES
4.1 Conventional Pole - Placement Problem
4.2 Generalized LO Problem

4.3 Optimal Pole Configuration

CHAPTER V. COMPUTER.ALGORITHMS
5.1 The Operator Norm II W0 W Q;1 II
5.2 Design Matrices G and K

5.2.1 Eigenvalue-Eigenvector placement Algorithm
5.2.2 Computation of the Matrix G

5.3 Numerical Integration Scheme
CHAPTER VI. APPLICATIONS
6.1 A Synchronous Machine
6.2 A 3 DOF Robot Manipulator
6.3 A Single DOF Gyroscope
CHAPTER VII. CONCLUSIONS, DISCUSSION AND FUTURE WORK
APPENDIX.A, MATHEMATICAL PRELIMINARIES
APPENDIX B. DERIVATION OF.A CIRCLE TYPE CRITERION

APPENDIX C. TWO USEFUL LEMHAS

LIST OF REFERENCES

viii

72
73
74

80

88

88

97

98

101

104

108

108

115

133

141

146

152

154

156

Figure

3(a).

3(b).

5(a).
_5(b).

6(a).
6(b).

6(c).

8(a).

8(b).

9(a).

9(b).

10.
ll.
12(a).
12(b).
13(a).

13(b).

LIST OF FIGURES

Illustration for precision tracking

Controller system configuration

Combined uncertain systems

Nominal systems

Configuration of Control system

Illustration of a, 3, v, and u

Illustration of a, 5, {(uo),

Geometric interpretation for
Geometric interpretation for

Geometric interpretation for

O

and £(u)

M - circle diagram for a - 0.5, fl = 3.5

y(t) and yo(t) for a = 0.5, fl = 3.5

The tracking error I (y - yo)2 dt for a = 0.5, fl = 3.5

r(t) and ro(t) for a = 0.5, 5 = 3.5

Input disturbance in II r - roll2 for

Control effort u(t) for a = 0.5, fl = 3.

M - circle diagram for a = -0.5, fl = 0.

y(t) and yo(t) for a - -0.5, fl = 0.5

The tracking error I (y - y)2 dt for a

r(t) and ro(t) for a - -0.5, fl = 0.5

Input disturbance in II r -

ix

roll2 for a

= 0.5, 5 = 3.5

-o 5, 5 = 0.5

-o.5, a = 0.5

Page
12
16
19
20
36

49

49

57
57
57
62
63
64
64

65

65

68

68

69

69

70

14.
15.
16(a).
16(b).
17(a).
17(b).
18.
19.
20.

21.
22.
23.

24.
25.

26.

27(a).

27(b).

27(c).

28(a).

28(b).

28(c).

29.

30.

31.

Control effort u(t) for a = -0.5, fl = 0.5
Boundary condition for free end point
Optimal pole configuration for n - r — 2
Optimal pole configuration for n - r = 3
Eigenvalue placement for n - 2

Norm configuration for n - 2

Norm configuration for n - 3

Flow chart

A synchronous machine

Actual output y(t) and nominal output yo(t)
Output deviation (y(t) - yo(t))
Input disturbance (r(t) - ro(t))

Control effort u(t)
A three DOF robot manipulator

y1<t> and y01(t>

Output deviation (yl(t) - yol(t))

Output deviation (y2(t) - y02(t))

Output deviation (y3(t) - yo3(t))

Control effort u1(t)

Control effort u2(t)

Control effort u3(t)

Input disturbance (r1(t) - rol(t))
y2(t) and y02(t)

The tracking error (y2(t) - y02(t))

70

76

84

84

85

85

86

103

109

113

113

114

114

116

127

128

128

129

129

130

130

131

131

132

 

32.

33.

34.

35.

36.

37.

Control effort u2(t)

A single DOF gyroscope

y(t) and yo(t)
The tracking error (y(t) - yo(t))

Required control u(t)

Input uncertainty (r(t) - ro(t))

xi

132

133

138

138

139

139

 

CHAPTERI

INTRODUCI ION

1.1 Literature Survey

During recent years a number of references have appeared that deal
with the design of controllers for uncertain dynamical systems, assur-
ing the proper dynamical performance with respect to stability,
regulation, and/or tracking. However, almost all of these deal with
qualitative aspects of system behavior, and not quantitative ones.
Here, on the other hand, the primary research objective is to further
develop a controller synthesis procedure proposed for treating uncer-
tain dynamical systems in a quantitative framework. In particular,
controller synthesis for "precision tracking" in systems that are both
uncertain and nonlinear is considered.

One of the popular methods of design incorporates state feedback
controllers that force the closed-loop poles to be at suitable loca-
tions in the open complex left-half plane, depending on the design
specifications. Typical criteria are relative stability, speed of
response, accuracy, and insensitivity to disturbance inputs or
parameter variations.

Various approaches have been pursued to develop controller
criteria, whether pertaining to tracking or not, based on this state

feedback concept. Some of these approaches are based on

( i ) Lyapunov stability theory [1, 7, 9, 19, 20, 32, 49],

( ii) Adaptive control [8, 29, 39],

(iii) Classical control concepts [21, 22, 23 37],

( iv) Geometric notions [6, 48, 52],

( v ) Servomechanism theory [12, 13, 46],

( vi) Functional analysis [3, 4, 5, 14, 16, 17, 24, 25, 26, 31,

41, 42, 43, 51, 54, 55, 56]

The state feedback schemes require the accessibility of system
states. When they are not available, a nominal observer structure is
used in the feedback loop to generate state estimates [2, 34, 44].

Criteria for uncertain systems that consider stability are derived
by Barmish, Corless, Leitmann, and Thorp [7, 9, 20, 32, 49]. They are
based on the constructive use of Lyapunov theory in which first a
suitable Lyapunov matrix is generated by solving either a Lyapunov
equation or a suitable optimal control problem, and then inventing a
control action which admits a suitable Lyapunov function for all admis-
sible uncertainties.’ The possible sizes of uncertain elements are
assumed to be in prescribed.compact sets. When information about the
possible sizes of uncertain elements is not available , adaptive con-
trol strategies are employed with the estimates of uncertain bounds [8,
29, 39].

Criteria by Horowitz and coworkers [21, 22, 23, 37] are essen-
tially in the frequency domain, and rely heavily on classical design
concepts. These controller criteria are developed for assuring system
performance specified by acceptable range of rise time, overshoot, and
settling time in the presence of parameter variations and disturbance
inputs. The Shauder's fixed-point theorem is used to establish the

validity of these designs. For higher order systems, controller

 

criteria are often based on the existence of a few dominant poles and
zeros which primarily determine the system transient response. Design
techniques for third and fourth order dominant systems are investigated
in [21, 37]. These procedures, although very intuitive, do not lend
themselves readily to time domain analysis. Nevertheless, Horowitz was
one of the early researchers to point out the importance of uncertainty
in shaping appropriate controllers.

In the geometric approach [6, 48, 52], it is recognized that the
properties of linear systems depend on certain linear subspaces of the
state space. The design problem then is to generate a linear subspace
which has desirable structure in the state space so that the design
specifications are met. By describing the design specifications of a
controlled system as a specific structure of the subspace generated by
the feedback controller, the design is treated more intuitively and
generally better insights are gained for the conditions of existence of
solutions for decoupled control or disturbance localization.

In the servomechanism problem [12, 13, 46], the controller con-
sists of two devices, namely, a servo-compensator and a stabilizing
compensator. It is assumed that the disturbance is unknown and un-
measurable, but satisfies a certain ordinary differential equation. The
reference outputs are also assumed to satisfy a similar differential
equation. A servo-compensator is constructed according to the charac-
teristic equation obtained from the differential equation, and then a
stabilizing compensator is designed so that the resultant closed loop
system becomes asymptotically stable.

The functional analysis approach which will play a central role in
this research was pioneered by Zames and Sandberg. They studied the

absolute stability of Lur'e type nonlinear systems [41, 42, 43, 53,

 

55] . Desoer and Wang [14] studied asymptotic tracking and disturbance
rejection properties within this framework for general nonlinear multi-
variable systems which consist of input as well as output channel
nonlinearities. They established the robustness of these controllers
with respect to linear perturbation of the plant. Lecoq and Hopkin
[31] developed a bounded-input bounded-output stability criterion for
systems with nonlinearities that do not satisfy sector conditions.
Recently quantitative controller criteria for precision tracking in
nonlinear uncertain systems were developed by Barnard and Jayasuriya
[3, 4, 5, 24, 25]. They formulated the tracking specifications in
terms of topological neighbourhoods in normed function spaces and
employed nonlinear state observers related to uncertain plants for im-
plementing state feedback controllers. Their approach is based on the
application of the Banach fixed-point theorem and equation comparison

techniques.

1 . 2 General Obj ectives and Approach

The need for quantitative controller criteria stems from a wide
spread demand for more stringent tracking specifications as opposed to
the classic notion of asymptotic tracking, (i.e. the output vector y(t)

-> yo(t)‘ as t -* co ), especially as required by high performance robot

manipulators, automotive engine and clutch control, and missiles with
ram jets where uncertainty becomes an important issue.

Although several attempts and approaches have been taken to solve
this very realistic problem of controlling systems with parameter un-
certainties, it is far from complete. The work of Horowitz and his

coworkers and that of Barnard and J ayasuriya constitute a design theory

 

for the direct satisfaction of design specifications. Although much
recent work has been done on robustness, a theory for the direct satis-
faction of design criteria for uncertain, nonlinear systems is yet to
appear. All these works however are a step in the right direction.
The research described in this thesis is an attempt to develop a formal
procedure for the quantitative pole-allocation that is pivotal to the
successful execution of the design methodology proposed by Barnard and
Jayasuriya [4] . In particular, the work described in here addresses
"precision tracking" in uncertain nonlinear plants with multiple inputs
and outputs. Information available about the uncertainties of the
model is restricted to be their possible sizes, i.e., the uncertainties
are assumed to belong to certain prescribed compact sets. ZIt is im-
portant to emphasize that these uncertainties are deterministic in
nature, i.e., they do not fall into the usual category of random vari-

ables.

In their problem formulation Barnard and Jayasuriya employed

( i ) weighted norms in Lm(the space of essentially bounded functions)

to measure tracking error and plant disturbances;
( ii) nonlinear observers of the Luenberger type to realize control-
lers;

(iii) operator equationszhilQDto represent combined plants and con-

trollers, and
( iv) nominal or average systems to define suitable command inputs and

to compare the actual and specified plant responses.

 

The author's main design criterion is stated as a quantitative
pole placement procedure for controlling the size of a certain linear
operator norm. The idea of pole placement appears frequently in con-
trol theory and has been around for quite some time now. The main
attempt here has been the arbitrary placement of closed loop eigen-
values. What is not clear, however, is how one should specify their
location for the satisfaction of design specifications. Some
guidelines are available, for example in terms of algebraic Riccatti
equations for a special class of linear systems (LQ problem). In the
present work the satisfaction of the tracking specifications is
directly related to the pole-placement. In particular, the tracking
specifications are met if an eigenvalue placement can be found so that
an operator which may be characterized by these eigenvalues falls

within a set of linear operators whose Loo - induced norms are upper

bounded by a threshold value. We therefore appropriately refer to this
"quantitative pole placement" notion as a sufficient condition for
"trackability in the sense of spheres". This is clearly a stronger

notion than controllability.

The main contributions of this thesis are

( i ) A pole-placement criterion developed through a generalized LQ
formulation, for the satisfaction of the norm condition.

( ii) An algorithm based on eigenvalue-eigenvector placement for the
computation of the operator norm in multi-input multi-output

(MIMO) systems .

 

(iii) An L2 - problem formulation for a class of uncertain systems, and

a graphical design procedure for the resulting controller

realization.

1.3 Organization

In Chapter II, the concept of tracking in the sense of input-
output spheres is introduced. Then design criteria for the precise
tracking for uncertain nonlinear systems are developed by employing the
Banach fixed point theorem together with the comparison of equations.
Finally, as a special case, a system with a sector bounded nonlinearity
is considered where simpler algorithms for accomplishing the design are
obtained.

In Chapter III, servo-tracking in Lur’e type nonlinear systems is

considered. In particular, the design criteria formulated in an L2 -

setting for single-input single-output(SISO) systems with sector
bounded nonlinearities offer a neat circle type geometric interpreta-
tion.

In Chapter IV, a pole placement based on a generalized LQ perfor—
mance measure is formulated, using classical variational techniques.
By means of a limiting process, an optimal pole pattern which is com-
patible with the Butterworth configuration is obtained. This provides
a.means:fin:selecting a set of eigenvalues for the satisfaction of
design criteria.

Computer oriented algorithms are developed in Chapter V in order
to utilize the theorem developed in Chapter LI. Especially an ex-

plicit expression for the operator norm of (II-16) is given in terms

 

(iii) An L2 - problem formulation for a class of uncertain systems, and

a graphical design procedure for the resulting controller

realization.

1.3 Organization

In Chapter II, the concept of tracking in the sense of input-
output spheres is introduced. Then design criteria for the precise
tracking for uncertain nonlinear systems are developed by employing the
Banach fixed point theorem together with the comparison of equations.
Finally, as a special case, a system with a sector bounded nonlinearity
is considered where simpler algorithms for accomplishing the design are
obtained.

In Chapter III, servo-tracking in Lur’e type nonlinear systems is

considered. In particular, the design criteria formulated in an L2 —

setting for single-input single-output(SISO) systems with sector
bounded nonlinearities offer a neat circle type geometric interpreta-
tion.

In Chapter IV, a pole placement based on a generalized LQ perfor-
mance measure is formulated, using classical variational techniques.
By means of a limiting process, an optimal pole pattern which is com-
patible with the Butterworth configuration is obtained. This provides
a means for selecting a set of eigenvalues for the satisfaction of
design criteria.

Computer oriented algorithms are developed in Chapter V in order
to utilize the theorem developed in Chapter II. Especially an ex-

plicit expression for the operator norm of (II-16) is given in terms

of eigenvalues and corresponding eigenvectors for MIMO systems.
Several programming strategies and integration schemes are discussed.
Chapter VI contains several examples including a 3 degree of
freedom(DOF) robot manipulator, a synchronous machine, and a gyroscope.
Computer simulation results confirming the validity of the theory are
also given. Finally conclusions and some Suggestions for future work

are given in Chapter VII.

 

 

 

of eigenvalues and corresponding eigenvectors for MIMO systems.
Several programming strategies and integration schemes are discussed.
Chapter VI contains several examples including a 3 degree of
freedom(DOF) robot manipulator, a synchronous machine, and a gyroscope.
Computer simulation results confirming the validity of the theory are
also given. Finally conclusions and some Suggestions for future work

are given in Chapter VII.

CHAPTERII

PROBLDI I-‘ORMUIATION FOR SERVO-TRACKING

In.this Chapter the design philosophy advanced by Barnard and
Jayasuriya [4, 26] is revisited. The notion of tracking that is
central to the formulation is carefully stated. Then the tracking
problem is formulated for a class of nonlinear uncertain systems with
external disturbances. The inclusion of the external disturbance ex-
plicitly in the plant model is a minor extension of the original
fanmflations NH 26]. The uncertainties allowed are assumed to be
varying only within the boundaries of certain prescribed sets, i.e.,
they belong to certain pre-specified compact sets.

The controller structure employed to realize the tracking
specifications is nonlinear and is of a feedback fornL. In order to
estimate the inaccessible states a Luenberger type nonlinear observer
is employed. The observer realization is based on the nominal model
corresponding to the actual uncertain plant. Deviations in tracking
that arise due to uncertainties are also quantified with respect to
this nominal model. A sufficiency theorem guaranteeing tracking in
the sense of Spheres is derived. This theorem essentially captures the

pole-placement nature of the primary design criterion.

 

10

2.1 Servo - Tracking
Conventionally, tracking is referred to as following a specified

trajectory in an asymptotic sense. That is the actual trajectory y(t)
b . . b .
e R approaches a reference or nominal trajectory yo(t) e R as time t

4 w . In this notion of tracking the initial deviations in transient
performance such as large overshoots are not significant as long as the
system exhibits a stable behavior and the actual output y(t) eventually

approaches the nominal output yo(t) in spite of undesirable transients.

Opposed to this classic notion of asymptotic tracking, a "precise"
servo-tracking in which the actual output y(t) follows the nominal out-

put yo(t) within an error bound So for all time t e [0, w) is adopted

in the work presented in this thesis. This concept of tracking is
known as "tracking in the sense of input-output spheres". The precise

mathematical definition of what this means is given below.

Definition 1 : A given output {y : T 4 RP } e L:[0, m), is said to

belong to an output sphere 0(y : yo, 50) of radius fio > 0 centered at

b b . ‘
{yo : T-rR } 6Lp[0, co) 1f Hy-yoll 550, where [

 

 

I is any norm
associated with the output function space T - (y I y : T 4 Y). yO is

referred to as the nominal output and
My : yo. 50> - (yl lly - yoll 5,90}

Here T - [0, w) is the time set.

Definition 2 : A given input ( r : T 4 Rm ) e L:[O, w) is said to

11
belong to an input sphere {2(r : r0, Bi) of radius ’Bi > 0 centered at

{ ro : T 4 RP} e L:[0, m) if IIr - roll 5 fii’ where I

 

 

I is any norm
associated with the input function space U - {r I r : T a 6}. r is

referred to as the nominal input.

0(r : r0, Bi) - {r I IIr - roll 5 Bi }.

With these concepts, we can now formalize the notion of tracking

alluded to earlier as follows.

Consider a system described by the operator equation

y,0 - QO rO

where the nominal output yo 6 T , the nominal input r0 6 U and the
operator $0 : U(£) 4 T. Let r : T 4 RP be any other input

function in the sphere 0 (r : r0, fii ) which generates y : T 4 RP as
the output satisfying the operator equation

y = $0 r
with specified constants fli > 0 and Bo > 0 . If the system output
y e 0 (y : yo, Bo ) with any r e O ( r : r0, fli ), then the system is

said to track yo in the sense of input - output spheres. This idea is

illustrated in Figure 1.
In view of the above definitions, it seems appropriate to define a

Sphere accounting for allowable external disturbances as follows.

 

12

   

/

 

 

 

 

 

 

 

 

 

 

 

* ‘ Y
r + C t 11 H Plant I k
on to er #_J—7 "
+ l
-I Feedback Element I::
Figure 1 Illustration for precision tracking
Definition 3 : Let w e Lg[0, 00) be an external disturbance with the

Specified constant fiw > 0 such that

0(w:0.fiw)-{weL:[0.°°)lIIW-Ollsfiw}
where IIoII denotes any Lp - norm associated with the function space

W - {w I w : T 4 RI }. Then the disturbance is said to be belongingto a

compact set of radius fiw centered at the zero element 0 e LSIO, m).

13

2.2 System Formulation
2.2.1 Plant
Consider the MIMO systems governed by the state equations of the

form

i(t) - A x(t) + B u(t) + D w(t) + f(x(t),7,t) (II-la)

y(t) - C x(t) (II-lb)
where the state x(t) 6 RP , the control u(t) 6 RP , the uncertainty 7 e
I‘CRa, the time t e T - [0, on), the external disturbance w(t) e WCRd,

the nonlinear function f : RP x R? x T 4 RP, and the output y(t) 6 RP.
A, B, C and D are constant matrices of dimension n x n” riacxn, b x n”
and n x d respectively.

