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SAMPLED-DATA CONTROL OF
SYSTEMS WITH SLOW AND FAST
MODES

By
Bakhtiar Litkouhi

_A DISSERTATION

Submitted to
Michigan State University
in partial fulfiiiment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Electricai Engineering
and Systems Science

I98d

 

 

,—
_. .4
l

ABSTRACT
SAHPLED-OATA CONTROL OF SYSTEMS'
WITH SLOW AND FAST MODES
By
Bakhtiar Litkouhi

The class of linear time-invariant singularly perturbed discrete-time
systems is considered. Different sources and typical representations of this
class of systems is surveyed. The infinite-time optimal regulator problem and
the asymptotic behavior of the resulting algebraic Riccati equations. as the
perturbation parameter tends to zero, are studied. It is shown that,
analogous to the continuous—time case, a near-optiomal solution can be
obtained by applying slow-fast decompositions. An iterative technique for
solving the full algebraic Riccati equation which uses the solution of slow
and fast modes is introduced. This technique has a high degree of convergence
and alleviates the curse of dimensionality by eliminating the stiffness and
reducing the order of the system.

Furthermore, feedback stabilization and control of this class of systems
is considered. The two-time-scale nature of the system is exploited to
decompose the design problem into two lower—order design problems. Moreover,
we address the important issue of “multirate measurements“ or “multirate
sampling." Composite control strategies are developed for the case of
single-rate measurements as well as for the case of multirate measurements.
Stability results and closeness of trajectories are shown under the

application of these composite controls.

 

 

 

Our findings are applied to the deterministic model of the longitudinal
motions of an F-8 aircraft and simulation results supporting the theory are

presented.

This Dissertation Is Dedicated
to
My Parents
Whom Through Their Love,
The Sun Always Shines For
Me.

ii

ACKNOWLEDGEMENT

I am deeply indebted to my major advisor Professor Hassan K.
Khalil for making this work possible.His help,guidence,patience,
understanding,and many hours of fruitful discussions were the brightest
sources of inspiration and encouragement.He offered me help anytime I
needed it and it has been always a real pleasure working with him.

I thank my Ph.D. committee members,Professor David H.Y. Yen,
Professor Ronald Rosenberg,Professor Robert 0. Barr,and Professor Robert
Schlueter who offered much help and advice.

I would also like to thank Professor John Kreer,the chairman of

the Department of Electrical and Systems Science,for his support and

encouragement.

TABLE OF CONTENTS

Chapter

l Introduction ....................

2 Structural Properties and Modeling of Two-Time-Scale
Discrete-Time Systems ................

2.1 Introduction ..................

2.2 Continuous-Time Singularly Perturbed Systems
and Decoupling Transformation ..........

2.3 Historical Review of Two-Time-Scale Discrete-
Time Systems ..................

2.4 Sources of Singularly Perturbed
Difference Equations ..............

2.4.l Inherently discrete-time singulary
perturbed models ............

2.4.2 Singularly perturbed difference
equations obtained by numerical
solution of stiff differential
equations ................

2.4.3 Sampled—Data Control of Singularly
Perturbed Systems ............

2.4.4 Two-time-scale discrete-time systems
which can be brought into the singularly
perturbed form by artificial introduction
of s ..................

2.5 Stability and Approximation Results ......

3 Infinite-Time Optimal Regulators For Singularly
Perturbed Difference Equations ............
3.l Introduction ..................

iv

Page

TB

26

26

27

30

36
38

65
65

 

 

Chapter

3.2 Related Background ...............
3.3 Problem Statement ........... ~ . . . .

3.4 Asymptotic Behavior of the Optimal
Solution ....................

3.5 Slow-Fast Decomposition and
Composite Control ...............

3.6 An Iterative Solution of Riccati Equation for
Linear Quadratic Singularly Perturbed

Systems .....................
3.7 Numerical Example ................
Composite Control and Multirate Measurement .....
4.l Introduction ..................

4.2 A Stabilizing Composite Control with State
Measurements in Fast Time Scale .........

4.3 Multirate Stabilization with Slow State

Measurements in Slow Time-Scale .........
Application , , , ................ ‘
5.l Introduction ..................

5.2 Longitudinal Equations of Motion for an
F-8 Aircraft ..................

5.3 Results for Infinite-Time Regulator .......
5.4 Multirate Stabilization .............
Conclusion and Recommendation ............
List of the Programs .................

List of References . .................

74

85

92
l0l
ll5
ll5

ll6

123
l58
l58

l58
I64
l7l
T79
182
208

 

LIST OF TABLES

Table Page
3.1 ........................... l03
3.2 ........................... lO4
4.l .................. ‘ ......... l45
4.2 ........................... T45
4.3 ........................... l46
4.4 ........................... T46
4.5 ........................... l47
4.6 ........................... l47
5.l ........................... l69
5.2 ........................... l69
5.3 ........................... 170
5.4 ........................... 171
5.5 ........................... l75
5.6 ........................... l76
5.7 ........................... l77
5.8 ........................... l79

vi

LIST OF FIGURES

OOOOOOOOOOOOOOOOOOOOOOOOOOO
OOOOOOOOOOOOOOOOOOOOOOOOOOO
OOOOOOOOOOOOOOOOOOOOOOOOOOO
OOOOOOOOOOOOOOOOOOOOOOOOOOO
OOOOOOOOOOOOOOOOOOOOOOOOOOO

l6l

 

CHAPTER 1

INTRODUCTION

Simplification of mathematical models for many physical and
engineering problems is a common practice of control engineers. In anal-
ysis and design crf large scale control systems the need for such simpli-
fications emerge quite naturally.

Methods of reduced-order modeling and control have received a
great deal of attention in recent years. Of these methods, aggregation
[Aoki, l978], and singular perturbation [Kokotovic et al., l976], seem
to be the most well-known. A typical simplification is to neglect some
small "parasitics” as time constants, moment of inertia, masses, capacitances
and inductances. Neglecting these small parasitics alleviates the ”curse
of dimensionality” by lowering the model order and exclusion of the fast
states which result in ”stiff” models. Approximated models using exclusion
of fast states, as in aggregation, may result in an unstable system or
a system which is far from its desired optimum. Singular perturbation
technique improves this approximation by reintroducing the fast states
as a "boundary layer" correction calculated in a separate time scale.

An important characteristic of singularly perturbed models is that the
structure of the system remains the same for time—varying and nonlinear
systems. This is established by a fundamental theorem by Tihonov [l952].

The singular perturbation approach is not only helpful in design

procedures but is a powerful tool for analytical investigations of the

properties of the system as behavior of optimal controls near singular
arcs, stabilizability, systems robustness, etc.

The singular perturbation method has attained a certain maturity
in continuous-time control systems [Kokotovic et al. l976]. The multiple-
time-scale property of these systems has been used in deriving the reduced-
order models which have been employed in approximation of some desired
objectives of the original high-order "stiff” models. More specifically,
the analysis and control design of linear time-invariant continuous-time
singularly perturbed systems has been well documented [Chow and Kokotovic,
l976, a.b.].

In spite of increasing flow of research directed in the area
of singular perturbation theory, many questions are still open as was
discussed in [Kokotovic et al. l976].

One area where on-going research is still in its earlier stages
is singularly perturbed difference equations. Hoppensteadt and Miranker
[l977] developed a multitime method for difference equations. Phillips
[l980] considered the singularly perturbed discrete systems in state
variable form and reduced-order models were obtained without considering
the initial value lost in the process of order reduction. Blankenship
[1980] developed a method of matched asymptotic expansion for a class
of singularly perturbed difference equations arising in optimal control
problems. Also, different applications of singular perturbation ideas
to discrete systems have been investigated by Mahmoud [1982], Naidu and
Rao [198l, a,b], Rajagopalan and Naidu [lQBO], and Sycros and Sannuti
[l983].

 

The objective of this dissertation is to investigate some open
problems for the class of linear time-invariant singularly perturbed
difference equations and employ the structural properties of singularly
perturbed systems to acheive the approximate control design for such

systems. The organization of the dissertation is as follows:

In Chapter 2, the continuous-time singularly perturbed systems
are, briefly, introduced and a decoupling transformation to separate
the slow and fast modes which is applicable to both continuous and discrete
systems is studied.

A historical review of singularly perturbed difference equations
is performed in Section 2.3 which discusses different model representations
and some structural properties of this class of systems. In Section 2.4
different sources of singularly perturbed difference equations are invest-
igated. In the last section of this chapter, Section 2.5, we introduce
a useful Stability criterion for the class of linear discrete-time systems.
Also, an initial value problem, in which solutions of slow and fast problems
are used to approximate the solution of the full problem, is investigated.

Chapter 3 deals with the problem of Infinite-Time Optimal
Regulators for singularly perturbed difference equations. First, a related
background is provided Unfamiliarize the reader with the problem and our
motivation. Asymptotic behavior of the optimal solution of linear quad-
ratic regulators is investigated. Conditions for independent design of
slow and fast subsystems is studied. A composite feedback control law,
which employs the slow and fast controls, is formed and applied to the

original system which results in a near-optimal solution.

Also, an iterative technique for solving the discrete-time stiff
Riccati equation is presented. This technique, by using the slow and
fast subsystems, overcomes the ill-conditioning and provides a fast
convergence. An illustrative example which supports the theory is given
at the end of this chapter.

Chapter 4 discusses the stabilizability of singularly perturbed
difference equations, in view of multirate measurements of the state
variables, using a composite feedback control law.

Different design procedures for forming a stabilizing composite
feedback control are investigated in this chapter,and it is shown that
the application of such control laws results in asymptotic stability of
the closed-loop systems and closeness of the trajectories to those pre-
dicted by slow and fast subsystems. Two different time-scales, slow and
fast, are introduced. The fast-time-scale has a period 1, while the slow-
time-scale has a period N = E%] (N is an integer such that _%--l < N f'é).

The composite feedback control law is formed by using the

stabilizing feedback controls of slow and fast subsystems when

i) Both evolve in the fast-time-scale n and their measurements are

available for all n (single rate), n = 0,l,2, ...N,... .

ii) The measurements of slow states are available only at slow-time
intervals, but the measurements of fast states are available for
all n. In this case we have a multirate measurements and slow and
fast controls are designed independently, "Parallel Design". Also,
the values for slow states for n # é-, K = 0,l,2,... are predicted

using their values at the beginning of the slow periods.

iii) There is a multirate measurement scheme but a pre-conditioning
feedback gain which stabilizes the fast states is designed first
and based on this gain the slow subsystem is designed, "Sequential
Design". 2

Finallysa numerical example for parallel design illustrates our
claims.

Chapter 5 is devoted to the numerical solutions of a more
realistic physical model. We have considered the deterministic model of
an F-8 aircraft with four state variables, two of which are slow states
(incremental velocity, pitch angle) and the other two are fast states
(angle of attack, pitch rate). Our claims about the near-optimality of
infinite-optimal regulator, iterative technique and multirate stabiliz-

ation is confirmed using this model.

Chapter 6 is the conclusion which precedes the list of the

programs used in solving our numerical examples and application.

CHAPTER 2

STRUCTURAL PROPERTIES AND MODELING OF THO—TIHE-SCALE
DISCRETE-TIME SYSTEMS

2.1 Introduction

The main objective of this chapter is to familiarize the reader
with two-time-scale linear time-invariant discrete-time systems and their
structural properties. Also different sources of such systems are dis-
cussed.

In Section 2.2 continuous-time singularly perturbed systems are,
briefly, discussed and an important decoupling transformation for
separating the slow and fast modes of such systems is introduced. Con-
ditions for existence of this decoupling transformation, which could
be applied to both continuous and discrete systems, are given.

Section 2.3 introduces the two-time-scale time-invariant discrete
systems. IX historical review, which includes different model representa-
tions of this class of systems, describes some structural properties as
pole clustering, and contains different methods for accomplishing slow-fast
decomposition. This, hopefully, provides the reader with a better
understanding of the subject.

In Section 2.4 different sources of singularly perturbed dif-
ference equations are investigated.

Finally, in Section 2.5, a useful stability criterion for dis-
crete-time systems is introduced. An initial value problem is discussed

which reveals an 0(6) approximation between solution of the slow system

represented by differential equations and the one obtained from difference
equations. Also approximation of the full states using slow and fast
states is investigated.

2.2. Continuous-Time Singularly Perturbed Systems and Decoupling Trans-
formation.

 

Control systems possessing slow and fast phenomena are frequent
in applications. A linear continuous-time invariant model of such systems
is

x(t) = A x(t) + A

ll 22(t) + B1u(t), x(0) = x0 (2.la)

l

e 2(t) = A2]x(t) + A222(t) + 32u(t), 2(0) = z (2.lb)

D

where the state vector comprises the m1- and m2- dimensional vectors

x and z, the control u is an r-dimentional vector and E is a small

positive parameter representing small time constants. All matrices have

compatible dimensions. Slow and fast modes correspond to small and large
eigenvalues, respectively.

For E = 0 in (2.l) the order (m1 + m of the system reduces

2)
to m], that is (2.1) reduces to

§=Ani+mgwsfi . Raw

0
II

A21x + A222 + Bzu, (2.2b)

where bar indicates that 6 is set to zero. If A is invertible,

22
then

-1 _. -1 _

z = -A22 A21x -A2282u, (2.3)

 

yielding the reduced model

x = on + Bou , . (2.4)

where

-1
' A12A22A21

= _ -l
80 Bl A12A2232 '

>
I

0 ‘ A11

The use of E = O is formal since then

could be unbounded. That is why systems presented in the form (2.l)
are called “singularly Perturbed Systems."

Unless x0 and 20 are such that 2(0) = 20’ the boundary

condition 2(0) - 20 will not be met by the approximation (singular
perturbation) (2.2).

If quantities on the right hand side of (2.1) are of the same
magnitude, 2 will be of the magnitude g-i. For this reason 2 is considered
a "fast" state and (2.4) which neglects the fast dynamics is considered
as "slow" system.

System (2.l) is said to possess a two-time scale property if it

has m1 small eigenvalues of magnitude 0(l) and m2 large eigenvalues

of magnitude 0(é). Singular perturbation exploits this property of

the system to approximate it with two lower order, slow and fast, sub-
systems. The approximate slow subsystem is justified by considering that

in an asymptotically stable system the fast modes corresponding to large

eigenvalues are important only during a short initial period and after
this period the behavior of the system could be represented by its slow
modes. Neglecting these fast modes is equivalent to assuming that they
are infinitely fast, that is pushing 6 -> O in (2.1). [Chow and Kokotovic,
l976]. A fast subsystem is derived by assuming that the slow variables

are constant during the fast transients. Now subtracting (2.2 b) from

(2.l b) yields

where

To separate the slow and fast modes of the singularly perturbed
system (2.l) a state transformation due to Chang [l972] is used which
completely decomposes the system (2.l) into slow and fast modes by trans-

forming it into a block diagonalized system as in (2.6)

F W r ‘ f \ f
I I

n] A0 + 0(6) 0

 

 

 

 

 

: l
L 6 n2 J L 0 A22 + 0(e)(Ln2 j L 82 + 0(e) ,

From (2.6) it is obvious that as e + O, the first m1 eigen-
values of the original system (2.l) tend bathe eigenvalues of the reduced

system (2.4), while the remaining m2 eigenvalues tend to infinity as the

A22
eigenvalues of —;;-

Let these eigenvalues be divided into two distinct sets which

are arranged in increasing order

 

where S and f represent slow and fast modes, then we have

u, MA, I << 1 . (2.7)
m1 l
A schematic representation of (2.7) is shown in Figure 2.1 where the

shaded areas indicate the locations for slow and fast eigenvalues.

 

Figure 2.l.

If the system 2 = AX satisfies condition (2.7), then it

possesses a two-time—scale property.

The decoupling transformation is a useful tool in decomposing

 

ll

the singularly perturbed systems into slow and fast parts and could be
applied to both continuous and discrete systems. A more general case

of block-diagonalization of ill-conditioned systems is presented by
Kokotovic [l975] which considers systems not necessarily in singularly
perturbed form. Due to importance of this transformation throughout our
present work, we give a brief explanation of the latter work.

Consider the following free system

r E r s r a

x All A12 x

2 A2] A22 2 E
L J L J L -

 

 

 

 

 

 

where x and z are m1- and mZ-dimensional state vectors.

Let n2 = Z + LX , (2.9)
Where L is a real root of
A22L - LAll + LAlZL - A2] = O . (2.lO)

If L exists, then substitution of (2.9) into (2.8) yields the block

triangular form

I l

l I

E _ E (2.11)
l

 

 

 

 

12

 

where
81 = All - A12L (2.12)
B2 = A22 + LA12' (2.13)
If A22 15 invertible.let
- -l _
Lo ' A22A21 ’ Ao ‘ A11 ' A12Lo- (2'14)
Now L is sought in the form
L= L0+D, (2.15)
where D is a real root of
DA0 - (A22 + LOA12)D - DAlZD + LOA0 = O. (2.16)
It is proved by Kokotovic [1975] that a unique real root of (2.16)
exists satisfying
211A 11 ML 11
f H H 5 0 O 1 u (2.17)
“ADM/11211 “to.
if the following condition on matrix norms is met
11A") < 11M +1'A 1 HL '1)“ (218)
221-3'0 112‘ 0" °
where H H is assumed to be a 2-norm.

He also proves that D in (2.16) is an asymptotically stable

equilibrium of the difference equation

 

13

-1

= A22

0 (L A +0 A -L A D -D A D ) E f(D

K+1 0 0 K 0 0 12 K K 12 K

Furthermore, using the change of variable
0] = X ' M02 9
where M is a real root of
81M - M82 + A12 = O ,

and substitution into (2.10) yields

n1 B1 0 n1

 

 

 

 

 

 

L "2 . L O 82 E L n2 J

K)'

(2.19)

(2.20)

(2.21)

(2.22)

It is proved that under the conditon (2.18) the solution M 'of (2.21)

is the asymptotically stable equilibrium of the linear difference equation

-1
A A22.

- 1
M - [(All-A12L)M -M LA

K+1 K K 123A22 + 12

The above two-stage transformation could be represented by

I

I M n]

 

1
1
— 1 Q
1
1

-L I -LM .
t 2 1(“2E

 

 

 

 

(2.23)

(2.24)

where I1 and I2 are the m1- and mz-dimensional identity matrices

respectively.

14

It is easy to see that

(2.25)

 

 

 

 

 

 

The above transformation is particularly convenient for singularly

perturbed systems and was introduced by Chang [1972] in the form

fir .1r
n1 IE—éML -tM x

(2.25a)

 

 

 

 

 

L n2 1 L 2 J L J

and was applied on many control problems of singularly perturbed systems.
[Kokotovic, Haddad, 1975], [Chow and Kokotovic, 1976].

In many applications we are dealing with continuous-time models '
vwruflipossessa two-time-scale property, while they are not, explicitly,
in the singularly perturbed form. The major problem in converting a
given system of equations to a singularly perturbed form is in grouping
the state variables into slow and fast states such that (2.18) is
satisfied. If the original system possesses a two-time-scale property
(2.7) but condition (2.18) is not satisfied, it may still be possible
to satisfy (2.18) by either scaling and regrouping the state variables
or by allowing linear combinations of certain fast states with the
slow state group. Readers who are interested in learning about this
modeling process are referred to Chow and Kokotovic [l976-b],$ain

et a1. [1977], Anderson [1978],Sycros and Sannuti [1983] and Chow [1983].

15

2.3. Historical Review of Two-Time-Scale Discrete-Time Systems.

The well known difficulties in dealing with high-order models
and the class of "stiff" systems plus the recent interest in optimization
and control algorithms for discrete systems operating on widely separated
time-scales has provided an impetus for applying singular perturbation
methods to order reduction and control of discrete systems.

The objective of this section is to describe previous efforts
to extend singular perturbation techniques to discrete-time systems
having a two-time scale property. Different approaches to characterize
two-time-scale discrete-time systems are presented.

The models considered earlier in the literature classify into
three groups.

1. Phillips,and Rajagopolan and Naidu

.2. Mahmoud

3. Hoppensteadt and Miranker,and Blankenship.

Group 1.

Philips [1980] considers linear time invariant discrete-time
systems. There is always a basis such that a discrete-time system takes

the form

 

 

 

16

 

 

 

 

 

 

 

 

r i r i f a F 1
xs(k+l) AS 0 xs(k) BS
= + u(k) , (2.26)
1 xf(k+l) 1 E o Af , L xf(k) J E Bf J
where
Af ‘ xs
Rf g mix Ixj(Af)E
A 9

mEn EAi(AS)|.

The system (2.26) is not necessarily in its modal form. However, multiple
and complex conjugate eigenvalues are naturally grouped together in either
As or Af. System (2.26) is said to possess a two-time-scale property

if there is a sufficient gap between the eigenvalues of As and Af,

i.e.
xf << Ks . (2.27)
Noting that
min EA1(AS)E 3 EM?“-1 (lower bound)
1
max [Aj(Af)| f HAffi (upper bound).

the two-time-scale property can be expressed as

1111311" >> 1:11.11. (2.28)

 

 

 

 

 

 

17

Phillips[l980], then considers a class of discrete-time system of the

form

x(k) + El'jA z(k) + B u(k), x(0) = x

12 1 (2.29a)

x(k+1) = A

11 0

ejA x(k) + EA z(k) + B u(k), 2(0) = z0 (2-29b)

z(k+1) 21 22 2

where x and z are m]- and mz-dimensional vector states, u is an
r-dimensional input, 6 is a small positive parameter, 0 f j 5 l and
A]: exists. He shows that for sufficiently small E, the system (2.29)

possesses a two-time-scale property. In particular, let

-1 l

e u - .
3 “A22 A21A11 A12)
1» -- 11A,, 1311 111,21
c = 1111311 ,
and
d = a + b
Then, if
E < “EL—~— ,
c(d +8ab)

the system (2.29) can be transformed using the decoupling transformation

of Kokotovic [1975] into the form (2.26) with

18

A5 = A11 ' eH’th
Af = 6A22 T QI-jL A12
135 = (I-ML)B1 - M82
Bf = LB1 + 32 .

The matrices L and M satisfy equations (2.10) and (2.21) with A
1-jA

12’

j .
A21 and A22 replaced by E 12, 6 A2] and 6A22’ respectively. .
Furthermore, he shows that L = 0(EJ) so that letting L = 63L,

the matrices AS and Af take the form

A A

A = A - E A L, Af = 6(A22 + L A

s 11 12 12X

Thus, the system satisfies the two-time-scale property (2.28) since for

sufficiently small 6

A

-1 -1 ~
[((AH - e A L) H >> equz + L A

12 1211 . (2.30)

Also, x is the slow state and z is the fast state.
A special case of (2.29) with j = 0 has been considered by
Rajagopolan and Naidu [1980]. It takes the form

x(k) + e A 2(k) + B

x(k+1) 12

A

u(k) (2.3la)

11 1

2(k+1) A

21x(k) + e A22 z(k) + 32 u(k), (2.31s)

 

 

 

 

 

 

19

Although (2.31) is a special case of (2.29), it is seen that in the
absence of inputs the two systems are equivalent in the sense that
z(k) = €jz(k).

For simplicity let us continue our discussion using the model
(2.31). Letting E = O in (2.27) yields a reduced (degenerate) system

of order of m].

'>Z(K+1) = Anx(K) + 13111110 (2.32a)
'2(1<+1) = A217“) + 826(K) . (2.32b)
We note that E10) = x0 and ‘E(O) f 20. This situation of order re-

duction and consequent loss of initial conditions is analogous to singular
perturbation in differential equations [O'Malley 1971].
The state variables of the full system (2.31) and the reduced

system (2.32) are shown in Figure 2.2.

 

 

if ppfiqu f E {ELJ‘FTTE
4 1 - . + .. E -
...1 a. _» , -’-‘-‘ “1)»- afetizw‘r—-_——~;’—— 299+ 41—«5—‘k—13E1Ewe—‘53
. 4 ' E i
. a?“ __ 1 g __
/ ' - - I 1 Q...—
\/ u \ U

 

 

. r .7 7—- - ‘ ~-.
\‘J
f*
-2
I

_ mm. aflgazL.-__. .-
(Cl) .,,.
Full System Reduced System

Figure 2.2

20

System (2.31) may be regarded as a system in slow-time scale, that is,

the slow state is varying on an 0(1) time-scale.

Group II
Mahmoud [1982] considers the system

x(K+1)

A]]x(K) + A z(K) + B]u(K), x(O) (2.33a)

12 *0

A

z(K+l) K) + A z(K) + B u(K), 2(0)

22 2 (2.33b)

21’“ 20

His work is essentially repetition of the work of Phillips [1980] and
Kokotovic [1975], except that 6 does not appear in his system explicitly.
By using the decoupling transformation, mentioned in Section 2.2, he

arrives at a similar condition as (2.30) so the system (2.33) possesses

a two-time-scale property.

Group III
Hoppensteadt and Miranker [1977] have considered a free system

of the general form

x(K+1) = Ax(K) + t f(x(K),€),x(0) = x (2.34)

0,

where A and f are time-invariant.

They assume that there exists an invertible matrix P such that

-1

P AP = diag(a,S), (2.35)

where the matrix 9 is oscillatory, that is, has all characteristic

roots on the unit circle, |AE = 1, and the matrix S is stable, that

 

 

 

21

is, has its eigenvalues insidethe unit circle, IA) < l.

‘ The matrix 6

is assumed to be diagonalizable.

