MSU

LIBRARIES
m

\—

 

 

RETURNING MEIERIALS:
Place in book drop to
remove this checkout from
your record. [1mgg will
be charged if book is
returned after the date
stamped be10w.

 

 

 

 

MULTIMODEL CONTROL AND ESTIMATION
OF LINEAR STOCHASTIC SYSTEMS

BY

Zoran Gajic

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Electrical Engineering and
Systems Science

1984

ABSTRACT

MULTIMCDEL CONTROL AND E
OF LINEAR STOCBASTIC 8'

BY

Zoran Gajic

The fixed point method is developed for obtaining
efficient numerical solutions of linear-quadratic Guassian
problems. As a crutial step in this method, a fixed point
algorithm for the solution of algebraic singularly
perturbed Riccati equations is derived. It is shown that
each iteration step improves the accuracy by an order of
magnitude, ime., the accuracy of 0(ek) can be obtained by
doing only k-l iterations. On the other hand, only low-
order systems are involved in albegraic manipulations, and
no analicity requirements are imposed on the system
coefficients.

The concept of multimodeling control is the subject of
consideration in the third chapter of this thesis for the
case of linear stochastic systems. We study this problem
without scaling the white noise input in terms of
singularly perturbed parameters. In this case, white noise
is affecting the system even in the limit as perturbation
parameters tend to zero so that no straight forward
extention of determinisric results is possible. It is

shown that well-posedness cannot be achieved without

Zoran Gajic

communication of decisions, which in fact allows us to
capture information lost through order reduction.

In the fourth chapter, we prOposed a new decentralized
estimation scheme. Instead of doing estimation with
locally optimal filters with necessary generation of
correction terms in order to construct global optimal
estimate, we propose a form for locally nonoptimal filters
such that the global optimal filter can be obtained without
any correction terms. In general, our scheme is much
simpler for both off-line and on-line computations.

Applied to systems with slow and fast modes, that have been
treated in Chapter III, this procedure gives further

advantages.

ACKNOWLEDGMENTS

I want to express my deep gratitudes to Dr. Hassan
Khalil, my thesis advisor, for all his patience, guidance,
and encouragement during the course of my research. His
expert advice, useful insight, and stimulating discussions
made this work possible.

I would like to thank my committee members, Dr. R.
Schlueter, Dr. R. O. Barr, and Dr. H. Salehi for their help

and valuable suggestions.

TABLE OF CONTENTS

LIST OF TABLES O O O O O O I O O O O O O O O O O O 0

LIST OF FIGURES. O O O O O I O O O O O O O O O O O O

I.

1.1

I02

II.

II.l

II.2

II.2.

II.2.
II.2.
II.2.

II.2.

11.3

11.3.

1103.

INTRODUCTION 0 O O I I O O I O O O O O I O O O O

Singular perturbations as time scale decompo-
sition technique. . . . . . . . . . . . . . . .

Singularly perturbed linear quadratic Guassian
problem . . . . . . . . . . . . . . . . . . . .
CONTROL AND ESTIMATION OF SINGULARLY PERTURBED
LINEAR SYSTEMS. O O O O O I O O O D O O O O O O

Linear-quadratic Gaussian problem for
singularly perturbed systems . Main results. .

The fixed point solution of LQG problem of
singularly perturbed systems . . . . . . . . .

l The fixed point solution of Riccati
equations. 0 O O O I O O O O O O O O O O O O

1.1 Zero order solution. . . . . . . . . . . .
1.2 Solution of higher-order of accuracy . . .
2 Fixed point solution of L and M equations. .
3 The iterative solution of LQG. . . . . . . .
Well-posedness of model order reduction for
singularly perturbed linear stochastic
systems. 0 O O O O O O I O I O O O O O O O O O

1 State estimation . . . . . . . . . . . . . .

2 LOG CODtrOl. o o o o o o o o o o o o o o o 0

i1

page

iv

17

18

24

24
24
26
34
36

44
46

III. MULTIMODEL CONTROL OF LINEAR STCCHASTIC
SYSTEMS. O C O C C C O O O I O C O C I O O

III.l Previous results in multimodel control. .

III.2 Problem statement and formulation of
stochastic multimodel strategies. . . . .