The following assumptions are made with regard to the plant
description (II-1)

( i) The pair {A, B} is completely controllable

( ii) The pair {A, C} is completely observable
(iii) The uncertain elements 7 6 PC Ra and external disturbances

w e WCRd where I‘ and W are compact sets with prescribed
boundaries.
The control problem to be considered is the synthesis of a con-
troller that assures tracking in the sense of spheres for the above
system irrespective of the uncertainties. Specifically we seek a

feedback controller which assures that every output y is in 0(r : y

9

0

50) for every input r in 0(r : r Bi) and any disturbance w in 0(w :

0’

WC, fiw), no matter what Specific value the vector parameter 7 takes in

the prescribed boundary.

14

2.2.2 Nominal Plant
By considering a hypothetical plant which is completely known
(i.e. , free of uncertain elements and external disturbances) we estab-

lish a nominal plant corresponding to equation (II-l) given by

x(t) - A x(t) + B u(t) + fo(x(t),t) (II-2a)
y(t) - C x(t) (II-2b)
where x(t) 6 RH , u(t) e Rm , time t e T - [0, an), the nonlinear
_ n n b
function fo . R x T 4 R and y(t) e R . A, B, and C are constant
matrices of dimensions n x n, n x m, and b x n respectively. It is

w)orth noting here that the nonlinear design function fo may be chosen

to be the nonlinear uncertain term, with the uncertain parameters re-
placed with certain nominal values.
Based on this nominal plant, a nonlinear observer is constructed

next for estimating the system states, which may be inaccessible.

2.2.3 Observer and the Controller
For the implementation of any state feedback controller for the

plant (II-1), a nonlinear observer based on the nominal plant is

employed. This observer is synthesized according to the following
equations

x(t) - A x(t) + Bu(t) + fo(x(t),t) + GC(x(t)-x(t)) + V1r(t) (II-3)

A

where x(t) e Rn, u(t) e Rm, c e T [0, co), r(t) e Rm, f0 : R" x T -» R".
G and V1 are constant matrices of order n x b and n x m respectively.

The State feedback control law given by

15

A

u(t) - v2r<t> + K x(t) (1x-e)

is postulated as a possible controller for servoaction, where t1(t) e

A

m

R , r(t) e RP and x(t) 6 RP, and K and V2 are constant design

matrices of order m x n and m x m respectively.

The feedback system represented by equations (II-1), (II-3) and

(II-4) are combined into the form

i A BK x sz D
. .. A + r(t) + W(t)
x -cc A+BK+GC x Bv2+vl o
I f(x(t). 7. t)I
+ A (II-5a)
f (X(t). t)
O
y(t) - [ c o 1 x I (II-5b)
;

This augmented system configuration is Shown in Figure 2.

Rewriting (II-5) in a combined state form gives

é(c) = R z(t) + Bor(t) + B1w(t) + 6(z(t), 7, t) (II-6a)
y(t) = c0 z(t) (II-6b)
PX
where z =
L_X
' A BK
R—
_ -cc A+BK+GC
' BV
B a 2
0
_BV +v

 

2 1

l6

 

 

U cer in Plant

 

 

O

 

 

 

 

 

 

r‘_—-—_*‘_- _-“ "““‘F'l
S

L________L__-__-_j
I K 1.

|L__.________._._____

 

 

I..._._..____...____._._._...__._...___._..__._l

Figure 2 Controller system configuration

17

D
Bl =

o
co — [ c o 1

f(X(t). 7. t)
0(Z(t), 7. t) -

fo<§<t>. c)

2.2.4. Operator Representation
The augmented system incorporating the plant and the observer
leads to an operator formulation that lends itself especially well to

tracking analysis. Taking the Laplace transform of equation (II-6)

yields
32(3) - 2(0) - R 2(5) + BOR(S) + B1W(s) + 5 (II-7)
where Z(s) = £ z(t)
R(s) - £ r(t)
W(s) - £ w(t)
5 - £ V(Z(t), 7: t)
and 2(0) -

x(O) is the initial combined state.
X(0)

Rewriting (II-7) yields
[SI - R] 2(s) - BOR(S) + B1W(s) + 5 + 2(0) (II-8)
Where 1 denotes the Zn x 2n identity matrix.
. . . -l -l .
Premultiplying equation (II-8) by P (s) - [SI - R] , we obtain

Z(s) = P-1(S)BOR(S) + P-1(S)B1W(s)+P-1(s)5 + P-l(s)z(0) (II-9)

Taking the inverse Laplace transform of (II-9) yields

l8

1 l 1

z(t) - £-

{P-1(S)BOR(S)} + £' {P-l(s)B1W(s)) + £‘ {P'1(s)5}

f'l{P-1(s)z(0)} (II-10)

By utilizing the convolution theorem, (II-10) can be written in the

integral form
t R(t - ) R
z(t) - I e r [Bor(r) + B1w(r) + 6(z(r),1,r)]dr + e tz(0)
0

(II-ll)

1 Rt

where £- {P-1(s)) - e

Remarks :

( i) f stands for the Laplace transform operator and

f- stands for the inverse Laplace transform operator

(ii) If £{fl(t)} - F1(s), £ {f2(t)} - F2(s) and the convolution

t
( f1* f2) - IO f1(t-r) f2(r) d1 , then

f {f1* f2} - F1(S)°F2(S)

.Assuming that the function 6(z(t),'y,tfl is continuous, and the
eigenvalues of the matrix R are in the open left half complex plane
(i.e., the augmented system represents a stable behavior ),vm:can
write (II-ll) in an equivalent operator form

z(t) - WNyz(t) + WBor(t) + WB1w(t) + q(t) (II-12)

where W and N7 are respectively a linear and an uncertain nonlinear map

from L:n(T) back into itself given by

19

C

(u z)(t) - I eR(t") z(f) dr
0

(N72)(t) - 0(Z(t). 7. t)

and q(t) - em: 2(0)

The closed loop system in this operator form is shown in Figure 3(a).

 

 

Figure 3(a) Combined uncertain system

Now by augmenting the nominal plant defined by (II-2) with the
nonlinear observer given by (II-3), a set of nominal operator equations
can be immediately defined as

zo(t)- W Noz°(t) + W Boro(t) + q(t) (II-13a)
yo(t)- Cozo(t) (II-13b)
where (r0, yo, 20) is a completely known triple of a specified nominal

output yo, and corresponding solutions 20, re, with ro serving as a

20

nominal command input relative to yo. The nonlinear map No is repre-

sented by
fo(x<t),t) I

(Noz)(t) - I A
f°(x(t).t>

The nominal system charaterized by operator equation (II-13) is shown

in Figure 3(b).

 

 

Figure 3(b) Nominal system

Remarks :

( i) It should be pointed out that the function space L:[0, 0) is used
rather than L:[0, w), or L3 [0, w), because the norm denoted by

II-II associated with Lb[0, m) can represent more precisely and
Q

naturally the physical constraints usually attributed to track—

ing. Thus, throughout the thesis II-H denotes L 6-norm

defined in the Appendix A, unless otherwise specified.

21

( ii) For MIMO systems where the nominal outputs yolnouo yob are to
be tracked within the Spheres of radii 501 ------ fiob’ an effec-
tive Sphere of radius Bo is considered such that

flo — m1n{flol coo... flob}°
Then the effective output sphere is defined by]
ll y - yo IILP s 50 .
m
so that the system can be treated via a single Sphere condition.

However due to the overly restrictive error bound So, the results

are usually very conservative.,

(iii) To obtain less conservative results, weighting factors 89,, :i -
floi

1, oo--, b, may be employed so that
|| 20(y - yo) IIL: S 30

Where 20 is the RP x b nonsingular diagonal weighting matrix given

by

‘bk

3
01 ° . (::)

o o

0'92
- 50b

2.3. Design Criteria

 

p- -
_

 

To develop controller criteria for servo-tracking in the sense of
spheres, a combined equation comparison and the local form of the

Banach fixed-point theorem are employed. First, the comparison of the

22

actual and the nominal systems, (that is, the comparison of any uncer-
tain combination (r, y, z, w) satisfying equation (II-12) and a

completely known combination (r0, yo, zo) satisfying equation (II-13))

leads to the error type operator equation

2 - zo - W(N1z - Nozo) + WBo(r - r0) + WBlw (II-14a)
y - yo = Co(z - zo) (II-14b)

The initial values of the actual and nominal systems are assumed to be
identical.
Rewriting equation (II-14) by introducing the nonsingular weighting

matrix W0 yields

W z - W W(N W-lW z - N W-1W z ) + W WB (r — r ) + W WB w + W z
o o 1 o o o o o o o o o o 1 o o

-l
and y - yo - COWO (Woz - Wozo)

which can be written as

z - o z
- w w(N w‘lé - N w ‘12 ) + V vs (r-r ) + w o3 w + Q (II-15a)
o 7 o o o o o o o o 1 o
-1 - -
and y - yo - COWo (z - zo ) (II-15b)

L2n
Q

where the nonlinear map Q : (T) 4 L:n(T),

and z - W 2

Equation (II-15) is a fixed point equation. In essence the discussion
of the tracking associated with the original problem has now been

reduced to a study of the solutions of this fixed point equation.

 

23

By applying the Banach contraction mapping theorem to equation
(II-15) , we obtain the following result which gives sufficient condi-

tion on design elements G, K, V1,'V2, and the design function fo that:

assure servo-tracking in the sense of input-output spheres. This
theorem is central to much of what is discussed in the remaining chap-
ters. Computation of operator norms involving the nonlinearities and
uncertainties are important to fix the linear operator norm pivotal for
the satisfaction of design criteria as well as effective evaluation of
this norm. This norm depends quite naturally on the eigenstructure of

a related linear operator.

Theorem 1 : Let f and fo be continuous, and let G and K be assigned

, so that the eigenvalues of matrix R are in the open left-hand complex

plane. Let (r0, yo, zo) be a known combination satisfying equation

(II-13) and (r, y; z, w ) be any combination satisfying equation (II-

12). Then for any input r in the specified sphere

0 (r : r0, Bi) - { r e L: I II r - ro II 5 Bi }

and for any external disturbances w in the specified sphere
0 ( w - 0 5 ) - { z e Ld | || w || < 5 }
' ’ w w ’ w

there exists a unique combined response 2 ixltflue specified flo-

neighbourhood
2n
0 (z : 20. Bo) - { z 6 Lb I II WO(Z ' 20) II S flo }
50
provided n s (II-16)

 

pofii + plfiw + 52 + p350

 

24

 

where n - II Wo W Qo'1 II

Po - ll QOBO ||

Pl ' II QoBl II

p: - :2? || Qo I N720 - N020 1 ||

|| Qo[ N1 2 - N12' ] ||
p3 = z z'fggzzz fl ) ll wo( z ' 2') II
’ 2 fl 2' o, o
7 e F

with respect to a nonsingular constant matrix Q0.

In order to prove this theorem we invoke the following.

Lemma 1.: let (x, II'II) be a Banach Space and let 0

B

sphere in X with center at xo 6 X.and constant B > 0 such that

(x0) be a closed

0B(xo) - { x I II x - xo II 5 fl }

Let W : X.4 X. be a linear operator satisfying the following

conditions : with constant K, 0 < n < 1
( i) II W (x) - W(y) II 5 5 II x - y II, V x, y e 0fi(xo)
(ii) || o (x0) - onI s 5 (1 - n)

Then it follows that

( i) W maps 0fi(xo) back into itself

*
(ii) W has a unique fixed point x e 0 (x0) such that

3
W (x*) - x*

Moreover,

25

. * . n
( i) x - nlimm xn — W (x0)

where xn+1 - W(xn), n - 0, 1, 2, ...

and x0 is any element in 05(xo)

.. * nn
(11) II x - xn II S ___ II W(xo) - xo II
l-n
Proof of Lemma 1 : See Martin ([33], Chapter 4 )

We follow the steps given below to prove the theorem.
( 1) First, we establish the conditions needed for the map W to be
a contraction.
(ii) Next, we use the fact that W must be a contraction in the

output sphere 0(2: 20, Bo) and that the closed sphere

ll W 20- i0 ll

1 - n }

 

_ 2 - -
01 - { z e Lmn(T) I II z - zoII 5

must be entirely contained within 0(2: 20, 50) which estab-

-*
lishes the existence of a unique solution 2 in the output

sphere 0(z: 20, 50).

Proof of Theorem 1 :

Consider two arbitrary points i, i' e O(z : 20, BO) and compute

II W é - W 2' II, yielding

l

e
N
I
e
N‘-
ll

-1 - - -1-, _
wowoo (20(va0 2 - vao z ) (11 17)

which on taking the Lm- norm on both sides yields

 

26

 

 

 

II W 2 - W 2' II 5
-1 -1-
_ IIQINWz NWz'II- _
IIWOWQolll sup ° “3 _”° llz-z'll
z.z'en<z z .5 > || 2 - 2' II
2 ” z' 0 II 18
7 e F ( ' )
- - - 2 - -
where 0(z : 20, BO) - { z e Lbn[0, m) I II z - 20 II 5 Bo }.
Remark : It is easy to Show that if z 6 Lin that z 6 Lin.
. 2n 2n . . . .
For the operator W . LCD 4 Lb to be a contraction map, it is required
that
Moi-oi'HSAIIi-EIL o<e<1
where
-1 -1-
llQlNWz-NW z'll
n - IIWOWleII_ _ sup_ _ o 1 ? _ 1 o < l
z,z:e0(z:z ,fl ) II 2 — 2’ II
2 i z' o o
7 e P
which leads to the condition
ll Q l N z - N 2' 1|!
5 - IIWOWleII sup ° 7 7 < 1 . (II-l9)
z,z'eO(z:zo,flo) II Wo(z - z') II
2 i z'
7 e F
Next, we consider
oi -é - w WQ'lQ (N w'lé - N w‘lé ) + w WQ‘IQ B ( r - r )
o o o o o 7 o o o o o o o o o o
+ w wo'lo B w (II-20)
o o o l '

Taking the norm on both sides of equation (II-20) together with the

triangular inequality gives

27

- - -l -l- -1-
|| o 20- 20 || 5 || wowoo || ( :3; IIQO(N7WO zo- NOWO zo)II
+ II QOBOII llr - roll + IIQOBIII IIWII}
which can be written as
- - -1
II W 20- zo II 5 II WOWQo II { sup ||Q0(N7 zo- Nozo)II
76F
+ II QOBOII ||r - roll + IIQOBIII llwll} . (II-21>

Since the output is required.to be in the sphere 0(z : 20, BO), we re-
quire
|| o 20- 20 || 5 50(1 - n) (II-22)

which follows from the Lemma 1. Therefore, from equations (II-l9) ,
(II-21) and (II-22) we obtain the inequality of the theorem given by

flO

n S
pofli + plfiw + p2 + 5350

with BO, 51, 52, and 53 defined as in the statement of the theorem.

This completes the proof of Theorem 1. D
Remarks : Inequality (II-16) will be referred to as the primary
design criterion for precision tracking in the sense of Spheres. Some

important design features of this criterion are

( i) Design elements G, K, V1, and V2 must be chosen so that the

eigenvalues of the matrix R e a?“ x 2n

are at suitable locations
in the open left-half complex plane and that the inequality (11-

16) of the Theorem 1 is satisfied. The upper bound on the

operator norm II WOWQ;1 II depends on the design specifications

 

28

such as tracking accuracy, the extent of the disturbances and the
size of the uncertainties. Thus for precise tracking in the
presence of large disturbances and large plant uncertainties, a
1
II

small value of operator norm II WOWQ; is typically needed

which might result in high gain feedback. It should be noted
however, that the norm bound requirement is only a sufficient

condition.

( ii) A larger upper bound for the linear Operator norm II WOWQ;1II

(iii)

can be allowed by a proper choice of a nonlinear design function

£0. The norm values 52, p 3 and n play a vital role in the

design procedure. 52 can be regarded as a measure of the maximal
difference of operator N7 and operator N0 at 20, 53 is a measure

of the severity of the nonlinearity and the uncertainty of the
system, and n is a measure of the "trackability" of the systeuL

The function fo Should be assigned so that the values of 52 and

53 will allow a larger upper bound for the linear operator norm

~1
II WOWQO II.

A Quantitative pole-placement is defined by (II-16). That is, a
proper selection of eigenvalues of the matrix R will potentially
enable one to satisfy the operator norm condition. To achieve

an optimum design the eigenvalues must be placed such that the

29

operator norm is as close as possible to the threshold value,

30 .
pofli + plfiw + P2 + P350

( iv) The nominal plant defined in equation (II-2) is an essential part

of the design criteria. That is, the nominal input ro must be

determined so that (re, yo) satisfies the nominal equations.

2.4 A. Special Case

Although it is theoretically possible to compute the norms 52 and
53 in the inequality (II-l6) for virtually any nonlinearity, it is very

useful to have special classes of nonlinearities which can be handled
rather easily by using simple algorithms. To this end, a special type
of nonlinearity, i.e. a sector bounded nonlinearity, associated with
the system given by (II-l) is considered in this section. It is

defined in precise mathematical terms as follows.

Definition 4 : If a nonlinear function ¢(., .) satisfies

( i) $(0, t) - 0 for all t e [0, m)

(ii) a 5 s B , for all $1 # W2 and

$1 - g02 for some a, B 6 R1.
then o is said to belong to the sector [a, B], or to be confined to the

sector [a, B].

30

The above definition implies that the nonlinear function.¢K-, .) lies
between two straight lines having slopes a and ,8 respectively and
passes through the origin.

Now consider the system given by (II-l)

i(c) - A x(t) + B u(t) + D w + f(x(t), 7, t) (II—22a)

y(t) - C X(t) (II-22b)

where f(x, 7, t) : RP x R? x T 4 RP satisfies Definition 4, and the
other variables are the same as those defined in (II-l).
Consequently we can establish a nominal plant corresponding to

the above system (II-22) of the form

i(t) - A x(t) + B u(t) + fo(x(t)) (II-23a)
y(t) - C x(t) (II-23b)
“ 4 RP.

where f0(x) : R

Due to the characteristics of the sector bounded nonlinearity, the

nonlinear design function fo(x) can be chosen to be linear with slope

equal.tx> the arithmetic mean of the lower and upper bounds a, B on the

nonlinearity. Thus,

fo(x) = % (a + 5) x(t) (II-24)

For the state feedback design scheme, the same type of observer given
by (II-3) based on the nominal plant (II-23) is employed.

Fbllowing,the same procedure outlined in section 2.3, we can.ob-
tain Simple formulae avoiding difficult computations for evaluating
each norm in the inequality (II-l6). Namely, 52 , the measure of the

maximum difference of N1 and NC at z , is

O

31

p2 - :2? II Q0 [N720 - N020] II

- % (a + 5) sup || Q0 20 ||

and the measure of the severity of the nonlinearity, 53 , is

 

H<%[le-Nf'1 H
p = SUP I
3 Z,Z'€0(Z:zo’fio) II WO( 2 - z ) II
2 fl 2'
7 e P
- max{ IaI, |5| } (II-25)

for a proper choice of the weighting matrices W0 and Q0.

Remark : Consider, for example, a system with the nonlinear function
0
f(x, 7) - , where I7I s 0.2, and choose the weighting
7 Sin x1

matrices W0 and Q0 as

l 0 l 0
W0 "‘ 1 i and Q0 3
0 I O 1
max
Consequently,
1 0 0
N - N ' -
Qo[ 7x 7x 1 . .
0 1 7(Sin Xl-Sln x1)
0

7(Sin Xl-Sln x1)

and W [x - x'] -
o

 

32

x1 ‘ xi
1 a
A ("2' x2)
max

Since N7 is only a function of the state x1, p3 is the maximum gradient
of N7 with respect to x1, or 53 — I7I. However, if Q0 and Wo are

chosen as

then QO[N1x - vi'] -
7(Sin xl-sin xi)

.1. .
A ( x1 ' X1)
and Wo[x - x'] - max
x2 ' x2
Hence, p3 = AmaxI7I which is not the same as max { IaI, IflI }.