They also assume

that f is a smooth function of its arguments. Applying the transformation

9Ku(K)
x(K) = P v(K) 9 (2.36)
yields
_ A-K-l K
u(K+l) - u(K) + E e g(a u(K),v(K),E) (2.37a)
v(K+l) = Sv(K) + eh(eKu(K),v(K),e), (2.37s)
where
_ 9
f - P(h).
Next it is assumed that there is a smooth function a(u,é)
(fast quasi-steady-state) such that
¢(eK u(K) .6) = sueKu<K1§1 +E h<eKuK.¢.61, (2.38)
with a = 0(6).
Letting v(K) = ¢( Ku(K),E) + V(K) yields
_ -K-1 K .
U(K+1) - u(K) + F 9 9(9 U(K),q + V(K),€) (2.396)
v<K+11 = sv<K1 + ethia u(K).¢ + V(K)se)-h(9KU(K),¢,E)}
(2.39b)

Due to the location of E, u will change little before V has reached

0(6). Thus the slow behavior

is found by solving

 

 

 

22

K

K“g(e'<u(1<).4(c-> u(K).e>.e) (2.40)

u(K+1) = u(K) + E 0-

By expanding u(K) in E, the solution of (2.40) is found in the form

u(K) = U(K,S,E) = u0((<,5) + e u‘((<,s) + 0(8), (2.41)
where S = 6K. Note that U is evolving in slow time scale S.
Solving for the terms in the expansion yields
U0(K+1,S) = u0(x,5). (2.42)
So U0 is independent of K, that is, in the slow-time scale the limiting

value of u is constant.

Equating coefficients on a fast time-scale and taking the limit

as 6 + 0 and assuming U1 is bounded yields

K 1

9.11.9.

lim «1 e-n-ig(enUO(S),0,O). (2.43)
s k+m x

"NI!

n 0

Thus, it is shown that the solution to (2.34) can be approximated by

0
eKU (6K)
x(K) = P K + 0(e), (2.44)
S vO
where
(10(0)
P = x .
0
V0

And the approximate solution to the original problem is determined by

the solution to the reduced equations (2.43) and (2.44).

P\v

.‘i

23

Blankenship [1980] analyzed the system

x(K)f e A z(K) + e B u(K) (2.45a)

x(K) I e A 12 1

x(K+1) 1]

K) + 6 A x(K) + ED z(K) + (B

A 21

z(K+1) + §H )u(K). (2.45b)

22“ 2

The above system is a fast time-scale model.

Here the fast eigenvalues are inside the unit circle but of
0(1). The eigenvalues corresponding to slow modes are assumed 0(6)
away from 1. This model thus assumes that the slow modes are almost
constant while the fast modes are approximately given by the boundary-

layer system.

z(K+1) = A z(K) + B u(K). (2.46)

22 2

It should be noted that system (2.29) assumes that the fast state 2:

is treated on a time-scale slow enough for its response to be deadbeat
while system (2.46) assumes that the slow state is treated on a time-
scale fast enough for the slow state x to remain approximately constant.

The presence of E in the above mentioned models classifes them
as singularly perturbed systems as the order reduction and separation
of time-scales are apparent by setting 6 = 0.

It should be noted that system (2.37) has the structure of (2.45)
where u(K) is the slow state and v(K) is the fast state. The two-
time-scale property of the system (2.45) will be shown in Section 2.4.
Blankenship examines a linear quadratic regulator problem subject to the
system equation (2.45).

In deriving the results in his work, the control input is assumed

24

tzca consist of two components, one which vanishes as K + w and the other

which is bounded. That is

u(K) = v(K) + u(EK) , K = 0,1,2,... (2.47)
1im v(K) = O .
K-mo .

For u(K) of this form a solution of (2.45) is sought in the form

X(K) = 300 1‘ x(EK). z(K) = b(K) + Y(€K) (2.48)
with

lim a(K) = 0, lim b(K) = 0 . (2.49)

K-m K...»

The terms (x(eK), Y(&K)) are "outer solution“ and (a(K),b(K)) are
" initial boundary layers" similar to those defined for singularly per-
turbed differential equations [Hoppensteadt, 1971].

The final-value problem

a(K+1) a(K) + EA a(K) + eA b(K) + e B u(K)

11 12 1
b(K+1) = A220(K) + 6A2]a(K) + 6D b(K) + (82%H)V(K) (2.50)
lim a(K) = 0 , lim b(K) -'-' 0 (2.51)
K-mo K—mo
c191“ines the boundary layer terms, and
X(€K+€) = (1+6An) x(eK) + 6A12Y(€K) + EB]U(EK)
Y(eK+e) = A22Y(€K) + 6A21 X(eK) + EDY(6K) +(82+6H)U(eK) (2.52)

25

defines the outer solution.

Now by assuming that A22 is stable and that u(K) is any

Function which satisfies (2.47) and

foo = 1 e"v‘”)(1<). ”4(5) = E e“u(”)(s1, 1 M”) (K)! < .,
n=0 n=0 K=0 (2.53)

where S = 6K (superindex (n) shows the nth derivative) and by taking
asymptotic expansions in a,b, X, and Y and matching the coefficients
iri E in the fast-time-scale K and slow time-scale S, he shows that

the solution of (2.45) satisfies

x(K) = xi0)(ek) + 0(6) (2.54a)
z(K) = v(o)(ex) + b(0)(K) + 0(e), I (2.545)
where
Efgéél = A]]X(O)(S) + [A12(I-A22)'182 + 81]u(0)(S), x(°)(0) = x0
(2.55)
v(0)(s) =( -A22)"Bzu(°)(5) (2.56)
a(0)(K) = 0 (2.57)

26

Picate that the system (2.55), (2.56) is the "reduced-order" system cor-
r~eesponding to (2.45). It evolves in the slow time scale S = EK con-
sistant with the analogous notion of reduced-order in the continuous-
time problems and the solution is obtained by solving the differential
Equation (2.55). Also it is interesting to note that this solution
parallels the expressions derived for the uncontrolled cases by Hoppensteadt
and Mi ranker [1977] since they use the method of matched asymptotic
expansion using a multitime method and, like Blankenship, they exhibits
a hybrid situation where the slow system is represented by a set of
differential equations and boundary-layer (fast) system is given by
difference equations, which is different from the case of Mahmoud and
Phillips where the slow and fast systems are both given by difference

equations.

In this thesis we adopt the model of Blankenship and Hoppensteadt

and Miranker. Justification for adopting this model is given in Section 2.4.

2 - 4. Sources of Singularly Perturbed Difference Equations.

There are four important sources of discrete-time models de-
SCr‘ibed by singularly peturbed difference equations of the form (2.45).

These sources are presented in this section.

m. Inherently discrete-time singularly perturbed models.

This class of systems results when the physical system is in-
hErently discrete. Such models are common in economic, biological, and
sociological systems. Some examples of this type of discrete-time
singularly perturbed systems are given in [Hoppensteadt and Miranker,

1977]. We breifly explain one of these examples.

 

 

27

Example 2.4.1: A population genetics model

In a large population of diploid organisms having discrete

generations, the genotypes determined by one locus having two alleles,

A and a, divide the population into three groups of type AA, Aa, and

aa , respectively. The gene pool carried by this population is assumed

to be in proportion Pn of type A in the nth generation. It follows

that [Crow and Kimura, 1970]

 

P = P + Pn(l-Pn)[(WH-W12)Pn + (NEZ-W22)(l-Pn)] (2 59)
n+1 n 2 2 ’ °
WHPn + 2w12Pn(l-Pn) + w22(1-Pn)

where W11. W12. and W22 are the relative fitnesses of the geno-

types AA, Aa, and aa, respectively. Now if the selective pressures

are acting slowly, i.e., if w11=1+ 601, 1112 =1

, )122 = l + 65,
where 6

is small positive number, then

P (1-Pn)[(a+B)Pn-BI

_ n I

This model is a special case of (2.45) where there is a slow state only.

w. Singularly perturbed difference equations obtained by numerical

30" ution of stiff differential equations.

This class of systems is ususally found as a result of numerical
SOUl ution of stiff differential equations where they are approximated

by corresponding difference equations, usually for the purpose of digital

Simulation. To clarify this we give two examples; the first one was

Considered by Hoppensteadt and Miranker [1977] and the second one by
Blankenship [1981].

28

Example 2.4.2.

The two-dimensional stiff linear differential equation

0.0.
HN

= w-+ec)z , B= [b 0E, (261)
0 0

which is written in the fast time-scale is considered. Let Z = (x,y)T,
Introducing a mesh with increment h and applying an r-step linear

multistep method to'the system leads to the following difference equations.

r r
. .- +6 .Z.=, =,+l,...
jZO 013 Zn-j h( B C) j=Z=1 BJ n-j 0 n r r
(2.62)
Let Y = (x x x ) and Y = (y y y )T
—n n’ n-l"'°’ n-r+1 n n’ n-l""’ n-r+1
and the r x r matrices
F - - E I
'“1 "' “r-1 “r E a] ... Br 1
R = 1 o and 5 = E , (2.63)
I O
L 1 .0 J

 

 

The difference equation for Zn may be‘written in the following form '

R in + thKn + €h8>(CHLn + C12 Yn) (2.6451)

II, |
+
u—l

.<
I

n+1 ' RYn + ehS<C215n + C22Yn)' (2'64b)

29

Equation (2.64) is of the form of (2.34) and by using a transformation
similar to (2.36) it is shown [H0ppensteadt and Mirankar 1977] that

the system can be brought into the singularly perturbed form.

Example 2.4.3. Blankenship uses the Euler's approximation for the fol-

‘I owing stiff differential equations,

d§(t)

dt = A §(t) + §E(t) + F U(t) (2.65a)
e 12—}? = “C x(t) + (Tat) + a a(t) (2.65b)
~ m1 ~ m2
x(0) = x0 6R , 2(0) = 20 e R , o 5 t 5 1,
where the matrices are constant, 6 > 0 is a small parameter and D
is assumed to be nonsingular.
Introducing the stretching time scale T = t/6 and
y(«:) = '26:) + 6467461). .4.) = me), u(-:) = (net),
we obtain
dig—)- = E AX(T) + E B y(t) + E FU(‘[), (2.66a)
dy(t) =

d1 S y(:) + ECX(T) + EDy(t) + (G + €H)u(t), (2.66b)
x(0)=x09Y(0)=yOsofo1/es

Where the coefficients A,B, etc., are simple combinations of A,B, etc.;

S=D.

 

 

30

Let {O,h,2h,...,Nh} be a mesh on [0, 1/6] and let

xn = x(nh), yn = y(nh), un = u(nh)

be a numerical approximation to X(T), y(t), and U(T). Using Euler's

aataproximation to the derivative we obtain

xn+1 xn + E hAxn + EhByn + éhfun (2.67a)
yr“.1 = (I+hS)yn + EhCXn + EhDyn + hGun + EhHun (2.67b)
x(0) = x0. y(0) = yo , n = O,l,...,N-1= o (l/Eh).

We note that system (2.67) is a singularly perturbed system
equivalent to (2.45) with h = l and I + hS = A22.
Also, note that for a discrete model obtained in this way,
( I + hS) is generally nonsingular since h must be small for a good

approximation to a continuous-time system.

.§1;;5L;§, Sampled-Data Control of Singularly Perturbed Systems.
Another source of singularly perturbed difference equations comes
1:<>rnm study of sampled—data systems or computer-controlled systems where
a continuous-time singularly perturbed system is driven by an input
sDecified at discrete-time points and has output and state variables
Sampled only at discrete-time points.
The standard example of sampled-data system follows when u(t)

is a piece-wise constant function of time, i.e.

 

31

u(t) = u(t t f t < t

K) ’ K K+1 ’

anr1d the state and output are sampled at discrete time points tK. Con-
sider the following singularly perturbed linear time-invariant continuous-

t ‘1 me system

x(t) + A Z(t) + B u(t), x(0) (2.68a)

X = A 12 1

ll
X
0

11

(2.68b)

I
N

e 2 = A21x(t) + A Z(t) + 82u(t), 2(0) -

22 0’

wanere x and Z are m]- and mz-dimensional state vectors and all the
matrices have compatible dimensions with A22 nonsingular.

The solution of the system (2.68 ) between (0,t) is given by

x(t) x
o t ~ , N
= eAt + E eA(t")B u(r)dt, (2.69)
Z(t) Z0 0 '
Where
1 1 , 1
A11 A12 B1
A = , E = , (2.70)
A21 A22 B2
L T T J L 1‘. J

 

 

 

 

For a piece-wise constant u(t) = u(tK), tK f t 5 tK+l and the sampling

Period ET = tK+1 - tK we have [Levis, et.a1., 1971], [Levis, Dorato,

19711

32

X(K+l) X(K)
= MET) + 1‘(€T)U(K) ,
Z(K+l) Z(K)
where
~ ET
a(ET) = eAET and r(€T) = E eAtht .
0
At

To evaluate e

diagonalize A and we get

6M A L 0

-L I -6LM o -?;-+ LA

12

and
(A -A L)t 0

EM e1112

= (A +6LA
-L I -6LM o e 22

IVEML

k

(A -A L)t (A +€LA]2)t/€

e (I -EML)+eMe 2'2

1 L

(A -A L)t (A +ELA )t/E
1‘ 12 (IE-EML)+(I2-€LM)e 22 12 L

 

L -Le

we use the Chang transformation (2.25a) to block

L

12)“6

..EM

(2.71)

(2.72)

 

33

(A

-A

L)t (A +€LA )t/6
M + EMe 22 12

12

(A
ELe

-A L)t (A +€LA

)t/e
M+(12-&LM)e 22 12

11 12

 

I1 and I2 are (111 and mz-dimensional identities, respectively, L

satisfies (2.10) and 14 satisfies (2.21) and could be approximated by

L A22 A2, + 0(6) = L0 + 0(6)
_ -1 -
M - A12 A22 + 0(6) - MO + 0(6).
_ -1 . . g . . '
Let A0 - A11-A12A22A21. For suff1c1ently small 6,4 (ET) lS g1ven by
the following: (1,1) element is
e[AO+0(E)]T 2 0122+ 0(t)JT
=3 [IeéML i'OK n +E[M +(N6De [L +(Nefl
1 O 0 0 O
2 2 - A22T
= [I + ETA0+0(€ )][I]-EMOLO+0(E )]+ EEMO+O(t)][e + 0(6)][L0+O(€)]
= I + GETA +M (eAZZT-I )L 1 + 0(62) (2 74)
0 0 2 0 ‘
_ 2
— I + EA + 0(6 ) ,
where A22T
= A -
A TAO + A0(e IZ)L0
(1,2) element is
€[AO+0(e)]T [A22 + 0(6)]1
= -ée [M0 + 0(6)] + €[MO + 0(E)1e

A T

.en1 + EAOT + 0(ez)1tMO + 0(6)] + EEMO + 0(6)][e 22 + 0(6)]

34

A T
=6M0(e 22 -Iz) + 0(62) =6 B + 0(62), (2.75)
where A22T
B = M0(e '12)
(2,1) element is
6(A0+0(€ )11 _2 2 [A22+0(€ )]T
=-LLO+0(E)]e [Il‘EMoLo+O(e )]+[Iz-ELOMO+0(6 )Je [L0
=-[L +0(e)1£1 +6111 +0(€2)1£I £11 L +0(62)1+r1 {L M +062) eA22T+0(e )1
0 1 0 1 0 0 ~ 2 0 0 3E
[L0+0(6)]
A22T
= (e -12)L0 + 0(6) = C + 0(6), (2.76)
where A
T
_ 22
C "' (e ‘12)L0o
(2,2) element is
EC A0+0(€ )] T 2 A22T
= [L +0(€)]e GE M +0(€ )]+[I -6L M +0(6 )][e +0(E)]
0 0 2 0 0
A22T
= e + 0(6) = S + 0(6) , (2.77)
where
A T
S = e 22
We have
r](ET) ET AAt~
[‘(ET) = =E e dt
3 (6T) 0

+OG:D

35

0 ll'AlZLO and F22 = A22 +'eL A12. then

6T [A +0(E)]t F t [A0 + 0(6)]t

1‘1(€T) =E [e O B] + Me 226— 82-e M82+0(6 )1 dt,

and

[A +0( 6) ]t

t
J. Fzze _ (A0+o(6)1t MB

F222 0
B1+e LB1+Le 2

F

-LMe 22% B + 0(6)]dt.

2

For sufficiently small E, F22 is invertible and we have

F

T _ eT [A +0(€)]t
r](6T) = 614(6 22 A E 0

-12)F2282 + 0 [e (BE-1182)+0(’:)]dt.

Using Householder theorem (see appendix 2.1) we obtain

F 1

_ 61 [A +0(e)1t _‘
22 A 1(3 +1 g-e O Bodt] + 0(62),

‘12) 22 2

1‘ (6T) = EEM (e
l O )0
where

80 = 31-M0 2,

A0 + 0(6)
Using the power series expansion for e we can see that

F](6T) can be obtained by

A
r,(ei) = EEMO(e

T

At -
_I 1 1

6T
22 - 0 E2
21A 3 + E0 E—e Bodt1+0( ).

22 2

By the same type of approximations we have

36

A T A T

_ 22 -l 22 -l
r2(sT) - (e -12)A2282 + E(e -12)A22LB1
ET [A 0(6)]t A T
0+ . 4 22 -1
- E0 Le L30 + 0(6)]dt-eLM(e .12)A2232
+ 0(62),
. . . 6At .
By us1ng power series expan51on of e . we obta1n
er + 0(62)
r(ET) = , (2.78)
G + 0(6)
where
A T
_ 22 -l
G — (e -IZ)A2282
and
F = ”CG + TB0 .

Now system (2.71) can be represented by

[1+EA + C(62)

x(K+1) 1 x(K) + EEB + 0(6)]Z(K) + ECF + 0(6)]u(K)

Z(K+l) [C+0(é)] x(K) + [S + 0 (6)] Z(K) + [G + 0(6)]u(K) , (2.79)

which is a singularly perturbed difference equation,

2.4.4 Two-time-scale discrete-time systems which can be brought into the
singularly perturbed form by artificial introduction of E.
In conclusion of this section we give a simple example of dis-

crete-time systems which are not in, so called, "explicit singularly

37

perturbed" form but 6 can be introduced artifically to transfer the

system into singularly perturbed difference equations.

Consider the following discrete-time system:

 

 

 

r 1 r 1 ' i r 1
X](K+1) 1.025 .0175 -.0075 X1(K) .0082
X2(K+1) = -.0232 .978 .0192 X2(K) + -.029 u(K),
Z(K+1) -1.325 .975 -2.1 Z(K) E 1

k a L J L g

 

 

 

 

 

with eigenvalues A] = 1.009643, A2 = -2.lO9263, and A3 = 1.002619. By
investigating the numerics of the above system it is seen that the system

could be put in the singularly perturbed form as follow:

 

 

 

 

 

 

 

 

, ._ 1 r 1 ' __ ‘ , E
)(_(K+l) I+6AH 67112 x(K) EB1
= + U(K).
K+1 A A Z
L Z( )4 L 21 22 1 L (K) J L 82 1
where
-— ‘ T
= = A
E .01 K_ [x1 x2.
2.5 1.75 -.75
All = , A12 - , A2] = (-l.325 .975)
2.32 -2.2 1.92
.82
A = -2.1 , B = and B = 1.
22 l E_2.9] 2

For a physical example one

ninth-order boiler problem

to the above example.

can refer to Mahmoud [1982] who has considered

and 6 could be introduced artifically similar

38

It should be noted that in most physical problems some scaling
and regrouping may be needed to obtain 6. This is shown in Chapter

five where we consider an F-8 aircraft model.

2.5 Stabilitygand Approximation Results

This section addresses some important tOpics which are useful
to understand the two-time-scale nature of systems described by singularly
perturbed difference equations. In particular, stability and approximation
results are presented.

Consider the following discrete-time system

x(n+l)

[1+EA11(E)]x(n) + E A12(€)Z(n) (2.80a)

Z(n+l)

421(6) x(n) + 422(612(n). (2.80b)

where (12-A22(0)) is invertible, x and Z are m]- and mzedimensional

state vectors, respectively,and all the matrices are analytic functions

of E with compatible dimensions. Unsing the transformation

y(n) = Z(n) + L X(n). (2 81)
we have
X(n+l) = [I + e411(61-6A12(€)LJX(n) + e412(6))(01) (2.82a)
y(n+l) = [A21(€)-A22(€)L + L + ELAH(€)-ELA12(E)LJX(A) +
[A22(€) + ELA12(E)Iy(n) (2.82b)

39

Using the implicit function theorem and nonsingularity of (12-A22(0))

it can be shown that for sufficiently small 6 there exists L

satisfying
A2](E)-A22(€)L + L + ELA11(E)-ELA12(€)L = O, (2.83)

and it can be approximated by

-1

L = -[I-A22(O)] A21(0) + O (E),

which reduces the above system to the following block triangular form

r q r ir \

 

 

 

 

 

 

x(n+l) 1 + 6A0 + 0(62) 6A]2(€) x(n)
= , (2.84)
L y(n+l)d ; 0 A2201) + 0(6)} L y(n)’,
where
_ -1
A0 ' A11 T A12(I'A22) A21'

Transformation (2.81) is the same one presented in Section 2.2.

From (2.84) it is seen that the eigenvalues of (2.80) are given
by the eigenvalues of [I + 6A0 + 0(62)] and [A22(c0 + 0(E )]. Using the
continuous dependence of the eigenvalues of a matrix on its parameters itibllows
that the eigenvalues of I + 6A0 + 0(62) are in the neighborhood of
the point z = l in the complex plane and the eigenvalues of A22(0) +

O(€ ) are in the neighborhood of the eigenvalues of A22(O). Since

IZA22(O) is nonsingular, A22(O) has no eigenvalues at the point

40

2 = l, or in other words, the eigenvalues of A22(0) are 0(1) away
from the point z = 1. Thus, for sufficiently small 6 the eigenvalues
of (2.80) are clustered into slow and fast eigenvalues as shown in Figure
2.3. There are m1 slow eigenvalues and 1112 faSt eigenvalues. Figure

2.3 is the discrete-time (or z-domain) version of Figure 2.1 which shows

Unit Circle‘“ I. '

e,
1 I _ p

1' I ' 1 l “ '1

1 ;A I I

 

 

Figure 2.3

the slow-fast clustering of eigenvalues in the continuous-time (or 5-
domain). If we denote the set of slow eigenvalues by mS and the set
of fast eigenvalues by mf, then, for sufficiently small E, the eigen-

values of (2.80) satisfy the condition

41

 

min El-A.E
iemf ‘
max E1-A.E >> 1 (2°85)
isms 3

Notice that (2.85) is more general than the eigenvalue separation condition
(2.27) which was used byFWfillips[l980] to define the two-time-scale
property of discrete-time systems. Notice, however, that if (2.80) is
asymptotically stable and the fast eigenvalues are well-damped,then (2.85)
implies (2.27). Phillips' definition cannot handle the cases of unstable
eigenvalues (outside the unit circle) or stable but oscillatory eigen-
values (inside the unit circle but close to it). These cases are important
since, in general, we deal with open-loop systems where the eigenvalues
could be of any of the above forms. Such eigenvalues will then be stabilized
by the use of feedback.

The tflock triangular form of (2.84) leads to the following stability

criterion.

Theorem 2.5.1. If the eigenvalues of A0 are in the open left-half

complex plane, i.e., ReA(A0) < 0, and the eigenvalues of A22 are inside
*

the unit circle, i.e., EA(A22(€)E < 1, then there exists 6 > 0 such

*
that for all 0 < 6 f E the system (2.80) is asymptotically stable.

Proof: From (2.84), the eigenvalues of (2.80) are given by the eigenvalues

of I + 6(A0 + 0(6) and A

A22(O) are inside the unit circle, it follows that, for sufficiently

22(O) + 0(6). Since the eigenvalues of

small 6, the eigenvalues of A22(0) + 0(6) will be inside the unit

42

circle. By a well-known theorem [Stewart 1973. pp. 266] the eigenvalues
of I + €(A0 + 0(6)) are given by 1 + 6A1 where A1 are the eigen-
values of A0 + 0(6). For sufficiently small 6 (the eigenvalues of

A0 + 0(6) have negative real parts. Let Ai = -ai + jei’ “i > 0. Then
2 . 2 2 2 2 2 2 2
(1 + 611) = (1 -Eai + tjsil f (1-eai) + e s, = 1-266i + 6 (a1 + Bi)’

which is less than one for sufficiently small 6. Thus all the eigen-

values of (2.80) are inside the unit circle.

Approximation Results:

 

Consider the linear time invariant discrete-time singularly

perturbed system

x](n+1) [I + 6A11(€)]x1(n) + EA]2(E)x2(n) + €81(€)u(n) (2.86a)

x2(n+l) A2](E) x](n) + A22(e)x2(n) + 02(e)u(n). , (2.86b)

x](0) and x2(0) are given and E is a small positive parameter .

All the matrices are analytic functions of 6 and [A(A22)E < 1. The
control input u(n) is assumed to be constant for K/é f n < K+1/é.
Where K indicates the sampling points of slow states (see Figure 2.4).
Matrices evaluated at E = 0 are denoted by deleting the argument

5, i.e., A = A(0). The solution of (2.86) will be approximated by the
solutions of slow and fast subproblems defined to describe the behavior

of the slow and fast states, respectively.