III.3 Proof of well-posedness of multimodel
Strategies. 0 O O O O O O O O O O O O O 0

111.4 Purther decomposition of multimodel
strategies. . . . . . . . . . . . . . . .

Appendix 3.1. . . . . . . . . . . . . . .
Appendix 3.2. . . . . . . . . . . . . . .
Appendix 3.3. . . . . . . . . . . . . . .
Appendix 3.4. . . . . . . . . . . . . . .

Appendix 3.5. . . . . . . . . . . . . . .

IV. DECENTRALIZED MULTIMODEL ESTIMATION . . . .
IV.1 New method for decentralized estimation. .

IV.2 Problem statement and formulation of
multimodel estimation. . . . . . . . . . .

IV.3 Finite time multimodel estimation . . . .

Appendix IV.l . . . . . . . . . . . . . .

V. CONCLUSIONS. 0 O O O O O O 0 O I O O O O O 0

iii

page
50

56

75

80
88
91
97
100
106

110
110

119

127

138

147

150

Table
Table
Table
Table
Table
Table

Table

2.2
2.3
3.1
3.2
3.3
3.4

LIST OF

iv

TABLES

page
33
34
42
74
97
98
99

LIST OF FIGURES

page
Figure 1.1 . . . . . . . . . . . . . . . . . . . . . 13
Figure 4.1 . . . . . . . . . . . . . . . . . . . . . 114
Figure 4.2 . . . . . . . . . . . . . . . . . . . . . 115
Figure 4.3 . . . . . . . . . . . . . . . . . . . . . 116
Figure 4.4 . . . . . . . . . . . . . . . . . . . . . 117
Figure 4.5 . . . . . . . . . . . . . . . . . . . . . 118
Figure 4.6 . . . . . . . . . . . . . . . . . . . . . 119
Figure 4.7 . . . . . . . . . . . . . . . . . . . . . 126

Figure 4.8 O I I O 0 I O O O O O O O O O O O O O O O 137

INTRODUCTION

INTRODUCTION

Motion of dynamical systems under influence of the
environment can be represented by the controlled

differential equation
im = f(x(t), u(t), t) (1)

where u(t) is the impact of the environment, t is time and
x(t) is the state of the system. In this case, the
environment has at any time t its program u(t) of effecting
the system. In more general case, the system and

environment are in a close tie so that
fit) = f(x(t), u(x(t)), t) (2)

represent a self-contained system-environment evolution
equation.

For a control engineer u(t), so called an open loop
control, and u(x(t)), known as a closed loop or feedback
control, are tools of driving the system in desired state
subject to optimizing its efforts (minimizing required
time, energy and so on). Those constraints can be

represented by introducing an additional functional

J = from», u(t), t)dt (3)

which has to be optimized.
In a more realistic case, random quantities are

present and equation (2) becomes
i<t) = f(x(t), u(x(t)), w(t), t) (4)

where w(t) is a random process.

Since a control engineer has to deal with the systems
of very different natures, he or she would be lucky if all
variables operate in the same time scale (approximately of
the same speed). Most likely variables will have different
speeds and the theory of singular perturbation might be
well suited to capture such situations by introducing time

scale separation corresponding to variables of a different

speed
il(t) = f1(x(t), ul(x(t)), w(t), t)
21§2<t) = f2(x(t), u2(x(t)), w(t), t) (5)
es_lxs(t) = fs(x(t), us(x(t)), w(t), t)

where (le, xTz, ..., x571. = x is a partition of the state
vector x subordinate to the time scales 1, ...., s, and
51. ..u, €S_1 are small positive parameters. In the most
general case, with multiple time scales, when partition of
x subordinate to the time scales is not possible, we are
faced with stiff differential equations describing the

motion of a dynamical system. Those equations are

extremely hard to handle and have not been studied so far in
the control framework.

In this thesis, we will study dynamical systems having
the form of (BL. Before we formulate our problems and
indicate the ways for their solution, we want to point out
some general features of (5) which are very well known in
the literature.

First of all, applying standard numerical methods for
solutions of singularly perturbed problems leads to
numerically ill-defined problems. In other words, when
ei-+ 0, slope of differential equations tend to infinity,
whereas Jacobian of corresponding algebraic equations is
close to being singular. Special numerical techniques are
necessary. Development of these techniques goes together
with studying the general theory of singular perturbation
which contains the main idea of decomposing a problem into
several subproblems in different time scales, where those
subproblems are well defined. On the other hand, presence
of random qualities which we assume throughout this work to
be Gaussian white noise makes those problems more difficult
to solve and some of the deterministic techniques cannot be
extended to the stochastic problems.