Finally, inequality (II-16) reduces to the following

3° (11 26)
pofli + plfiw + 52 + p350

 

where n - II WOWQO-1 II

50 || Q08o ll

p1 - II QoBl II

yaw) sup n Qozo H

p, - max { Ial . lfil }

 

33

Inequality (II-26) is the same as (II-16), however, we can eliminate

involved computations for evaluating the norms 52 and 53 in this Spe-

cial case. A reasonably large class of physical problems can be
brought into this form. A single DOF example is considered in Chapter

VI to illustrate the use of this special form.

 

CHAPTER III
SERVO-TRACKING IN A LUR'E TYPE SYSTEM

In Chapter II, we considered the input-output tracking problem

from an LCD point of view. Design criteria were established for this

servoproblem that were primarily numerical in character. It is in
general difficult to generate explicit closed form results and/or

elegant geometric interpretations in the Lao - setting. The latter ob-

servation was motivation for the study of servotracking in a so called
Lur'e type system.
At a general level we are concerned with the servotracking in un-

certain nonlinear systems in a L2 - setting. The often considered

asymptotic tracking problem can be captured in such a formulation. In
asymptotic tracking one typically considers the behaviour of the error
vector as time gets infinitely large. No global error measures are
employed. In our present formulation however we impose a global
tracking error bound in addition to the asymptotic requirement, i.e.,
if yoi(t) is a desired nominal trajectory and yi(t) is the actual
trajectory, then we require
( 1) | yi(t) - yoi(t) | e o as t e w, v 1 = 1, ..., n

and (11) || y - yo ||L2 5 fi

0

The general servomechanism problem has been addressed by Desoer

and Wang [14] , Solomon and Davison [46], and Barnard et.al. [5].

34

 

35

Except in [5] the notion of conventional asymptotic tracking was
treated . inn [5] the notion of tracking employed is similar to our
present work.

The class of systems whose forward loop has a linear, time-
invariant subsystem and whose feedback loop contains a memoryless time
varying nonlinearity is what is typically known as a Lur'e type system.
This configuration, though simple, includes a fairly large class of
important feedback systems and has been studied quite extensively from
an absolute stability view-point [36, 41, 42, 50, 54, 55]. .A classic
example of a system in this class would be a set of coupled nonlinear
oscillators where the restoring force is nonlinear. In what follows we
formulate the servotracking problem for this class and give a methodol-
ogy for the direct design for tracking specifications. We also give a
geometric interpretation of the design criteria in the case of a 8180
system where the nonlinearity is assumed to be sector bounded” This
interpretation is given in the frequency domain and is similar to the
Nyquist stability criterion and the circle criterion for the absolute

stability problem.

3.1 Statement of Prdblem
We consider the system shown in Figure 4 given by the state equa-

tions

i(c) - A x(t) - B ¢(y(t),t) + B u(t) (III-la)
and the output equations

y(t) - C x(t) (III-la)

where the state x(t) 6 RP, the control u(t) 6 RE, t e T - [0, m), the

output y(t) 6 RP, and the nonlinear function ¢(~, .) 6 RP x T 4 RE is

 

 

36

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4 Configuration of control system

37

continuous in both its arguments. A, B, and C are constant matrices of

order n x n, n x m, and b x n respectively.

Remark : Note that the above system description incorporates a
linear, time-invariant subsystem.and a nonlinear, time varying element

in a feedback path.

We make the following assumptions regarding the system.
( i) The pair {A, B} is completely controllable
( ii) The pair {A, C} is completely observable

(iii) The nonlinearity W(‘,') satisfies the memoryless condition

u(o, t) = o 6 R? v t e [0, m)
and the generalized sector bound condition

||[W(Wl. t) - vwll- [W(¢2. t)- vwzlll S €(V) Ilwl - ¢2II
(111-2)
1 b
u e R., v c e [0, m) and v 51, ¢2 e R , where §(u) - ( |5 - u|, |u -

aI } for real numbers a, 5 satisfying 3 2 a and 3 > 0.

Control Objective : The primary design objective here is to synthesize
a controller that tracks a specified nominal output yo 6 L3 [0, m)
within a pre-specified tolerance 30 with respect to the L2 - norm i.e.,
we require

ll y - yo IIL2 S fio

despite the uncertainties in the system, nonlinearity and the reference

input. The input uncertainties are assumed to be of the form

38

and may be viewed as a disturbance with finite energy. We say it is

of finite energy primarily because the measure employed is the L2 -

norm.

3.2 Problem Formulation
Control : We consider the state feedback control law

u(t) = r(t) + K x(t) + ¢o(y(t), t) (III-3)
as a means of satisfying the design Specifications, where u(t) e R? is
the control, r(t) 6 R9 is the reference input, ¢o(y(t), t) e R? is a

nonlinear design function in the class of generalized sector bound
functions and a constant feedback gain matrix K of order m x n .

Combining the system equations (111-1) and (III-3) yields
r(t) = (A + B K)x(t) - B¢(y(t),t) + Br(t) + B¢o(y(t),t) (III-4a)

y(t) = C X(t) (III-4b)

which can be written as

i - A x - B w(y,c) + B r + B ¢o(y,t) (III-5a)

y C x (III-5b)

where A - [A + B K] 6 R? x n.

Next, we formally transform equation (III-5) into the integral

form following the procedure of Chapter II given by
t A(t-r)

X - I e [-B¢(Y(T). f) + B¢o(Y(T).T) + Br(r)] d? + q(t) (III-6a)
o

with the output equation

y — C x (III-6b)

39

where q(t) = eAt x(0), and x(O) is the initial state. Since the

function $(y, t) and ¢o(y, t) are continuous, equation (III-6a) can

be written in the standard operator form

x - -WBN7y + WBNoy + WBr + q (III-7)

where W is a linear operator and N7, N0 are nonlinear operators mapping

L3[0, m) back into itself given by

t -
(W x)(t) = I eA(t-T) x(r) dr
0

(NVY)(t) = ¢(Y(t).t)

(Noy><t> = wo<y<t).t)

To Show that N7, N0 : L§[0, m) 4 L;[0, m) it suffices to consider
IIN y - uy II - IN y - uy I2 at
7 l l 2 0 7 1 l

2
- I: I¢y1 - ule dt

From the sector bound condition it follows that
2 2
I” Iwyl - vyll at s max {I5 - vl. Iv - al) Ilylll
0

Hence (15y1 - vyl) e L? since y1 e L3[0, m). Now it follows that wyl e

L§[0, m) since L§[0, m) iS a linear space and vyl e L$[0, m). Thus

°moo-+moo
N7 . L2[0, ) L2[0, ).

On premultiplying (III-7) by the matrix C, we obtain

0 x - -CWBN7y + CWBNOy + CWBr + Cq (III-8)

40

which can be written as

y - -LN7y + LNoy + Lr + q1 (III-9)
where L - CWB

q1 - Cq .

The actual servosystem is now characterized by the input-output
representation (III-9). In order to estimate the tracking error we now
compare this system with a hypothetical reference model characterizing
the nominal system corresponding to (III-9). This nominal system is

considered to be given by

yo - Lro + q1 (III-10)
where (yo, r0) is a completely known combination of the prespecified
nominal output yo, and the corresponding solution ro serving as a

nominal command input.

Remark : In the above we have assumed the same initial conditions for
the actual system and the nominal system . If they are not the same
it can.be accounted for by adjusting the radius«of the output sphere.

For a treatment of this notion see Jayasuriya [25].

3.3 Main.Result

To establish design criteria under which servo-tracking in the
sense of sphere is guaranteed, comparison of equations and the Banach
fixed point theorem are employed. By comparison of any combination

(r, y) satisfying equation (III-9) and known combination (r0, yo)

satisfying equation (III-10), we obtain the difference

y - yo = -LN7y + LNoy + L(r - r0) (III-ll)

41

which governs the tracking error.
Now we rewrite equation (III-ll) in the Hammerstein form
y - -LN7y + LNoy + L(r - r0) + yo
- -LN7y + r1 (III-12)
where r1 - LNoy + L(r - r0) + yo .
Invoking the local form of the Banach Contraction mapping theorem

with the fixed point equation (III-l2) obtained via equation com-

parison, forms the basis of the following theorem for servotracking in

 

 

 

 

the L2 - sense. For the rest of this chapter . denotes the L2-
norm.
Thegge; 1 : Let the eigenvalues of the matrix A be in the open left

half complex plane. Let r e Lm[0,W) and y e Lb[0,m) be an combination
. 2 2 3’

satisfying (III-9) and r0 6 L§[0, m), yo 6 L3[0, m) be a known combina-
tion satisfying (III-10). Then for any input r(t) in the specified.

input Sphere

0(r: r0, Bi) - { r e L§[0, m) I II r - ro II 5 Bi }

and for any nonlinearity in the sector [a, ,9], there exists a unique

response y(t) e Lg[0, m) in the specified output Sphere

n<y= yo. 5,) - 1 y e L310. w) I II wo<y - yo) II s A, 1

pofio

 

provided 0 (III-l3)

I31 + 51 + (102 + p3)flo

where n - IIW L Q

42

 

 

-1
Po - || Qo ||
"1 ' ”0 ,Stpr II Q0 ( N770 ' Noyo ) II
II Qo (va - va' ) II
P - p sup _ l
2 ° y.y'eo(y:yo.flo) ll wo<y y ) II
y r y'
7 e F
ll Q, (no, - Noy' > ll
P3 - P SUP W _ I
o y’y'eo(y:yo’flo) II 0(y y ) II
y r y'

Proof of this theorem follows the same reasoning given for the main

theorem of Chapter II.

Proof of Theorem 1 : Consider the operator equation (III-12) which can

be written as the fixed point equation

y - W y (III-l4)

1 1

- -1 - - -1 -
y + WOLQO Q N w y + WOLQO Qo(r - r0) + yo

-1 -
- 'WoLQo Q N W o o o

o 7 o

where y - Woy, y - Woyo, and W0, Qo are nonSingular weighting

0

matrices.

Consider two arbitrary points y, y' e 0(y : yo, Bo) and compute

- -, -1 -1- -l-,
WY - WY -(WOLQo ) (QOIN7Wo y - NVWO y 1)
+ (w LQ'l) (Q [N w‘l‘ - N w’1" 1) (III-15)
o o o o o y o o y
which on taking the L - norm on both sides yields

2

43

 

 

' -l- -l-,
_ _ _1 supA ||Q0(N7Wo y - N711o y >||
lley - ey'll s IIWOLQO II 19,9:en<g:yo.50> II , _ ,' ll
y g y' Y Y
7 e F
-l- -l-,
sup IIQ0(Nowo y ' Nowo y )II
+ - -' Q ; ’ - - 7' II-l6
y,y e (1 yo fie) || 9 _ 9' I! l lly V II (I )
y r y'
Thus, for W to be a contraction we require that
||®-o¥||s~||§-y|L 0<n<1 Guam

Consequently from (III-l6) and (III-l7) we get

ll Qo (Nyy - Nyy' ) ||

 

 

-l
K ' IIW LQ II { sup _ '
o o y,y'efl(y:yo,fio) II W0(Y Y ) II
y # y'
7 e P
ll Qo (Noy - Noy' ) ||
+ sup - . } < 1 (111-18)
y,y'60(y:yo,flo) II wo(y Y ) II
y r y'

Next, we consider

1- -1-
yo - Nowo yo) + Qo(r - r0) }

- - -1 -
Wye - yo = - (WOLQo ) { QO(N1Wo
and take the L2 - norm on both Sides along with the triangular ine-

quality to get

- - -1 -1- -1-
Ilry, - yell s IIWOLQO II { jug I,Imowo y, - Now, ,0, II

+ llQo || || r - r0 || } (III-19)

Since the output is required to be in the sphere 0(y : yo, 30) we re-

quire

IA

II Rio - 90 ll flo ( 1 - n ) (III-20)

44

Finally, from (III-18), (III-19), and (III-20), we obtain the ine-

quality (III-13)

pfi
HwoLtlus °°

 

fii + P1 + (P2 + P3)flo
with 50, pl, 52, and 53 as given in the statement of the theorem.

This completes the proof of Theorem 1. D

3.4 A.Special Case

The above result when specialized to the SISO case admits a very
elegant geometric interpretation. This geometric interpretation is
brought about by minimizing the contraction coefficient that results
from the application of the fixed-point theorem. This minimization in
a sense leads to the least conservative design obtainable via the
operator methods employed here.

In what follows we will first establish conditions for the minimum
contraction coefficient of a Hammerstein type equation with the general
nonlinearity considered above. Then, we will interpret this minimum
contraction coefficient in terms of the frequency response of the
linear time-invariant portion of the overall system. Finally, we will
combine this with the SISO version of the main result given earlier
to yield the geometric interpretation.

First write the general Hammerstein equation given by (III-12)

y---I.ley+r1

where L is a linear operator mapping a Banach space Y into itself and

r1 is a term accounting for both independent energy sources and initial

conditions . We assume the nonlinearity to be of the wider class

45

N : Y 4 Y such that for any real constant v and for all y1, y2 6 Y

lleyl - vyl - (N7y2 ° vy2)|| S £(v) lly1 - Yzll (III-21)
where £(u) - max { Ifl - VI, Iv - aI } for real numbers a , fl satisfy-
ing fl 2 a and B > 0 as defined earlier. Sector bounded nonlinearities
clearly belong to the class (III-21). This can be shown by consider-
ing

W - NVY

- ¢(y(t). t) : LZIO, an) 4 L2[0. 00)

satisfying the following conditions
( i) u(y(t), t) is measurable when y(t) is
( ii) w(O, t) - 0 for all t e [0, a)
(iii) For two real numbers a and B
vs (elm. t) - m2“), t)

¢1(t) - ¢2(t)

 

IA
7(1)

where W1 # m2 , and t e T - [0, m).

Condition (iii) implies that the nonlinearity is confined to a sector

[a, B] whose lower and upper bound are a and B respectively. At any

fixed t e [0, m), the function u(y(t), t) - v y : RP x T 4 RP has a
Slope with magnitude not exceeding max { I B - VI, Iv - aI }. Hence,

||N -u -(N -v )||2 - |¢( t)-v -(¢( t)-V )|2 dt
7Y1 y]. 7Y2 3'2 0 3’1 9 Y1 yz v YZ
2
|

s(max{Ifl-vI,Iu-al})2 IZIy1 - y2 dt

2 2
-(max{Ifi-v|,Iv-al}) Ilyl ‘ Y2I|

 

46

so that (III-21) is satisfied.
Next, we assume that the linear operator L can be defined by the
convolution
y(t) - In h(t - r) u(r) d1
0
- L u

where h e L1[0, m) and L : L2[0, m) 4 L2[0, m)

Remark : This is obviously true if the linear time-invariant subsys-

tem of the forward path is asymptotically stable, which is equivalent
to requiring that A be a Hurwitz matrix. The latter condition be

clearly met since A - A + BK where {A, B} is controllable.

Moreover, the Fourier transform of h, h(jw), is assumed to satisfy

[1 + l (a + 5) h(jw)] 4 o , v w 6 R1

2

In Theorem 2 we develop the minimum contraction constant for the

Hammerstein equation.

Theorem 2 : Let the eigenvalues of the matrix A be in the left half
complex plane, then the minimum contraction constant of the equation

(III-12) for the SISO case is

l (5 - a) sup 1 | [ 1 + l (a + 5) h(jwn'1 h(jw) | < 1 (III-22)
2 w e R 2

 

47

It is worth noting that (,6 - a) is a measure of the deviation of
the nonlinearity from linearity. When a - fl , the contraction constant

becomes zero and the Nyquist stability criterion is recovered.

To prove Theorem 2, the following lemmas are needed.

Iemna.l : Let u be a real number such that ( I + u L )'1 exists and

suppose that

x(u) - ||(1 + uL)'1L|| {(v) < 1 .

Then for any r 6 Y, there exists a unique y 6 Y satisfying (III-12).

1

Note that x(v) is the contraction coefficient.

Proof of Lemma 1 : Before applying the contraction mapping theorem to
(III-12), it is modified to another mapping whose fixed point is also
the fixed point of the original mapping. This modification is used to

facilitate the idea of minimizing the contraction coefficient.

Let the operator N7 be defined by

N - + N . III-23
7y uy 7y ( )
Combining (111-12) and (III-23) yields

y - -L ( vy + Nyy ) + rl
which can be written as

( I + VL ) y - -LN7y + r (III-24)

1 .
Since u can be chosen so that (I + 11L).l exists, (III-24) becomes

(III-25)

y = -(I + uL)'1 LEVY + (I + uL).1 r1

48

- ¢ y
whose solution is clearly the same as that of (111-12).

Now consider
-1 -
¢yl - ¢y2 - -(I + vL) LN7(y1 - yz) (III-26)
where yl, y2 6 Y.

Taking the L - norm on both sides of (III~26) yields

2

-1 ~ ~
Il¢yl - ¢y2II - I|< I + uL) L(N,y1 - N7y2> II

-1
||( I + uL) L(N7y1 - va2 - u(yl - y2)II

 

II I L '1L N7y1 - vaz II II

s + - -

< u > < Y1 _ Y2 v) Y1 yzll

5 ||( 1 + .4)1 LII f(u) ||yl - y2II . (111-27)

If the coefficient x(v) - || (I + uL)‘1L || 5(u) < 1 in (111-27),
then the operator ¢ is a contraction. Thus there exists a unique y

6 Y satisfying (III-12) and (III-25), according to the contraction map-

ping theorem. D
Lemma 2 : Let S be the set of real V such that (I + uL)'l exists,
i.e.,

S = { I u e R, (I + 11L)‘1 exists }

K

and if there is a real number v e S such that x(u) < 1 , then

x(uo) inf x(v)

i.e., x(uo) S x(v)

where u = E (a + 5).
° 2

49

Proof of lemma 2 : We first Show that (I + Vow-1 exists. If L - 0,
the inverse obviously exists. If L vi 0 and u e S , then II( I +

uL)'1L|| a o.

I + v L - I + VL + (v - V)L
O O
- (I + vL) [ I + (”c - V)(I + uL)'1L] . (III-28)

In equation (III-28), (I+ 1/OL)-1 exists if Ivo - uI II(I + vL)-1LII <1.