43

 

 

u(n)
(""""l
1 I l "-"r
F"“1 . I n
.-.. 1 ;
0’1234* 1 g_ ;_ :L "'
Z a a C
Figure 2.4

Slow subproblem

Assume that x2(n) has reached its steady state, then system

(2.86) reduces to
11(n+1) = [I + EAH(€)]x1(n) + EA]2(€)3<‘2(n) + 681(t)U(n) (2.87a)

1201) = A21(5)'x'1(n) + A22(€)§2(n) + 32(emn). (2.876)

where bars show the steady state case and u(n) = U(n). From (2.87b)

we have

44

gm =[12-A22(6)1' [A21(€)x_](n) + 82(&)'u‘(n)1. (2.88)

So

i5<n+1> = LI + 6(A,,(61 + A,2(E)(12-A22(611“Az,(6)1x,(n)

+ 6t8,(e) + A12<e)(Iz-A22(E))

Since the matrices Aij(e) and Bi(€) are analytic in 6,
they can be approximated, up to 0(6), by their values at E = 0. Further-

more (IZ-Azz(é))"1 can be approximated as

_ -1 _ -1 _ -1

Employing these approximation in (2.89) we obtain

x1(N+l) = (I + 6A0)x](n) + €80u(n), x1(0) = x](0), (2.90)
where
_ -1
A0 ' A11 + A12(12‘A22) A21
_ -1
Bo ‘ B1 A A12(I2'A22) 82-

Note that systems (2.86) and the reduced system (2.90) evolve in the
fast time-scale n.

Since U,(n) is constant over the cycle -E f n < '67' we

can express §1(£El). in terms of ;1(é) and u(K/6)

45

K+1 1 K+1 1 .
~ K+1 l/6~ ‘6'" “‘6" 'J _ _
x1 (T): (11 + 6A0) x](K/E) + jZK/E [(I1 + 6A0) ,E-BO]u(K/6).
(2.91)
Letting i = 5&1 -1-j we get
~ (K+1) ' (I + 5A )1/E~ (K/E) + 6 1/671 (I + EA )18 'lK 5)
X1"E‘ ‘ 1 0 X1 120 1 0 0u / '
Now let
xs(1<) = Imus). u (K) = ink/é),
and
A0 1 A0(1-t)
AS = e , BS = E0 e dtB0 . (2.92)
We define the slow subsystem to be
xS(K+l) = Asx (K) + BSuS(K), xs(0) = x](0). (2.93)

Fast Subsystem:

 

Consider x](n),'x2(n) to be constant during the fast transients.

Let
xf(n) = x2(n) - x2(n), (2.94)
where

[A2]§q(n) + B U(n)]. (2.95)

46

The fast subproblem is defined to be
xf(n+l) = Azzxf(n) . xf(K/€)= XZIK/El-RZW/E). (2.96)

07‘

xf(K/6) = x2006) -(12-A22)'][A271(K/E) + (320%)] for x = 0,1,2,...
' We note that for every period of slow-time-scale the fast sub-
system has a different initial condition due to 72(K/E) which depends
on §E(K/€). Figure 2.5 shows a typical shape of the response of 7,
and xf. At the beginning of every 1/6 cycle, the fast modes are excited

by the jump in u. Then xf dacays exponentially towards zero.

px1(n)

 

 

 

 

w
I ’ 3
I 1 i ==n
0 It Z/é 31";
x (n)
I f
1
\\ X
I 1 L : n
0 V: 1:- 5%

Figure 2.5

47

Let us apply the Chang transformation (2.25a) to the system (2.86) i.e.,

f l f E r q

01(n) 11'EM(E)L(E) -€M(€)

 

 

 

 

 

 

02(01J L [(6) 12 X201) , (2.97)

L

where L(E) and M(€) satisfy

 

 

 

 

 

 

L(€) = A22(€)L(s) -€L(€)AH(E) + €L(€)A12(6)L- A2](E) (2.98)
o = M(E)A22(E) - (1(6) 6:4,,(6) -A,2(6)L(6)1M(6) +6 M(6)L(6)A,2(e)
”A (e): .
‘2 (2.99)
and could be approximated by
L(6) = - (12-A22)'AA21 + 0(6) (2.100)
11(6) = - A12(I - AZZ)‘1 + 0(6). (2.101)
We get
An](n+1)1 r I] + 6A0 + 0(62) O A 01(n)
L02(n+])E E O A22+O(E) J L 02(n)J
680 + 0(62)
+ u(n), (2.102)
L 82 + 0(6) E

 

 

48

where
= -1
A0 A11 T A12(I‘A22) A21
_ -1
B0 ‘ 81 T A12(I'A22) B2
The solution for (11 and n2 for K/E 5 n < 5%9- is
n1(n) = [I + 6A0 + 0(6 )1 n](K/€) + 6 z [I + 6/10 + 0(6 )1-
i=0

[30 + 0(6)] U(K/€) (2.103)

n-K/E n-l .
= 1 6 6
n2(n) [A22 + 0(5)] n2(K/€) + 120 [A22 + 0(6)] [82 + 0( )1u(K/ ).
(2.104)
From (2.103) we obtain
%- 1
1/6 ' .
K+1 _ -2 K 2 4
n1(-;;9 — [I + 6A0 + 0(6 )1 01(g1 + E 120 [I + 6AO + 0(6 MBO + 0(6)].
u(K/E). (2.105)
Using the following identities
[1 + 6A + 0(62)11 = [I + EA]T + 0(6) , 0 5 i 5 1/6 (2.106)
1/6 A
[I + 6A] = e + 0(6) (2.107)
%4-1
. 1
S I (I + 5A)1 = E eA(T't)dt + 0(5) (2.108)

i o «0

49

ka*proofs see Appendixes 2.2, 2.3 and 2.4, respectively)and the continuous

dependence of the solution of difference equations on parameters (see

Appendix 2.5) we get
n](K/€) = Xs(K) + 0(6)-
Now Chang transformation yields

x](K/E) = xS(K) + 0(6).

(2.109)

(2.110)

(2.110) shows that the slow trejectories x1 could be approximated up

to 0(6) by the solution of the slow subproblem.

We define the steady state of n2(n) as follow

62 = (42249611712 + [82+0(6)IU(§), g; n < A} .

Subtraction of (2.111) from (2.102) (n2 - equation) and letting

~

n2 = r12 - :2 yields

52(n+l) = [A22 + 0(6)]n2(n) , -E 5 n < EEl-,

with initial conditions

0,l,2,...

fi2(K/6) = 82(K/6) - 62(K/6). K

= - (I-A22)'T[A21x](K/€) + 820(K/6)3 + x2(K/€) +

(2.111)

(2.112)

0(6).

(2.113)

50

Comparing (2.112), (2.113) and (2.96) and using again the continuous-

dependence of solutions on parameters we obtain

K K+1

n2(n) = xf(n) + 0(6) , 6'5 n < 7§—-. ’ (2.114)
But for K/6 5 n < 5E1. we have
x2(n1 = (1-A,,)"n1(n) + n2(n) + 0(6). (2.115)

or

(I-A 1"

X
N
A
3
V
II

22 A2];1(n) + xf(n) + Eé(n) + O(&)

l

(I-A22)'TA2131(n) + xf(n) + (1-A22)' 820(K/6) + 0(6),(2.116)

where 01(n) is an 0(€)-approximation of 01(n) which is given by

n-l
n](n) = (1 + 6A0)n-K/EnE(K/€) + 6 ,2
1:

0 (I + 6A0)‘Bou(K/6).-é g n < 521.

(2.117)

Thus, we can express x2(n) in terms of the fast and slow states and

control input.

Based on our discussion above, the following theorem holds.

Theorem 2.5.2: If (A(A22)| < 1, then for all finite K 3 0 the solution

 

of (2.86) can be approximated by

51

x (K) + 0(6)

S
- (I-A )‘TA

X
—l
A
.7<
\
(1"
v
II

n-l -
n-K/6XSEK) + e I (I + 6A0)‘800(K/6)1

i=0

I + 6A

X
N
A
3
v
I

22 21[( 0)

221'182u(1</6) + 0(6), E: n < 521.

+ xf(n) + (I-A
where xS and xf are solutions of the slow subsystem (2.93) and the
fast subsystem (2.96), respectively. Moreover, if Re A(AO) < 0, the

above theorem holds for all K 3 O.

APPENDIX 2.1

Householder Theorem:

 

If A is a nonsingular matrix then [Householder, l964l

(A + Bco)“ = A‘1 - A"B(I + coA'1 )']CDA'],
For C = V'1 we have
(A + BV”D)‘1 = A"1 - A-IBEV + DA"BJ"DA“ .

In particular when 8 = EI and V = I we have

1"]

'1 [I + €DA‘ ] DA

(A + ED)'1 = A - 6A.1 '1.

If all the matrices are 0(l), then we obtain

-l

(A+mr‘=A +mq.

52

APPENDIX 2.2

Wish to prove that for any positive integer j 5 l/6
1 2 J' J'
ll(1+EA+€B(E)) -(I+EA) U =0(E). (I)
Let L.H.S. of (I) = T. , H
J 1 : j i
1. < .2 .e (;) (1 + ea) (Eb) . (2)

(2) can be shown by mathematical induction. Apparently, T1 fezb.
Suppose that (2) is valid for (j-l). Then

j-l
“(I + EA + €23)(I + EA + e23)

.4
ll

3-1
- (I + EA)(I + 6A) H

j-l
+ €2b(] + Ea + 62b)

IA

(1 + ea)1j_1

j l
(l + 6a)

1.

6‘ (j;]) (1 + Ea)j']-i(€b)i
1

IA
IIIV’] I

+

'-l . .
Ezb 3: 6r (J;‘> (1 + 6a13“"(Eb)’
r=O
J." ' °_‘l ._. _ .
E 6‘ (31 ) (1 + ea)J ‘ (ab1‘
i=l

j
+

-1 . .

X0 6r*‘(3;‘1(1+6a13"'r

r:
(eb)r+l

j-l

Z

e‘(j;‘)(1 +.6a)j"(6b1‘ +

e‘(4“)(1+6a)j"(eb)
1 1 1'1

1

II [‘49.

l

53

 

54

j-l .

= jg] 6‘t(j;1) + (311)1(1 + 6a1j"(6b)‘ + 63(611)j
j-l 1.3. ._1. . . . i . . ..
' .21 6 (1>(‘ + 6a)‘ <6b)‘ + 63(6b1J = z 6‘ (31(1 +6a>"‘
1= . i=l

(6b)‘.

which proves (2). Now, using (2) we have

...
A
ll [WC—I-

j , 6‘ <3) (1 + 6a>"‘(6b>‘
1' l
,2] (.1 (T1339

1

IA

j.. .
(1 + 6a)”6 ,2] 6‘ ($1 <6b1‘.
1:

. . l/G _ a . *
S1nce l1m (l + 6a) _ e , there ex1sts 6 >0 such that for all

(l+€a)1/€<K] eaAK.

So

.—I
A
x
do
ll MU
m
—‘

A

u—Jo L...
V
A
m
U-
v

—I

(A.

I A
—J
\
m

II
7<
II MLJ
A
m
N
CT
v
.1
A
(J
v

.i

ll
7?
{—1
ll MO
0
A
m
N
0‘
V
—l
A
.1 (.4
V
I
A
(.1
V
L.»

i

II
x
r...

A
—J
. +
m
N
U-
v

I
—J
L.._l

IA
7Q
{—.1

A
._a
+
m
N
U-
V

l
(-l)
KEl +1g€2b+§—E———€4b2+... -l]

IA

KEl + 6b + (1%:

KEl + 6b + 62b2

55

) Ezbz + (1'63,1'2€ €3b3 + ..hq}

+ €3b3 +-.. -1]_

K 6 b[l + (6b) + 62b2+ ... J

Q.E.D.

 

APPENDIX 2.3

 

Wish to prove “(I + 6A)”e - eAH = 0(6) .
Let 11A11=a.
and let v = Ln(I + 6A)1/e = Je— Ln(I + 6/1)
=é£6A--‘2-(6A)2+-‘3—(6A13 ----1
= A -.% e A2 +.% 62A3 _ ...
=A+€[-%—A2+%—€A3-%—62A4+... 1,

Inside the bracket, by ratio test, is convergent series and bounded since
“_l§A2+e§3 _e224 +...1; 5%.a2+‘§6a3+};62a4+m
< a2 + ea3 + eza4 + ... = 32 (1 + ea + (ea)2 + ... )

=(.112 1 = a2
T565 173
So we have
Y = A + 0(6),
and

57

(1+ 6/11“6 = eY = eA‘O“) = 1+ (A + 0(6) + ‘. (A + 0(6112 +
Applying norms we obtain

“(I + 6A)”6 - eAH

2 3
= {I I+A+0(6 +-(—ZJ:—-2-2—2—MOE +———A—l-LA+06+- -I(+A+-é—.—+3B— +---)“
2 3
- 2a 3a 4a
50(E)[l+-2—.+ TE+T+ H]

il l i+l l 1+2
+0(E)[l'+W(l) m(1)+ I
+

2 3
_ i__ a
- o(&)[] + a + 2: +.§T + 1
2 K-2
2 l a a
+ 0(6 ) ET-[l + a + éT-+- K-2)! + . j
2 3
3 l a a .
+ 0(6 ) 3: [1 + a + §T'+ 3T'+ J
+
2 3 K-1
+0(61)1[]+a+a—,—+a—,—+ +a l+ J

N
(A)

58

= 0(6) ea + 0(62) 522—?— + 0(63) e—a— + 0(e),

Q.E.D.

APPENDIX 2.4

Wish to prove that for sufficiently small 6

1/6 -1
52

. 1
(1 + 6A)‘ = J a(“T)AdT + 0(6) . (1)
1 0

0

We have

J1 A(1-T)dT _ 1

E 6'
J I eA("T)dT. (2)
0 i

1 e(1-1)

I
II M\

Let t = 6i - r, then the above is equal to

E

 

 

"E e A(l-Ei + t) _ ‘/ A(l-Ei) At
2 e dt - 2 e e dt (3)
i=1 0 i=1 0 '
But
2 2 2 3 E
eAtdt = [I + At + Lflil— +--- Jdt = It + A? + A I +
2. 2 3
0 0 0
-EI+€2—A-+€3AZ—+
23 31
2
=EI+EZEEAT+i+ 3
Using the ratio test on the norms and letting “A“ = a yields
2 2 2 3
“g¥-+ g6 +- H j §;-+ :? + 64? + ... < aEl + Ea + (ea)2 +,,, 3

.Ji__
l-Ea '

59

60

Thus
6
eAtdt = 61 + 0(€2)-
«O
. ATso,
1/e .
X eA(1'El) = I + e6A + e2EA + ... + 66(1/6-l)A .
i=l
Thus
1 A(l— ) eA e(é -l)A
A e T d1 = EEI + e + --- + e 3E1 + 0(e)3’
0
1/e-1 . A
.e; Jmu+o(o1-
i=0

0n the other hand [Hoppensteadt and Miranker, l977]
(I + 6A)‘ = eeiAEI + 0(6)] .
So
1 1 1/6 1
E - . ..

e g (1 + EA)1 = e g eEiAEI + 0(6)] .

Thus

(8)

Q.E.D.

APPENDIX 2.5

Lemma: Consider the difference equations

x(k + l)

A x(k), x(0) = x0, (1)

and

Y(K + I) (A + E) Y(k), Y(0) = yo. (2)

where “E“ = 0(e) and “xo-you - 0(e).
The following assertions are true

(1) “x(k) - y(k)” = 0(e) for all finite K

(11) If Ix<A>I s e < 1. then 'ux<k) - y(k)n 5 c e a“ for all K.

where 0 < a < l, Hence “x(K) - y(K)U + 0 as K + w .

Proof: The assertion is obvious in the case of finite time. We prove

it only for the infinite-time case. From the continuous dependence of
*

the eigenvlaues of a matrix on its parameters there exists 8 > O

*
such that for all e < e the eigenvalues of (A + E) satisfy
|A(A + E)! S B] < 1 .
Moreover, for any 5 > 0 there exists a matrix norm |-| such that

I(A+ E)! 5

 

A(A+E)]+5581+o.

6l

62

Choosing 6 = (l-81)/2 we get

where 82 = (l + Bl)/2

Subtracting (T) from (2), letting gk = |y(k) - x(k)| and using the

equivalence of norms we get

AK,1 = ly<K+1) - x(K+1)! = l(A+E)y(K) - Ax(K)!
=HMDUW)-MM)+EMMI
f 82 9K + IE1 MK)!
5 82 9K + C] e IXOIBK .
So we have

K
gK+1E829K+C268 3 8(1982<1.

From above we get

(7‘

+ C

(Q
._I
A
m
N
(D
O

(.0
N
A
m
N
(D
—-I
+
('5
N
n
U)
IA
U)
NM
to
O
+
m
N
('5
N
m
+
('3
N
m
00

But

 

- l-(B /s)n
= B" 1 g (4)
2
1- ___
e

If 82 < B , then we have

 

n n n
8n_1 1 ' (82/8) = 8 ‘ 82 < K B“
1 _ 82/8 8 -82 -
If 32 > 8, Then
"_1 (2)" -1 8n _ Bn
(4) = a B = 28 _ 3
.82. -1 2
B
_ 1 -(e/e )"
= .31 2 <ng
1 ' 8/82
If 82 = 8 , then
n;l n-l n-l n-l n-l n-l
(4 = g B = n B - n(/§) (/§) 5 K (/3) since 8 < l,
=0
So we have
gn < 83 go + 6 CI an , u < I,

64

where C is a constant of 0(l).

So 9n + 0 as n + m.

Q.E.D.

CHAPTER 3

INFINITE-TIME OPTIMAL REGULATORS FDR
SINGULARLY PERTURBED DIFFERENCE EQUATIONS

3.l. Introduction

 

Optimal control of infinite-time regulators for singularly per-
turbed continuous-time systems have been discussed by Chow and Kokotovic
[1976] and the role of slow - fast decomposition and composite control
in approximating the performance index are thoroughly investigated.

Also, Blankenship has considered the optimal control of finite-time
linear quadratic regulators of singuTarly perturbed difference equations
and showed that the basic features of singular perturbation theory can

be extended to the class of discrete systems as well.

Section 3.2 is devoted to discussing their work and, hopefully, pro-

vide the reader with a better understanding of the problem and our motivation.

In Section 3.3 the problem statement is introduced and, more
specifically, the source of difficulty in applying Blankenship's approach
is addressed.

Section 3.4 deals with asymptotic behavior 0f the optimal solution
of linear quadratic regulators and some improtant theorems are proved
which removes the difficulty in dealing with the problem mentioned in
Section 3.3.

In Section 3.5 it is shown that a near-optimal solution can be

obtained by applying slow-fast decompositions as-in the continuous-time

65

66

work of Chow and Kokotovic [l976]. In our case the slow optimal control
problem will be a continuous-time problem, while the fast optimal control
problem wilT be a discrete-time problem. -

In Section 3.6 we present an interative technique to slove the
discrete-time stiff Riccati equations which avoids the ill-conditioning
problem and provides a high convergence rate.

Finally, our claims are illustrated, by considering a simple

example, in Section 3.7.

3.2. Related Background

Chow and Kokotovic [l976] investigated a near-optimum state re-
gulator for singularly perturbed continuos-time systems which is composed
of the slow and fast subsystem regulators and showed that a second-order
approximation of the optimal performance is achieved. Also, they for-
mulated a complete separation of slow and fast regulator designs.

Due to close anology of our problem with the continUous-time
problem we give a brief explanation of their work.

They consider the continuous-time singularly perturbed system

x1 = Anx.| + Amx2 + Blu, x](0) = x10 (3.1a )
6x2 = A21x1 + A22X2 + Bzu, x2(0) = x20 (3.lb )
y = C1x1 + sz2 , (3.lc )

where E is a small positive scalar, the state x is formed by m1
and 1112 vectors x1,x2. The control u is an r vector and the out—

put y is a K vector.

67

The performance index is

min J = J (yTy + uTRu)dt, R > 0. (3.2)
0

l\.>|--J

Optimal control is given by

_ -1 T
110pt - -R B Kx, (3.3)

where K is the stabilizing solution of the Riccati equation

T T

0 = -KA-A K + KSK-C C (3.4)
. _ _ -l T
With C - [C1 C2], S - BR B , and
IA A l T l
11 12 B1
A: ’ B =
A21 12; B2
(T E J L?J

 

 

 

 

By assuming A22 is nonsingular, the slow and fast subsystems are defined

(see Section 2.2) with their performance indexes.

Slow regulator problem:

For the slow subsystem

)(0
II

S onS + BouS , xS(O) = x10 (3.5a)

'~<
U)
1|
0
o
x
m
+
O

0us, (3.5b)

68

where
_ -1 _ -1
A0 ‘ A11'A12A22A21 ’ Bo ‘ B1‘A12A2232
_ _ '-l _ -l
Co ’ C1 C2A22A21 ’ Do"‘C2A2232’ (3°56)
find uS to minimize
= l. m T T
as 2 To (ysys + usRuS)dt, R > o . (3.6)
In terms of xS and us, (3.6) becomes
= 1_ m T T T 1 1
JS 2 J0 [XSCOCOXS + ZuSDOCOXS + uSRouSJdt , (3.7)
where
_ 1
R0 - R + 0000.

They prove that if the triple (AO’BO’CO) is stabilizable-

detectable, then the Riccati equation

_ -1 1 -1 1 T -1 T
o - - KS(A0-BORO ooco) - (AD-30R0 poco) KS + KSBORO BOKS
1 -1 1
- CO(I-DOR0 ooh:0 (3.8)

Dias a positive semidefinite stabilizing solution KS and the optimal

Control for (3.5) and (3.6) is

Fasst regulator problem:

 

For the fast subsystem

69

6%

f Azzxf + 3211f , xf(0) =x20 - x2(0) (3.lOa)

where '

find uf to minimize

1 m 1 1
Jf = 2-A0 (yfyf + ufRuf)dt, R > 0. (3.11)
It is a1$o shown that if the triple (A22,BZ,C2) is stabilizable-
detectable, then the Riccati equation
0 = -K A -AT K + K B RTABTK -cTc (3.12)

f 22 22 f f 2 2 f 2 2

has a positive semidefinite stabilizing solution Kf and the optimal

control for (3.10) and (3.11) is

- -1 T .
uf - -R Bszxf. (3.13)

It is shown that, under the stabilizability-detectability of slow
aand fast subsystems, the solution of the Riccati Equation (3.4) possesses

a power series expansion at E = O that is,

 

 

 

 

f 1 f a
(i) (1)
K1 6K2 m . K1 6K
K = + Z 31_ , (3.14)
'=1 i1 . T .
T 1 (1) (1)
L 6K2 EK3 1 L eK2 6K3 J

311:!

(3.15)

70

Furthermore, it is proved that for the composite feedback control

-1 T -1 T 1 -1 1

_ -l T -l T -
uC - -[(I-R BZKfAZZBZ)RO (DOCO + BOKS) + R BszA22A21JX1 - R BZKfXZ’
(3.16)
uOpt = uC + 0(6), (3.17)
and
J = dc + 0(62), (3 18)

where JC is the value of performance index J of system (3.1) with
uC and hence the composite feedback control (3.16) is an 0(E2) near-
optimal solution to the complete regulator problem (3.1), (3.2).

Blankenship [1981] studied linear quadratic optimal control problems
for singularly perturbed difference equations when the cost function is

defined on a finite-time period. In particular he considered the system

x(nil) = x(n) + EAx(n) + EBZ(n) + EFu(n) (3.19a)

Z(n+l) = SZ(n) + ECX(n) + EDZ(n) + (G+€H)u(n), (3.19b)
where

x(0) = x0 and 2(0) = 20.

x and Z are m]- and mZ-dimensional state vectors and the control
iriput u is r-dimensional. All the matrices are constant matrices of ap-
Pr‘opriate dimensions. 6 > 0 is a parameter and n = 0,1,2,...,N-l.

The performance index to be minimized is

N-1

N)K X(N) + 2XT(N)K Z(N) + ZT(N)K3Z(N) + K) [uT(K)Ru(K)
=r

r l 2

+ XT(K)Q]X(K) + 2xT(K)022(k) + ZT(K)Q3Z(K)1, 0 5 r 5 N-l,
(3.20)

where

Q=QT

Slow-fast decomposition

2. He, essentially, showed tn;

perturbation approach to czntfn

time ones. We extend on his a?”

(I)

control problem and study *1
algebraic Riccati equations as

It is shown in Section

finite-time problem does not is?

time problem. A special scai‘

is employed and is shown to

(7
(I)

in the perturbation parameter

3.3. Problem Statement

 

Consider the linear tir

x(n+1)

z(n+l)
y(n) it

Where ‘5 > 0 is a small ;-::i:‘

the m1 and m2 dimension?"

> D with Q define: for”

Qi’ i = 1,2,3 in the same way.
of system (3.1) was discussed in Chapter
: the basic features of the singular

.aus-time can be extended to discrete-

: and consider the infinite-time optimal

at) ototic behavior of the resulting

.ne perturbation parameter tends to zero.
;.3 that the asymptotic behavior of the in-

T w as a limiting case of the finite-

()

7? :f the solution of the Riccati equation

a;0ropriate to expand solution as a series

e-invariant discrete-time system

~‘n) +3 Bz(n) +5 Fu(n) (3.21a)
- SZ(n) + Gu(n) (3.21b)
- * 022(n) + Mu(n), (3.21c)

a: parameter, the state vector comprises

3:20rs x and z, the control u is an

72

r-dimensional vector and the output y is a k vector. The initial
states x(O) and 2(0) are given. The controls u(n) are to be selected

to minimize the performance index

a = c [yT(n)y(n) + uT(n)R u(n)], R = RT > 0. (3.22)

n

IIMB

O

For simplicity the matrices A, B, C, F, G, S, 0], DZ, M and R are
taken to be independent of e but they could be analytic functions of
e and the problem would be treated in the same way. The finite-time
version of this problem was considered by Blankenship who investigated
the asymptotic behavior of the optimal solution as E + 0. The asymptotic
behavior of the infinite-time problem we are discussing here does not
follow form Blankenship's study as a limiting case when the terminal
time N tends to m. To see this observe that he gives the solution
to the Riccati equation by defining

V(x(n),Z(n), n, E) = min [Jn(u)l ,
1vhere V is the "cost to go” from the point (X(n),Z(n)) at time n
in the problem (3.19), (3.20).