In the first chapter of this thesis, an introduction
to the theory of singular perturbations with application to
control problems is given. The emphasis is on
linear stochastic systems with quadratic performance

indices which are the subject of our study.

In the second‘chapter, a fixed point method is
developed for obtaining efficient numerical solutions of
linear-quadratic Guassian problems following its analytic
solution presented in [4]. As a crutial step in this
method, a fixed point algorithm for the solution of
algebraic singularly perturbed Riccati equations is
derived. It is shown that each iteration step improves the
accuracy by an order of magnitude, iae., the accuracy of
0(8k) can be obtained by doing only k-l iterations. On
the other hand, only low-order systems are involved in
algebraic manipulations.

The concept of multimodeling control which has been
formulated and studied in [14] for deterministic systems is
the subject of consideration in the third chapter of this
thesis for the case of linear stochastic systems. To the
contrary of [17], [18], we study this problem without
scaling the white noise input in terms of singularly
perturbed parameters. In our case, white noise is
affecting the system even in the limit as perturbation
parameters tend to zero so that no straight forward
extention of deterministic results is possible. It is
shown that well-posedness cannot be achieved without
communications in decision, which in fact allows us to

capture information lost through order reduction.

5

In the fourth chapter, we proposed a new decentralized
estimation scheme. Instead of doing estimation with
locally optimal filters with necessary generation of
correction terms in order to construct global optimal
estimate [29] - [32], we propose a form for locally
nonoptimal filters such that the global optimal filter can
be obtained without any correction terms. In general, our
scheme is much simpler for both off-line and on-line
computations producing the same results as [29] - [321.
Applied on systems with slow and fast modes, that have been
treated in Chapter III, this procedure gives further
advantages leading to something that we call decentralized
multimodel estimation (its final result is of multimodel

type).

1.1 SINGULAR PERTURBATIONS AS TIME SCALE DECOMPOSITION

TECHNIQUE

Singularly perturbed systems of differential equations
are characterized by the presence of a small positive

parameter e multiplying some derivatives

>6
F;
I

- f(xl, x2, 6, t). xl(to) x10 (1.1a)

m

x
M
I

- 9(x1, X2! E, t) I X2<to) X20 (lolb)

This type of systems display slow-fast time structure.
Because of smallness of e the derivative of x2 is
relatively much larger than that of XI, at least at the
initial time to, so that x1 is called a slow variable,
contrary to x2 which is a fast variable.

Singularly perturbed systems of differential equations
were studied in mathematics starting in 1952 by Tikhonov,
Vasileva, OPMalley, Hoppensteadt, and others. They were
brought to the control literature in 1968 by Kokotovic and
since have been studied'extensively by him and his
coworkers, Sannuti, Khalil, Chow, and others.

There are simple reasons for the interest in singular
perturbation methods in control. First, dynamics of

physical systems display slow and fast phenomena. In many

7

cases, their models either appear in the explicit
singularly perturbed form (5), e4}, when the singular
perturbation parameters represent small time constants,
masses, moments of inertia, inductances, capacitances,
resistances, viscosity coefficients, etc., or they can be
brought into that form using insight gained from
engineering experience. Second, singular perturbation
methods avoid the stiff numerical problems associated with
the interaction of slow-fast phenomena. Third, singular
perturbation methods lead to systematic procedures for
model order reduction and enable analytical studies of the
impact of using reduced—order model in optimal and feedback
control.

The time scale decomposition in singular perturbation
methods starts by defining a reduced system which is

obtained by setting 8 = O in (1.1).