This condition is satisfied because Iuo - VI 5 £(v) (See Figure 5(a))

and ||(: + uL)’1L|| 5(u) < 1 . Since g(uo) - 5(u) - [yo - u| as shown

in Figure 5(b).

a u go - 2 (a + 3)

 

how:

 

 

 

 

 

€(u) - max [Ifl - yI, Iv - al}

Figure 5(a) Illustration of a, B, u, and V0

0 u Vo- %(a+fi) ‘5

 

soc)

 

 

€(V)

 

 

 

Figure 5(b) Illustration of a, B, €(Vo), and ((v)

50

Now consider

||<I+voL>'1L|| 5(vo>

II(I+voL)-1LII€(u)-Iu-VOI II(I+VOL)-1LII

||(1+uL)‘1L+(1+VOL)'1L-(I+VL)L||§(u)-|u-uo| ||(I+uoL)‘1L||

IA

||(1+uL)’1L||§(u)+||(I+uoL)'1L-(I+uL)'1L||g(u)
-|u-uo| ||(I+uoL)'1L|| . (III-29)

Here, (1+uoL)'1- (In/L)“1 = (1+uoL)‘1[(I+uL)-(I+uoL)](I+uL)'1

- (1+uoL)'1(uL-VOL)(I+VL)’l

a (u-VO) (1+uoL)'1L(I + uL)’1. (III-30)

From (III-29) and (III-30), we obtain

||(I+uoL)'1L-(I+VL)’1L||g(u) - ||((1+uoL)'1-(1+VL)'1)L||g(u)

Iu-VOI II(I+VOL)-1L(I+VL)-1LII€(V)

IA

ly-uo| ||(I+uoL'1LI| ||(I+uL)'1L||§(u)

Iv-VOI ||(1+uoL)'lL|| x(v) (111-31)

Finally, substituting (III-31) into (III-29) yields

x(uo) II(I + VOL)’1L|| 5(uo)

1

IA

x(v) - (I-n(v)) lu-uol ||(1+uoL)' L ||

Since x(v) < l , x(uo) s x(u). This completes the proof of Lemma 2. D

51

Hence, the minimum contraction coefficient is obtained when the
real number u is the arithmetic mean of the maximum and minimum slopes

of the nonlinearity ¢(y(t),t).

lemma 3 : The L2 - induced norm of the linear operator is given by

IILIIZ - sup 1 |E(Jw)l
w e R

Proof of Lemma 3 : Let u e L2[O, 00) and y - L u , then with Fourier
transform of u, u(jw),

IIYIIZ = %; Jo |E<jw>|2 Ifi(jw)|2 dw (from Parseval's theorem)
~Q

. 2 2
s supR1|h(Jw)| Hull

0) 6

which shows that

||L|| s sup 1 |E(jw)| . (III-32)
w e R

Since lim h(jw) = O (Riemann-Lesbegue Lemma), and from the continuity
w»iw

of Ih(jw)|, there exists an wo such that

|E<jwo>l - sup 1 Ifi<jw>| -
w e‘R

Now consider a sequence of functions
u (t) - J «n 2 cos(w t) sin( l— t) / (wt)
n 0 2n

whose Fourier transforms are

. — 1_ 1_
Fn(3w) = '- J «n for w e [ w — 2n , w + 2n ]

52

and for w e [-wo - %; , —wo + %;
0 otherwise,
and Ilunll = 1 .
Then for yn = L un
l_
o+ 2n
~ , 2 2 ~ . 2 .
I lh<on>l - llynll l - I n <|h<on>l - lh<3w>l2> d... I
;_
wo- 2n
. 2 .
s max I [h(on)| - Ih(Jw)|2 I
we[w l— w + l‘]
2 ’ 0 2n
(III-33)
In equation (III-33), Ilynll can be made arbitrarily close to [h(jwo)|
by choosing n large enough. Since Ilyll S IHJI , the inequality
(III-32) can be replaced by equality. D
Proof of Theorem 2 : From Lemmas (1), (2) and (3) it follows quite

easily that

n<u> = l (3 - a) sup 1[ 1 + % (a + 3) E<jw> 1'1 E<jw) < 1
2 w e R

Having established the minimum contraction coefficient we runv in-
‘voke cnxr main result to develop a theorem for the SISO case leading to

a geometric interpretation.

Theoren.3 : Let K be assigned so that the eigenvalues of the matrix A
are in the open left half complex plane. Let h(s) be the frequency

response transfer function of the linear, time invariant portion of the

53

forward path. Let (r,y) e L2[O,w) be any combination satisfying equa-
tion (III-9) euui (ro,yo) e L2[0, m) be a known combination satisfying

(III-10). Then , for any input r(t) in the specified input sphere

0 (r: r0, fli) - { r e L2[0, m)| Ilr - roll2 s fii }

and for any nonlinearity in the sector [a, fi], there exists a unique

response y(t) e L2[0, w) in the specified fio-neighbourhood such that

 

0 (y: yo. fio) = { y e L2[0. w) I Ily - yollz 5 Bo }
if
h(jw) 1
sup1|h(jw)l a s 1 - supll 1 I 3(fl - a) (III-34)
weR. weR. 1 + 2 (a + fl) h(jw)
p + .5
where 5 - o l
flo
p - sup II N y - N y ||
0 7 e F 7 o o o 2
5 ' II I ‘ r0 ||2
W; : Consider the operator equation (III-12) which

can be written as

y = -LN7y + L Noy + L( r - ro ) + yo

_ Qy (III-35)

From Theorem 1, the minimum contraction constant K of (III-35) is

 

h(jw) 1
n - sup 1 1 - (B - a) < l (III-36)
w e R 1 + 2 (a + fl) h(jw)
Now from equation (III-35)
@yo - yo - -(LN7yO - LNoyo) + L(r - r0) (III-37)

Taking the norm on both sides of equation (III-37) gives

54

Iléyo - yoll s IILII I wstpR}||N7yo - Noyoll + llr - roll] (III-38>

Since output y(t) is required to be within the 30 - sphere , from

Lemma 1 of Chapter II, we require
||<1>yo - yo II 5 50 <1 - n) (111-39)

Now, by using equations (III-38) and (III-39), inequality (III-34) is

obtained, thus establishing Theorem 3. D

Geometric Interpretation :
The design criteria advanced in Theorem 3 have an interpretation
similar to the circle criterion known for absolute stability . First

of all note that the constant p0 is a measure of the deviation of the

nonlinearity from linearity. Thus the proper selection of the non-

linear design function Ibo which eliminates or averages out as much of

the nonlinearity will allow a larger upper bound for sup Ih(jw)| in
(III-34). Since the nonlinearity is confined to a sector [a, [3], a

favourable design function $0 is simply the arithmetic mean of the up-

per and the lower bounds of the nonlinearity zp(y(t) , t).

Next,the minimum contraction coefficient K.

 

h(jw) 1
n=sup1 '-

1 (fl-a)<1
weR l+§(a+,3)h(jw)

is evaluated similar to the M - circle concept known in the design of
classical compensators.

First write

h(jw) = a + jw where a, w 6 R1.

Then M is given by

55

a + jw

 

M ' 1
l + E (a + fl) (0 + jw)
2+ 2
2 a w

and M = (III-40)
{1+%<a+fi>a}2+{%<a+fi>w}2

From equation (III-40), we obtain

 

 

2 2
02{l - % (a+fi)2} - M2(a+fl) a - M2 + w2 {1 - % (a+fl)2} - o . (III-41)
M 2 . -l . . . .
If 4 (a + 3) - l - O, we obtain a - which 18 a straight line
a + fl
parallel to the w - axis and passing through the point ( a-i 5 , O ).

If % (a + ,B)2 - l 7‘ 0, from equation (III-41), we obtain an equation
of a circle which is given by

2M2 (a + fl)

 

 

 

 

 

 

4 M
{ a - 2 2 )2 + w2 - ( 2 2 )2 . (III-42)
4 - M (a +5) 4 — M (a + B)
4 M
This circle has a radius of 2 2 and is centered at
4 - M (a + fl)
2M2(a + fl)
( ,0).
4 - M2 (a +19)2
In interpreting the design criterion
h(jw) fl - a
sup Ih(jw)| 6 s l - sup 2

l .
1 .—
w e Rl w e R. l + 2 (a + fl) h(Jw)

the quantity on the right hand side of the inequality can be readily
evaluated by the M - circle approach. Consequently the above condi-

tion reduces to a simple circle condition

56

sup |h(jw)l S 6
1
w e R

where 6r depends on M, 6, a, and 6

We also require that

how) I9 - a
1 1+% (aw) h(jw)

 

 

sup

w e R

< l (III-43)

which may be interpreted as outlined in Appendix B.
'Phe overall.design criterion therefore has the following ultimate
interpretation. The design criterion is met if one of the follxnving

three conditions is satisfied.

Case (a) : a > O

 

The locus of h(jw) for co 6 (-00, ac) lies outside the circle Clof

(

radius 0]

mud

l l . l l' l
a - ) centered in the complex plane at [-2 (a + fi)’

fl

and inside the circle C2 of radius 1%- (l - rc) centered at the

origin. This is depicted in Figure 6(a).

Case (b) : a = O

 

Re [h(jw)] > - i for all real w and therefore h(jw) should be

inside the circle C3 of radius % (l - n) centered at the origin as

shown in Figure 6(b).

57

 

 

 

 

 

 

”@W

 

 

(b) a -

Figure 6

58

Case (c) : a < 0

 

The locus of h(jw) for w e (-m, w) is contained within the inter-

. 1 1 , .1. 1 1
section of the circle C4 of radius 2 (B a) centered at [-2 (B +
i), 0] and the circle C5 of radius %(1 ' K) centered at the
origin. This is shown in Figure 6(c).

Remark : Unlike the Lao - problem formulation where an algorithm for

determining the feedback gain matrices is difficult to obtain , here we
can utilize the Butterworth pole-patterns for satisfying the design

criteria. It should be reemphasized that L2 design criterion too is a

quantitative pole placement. This however can be achieved quite

easily by monitoring the frequency response.

3.5 Design Procedure

The magnitude of the closed loop frequency response Hujw)l

depends on the augmented system matrix 8, thus an algorithmic procedure
is proposed for the satisfaction of the design criteria. The following

procedure is found to be effective.

( 1 ) Specify a, 6, fio’ 51’ and y0 .
( ii) Select $0 as the arithmetic mean of the sector [a, B] and
evaluate p0.

(iii) Evaluate 6
( iv) Find K based on the algebraic Riccatti equation and

evaluate r0. The input weighting matrix is taken to be the

59

identity and the state weighting matrix is iteratively
adjusted.

( v ) Find x by using the M - circle diagram .

( vi) Check the inequality (III-34) based on the circle inter-
pretation .

(vii) Iterate steps (iv) through (vi) until the inequality Ls

satisfied.

3.6 An Illustrative Ema-ple
To demonstrate the applicability of the design criteria above,

consider the linear, time invariant system with a nonlinear feedback

element given by

x1 0 1 x1 0 0
- + u - ¢(y(t).t)
i2 0 -1 x2 1 1
y - [ l 0] [ x1 J
x2
where the nonlinearity w(y(t), t) is confined to a sector [(2, 6]. We

consider two cases below.

Case 1 : a - 0.5, fli- 3.5
Consider the above system with the nonlinearity
w(y(t), t) - 2 y(t) + 1.5 y(t) sin(5 t)
which certainly belongs to the sector [0.5, 3.5].

Let the input sphere fli - 1.0, the output sphere flo - 0.1, and the

desired output y - t e

60

Design.procedure :

Choosing the nonlinear design function $0 as

po-i- (a+fi) y(t)
- 2 y(t)
we get, p0 - sup II vao ' NOYO ll
_ max || 2 yo + 1.5 yo sin(t) - 2 yo ||
- max || 1.5 yo sin(t) ||
5 1.5 II Yo II

According to the definition of L2 - norm and the nominal output yo ,

the norm of y0 is

H y, II - [ fl tze lzdt 11”-
0

— 0.866

which yields p0 = 1.299.

Consequently 6 can be computed as

— 22.99

INow the eigenvalues of matrix A - (A + BK) are obtained via the
Butterworth pole pattern to be

A - -3.535 t j 3.464

1,2
yielding the feedback gain K
K - [ -24.495 -6.07 ]

Thus the transfer function h(s) becomes, with s = a + jw,

 

61

h(s) - 0 (SI - R)“1 B
1
52 + 7.07 s + 24.495

where I is the n x n identity matrix .

With K and the nominal output yo, the nominal command input'ro can now'

be evaluated as

re - e't (2. + 10.14 t + 18.424 t2)

Next consider the contraction coefficient 1c and sup1|h(jw)|. Using the
weR

M - circle diagram as shown in Figure 7, we obtain
M - 0.0383
and max |h(jw)| - 0.0408

yielding

~= %(fl-a)(M)

- 0.0287 .

Finally, the inequality (III-34) is computed as

sup 1|h(jw)| s % (1 - x)

w eIR
- 0.04225.

Thus the design criteria for servo-tracking are satisfied.

Remark : According to the geometric interpretation (Case (a) of sec-
tion 3.4), tflmis result can be easily checked directly as in Figure 7.

Since the frequency response h(jw) remains outside the circle C1 of

.111, _111
radius 2(a - fl ) 0.857 centered at ( 2(a + 6

circle is not shown in Figure 7, since it is located to the far left of

), 0) = (-1.142, 0) (this

62

the locus h(jw)) and inside the circle C2 of radius %(1 - x) - 0.04225

centered at the origin, the same conclusion as above is drawn .

 

 

Inn
. 1 0.05
C1rc1e C
2
M -
circle 0.05
1 l r ::
I , H R
-0.05 Frequency Response e
h(jw)
I . 0.0383 {-0 05
HUAX I 0.040!

 

Figure 7 M - Circle diagram for a - 0.5, fl - 3.5

Computer Simulation :
In order to verify that the design specifications are satisfied,
the resulting system was simulated on a digital computer. Figure 8(a)

shows that the output y(t) and the nominal output yo(t) resulting from

the command input r(t) shown in Figure 9(a) and given explicitly by

r (t) if r (t) s 12.1
r(t) - ° 0
12.1 if ro(t) > 12.1

so Ilr - rOII S 1.0 . The input disturbance Ilr — roll is shown in

Figure 9(b) and is clearly in the input sphere of radius fii - 1.0 .

63

The error ||y - yoll computed from Figure 8(b) is of the order of 10-2

which satisfies the output sphere 6° - 0.1 . The control u - r + K x

+ $0 is shown in Figure 10.

 

 

0.6 —
yo(t)
I?
>~ 0.4-
- y(t)
13
0
>3
OHZF
0.0 1 1 1 _ 1 1
0.0 2.5 5.0 7.5 ~10.0 12.5 15.0

Time (Sec)

Figure 8(a) y(t) and y°(t) for a - 0.5, fl - 3.5

 

64

 

 

 

0.003 ,
U
'0
°14 0.002 _
0
>5
3?
%
0.001 -
0.000 1 l l 1 l I
0.0 2.5 5.0 7.5 10.0 12.5 15.0

Time ( sec )

Figure 8(b) The tracking error I (y - yo)2 dt for a - 0.5, fl - 3.5

15 r

ro(t)

 

ro(t)

10

r(t),

 

o J J l I l

0.0 2.5 5.0 7.5 10.0 11.5 15.0

Time ( Sec )

Figure 9(a) r(t) and ro(t) for a - 0.5, fl - 3.5

 

65

 

 

 

1.00 -
N
3:: 0.75 -
A0
14
l: 0.50 -
0.25 -
j I I I I l I
5.0 7.5 10.0 12.5 15.0

 

0.00
0.0 2.5
Time (Sec)

Figure 9(b) Input disturbance in llr - roll2 for a - 0.5, fl - 3.5

2.0

u(t)
1.5

1.0 P

 

 

 

 

 

0.0 -
I I L j I
5.0 7.5 10.0 12.5 15.0

 

 

Time (sec)

Figure 10 Control effort u(t) for a - 0.5, fl - 3.5

 

66

Case 2 : a = -0.5, fl - 0.5
Now consider the same system as in Case 1
linearity
¢=%ya>umo

which is confined to a sector [-0.5, 0.5], i.e., a - -0.5, B - 0.5

Let Bi = 1.0, flo = 0.1, and y0 = t2 e as in the previous case.

Design Procedure :

, but with the non-

Similar to the previous case, the nonlinear design function $0 - 0

is chosen. Hence the norm po can be evaluated

‘0
1|

0 sup || Nyyo - Noyo ||

max II % yo sin(t) II

1
5 || yo I!

[ I” It2 e-t |2 dt 11/2
0

0.866

Here. llyoll

Therefore, po = 0.433

fiJJL

Consequently, 6 = B o = 14.33

0

Now we choose the eigenvalues of A as

_ - +
11.2 2.985 _ j 2.900

from the Butterworth pole pattern yielding
K - [ -17.32, -4.97 ]

and the frequency response transfer function

67

1

 

h(s) - 2
s + 5.97 s + 17.32

With K and yo, ro is evaluated as

re — e't (2. + 7.94 t + 12.35 t2)

From Figure 11, M - 0.059 and sup 1 |h(jw)| = 0.0577.

w G

Consequently, with the contraction coefficient n

n-%<fl-a><M)

=0.0295

the design criterion (III-34) is verified, yielding

. 1
sup 1 [h(Jw) | s 3 (1 - n)
w e R

- 0.0677
This result is also verified directly by the geometric interpretation
given in the previous section (Case (c)) which shows that the locus of

h(jw) lies in the intersection of the circle C4 of radius 2.0 centered

at the origin (this circle is not shown in Figure 11) and circle C5 of

radius 0.0667 centered at the origin.

Computer Simulation :

The cxnmnand input r(t) which satisfies the input sphere condition

IIr - ro II 5 1.0 given by
r(t) - ro(t) + 1.1 e'0‘3 t sin (10. c)
is used. fflmzreference command input ro(t) , actual command input

r(t), and corresponding input disturbance' IIr - roll are shown in

Yo(t). y(t)

68

     
    
   

Circle Cs

 

Frequency Respon

M- 0.0590
HMAX= 0.0577

 

Figure 11 M - circle diagram for a - -0.5, fl - 0.5

 

 

 

0.5 , yo“)

0.4 - y(t)

0.1 -

0.0

-0.2 1 ‘ J 1 . '
0.0 2.5 5.0 7.5 10.0 . 12.5 15.0

Time ( Sec )

Figure 12(a) y(t) and yo(t) for a - -0.5, fl - 0.5

 

 

 

 

 

69
x10"

0.: ,
N
no 0.4 -
>5
5':
‘-—0

0.2 -

0.0 J L. l J L I

0.0 2.5 5.0 7.5 10.0 12.5 15.0

Time (Sec)

Figure 12(b) The tracking error I (y - yo)2 dt for q - -0.5, fl - 0.5

10.0 "' .ro(t)

 

7.5

5.0

 

2.5

 

0.0- - -

 

-2e5 l L l J J. 1

0.0 2.5 5.0 7.5 10.0 12.5 15.0

Time ( Sec )

Figure 13(a) r(t) and ro(t) for a - -0.5, fl - 0.5

70

1.00

 

0.75

0.50

II(r - to)”,

0.25

 

 

0.00 1 ‘ 1 1 _L 1

0.0 2.5 5.0 7.5 10.0 12.5 15.0

Time (Sec)

Figure 13(b) Input disturbance in Ilr - roll2 for a - -0.5, fl - 0.5

u(t)

 

 

 

 

 

 

 

-‘ l l 1 I I I

0.0 2.5 5.0 7.5 10.0 12.5 15.0

 

Time (sec)

Figure 14 Control effort u(t) for a - -0.5, fl - 0.5

70

 

O

‘2

U!
I

O

U‘

D
I

l|<r- r0)”2

 

0.00 l I I I I l
0.0 2.5 5.0 7.5 10.0 12.5 15.0

Time (Sec)

Figure 13(b) Input disturbance in Ilr - roll2 for a - -0.5. fl ' 0-5

u(t)

_e
I

 

_1 I I J I I

I
0.0 2.5 5.0 7.5 10.0 12.5 15.0
Time (sec)

Figure 14 Control effort u(t) for a - -0.5, fl - 0.5

71

Figure 13(a) and Figure 13(b) respectively. Figure 12(a) shows the

reference input y°(t) and the actual output y(t). The error Ily - yoll

is shown in Figure 12(b). This clearly satisfies the output sphere

specification 60 = 0.1 . On comparing the actual output error which

1

. -2 . . . . . . . -
15 of order 10 With the de51gn spec1f1cat10n which Is of order 10
it is clear that the result is conservative. This characteristic is

primarily due to the generality of inputs and the nonlinearities that

are admissible in L2[0, w) . The required control u(t) is shown in
Figure 14. This control effort is smaller than that of the previous
case for a - 0.5 and fl - 3.5 shown in Figure 10. It is expected,

since this nonlinearity is less severe than the previous nonlinear ele-

ment .