He proves that V has a Hamilton-Jacobi equation which has

Solution

V(X,Z,n) = XTPlX + 2xTP§z + szgz , (3.23)

Where P = (P1, P2, P3) satisfies
n n n n
3 _ 3 3
Pn - Q3 + F (Pn+l) + 0(6) (3.24)
2 _ 2 2 3
Pn — 02 + F (Pn+], Pn+1) + 0(6) (3.25)

73

l _ l l 2 3
Pn - Pn+1 + Q1 + F (Pn+], Pn+1) + 0(6) (3.26)
PN = Ki’ i = 1,2,3 , n = 0,l,...,N-1. (3.27)

Under proper conditions, he gives the solution to Pn up to 0(62)

for sufficiently small 6. In particular the solution for P; is given

by

—2

pl = p‘°(en) + 2;0 + (N.n)[01 + F](P 3)] + 0(6), (3.23)

, T5

l

where P10 indicates the zero order term of P and 2;0 represents

the zero order term of the boundary layer solution. Note that (3.28)
gives an asymptotic formula for the solution of the associated Riccati
equation which is proportional to N, so it blows up as N + o. By
appropriately scaling the solution of the Riccati equaiton, similar to
(3.14), in the next section we will be able to overcome this difficulty.
The infinite-time regulator problem (3.21) (3.22) could be a
result of discretization or sampled-data control of infinite-time regulators
‘for singularly perturbed continuous-time systems using the method of
[ILevis and Dorato, 1971];the details are similar to the finite-time
example presented by Blankenship [1981] and explained in Section 2.4.
TWie form of the performance index is a little bit more complicated than
tflie one studied by Blankenship because of the presence of the matrix M.
Mflien M = 0, the performance index J reduces to the one studied by
Blankenship. The current form is chosen to accommodate the sampled-
data control case where J is obtained by discretizing an integral per-

‘Ftarmance index of a continuous-time system.

74

It was shown by [Levis and Dorato, 19711, (See Appendix 3.1),
that discretizing an integral performance index with quadratic terms in
the state and control (and no cross product terms) results in a discrete-

time performance index of the chosen form with M'# O.

3.4. Asymptotic Behavior of the Optimal Solution
The optimal control of the system (3.21) with performance index

(3.22) is given by [Levis and Dorato, 1971]

1 T T
T <

[a PA+M 01 x( 1, (3.29)
Z

M+BTPBJ' n)
n)J

”opt.(n) = -[R+M

where P is a stabilizing solution of the discrete-time algebraic Riccati

equation
P = DTD+ATPA-CATPB+DTM1[R+MTMFBTRB1~A[BTPA+MTDJ , (3.30)
and where
IfiA 68 6F
A = , a = , 0 01,02
1 c s G

Ir1 studying the asymptotic behavior of the Riccati equation (3.30), we
Seek the matirx P in the form

r 3
P1/6 P2

P = . (3.31)
T
2 P

 

 

L P

75

.-

The form (3.40) plays a crucial role in studying the solution of the
Riccati equation (3.30). It is different from the form used by Blanken-

ship since he used

P = . (3.32)

 

 

N—I

Substituting (3.31) into (3.30) and partitioning the Riccati equaiton

yields
0 = f1(P],P2,P3,6), (3.33)
P2 = f2(P],P2,P3,g), (3.34)
P3 = f3(P],P2,P3,€), (3.35)

where the functions f1, f2 and f3 are defined in Appendix 3.2. To
study the solution of (3.33)-(3.35) near 63: 0, it is natural to start
by'setting E = 0 in (3.33)-(3.35). This yields

_ T T T T T
0 - P](O)A+P2(0)C+A P](0)+C P2(0)+C P3(0)C+D]D]

- [P1(0)F+P2(0)G+cTP3(0)G+0]TM1[R+MTM+GTP3(0)01‘1

x [FTP](0)+GTP;(0)+GTP3(O)C+MTD]1, (3.36)
P (0) = P (0)B+P (O)S+CTP (0)s+010
2 1 2 3) 1 2

4P](0)P+P2(0)G+cTP3(0)G+DIM1[R+I~TTM+GTP3(0)GJ’1

[GTP3(0)S+MTDZJ. (3.37)

76

_ T
P3(O) - S P3(0)S+DT D

T T T -1
2 2- -[3T P 3(O)G+DZM3[R+M M+G P3(0)G]

[GTP3(O)S+MTD (3.38)

2].

Equation (3.38) is a discrete-time algebraic Riccati equation. It is

well known[Kwakernaak and Sivan, l972] that if the pair [S-G(R+MTM)"1

MTDZ,G]T is stabilizable and the pair [S- G(R+MTM)-1M TD 02

 

JDZD 2- 02M( M(R+MTM)'1MTDZ ] is detectable then (3.38) has a unique positive
semidefinite solution. It is obvious that the stabilizability of
[S-G(R+MTM) IMTDZ,G] is equivalent to the stabilizability of [5,6].
Moreover, using the matrix identity I- M(R+M TM)'1MT= (I+MR MT)'1 (For
proof, see Appendix 3.3), it can be shown that the detectability of
'1MTDZ, JDZDz-D;M(R+MTM)'1MTD2 ] is equivalent to the detecta-
bility of [s-G(R+MTM)"MT

 

[S- G(R+M TM)

02,02] which is equivalent to the detectability

 

 

 

of [5,02]. Note that
T T l T _ , T . -l MJDT
VDZDZ- DZM( (R+M M) M 02 - WDZEI -M(R+.4T M) D2
= JD;(I+MR'1MT)'ID2 = {QTQ, for some matrix Q.

Thus we assume that the triple [S,G,D ‘ is stabilizable-detectable which

2.1
guarantees the existence of P3(0) 3 0. Furthermore, from the properties

of Riccati equations [Kwakernaak and Sivan, 1972] we have the stability

property

New <1 (3.39)

,

77

T T

where a3 = S-G[R+M M+G P ‘

(0)63‘ [GTP3(0)S+MTD

3 2].

We turn now to equation (3.37) and notice that P2(O) can be

expressed in terms of P](0) and P3(0) as

92(0) = L1 + P](0)L2 , (3.40)
where
L1 = {0102+CTP3(0)5-[CTP3(O)G+DIM3[R+MTM+GTP3(0)G]-]
[GTP3(0)S+MTDZJ}L3",
L2 = {B-F[R+MTM+GTP3(O)GJ-1[GTP3(0)S+MTDZJ}L3-],
L3 = I-a3 ,

and where the nonsinguiarity of L3 follows from the stabiiity property

(3.39). Substituting (3.40) into (3.36) yieids

. “T . “-1“T

o = P](O)A+A P](O)+Q-P](O)BR B P](O), (3.41)
where

B = F+LZG,

i = R+MTM+GTP3(O)G,

A = A+L C-éfi'ltGTLT+GTP (O)C+MTD J,

2 i 3 i

and

. _ T T T T

Q - D101+L1c+c L1+C P3(O)C

- [L1G+CTP3(0)G+DIMJ§'1[GTL{+GTP3(0)C+MTD]3.

78

Equation (3.4l) is a continuous-time algebraic Riccati equaiton. We
assume that the triple (A,é,/6) is stabilizable-detectable. This
guarantees [Kwakwenaak and Sivan, l972] that (3.4l) has a unique positiVe

semidefinite solution P](0) 3 o and
Re {A(a1)} < 0, (3.42)

where

a] = A-éfi-IETP](0).

Thus we have established the existence of the solution of (3.33)-(3.35)

at E = 0. For 6 near zero, let
Pi = Pi(0) + e Ei for i = l,2,3, (3.43)

where E1 indicates the non-zero-order terms. The existenc of Ei’

i = 1,2,3 is established by applying the implicit function theorem where
the nonsingularity of I-a3 and a], (which follow from the stability
properties (3.39) and (3.42), respectively), are used to show that the
Frechet derivatives of E1 for i = l,2,3 at E = 0 is invertible.
This is shown in Section 3.6 and the existence of Pi for i = 1,2,3
follows immediately.

It remains now to shdw that this solution is stabilizing. THis,
however, follows immediately by applying the stability criterion, derived
in Appendix 2.l, to the closed-loop system where the stability properties
(3.39) and (3.42) are used. Our conclusion is summarized in the following

theorem.

79

Theorem 3.l: Assume that

 

Condition a: The triple (4,3, 6) is stabilizable-detectable

in the continuous-time sense; i.e., every eigenvalue of A which lies

in the closed right-half complex plane is controllable and obserable,
Condition b: The triple (S,G,DZ) is stabilizable-detectable

in the descrete-time sense; i.e., every eigenvalue of S which has modulus

greater or equal to one is controllable and observable.

* *
Then there exists 6 > 0 such that for all 0 < €< 6 , the Riccati
equation (3.30) has a unique positive semidefinite stabilizing solution.

Furthermore, the solution has a power series expansion at E = 0, that

 

is
' (1') m‘?
m 1' P.| /6 P2 3
P = iZO I§T
’ UN (1')
L P2 P3

One unpleasant feature of Theorem 3.l is that Condition a is
dependent on P3(0), the solution of the discrete-time Riccati Equation
(3.38). It will be shown later that the matrices 4,3 and 6 are in-
deed independent of P3(0). For the time-being, however, let us assume
that Conditions a and b hold so that Theorem 3.l provides us with a
reasonable way to approximate the optimal control for small 6 . An

approximate state feedback control is defined by

u(n) = -[R+MTM+BT5 BJ-ICBT5.A+MTDJ (x(n))

;z(n)"

k

(3.44)

where 5 is obtained by truncating the expansion of P; i.e.,

80

S = .2 4%— . . (3.45)

 

 

The near optimality of the control law (3.44), (3.45) is established in

the following theorem.

Theorem 3.2: Under conditions (a) and (b), the use of feedback control

(3.44), (3.45) is 0(62N) near-optimal in the sense that

J-J = 0(52N)

opt (3.46)

where J is the value of the performance criterion when (3.44), (3.45)

is used, while JO is its optimal value.

pt
Proof
Let us represent the optimal feedback control as
x(n)
_ 0 O . 0 . _
u - -(F F )X(n)g -I X(n), X(n) - , (3.47)
opt l 2 -
z(n)
then
a = €xT(0)PX(0), (3.48)

opt

where P satisfies equation (3.30). Similarly, the approximate feed-

back control is represented as

u = -(F F

1 2mm 4 -IX(n). (3.49)

8)

It can be easily verified that

F? - F1 = 0(6”) , i = 1,2 . (3.50)

The closed-loop system under the feedback control (3.49) is

X(n+l) = (A~BI)X(n), (3 51)
and the performance index is

a = 5XT(O)P'X(O), . (3.52)

where P' satisfies the Lyapunov equation

P' = (A-ny)TP'(A.ax)+DTD+3T(R+MTM)3-DTMI-3TMTD . (3.53)

Subtracting (3.30) from (3.53) and letting h = (R+MTM+BTPB) yields

P'-P T

(ms) (P'-P)(A-m) + (mafia-m)

ATPA+IT(R+MTM))I - DTMI - ITMTD

+(ATPB+DTM) fi" (BTPA+MTD)

(A-BI)T(P'-P)(A-BI) - flips—Hm + 3:723me

T( T T T T T T

+ I R+M M): - D M345 M D + (ATPB+DTM)§'](B PA+M D)

(A-B§)T(P'-P)(A-flI)-IT fl fi'13TPA

l T

- ATPB fi' fi 3 + 3 h a

-DTMI-$TMTD + (ATPB+DTM) fi'1 (BTPA+MTD)

82

(A-BI)T(P'-P)(A-BI) ~IT~R T2"1 (BTPJHMTD)

(ATPB+DTM) 'fi’] '1? a + IT R3 + (21s.T T )fi'1

PB+D

(BTPA+MTD)

(A-BZF)T(P ' -P) (st-m) + [33- (ATPa+oT14)§" 1

fits-fl" (BTPMMTDH .

So we obtain

T( 0 T

P'-P)(A~fl$)+[I-I J (R+MTMtB 0

P'-P = (it-m) T

But the term inside the last bracket is

O _ O 0 _ N
[I-I 3 ‘ [Ful'F], FZ‘FZJ " 0(6 ).
Letting P'-P = V, (3.54) becomes
v = (A-m)TV(A-m)+0(-22N).

PB)[I-I 3.

(3.54)

(3.55)

By application of the Implicit Function Theorem we can show that P'

possesses a power series expansion at 45= 0. So V can be expanded

as follows:
1))
,i
fr

<
ll
11 15-4 8

o .
L V2 V31) )

 

 

Partioning (3.55) and matching the zero order terms yields

(3.56)

83

= (0) -l T (O) -T T T (0)
0 v1 (A -FR0 N0)+v2 w+(A-FR0 NO) v1
+ wTv£0)T+wTV§O)w, (3.57a)
véo) = v§°)J+v§°)a3+wTv§°)a3 , (3.575)
véo) = egv§°)a3 , (3.57c)
where
_ T T
R0 - R+M M+G P3(O)G ,
_ T T
DO - s P3(0)G+DZM ,
N = P (O)F+P (0)G+CTP (O)G+DTM
o 1 2 3 1 ,
= _ -1 T
J 8 FR DO ,
. _ _ -1 T
w - c GR 0 .
. . . . (0) _
Stabithy of a3 1mpl1es V3 - 0.

Evaluating Véo) from (3.57b) and substituting into (3.57a) and using

(3.57c) yields

(0) T (0) -
V.l 011+a1V.I - 0 .

But a] is stable which implies V(O) O.

For higher order terms up to (ZN-l) we prove, by induction,

that V§1) = 0 for j = 1,2,3 and i = l,...,2N-l .

Let A-FR61NS = a, then partitioning (3.55) yields

0 = V18
+ 0
- .T
. V3 - a3

84

T T T

+ w' v + w'

+ vzw' + B' 2

v1

(GZN)

. . .T . T . .T' .
+ V a + N V3a3 V1J + 8 V2a3

2 3 + ”'

+€E(3'

. .T T T T
V3a3 +'E(J V

l 2 3 2

I IT I I I IT
v3w +-6(81 vzw +3 V18 +w v

J' + J' V a' + as V J') + 0(62N)a

T
28

4') we”).

(3.58)

where B',W',J', and a5 are matrices analytical on E with their zero-

order terms

given by B,W,J, and a3, respectively.

Now assume that for l f i f K-l we have

and we prove
Note that we
Matching the

v(i) = 0

J 3 J =1’233’

that V§K) = 0, j = 1,2,3.

already proved that (3.59) holds for K = l.

th

K -order terms in equation (3.58) yields

0 = v(k)B + VéK)w + BTng) + 91(Vgi)) ,
VéK) = V(K)J + V900!3 + wTng)a3 + 92“,?) ,
where g], 92’ and 93 are functions of vgi) for
0 f i f K-l and j = 1.2.3.

(3.59)

(3.60a)

(3.60b)

(3.60c)

')

85

In view of (3.59), (3.60) reduces to

o = V(K)3 + véK)w + Bvag), (3.61a)
VéK) = V(K)J + VéK)a3 + NTng)a3 , (3 61b)
ng) = agng)a3 (3 61c)

But (3.61) is similar to (3.57) and we can repeat the same argument which

completes the proof of theorem 3.2.

3.5. Slow-Fast decompostion and composite control

System (3.21) is singular as a function ofe ; i.e., we can
observe order reduction and separation of time scales as e + 0. Blanken-I
Ship [1981] showed that for small 6 the variables can be decomposéinnto
slow and fast variables. Using Blankenship's time decompostions, slow
and fast subproblems are defined in a way similar to that of Chow and

Kokotovic [1976].

Slow Subproblem: The slow variables, evolving in slow time scale

 

en, satisfy the outer solution

X(€n+€)-X(en) =6 AX(6n) + eBZ(en) +5 FU(e-n) (3.62a)
Z(en) = CX(en) + SZ(gn) + GU(.»:-n) (3.62b)
Y(en) = D1X(en) + 02Z(en) + Mu(en). (3.62c)

Dividing (3.62a) by E and letting e + 0 yields

= A §(t) + B 2(t)+ F U(t) (3.63a)

0.0.
c-i-xl

86

Z(t) = c X(t) + S Z(t) + G U(t) (3.63b)

D1 x(t) + D2 Z(t) + M U(t), (3.63c)

‘<
A
('I"
V
II

Assuming that (I-S) is nonsingular, (3.63b) can be used to eliminate

2(t) from (3.63a) and (3.63c) resulting in the slow subsystem

dx
—-—-—S=
dt Asxs(t) + BSuS(t) (3 64a)
= \
ys(t) Csx(t, + Dsu (t), (3.64b)

where x = 2, y = y, u = U' and x (0) = x(O), and where

AS = A+B(I-S)-]C, BS = F+B(I-S)-]G,
_ -1 _ -1
Cs - D]+Dz(I-S) C, 05- M+02(I-S) G.
We define a slow performance index JS as
a = . ohm (t) + uTma u M) at (3 65)
s 0 s s s s ' ‘

where in obtaining (3.65) we used the limiting relation

lim 6 T U”T
5+0 n 0

III.

(€n)R U(en) = J: u (t)Ru(t)dt.

The slow problem defined by (3.64), (3.65) is a continuous-time regulator
problem identical to the slow problem of Chow and Kokotovic [1976].

Following their work we assume that

 

Condition a': the triple (AS,BS,CS) is stabilizable-detectable
in the continuous-time sense; i.e., every eigenvalue of AS which lies
in the closed right-half complex plane is controllable and observable.

Under Condition a', the optimal feedback control law of the slow problem

87

(3.64), (3.65) is given by

_ -1 T T A
us(t) — -RS [DSCS + BSPsle(t) = -F x (t), (3.66)

where RS = R+DIDS and PS is the unique positive semidefinite stabilizing

solution of the continuous-time algebraic Riccati equation

_ _ -1 T _ -1 T T
o - PS(AS BSRS oscs) + (AS BSRS oscs) ps

-1 T T -1 T
- PSBSRS BSPS + CS(I-DSRS os)cs, (3.67)

Fast Subproblem: Assume that the variables x(n), z(n), y(n)

 

and u(n) decompose as

X(n) = xf(n) + xs(t) ‘ (3 686)

Z(n) = zf(n) + 25(t) (3 68b)

y(n) _ yf(n) + ys(t), (3.68c)
and

U(n) = uf(n) + us(t). (3 68d)

Substituting (3.68) in (3.21), taking the limit 8 + 0 and using (3.63b)
and (3.63c) we get

zf(n+l) = Szf(n) + Guf(n), (3.69a)
yf(n) = Dzzf(n) + Muf(n). (3.69b)
We define a fast performance index Jf as

f :y;(n)yf(n) + u;(n)Ruf(n)1. (3.70)

C;
II
II P“! 8

n 0

88

The fast discrete-time regulator problem (3.69), (3.70) is a standard
problem and it is well-known [Kwakwenaak and Sivan, 1972] that if the
triple (S,G,DZ) is stabilizable-detectable (condition b), then the

optimal solution is given by

T T T

uf(n) = -[R+M M+G P T

- T
.61 [G PfS+M 0212f(n) g -Ffzf(n),(3.71)

where Pf is the unique positive semidefinite stabilizing solution of

the discrete-time algebraic Riccati equation

T T
DZ+S PfS-[G

T T T T

021 [R+M M+G P T T

[G P T

_ T
Pf - 02 P S+M S+M 021. (3.72)

f G]

f f

Inspection of (3.38) and (3.72), together with the uniqueness of the solution

of the Riccati equation, shows that
P3(0) = Pf . (3.73)

Motivated by the results of Chow and Kokotovic [1976] in the continuous-
time case it is natural to ask the question: Is there a similar relation
between P](0) and PS? The answer is yes, as it can be seen from the

following lemma.

Lemma 3.1. If (I-S) is nonsingular, then the matrices A, 8 and 6

appearing in (3.41) are given by

. _ _ -1 T

A - AS BSRS DSCS , (3.74)
‘“-1“T _ -1 T

BR - BSRS Bs , (3.75)
. _ T -1 T

Q - CS(I-DSRS os)cs. (3.76)

The lemma is proved in Appendix (3.4)

As a consequence of the lemma we have

89

P](0) = P . (3.77)

Also, condition (a) is equivalent to condition (a') which is independent
of P3(O). Therefore Theorems 3.1 and 3.2 hold under conditions (a')
and (b).

Composite Control: With the solutions of the slow and fast problems

in hand, a composite feedback control is formed as

=u +
uc s uf

-sts(t)-Ffzf(n).

Approximating zf(n) by z(n)-z(t), expressing Z(t) in terms of

xs(t) and us(t) and approximating xs(t) by x(n), we get

C
II

-st(n)-Ff[z(n)-(I-S)'T(C-GFS)X(")J

f(I-S)'T(C-GFS)1x(n)-Ffz(n). (3.78)

-[FS-F

The composite feedback control law (3.78) is near-optimal as established
in the following theorem.
Theorem 341:Under Conditions (a') and (b) the composite control

(3.78) is C(62) near-optimal in the sense that

_ 2
J-Jopt — 0(6 ) .

Proof: The main step in the proof is verifying that the feedback co-
efficients in (3.78) are 0(6) perturbations of the feedback coefficients
in the optimal control (3.29). To see this, note that from (3.29) we

have

90

F0 = [R + 0(e)T‘TTNT + 0(6), 0;

0 o + 0(6)] .

Using Householder Theorem we obtain

F0 = [R6T + 0(E)1£Ng + 0(6) . DB + 0(6)]
= [RaTNg + 0(6) , R6106 + 0(6)]
= [FT , Pg].
On the other hand
Ff = R610; .

So by comparing with (3.29) when it is partioned we get

0 -
F2 — Ff + 0(6),
Let
_ -1
F] - FS - Ff(I-S) (C-GFS)
= [I + F (1-5)'TG1F - F (1-5)’Tc
f s f '
Using
_ -1 T T
. FS - RS (DS(CS+ BSPS).
yields
_ -1 T -1 -1 T -1 T -1
F] - [1 + R0 00(1-5) Gle (DSTCS + BSPS) - R0 00(1-5) C.

Using (1) from Appendix 3.4 and defintion of H'1 we get

_ -1 -1 T T -1 T -T T -1
F1 - H RS (DSCS + BSPS) - R0 H H 00(1-5) c
_ -1 T T -1 T T -1 T -1 T
- R0 H BSPS + R0 H [05cs - (I+R0 00(1-5) G)
(GTPfS + MTDZ)(I-s)‘TCJ.

91

Inside the bracket is

GIGS - GT T( )‘To R‘TDT(I-S)‘Tc.

P 3(1-3)‘Tc-MTD (I-s)‘Tc-G 0 0 0

f 2 T‘5

Using (3.72) yields

GIGS-GT( )'T[(I-5)TPfs - P +DTD + sTP

f 2 2 f

l

I-S Sl(I-S)' c

- MT02(I-s)‘Tc

T

T T( 2

= 05cS + G I-S)‘TP c - GT(I-S)‘T D

"T T -l
f 02(1—5) C-M 02(I-S) C.

Substituting for DECS yields

F = R‘THTBTPS + R6THTEMTD

T
1 0 s (

I-S)‘TDTD + GT(

-T
+ G 2 1 1-5) P C].

l f

Using (2), and (5) of Appendix 3.4 we obtain

= R‘T (F + L G)TPS + (L16 + CTPfG + DTM)T

o T 2 T

l T T(

R6 [F PS + G L + L )T + GTP c + MTD 1.

1

Ps f

2 1

So

F

-l T
R O .

1 o N

By comparing with (3.29) we obtain

F0 = F + 0(6).

1 l

The rest of the proof is similar to the proof of Theorem 3.2.

92

3.6 An Iterative solution of Riccati Equation for Linear Quadratic
Singularly Perturbed Systems.

One of the main difficulties in dealing with optimal control of
infinite-time regulators of singularly perturbed systems is solving the
stiff Riccati equation arising in this class of systems. In this section
an efficient iterative technique for solving the stiff algebraic dif-
ference Riccati equation (3.30) is developed and it is shown that the
accuracy of 0(6T) can be obtained by performing only (i-l) iterations.
Also, since only the lower order systems are employed, the algorithm is
very efficient from the computational point of view.

Let Pi = P1(O) + 6E1 for i = 1,2,3 as in (3.43), where

Pi(0) indicates the zero-order terms and let

N1 = ATP1A + ATPZC + CTPgA,
N2 = ATP1B + ATst + CTPEB.
N3 = BTP]B + BTPZS + STPEB,
N4 = FTP1F + FTPZG + GTPgF.
N5 = ATP1F + CTPgF + ATPZG,
N6 = BTP1F + sTPgF + BTPZG,
and
Ne = E1F + E26 + CTE3G.