51 = f(.}ElI £2! or t) 3'1‘t0) = X10 (1.23)

{D
I

" 9(XI’ 2‘2, 0, t) ‘9': £2 = '¥ (31! t) (1o2b)

The reduced system is used to generate satisfactory
approximation of x1 and x2. Obviously in order to be able
to solve (1.2b) we need nonsingularity of corresponding

Jacobian, i.e.,

'39
5;;(31' 32: or t) + 0 (1'3)

On the other hand, since

x2(to) = v«x1(to), to) i x20 (1.4)

we should not expect to have uniform convergence for the

variables x2. i.e.,

x2(t) + x2(tY uniformly VteItop T] (1.5)

6+0

where T 3}» determines the interval of interest. Thus, the
discrepancy between the time behavior of reduced system
(1.2) and its original form (1.1) full system, comes from
the fast transient. Since we neglect 0(a) quantities in
(lJ ), the best approximation that we can expect is to

achieve

x1(t) = x1(t) + 0(a) , Vtetto, T] (1.6a)
x2(t) = x2(t) + 0(a) , Vtettl, T] t1 > to (1.6b)

In order to improve the approximation of x2. we have to

study a boundary layer correction which is given by

diz ~ t-to ~
5;- 3 9(X10' 32(1), 0, to); T: ’::-3 X2(to) = X20 (1'7)

Then the following is valid

X2(t) = git) + (i2(t) ‘ 32(to)) + 0(8) Vksltol T] (1.8)

9
Of course, we need additional assumptions that guarantee
the existence of (1.7). They are given as a part of the

following Lemma.

LEMMA.1.1: If the equilibrium 32(to) is asymptotically
stable uniformly in x10 and to and x20 belongs to its .
domain of attraction then the solution of (1.7) exists for
all t z 0.

Adding analicity assumptions on the right hand sides
of (1.1),the procedure can be generalized by introducing
Taylor series expansions in e and forming corresponding
reduced and boundary layer systems by matching terms of
same power of e; In that case, the approximation up to
required order, let us say k, is constructed by forming k-
families of slow-fast subproblems. Then we achieve

approximations as

1,2,... (1.9a)

- k
xlappu) - x1(t) + 00: > . Vtelto, T] k

(t) = x2(t) + 0(sk) , Vtetto, T] k 1,2,... (1.9b)

2‘-2app

I.2 SINGULARLY PERTURBED LINEAR QUADRATIC GAUSSIAN PROBLEM

Since in the following chapters we will be dealing
with linear stochastic systems, we want to summarize the
basic results known so far in the control literature. The
steady state linear-quadratic Gaussian problem is defined

by the state equation

10

i = Ax + Bu + Gw E{x(to)} ='§o (1.10)
with measurements
y = Cx + v (1.11)

and criterion that has to be minimized

t
J = lim --$-- E j 1th(t)DTox(t) + uT(t)Ru(t)]dt , R > o
to 7"” tl-t to

where xeRn is state vector, ueRm is the control input, yzaRz
observed output, weRr1 and Veer are zero-mean, stationary,
white Gaussian noise with intensities W > 0, V > 0.

The optimal solution obeys the so-called separation
principle dividing our task in two independent problems:
1) the optimal estimation, and 2) the optimal control

problem. The optimal estimate is given by the well known

Kalman filter

i = Ai + Bu + K(y - Ci) i(to) = i5 (1.13)
where K is a filter gain given by

K = QcT'v"1 (1.14)
and Q satisfies algebraic Riccati equation

QAT + AQ - Qch'lco + GWGT = o (1.15)

Then the optimal control is a linear combination of the

optimal estimates, i.e.,

11

u = -F§ = -R’13Tpi (1.16)

where F is a regulator gain, whereas P is a solution of the

algebraic regulator Riccati equation
PA + ATP - PBR'IBTP + DTD = o (1.17)

A singularly perturbed version of this problem was
studied in [1] - [4]. Having slow and fast variables

preSent (1.10) - (1.11) can be represented as

x1 = Alxl + AZXZ + Blu + le (1.18a)
.12 = A3x1 + A4x2 + Bzu + sz (1.189)

y = Clxl + C2x2 + v (1.19)
and
J = tim ;-£-- B{JF1[sz + uTRuldt (1.20)

+-a> -
ti"”” 1 to to

where
z = Dlxl + 02x2 (1.21)

The optimal solution (1.13) - (1.17) is now parameterized
in the small parameter a in such a way that the problem is
not trivial anymore. Special form of matrices A, B, and G

are

12

A A B G

A = l 2 I B = l , G = l (1.22)
A3 A 4 B 2 G 2
1:" 2' '2' 2

so that in the limit as e + 0 these matrices blow up
ahd so do the variance of the fast variable and optimality
criterion making the problem ill-defined.