CHAPTERIV

AN APPROACH FOR SELECTION OF EIGENVALDES

In Chapter II, design criteria for precision tracking were

developed by embedding the problem in Lw— space of functions. With

respect to the satisfaction of those design criteria, a quantitative
pole-placement was identified. In this chapter an algorithm is
developed for the selection of eigenvalues which makes the operator
norm 1; a minimum. The approach to this eigenvalue placement is via a
generalized LQ formulation.

Although the idea of pole-placement is central to much of linear
control theory, it is not very well understood how eigenvalues should
be selected to satisfy performance specifications. One of the inter-
esting features associated with the design criteria contained in here
is that the placement of eigenvalues is directly related to the ability
to effect specified tracking. Namely, the tracking specifications can
be met if the eigenvalues can be placed in such a way as to obtain min-
imal values for 17. Hence it would be appropriate to refer to this
quantitative pole-placement idea as a sufficient condition for
"trackability". Thus it is clear why it would be important to seek
algorithms for pole-placement that yield minimal values for r]. A
criterion which resembles the Butterworth pole configuration is
developed for the SISO case. MIMO case, however, still remains an open

issue.

73

4. 1 Conventional Pole - Placement Problem

Consider a linear time invariant dynamical system of the form
x(t) - A x(t) + B u(t)

where, x(t) e Rn, u(t) e Rm, and A and B are constant matrices of
appropriate dimensions. Applying the state feedback u(t) - K x(t) to
the system above, a closed loop system with a design parameter K is
obtained. The question then is can K be adjusted so that this closed
loop system exhibits desirable features such as stability and
reasonable transients. It is well known that if {A, B} is control-
lable then arbitrary eigenvalues for the closed loop system can be
achieved. This idea of arbitrary placement of closed loop eigenvalues
is what is typically referred to as pole - placement.

Thus if {A, B) is controllable then the design problem reduces to
the question whether it is possible to capture the performance
specifications in terms of specific pole locations. Typically this is
achieved by a trial and error procedure : First a set of eigenvalues
is selected and the design completed followed by an actual simulation
study. If simulations show undesirable performance pole locations are
changed and the design carried out once more. This process is
repeated until a satisfactory performance is achieved.

The LQ formulation however selects appropriate eigenvalues once a
performance index(PI) is chosen. In this case although the eigen-
values are not picked arbitrarily their locations depend on the
weighting matrices of the PI. The problem with this is that it is not
known how one should select the weightings to satisfy a specified per-

formance. Nevertheless it affords a way of selecting the closed loop

74

poles in somewhat of a definitive form. This is the spirit in which

we develop the generalized LQ problem.

4.2 Generalized.LQ problem
We observe that the operator norm n — II WOWQ31 II depends only on

the linear structure of the uncertain nonlinear plant. The linear part

of the plant (II-l) is

{:(c) = A x(t) + B u(t) (IV-la)
y(t) = C X(t) (IV-1b)
n m b
where x(t) e R , u(t) e R , y(t) e R , and A, B , C are constant

matrices of order n x n, n x m and b x n respectively.

Since L0° - measures are used to describe tracking errors the quad-

ratic performance index of the LO problem is modified as

Tf 1 T p l T p
Jp’l [2—p1<y<t)-yo<t)> 01(y(t)-yo<t)>1 + E 1u<t> Q2u(t)) 1d: (IV-2)
0

to reflect such a measure, where the nominal output yo(t)e RP, t e [0.,
Tf], and Q1, Q2 are, respectively, symmetric positive semidefinite and

symmetric positive definite weighting matrices of order b x b and m x
m. In (IV-2) the limit as the positive integer p 9 w corresponds to

the L - case.
a)

Next, we determine the optimal control u(t) that steers the system

output given by (IV-l) so as to track the nominal output yo(t) which

simultaneously minimizes the performance measure given by (IV-2). By

75
augmenting the performance measure (IV-2) with the state equations via

Lagrange multipliers v(t) 6 RP, we obtain

T
Jg - I f 1 i; 1(y(t)-yo(t)>T01<y<t)-y°<c))1P + %; (u(t)TQ2u(t))p

+ vT(t) (A x(t) + B u(t) - {1(a) ] dt

which may be written as

T
- f l— _ T _ p 1_ T P
Jg I [ 2p {(0 x yo ) Ql (C x yo )1 + 2p 1 u Q2 u }

0
T O
+ v (A x + B u - x ) ] dt
Tf .
- I ¢ ( x, x, u, v, t) dt (IV-3)
0

A
e 1— T p
where ¢ ( x, X. u. V, t) - 2p{(C X - yo) Q1(C X - yo?)
1_ T p T _ -
+ 21){ u Q2u ) + v (A x + B u x)
Now we consider the extrema of the functional (IV-3) under the follow-
ing assumptions.
( i) The state and the control are not constrained.
( ii) The end point is free, i.e., Tf is free.
(iii) The initial condition of the state x(0) = x0. The final
state x(Tf) — xT 1S free.
With the above assumptions, we now take the first variation of Jg
denoted by AJg yielding

T
AJg - I f{¢(x+6x, x+6x, u+6u, v+6v, t)-¢( x, i, u, v, t)) dt (IV-4)
0

 

76
By employing the Taylor expansion theorem, equation (IV-4) becomes

Tf 8¢ . T 8¢ _ T .
AJ - I ([ —— (x, x, u, v, t)] 6x + {[ -— (x, x, u, v, t)] 6x
8 0 3X 8%

8¢ . T a¢ e T
+ [ 5; (x, x, u. V. t)] 6n + I 5; (x, x, u. V. t)] 6v

+ [ higher order terms in (6x, 52, 6u, 6v) 1 1 dt (IV-5)
Neglecting the higher order terms of (6x, 6x, 6u, 6v), integrating by
parts the term involving 6x , and relating 6x(Tf) to 6xT and 6Tf by

( See Figure 15)

6x(Tf) - 6xT - §(Tf) 51f

yield

x(t)

_l:j_

6x(Tf) 5x1

I__I

          

 

, T

l l
|

l I

I l

I l

l I

 

o Tf Tf+6Tf

Figure 15 Boundary condition for free end point

 

77

Tf 6¢ ~ L ~ ~ T
mg -I 1115; (x(t). x(t). u(t), v(t). t)]

 

0
d 695 ~ .1 .. .. T
- 3E . (x(t), x(t). u(t). V(t). t)] ] 6X(t)
6x
86 °

+ [ aT. (at). 32m. 6(c). 170:). t)]T 6u<t>

a¢ ~ .1 ..., a. T
+ [ a—v (x(t), x(t), u(t), v(t), 11)] 5V(t) } dt

8¢ ~ .1 ~ ~ .
+ I —, (xcrf), x(TfL u(Tf>. v<1f>. Tf>1T 6xT
8x
+ {[ ¢(§(Tf>, inf). inf), G<Tf>. Tfn
8¢ ~ 1 ~ - T L
- [a_° (x(Tf). x(Tf). u(Tf). v(Tf>. Tf>1 x(Tfn 51
X

(IV-6)

where i, i, E, and G are respectively the state, the time derivative of

the state, the control and.the Lagrange multipliers along an extremal.

Since the variation AJg vanishes on an extremal, each coefficient of

the independent variables must be zero

(IV-6), we obtain the following governing equations.

( i) The differential constraints

343

87 (Em. 320:). Mt). iHt). t)] = A 32 + B u -

~

>410

-0.

are the state equations

N10

A§+BE

Therefore from equation

(IV-7)

 

78

( ii) The coefficients of 6x(t) are

3¢ 2 L 2 ~ T d a¢ ~ 1 ~ ~ T
[5;*(x(t).x(t).u(t).V(t).t)] - a; ‘7 (x(t),x(t),u(t),v(t),t)] - 0
6x

which leads to costate equations

3 - {(Ci - yo)TQ1(C§ - yo) T

1P'1 cT Q (Ci - yo) - A G (IV-8)

1

(iii) The coefficients of 6u(t) becomes

3¢ 2 2 2 2 2 2 - 2 2
5; (x(t). x(t). u(t). v(t). t) = {uT 02 uIP 1 02 u + BT v

- o, (IV-9)

( iv) The coefficients of 6Tf are

if 2 L 2 2 T .
ar <x<Tf), x(Tf). u(Tf>. v(Tf). Tf)1 6xT

+ {[ ¢<§(Tf). §<Tf>. 6(Tf). 5(Tf). Tf>1

805 9 e
__ ~ ~ ~ ~ T ~
- [ai (x(Tf). x(Tf), u(Tf), v(Tf), Tf)] x(Tf)} 6Tf
- 0

which give the boundary conditions

§(T T 6T - 0

5T<Tf> 6XT + H<§<Tf>, f). G<Tf>, G<Tf>. 1f) f

where the Hamiltonian

A
1 T l T T
H - §E{(C x - yo) Q1(C x - yo))p + E;{ u Qzu )p + v (A x + B u).

Therefore, the boundary conditions for the free end point become

3(Tf) — 0 (IV-10a)

 

79

H(§(T 3201f), Gaf), $(Tf), Tf) = 0. (IV-10b

f).

Equation (IV-7) - (IV-10b) constitute a set of necessary condi
tions for an extremal of the generalized LQ performance index (IV-2)
If p - l we recover the LO results in the form of a state feedback wit
gains given by the Riccatti differential equation. In order to obtai
a.perturbed form of this LQ solution or equivalently the LQ pole

patterns we start by defining the positive quantities

~ T ~ p-l
{(Cx - yo) Q1(Cx - yo)} = 61 (IV-11a
~T ~ p-l
{u Q u} - e (IV-11b
2 2
which are then substituted in (IV-8) and (IV-9).
Then the costate equations (IV-8) become
L T ~ T ~
v — - 61 C Q1 (Cx - yo) - A v (IV-12
and the control E(t) from (IV-9) is
5(t) - - l— Q '1 BT G (IV-l3
62 2

substituting (IV-13) into (IV-7) and augmenting it with (IV-l2) yield

- A i - l— B 02'1 BT G (IV-14a

X10

T ~ T ~ T
61 C Q1 Cx - A v + 61 C Ql yo (IV-14b

<10
ll

Rewriting equation (IV-14) in a matrix form, gives

= A E + 8 EC (IV-15

N1.

where E = e 8?“

<1

80

A - B 02‘1 ET
A _ T 2 T e R2n x 2n
- 61 C Ql C - A
0
B - 6 R?“
I
.. T m
uc- e1 C Q1 yo 6 R
Remark : In (IV-ll) when p a m, we consider 61 a 0 to be a situation

when tracking occurs quite accurately and 61 + w to correspond to a

situation where no tracking is apparent. This motivates the limiting
analysis given below in which one would initiate the design by letting

£1 4 m to correspond to the worst case and then subsequently decreasing
61 so as to correspond to the specifications dictated by the operator

norms mentioned previously. These limiting values would basically
guide the selection of pole-configurations for initial start up of the

primary quantitative pole—placement algorithm.

4.3 Optimal Pole Configuration
Now we investigate the pole patterns of the augmented system (IV-

14) with respect to parameters 6 62 that determine respectively how

1,
much significance is attributed to the output error or the control ef-
fort. From an algebraic Riccatti type of equation for a steady state
operation, the optimal feedback gains associated with the optimal pole
configuration may be determined. The present method, however, allows

us to study the structure of the optimal pole configuration more easily

as a function of £1 and 62 . Namely, by using the root locus method,

 

81

we gain more insights to the solution. Results are obtained in the

limiting cases for SISO systems. We first derive the characteristic

poLynomials of A as a function of £1 and e observing that the optimal

2’

closed loop poles which are the eigenvalues of the matrix A should lie
in the open left half complex plane. Next, the migration of poles

with respect to parameters 6 62 is studied. Since the tracking

1’

performance is considered most important, we arbitrarily set % Qél = l

2

for simplicity, i.e., the control effort is allowed to take any value.

The characteristic polynomial of A is computed by employing Lemmas

Cl, C2 in Appendix C.

~ 31 - A B BT
det(sI - A) - det T T
51C Q1C ST + A
T T -l T
- det(sI -A) det{(sI +A )-elC Q1C(sI -A) BB } (by Lemma Cl)

1 1

BBT(sI +AT)' 1

det(sI -A) det{(sI +AT)(I -61CTQ1C(sI -A)'

det(sI -A) det(sI +AT) det{I -elBT(sI +AT)-1CTQ1C(SI -A )-1B}
(by Lemma C2)

-(-l)ndet(sI -A)det(-sI -A)det{I +el(C(-sI -A)'1B)TQIC(sI -A )‘181

= (-1)n a(s) a(-s) det(I + 61 h(-s)T Ql h(s) ) (IV-16)
Where a(s) - det (sI - A )

h(s) = C(sI - A)'1 B

and I's are the identity matrices of appropriate dimensions.

 

82

Thus the eigenvalues of the closed loop system are the zeros of (IV—16)
which are in the open left half complex plane. let.the open loop

transfer function h(s) of a SISO system be represented by

 

 

b(s)
h(s) " a(s)
r
bo .H (s - pi)
= 1'1 (IV-l7)
n .
H (s - A.)
1-1 1

where bo is a nonzero constant, pi, i = 1 ... r , are the zeros of the

open loop system and A1, i - 1 00- n , are the poles of open loop sys-

tem. Then with Q1 = l for simplicity, (IV-l6) becomes

n n-r 2 r

H (s - A.)(s + A.) + (-l) e b H (s - m.)(s + p.) - 0 (IV-l8)
i=1 1 1 l o i-l 1 1

The asymptotic behavior of the closed loop poles given in (IV-18) as a

function of 61 is outlined in the following theorem .

Theorem 1 : Suppose that the open loop system whose transfer func-

tion is represented by (IV-l7) is controllable and observable then

( i) for £1 4 0, the eigenvalues of the closed loop system approach

asymptotically the numbers Xi’ i = 1 ~00 n, where

{ Ai if Re (A1) 5 0
1

A. if Re (A.) > 0
1 1

 

83

(ii) for el-+<n , r eigenvalues of the closed loop system approach

asymptotically the values $1, i - l ... r, where

~ { mi if Re (mi) 5 0
81‘ .
- mi 1f Re ($1) > O
and the remaining (n - r) eigenvalues approach the asymptotes

through the origin and make angles 0 with the negative real axis

 

 

 

of
2 x n - r - l
a + - .0. o o
(a) 6 _ n _ r , 2 0 2 , for (n r) 18 odd
1
(£+2)“ n - r - 2
(b) 0 - i E‘T—?_ , 2 - 0 ... 2 , for (n - r) is even.

The distance from the origin for the far away eigenvalues are
l

asymptotically ( 61 bi ) 2(n-r)

Figure 16(a) and 16(b) show the pole configurations for (n-r) = 2,

3 respectively.

Illustrative examples : In order to verify the theorem above, the
sensitivity'of the norm 0 with respect to eigenvalue selection is com-

puted for the following second and third order systems.

( i) Second order system

A system given by (IV-l9) is simulated to compute the norm n.

 

84

 

 

 

 

 

II
Im
o
r?
I
\ 9
t
R 0
e
(a) n - r - 2 (b) n - r - 3
Figure 16 Optimal pole configuration
{:1 0.23 0. x1 0.
_ + u(t) (IV-l9)
x2 ~0.57 1.42 x2 1.

The set of eigenvalues for the closed loop
the distance from the origin p, and the angle
tive real axis shown in Figure 17(a) are

p : 0.5 1.0 1.5 2.0 3.0 5.0 10

9 : 10. 20. 30. 40. 50. 50. 70.

The results shown in Figure 17(b) verify that

system are chosen so that

0 measured from the nega-

.0 20.0
80.

the minimum norm values

are obtained when the set of eigenvalues for the closed loop system are

chosen in the vicinity of a 45° line. This is quite consistent with

the predictions of the theorem given previously.

 

‘—__.
v

 

 

 

Figure 17(a) Eigenvalue placement for n - r - 2

Il-IIH

 

 

 

 

 

 

Figure 17(b) Norm configuration for n - 2

 

86

(ii) Third order system

The system given by (IV-20) is simulated for computation of the

operator norm 0 with the set of eigenvalues,

that is
p - 1.0 5.0 10.0
a - 0.0 30.0 45.0 60.0 85.0
x1 0 l 0 x1 0
i2 - 0 -1 ~72.464 x2 + o u (IV-20)
x3 0 0.027 -10. x3 1

Figure 18 shows the norm configuration we expect. Namely, the minimum

norm value of'n is obtained when a pair of complex conjugate eigen-

values lie in the vicinity of a 600 line together with a real pole.

||°|lA

P-IHO

P~5.0

P — 10.

 

60

¢

Figure 18 Norm configuration for n - 3

 

87

For MIMO systems too, the asymptotic behavior of the closed loop
eigenvalues can be obtained from equation (IV-l6). However, it is not
as simple to determine the pole configuration because the eigenvalues
that tend.to infinity generally form several clusters of different or-
der and radii. For the limiting cases, the theorem applies with

varying asymptotic distances from the origin.

 

CHAPTER.V
COHPUTER.ALGORITHHS

In order to accomplish the design task computational algorithms
are required so that the operator norm needed in the design inequality
(II—l6) can be evaluated. In this chapter, explicit computer oriented
formulae necessary for the execution of Theorem 1 of Chapter II are
developed. In particular, algorithms for computing the design

matrices G and K that specify the closed loop eigenvalues of R and for
computing the critical operator norm II W0 W le II are given. Several

programming concerns associated with the numerical schemes are also

discussed in here.

5.1 The Operator Nbrn II “6 i le II

In this section computer oriented formulae for evaluating the

operator norm [I W0 W Q;1|| are developed. The norm of the linear
operator W
t R(t - r)
(0 z)(t) - I e z(r) d1 (V-l)
0
is first considered.
2n 2n . .
Theoren.1 : Let 0 : L00 [0, w) 4 L00 [0, m] be a linear operator given

by

88

 

89

t

(Q z)(t) - I H(t - 1) 2(1) dr ,
0

and let each element Hij of the matrix H be such that Hij e L1[0 , w)

 

II‘I'zII
Lin t
Then, IIWII - sup IIZII s I II H(r) II dr
n 2n 0
z 6 Lb [0,m) L00
2 fl 0

2n
where IIH(r)II - max 2 I Hij(r) I .
i

t
Proof of Theoren.1 : Let g(t) - I H(t - 1) 2(1) dr (V-2

where g, z 6 Lin and Hij e L

Equation (V-2) can be written in component form as

t

gi(t) = I0 § Hij(t - r) zj(r) d1 (V-3

which on taking absolute values on both sides yield

t
I gi(t)| 5 J0 p | Hij(t - r)I | zj(¢)| d, (v-4

By defining the induced matrix norm

|| H(t - 7) HL - max [ s | Hij(t - r) | ]
m 1 J
t E H.. t - S H t - V-S
we ge j I ,,< .) I II < r) II,” <
and similarly, I zj(r) I s max I zj(r) I = II z(r) IIL (V-6
j co

By substituting (V-S) and (V-6) into (V-4) , we obtain

t
I gi<t>| s Ilzll [0 || H<t - r>l| dr

 

90

- IIZIII: || H<r>ll d.

which implies
II g ||L2n s l|z|lI: ll am” a. (V-7)
Q

Equation (V-7) can now be written as

ll 3 ||
5 II H(r)II dr , for 2 fl 0
II z I] 0

which implies II W II 5 Im II H(r)II dr
0

This completes the proof of Theorem 1 . D

From Theorem 1, an upper bound of the operator norm II W II can be
estimated by evaluating an integral if H(t) = eRt can be explicitly
represented. In what follows , an explicit expression for eRt is
given.
Lanna 1 : Suppose matrix R has distinct ei envalues A1, 1 - l 00- 2n ,

 

and pi, i.== l ... 2n, are the corresponding eigenvectors, then the

matrix R can be diagonalized in the form of

_1 *1. Q

where P is the modal matrix such that

P = [ p1. p2. ---'- . Pzn ]

 

91

Proof of Lanna l : See Strang ([47], Chap 5)

Remark : Distinct eigenvalues are assumed for the matrix R since a

specific spectral decomposition corresponding to this case is used

later in the chapter.