Note that

93

P‘T = (R + MTM+33Tpa)’T = {R + MTM + GTP (0)G + e(GT

3 3

and by Householder theorem (see Appendix 2.1) we obtain

P‘T = (R + HTM + nTPa)‘T = RaT + e K

where

_ T -1 T -1
K - -[R + E (G E36 + N4)] (G E36 + N4)R0 .

0

Dividing K into zero-order and non-zero-order terms we get

K=K +EK

O 1’
where
_ -1T T T TT -1
K0 — - R0 [G E36 + F P](0)F + F P2(0)G + G P2(0)F]R0 ,
and
_ T -l -l -1
K1 - -K(G E36 + N4 )R0 - RO N4eRO ,
where
N4 - N4(0)
N = .
4e E
Now subtracting (3.38) from (3.35) yields
_ T -1T T T
6E3 - 6(5 E35 + N3)-€LDOR0 (G E35 + N6) + (S E3G + N6)
-1T T.2 T ~-1T T
0 D0 + DOKD0 J-E [(S E3G + N6)R (G E35 + N6)
T T T T
+ DOK(G E35 +‘N6) + (S E3G + N6)KDO].

Eliminating 6 and factoring yields the discrete Lyapunov equation

-1
EG+N4)] ,

94

E3 - £133.33 = 03 + e «)3 (51,52,53e) (3.79)
where
03 = N3(0) - DORBTN;(O) - N6(0)R5T05 + 00R5TN;(0)R5T05T
Y3 = N3e ‘ DORGTNge ‘ NGeRGTDg ' D0K1Dg ' TTSTEaG + N6)§ T
(GTE3S + N2) + 06K(GTE3s + NE) + (STEBG + N6)KDg].
Note that

By the same way,subtracting (3.37) from (3.34) and (3.36) from (3.33),

respectively yields

52 - E18 - E25 -CTE3S + N0R6TGTE33 + NeRéTDg - NORaTGTE3GRBT03
= C2 + 6 ~72 (E],E2,E3,&), (3.80)
and
E1A + ATE1 + £20 + GTE; + CTE3C—N0R61N; - NeRaTNg + NORéTGTE3GR6TNg
= G1 + e y] (51,52,53ss), (3.81)
where
02 = N2(O) - N0 6TNT(0) - N5(0)R6Tog + N086T 4(0)R5Tog
Y2 = N2e - NORaTNge - NSenéTog - NOKIDg - (Ne + NS)P‘T(GTE35 + N2)
- N0K(GTE3S + N2) - (Ne + N5)k0$

95

_ _ -1 T -1 T _ -1 -1 T
c1 - N1(O) + NORO N5(O) + N5(O)RO N0 N080 N4(O)R0 NO ,
and
_ -1 T -1 T T ' ~-1 T
Y1 ' 'N1e T N0R0 N5e + N5eR0 N0 + N0K1N0 + (Ne + N5)R (Ne T ”5)
T T
+ N0K(Ne + N5) + (N6 + N5)KNO.
. _ -1 T _ -1 T
Letting w - C-GR0 N0 and J - B-FR0 D0 from (3.80) we get
E = (E 0 + wTE a + c + e )L'T (3 82)
2 1 3 3 2 Y2 3 ' '

Substituting (3.82) in (3.81) and using (3.79) yields the continuous

Lyapunov equation

tTT' -1 T T
E](A-FRONO + L2H) + (A-FR0 N0 + LZW) E1
_ ‘ -l T -T 2‘ T T -T _ T -l
— -(C2 + e{2)L3 W—W L3 (C2 + ~{2) -W L3 (E3 a3E3a)L3 N
+ C] + E Y]. (3.83)
But
-1 T = -1 T _ —l T
A-FRO N0 + L2H A + L2C - FR0 NO LZGR0 NO
- r‘ 'TT
- A + L2G - (F + L20)R0 NO ,
Using the proof of Theorme 3.3 we have
T _ T T T T
N0 - (F + L26) P1(0) + (L1G + C P3(O)G + D1 ) ,

mwiby recalling that 8 = F + LZG we obtain

96

-1 T _ ~ -1‘T ~ -1 T T
T T _ ~ - -1“T _
+ G P3(0)C + M 01) - A - BR0 B 1(O) - a].
Now,(3.83) reduces to
T _ -1 T -T T
E1°‘1 T alEl ’ '(Cz T EY2TL3 w ‘ w L3 (C2 T 6Y2)
T :T -1
‘ w L3 (C3 T EY3TL3 w T C1 T 671’
OY‘
E a + aTE = 3 + e (3 84)
1 1 1 1 Y ' °
where
_ -1 T -T T T -T -1
a - -C2L3 w-w L3 cz-w L3 C3L3 w + C1
- - -T -
Y T ‘72L3TW'WTL3TYE'WTL3 Y3L31w T Y1'

Equations (3.79), (3.82), and (3.84) have an interesting form since all
non-linear terms and cross-coupling terms are multiplied by a small
parameter E. This suggests that a successive approximation algorithm

can be efficient for their solution.

Let us propose the following algorithm:

E§T+T)a] + aTETTTT) = 4 + & y(i), (3.85a)
(i+1) T (i+1) _ (1)

E3 - o353 a3 - C3 + 6 Y3 9 (3.85b)
(i+1) (i+1) T (i+1)” -1 -1 ‘(i) -1

E2 E1 L2 + w E3 a3L3 + C2L3 + 6 {2 L3 .

(3.85c)

97

6(0) = 6(0) = 6(0) = 0 . (3.86)

Theorem 3.4:

 

The algorithm (3.85), (3.86) converges to the exact solution E with

the rate of convergence of 0(6), i.e.,
llE-ETTTTTH = 0(6)llE-ETT)ll . i = 0.1.2.... (3.87)
or equivalently

HE-ETTTH = 0(eT), i = 1,2,... . (3.88)

Proof:

Let us represent equation (3.85) by

Qi(E1,E2,E3,€) = 0 , i = 1,2,3. .- (3.89)

As a starting point we need the existence of bounded solutions of

E E2, and E3 in the neighborhood of E = 0. This is established by

"’
applying the implicit function theorem to show that the Frechet derivatives

of Q], 02, and 03 with respect to E], E2, and E3 at E = O are

invertible. i.e.,

d 01(E,€) is invertible for i = 1,2,3,

where

l . (3.90)

 

But

So

a] is
exists

(3.91).

i
a3 S
exists

(3.92).

L3 15

(3.93).

given

98

= T
Q1 E10;1 + a1E] - ¢ - eY(E],E2,EB,E).

. 1 T
do = 11m —-L(E + A6E )a + a (E + A8E ) - 4
1 6:0 A+O A 1 1 1 1 1 1
E=O
- GT(E1 + A6E1,E2 4 A6E2, E3 + A6E3) + 4
T
‘ E1°‘1 ‘ “151
= (6E])a] + 6T(3E1). (3.91)

stable matrix in the continuous-time sense, so given do], there

a unique E1 satisfying the continuous-time Lyapanov equation

By the same way we obtain

dQ3 = 6E - 6T

3 3(6E

3).,3. (3.92)

stable matrix in the discrete-time sence, so given dQ3, there

a unique 6E3 satisfying the discrete-time Lyapanov equation

For 02, using the same approach, we get

602 = 5E L - (6E )J-wT(5E

2 3 (3.93)

1 3Ta3°
invertible, so given sz, there is a unique 6E2 satisfying
The existence and uniqueness of 6E], 6E2 and 6E3 for any

60], 602 and 603 establishes the invertibility 0f the Frechet

derivative.

99

.-

For i = 0, subtracting (3.85b) from (3.79) yields

(E3'E§])) ' a;(E3‘E§]))a3 = EEY3(E]9E29E336) 'Y3(0503096)]'

By stability of a3 and existence of bounded solutions of E1.E2, and

E3 we obtain

(TTH = 0(6). (3.94)

“Es'E3

Similarly, subtracting (3.85a) from (3.84) yields

fi-fi”n,+fluré”)=éi(afl.

1

By stability of a] and existence of bounded solutions Ei’ i = 1,2,3,

we get
HE1-E§TTH = 0(6). (3.95)

and by subtracting (3.85c) from (3.82) and using the same approach we

have
11E -E(T)ll = 0(6) (3 96)
2 2 ° °

For the next iteration step we have

and subtracting the above from (3.79) yields

(2)) T E(2)

(E -53 + 63(53- 3 >63 = ELY3(E,e)-v3(E(T).E)l-

3

The term [Y3(E,E)-v3(E(T),E)l satisfies a Lipschitz condition uniformly
in E for E sufficiently small. Hence its order of magnitude is the

same as (Ej-E§T)), j = 1,2,3. So from (3.94) we obtain

100

uEs-E§2)H = 6 - 0(6) = 0(92). (3.97)
Repeating the same arguments we can conclude that

1153-4911 = ow).
and by analogy

o<eT) .

1151-59)“

MHz-séTTH = 0(6‘),
or

(IE-Emil = 0(5),

which proves the theorem.

Clearly, since we are doing iterations of the same system of
equations, this algorithm is efficient not only for obtaining required
accuracy, but for finding an exact solution.

It should be pointed out that another way to overcome the stiff-
ness problem in solving the Riccati equation (3.30) is by using a power
series expansion with respect to small parameter E and matching the
corresponding coefficients. This provides us with a family of well-
defined problems for which standard techniques are applicable. However,
if we are interested in high order of accuracy the amount of required
computations can be substantial, even though we are faced with solving
low order problems.

The steps in our alogrithm are as follow:

lOl

Algorithm

Step 1.

Step 2.

Step 3.

Step 4.

Step 5.

Step 6.

Step 7.

Riccati

Solve (3.36)-(3.38) i.e. find P](O), P2(O) and P3(0).

c c c and A.

Find 1: 29 3

a a
1’ 3’
Let i = 0, £§°T = 0 for j = 1,2,3.
(1) (i) .
s P. = P. + e . , = , , .
et J J(O) EJ J l 2 3

Find YTT). réT). Y§TT. and r

(i).
Solve (3.85).

Check the required accuracy. If it is not satisfied, set

i = i + l and go to step 4, otherwise stop.

Thus, the overall solution of (m1 + m2) dimensional algebraic

equation (3.30) can be found by solving two lower-order Riccati

and two lower-order Lyapanov equations which can bring considerable

saving in computations and high rate of convergence.

3.7. Numerical example

 

In this section by means of a simple example we illustrate the

points in Theorem 3.2 and the iterative technique mentioned earlier.

102

Consider the difference equations

x(K+1) (l-ZE)x(K) + E z(K) + l.5€ u(K)

-.7x(K) + .452(K) + .8 u(k)

z(K+1)
y(K) = .6x(K) + .752(K) + .4 u(K),
with the performance index
min J = e 2 [y
Slow subsystem:

dxS
T755 ‘ -3.272727xs(t) + 2.954545

ys(t) = -.3545xs(t) + 1.490909 us(t),

with the performance index
_ 2 2
J5 — (Z [ys(t) + uS(t)]dt .

Fast subsystem:

zf(K+1) = .452f(K) + .8 uf(K)

yf(K) = .752f(K) + .4 u K)

f<

with the performance index

00

2 2
a, = K20 [yf(K) + uf(K)1.

(3.98a)

(3.98b)

(3.98c)

(3.99)

(3.100a)

(3.lOOb)

(2.101a)

(3.101b)

(3.102)

Using "LAS package" [Bingulac et al., 1982] the programs for solving the

example are written and-run on Prime Computer at Michigan State University.

103

Solving for slow and fast Riccati equations yields

Pf = .508377,
PS = .006971,
and
P2(0) = L1 + PSL2 = .380134.

Solution of the approximate feedback I is found to be
I = (.191039, .325184).

In the next table values of V = P-P' and [J-IOI, for different values

of E, are given.

 

 

 

 

 

 

Table 3.1
Fl-F? F2-Fg Pi-P] Pé-Pz Pé-P3 T

e = .1 .062091 .019034 -.020479 .001304 -.OOO626 1

e = .05 .035648 .009489 -.Oll813 .000323 -.000148

e = .025 .022486 .004739 -.008277 .000080 -.000036 i

 

As we see the numerical values agree with the theoretical results.

Hi
4.
II
C)
25

i = 1,2 ,
and

Pi-Pl = 0(e2) , i = 1,2,3

and consequently

On the next table we give the numerical results using the iterative

J-J

104

opt

= 0(e2).

 

 

 

 

technique for 6 equal to .1, .05, and .025, respectively. Also the
exact solution is evaluated for comparison.
Table 3.2
l e = .025 e = .05 e = l
Ttgiation P1 P2 P3 P1 P2 P3 P1 P2 P3
1 T .016971 .375847 .510311 .026655 .371657 .512107 .045149 .363768 .51576
2 .0172§l_.37524§_.510§§§_ .027§§Z_.37124§_.512§gfl_.O49§§QJ.36229§_.516§§3
3 .Ol7291_.375732,.51035§_ .0279§Z_.3712§§_.5123§9..050239..362919 .516443
4 .017291 .375739 .510355 .02794g_.371254_.51234Q_.050319_.3620§2_.5164§§_
5 T " " " .027942 .371254 .512340 .050319_.36204g_.516454
6 " " " ” " T .050329..36204§_ "
7 .. .. u n u .. .. u u
Exact
Solution .017291 .375739 .510355 .027943 .37l263 .512360 .050320 .362048 .516455

 

 

 

By investigating the results in Table 3.2 we observe that the iterative

technique has a convergence rate of E, or even less, at each iteration

and tends to the exact solution.

It is interesting to note that this

algorithm gives better result as the value of 6 decreases while the

standard methods for solving discrete-time Riccati equations exhibit

105

worse results for smaller values of 6. In Table (3.3) the number of
iterations to obtain convergence in our iterative technique is given for

different values of 6.

No. of iterations for Convergence

 

6 up to 6th digit after the decimal point.
.025 3
.05 4
.1 6

 

It should be noted that, due to the round-off errors of the

computer system, there are some small errors on the 5th or 6th digit

after the decimal point. (As in the case of values of Pi’ i = 1,2,3,

for E = .05).

APPENDIX 3.1

Consider the sampled-data control of the regulator problem
x(t) = Ax(t) + B u(t) , X(t
y(t) = C X(t).

with the performance index

J = J (yTy + uTRu)dt

T T T

= J [x C Cx + u Ruldt , R > 0.
t

Let u(t) be a piecewise constant function of time, i.e.,

By sampling the above system with period T, see Section 2.4, we obtain

Xi+1 = TTTTXi + F(T)ui , X0 = X(to)
y.i = C Xi + D U, ,
Where
4(T) = eAT ,
and

106

107

with the performance index

{[P(T)Xi + I‘(T)u1.1T

J = T. CTC[P(T)X1. + I‘(T)u1.] + u-{Rui}.

1 0

So, J will take the form

J = .2 [ngXi + ZXTMui + u: 8 ui] ,
1-0
where
T
T
0 = j 4 (T)cTci(T).
0
T
M = ] ¢T(T)cTcr(T).
0
and
~T TT
R = j [R + r (T)C Cr(T)]dt .
0

We note the appearance of the cross product matrix M when a continuous-

time regulator without a cross product is sampled.

APPENDIX 3.2

The functions f1, f2 and f3 in (3.33)-(3.35) are defined by

' _ T T T T T
f".I - PTA+PZC+A P1 + C P2 + C P3C+DTD1

TP A+ATP 0+cTPT

+ SEA 1 2

A]

F+P G+cTP G+0T TP F+ATP G+CTPTF)]x

1 2 3 1 1 2 2

[R+MTM+GTP3G+e(FTPTF+FTPZG+GTP; )l'1 x

+GTPT+GTP c+MT0 +e(FTP A+GTP;

2 3 1 1
T T T
1B+A PZS+C P2

- [P M+e(A

[FTP1 A+FTP2C)] ,

f = P B+P s+cTP s+0T

T
2 1 2 3 1 TETA P

D B]

2

T T
TF+P2G+C P3G+D1

M+3(ATP F+ATP G+cTPTF)Tx

‘ TP 1 2 2

T T

[R+M M+G P T T T

T -1
3G+e(F PTF+F PZG+G PZF)] X

T T02+e(FTPT8+GTPT8+FTP 5)] ,

P S+M 2 2

[G 3

and

= sTP s+0T0 +eLBTP 8+8TP s+sT

f3 3 2 2 1 2 P

B]

G+DTM+e(BTP F+8TP G+sTP

T
' TS P3 2 1 2

F)l x

—i na-i na—4

N—l

T

[R+M M+GTP T T

G+e(F P F+F

1 P G+G P )J'T x

3 2

T T T
[G P3S+M 02+e(F P

T T T
1B+G P28+F P25] .

108

APPENDIX 3.3

Wish to prove that

I-M(R + MT )‘TMT = (I + Ma‘TMT)'T ,

Note that

T )-1 T 1 T

[I-M(R + M M ][I + MR’ M 1

1 T

= I + MR' M - M(R + MT )‘T T

M MR" M

TMT - M(R + MTM)‘TMT - M(R + M

I + M R‘ T )‘T(R + MTM-R)R‘TMT

1 T

M - M(R + M T

T )-lMT _ MR- T )-1 T

= I + M R’ M + M(R + M M

By the same way

[I + MA'TMTTTI - M(R + MT )‘TMTT = 1 ,

Q.E.D.

109

APPENDIX 3.4

Proof of the Lemma: Define

 

_ -l T -l T -l
H - I-R0 00(I-S+GR0 00) G,
then
-1 _ -l T -l
H - I+R0 DO(I-S) G.

H and H'1 are well defined if (I-S) is nonsingular. Consider

-T -1 T(I-s)'TD R'10T(1-5)'TG.

-1
(I-S) 6+6 0 0 0

= R +GT(I-s)'T0 +0T

H 0 0 0

ROH

Using equaiton (3.38) (or equivalently, equation (3.72), to eliminate

-1 T
DORO 00 we get

)-T

I

23

I
I

_ T T -1
RO+G (I-S 00+00(1-5) G

+sTP

+

T -T T -1
G (1-5) [0202 fs-PT] (I-S) 6.

Substituting for R0 and D0 using their defining expressions, given

after (3.57c), we have

-T -1 _ T T

R+M M+G T T

T —T T T -1
PTG+G (I-S) (s PfG+D2M)+(M 02+G PfS)(I-S) G

I

m

I
l

I-S)‘T0T0 (I-S)‘TG+GT(i-S)‘

T T T -1
+ G ( 2 2 (S PfS-Pf)(I-S) G

I-S)'T0TM+MT0

I-S) 0 0 (I-S)‘TG

T T(
2 2

R+M M+G I-S)‘TG+GT(

110

+ GT(I-S)‘T(I-S)TPT(1-5)(1-3)‘TG+GT(I-s)'TsTPf(I-s)(I-S)'TG+GT(I-S)’T
(I-s)TPTS(I-5)'T + GT(I-S)‘T(sTPTs-Pf)(I-S)‘TG
_ T T -T T T T T -l
- R+DSDS+G (I-S) [(I-S) Pf(I-S)+S Pf(I-S)+(I-S) PTS + S PfS-Pf](I-S) G.
Inside the bracket is zero, so we obtain
-T -l _ T
H ROH - R+DSDS .
Hence
-T -l _
H ROH - Rs (1)
Consider next
8 = F+L G = F+(8-FR'T0T)(I-S+GR‘T0T)'TG
2 0 0 0 0
_ -l T -1 T -l -1 T _ -l
- F+(B-FR ODO)[I-(I-S+GRO 00) GRO 001(1 S) G
_ -l -l T -1 -1 T -1
- F+B(I-S) G-FR0 DO(I-S) G—LZGRO 00(1-5) G
= B -(F+L G)(H'T-1) = B -8(H'T-i)
s 2 5 °
Hence
B = BSH (2)
Using (1) and (2) we get

111

112

. ~-1 “T _ -1 T T =
B R B - BSHRO H 85 BS AS BS ,

which proves (3.75).

Consider now

A = A+L c-é R’TLGTLT+GTPTC+MTDTJ . (3)

2 O

The second term LZC is given by

L2G = (B-FR6TDB)(I-S+GRBTDS)'TC
= (8-FR6TDT)(I-s+enaTog)’T(I-s+GR6Tog-GR6T08)(I-S)’Tc
= (8-FR5TDT)TI-(I-s+GRBTog)'TGR6TogJ(I-S)‘Tc
= B(I-S)‘Tc-(FR5Tog+LZGR6T03)(I-S)‘Tc
= 8(I-S)'Tc-8 R6Tog(I-S)‘Tc , (4)
and the third term of (3) is given by
LTG+cTPfG+0TM = [DTDZ+CTPfS-(CTPfG+DTM)R6TDg](I-S+GRBTDg)-TG
+ GTPTG+0TM.
Using the matrix identity
(I-s+GR5Tog)'TG = (I-S)’TGTI-RéTog(I-s+686Tog)'TGJ ,

and the definition of H, we get

T T _ T T -1 T T
LTG+c PTG+0TM - (0102+c PfS)(I-S) GH+(C PfG+D TM)H
= 0T02(1-3)'TGH+0TMH+CTPT(I-S)’TGH. (5)

113

Now (2), (4) and (5) yield

-1 T -T T

-T T T
0 H [H 00 0

(I-S)'TC+GT(I-S) D D + M

T -T
2 1 1 + G (I-S) PfC].

A = AS-BSHR
Substituting for H'T and D0 and using (1) yield

. _ -1 T -T -1 T -1 T -
A - AS-BSRS [G (I-S) 0080 00(1-5) C+M 02(I-S)

Tc+GTPTS(I-5)'Tc

+ GT(I-S)'T0T0 +MT

-T
2 1 T

P

0T+GT(I-s C].

f
Using (3.72) to eliminate Pf, we get

-T T

. 1 T'
(I-S) 02

A AS-BSRS [G

02(1-5)'Tc+MT0 (I-S)’Tc+MT0

2 l

T(I-$)'T0ToTJ

+6 2

1 T
Dscs ’

As'BsRs
which proves (3.74)

Finally.for proving (3.76) we write L1 as

LT = LDT02+CTPTS-(GTPTG+0TM)R6TDTTLI-s+GR6TogJ‘T U-S+GRaTDg
- GR6T031(1-5)‘T
= [DTDZ+CTPTS-(CTPTG+DTM)R6TDTJTI-(I-s+GR6Tog)’TGA6TogJ(I-S)'T
= (0T02+CTPTSJ(I-S)‘T-(LTG+cTPTG+0TM)R6Tog(I-S)‘T.
Now we get
0 = DTDI+DTDZ(I-Sl-TC+CT(I-S)'TD;D]+CTPfS(I-S)'TC+CT(I-S)-TSTPfC

T T T -1 T -1 - T -T —1
+ c PTc-(LTG+c PTG+0TM)A0 00(1-5) c-c (I-S) DORO (LTG

114

+ GTP T 61 6+0 P G+D M

T T )T
l f 1 '

P G+DTM)R

T T
G+DTH) -(L G+C f 1

1 (L

f
Using (5) and simple algebraic manipulations we get

T T -T T
cScS-c (

-1 T
1-5) 0202(1-5) C+C (

0 I-S)’TTI4S)TPTs+sTPT(I-S)
T02+cTPT5)(I-5)‘TG+DT

1

1 T T

H D0

+

(I-S)TPf(I-S)J(I-S)'TC-[(D MIR;

-1 T -T -1 -1 T T -1 T T
(1-5) 0-0 (I-S) 00H Rs [(0102+c PfS)(I-S) G+0TMJ

T T

[(0T02+c P T

T
l

1 T T

0 +0 P S)(I-S)'TG+DTM1 .

S)(I-S)' 2 T

G+D

- T T
MlRS [(01

f
Using (3.72) we obtain

I-S)'TD;DZ(I-S)'TC+C

-T T

I-S) [02 R‘TDT (I-S)‘Tc

T
( 0 01

T<

O>
II

T
CSCS-C DZ—D0
T

1

(I-S)‘TG+cTPTS(I-5)‘T 0 H'T ;T

T
G+D M+G 0

D 1

[0 T(I-S)'T JR

2

[DTDZ(1-3)‘1e+chfs(I-s)'TG+0TM+cT(1-5)'T00H‘T1T

-1 -1 -T T

T -T 1
c (I-S) 00H RS H 00

+

(I-S)- C,

After eliminating similar terms, the manipulations are just repetitions of

what was done to prove (3.74) and (3.75). First the term (L G+CTPfG+DTM)

1 1
is substituted, using (5). Second, (3.72) is used to eliminate Pf. The
remaining expression, which is independent of PT, can be easily shown to

T -1 T
be 05(1-058S os)cs.

CHAPTER 4
COMPOSITE CONTROL AND MULTIRATE MEASUREMENT

4.1. Introduction

 

In singular perturbation theory' the use of feedback control
input which is obtained by composing the control inputs of slow and fast
subsystems is considered frequently. By employing such feedback controls
on the full system a close approximation of the design objective has
been acheived. [Chow and Kokotovic, 1976], [Phillips, 1980].

In chapter three an infinite-time regulator problem for difference
equations was studied and the role of composite feedback control in
achieving an C(62) near-optimality was discussed. In this chapter we
extend on our discussion on the role of composite feedback control in
the context of stabilization. Also, the problem of stabilization in view
of multirate measurements of the state variables using a composite feed-
back control is investigated and different designs are proposed.

In section 4.2 we consider a discrete linear-time-invariant
system and by USing the stabilizing feedback controls of slow and fast
subsystems, which evolve in fast time-scale, we introduce a composite
feedback control for stabilizing the full system and we show the close-
ness of trajectories.