The first attempts of solving this type of problem can
be found in [l] and [2]. Both papers extend the ideas
developed for deterministic singular perturbations to the
stochastic problem. We will give the main results of [2]
where the control problem has been treated.

Formally plugging 2:: 0 (1.18b) and accepting the fact
that in the limit the approximation for the fast variables

is a linear combination of white noise, i.e.,
-1
x2 = A4 (A351 + B211 + G2W) (1.23)

they derived nice slow-fast decomposition scheme, using low
order Kalman filters operating in different time scales.
Obviously this was done under assumption that A4 is a
nonsingular matrix. This scheme is represented .in

Figure 1.1.

13

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

U Y

System

u=uS + uf u u

+ 2LT 5 FT Slow filter
Fl 0 (1.32) J.

Y

Co
u _
f Fast filter

 

 

 

 

(1.33)

 

 

 

 

Fig. 1.1 Formal slow-fast decomposition scheme for LQG.

In fact, the overall optimization process is replaced
by two low order slow and fast LQG problems. The slow LQG

problem is defined by

x5 = ons + Bens + Gow (1.24)
y = Coxs + NouS + How + v (1.25)
. . 1 t1 TDT T TD TR -
Js = 11m ----- E j” [xs oDoxS + szEo ous + u8 ouslot
t +"' 00 tl'to t0
t1... (1.26)
where
A A A A-lA B B A A-lB
o ’ 1 ‘ 2 4 3 o = 1 ‘ 2 4 2
G G A A'lc c - c c A-lA
o = 1 ' 2 4 2 o ‘ 1 ‘ 2 4 3
N - c A-lB H - c A'lc
o ' ‘ 2 4 2 o ’ ' 2 4 2

14

1 1

T
R0 = R + EOEo

On the other hand, the fast LQG problem is defined by

elf = A4xf + Bzuf + 62w (1.27)
yf = C2xf + v (1.28)
1 t1 T T T
Jf = 1im ----- E j” [foznzxf + ufRufldt (1.29)
to+-°° t1‘t to
tl-F 00

-inf (1.31)

“f
with corresponding slow filter

4 = ons + Bou + Kofy - Nou - c0451 (1.32)
and fast filter

eif = A4xf + Bzuf + K2(yf - czfif) (1.33)

where regulator and filter gains are obtained from

T T
o - R0 (EODO + BOPO) (1.34a)

'13
I

- T
R 13292 (1.34b)

u:
M
II

15

and
- cT + w ) '1 (1 3= )
T -1 I
with

T
Vo = V + HOWHO

P0' P2 and 00' 02 are solutions of regulator and filter

slow and fast Riccati equations

-1 T -1 T T
Po‘Ao ' B0R0 EoDo) + (A0 ’ B0R0 3000) P0
T -1 -1 T
+ DO(I - EORo EO)Do - POBORO BOPO = 0 (1.36a)
T T -1 T
and
T - T -
(A0 - cownovolcomo + Q°(Ao - Gowaovolco)T
(1.37a)
T -1 T T -1
+ G0(W - WHOVo H0W1Go - QOCOVO CoQo = 0
T T T -1
A402 + 92A4 + czwc;2 - ozczv2 c202 = o (1.37b)

Under the conditions that guarantee the existence of
positive semi-definite stabilizing solutions for Riccati
equations (1.36), (1.37) the suboptimal control is defined

as

A

u = uS + uf = -Foxs - szf (1.38)

16

It is interesting to point out (see Figure 1.1) that this
control drives both system and slow filter. Under this
suboptimal control, the relative error in J is 0(8), the
errors for the slow trajectories are 0(6), whereas the
trajectories of the fast variables are 0(61/2) apart.

Even though this approach leads to a nice slow fast
decomposition reducing both on and off-line computations,
it does not allow us to achieve approximations of order
higher than 0 (E).

A different approach has been taken in [41. It can be
used to get the approximation of an arbitrary order of
accuracy 065k), k = 1,2,".. We summarize its main ideas
and results at the beginning of the next chapter, since it
is the basis of a numerical algorithm developed for solving
singularly perturbed LQG which we are going to present in

the next chapter.