Lanna 2 : If a matrix R is diagonalizable, then its exponential
Rt _ P-l eA P
Alt
-1 e , <::>
=P -, P
O - A2,;
e

where Ai’ i =- l ... 2n, are the eigenvalues of R, and P is the

corresponding modal matrix.
Proof of Lanna 2 : See Strang ([47], Chap 5)

In order to facilitate the evaluation of the operator norm IIWII,

eRt is modified as follows. By defining the matrix Ei’ i = l ... 2n,

of order 2n x 2n which has 1 in the ith diagonal position and 0's

everywhere else, or

l,ifi-—-j=k

0 , otherwise

the diagonal matrix eAt can be written as

At 61. O

 

92

Zn Ait

- 2 E. e (V-8)
. 1
1-1

where A - -

Thus, using Lemmas 1, 2 and (V-8), for matrix R widnéfistinct

eigenvalues Ai’ i - l ... 2n,

eRt _ P-l eAt

P
-1 2n Ait
- P 2 E. e P
. 1
1-1
2n A t
= 2 P.1 Ei P e 1
i=1
2n A.t
1
- 2 81 e
i-l

where s. - P'1 E. P .
l 1

From spectral theory in a finite dimensional space ([JJY], Chap VII)

,

the matrix Si can be represented by

2n
H (R - A. I)
j¢i J

2n
.H. (Xi - Aj)
J#1

 

which leads to H(t) = eRt

2n Ait 2n
2 e H (R - A. I)
1-1 jfli 3
2n
n (A. - A.)
jfii J

(V-9)

 

 

93

Now following Theorem 1, an upper bound of the weighted operator norm

II W0 W le II may be computed as given in Theorem 2.

Igegggn 2 : Suppose the eigenvalues A1, i - 1 ... 2n, of matrix R,

are distinct. Then the weighted operator norm
||wmq'1||s maxtzlfi<t>|1dt
o o 0 i j ij

t
where (W z)(t) - I eR(t ‘ 7) 2(1) dr ,
0

A 2D. Aft
Rt -1 1
and H(t) - W0 e Qo - 2 e We S. Q

i-l

Proof of Theoren.2 : By Theorem 1,
-1 A
II wow, H s IZII um || .1.

where H(r) = W e Q

A

H can be written in the matrix form

H(r) - wo eR' le
2n A r -1
- W E e S Q
o . 1 o
1-1
2n Air -1
- z e w s Q
. o 1 o
1-1

Finally combining (V-10) with the induced matrix norm

IIH(r)II - max [2 IHij(r)I]
i j

we obtain

(V-lO)

(V-ll)

 

94

|| wo v Q;1 || 5 Im [ max r | Hij(t) | ] dt . D
0 1 J
Consequently the weighted operator norm II WOW Q gill is bounded

above as follows.

 

|| w 0 0'1 || 5 I0° [ max 2 | H..(t) | 1 dt (v-12)
o o 0 . . 13
1 J
2n Ait 2n -1
2 e we [ n (R - A. I) 00
‘ i=1 jfli 3
Where H(t) -
2n
n (x. - A.)
jfli 1 J

It must be noted that the formula (V-12) is valid only when the

eigenvalues Ai’ i - l ... 2n, of the matrix R are distinct. This

suffices for our purposes since in Chapter IV it was argued that the
optimal pole locations were of a Butterworth form which does not allow
repeated eigenvalues. Next the true norm of the operator, is

developed in the following theorem.

Theoren.3 : Let W : E: [0, m) » L:n[0 , m) be the linear operator
given by
t
(W z)(t) - I H(t - 1) 2(1) dr
0

and let each element Hij of the matrix H be in L1[0, m).

Then, || 0 || - max I‘0 2 | Hi.(r) | d1
1 0 j J

 

95

Proof of Theorem 3 : This proof consists of two parts. The first
part is to show the norm II W II is bounded from above by
max I” 2 I H..(r) I dr ,
i 0 j 13
and the second is to show that II W II is bounded from below by

max I” Z I H..(r) I dr
1 0 j 13

‘ From Theorem 1, we know that

|| 0 || 5 max I 2 | H. (r) | dr . (v-13)
. . ij
1 0 J

So it suffices to show that

IIWIIZmaxJ‘DZIHi(r)Idr
1 0 j 3
t
Consider g.(t) - I E H. (t - 7) z.(r) d1 (v-1a)
_ 1 0 j 1j J
where the index i is arbitrary but fixed.
A
Next, def1ne zj,t0(t) - Sgn [ Hij(t0 - t)] (V-lS)
and z. t - max 2 t - 1
II ,, to<>|| 3 || ,, to<>||

where t0 is arbitrary but fixed.

From (V-lh) and (V-lS) , we obtain

t

, - 2 H.. - . d
81(t0) Io j I 13(t0 7) 23(7) I T
but I gi(t0) I 5 II Hij ll 5 ll g ||L2n

Q

 

96

to II 8 II
Therefore, I 2 I Hi'(t0 - r) I dr

0 j J

I
I6

t
which yields I 0 2 I Hij(r) I dr

0 3

Since to in (V-l6) is arbitrary, it can be written as

IA
'6

(V-l6)

Ia z | Hij(r) | a, 5 || w ||

0 3

IA

and max I” z | H..(r) | a, || 0 || (v-17)
0 . 13

i J
Finally, by (V-13) and (V-l7), it follow that
|| 0 II = max I” z | H..(r) | a.
1 0 j 13

This completes the proof of Theorem 3. D

With Theorem 3 and (V-12), the operator norm II W0 W Qo II is

defined as follows :

|| we 0 Q0 || - mix I: [ p | Hij(t) | dt ] (V-18)

2n Ait 2n -1
2 e W [ H (R ' A- 1)] Q

A 1-1 0 'fli J o
where H(t) - J

 

2n
H (A. - A.)
j¢i J
Now we develop a lower bound of the norm II W0 W le II . From

equation (V-18),

 

97

-1 A
II W, W Q0 II = mix I:[ i I Hij<t> I dt 1
a max Im[ I 2 Hi°(t) I dt ]
i 0 j J
a max I I” Z Hi‘(t) dt ] I
i 0 j 3
Hence, the norm II WOW Q :31” is bounded from above and below

according to

 

x _ fl
max | I” 2 Hi.(t) dt 1 | 5 || wo.w 001 || 5 °
1 0 j J pofii+plflw+p2+p3fio
In the above derivations the operator norm II W0 W le II is

computed in terms of the closed loop eigenvalues of R which makes it
clear that II W0 W Q;1 II depends on the assignment of the spectrum of

the overall closed loop system. This reemphasizes that in a design
context it is useful to know special pole locations in the open left
half complex plane which minimize this operator norm. The pole
patterns developed in Chapter III provide a means byxflfich muflxan

assessment can be made.

5.2 Design. Matrices G and.K
In equation (V-18), the norm II W0 W le II is expressed in terms

of the matrix R and its distinct eigenvalues. Thus, it is convenient

to select a set of eigenvalues for matrix R first and compute

98
A B K

—G C A + B K + G C
in order to evaluate the operator norm. G and K are design matrices of
orders n x b and m x n respectively. These design matrices can be
computed by exploiting the separation property associated with the
linear portion of the overall closed loop system represented by R.

That is the characteristic polynomial of the overall system AR is

AR " AA+BK x AA + 00

The pole placement of R can be viewed as two subproblems since
a( R ) - a( A + BK ) U a( A + GC )

namely,

( i) For given a(A + BK) or AA + BK’ compute the controller gain K

(11) For given a(A + CO) or AA + 00’

compute the observer gain G .
To tflris end” the eigenvalue - eigenvector placement algorithm for MIMO

systems given in [35] is employed.

5.2.1 Eigenvalue - Eigenvector Placenant Algorithn

Unlike in the SISO systems, specifying the closed loop eigenvalues
for MIMO systems does not define a set of unique closed loop feedback
gains. (Hyen below is the algorithm [35] used for evaluating the
feedback K. By considering an augmented matrix ( A + B K ) and an

associated matrix RA given by
i

RAi - [ A1 I - A : B ], for 1 - l ... n

 

 

 

99

we can define a matrix Q" whose columns constitute a basis for the

i
kernel of RA denoted by ker [ RA ] such that

i i

SA.

1

Q a
A.
1 TA.
1

where the partitioned matrices SA and TA are of order n x m and m x m
i i

respectively. In the following theorem [35], necessary and sufficient
conditions are given for the existence of the feedback matrix K for a

given set of distinct eigenvalues.

Theorem 4 : Let A1, 1 - l . . . n, be a self conjugate set of distinct

complex numbers. Then, there exists a matrix K of real numbers such

that

( A + B K ) pi - Ai pi , i - 1 ... n (V-l9)

if and only if the following three conditions are satisfied for i = l,

., n .

( i ) Vectors pi are linearly independent vectors in Cn

* x
( ii) pi = pj whenever Ai = Aj where * denotes complex conjugate

(111) pi e span { Sxi }

If (i) - (iii) hold and rank B a m, then K is unique.

Proof of Theorem 4 : See Moore [35]

100

Based on Theorem 4, the following steps provide a systematic way

of computing the feedback gain K.

Step 1 : Select a self conjugate set of desired distinct eigenvalues

A1, 1 - l ... n for the closed loop system, that is A 1’ A jif
i i j
Step 2 : Form the matrix RA - [ A1 I - A : B ] of order n x (n + m)
i
for every A1, 1 - l ... n .
Step 3 : Find the matrix QA - [q1, q2, ... , qm] of order n x (n + m),
i
where qi, i - l ... n , are linearly independent vectors and
span the null space of RA , and partition QA as
i 1
SA.
1
QAi '-
TA.

where SA and TA are of orders n x m and m x m respectively.
' i

1
Step 4 : Select a corresponding set of closed loop eigenvectors pi,
i - 1, ..., n which meet the three conditions defined in
Theorem 4. That is,

( i ) Vectors pi are linearly independent vectors in Cn

* x
( ii) pi - pj whenever Ai - Aj where * denotes complex conjugate

(111) pi e span I SAi }

Step 5 : Compute a vector {i e C111 such that

 

101

P1 ' SA 51

i
and form wi - TA1 (i , for every Ai , i - l ... n
Step 6 : Form the matrices PA and WA as
PA - [p1, ... , pm] and WA - [w1, ... , wn]

Step 7 : Compute the feedback gain K

-1
K - WA PA

Remark : In step 3, to find a basis for the null space °f(;A , a
i
systematic procedure is given below.
( i ) Form the augmented matrix Ri given by

A. I - A : B

R - , for A1 , i - l ... n

where Ri is of order (2n + m) x (n + m), and Ia and la are

identity matrices of order n x n and (n + m) x (n + m)
respectively .

(ii) Obtain the m zero columns of [ AiII- A : B] by
performing elementary column operations on the matrix Ri'
(iii) The columns below the m zero columns of R1 span the null

space RA.
1

102

5.2.2. Computation of the Matrix G
In order to compute matrix C the same logic outlined in section
5.2.1 can be used by exploiting the duality of controllability and

observability.

Since a(A + GC) - a(A + GC)T

- a(AT + cTcT)

in Theorem 4, we can replace (V-19) by

T TT
(A + C G ) pi = A1 p

i
That is, the algorithm outlined in previous section is valid for

T

computation of observer gains G by replacing A by A , B by CT , and K

by CT of order m x m, n x b and b x n respectively.
Based on the eigenvalue - eigenvector placement algorithm outlined
above a general computer program is written for evaluating the design

matrices G and K , and for forming the matrix R. Finally the operator
norm II WOW Q :31” , for a given set of distinct eigenvalues is

computed as outlined earlier. The flow chart of Figure 19 outlines
the solution procedure employed for executing the design criteria of
the main result of Chapter II. Given below is a brief description of
the computer program. The main program 'CONPT' (Computation of

_Qperator Norm for Precision Tracking) has six high level routines.

Subroutine EIGSEL computes the spectra a(A + BK) and a(A + CO)
based on the optimal pole configuration developed in

Chapter IV.

103

 

C ”I“ )

 

 

( AthcopooplIpzvp3'fioifii’flwJ

 

 

 

 

 

 

I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Choose Choose
0(A + BK) a(A + GC)
A11, i-l..n A21, i-l..n
. I
Compute Compute
G
Compute Upper Bound
Form fio
u -
R b pofii+p1pw+p2+p3flo

 

 

Compute Norm

-1
n - IIWo‘I‘Qo ||

 

No

 

 

 

Yes

 

  

Write A B C G K
a(A+GC), a(A+BK)

    
 

 

c 0

Figure 19

Flow chart

 

 

104

5

Pofli+P15w+P2+P350

Subroutine SPHERE calculates the upper bound 0

-1
on the operator norm II WOWQo II

Subroutine PLFEED calculates the feedback gain matrices K and G.

subroutine EIGCC computes the eigenvalues of the augmented matrix

R to validate the accuracy of K and G.

subroutine PROJ computes the projection given by

 

2n A.t 2n 1
2 e 1 w [ n (R - A. I) 0'
eRt a 1-1 ° jfli 3 °
2n
n (A. - A.)
j¢i J

Subroutine OPNORH computes the operator norm by employing the

Trapezoidal rule for integration [10].

5.3 Numerical Integration scheme

The operator norm represented by the integral

I - max Ino [ z | Hi.(t) | ] dt (v-20)
i 0 j J

poses three difficulties. The first is the range of integration, which
is infinite; the second is the oscillatory nature of the integrand due
to complex conjugate pairs of eigenvalues; and the third difficulty is
the stiffness of the set of ODE , i.e., the eigenvalues of the system
matrix may be widely separated. If the equations are stiff, then very

small time steps need to be used for integration when standard

 

105

algorithms such as Simpson's scheme or the trapezoidal integration

scheme are employed.

( i ) Infinite range of integration : In order to evaluate the

integral it is necessary to write

T
I-F-If+J“
0 0 Tf
where the time interval [0, w) is truncated at Tf so that the

contribution of r is negligible. In order to estimate Tf ,
T
f

the settling time for a first or second order dominant pole may be used
depending on whether the slowest mode is due to a real pole or a

complex conjugate pair. The following conservative estimate for Tf is

used.

5

 

max I Re (A1) }

Then the integration is carried in two parts, i.e.,

A T 1.5 T
Ia If+I f
O Tf

1.5 T Tf
is very small, I is taken to be I = I ,

If the integral I
0

Tf

1.5 1%

T

'howevery if I is not negligible, then the remaining

f
interval is again divided into subintervals , until the

contribution from the tail end of the integral is insignificant.

 

106

( ii) Oscillatory nature of the integrand : This creates a major
problem, when using the standard schemes mentioned above. In
particular, if only a few function values are considered, or a
large time step is used, the integrand sometimes appears to be
quite a different function than what it actually is. Thus, the
usage of small time steps is inevitable with these schemes, which

is aggravated if the eigenvalues have large oscillation

frequencies 0:. A time step Ts such that
Zn 1
Ta 5 < —w > ( 50 >

is typically used when standard integration schemes are employed.

Here we use the time step TS,

2x 1
TS - 0.001 if (_w) (50) 2. 0.001
2« l 21r l
-(—w)(§0)1f(—w)(§0)50.001
Remark : To overcome this difficulty, the subprogram DCADRE of the

IMSL package that is based on adaptive integration [11] may be
used. Here, however, due to some programming difficulties in
linking DCADRE to our main program CONPT, the trapezoidal rule
is employed, thus making small time steps inevitable for

accurate results.

(iii) Stiffness of the Equations : The eigenvalues of R play a
crucial role in evaluating the integral given by (V-20) . For

example, if we let A3 and AS be two eigenvalues of R such that

 

107

A = max { I Re(Ai) I I A1 6 a(R) , i = 1 .... 2n }

.2

As - min { | Re(Ai) | | A1 6 a(R) , i = 1 .... 2n }

then the numerical integration must be carried out until the
slowest decaying exponential in the transient part, (i.e., the one

A t
corresponding to e s ) is negligible. Thus the smaller the

I Re A s I, the longer will be the range of integration. On the

other hand, if A is far out into the left half complex plane,

3
then the usage of excessively small time steps are prompted. If
I Re A2 I >> I Re AS I, then the highly undesirable computational

situation occurs, that is integration over a long range using a
small time step which is everywhere excessively small relative to
the interval. This suggests a procedure for selecting the
eigenvalues of the closed loop system to avoid numerical problems.
Namely, the eigenvalues for the closed loop system may be chosen

to be in one or two clusters where the separation is small enough.

 

CHAPTER“

APPLICATIONS

In this chapter, we give several examples to illustrate the ap-
plicability of the theory developed in Chapter II. These examples
include a synchronous machine, tracking in a robot manipulator
problem, and a single DOF gyroscope. In a synchronous machine only
uncertainty in the input disturbance is considered, whereas in the
robot example uncertainties in both the input and disturbances are al-
lowed. The gyroscope problem has a sector bounded nonlinearity and

input disturbance .

6 . l A Synchronous Machine
A synchronous machine with a conventional velocity governor con-
nected to an infinite bus as shown in Figure 20, is considered. The

dynamic equations of this machine represented in state space form are

 

 

 

 

 

 

 

 

 

 

'3' '0 1 0] "a“ '0‘ '0]
-c
a. 0 _§ ‘1
- M M 3 + 0 u+ f(o) (VI-la)
o - Kg .1
LPm‘ *0 wT T‘ ~Pm‘ ~1‘ -0 d
08 g
o -[1 0 0] o (VI-1b)
79
P

where 0 : rotor angle

$0

: rotor angular velocity

108

 

109

P : input power (p.u.)

Ce: machine damping coefficient (p.u. power second per electri-

cal radian)

M : generator inertia constant (p.u. power second2 per electri-

cal radian)

Tg: equivalent time constant (p.u. second)

Kg: loop gain of governor system

f(0) - 'fi I 0.317 sin(0+63.7) - 0.035 sin 2(0+63.7) - 0.15)

 

 

 

 

 

 

mx
S\\\\\\\\\‘.

Generator

Governor

Figure 20 A synchronous machine

To obtain numerical results the following parameter values given in

[53] are used. They are :
M - 0.0138
C - 0.0138
e

 

 

110

T - 0.1
g

K = 1.0
8

w = 120 x .
0

With these parameter values the equation (VI-l) becomes

x1 0 1 0 x1 0 0
x2 - 0 -1 -72.464 x2 + 0 u + f(x) (VI-2a)
0 0.027 -10 x 1 0
x 3
3
X
y = [ 1 0 0 1 1 (VI-2b)
x2
x3
where x = [0, 5, Pm]T 6 R3 , u e R; , y e R1 , and

f(x) - -9.928 sin(x1 + 63.7) + 2.536 sin 2(x1 + 63.7) + 10.915

The problem is to achieve a specified tracking performance for the
rotor angle 6 against an input disturbance.