Section 4.3 deals with the same problem as section 4.2 but the
composite feedback control employs multirate measurements. By letting

{0,6,2€,...,N€} to be a mesh on [0,-é], the fast states are measured

115

116

f0r EVERY us while the slow states are measured periodically

with a period of %-. To compute the slow states for any n, we solve
their dynamic equations in terms of their meaSurements at the beginning
of the slow measurement periods.

This way of forming the composite control is quite natural be-
cause one, usually, expects to acquire the measurements of slow states
at slow-time intervals, while for fast states the measurements are ob-
tained for each mesh point n.

A parallel design procedure for slow and fast subsystems is
introduced and a composite feedback control is formed, using these sub-
systems. By applying this composite feedback control to the full system,
the aysmptotic stability and closeness of trajectories is shown. Also,

a numerical example to illustrate the claims is given at the end of this
section.

T Finally, in Section 4.4 a sequential design is studied where a
pre-conditioning feeback gain is designed first. The role of this gain
is to stabilize the fast modes and allocates the corresponding eigenvalues,
appr0priately. Based on this pre-conditioning gain the slow subsystem
is designed. A composite feedback control is formed and a similar in-

vestigation as in Section 4.3 is performed.

4.2. A Stabilizing Composite Control with State Measurements in Fast

 

Time Scale
In this section we discuss the application of composite feed-
back control in the context of stabilizability and closeness of trajectories

for the case that slow and fast subsystems evolve in the fast time-scale (n)

117

and the measurements of states are available at all values of n. Con-
sider the linear time-invariant discrete-time singularly perturbed system

as in (2.86)

x](n+l) [IT+€A1T(€)]XT(n) + EATZ(E)x2(n) + 6B1(€)u(n) (4.1a)

x2(n+l) A21(€)x1(n) + A22(€)x2(n) + Bz(E)u(n). (4.lb)

The initial values are given and all the matrices are analytic functions
‘of E and (12-A22) 1s nonSTngular.

We assume that the control input decompses as
.u(n) = uT(n) + u2(n), n = 0,l,2,...
where u2(n) is exponetially stable i.e.,
Iu2(n)| f Kan , a < l.

The procedure for finding the slow and fast subproblems is
similar to the "approximation result" discussed in Section 2.5. So

in this section we briefly reintroduce them.

Slow Subsystem

 

Assuming x2(n) has reached its steady state (u2(n) =- )and

repeating the same steps as in Section 2.5, we arrive at
x](n+1) = (I+EAO)xT(n) + eBOuT(n), xT(0) = xT(0) (4.2)

where

118

I -A )‘TA

A T A 12T 2 22

+ A

11 21

B B + A

-1
0 1 12(T2'A22) 32'

Note again that matrices evaluated at E = O are denoted by deleting the
argument 6, This will be used through the text. Also, bars indicate
the steady state case. Now letting xs(n) = xT(n) and us(n) = GT(n)

we define the slow subsystem to be

xs(n+1) = (IT+€A0)xS(n) + EBOuS(n), xs(0) = X1TOT' (4.3)

Condition a.

 

Suppose that the state feedback control law for us(n) is

designed as

us(n) = FSxS(n) (4.4)

where Fs is chosen such that the Re A(A0 + BOFS) < O or equivalently

the closed-loop system

xs(n+l) = [I+E(AO+BOFS)]xs(n) (4.5)

is asymptotically stable in the discrete-time sense and meets some de-

sign objectives as pole-placement, linear quadratic, etc.

Fast subsystem

 

Following the same procedure as in Section 2.5 and letting

xf(n) = x2(n) - §2(n) and uf(n) = u2(n) and

119

~ -1 ~ ._
x2(n) = (I-Azz) [A21x1(n) + B2u1(n)] . (4.6)
The fast subsystem is defined to be

~

xf(n+1) = Azzxf(n) + Bzuf(n),xf(0) = x2(0) - x2(0) . (4.7)

Condition b.

 

Suppose that the feedback control law uf(n) is designed as
uf(n) = fof(n) (4.8)
where Ff is chosen such that the closed-loop system

xf(n+1) = [A +B F ]x (n) (4.9)

22 2 f f

is asymptotically stable and meets some design objectives. Again the

design method is not crucial.

Composite Control
With the solutions of slow and fast problems,a composite feed-

back control is formed as

uc(n) = us(n) + uf(n) = sts(n) + fof(n). (4.10)

Substituting for xf(n) = x2(n) - 22(n), using (4.6) and (4.4), and
approximating xS(n) by xT(n) yields

1

”C(n) " [F5 ' Ff(12-A22)- (A2] + BZFS)TXT(n) + FfX2(n) (4°11)

III)

lelTn) + F2x2(n),

120

—‘

Applying the composite feedback control (4.11) to the full svstem (4.1)

yields
xT(n+1) = [IT+6ATT)E)]xT(n) + EAT2(E)x2(n) (4.12a)
x2(n+1) = Aé1(€)xT(n) + Aé2(€)x2(n) (4.12b)
where
ATj(€) = ATj(6) + 8T(6)Fj , i,j = 1,2. (4.13)

Using a decoupling transformation similar to (2.25a), i.e.,

c 1
IT-EM](6)L1(E) ‘€M](e)

n = x (4.14)
LT(€) I

 

 

where the matrices L1 and M1 satisfy

0 = 421(6) + L1161-422191L16) + EL](€)£71'H(El-412(€)L1(€)1
(4.15)
0 = AT2(€) + M1(e)-M1(E)Aé2(€) + GTATTe)-AT2(€)LT(€)J M1
" EM](E)L](6)TT12(E) (4016)

In fact L1TE) and MlTE) can be approximated by

_ —- -1—- _ -1
L](E) — - (12-422) AZT + 0(6) - -[12-A22-B2F21 [A21 + BZFTJ + 0(6)
(4.17)
_. ._ -1 _ _ -1
MT(6) - - A1ZTTZ'A22) + 0(6) - -[AT2 + B1F2TT12'A22 BZFZJ + 0(6).

(4.18)

121

The system (4.12) becomes

_ - - - -1—- 2
nT(n+l) - [II + €(ATT+AT2(I-A22) A21) + 0(6 )lnT(n) (4.l9a)

n](n+1) = [£22 + 0(6)]n2(n) T (4.19b)

where we have used Householder Theorem to show 0(6) approximation
of matrices.

But (see Appendix 4.l)
=A +BF. (4.20)

Now the asymptotic stability of (4.19) and consequently (4.12) follows
by using

Re A(A + BOFS) < 0

0

[A(A + BzFf)l < 1

22

and Theorem 2.5.1.
By comparing (4.5) and (4.l9a) and using (4.20) and (4.14) it

is obvious that
xT(n) = xs(n) + 0(6). (4.21)

Similarly, comparing (4.9) and (4.19b) and using (4.14) it follows that

x2(n) xf(n) - L] x (n) + 0(€)

S

xf(n) + (IZ-Azz-BZFZ)‘T(A2T+82FT)xs(n) + 0(6).
(4.22)

122

But

-1
(Tz'Azz‘Bze) (A21T32F1)

= (I -A -B F )‘TTA

2 22 2 f TB T “B F

21 2 s 2 iTT 2 A22) T
-1

A2 T2+BFS)1

= (I -A F)'T(Iz-A

2 22 B2 Ff A

A22'32TfTTTz‘A22) 21

+

(I -A -B F )'T(T -A ) T8

2 22 2 f 2 22B MfTTTz 22 F

2 s

(I Z'AZZTTTAH1TBZF ).

Hence

x2(n) = xf(n) + (I2-A22)'T(A21+BZFS)xs(n) + 0(6) (4.23)

which agrees with the intuitive decomposion of x2 as the sum of x2
and Y2, where Y2 is given by (4.6).

Based on our discussions above we conclude the following theorem.

Theorem 4.2.1

 

Under the conditions a and b, and for sufficiently small 6,
the application of the composite control (4.11) to the system (4.1)
results inan asymptotically stable closed-loop system. Moreover, the

solution of (4.1) can be approximated by

x (n) + 0(5)

S

X
_4
A
3
v
II

x200 = xf<n1 + (I-A221‘1182F +Am>x (n > + 0(6)

where xf and xS are solutions of the fast subsystem (4.9) and the

slow subsystem (4.5), respectively.

123

4.3. Multirate stabilization with slow state measurements in slow

time-scale.

 

The practical need for multirate measurement is a basic con-
sequence of the finite computing capabilities of onboard digital computers
and the common goal of reducing the operating cost. Functions associated
with fast modes typically demand measurements an order of magnitude higher
than the rate which is necessary for suitable control of slow modes of
the system. Faced with widely varying measurement requirements among the
dynamic modes of the system, a multirate feedback control structure is
the solution to computational and cost limitations.

Synthesizing a multirate control system to meet desired objectives
has been a difficult task. As an example, the problem of multirate
sampled-data control of optimal regulators for singularly perturbed systems
has been an open subject.

In this section we investigate the application of a multirate
stabilizing composite feedback control on system (4.1) when the measure-
ments of the slow states are available only at slow time-scale (E),

K = 0,1,2,..., and the control input consists of slow and fast parts.

A parallel design procedure for designing control inputs of slow and

fast subsystems is introduced. Also, an overall control input which is
composed of control inputs of slow and fast subsystems is evaluated which
results in stabilizing control for the full system.

Again, consider system (4.1). We assume that the control input

decomposes as

u(n) = u](n) + u2(n), n = 0,1,2,...

124

where u](n) is constant for -E f n < 521 and u2(n) is exponentially

stable, i.e.,
|u2(n)| 5 Kan , a < 1

Slow subystem
Assume that x2(n) has reached its steady state (u2(n) z-0).
Repeating the same steps as in Section 2.5, we arrive at (4.2) which is

§T(n+l) =(IT+€A0);T(n) + EBOUT(n), 21(0) = x1(0)

and A0,B0 are given as before and UT(n) = uT(n). Since GT(n) is

constant over the cycle é-f n < K21 we can express 'x1(5zl) in terms

of §T(K/€) and GT(K/é).

 

K+1 -1

K+1 .
1 ’1E— -—- -1-J
~ K+1 _ -— ~ K e - K
X] (T) " (I]+&A0)e X] (E) + E jEK/E (11+EA0) BOUT (E) -
Letting i = 5E1--1-i we obtain
xT(-§¢) = (IT+ A0) xT(K/6) + e 1&0 (T1T A0) BouT(K/ ).(4.24)
Now let
u (K) = U (5) (4 25)
s 1 6 °
xS(K) = §T(k/E) (4.26)
A
AS = e 0 (4.27)

and

125

l AO(1-t)
BS = TO e dtB0 (4.28)
We define the slow subsystem to be
xS(K+l) = AS xS(K) + BS uS(K), xs(0) = X1TOT' (4.29)

Suppose that the state feedback control law for uS(K) is designed as
u (K) = FS x (K) (4.30)
where FS is chosen such that the closed-loop system

xS(K+l) = (AS+BSFS)XS(K) (4.31)

is asymptotically stable and meets some design objective like pole-

placement, linear quadratic, etc.

Fast subsystem

 

Following the same procedure as previous section we define the

fast subsystem to be

xf(n+l) = A22xf(n) + 820T(n), xf(§) = x2(§) - ;2(§) (4.32)
where

~ -1~ —.

x2(n) = (12-A22) [A21x1(n) + BzuT(n)J. (4.33)

From (4.32) and (4.33) the initial conditions for fast subsystem
are

xf(€) = x2(g) - (iZ-Azz)’TTA2TxT(§) + B2u5(K)] (4.34)

126

where we have used (4.25)

From (4.34) we obtain

xf(0) = x2(0) - (IZ-Azz)‘TTA2TxT(o) + B2u5(0)]. (4.35)

Now, let us rewrite the equation for x2. Applying (4.33) during the
(k-l)th cycle we get

 

~ -1r ~ K-1 .5
Thus
K _ -1 .5

Now, the.initial condition of equation (4.34) reduces to

xT(5) = (I -A )‘TB

6 2 22 (K-l) - u (K)] , K > 0. (4.35)

2Tus 5
Again, assume that condition 'bk as in section 4.2, is satisfied

so we have the asymptotically stable closed-loop system

xf(n+l) = (A + B Ff)xf(n) (4.37a)

22 2

with

xf(0) = x2(0) - (IZ-AZZ)‘TTAZTXT(O) + 9205(0)] (4.37b)

K _ -l
xf(g) - (IZ-AZZ) 82[uS(K-1) - uS(K)] , K > 0. (4.37c)
We notice that the initial conditons for the fast subsystem depend on the

slow control. This means that any abrupt change in the slow control will

excite the fast modes and causes fast transients for a short period.

127

This observation is particularly important in our scheme since for every
%- time-intervals there is an abrupt change in the slow control so that
the fast subsystem has to be resolved at the beginning of each cycle of

the slow control.

Composite Control:

 

With the solutions of slow and fast subsystems in hand, a composite

feedback control is formed as

uC(n) = us( n) + uf(n) = FSxS(K) + fof(n), g-f n < fg-u(4t38)

So we have

u (n)

c F x (K) + Ff[x2(n) - ;2(n)l

S S

_1 _—
FSXS(K) + FfX2(n n) ' Ff(I2-A22) [A21X~1(n n) + BZU-‘(n)] °

(DIX

We approximate uT (n) by F ”X1T

polating (4.2) for '%<:n 5 KET , i.e.

) and also approximate §T(n) by inter-

 

.E
"‘e

§T(n) = [(ITieAO) +6 :2 (I T+eAO)" T JBOFS TXTT‘eK’T- (4.39)

ml7< —'

K . ~ _ .5
Note,for n -é we define xT(n) — XTTET

Thus we have the following form for the composite control law

K+1
E

 

nfl7<

u (n) = E(n)xT(Té-) + fo2(n) ,

c (4.40)

<n<

where

T T 6:]. (4.41)
FS- FT(I 2 -A22) TA21V (n ) + BZFS], K/e < n < .E

i
‘1 _
Mk -TF 12 A22) (A2T+B2Fs) , n - K/e

128

and V(n) satisfies

V(n+l) = (I +eA )V(n) + eBOFS, veg) = I (4.42)

l 0 l

The use of the composite feedback control law (4.42) is justified

by the following theorems.

Theorem 4.3.l: Stability Result

 

If the feedback control input uc, defined by (4.40), is applied
to the system (4.1) and if the closed-loop systems (4.3l) and (4.37) are
asymptotically stable in the discrete-time sence, i.e., all their eigen-
values are withing the unit circle, then for sufficiently small 6,

X(n), XT(n) = [XI(n), x;(n)], is asymptotically stable, i.e.,

X(n) + 0 as n + m.

We note that while the measurements of x1 are available at only
slow cycles (£9, K integer, Theorem 4.3.1 gives the asymptotic stability

of x1 at all fast mesh points n.

Theorem 4.3.2: Approximation Result

 

If the conditions of Theorem 4.3.l are met and if the initial con-

ditions

xS(0) = x1(0), (4.43a)
xf(0) = x2(0) - (12-A22)-1(A21+BZFS)x1(O) (4.43b)
xf(|—:) = (IZ-Azz)‘TBZFS[xS(K-i) - x5001, K > o (4.43c)

are satisfied, then, for sufficiently small E, the state trajectories

could be approximated as

l29

xfiWQ=xgm+0fi)

xf(n) + (IZ-A22)'1[A21V(n) + BZFSJXS(K) + 0(e)

X
N
A
3
V
II

We note that

(1) Theorem 4.3.2 gives an approximation result for x], only, at the
points K/E: while it gives an approximation result for x2 for all n,

which is the best we can expect in view of using multirate measurements.

(2) The value of the theorem as a design tool can be seen if we desire a
specific case, pole-placement design say. A designer would choose FS

to locate the poles of the closed-loop system (4.3l) at certain locations
inside the unit circle. Next, Ff is chosen to locate the poles of the
closed-loop system (4.37) at certain locations inside the unit circle.
Finally, (4.3l) and (4.37) are solved for the initial conditons (4.43)
to obtain x (K) and xf(n). The actual response of the system is

s
predicted using the relations

K - ,
X-l ('é) - XS(T\)
and
- -l
x2(n) — xf(n) + (IZ-AZZ) [A2]V(n) + Bstle(K).
If the designer is not satisfied with the response, the choice of
FS and F1, is iterated until a satisfactory choice is reached.

(3) The solutions of (4.3l) with initial conditions (4.43a),and of (4.37)
with initial conditons (4.43b,c) can be obtained simultaneously. At K = 0,

given x1(0) and x2(0) we can compute xf(n) for all 0.: n <-% .

130

At K = 1, given xs(0) we compute xS(l) which together with xs(0)
providesthe initial conditions xf(%) so that we can compute xf(n) for
l 2

—<n<

e-

'E . In general, at any K, xS(K) and xS(K-l) will be available
so that (4.37) can be solved in the K

th cycle.

Proof of the Theorem 4.3.l

First, we prove the asymptotic stability for x](§) as K + m.
Second, we prove that x1(n) and x2(n) are linear combinations of
x](é) and x2(§) with bounded coefficients and so they are asymptotically
stable as n + m.

Applying the composite feedback control (4.38) to the full system
(4.l) yields

x](n+l) = [11+ A]](E)]x1(n) +éatA12(é)+ B](€)Ff3x2(n) +€iB1(€)f(n)x1(é)
(4.44a)
x2(n+1) = A2](E)x](n) + [A22(&) + 82(€)Ff]x2(n) + 82(€)F(n)x](-é—). (4.44m

Using the decoupling transformation

T . l
11-EM2(E)L2(E) - M2(c)

n = x (4.45)

 

 

where the matrices L2 and M2 satisfy
0 = A2](&) + L2(E) - [A22(E) + 82(€)Ff]L2(E)

+eL2(e)tAn<e) - (A156) + aneufnzw (4.46)

l3l

0 = A]2(e) + B](e)Ff + M2(e) - M2(e)[A22(e) + Bz(€)FfJ
+ [A1](&) - (A12(6) + B1(E)Ff)L2(E)]
- M2(€)L2(€)E(A12(€) + B1<e)Ff)L2(e)1
yields
n](n+l) = [I]+EA+0(62)]n1(n) + €[§+0(€)]F(n)x1(K/E)

nz<n+i) = £422+82Ff+0(e>inz(n) + caz+o<e)3%<n)x1(K/6).

where

>2
I

‘ -l
- AH + (A +B F )(Iz-AZZ-B F ) A

12 l f 2 f 21

_ B B F )-13

002
I

1 + (A12+BlFf)(IZ'A22' 2 2'

From (4.48a) we have

n—K/E

n1(n) = [I]+€A+0(€2)J n](K/e)

n-l

4-6 2 [11+ A+o(62)1”’1'jt§+0(6)3%(j) x](K/€),

j=K/€

< n E K21.

 

n47:

In particular

5:1

n]( e +€:A'7+0(€2)JVe n](K/&)

)=[11

5E1 -l K+l

+ e E [11+6A+0(e2)1
j=K/e

(4.47)

(4.48a)

(4.48b)

(4.49)

(4.50)

(4.5l)

132

using the inverse of transformation (4.45) we get

 

K+1 K+1
K+1 21 “T ~ ZT‘T‘~ 4. K
n1(—E—) = {Uri-END“ )1? + e .ZK [HTEAHNE )J [3+0(E)]F(J)}n1('€)
J=-' '
, K
+ 0(e)n2(g).
It is shown in Appendix 4.2 that
K+1
1 T“ 6521-14-. ; ‘g-i $1.2.
(11+efi)e + e .ZK (11+ A) §F(3) = (11+eAO), + E E (11+EA0) BOFS.
J‘g £-0
On the other hand,using (2.l07) and (2.l08) we have
we A0
(11+EA0) = e + 0(E) = A5 + 0(6) (4.53)
‘16" J. 1 A0(l-t) '
e - ; (11+tAO) BOFS - ( e Bodt F5 + 0(a)
J-O 0
= BSFS + 0(t)‘ ‘ (4.54)
Noting
JE' '1 1 _1-£ ‘16“] .
Z (11+sA0)‘€ Z (114-MOW , (4.54)
[=0 j=0
we get
n1 (£21) = [AS+BSFS+O(t)]n1(~E-) + 0(s)n2(§). (4.55)
Similarly, from (4.48b) We have
_ n-K/t
n2(n) - [A22+82Ff+0(t)] n2(K/F)
+ “‘21 [A +B F +0( )Jn-]-jTB +O( )“F(°)x (KE) 1(- < n < K+1
ng/tzzsz ~2€T31 ’e -e

(4.56)

133

. and

ET] 1 K+1
(K+1) - A +3 F +0 e %' (K) + g_ - A +3 F + E J B +o(e
n2 ‘E" ‘ T 22 2 f T )3 “2‘? jék T 22 2 f 0‘ )3 T 2 )T

Ffikfié. (4w)
Using the asymptotic stability of (A22+82Ff) it follows that

l
+3men€=mq

TA22 2 f

Using above and (4.41) yields

5&1 '1 Kil--l-j

K+l _ K 6 -l K
K+l
T -1 e K—zl -Il-J ‘1 e
+ 0(€)x](K/E), (4.58)
where V(n) satisfies
- .K -
V(n+l) - (I+EA0)V(n) + EBOFS, V(E) — I. (4.59)
Noting that
%--1
f (A +8 F )j = (I -A -B F )"TEI -(A +3 F )1/e]
i=0 22 2 f 2 22 2 f 2 22 2 f
_ -l
- (Iz'Azz‘BzF ) + 0(6),

the second term on the right hand side of (4.58) simplifies to

[(I -A )‘TB

2 22 FS + 0(6)]x](K/E) (4.60)

2

134

To simplify the third term on the R.H.S. of (4.58) we need the following

 

lemma
Lemma 4.3.1: 1
n- .
Let Y(n) = 2K An'T‘JB V(j), §-< n 5 5E1-, (4.6l)
3::

where V(n) satifies (4.59) and “A“ = a < l(A is an asymptotically

stable matrix).

Then ,
"'1 n-l-j K K+l
Y(n) = .XK A B V(n) + 0(6), E'< n 5 _ET" (4.62)
3"6

or, equivalently,

K
Y(n) = (I-A)“(1-A"‘€)BV(n) + 0(6). (4.53)
Proof:
(4.6l) can be written as
n-l . n-l .
Y(n) = {K A"‘T‘JBV(n) + )K An'T'JBEV(j)—V(n)1,§< n 5 1%. (4.54)
j=€ j=€

From (4.59) we observe that
V(n+l) - V(n) = 0(6).
SO we obtain
”V(n) - NJ)“ 5 c I. E .

where 2 = n-j.

135

Now,for the second summation in (4.64) we have

-1 . -l .
HTZK A"'T'J BEV(j)-v(n)iu s TXK HA““'JH HBH (V(J)-V(n)u

 

J‘e 3‘:
K
”‘1 . n-
= c' {K an'T‘J 2 e = c' f a“ 2 e, (4.65)
j=E' £=l
where
C' = CUB“.
Let
K
n- K
S = ...? a£-] Z = 1 ‘1' 2a '1' 3o:2 +....+ (n-éth-E ,
£=l
then
K
K n-
S-So: =1 + on + 0:2 +---+ dn-l-E - (n-éh E-
K
n.
_ 1'61 E ( K n-E
""‘"r::“ ' "' 59 Q
So
"”2— n-—K—
S - 1TQTTT-1-a - (D-é90 61"g < n f'EEl a

which is bounded.

Now using (4.65), equation (4.64) reduces to

n-l .
Y(n) = .XKlAn-T'J BV(n) + 0(6), §-< n 5 5%?“

J:
IE Q.E.D.

136

Using (4.60) and Lemma 4.3.1, (4.58) can be rewritten as

K+1 _ _ -1 'ET" ‘ -1 - K+1
”2T‘Eri ' TTTz A22) Bst ’ jZK (A22+82Ff) BzFfTTz‘Azz) A21VT‘E—9

+ 0(6)]x1(K/€) + 0(€)n2(é). (4.55)

But
K+1. -1 K+1
E 2
.ZK (A22+82Ff) = E (A22+32Ff)
J~§ 2—0

3 F )‘1 + 0(5) (4.57)

A22‘ 2 f

= (12-

and
K+1
)1/e + T 4 (I+EA )5?— +3. 6 3 1:
.EK 0 0 s
3‘?
1/6 T'-]
K
0) + £20 (I+EA0) E B

(I+EAO

e .
(1+ A OFs’

which, in view of (2.106)-(2.l08), is given by
K+1 _
V(T) - AS '1' BSFS + 0(6). (4.68)

Using (4.67), (4.68) and x1(n) = n](n) + 0(6)n2(n) in (4.66) yields
41205-2) = :H + o<e)151 (E) + 0(6)nz(é), (4.69)
where

- -1 -l
H - (Iz-AZZ) [A2](AS+BSFS) + BZFS] + (Iz-Azz-BzFf) A2](AS+BSFS).(4.70)

137

Combining (4.55) and (4.69) and using transformation (4.45) yields

 

 

 

 

 

 

( 1 1
x(K+1) (45+85F5) + 0(e) 0(6) rxxé)
K+1-1A K
(4.71)

The above system can be represented as

x(TEl) = (A+E)x(é) . usu= 0(6) (4.72)
where
AS + BSFS 0
A = . (4.73)

-l

(IZ'AZZ) TAZlTAs+Bst) + BZFS] 0
Using the asymptotic stability of (AS+BSFS) and the continuous dependence
of eigenvalues of a matrix on its parameters, it follows that for suf-

ficiently small 6 the system (4.71) is asymptotically stable. Hence
x1(g) + o and x2(§)-+ o as K + m . (4.74)

To show that the asymptotic attractivity holds for every n, let us
rewrite (4.51) as

n-l
m(n)=Hhkmnwe+égéUfiaw41fiU)+Nflhflé

+ 0(€)n2(K/E).§-< n 5 551 , (4.75)

138

where (2.106) has been used.
Using Appendix 4.2, (4.75) can be written as
”'6 "'1 n-l-' K K
n1(n) = [(11+6A0) + 6 32K (11+EAO) JBOFS+ 0(6)]n1(§) + 0(€)n2(g).