II. CONTROL AND ESTIMATION OF SINGULARLY

PERTURBED LINEAR STOCHASTIC SYSTEMS

II. INTRODUCTION

We present the main results for the estimation and
control of singularly perturbed linear stochastic systems.
Based on these results, a new fixed point iteration
algorithm is developed to overcome the main problem of
solving corresponding Riccati equations. It is shown that
each iteration step in the proposed method improves the
accuracy by an order of magnitude, i.e., the accuracy of
O(ek) can be obtained by doing only k-l iterations. On the
other hand, only low-order systems are involved in
algebraic manipulations, so that, from a computational
point of view, the algorithm is very efficient. Solutions
of Riccati equations and of overall LQG problems are
illustrated by realistic examples.

At the end of this chapter, we study the use of the
reduced-order model, obtained by neglecting fast variables,
in solving the estimation and control problems for
singularly perturbed systems. It is shown that the use of
reduced-order models leads to near optimal solutions in the
estimation problem but not in the control problem. The
fact that the reduced-order models do not result in near-
optimal control strategies plays a crucial role later on in

developing multimodel strategies.

18

II.1. LINEAR-QUADRATIC GAUSSIAN PROBLEM FOR SINGULARLY

PERTURBED SYSTEMS. MAIN RESULTS.

Singularly perturbed linear stochastic estimations and
control problems were studied in the past decade by a few
researchers [1-4]. The recent paper [4] seems to be the
most complete one. It alleviates the difficulties of the
previous approaches and is conceptually simple. We
briefly summarize the main results of [4]. Consider the

singularly perturbed system

x1 = Alxl + A2x2 + Blu + le (2.1)
6x2 = A3x1 + A4x2 + Bzu + sz (2.2)
y = Clxl + C2x2 + v (2.3)
n1 n2 m .
where xleR and xzeR are state vectors, usR IS a

r r
. l 2
control input, weR and veR are zero-mean,

stationary, white Gaussian noise with intensities W > 0 and
V > 0 respectively, and 6 is a small positive parameter.

In the following Ai' Bj, Gj, Cj, i a l, ...., 4, j a l, 2
are constant matrices; in general, they are analytic
functions of s (41. With (2.1) - (2.3) consider the
performance criterion

T
t x x
J = lim --l-— E jfl 1 R1 1 + uTRzu dt
t »~m t1-to t

x x
o 2 2
t1*‘” 2.4)

 

19

with positive definite R2 and positive semi-definite R1:
which has to be minimized.

The optimal control is given by
u = -Fl(e)§1 - r2(e)§2 (2.5)

where £1 and £2 are optimal estimates of the state vectors

x1 and x2

- Alil + A2X2 + Blu + Kl(€)(y - Clxl ' C2x2) (2.63)

X
p.)
I

"’ A3X1 + A4§K2 + 3211 + K2(€) (y " Clxl " szz) (2.61))

m
x

N
I

The matrices F1, F2 and K1, K2 are regulator and filter

gains respectively.

-1 T T T -l T T
T T -1 T T T -1
K1 3 (Qlcl + 02C2)V I K2 = (Echl + 03C2)v (2.7b)

where Pi' Qi' i = 1,2,3 are solutions of corresponding

regulator and filter Riccati equations
AT(s)P(s) + P(e)A(s) - P(s)SR(e)P(€) + R1 = o (2.8a)

A(s)Q(e) + Q(c)AT(s) - Q&:)'SF'Q(6) + G(e)WGT(s) = o
(2.8b)

with scaling compatible to the nature of their solution

P 6P Q Q
2(a) = % 2 0(6) = i 1 2 (2.9)
8P2 8P3 Q2 :03

20

and newly defined matrices as

B G
A(s) = A1 A2 , 8(6) = l r G(€) = l
E E , E E

(2.10)

1

- - _ - T . _ T -1
c - (C1, C21 , sR(e) — B(€)R2 B (-) , sF - c v c

Eliminating u from (2u6), by using (2.5), the optimal
filter can be represented as a system driven by the

innovation process v = y - Clxl - szz

A

X1 = (A1 ‘ B1F11x1 + (A2 - BlF2)X2 + Klv (20113)

6:22 (A3 - 3291):“:1 + (A4 - 3292):“:2 + K26 (2.llb)

As was shown in [4], for the purpose df achieving
decomposition on the slow and fast variables, this filter
is transformed via the use of a nonsingular transformation