Let the desired behaviour of the rotor angle 0 be given by
yo = 25 ( l. - cos 2t )

with the acceptable error bound i 1.0 % . This gives the output sphere
specification

I y - yOI S 0.5 , or 60 = 0.5

Next, let the input sphere specification be

I u - u S 1.0 , or Bi = 1.0

o l
The weighting matrices chosen primarily to yield favourable norm values

are

 

111

 

 

_ l 0 0 I
l
0 0
where Z - A , A = max I IAI A 6 a(R) } ,
max max
1
10 0 A.
max

and 0's are the zero matrices of appropriate dimension.

Design Execution :

It is easy to compute

and

Due to the absence of any uncertainty in the nonlinear term, the non-
linear design function can be chosen as
fo(x) - f(x)
y1eld1ng N72 - Noz
This gives

P2 - sup II N Z

..-N.2.II

= 0
p3 can be computed by finding the maximum gradient of f(x) with respect

to each state variable. Thus,

p3 - max I Vx f(x) }
- max I O.,I-9.928 cos(x1+63.7)+4.114 cos 2(x1+63.7)I. 0 }

a 15.0

Now with the above norm values, the upper bound given by (II-l6) is

0
”0’9: + P1fiw + ”2 “I ”3‘90

 

19 0.5
— (VI-3)
+ (0.5)(15.0)

>4h‘

max

 

112

That is if a set of system eigenvalues is chosen so that the operator

norm II WOW Qo II is less than the upper bound given in (VI-3) , the

conditions of Theorem 1 of Chapter II are satisfied.
Based on the pole configurations of Chapter IV and algorithms of

Chapter V, we obtain the set of system eigenvalues

a( A + BK ) - I -90. , -45. i j 77.94 1

yielding K - [ 10059.73 221.057 -l69.0 1
and a( A + cc ) - I -150. -75. i j 129.90 }

yielding 0 - [ -289.0 -41808.05 40755.61 1T

With the set of spectra above, we obtain the threshold

p0

= 0.066
pofli + plflw + p2 + p3flo

and the critical value of the operator norm
|| wo w Qo || - 0.064

which satisfies the inequality.

Next, with the above feedback gain K the nominal command input ro(t)

for the closed loop system is computed. It is

ro(t) - -251493.017 + 251244.69 cos 2t - 11174.955 sin 2t

Computer Simulation : The augmented system for computer simulation is
Plant :
x 0 1 0 x1 0 0
x - 0 -1 -72.464 x2 + 0 u + f(x)
x 0 0 027 -10 x3 1 0
y = [ 1 0 0 ] x1
x2
x3

where f(x) = -9.928 sin(xl+63.7) + 2.536 sin 2(x1+63.7) + 10.915

y(t). yo(t)

20 -

 

Figure 21

0.04 r
0.03 - I

0.02

(y(t) - yo(t))

0.01 -

 

 

 

 

113

I 2 J 4 5
Time ( Sec )

Actual output y(t) and nominal output yo(t)

 

 

 

 

 

 

I

-0.01«
0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I I IWI I

Time ( sec )

Figure 22 Output deviation (y(t) - yo(t))

114

 

 

C

(r(t) - ram)
J

. A]

j

l

:3

]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Time (Sec)

Figure 23 Input disturbance (r(t) - ro(t))

I00 -

J. III” I I] I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

‘I00 1 1 1 1
0 I 2 3 4

Time (Sec)

Figure 24 Control effort u(t)

 

115

Observer :
. 0 1 0 0 0
x - 0 -1 -72.464 x + 0 u + f(x) + G [1 O O] [x - x]
0 0.027 -10 1 0
A 3
where x e R and
c - [ -289.0 -41808.06 40755.61 1T.
Control : u(t) - r(t) + K x(t)
K - [ 10059.73 221.057 -169.0 ]

with the initial conditions x(O) - x(O) - 0 e R3 .

The time responses of the system are shown in Figure (21 - 24).

Figure 21 shows the actual output y(t) and the nominal output yo(t) to

a square pulse input disturbance shown in Figure 23. Figure 24 shows

the control u(t). The error (y - yo) shown in Figure 22 is within the

given tolerance for all time, and it is much less than the desired er-
ror bound 0.5. Although the output sphere specification has been
satisfied it is clear that this scheme is quite conservative. This is
primarily due to the extent of the disturbance that can be allowed
within the specifications. Namely, an infinite number of admissible

functions in Lm[0, do) are allowed for the disturbances.

6.2 A 3 DOF Manipulator

We consider the three DOF manipulator shown in Figure 25. This
manipulator has a rotational joint and a translational joint in the (x,
y) plane. Moreover the arm can be lifted along the vertical z-axis

thus defining the third degree of freedom. The dynamic equations for

 

116

 

 

 

Vertical z - axis

// Force F
V 9 2
0"

Figure 25 A three DOF robot manipulator

this robot configuration follow directly from an application of

Lagrange's equations and take the following form [18]

M<¢(t). 1) 72c) - -f<¢<t). x(t). 1) + u(t) (v1-4)

where w(t) - [r(t), 0(t), z(t)] Tspecifies the configuration at time t
in a cylindrical frame of reference, 7 is the payload uncertainty and
the dot denotes time derivatives. u(t) represents the generalized
forces and is given by

T

(t) - [Frt To: F I

Z

where Fr is the radial force, T0 is the torque and Fz is the vertical

force associated with the coordinates r, 0, and 2 respectively.

H(¢(t),1) and f(¢(t),¢(t),7) are given below.

117

(M2 + M1) 0 0
M(1/2(t),'y)- 0 (M1 + M2) r2(t)-Ml 12 r(t)+J 0 (VI-5a)
0 0 (M1+M2)
( -(M1+M2) r(t) +§ M1 12 ) 272(6)
f(¢(t).W(t).7)= < 2<M1+M2> r(t) -M1 2 > £<t> Mt) (VI-5b)
0

where M1 and M2 are the arm mass and the payload mass respectively, and
2 is the length of the arm AB. The net moment of inertia of the arm

and the swivel joint J is given by

J = JM + JM

where, JM and JM are the moments of inertia of the swivel and the arm,
1 3

respectively, about the z-axis. M3 and rz are the mass and the radius

of the swivel.
Equations (VI-5) depict a highly coupled nonlinear set of equa-
tions. By employing the state dependent transformation

11(t) - M(¢(t),1)'ut(t) (VI-6)
on the input u(t), the equations of motion (VI-4) are transformed into
Wt) = -M(W(t).7)'f(¢(t).W(t).7) + ut<t> (VI-7)

The inertia matrix M(¢(t), y) is clearly invertible for all t e [0, m),
which follows from the positive definiteness of the mass matrix of a
r“nipulator.

0w equations (VI-7) can be rewritten in the usual state space

xielding

 

 

118

. 0 13 0 0
x(t) - 0 0 x(t) + 13 ut(t> + fN(x(t)’7) (VI-8a)
y(t) = 1 I3 0 1 x(t) (VI-8b)

where, I3 and 0 are 3 x 3 the identity and the null matrices respec-

tively,
¢(t)
x(t) = 0(t)
-[m 9,2,}, 79,211? .116,

ut(t)- M-1(1,b(t), 7) u(t) 6 R3, and the nonlinear term

fN - - M‘1(¢(t).v>-f<¢<t).¢<t>.7)

' (M2 + M1)'1 (-(M1+M

2)

r(t) + l M1 2 ) 52(t)
2

- ((111 + M2) r2(t)-M1 2 r(t)+J)-1(2(M1+M2) r(t) -M1 2 )i(t)é(t)

 

 

O
le

' sz

fN3
Equations (VI-8) are decoupled with respect to the linear parts
and are used in executing the design procedure previously outlined in

Chapter II. This form clearly allows the arbitrary placement of

eigenvalues of each decoupled subsystem.

Design Objective :
Our basic design objective is to synthesize a control u(t) in or-
der to achieve the tracking performance specified by the output

constraints

119

ll yi - yoill s 90,. floi> 0. i = 1. 2. 3

despite the input disturbance and the payload uncertainty. 1 =

yoi’

l, 2, 3 are the three nominal outputs to be tracked and yi, i- 1, 2, 3,

are the three actual outputs.

Input Sphere :
Let the input sphere be given by

fli - 1.0 , i - l, 2, 3.

Nominal Output :

The nominal outputs to be tracked are

yols 0.8 - 0.8 e'3t(eos(t) + 3. sin(t))

for the radial displacement of the arm,

for the angular rotation of the arm, and

yo3 - 0.5 - 0.5 cos(t)

for the vertical motion of the arm.

Output Spheres

Output sphere specifications are
flol - fioZ - flo3 - 0'1
Thus the tracking specifications call for precise tracking of the

nominal outputs given above upto an accuracy of 0.1 m in yl, 0.1 rad in

y2 and 0.1 m in y3.

 

120

Bounded Uncertainty :

We consider the payload M2 to be the primary uncertainty and as-

sume that

M2 6 r - [0, 20] kg.

In order to design a controller as outlined previously, the
threshold value specified in equation (II-l6) needs to be computed
first. This requires the computation of several norm quantities as
described in the main theorem of Chapter II. We use the following

data for all computations.

M1 - 40 kg
M2 - [0, 20 ] kg
M3 - 100 kg
2 - l m
r(t) - [0.0, 1.0 ] m

z(t)
0(t) - [0., fl ] rad

[ 0.0, 1.0 ] m

r = 0.1 m.
2

Let the design matrices V1 - 03 and V2 - I3, then

80 = [ 0

The weighting matrices W0 and Q0 chosen primarily to yield favorable

norm values are

121

I3 03
where 2 - l I and IA.I is the maximum absolute
0 3 1 max
3 IAilmax
value of the eigenvalues of matrix R. I3 is the 3 x 3 Identity

matrix, 03 and 06 respectively are 3 x 3 and 6 x 6 null matrices.

Remark : The selection of the weighting matrices is rather ar-
bitrary. For example they may be set to the identity matrix. This

however will not yield favorable norm values.

Then, 80 - ll QOBOII
E 0
6 T
-II IO :I'Io3 I3 03 I3] II
6 X
l
_ ’ i-l,oooo,12
I)‘ilmax
and P1 ' II QOBIII
-0.

since B1 = 0 due to the absence of external disturbances.

To calculate
p2 - sup II Qo(Nyzo - N020) II
7 e P
we need to select a nominal nonlinear function fo(x) to cancel as much
as possible the uncertain effects of fN' We choose fo(x) to be of the

same form as fN(x, 1) with 7 replaced by 70, where 70 are the arithe

metic means of the uncertain parameters

 

122

In this case

7 - M2 - [0. 201 kg

thus yielding

- 10. kg

where I712, L12 and 112 respectively are the mean value, lower bound and

the upper bound of M2.

 

 

 

 

 

f1(X)
Thus, fo(x) - f2(X)
0
where f (x) - (x - M1 1 ) x2
l l _ 5
2(M1 + M2)
-2 (M + E ) x — M 2

and f2(x) = 1 2 1 1 x4 x

J - M1 2 x1 + (M1 + M2) x1
Thus, p2 - sup II QO(N72o - Nozo ) II

I3 03 03
-sup|| O 1 ,3 f H) II
T——T - x
3 Ai max N o
-M 2 M 2 1
l l 2
- max { I( + u. ) X5 I TKTT ,
2(M +M ) 2(M + E ) 1 max
1 2 l 2
-2(M1+M2) x1 - M1 2 2(M1+M2) x1 + M2 2
( 2+ )xaxs
- 2
J - M1 2 x1+ (M1+M2) x1 J - M1 2 x1 + (M1+M2) xl

 

123

 

1
IA.I I
1 max
On substitution of numerical values, it follows that

6.0

p -
2 IAiImax
Computation of p3 involves the calculation of gradients of the

nonlinearity with respect to the state vector x, and is given by

p3 - max I G1, G2}

 

T
where, Gl - max II Vx le II
{ I I 1. M1 2 I I
- max x T——T , 2 (x - ) x }
5 Ai max I 1 2(M1 + M2) I 5
- 1.34

T
G2 max II Vx fN2 II

{I -2( (J -M12xl+(M1+M2)xi) (M1+M2)+(2 (M1+M2)x1+M12) (41124.2 (M1+M2)x1
-max 0

 

2 2
(J - M1 2 x1+ (M1 + M2 ) x1 )

-2(M1 + M2) x1 - M1 2

 

l.
I X4 X5 I IAiImax’ I 2 I. IXSI I
J - M1 2 x1 + (M1 +M2) x1

 

.2(Ml + M2) x1 - M1 2 . Ix I
2 4

 

J-MlAthl+(M1-1-Mz)x1

- 21.0

Hence, we obtain p3 - 21.0

 

124

Ragggk : In computing p2 and p3 as above it is implicitly assumed that

1

 

< 1. At the end of the design this condition needs

IAiImax

to be verified. It will clearly be satisfied in this case.

Now assembling all of the above computations we compute the
threshold given in equation (II—l6)

B 0.1

o -
pofli + plfiw + p2 + p330 1. + 6.0 + (21.0)(0.1)

IAiImax IAiImax

 

(VI-9)

Now it only remains to find a set of eigenvalues for the system

matrix

A B K

- G C A + B K + G C

so that the norm II WOWQo-III is less than the upper bound (VI-9).

Based on the numerical scheme previously outlined, we obtain the
spectra
a(A +B K) -{ -47.0 i j 49.0 , - 50.0 i j 53.0 , - 53.0 i j 51.0 I
and
a(A + CC) -I-110.0 i j 111.0, -1l3.0 i j 114.0, -115.0 i j 113.0 I
yielding
I - 4160. O. O. - 94. 0. 0. I
K = 0. - 5309. 0. 0. -100. 0.
0. 0. - 5410. O. 0. -106.

and

0. -226. 0 0. -25765. 0.

-220. 0. 0. -24421. 0. 0. T
c - .
0. 0. -230. 0. 0. -25994.

 

125

With the above spectra, we obtain the upper bound

Bo
pofli + plflw + p2 + p330

 

- 0.047

and the critical norm of the operator
||w wQ '1|| - 0 041
o 0 °

which clearly satisfies inequality (II-16).
With K known we can now compute the nominal command input func-

tions roi(t), i = 1, 2, 3, as follows.
rol(t) - 3688. - e'3t (3680. cos(t) + 10336. sin(t))

r02(t) - e't(2. + 196. t + 5210. :2)
ro3(t) - 2705. - 2704.5 cos(t) + 53. sin(t).

Thus it follows that the design specified by matrices G, K, the non-

linear function fo(x) and the nominal inputs roi’ i — 1, 2, 3,
guarantee the rec- 1 tracking performance according to the theorem.
The validity of th rem is also confirmed by simulation results.

Simulation of the Closed.Loop System : The system dynamics for simula-

tion are
Plant 2
i(t) = [0 I3 ] x(t) + [2 ]ut<t> + [f (th) ]
0 0 3 N ’7)
y(t) = [ I3 0 1 x(t)

Observer :

 

 

126

1 0 13 . 0 0 .
x(t) - 0 x(t) + I3 ut(t) + fo(x(t)) + CO [x(t)-x(t)]

0
-220. 0. 0. -24421. 0. 0. T
where 0 - 0. -226. 0. 0. -25765. 0.
0. 0. -230. 0. 0. -25994.
Control :

A

ut(t) - r(t) + K x(t)

- 4160. 0. 0. - 94. 0. 0.
where K - 0. - 5309. 0. 0. -100. 0.
0. 0. - 5410. 0 0. -106.

and the initial conditions x(0) - x(0) - 0. 6 R

Figure (26 - 29) show simulations for M2 - 20 kg. Figure 26 shows

the nominal output y01 and the actual output yl. There is hardly any
difference in the two graphs. This clearly demonstrates the tracking
accuracy. Figure 27(a), (b) and (c) show the errors ei - yi - yoi’

i = 1, 2, 3, respectively resulting from the input disturbance shown in

Figure 29. These errors are of the order of 10'3 which is quite con-

servative in comparison with the imposed output sphere flo - 0.1

This conservativeness is not surprising due to the generality of the

inputs and the nonlinearity admissible in.LhJ()., w). The required

control inputs are shown in Figure 28(a). (b), and (c). Figure (30 -

32) show simulations for a sinusoidally varying uncertainty M2 =
10. + 10 sin(lOt). Figure 30 shows the nominal output y02 and the ac-

tual output y2 . Figure 31 shows the error y2 - y02 and the control

 

 

127

input u2 is shown in Figure 32. The latter uncertainty is considered

just for the sake of demonstrating that the methodology is valid for

any uncertainty in a given band.

 

 

 

1.00 P
a 0675'-
’3
H
>0 0.50-
E
g 0.25-
0.00 ‘ L ' ' _’
0 2 5 8 10

Time (Sec)

Figure 26 y1(t) and yol(t)

 

 

so
u2(t)
(kg-m)
20

4o
u3(t)

(kg)
20

 

 

130

 

 

 

l L 1 l I
2 4 I 8 10
Time (Sec)
Figure 28(b) Control effort u2(t)
1 L l l I
2 4 I 8 10

Figure 28(c)

Time ( Sec )

Control effort

u3(t)

 

<r1(c> - r°1(c)) (kg)

y2(t). y02(c) (rad)

131

 

 

2 _
I
0 |-
-1 _
_2 J l l l I
0 2 4 0 I 10
Time ( Sec )
Figure 29 Input disturbance (r1(t) - r°1(t))
0.6 —
0.4 -
0.2 -
0.0 l L I l

 

 

Time ( Sec )

Figure 30 y2(t) and y°2(t)

 

132

xm'3

°
*
I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(Y2(c) ' Yoz(t)) (rad)
N

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1W1 W W 1

 

 

I
O
N
r
fl

 

 

-o.4 ‘ ‘ *‘ ‘ '
0 2 4 0 0 10
Time (Sec)

Figure 31 The tracking error (y2(t) - y02(t))

 

 

 

30 F
u2(t)
(kg-m)
20 -
l
10 -
o .
-‘0 i l L L l
o 2 4 I a 10

Time (sec)

Figure 32 Control effort u2(t)

133

6.3 A Single DOF Gyroscope
In this section a structurally rigid model of a gyro as shown in

Figure 33 is considered to illustrate the special case of Chapter II.