é-< n 5 5il-. (4.76)

Inside the bracket above is bounded uSing (2.106)-(2.108). Now,
asymptotic stability of . n](é) an n2(é) implies n](n) + 0 as
n4+ m.

Using the inverse of transformation (4.43), equation (4.56)

can be written as

K .
_ "‘E’ K "‘1 n-l-j . .
n2(n) - [A22+32Ff+0(€)] n2(g) + jEK/E [A22+82Ff+0(€)] [82+0(€)]F(J)
(51(2) + 0(e)nz(§)) , £5 n 5 521—. (4.77)

Using Lemma 4.3.l it can-be shown that

n-l .

52K [A22+52Ff+0(6)1”'1‘jIBZ+0(e)1F(j)
3:3

n-K
is bounded while the boundedness of [A22 + BZFf + 0(6 )] 6 follows

from the stability of [A22 + BzFf].
Again, by asymptotic stability of n1(é) and n2(é) we can
see from (4.77) that n2(n) + O as n + m.
Q.E.D.

Proof of the Theorem 4.3.2:

From (4.7l) we have

K+1 _ 4 K K
X1(—;79 - [AS+BSFS+0(€)JX1(E) + 0(6)x2T€)°

139

Comparing above with (4.3l) it is obvious that

K)

X-I(€ = X (K) + 0(6).

S

To prove the rest of the claim, first we establish a relationship
between n2(n) and xf(n).

We rewrite (4u48b) as

_ -l
n2(n+1) - [(A22+32Ff) + 0(€)Jn2(n) + [82+ 0(6)]{I(Iz-Ff(IZ-A22) 32)

K -1 ~
FS ‘1' 0(6)]X1(€)'Ff(12'A22) A2]}X](n);

where §](n) is given by (4.39).

Define ‘né (K) 35

-— = -— -1 -1

K .
A21+0(€)}x](§). (4.78)
Subtracting fié(K) from n2 and letting y(n) = n2(n)-Eé(K) yields
= 5 _ -1 ~ - K
Y(n+1) CA22+32Ff + O(€)J{(n) [32+ 0(€)IFf(I2-A22) A21EX](n)-X1(g)].

5:1

6 (4.79)

y(é) = 52(é)-35 (K), 5.5 n <

Now,we represent y(n) as the sum of a zero-input response

y] and a zero-state response 72.

y(n) = 51(n) + 42(n) . (4.50)
where

y](n+1) = [A22+82Ff+ 0(6)]v1(n) , -§ 5 n 5 521 (4.8la)

with the initial conditions

140

(71(0) = y(0) = 52(0) - 32(0)=-(IZ-A22)‘T(A21+32Fs)x1(0) + x2(0)+ 0(5)

K 1

«71%) = 52(5) -TiZ(K)=(12-Azz)' BZFS[xS(K-l)g- xS(K)] + 0(5). (4.8lb)

where (4.71) has been used to substitute for x2(§gl- and

_ -1 ~ - K
Y2(n+]) "' [A22+82Ff + 0(6)]Y2(n) " [82+ 0(6)]Ff(12‘A22) A21[x](n)-x1(g)].

Y2(lé) = 09 K > 0, g: n < 1%1 . (4.82)

From (4.82) for é-< n 5 5E1. we get

n-l .
5 _ n-l-J -1 K
(z(n) - jzé,TA22+32Ff + 0(6)] [32+ o(€)JFf(12-A22) A21x1(g)

n-l -
n-l-J '1" -
- J.=_):',E('[A22+52F1, + 0(6)] [82 + 0(E)IFf(Iz-A22) X1(J).
(4.83)
K K+1
'5 < " $‘7€‘-

Using the asymptotic stability of (A22+B2Ff) and Lemma 4.3.1
(Note that from (4.39) §](j) satisfies a similar equation as (4.59)),

(4.83) reduces to

K

(n) = (I -A -B F )‘TEI -(A +B F )nUEJB F (I -A )']A x (K)

Y2 2 22 2 f 2 22 2 f 2 f 2 22 21 1'€
n-l

n-l-j -1 ~
- J.EéEAzzszf + 0(6)] [32 + 0(6)]Ff(12-A22) A21x](n) + 0(6),

where we have used the Householder Theorem.

Or, equivalently,

 

K
_ -l "'E' -1
Y2T") ‘ (Tz'Azz'BzFf) TT2'TA22+32Ff) JBzFfTTz'Azz) A21“
~ +
[X] (é)-x1(n)], é< n 5 K6] . (4.84)

141

~ K _ K ~ _ ~
From (4.39) we have x1(€) - x](§) and x1(n+l) - x](n) + 0(6)
and we obtain
TTX1TE)O'X")TT=£ 0(€)sc2€.
where C is a constant and 2 = n - é-.

Now,(4.84) reduces to

y2(n) = (IZ- A22-B2 Ff) B 22Ff(I -A22) HA2]LX (E) - x (n )]
nK
-1"'€ K ~
' TTz‘Azz'BzFf) TA2 2+32 F f) BzFfTT 2 A22) A21TX1TET ' X1T")T
+ 0 (6) . ' (4.85)

We also note that

)"'E’ -1 K ~ ""K‘ K
“T HAZZ'T'BZFf BzFlez‘Azz) A21EX1T§)'X]TH)JHE COT Tn‘ é’) =0TE) (4.85)

where
14 +5 Ffu=

Using (4.86), (4.85) reduces to

72(n) = (Iz-Azz-BzFf)'1BzFf(Iz-A22)'TA21[x1(é) _ 21(n)1 + 0(5)

-1 -1 K ~
(4.87)

Now, by comparing (4.81) with (4.37) we observe that

i1<n) = x.(n) + 0(5) . (4.88)

142

From the inverse of the transformation (4.45) we get
szn) ='L271~1Tn) + ”z(n) + 0T6)
-1 '

= (IZ'AZZ'BZFf) A21n1(n) + n2(n) + 0(6). (4.89)

But
I12(I'I) = Y(n) + T12 (K) = ‘{](N) + ~{2(n) + ;2(K)’ (4.90)

where Hé(K) is obtained from (4.78) to be
_'(NF(I -A -B F )‘TIEI -B F (I -A )‘TJB F -B F (I -A )‘1A 1x (5)+0(e)
n2 , 2 22 2 f 2 2 f 2 22 2 s 2 f 2 22 21 1 e

_ -1 -1 K
- [(Iz-Azz) (BZFS+A21) - (12-A22-82Ff) A211x1TEJ + 0(5) (4.91)

In view of (2.l06)-(2.108) and Appendix 4.2, comparison of
(4.39) with (4.51) yields

mo>=flm>+me. ‘ (4%)

Now, using (4.87), (4.88), (4.9l) and (4.92), equation (4.89)

reduces to

_ ' -1 ~
x2(n) - (Iz-Azz-BZFf) A21x](n) + xf(n) +

1

- -1 K N
[(IZ-Azz-BZF ) A21-(IZ-A22) A211£x1(g) - x](n)]

4 [(1 -A )‘T(8

-1 K
2 22 2Fs+A21T ‘ TT2'A22'32Ff) A21TX1T€T +

5 n 5 K21 (4.93)

m|7<

0(6).

Or

143

x2(n) = xf(n) + (Iz-A22)'TA21§1(n) + (12.422)'182F5x1 (Q) + 0(5) . (4.94)

Approximating x](é) by xS(K), using (4.39) and (4.42) we

obtain

x2(n) = Xf(n) + (IZ-AZZYT[BZFS+A21V(n)]xS(K) + 0(5), §< n 55-21-4495)

Q.E.D.

Equation (4.95) states that x2(n) can be approximated for all

n, using the solution of slow and fast subsystems.

Examgle: To illustrate our claims,we apply the above design procedure
to the example of Section 3.7 when the design criterion is the pole-
placement.

Consider the difference equations

 

x](n+l) = (l-2€)x1(n) + €x2(n) + l.5€u(n), x1(0) = .5 (4.96a)

x2(n+1) = -.7 x1(n) + .45x2(n) + .8 u(n), x2(0) = -.5 (4.96b)
Slow subsystem

xs(K+l) = AS xs(K) + BS uS(K), xS(O) = .5 , (4.97)

where

AS = .038073 and BS = .868406

The gain FS in uS(K) = FSxS(K) in chosen such that the closed-loop

slow subsystem has eigenvalue located at .5,i.e,, AS + BSFS = .5 which

results in FS = .532026.

144

Fast subsystem
xf(n+l) = Azzxf(n) + Bzuf(n)
where A22 = .45 and 82 = .8.

The gain Ff in uf(n) = fof(n) is chosen such that the closed-loop

fast subsystem has eigenvalue located at .5, i.e., A22 + BZFf = .5.

which yields

Ff = .0625.

The programs are written, using 'LAS package' [Bingular et al.,
l982]. and run on the Prime computer at Michigan State University.

The results are evaluated for three different values of E
(.l, .05, .025) and four slow periods.

The results are tabulated in following tables. For the sake of
compactness we do not give all the values of x2(n) and xé(n) (which
is the predicted value of x2(n) given by (4.95) and should be within

0(6) from x2(n) for each n). although these values are available.

 

145

 

 

 

 

 

 

 

 

 

 

 

5
Table 4.l
n x2(n) XZTH)
o -.5 -.5
1 1 -.3747l8 -.266483
5 1 .031075 .027885
10 E .098466 .074759
11 ; -.o12021 .025102
15 T -.004897 .028341
20 .038l96 .037597
21 f -.00854l .013053
25 j -.oo1539 .014173
30 1 .018002 .018352
31 ‘ -.oo3950 .006528
35 i -.ooo779 .007088
4o 3 .008434 .009428
I
Table 4.2
K K ._5
x1(g) xS ( K) x205) x2( 8)
o .5 .5 -.5 -.5
1 .235551 .250044 .098966 .074759
2 .ll0806 .125044 .038l96 .037597
3 .051924 .052533 .018002 .018352
4 .024331 .031272 .008434 .009428

 

 

 

 

 

 

 

 

 

 

146

 

e - 05 ~
Table 4.3

n x2(n) xé(n)

o -.5 -.5

1 - 374718 -.320600

10 1 .03l778 -.000245

20 1 .075225 .072005

I

21 -.022511 -.000951

30 -.000947 .013139T

40 .035731 .035010

41 T -.o11732 -.00048l

so 1 .055255 .005570

60 T .017445 .0l8008

i

51 ' -.0057l8 -.000240

70 .032341 .003286

80 .0085l4 .009005

Table 4.4

K\\\ x1(§) xS(K) x2(K/e) xé(§)
o .5 .5 -.5 -.5
1 .243044 .250044 .075225 .072005
2 .118522 .125044 .035731 .035010
3 .057894 .052533 .017445 .0l8008
4 .028255 .031272 .0085l4 .089005

 

147

 

 

 

 

 

 

 

 

.§;:_Jli§
Table 4.5
n\\1 x2(n) xé(n) 1 n x2(n) xé(n)
o 1 -.5 -.5 1 81 -.o12994 -.007248
1 T -.3747l8 -.347559 T 90 -.023584 -.014802
10 -.057701 -.059808 1 100 .002864 .005327
20 .022807 .02l298 1 110 .0l3l88 .013939
30 .058087 .055735 1 120 .017200 .017507
40 .072015 .070402 1
41 -.025451 -.014493 1 121 -.005423 -.003525
50 -.047775 -.029598 1. 130 -.o11550 -.oo7402
50 .005755 .010551 ET 140 .001415 .002554
70 .025552 .027873 11 150 .005520 .005971
80 .034785 .035207 11 150 .008504 .008805
Table 4.6
N x1 (5) xsm x2046) x'z(K/e)

* .5 .5 -.. -..

E .245391 .250044 .072015 .070402

f .l2l82l .125044 .034785 .035207

1 .050230 .052533 .017200 .017507

T .029778 .031272 .008504 .008805

148

By noting the results in Tables 4.l-4.6 we can observe the
following:
l - Asymptotic stability of x](é) and x2(é) as1 K increases.
- Asymptotic stability of x2(n) as n increase.

The closeness of x1(é) and xS(K), K = 0,l,2,..., up to 0(6).

h b.) N
l

- The closeness of x2(n) and xé(n), which is the approximated value
of x2(n), up to 0(6).

5 - The abrupt change in x2(n) and xé(n) at the beginning of each

slow cycle which is a result of the abrupt change in the slow control

that excites the fast modes.

4.4. Sequential Design

 

The composite control of Section 4.3 has a slow component, which
stabilizes the slow modes, and a fast component, which stablizes the fast
modes. Suppose, however, that the open-l00p fast modes are already
asymptotically stable with acceptable transient response, i.e., the
eigenvalues of A22 are appropriately located inside the unit circle,
then the fast component of the composite control may be omitted and only
the slow control is used. If this is possible, a considerable reduction
in the on-line computations will be achieved since implementation of the
slow control does not require the solution of the slow equations.

Even if A22 'is not asymptotically stable,or it is so but
its eigenvalues are not sufficiently well damped,the above idea might
still be useful by using feedback from the fast variable to pre-condition
the matrix A22 to have the desirable stability property and then a slow

control can be designed as in Section 4.3. Such a design procedure will

149

be sequential since the design of the slow control will be dependent on
the pre-conditioning feedback gains.

In this section we investigate this sequential design procedure
and give a similar approximation results as Section 4.3.

Again, consider the singularly perturbed difference equation
(4.l) where (IZ-AZZ) is nonsingular and let the input control decomposes in-
to two parts as in (4.98).

u(n) = u1(n) + u2(n) (4.98)

where u1(n) is constant over the cycle é-f n < 5E1-. Let -us choose

the feedback control
u2(n) = F x (n) (4.99)

such that the matrix A22+BZF2 is asymptotically stable and meets some
desired objectives or has appropriate eigenvalue locations. Now,system

(4.l) becomes

x](n+l) [11+EA11(E)lx](n) + EIA12(E)+B](E)F2]x2(n) + 681(6)u](n)

(4.l00a)

x2(nfl) A21(€)x](n) + [A22(E)+B2(E)F2]x2(n) + Bzu1(n) (4.100b)

Slow subsystem
Following the same method as in previous section the slow

subsystem is defined to be

xS(K+l) = ASXS(K) + Bsus(K), xs(0) = x1(0) (4.l0l)

150

~

'1 ~

where AS = eA , BS = J eTT-t)AdtB , A and B are defined by (4.49)
O

and (4.50)

Suppose that the state feedback control law for us(K) is de-

signed as

u (K) = F x (K), (4.102)

where FS is chosen such that the closed-loop system
xS(K+l)=--(AS+BSFS)XS(K) (4.103)

is asymptotically stable and meets some-design criteria.

Composite Control

 

With the pre-conditioning feedback gains and slow control in

hand, a composite feedback control is formed as

uc(n) = F2x2(n) + uS (n) = F2x2(n) + FSxS(K). (4.104)
By approximating xS(K) with x](é) we obtain
K K K+
uc(n) = F2x2(n) + st1(g) , 5’: n <._€l . (4.l05)
Applying (4.l05) to system (4.l) yields
_. r ’ E.
x1(n+l) - [11+EA1](E)JX1(n) +ELA12(E)+B](C)F2]x2(n) + €B1(E)st](e)
(4.l06a)
.. r ’ T_(_
x2(n+l) - A21(E)x](n) + LA22(6)+BZ(C)F2]x2(n) + B2(&)st1(e). (4.106b)

We define the fast subsystem to be

151

xf(n+l) = (A22+BZF2)x f(n n) (4.107a)

xf(0)= V(O) (12-A2282F2)'](A2]+BZFS)X1(0) (4.1075)

xf(é) = (Iz-AZZBZ BF 2) 82 F SS[x (K - l)-xS(K)], K 7 1,2,...
(4.l07c)

Theorem 4.4.1

 

If the feedback control input uc, defined by (4.l05), is
applied to the system (4.l), if the matrices (A22+B22 F ) and (AS+BSFS)
are asymptotically stable in the discrete-time sence, i.e., all their

eigenvalues are inside the unit circle, and if the initial conditions

xs(0) = x1(0)
xf(0) = x2(0)- (12-A22-B SF2)'T(A21+82Fs)x1(0)
XfTé) = T12 A22-82F2)182FSEXS (K-1)-XS(K)], K = 1,2,...

are satisfied, then for sufficiently small 6, x1(n) and x2(n) are

asymptotically stable solutions and
x (5) = x (K) + 0(6) K = o 1 2
1e S 9 9 9 9".
and
_ -l
x2(n) - xf(n) + (12-A22-82F2)H82F +4217‘(n )le(K),+ 0(E),

where

152

Erggfi

This is a special case of Theorem 4.3.l with A22 and A12
replaced by A22+BZF2 and A12+81F2, respectively and Ff = 0.

Q.E.D.

To point out the computational difficulties for this type of
design procedure let us, for example, choose the design criterion to be
pole-placement.

The designer first evaluates the pre-conditioning gain F2.
Using F2, the poles for slow subsystem are selected. .On applying the
composite feedback control,if the designer is not satisfied with the
response of the system, the whole procedure should be repeated. If
the design of the pre-conditioning gain is costly, then the sequential de-
sign procedure is not desirable. Based on the specific problem, the designer

may wish to choose the parallel design, discussed in previous section.

APPENDIX 4.1

By letting F2 = Ff and F1 = f we have

__ __1____ ... -1 ...
A11+A12(12-A22) A A +B F+(A +B F )(I -A -BZFf) (A2]+BZF)

21 11 1 12 1 f 2 22

_ . -1 ~-~m
- A]1+(A12+B1Ff)(12-A22-82Ff) A21+EBTKA12+B1Ff)(Iz-Azz-BzFf) BZJF - A+BF.

Now wish to prove

K+§T5 = A +8

0 oFs

_ -1 -1
L.H.S. - A]1+(A12+BlFf)(IZ'AZZ'BZFf) A21+EB1+(A12+81Ff)(Iz-A22-82Ff) 82].

-1 -1
{[IZ‘Ff(12-A22) BZJFS-Ff(I .A22) A21}

‘ A11+("12+B1FF)“2"”‘22'32F1=fl“2“32F1=“2"‘22)-13’3‘21“31':1=(12‘A22).1
A21+(A12+B1FF)(Iz'Azz'BzFF)-1(Iz‘Azz'BzFF)(12‘A22)-132Fs
+ B][12-Ff(Iz-A22)'182]FS
= A11+A12(12‘A22)4A21+31FF(12‘A22)-1A21‘81FF(12'A22)-1A21
+ A12(12'A22)-132Fs+B1Fs
= A11“’1‘12“2‘A22).1 21+[B1TA12(12'A22)-1323Fs
= A0"BoFs
Q.E.D.

153

APPENDIX 4.2.

Wish to prove by induction that

K
n-l . . n- n- -l
(I +6K)"'K/e+ e (I +€A)n'1'J§F(j) = (I +EA ) + e jg (I +€A 0)n J 1B F
1 ._ 1 1 0 K( 05
J-K/e F';
for é-< n < 5&1. . (1)
For n = é-+ l, (1) reduces to
I1+E(A+BF) = I1+&(A 0+BOFS ),
which is true (see Appendix 4.1) where
F ‘ Fs'Ff(I 2 A22) (A 21+82Fs ) °
Now let the assertion be true for n = é-+ m, 0 < m f-%, i.e.
~ m ém-l m+lé--l-j~. K+m-1 ~2+m-l-j
(Iva/1) + 1K (Ive/1) em) = (I +eA 0'“) + 6 2K (Ive/10)
3:? 3:?
BOFs
Or by 2 = j—~E we have the following equality
efi m e - 1 +eA m 1 £8F(( + K) - (1 +eA )m + e m3(1 +eA )m‘I'ZB F
(11+ ) + E ( 1 1 K E" ‘ 1 0 £_ 0 o o s'
(2)
For n m + l + E- we should have
m+é m+é j m+é m+é4
m+l « ' c _ m+1
(I +EA) + e jig (11+6A)m BF(j) - (11+6A0) + E jg; (I1+&A0)BOFS.

154

155

Or by Z = j -K/€ we get

m

m
e m+l m- £~ _ m+l m-£
(11+ A) +e 20(11+eA) 3112+ £1-111+eAO) +6 ; (1116A0) BOFS ,
2 £-0
(3)
where
r 1 -1 2 1 l-j
jth‘FF112‘A221 B2Fs1'F1112'A221 “21111116110111e 1111116AM1§FF511
5( +-K) -1 j: 0 1
3-‘e " 2 f o 1
5 for Z = 0 . 1
J
Let
0(1) = (1+eA )1 + 61§1( I+EA011 1 is F o < ' K
0 2:0 80 S 1 1 SE.
So
E K _ -l
where
_ -l
C ' Fs‘FF112'A221 Bst
~The L.H.S. of (3) is
m+l m:l m- ME ~‘ K
(11+6A) + 62 (11+EA) BF (2+?) + EBF(111+E'1
£=0
m-l
_ — ~ m "Z‘ 1'” K N \‘1
- (11+2K)[(I1+EA)m+ e 220(I +EA) §F(21~€)1 + EBCC-Ff(12-A22, A2]Q(m)],

where we have used (4). Inside the first bracket above is equal to

Q(m) by (l), so we get

156

_ ~ ~ -1 ~
L.H.S. of (3) - [I]+€A-€BFf(I2-A22) A2]]Q(m) + EBC . (5)
But

~ ~ -1 _ ’ -1
I]+e[A-8Ff(Iz-A22) A21] - 11+ECA]]+(A12+B]Ff)(Iz-A -B F ) A

22 2 f 21

-1 -1
(81+(A12+B1Ff)(IZ'A22'BZFf) 32)Ff(12‘A22) A21J

. -1 -1
I1+E{A11+(A12+B1Ff)(Iz'Azz'BzFf) [Iz'BzFf(Iz'A22) 3A21

-1
B1Ff(Iz-A22) A

21}

-1

I +ECA A

-1
+(A +3 Ff)(I -A 2) A21-B1Ff(1

1 11 12 1 2 2 2'A22) 21J

I1+6AO . (6)
A150
~ _ ~ -1

- [31+(A12+B]Ff)(Iz-A22-82Ff) BZJEIZ Ff(12-A22) BZJFS

-1 -1
B1[12‘Ff(12'A22) 323Fs+(A12+31Ff)(12'A22) Bst

-1 _
[81+A12(I2'A22) B23Fs ‘ BoFs ° (7)

Using (6) and (7), the R.H.S. of equation (5) becomes

m
)m+1 + e Z (1 +eA

F
2:0 1

0 s

(I]+EA0)Q(m) + EBOFS (I]+EA )m-zB

0 0

= R.H.S. of (3) .
Q.E.D.

APPENDIX 4.3

Wish to prove

n11 [A22+82Ff+0(e)Jn'1'jEBz+0(e)J?(j) = 0(1). (1)
jig

n-
L. H. s. of( ZKZEA22+B Ff+0(€)3 3'" 1'jl:82+0(e)3

i=5

.[Fs-Ff(IZ-A22) l(A21v(j)+Bst )1:

where §(j) has been substituted for using (4.41).

Using Lemma 4.3.1 we get

n- -1
L.H.S. of (1) = Z K[A22+82Ff+0(€)]n 1 J.[Bz+0(€ )]

3:2
-1
[FS-Ff(IZ-A22) (A21V(n)+BZFS)]+O(€)

n-K
= E12-A22"82F11“"0(€)2'(2‘L\22+B Ff+0(e)J 5 J.

[82+O(€)][FS-Ff(I2-M221(25A21V(n)+B F )J+0(€ )
= 0(1).

Q.E.D.

157

CHAPTER 5
APPLICATION

5.1 Introduction

The main objective of this chapter is to use a more realistic
physical model to demonstrate our results about near-optimality of the
composite feedback control for infinite-time regulators and the iterative
technique for solving the discrete-time stiff Riccati equations which
were presented in Chapter 3, and also the asymptotic stability of the
solutions and closeness of trajectories in multirate stabilization pre-
sented in Chapter 4. For this purpose, we consider the deterministic
model of an F-8 airCraft. Different control problems of this aircraft
are investigated by different authors [IEEE Transaction on Automatic
Control, Mini issue on the F-8 aircraft, Oct., T977].

In particular, we consider the model considered by J. Elliott

[1977].

5.2. Longitudinal Equations of Motion for an F-8 Aircraft

The linearized aircraft equations of motion is given by

 

 

 

 

 

 

 

 

I 1 r w 1 1
u x -g x O u I X: X
1.1 01 Ce 5T
6 0 O O l a 0 0
S
d = e
at +. [ ’
a Zu 0 Za 1 z a 1 ZS 0 6T
1 i e
M 0 M M ‘ M, 0
LqJ LU anLq‘ A08

158

159

where

u = incremental velocity, ft/s.

e = incremental pitch angle, rad.

a = incremental angle of attack, rad.

q = incremental pitch rate rad/s.

5e = incremental elevator position, rad.

5T = incremental throttle position, nondimensional.
g = gravity acceleration
M( ), X( ), Z( ) are longitudinal dimensional stability derivatives
reffered to stability or wind axes.
By experience with this model, it is known that u and e are

slow while a and q are the fast variables.