[10] into new coordinates

n I - eML -€M i
.1 = “1 (2.12)
112 L I X2

so that filter becomes

”1 = [(A1 - BlFl) - (A2 ‘ 81F2)L]n1 + (Kl - MKZ - EMLth)
(2.13a)

(2.13b)

21
with the innovation process v = y - (Cl - CZLJnl - [C2 +

The optimal control is now given by

u = -(F1 - F2L)nl - [F2 + e<Fl - FZL)h]n2 (2.14)
Matrices L and M satisfy

(A4 - BZF2)L - (A3 - B2F1) - eLHA1 - BlFl)

Thus, in order to find the optimal solution in the
decomposed form above, we have to solve two Riccati
equations (2.8), a weakly nonlinear equation (2.15a) and a
linear equation (2.15b).

The following Lemma is summarized from [4], [61.
t£MMAi 2.1: If A4 is nonsingular and the triples (A0, 30'
D0), (A0, 60' C0), (A4, 82, D2), (A4, G2, C2) are
stabilizable and detectable then, for a sufficiently small€
(2.8) have unique stabilizing solutions which possess power
series expansions at e = 0.

All matrices appearing in Lemma 2.1 are defined in the
previous chapter.

Using results of this Lemma, the approximate

stabilizing control is defined as

22
(k) _ (k) (k) (k) . (k) (k) - (k)
-(Fl - F2 L )nl ' [F2 +c.(Fl

uapr ’
_ F2(k) L(k))M(k)]fiz(k) (2.15)
with approximative filters

31(k) = [(A1 - BlF1(k), _ (A2 _ BlF2(k),L(k)]gl(k)

582(k) = [(A4 ’ BzF2(k)) + €L(k)(A2 ' BlF2(k)) 1fi2(k)
+ (K2(k) + €L(k)Kl(k))v(k) (2.17b)

where V(k) = Y " (Cl " C2L‘k))fil(k) " [C2 +8 (Cl

- CZL“”)M‘k’1fi2(k’ and

(k) -1 T (k) T<k) _ k
F1 R2 (Blpl + Bsz ) - Fl + 0(5 )

(k) '1 T (k) T (k) _ k
F2 R2 (€B1P2 + BzP3 ) - F2 + 0(8 )
(k) - (k) T (k) - k
Kl - (Ql C1+02C2)V1- Kl + 0(€ )

(k)
K2

(eQT(k)CT + Q 3(k’c2wl = K2 + o<ek)

The main result from [4] can be now summarized in the

following thereom.

THEOREM 2;1: Suppose that conditions of Lemma.2.l hold.
Let x1 and x2 be the optimal trajectories and J be the

optimal value of the performance criterion. Let x1(k),

(k)
I

x2 and J‘k) be the corresponding quantities under the

23

approximative control law u(k) then

apr

2551:! g 0(Ek) (2.18)
J

var(xl - xl(k)) = 0(s2k) (as t+-m ) (2.19a)

var(x1 - x2(k)) = 0(s2k'1) (as t-+w ) (2.19s)

Note that by choosing appropriate initial conditions for
(2.17) as fil‘k)(0)==(1 - eu‘k’r‘k’)il(0) - am<k’£2(0) and
82"”(0) . L(.k)xl(0) + 3:2(0), then (2.19) holds for all
.t z 0.

Using standard techniques [5] for solving Riccati
equations (2.8) can be inappropriate in this case. They
can be unefficient - very slow convergence or even worse,
when a is very small and Jacobian close to be singular,
they can produce wrong results.

One way to overcome this problem is a power series
expansion with respect to small parameter 2. Matching
corresponding coefficients we can get a family of well
defined problems for which standard techniques are
applicable. However, in the case when we are interested
in high order of accuracy, the size of required
computations can be considerable, even though we are faced
with solving low order problems. Then advantage of doing

series expansion is questionable. Presence of small

parameters: gives us another way of solving these Riccati

24

equations, which we are going to explore in the next

section.