Y

 

 

Figure 33 A single DOF gyroscope

Let X, Y, Z be a set of axes attached to the vehicle. The rotor is
mounted in a single gimbal, in which it can turn about axes, so that a.
rotation about that axis leads to the gimbal axis x, y, 2. By employ-

dw

ing the Lagrangian approach, for small 0 and ‘3}: - 0 , the equation

of motion for gimbal rotation 0 about the output axis is obtained :

2
11.9. 12 _ _
(Jr + JS) 6,2 + Cd dr + (Kc + anz) 0 anY

(VI-10)

 

 

134

where Jr and JE denote the moments of inertia of rotor and gimbal,

respectively, about the axis x, w w and w Zdenote the angular

X’ Y’

velocity components of vehicle along X, Y, Z, and Kc and .Cd represent

the torsional spring constant and torsional damping coefficient of the
torsional spring and dashpot, respectively. Defining new variables

relating the gimbal rotation 6 and the time scale 1 in (VI-10) by

x - (KC / 0n) 9

1/2
t - (KC /(Jr + Jg)) 7

yields i + 2 5 i + (1 + 7) x - u (VI-ll)
where . denotes the derivative of x with respect to t and

7 - (Cn wz) Kc

-1/2
Cd (Kc (Jr + Jg))

 

2

113-(DY

Rearranging equation (V-11) in the state space form yields

x1 0 1 x1 0 O
= + u + (VI-12a)
x2 -1 -26 x2 1 7 x1
y = [ 1 0 ] x1 (VI-12b)
x2

For computer simulation the numerical values of the parameters of the

gyro given in [45] are used. They are :

2
(Jr + Jg) - 54 dyne-cm-sec

Cn = 10.8 x 104 dyne-cm—sec

 

 

135

Kc = 54 x 104 dyne-cm-rad'l

Cd - 324 dyne-cm—sec
It is assumed that the uncertain angular velocity wz is bounded, and
that the uncertainty I7I S 0.2

The problem is to determine a control u(t) which guarantees that

the system output is within a given tolerance, (that is Iy - yol s flo),

despite the uncertain parameter and the input disturbance.

Nominal output : The nominal output of the gimbal rotation to be

tracked is

yo = 0.25 (1 - cos 3t )

Output sphere specification : Let the controller requirement be to
maintain the fluctuation in gimbal rotation 0 within 1 0.7 % of

y°(t) for all t e [0., w). That is

Iy - yol s 0 0035 , or 60 = 0.0035
Input sphere specification 2 Let the input wY be of uncertainty such
that

Iu - uol s 0.1 , or 61 = 0.1

Triiee weighting matrices W0 and Q0 which give favourable norm values are

E 0
W0 = , and
0 Z

 

136

, Amax - max { [AI A 6 a(R) ) , and

where 2 -

YIH O

max
I is the identity matrix of order 2 x 2
Then p0 - ||Qo Boll - 1.0
Since there is no external disturbance
P1“ HQO 81" ‘0.

The uncertain nonlinear term

0
f(X, 7. t) '
7X1

can be considered as a sector bounded nonlinearity with respect to

state xl . Since the uncertain element [1| 5 0.2 defines the lower and

the upper bounds of a sector bound nonlinearity with a - -0.2 and B =

0.2, the nominal nonlinear function fo(x) can be chosen as the zero
function.
Consequently, p2 - % (fl - a) ”Q0 20]]
-0.15
p3=max( lfll, lal}
= 0.2

chrw with all of the above computations, the threshold given in (II-26)

b e comes

'60

= 0.014
pofli + plfiw + ”2 + P3flo

 

137

Next, solving the pole - placement problem using the algorithms of
Chapter IV, we get
a(A + BK) - { -9.0 i j 9.0 )
a(A + GC) - [ -l8.0 i j 18.0 }
yielding K - [ -l6l.0 -l7.94 ]
G - [ -35.94 -644.84 ]
and the critical norm of the operator

'1|| - 0.013

I lwomo
< 0.014
satisfies the inequality (II-26).

Now, with K the nominal command input ro(t) is computed as

r0 = 40.5 - 38.25 cos 3t + 13.5 sin 3t

Computer Simulation : The overall closed loop is
Plant :
i 0 1 x 0 0
l = 1 + u +
x2 -1 -26 x2 1 7 x1

Where |7|S 02

Observer :

x = x + u + G [ 1 0 ] [ x - x ]
-l -26 l

‘Vlieere x e R2 , and c - [ -35.94 -644.84 ]T

 

138
0.5 r
i?
V
o 0.4 -
>5
7.?
V
x
0.2 -
0.0 ' ' ‘ '
O 1 2 3 4 5
Time ( Sec )
Figure 34 y(t) and yo(t)
x10"z
0.2 ~
;: 0.1 -
it
0
>5
,\ 0.0
3
3:
-o.1 _
-o.2 ‘ ‘ 4 l J
0 1 2 J 4 5

 

 

 

 

 

Time ( sec )

Figure 35 The tracking error (y(t) - y°(t))

 

139

 

 

 

 

4
u(t)
2 h
o -
_2 l I I I l
0 1 2 3 4 5
Time ( Sec )
Figure 36 Required control u(t)
0.2 -
2: 0.1
i",
o
u
A 0.0 -
U
3:
-0J -
-0.2 1 ' 4 ’ '
0 1 2 3 4 5

Time (Sec)

Figure 37 Input uncertainty (r(t) - r°(t))

 

140

Control :

u(t) - r(t) + K x(t)
where K =- [ -161.0 -17.94 ]

with the initial conditions x(O) - x(0) = 0. 6 R2.

For computer simulation the uncertain element 7 is assumed to be a

function of time given by 7 = 0.2 sin 5t and the input uncertainty

is a square pulse with magnitude i 0.1 and in Figure 37. Figures (34
- 36) show the simulation results. The actual output y(t) and the

nominal output yo(t) are shown in Figure 34, and control u(t) is shown
in Figure 36. The error (y - yo) shown in Figure 35 has a maximum of

0.002 rad, and is less than the given tolerance 0.0035 rad.

Remark : It is worth noting that systems with sector bounded non-
linearities lend themselves well to straightforward norm calculations

as outlined in Chapter II and employed in the above example.

 

CHAPTER VII
CONCLUSIONS, DISCUSSION AND FUTURE WORK

The research presented in this thesis in a broad sense addressed
the issue of "robustness" of control systems. In particular, precision
tracking of specified outputs in the presence of uncertain parameters
and external disturbance was considered. Specifically there were three

objectives for this research :

(a) To study different formulation of the tracking problem for uncer-
tain systems incorporating a more global concept of tracking than
is typically considered in asymptotic considerations. This global
form allows one to precisely capture the physical nature of track-

ing.
(b) Consider different controller structures that allow precision
tracking and are simple. In particular, investigate a pole-

placement technique that is quantitative in nature.

((1) Apply the results of (a) and (b) to the currently active and im—

portant area of robotic manipulators.

Given below is a discussion of the contributions made under each

Obj ective .

141

 

¥

142

(a) Problem Formulation :
Two basic formulations based on functional analysis of poorly
defined systems were studied. In the first formulation given in

Chapter II the tracking problem was embedded in the Banach space of es-
sentially bounded functions L:[0, on). In this formulation the

physical requirements of tracking are quite naturally captured and the
necessary design criteria are given in the time domain.

In the second formulation developed in Chapter III the problem was
embedded in the Hilbert space L§(T) and. frequency domain interpreta-

tions were sought for the precision tracking problem. This
formulation allows one to capture asymptotic features of tracking with

an L2 bound on the overall error function. A circle type criterion

with a transparent geometric interpretation was developed. The latter
was possible due to the frequency domain interpretations manifested by

L2 - functions. The latter interpretation however is valid only for

S ISO systems.

(b) Controller Structure :

The basic controller structure investigated consisted of a non-
linear Luenberger observer based state feedback control. This
eSsentially gives rise to a design criterion that requires nothing but
a clever way of assigning the spectrum of a linear operator so that a
Certain upper bound on a crucial operator norm is satisfied. This is
What is referred to in here as a "quantitative pole-placement".

For the Lao - formulation a design procedure leading to a certain

perturbed form of the well known Butterworth pole-patterns was

 

143

developed. These were arrived at by setting up a minimization problem

in LP(T) and considering the limit as p -2 c0. To obtain the pole pat-

terns a modified Riccatti type equation needs to be solved. To get a
quick idea however we start with the Butterworth patterns for co radius
and then gradually decrease the radius maintaining the patterns until
design criteria are satisfied in an "optimal" sense. Admittedly this
approach using Butterworth pole-patterns lacks rigour but accomplishes
the task adequately.

It is also interesting to note that the results contained in here
seem to suggest that high - gain plays an important role in uncertain
problems. It should be pointed out however that the results here

provide a rational scheme for selecting such high gains.

(c) Applications :
Several examples including a robotic manipulator were reported in
Chapter VI. Algorithms needed for design execution were developed in

Chapter V.

The work presented in here together with the "Quantitative
Feedback Theory" of Isaac Horowitz [22, 23] constitute the only design
theory available to the best of our knowledge for directly satisfying

the design specifications for uncertain systems. Despite much recent

Work on robustness (especially H paint of View) With Significant con-

tributions, direct design for specifications remains incomplete.

The conservative nature of the results obtained thus far remains a

IIlajor concern. Consideration of time dependent weighting schemes may

 

144

prove useful to answer the latter. Future work may also include feed-
back linearization techniques in the problem formulation. The
geometric interpretation given for the SISO sector bounded case appears
to be extendable to nonlinearities with more structure, for example,
monotone nonlinearities. Different classes of nonlinearities and more
structured uncertainties should also be considered in future investiga-
tions. Also it seems plausible that frequency domain interpretations
leading to transparent design criteria similar to the one obtained for

the L2 - case can be obtained for the Lao - tracking problem, by intro-

ducing the notion of exponential weighting which is predicated on the
two facts given below :

(1) If Y(t) - g(t)*U(t)

then V a e R

y(t) exp(at) - g(t) exp(at) * u(t) exp(at)

(2) £ [f(t) exp(at)] - f(s -a) when f(s) = £ [f(t)]

( * denotes convolution )

To gain additional insight to the problem future studies should
also include linear problems with uncertainties for which more explicit
results can be anticipated. Recently Vidyasagar [51] reported some in-
teresting results based on factorization methods for a somewhat related
Problem dealing with linear systems. Specifically, regulation and
asymptotic tracking in the presence of persisting disturbances were

Stu-died. The notion of persisting disturbances is quite naturally
Cap tLII-ed when the problem is embedded in the L:[O, 00) space of func-

t_1°n8 as done in our work. Same thought should also be given to

 

145

incorporate the factorization approach to investigate continuous track-

ing studied in this thesis. Establishing a connection between the L2

— theory presented in here and Han methods appears feasible.

 

 

APPENDIX A

HATHDIKI'ICAL PRELIMINARIES

The study of nonlinear differential equations in general requires
rather sophisticated mathematical tools. The work reported in this
thesis is primarily based on properties of linear operators. Some
useful definitions are collected in this appendix to give the reader an
idea of what is involved. Any standard text on functional analysis
such as Rudin (Functional Analysis, McGraw—Hill, 1973) can be consulted

fo 1: details .

Definition A1 : A set X over a field F (R or C suffices for our
purposes) together with two operations + termed addition, and . termed

sc‘alar multiplication is a linear vector space if the following axioms

hold

( i ) x1 + x2 - x2 + x1 , V x1, x2 6 X
(Commutativity of addition)
( 11) (x1 + x2) + x3 - x1 + (x2 + x3), V x1, x2 6 X

(Associativity of addition)

(iii) There is an element 0 e X such that

x+0-0 , VxeX

( iv) For each x e X there is an element -x e X such that

x+(-x)-0, VxeX

146

 

 

147

(v)a(fix)-(afl)x,foreacha,fleFandforeachxeX

(vi) (a+fi)x-ax+,6x,foreacha,fleFandforeachxeX

(vii)1x-x,foreachxeX

The set of all n-tuples of real numbers denoted by R1'1 is a real linear

vector space, similarly, Cn consisting of all n-tuples of complex num-

bers is a complex linear vector space. In what follows we assume that

the field F is the reals R.

X is a linear subspace if XS

Definition A_Z : A nonempty subset XS

is closed under the operations of addition and scalar multiplication in

X- That is, (i) x1 + x2 6 XS, V x1, x2 6 XS

(ii)axeXS , V xeXSandaeR

One of the important subspaces is the kernel of a linear map

‘1' = x1 -* 12
Where X1, X2 are vector spaces. The kernel also termed the nullspace

0f ‘1! is the set given by
Ker‘F-{xexllitx-O}

I) where X is a linear

 

 

A normed linear space (X, I

o I] is a real valued function in X called the norm

\De finition A3 :

vector space and

 

 

Such that
( i)||x||20, V xeX,and||x||=Oifandonlyifx=0.
( ii) [I ax I] 5 la] ”x'] if x e X and a is a scalar.

148

(iii) II x1+x2 II S IleII + IIx2II , V x1, xzeX

The "norm" on the linear space X is used to denote the real valued

function that maps x to IIxI I. Hence, it can be considered as a

generalization of the concept of the length of a vector in R2 or R3.

Namely, given a vector x in a normed linear space, the nonnegative num-
ber II x II can be thought of as the length of the vector x.

Similarly, given two elements x1 and x2 in X, II x1 - x2 II can be con-

5 idered distance in a sense between the two points xland x2.

Definition A4 : A sequence { xn } in a normed linear space (X,

I ' ° II) is said to be a Cauchy Sequence if, for every 6 > 0, there is an

integer N(e) such that

II xn - xm II < 6 whenever n, mzN(e).

Definition A5 : A normed linear space is said to be complete if

 

every Cauchy sequence in the space converges to a point _in the space.

That is, if for each Cauchy sequence { xn } in the space, there is an

elenient x in the space such that xn -+ x. A complete normed linear

Space is called a Banach space.

Some examples of Banach spaces are (i) continuous functions

defined on compact intervals C[a, b], (ii) Lebesgue spaces Lp, (iii) R

n
n
(C ) , and (iv) sequence spaces 2p

 

149

Definition A6 : Let X and Y be linear spaces, then III : X -* Y is said
to be a linear map if

( i) W( axl + 3x2 ) = anl + 60x2,

(ii)\II(ax)=-aII!x , Vx x eX,andVa,fl 6R1.

l 1’ 2’

The notation \II : X -* Y implies that 1F is mapping from X into Y.
Hence a linear mapping can be thought of as a function whose domain and
range are both linear spaces. Linear mappings from X into the scalar

field R are known as functionals.

Definition A7 : For all p e [1, 00), Lp [0, an) is defined as the set

containing all measurable functions f(-) : [0, cc) -> [0, 00) such that

I00 | f(t) |p dt < w
0

Lac [0, an) denotes the set of all measurable functions f(-) : [0, 0°) -+

[O , on) that are essentially bounded on [0, 00).

Rel-ark : A function is said to be essentially bounded if it is

 

bounded everywhere except possibly on a subset of measure zero.

Thus, for l s p < ac, LP[0, 00) denotes the set of measurable

functions whose pth powers are absolutely integrable over [0, 00),
Whereas LOOIO, 00) denotes the set of essentially bounded measurable

flu-lotions. Furthermore, for all p e [1, C0], the set Lp[0, 00) is a

real vector space in the sense of Definition A3

150

 

 

 

Definition A8 : For p e [1, co), the norm function I - I : Lp[0, co) —>
[0, 00) is defined by
I|f<'>ll - 1 I”0 I f(c) Ip dc ll/P <A-1)
P 0
And the function IIoIIm: Lm[0, m ) -> [0, on) is defined by
IIf(°)I|¢, - ess. sup If(t)I (A-2)
t 6 [0. °°)

 

 

- I , for 1 S p S on, maps the linear space Lp[0, 00)

The function I

into the interval [0, ac). According to Definition A7, the right hand

sides of (A-1) and (A-2) are well defined and finite. For each p e [1,

 

 

co ] , the normed linear space (Lp[0, 00) -2 I I ) is complete and hence

is an inner product. As

 

 

a Banach space. For p - 2, the norm I I2
a matter of fact L2 is a complete inner product space and therefore is

a Hi lbert space.

Definition A9 : Let L;[ 0, an) denotes the set of all n-tuples f —

 

[f1, °-~-o-,f 1T, where f. e L [0, on) for 1 - 1------n. Then the L -
n l p p

norm of f 6 LEW, an) is defined by

II
II f<-> IIp - I) II fi<-> II; 11/2 , p = [1, m)

i=1

 

 

151

In other words, the norm of a vector valued function f(-) is the square

root of the sum of the squares of the norms of the component function
fi(-) for i - l-Hon. For f e L:[0, 00) however

IIf(.)II0° - mix { ess. sup Ifi(t)I

be a given norm on CD, then for each matrix

 

 

 

 

Defiigtion A10: Let

A 6 CI1 x n, the quantity IIAI Ip defined by

||A|| "Ax”?
, :31; mm;
x e cn

IIAXIIp

”141-1|le

||le<1

is called the (matrix) norm of A induced by the corresponding vector

norm II'II.

 

APPENDIX B
DERIVATION OF A CIRCLE TYPE CRITERION

Consider the term
h(jw)

‘_-f—_——_—‘————_ % (a - fl) < 1
1 +5 (a+fl) h(jw)

appearing in the inequality (III-34)
The complex quantity h(jw) can be written as

h(jw) - a + jw , where a, w e R1

Combining (B-1) and (B-2) yields
a + jw 1
‘ (a - fl) < 1

1 + % (a + fi)(a+jw) 2

which on rewriting explicitly gives

(02 + w2)1/2

 

l
- (fl - a) < 1
[{1+%(a +fl)o)2 + { % (a + p>w 1211/2 2

NOW. squaring both sides of (B-4) and rearranging it gives

(aa+1)(fia+l)+afiw2

 

{l+%(a+fi)a}2 + (% (a+fl)w}2

(B-l)

(3-2)

(B-3)

(5-4)

(B-5)

If 1 -+ % (a + fl) h(jw) u 0 , then (3-5) can be simplified further to

Yield

( a0 + 1) ( 60 + 1) + afi 62 > 0 , v w e R1

152

(B-6)

 

153

from which we get the following :

( i) When a > 0, the equation (B-6) becomes

( a + l ) ( a + l ) + w2 > 0 , v w e a; .
a 5
( ii) When a - 0 , (B-6) reduces to
1

60 + l > 0 , V w e R
(iii) When a < 0 , equation (B-6) yields

(a+§)(a+l)+ w2<0,VweRl.

3

Next, by considering the inequality (III-34)

sup 1Ih(jw)| 6 s (1 - x) (B-7)
weR

it: is clear that h(jw) should lie inside the circle of radius % (l - K.)

centered at the origin. Cases (i) - (iii) and (B-7) form the basis of

the geometric interpretation given in section 3.4 .

 

 

 

APPENDIX. C

TWO USEFUL LEHHAS

Lemma Cl : Let P be a square partitioned matrix of the form
P P
P = 11 12
P21 P22
where P11 6 Rm x Rm, P12 6 Rm x Rn, P21 6 Rn x Rm, and P22 6 Rn x Rn.
Then
. -l .
( i) det(P) - det(Pll) det(P22 - P21P11P12) if det (P11) ¢ 0
.. -1 .

(ii) det(P) - det(P22) det(P11 - P12P22P21) if det (P22) # 0
Proof of Lemma C1 : This proof is straight forward by using elemen-
tary row and column operations.

( i) When det(Pll) # 0, P can be written as
I 0 P 0 I P P'1
P _ m 11 m 11 12

_ -1 _1
P21 P11 In 0 P22'P21P11P12 0 In
-1

Thus det(P) a det(Im+n) det(Pll) et(P22 - P21P11P12) det(Im+n)

= det(P ) det(P - P P'lp )

11 22 21 11 12

(ii) When det(Pzz) # 0, then

154

 

155

-1 -1
P _ Im P12P22 P11'P12I’22 P21 0 In 0
-1
0 In 0 P22 P22P21 In
. -1
which reduces to det(P) - det(P22) det(P11 - P12P22P21).
This completes the proof of Lemma Cl. D
Lemma CZ : Let G and K be matrices of orders m x n and n x m

respectively, then

det(Im + G K) - det(In + K G)

where Im and In are the Identity matrices of orders m x m and n x n.

Proof of Lemma 6; : See Plotkin [40].

10.

11.

12.

13.

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