 

u at
9
\
_a.! .
i {l K‘
h mt" tal “if“
0 zen In.“- .\
\\ \K/X k?
‘ ‘4

\Q .4...-

-‘

\. \‘ .‘x‘fl
q=é was.
a

Figure 5.1. Aircraft longitudinal variables.

160

Representative numbers for flight of the F-8 at 20,000 ft. with

total equilibrium velocity = 620 ft/s (Mach number = .6) and do .078

 

 

 

 

 

 

rad.are
1
I u‘ T'-.o15 -32.2 -14.0 0 ul I-1.1 8.9
e o o 0' 1 e o o
gfv a = -.ooo19 o -.84 1 a + -.11 o
L_qJ L. .00005 0 -4.8 -.49 q -8.7 o
L 1
J (5.2)
Scaling

First scaling is to bring the system into the normal singularly

perturbed form. This system takes the form

A11 A12/e
A12 A22/611
. I ex 0 . .
The transformat1on l brings the system 1nto the form
0 12
F 7
A11 A12
5.21 522
e e J
;

 

 

161

We take t = %5- which is the ratio of the magnitude of slow to

fast poles. The first scaling is

, 1, 1]. (5.3)

Second scaling is to balance the outer diagonal elements of

-1
(A11 ' A12A22A21)

_-_1_
$2 - d1ag [400 , 1, l, 1]. (5.4)

Now, total scaling is

1 1.

 

 

 

 

S = $25] = diag [Til—0'0 , 30 ,1,1], (5.5)
and we have
' -.o15 -.0805 -.001l666 o 1
Ac = 0 0 0 .03333
-2.28 0 -.84 1 (5.63)
L .6 0 -4.8 -.49 J
r 1
-.0000916 .0007416
0 0
BC = -.11 0 . (5.6b)
L -8.7 0 j
T

The initial values are [-l, 0, .08, 0] .

162

we use the sampled-data to discretize the system. Two different
sampling rates are chosen, T = .05, which is a typical value [Elliott,
1977], and T = 1 which is much larger, and we observe that our claims
hold for both choices. The choice of sampling period T is based on a

theorem [Kalman et a1. 1963] which states that:

If the continuous time-invariant system is completely controllable, then

the time-invariant discrete-time system is completely controllable if

Im{Ai(A) - x.(A)} # n §§¥, (5.7)

J

whenever

- = = t t
Re{Ai(A) Aj(A)} 0 and n . 1, 2...

The open-loop eigenvalues of system (5.2) are
-.006852 1 j .076519 , -.665648 f j 2.182122,

so that a choice of sampling period which satisfies T < 1.44 will
preserve controllability.

Sampling the system (5.2) yields

A T T A (T-t)
an+1) = e C X(n) + j e C Bcdt u(n)
0
0r
x(n+l) = A x(n) + B u(n) . (5.8)

By equating (5.8) with our form of system

163

x](n+l) (I +EA11)x1(n) + 6A12x2(n) + EB]u(n)

l

x2(n+1) = A21x](n) + A22x2(n) + Bzu(n) , (5.9)

j’ Bi (i,j = 1,2) are determined.

We have evaluated the eigenvalues for slow and fast subsystems,

the matrices A1

i.e., eigenvalues of A0 = AH + A12(12-A22)’1A2] and A22. It is
seen that the eigenvalues of A0 are close to the slow eigenvalues of
A and the eigenvalues of A22 are close to the fast eigenvalues of A
which guarantees the existence of two-time-scale property of the full
system.

All the programs for performing the computations are written using
the Prime Computer and "LAS" package [Bingulac et al., 1982] at Michigan
State University. The programs are attached at the end of this chapter.

164

5.3 Results for Infinite-Time Regulator:

For computational purpose of this part we choose the output to be

y(n) = DX(n).

where

D = diag [.1,.1,.1,.1] ,

with the performance index

00

 

 

 

 

 

 

 

 

min .1 = e z] [yT(n)y(n) + uT(n)Ru(n)] .
n:
where
_ T =
R ‘ R I4x4
Slow subsystem:
For T = l we have
, r 1
-.530438 23934901 .107593 .022050
dxS
"E’= xS(t) + us(t)
1L2.178240 -.087888 J L-l.297632 1000810,
I 1 o 1 r o 0 1
ys(t) = 0 .1 xs(t) + o o us(t),
-.009481 .002009 -.l68058 -.000019
L .217693 -.009422) \ -.129740 .000087 J

 

and for T = .05 we have

 

 

 

165

 

 

 

 

 

.001112

.000002 J

0
0
0

.000004 ,

-.000414‘

 

' -.022539 -.120705 ‘ ' .002932
dx
.a‘é' = xs(t) +
. -.109791 -.000221 J L-.065042
r \
.1 o r 0
ys(t) = 0 .1 xs(t) + 0
-.009920 .000024 ‘ -.167970
L-.219581 -.000443, _ -.130084
with the performance index
min 1 = {m [yT(t)y (t) + uT(t)u (t)]dt
s «0 s s s s '
Fast subsystem:
For T = l we have
-.329907 .193177‘ -1.984393
zf(K+1) = zf(K)+
-.924546 -.263252, -3.192726

.000925;

‘

u (t)

 

 

166

r 1
0 0
D2 = 0 0
1 0

L0 1]

 

 

For T = .05 we have

' .953092 .048269 -.016002 -.000002
zf(K+l) = zf(K) + uf(K),
L-.231691 .969983 -.428220 .000001

 

yfuo = 112 .(K).

with the performance index

00

min of = K20 [y;(K)yf(K) + 111100114101 .

In the following two tables we give the values of V = P-P'

for tWo sampling periods T - l .and T = .05. It is observed that the

numerical values agree with the theoretical results (Theorem 3.2).

 

( *' - -'—_‘\

T = .05'

-0.068998
-0.008956

0.000213
—0.000029

-0.027394
-0.005159

0.000174
-0.000010

167

Table 5.1

V = P-P'

.008956
.094044
.002944
.000406

Table 5.2

v = P-P'

.005159
.031458
.000988
.000177

0.000213
0.002944
-0.000162
0.000011

0.000174
0.000988
-0.000036
0.000007

-0.000029
-0.000406

0.000011
-0.000009

-0.000010
-0.000177

0.000007
—0.000002

 

 

168

On the next two tables we give the numerical resutls using

the iterative technique and also the exact solution for T = .05

 

 

 

and T = 1.
Table 5.3
T = .05
Solution of the Riccati equation
Iteration P1 P2 P3
1 1.405989 -.001l67 -.295408 -.156022 .398268 -.048487
-.001167 1.552002 -l.361687 .250796 -.048487 .094758
2 1.413724 -0.001904 -0.297776 -0.157317 0.401409 -0.049061
-0.001904 1.558631 -l.367825 0 252564 -0.049061 0.004061
3 1.414150 -0.001984 -0.297944 -0.157413 0.401629 -0.049103
-0.001984 1.558902 l-1.368078 0.252672 -0.049103 ‘0.094980
4 1.414172 -0.001989 -0.297953 -0.157418 0.401640 -0.049105
E-0.001989 1.558913 -l.368088 0.252678 -0.049105 0.094981
5 1 1.414173 -0.001989 -0.297954 -0.157418 0.401641 -0.049105
!-0.001989 1.558913 -1.368089 0.252678 -0.049105 0.094982
Exact 1.414093 -.001990 -.297974 -.157426 .401665 -.049107
Solution -.001990 1.558813 -l.367995 .252662 -.049107 .094990

 

 

 

 

169

.

 

 

 

 

Table 5.4
T = 1
Solution of the Riccati equation

iteration P1 P2 4 P3
1 .065598 -.000306 -.012264 -.010470 .022148 .000990
-.000306 .072273 -.0639441 .012474 .000990 .011486
2 0.069828 -0.000140 -0.013135 -0.010497 0.022555 0.000914
-0.000140 0.076873 -0.067946 0.013283 0.000914 0.011499
3 0.070894 -0.000126 -0.013362 -0.010509 0.022676 0.000891
1-0.000126 0.078008 -0.068924 0.013483 0.000891 0.011504
4 1 0.071153 -0.000124 -0.013417 -0.010512 0.022706 0.000886
1-0.000124 0.078278 -0.069158 0.013531 0.000886 0.011505
5 g 0.071215 -0.000123 -0.013431 -0.010512 0.022713 0.000884
g-0.000123 0.078343 -0.069215 0.013542 0.000884 0.011505
6 5 0.071230 -0.000123 -0.013434 -0.010512 0.022715 0.000884
1-0.000123 0.078358 -0.069228 0.013545 0.000884 0.011505
7 0.071234 -0.000123 -0.013435 -0.010512 0.022715 0.000884
-0.000123 0.078362 -0.069231 0.013546 0.000884 0.011505
8 = 0.071245 -0.000123 -0.013435 -0.010513 0.022715 0.000884
1-0.000123 0.078363 -0.069232 0.013546 0.000884 0.011505
Exact E .071240 -.ooo123 -.o13435 -.010513 .022716 .000884
Solution 1 .000123 .078369 -.069237 .013547 .000884 .011505

 

 

 

8y investigating the results in Tables 5.3, 5.4 we observe that

the iterative technique has a convergence rateof0(e)=-

30 ’

or even less,

170

at each iteration and tends to the exact solution.

It is important to mention that the saving in computational
efforts for this fourth-order example was quite substantial. Solving
the Riccati equation using iterative technique was very much faster
than solving the full Riccati equation. This confirms the excellent
efficiency of iterative technique,specia11y for high-order stiff systems.

It should be noted that due to round-off errors there are very

small differences in some of the numerical figures.

171

5.4. Multirate stabilization:

 

Our criterion for this part is pole-placement. The eigenvalues

of slow and fast subsystems; for the continuousetime system are given as

s1ow, -.007442 * j .076413

fast -.665 i j 2.18389

Real parts of slow eigenvalues are small which results in very oscillatory
response. To avoid this problem, we locate our closed-100p slow eigen-
values at -.l i’j .1 with magnitude .141. Also, we locate the closed-
loop eigenvalues of fast subsystem at -l f j 2 with magnitude 2.236.
As we can see, the ratio of magnitudes of slow and fast is about 16
which provides a proper gap between two sets of eigenvalues and has the
desired two-time-scale property (2.7).

ST

Now, the mapping 2 = e from S-domain to Z-domain [Franklin

and Powell, 1980] will provide us with discrete-time eigenvalues.

Closed-loop fast eigenvalues

z = eST - e(']i j2)T = rf e738; ,
where rf = e"T and 9f = 2T (rad) .
For T = l, we get E.V. :_-.36 f j.78
For T = 0.05, we get E.V.:1 .95 I j.095

Closed-loop slow eigenvalues

E
:-
m
1
m
1
u
(0
DJ
:1
o.
(D
u
___l
_'
A
1
m
a.
v

172

1, we get E.V. 2. .9 t 3309

For T

For T .05, we get E.V. :..995 f j.005

In view of (2.85), we observe the two-time-scale property of our

system.

It should be noted that the eigenvalues of the slow subsystem,
evaluated above, are found in fast-time scale, while for our design
procedure they should be determined in slow-time scale. .Thus, the actual

locations of slow eigenvalues are found by raising their values to the
power %-.
This is equivalent to the mapping

+ .
z = eST/E = e30(-.1 - J.1)T .

And the locations of slow eigenvalues are as follow:

For T = l, we obtain E.V. :_.05 f j.007
.05, we obtain E.V. :..85 i j.13.

For T

173

 

 

Table 5.5
T = .05
The values of x and x' .The values of x and x'
at the end of tie slow2 at the beginning of eagh
periog+1 K+1 slow period K
K x1(K/§) xS(K) x2(—€—) xé(-—e—-) x2(-€ +1) xég-é-H)
0 -.1 .1 -.592194 -.585282 .083056 .083045
0 0 -.769791 -.801221 .136725 .136494
1 -.095030 .095192 -.075732 -.211630 .593010 .465931
-.043437 .029220 -.193325 -.344752 .665538 .529259
2 -.086633 .087887 -.097644 -.057633 .072933 .258245
-.049281 .049674 -.272114 -.209182 .200351 .361502,
3 -.077950 .079022 .057672 .058503 .097439 .094507
-.062357 .062841 -.095404 -.100700 .240697 .223220
4 -.068453 .069354 .106949 .142069 .058769 .030285
-.066155 .070100 -.052787 -.016520 .086139 .112179
5 -.059030 .059473. .167037 .198260 .107321 .121362
1-.068390 .0727061 .020077 .046373 .040502 .025655
6 §-.049827 .049824 .196989 .231996 .167449* .183923
g-.067180 .071768 .059421 .091049 .026283 .039331
7 -.041164 .040726 .217028 .247800 1 .197162 .222935
-.064227 .068247 .093410 .120495 g .063894 .085832
8 -.033179 .032395 .223482 .249722 .217118 .242996
-.059729 .062954 .113773 .137520 .095218 .116833
9 -.025988 .024958 .221422 .241304 .223464 .248255
§-.054327 .056561 .126335 .144690 .113949 .135152
10 §-.019640 .018476 .212184 .225573 .221340 .242362
-.048374 .049605 ‘.131292 .144290 .125091 .143372
11 -.014l49 .012955 .198085 .205053 .212048 .228456
‘-.042219 .042507 .130771 .138310 .129074 .143802
12 -.009494 .012955 .180653 .181802 .197915 .209173
-.036112 .042507 .125918 .128439 .127862 .138453
13 -.005633 .008363 .161315 .157447 .180464 .186673
-.030250 .035584 .118002 .116079 .122591 .129043
14 -.002505 .004638 .141162 .133236 .160749 .162682
-.024770 .029064 .107996 .102368 .111169 .117001

 

 

 

 

 

 

 

174

Table 5.6

The values of x and x2

The values of x2 and x5
at the end of t e slow 7

.at the beginning of

 

 

period 1 the slow period

K mg) x500 x2023) xéd‘i‘) x2(-."- +1) xére'S +1)
0 -.1 -.1 .155724 .156616 .292168 .291966
0 0 .065083 .069128 .084391 .087832

1 -.023462 -.025672 .031551 .035121 .036552 .027735
-.017459 -.013225 .048898 .037227 .237570 .228585

2 .010426 .002822 .014661 .004359 .041094 .027036
.006065 .001322 .014912 .003887 .070729 .067993

3 -.001649 -.000217 .002591 .000346 .012494 .002774
-.000042 -.000099 .000526 .000294 .028602 .007382

4 -.000611 .000014 .000794 .000024 .000198 .000209
-.000565 .000006 .001385 .000019 .003709 .000565

5 .000333 -.000001 .000473 .000001 .001206 .000014
.000176 -0.000000 .000458 .000001 .001899 .000038

6 -.000043 -.000000 .000068 .000000 .000377 .000001
.000002 -.000000 .000005 .000904 .000002

 

.000000

 

 

175

 

 

Table 5.7
T = .05
n x2(n) xé(n) I n x2(n) xé(n) '[ n x2(n) xé(n)
0 .08 .08 91 -.097439 -.0945071 190 .178667 .200476
.0 .0 -.240697 -.223220§ .063017 .099918
1 .083056 .083045 100 -.048251 -.o41858? 200 .192377 ..223292
.136725 .136494 -.049784 -.045698j .068320 .108500
10 .111319 .111161 110 .026399 .030527f 210 .196989 .231996
.983884 .981730 -.027544 -.032584; .059421 .091049
20 .441535 .440753 1 120 .057672 .058503?
- .089893 '.086384 1 -.095404 -.100700; 2“ "97‘52 '222935
30 .592194 .590487 1 : '053894 -°85332
.769791 .766151 1 121 .058769 .030285' 220 .204229 .231503
1 -.086139 -.112179; .091511 .120050
31 .593010 .465931 3 130 .079053 .060754E 230 .213510 .243348
.665538 .529259 E -.035078 .019751? .097744 .127159
40 .436467 .378445 ' 140 .100778 .121672‘ 240 .217028 .247800
.027074 .241178 { -.034331 .031406; .093410 .120495
50 .185518 .258244 E 150 .106949 .142069;
.045237 .225835 i -.052787 -.0165205 24‘ '2‘7118 '242996
60 .075732 ..211630 E '0952‘8 '1‘5833
.193325 .344752 151 .107321 .1213521 250 .219593 .245323
-.040502 -.025555; .107060 .130206
61 .072933 .258245 : 160 .127713 .147831: 260 .222570 .248590
.200351 .361502 1 .033415 .0673663 .112065 .135945
70 .067100 .189223 ~ 170 .156057 .184268: 270 .223482 .249722
.258973 .131780 1 .042931 .077483 .113773 .137520
80 .083026 .094362 1 180 .167037 .198260:
.284971 .117392 1 .020077 .046373j 27‘ '223454 '248255
90 .097644 .057633 3 g '1‘3949 "35152
.272114 .209182 181 .167449 .183923; 280 .222807 .245876
.026283 .039331: .116424 .132585

 

 

 

 

 

 

 

 

 

n x2(n) xé(n) n x2(n) xé(n) n x2(n) xé(n)
290 .221877 .242672 331 .212048 .228456 391 .180464 .186673
.121006 .137086 .129074 .143802 .122591 .129043
300 .221422 .241304 340 .207110 .220415 400 .173644 .176626
.126335 .144690 .117455 .121471 .104289 .098934
350 .200586 .209417 410 .164687 .162857
30‘ '22‘340 '242352 .120705 .123936 .106410 .099952
"2509‘ '143372 360 .198085 .205053 420 .151315 .157447
310 '218155 '236597 .130771 .138310 .118002 .116079

.119201 .129120

320 .213880 .228734 361 .197915 .209173 421 .150233 .162682
.123042 .132527 .127862 .138453 .108133 .117001
330 .212184 .225573 370 .191787 .199765 430 .153998 .152558
.131292 .144290 .112111 .111030 .094816 .086093
380 .183715 .186882 440 .144661 .138676
.114762 .112700 .096457 .086588
390 .180653 .181802 450 .141162 .133235
.125918 .128439 .107996 .102368

 

 

 

 

 

177

 

 

 

 

Table 5.8
T = 1.0
n x2(n) xé(n) n x2(n) xé(n) n x2(n) xé(n)
0 .08 .08 100 .001886 .000185 210 .000068 -0.0
.0 .0 .003536 .001417 .000005 -0.0
1 .292168 .291966 110 .003093 .000496
.084391 .08782 .000696 .000636
10 .176842 .179724 120 .002591 .000346
.00644 .019437 .000526 .000294
20 .162823 .165221
.018140 .029970 121 -.ooo198 -.000209
30 .155724 .156616 ' 003709 ‘ 000555
.065083 .069128 130 .000852 .000011
-.000003 -.000108
31 .027735 .036552 140 .000719 -.000035
.228585 .237570 .001238 -.000046
40 .036730 .040747 150 .000794 -.000024
.019060 .030204 .001385 -.000019
60 .027406 .031095 -
.038055 .021736 151 -.001206 .000014
60 .031551 .035121 '°°°‘899 °°°°°38
.048898 .037227 160 -.000415 .000001
-.000382 .000007
61 .041094 .027036 170 -.000503 .000002
.070729 .067993 -.000444 .000003
70 .012468 .002888 180 -.000473 .000001
.014228 .013399 -.000458 .000001
80 .015736 .005756
.015115 .006912 181 .000377 -.000001
90 .014661 .004359 '000904 '°°°°°°‘
.014912 .003887 ‘90 '000045 '0'0
L .000111 -o.o
91 .012494 .002774 200 .000084 -0.0
.028602 .007382 .000013 -o.o

 

 

 

 

 

 

178

By investigating the results in Tables 5.5-5.8 we can observe

the following:

1.
2.
3.

Asymptotic stability of x1(é) and x2(é) as K increases.
Asymptotic stability of x2(n) as n increase.

The closeness of x1(é) and xs(K), K = 0,1,2,..., up to 0(6).

The closeness of x2(n) and xé(n), which is the approximated value
of x2(n), up to 0(6).

The abrupt change in x2(n) and xé(n) at the beginning of each
slow cycle which is a result of the abrupt change in the slow control
that excites the fast modes.

The settling-time of x2(n) is longer for the sampling period

T = .05 than T = 1. This is expected by the following estimate

for the settling-time

S.T. = max -——£LJi—- ,

1' lRe(A,-)|

where for the case 1 = .05 our settling-time is equal to

4.6 ___

T17"

S.T. = 46 (sec).

CHAPTER 6
CONCLUSION AND RECOMMENDATION

In this dissertation we investigated some problems which have
been open for the class of linear time-invariant singularly perturbed
difference equations. Indeed, we arrived at approximate control designs
by employing the two-time-scale property of such systems.

After providing a historical review which reveals different model
representations and sources of singularly perturbed difference equations
along with some structural properties for this class of systems, we in-
troduced a stability criterion for our system and obtained an initial
value result which approximates the solution of the full system by using
the solutions of slow and fast subproblems.

Ne, also, investigated the asymptotic behaviour of infinite-time
optimal regulators (linear quadratic) and showed that it does not follow
as a limiting case of the finite-time problem considered by Blankenship
[1980] and a Special scaling was employed to remove this difficulty.
Furthermore, conditions for independent design of slow and fast subsystems
were derived and by applying slow-fast decomposition, as in continuous-
time work of Chow and Kokotovic [1976], we acheived an 0(62) near-
optimal solution.

In contrast with continuous-time work, a priori knowledge of

perturbation parameter e is necessary for our design procedure.

179

Fin

180

The well-known difficulty in solving discrete-time "stiff"
Riccati equations was overcome by providing an iterative technique with
a fast convergence rate which avoids the ill-conditioning by dealing with
lower-order, slow and fast, subsystems. The efficiency and value of this
technique is more appreciated as the order of the system increases and

E gets smaller. We achieve off-line computational savings by solving

two lower-order, slow and fast, models and also, by avoiding the ill-

conditioned numerical problems.

Multirate control design of singularly perturbed systems to meet
desired objectives is an important problem. He studied the stabilization
of this class of systems using single rate and multirate measurements of
the state variables. Different design procedures for forming a stabilizing
composite feedback control were investigated and we showed that the 1
application of such controls results in asymptotic stability of the closed-
loop system and closeness of trajectories to those predicted by slow and
fast subsystems.

The proposed scheme has several computational advantages. The
off-line computational effort is reduced because design and simulations
are performed for two lower-order models instead of the full model. There
is computational savings because of the order reduction and avoidance of
stiff numerical problems.

The on-line computational effort is reduced because the slow
feedback signal has to be processed only at slow-time intervals rather
than fast-time intervals as in the single rate case. Of course, we have

the computational cost of predicting slow states, between the slow-time

181
intervals, in terms of their values at the beginning of each slow-period.
By means of numerical examples, our claims were confirmed.

The results obtained in this dissertation show that many of the
phenomena normally associated with continuous singularly perturbed systems
are also present in discrete systems. These results provide a foundation
for further research. In particular, our multirate stabilization results
can be extended to the class of output feedback control systems. Also,
with regard to the nonlinear continuous work [Peponides, et al., 1982]
and along the lines of multitime method of Hoppensteadt and Miranker
[1977] for difference equations, the extension of multirate stabilization
results to nonlinear case is.feasible. Furthermore, it seems evident that
our results could be extended to time-varying case in view of usual
features of time-varying systems.

An interesting research topic is the establishment of bounds on
e and deriving sufficient conditions, usually using matrix norms, for
validity of approximation results as in multirate stabilization and in-
finite-time regulator.

As was mentioned earlier, many systems possess a two-time-scale
property, while they are not, explicitly, in the singularly perturbed
form. In spite of some efforts for converting a given system of equations in-
to a singularly perturbed form as in [Phillip's, 1980], [Sain et.a1.,
1977], and recently Sycros and Sannuti [1983], more work is still needed

in that direction.

CTD:
PRT:
CHK:
FULMAT:
INIT:

CIT:

CITC:

DISRIC:

SRIC:

SF STAB:

STFUL:
FRIC:

FF3:
F12:
FES:
FXS:

THM2:

182

List of the Programs

Sampled-data of the continuous system.

Partions the full matrices A and B to find the block matrices.
Evaluates the eigenvalues of slow and fastsmatrices AS and A1,.
Knowing the block matrices finds the full matrix.

Initilizes the matrix Riccati P to its zero order terms and
sets the errors to zero for the iterative technique.

Finds the constant matrices of the iterative techniques.
Continuation of CIT.

Solves a discrete Riccati equation by fixed point method and
evaluates the gain F0.
Solves the slow subsystem Riccati equation and evaluates PS.
Demonstrates the stability of slow and fast subsystems and
evaluates xé(n) which is 0( ) close to x2(n).

Demonstrates the stability of the full system.)

Solves the fast subsystem Riccati equation and evaluates the
values of 03, L],L2,L3, and Pf (fast Riccati matrix).

Note that this program has to read the slow Riccati matrix PS.
Finds the matrix Y3 used in iterative technique, equation (3.79).
Finds the matrics y] and ya in (3.80) and (3.81). .
Finds the errors E],E2, and E3 at each iteration.
Finds the full system Riccati matrix (3.33)-(3.35) using fixed
point method.

Demonstrates theorem 2 of chapter 3.

183

Remarks:

i) In the iterative technique the sequence of execution of the programs
should be as follows:
CTD, PRT, INIT, CIT, CITC, SRIC, FRIC, FF3, F12, FES, and FXS.

ii) For obtaining an extra 0(6) accuracy in each iteration the sequence

FF3, F12, FES, and FXS should be executed.

184

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