11.2 THE FIXED POINT SOLUTION OF LQG PROBLEM OF

SINGULARLY PERTURBED SYSTEMS

As a summary of the previous section, we can notice
that the complete solution of LQG for singularly perturbed
systems is based on the solution of the algebraic Riccati
equations (2.8) and algebraic equations for the
transformation matrices L and M, i.e., (2.15). Thus, the
nature of their solution determines the nature of the
overall LQG problem. We are going to explore the use of
fixed point iterations to solve those equations in an

efficient way.
II.2.1 THE FIXED POINT SOLUTION OF RICCATI EQUATION

Let us consider a regulator Riccati equation (2.8a)
only. Filter Riccati equation (2.8b) can be solved in
exactly the same way by using analogies which we will point

out later.
11.2.1.1 ZERO ORDER SOLUTION

Partitioning (2.8a) we have the following set of
equations

T T T T m

T T

25

T T - T
0 3 -P1A2 - P2A4 - EAlPZ - A3P3 + Eplslpz + PlSP3 + thS P2

' T

AT T T 2T T
0 = —cP2A2 —EZA2P2 - P3A4 - A4P3 + E P231P2 + €P2$P3 +

2:155sz + p3szp3 - 033:2 (2.22)
where

T T -1 T -1 T
R 01 01 01 p2 S1 = Ble B2 S2 = B232 B2
1 " T T -1 T

setting 25: O in (2.21) - (2.22) results in
T T T T
0 = ‘BlAi ' A121 ‘ B2A3 ‘ A332 + 215131 + 31522 + stTfl

T T
225222 ’ 0101 (2.23)

+

T T I
0 = .21A2 ’ £2A4 ' A323 + 21823 + £28223 - 0102 (2.24)

T T
’B3A4 ' A423 + B38223 ‘ 5292 (2.25)

O
N

This 'zero' order system has decoupled form and can be
solved like two low-order Riccati equations. Equation
(2225) is one of them whereas the second one is obtained
after eliminating 22 from (2.23), by using (2.24) and
(2.25), [6],

26

A A

0 = ’21A " AAT‘El + '21 . SR . El " R1 (2026)
A - A - B R-1 T o
’ o o o r o
‘ -1 T
_ T -1 T
R1 - 00(1 - rRo r )DO
A - A A A'lA B - B A A-lB
0 - 1 - 2 4 3 o ‘ 1 - 2 4 2
-1 -1 T
p0 = 01 " 92A4 A3 I 8 -pzAd Bz R0 = R2 + r r

The remaining unknown 22 is given by

22 3 £121 ‘ 22 (2.28)

where

21 ($23 - A2)LA4 - $223)'1
_ T T -1

Our zero order solution has a form

£1 522

B 3 T
822 6 B3
II.2{L2 SOLUTION OF HIGHER-ORDER ACCURACY

Zero order solution 2 is 0(8) close to the exact one.

Let's relate them through the error term E
SE = P - E (2.29)

or by using a compatible partition

27

.~2 _ ..
2 T q = T (2.30)
8 52 €453 €(P2 ' 22) 6 (P3 " 23)
Clearly 0(ek) approximation of B will produce 0(ek+l)

approximation of the sought solution P. That is why we are
interested in finding a convenient form for the error term
and an appropriate algorithm for its solution.

Subtracting (2.25) from (2.22), (2.24) from (2.21) and
(2.23) from (2.20) and after doing some algebra we get the

following set of error equations

T
31121 + 12151 = DTHI + 319 + DTH3Q + 532 (2.31)
T - ( 2)
E2123 + E1921 + 92253 “ H1 2-3
T -
2393 + 2323 - 33 (2.33)
where

D3 = (A4 - 32.23)

T T
222 (A3 ‘ 5 31 ’ 5232’

‘1
D 3 D3 1222 (2.34)

28

U
H
I

T - ..
‘A1 ' S1B1 ' 522) ’ 221133 22 = 911 ' 92193 E22
1 T‘ .. ,
T T T
T T

T T '1‘

Note that under bar denotes matrices (evaluated ate.= 0)
and that right hand sides of (2.31) - (2.33) are known up

to 0(a). Then (2.31) - (2.33) can be represented as

T

EBQB + D333 = Y3 ‘l" €h3(E2,E3'€) (2.35)
T _ E (

8223 + EIDZI + D2233 - Y2 + h2 El'EZ'E3’E) (2.36)

E + TE = v + 8h (E E E e) (2 37)

191 D1 1 1 1 1' 2' 3' -

where Y1, Y2 and Y3 are known constants. These equations
have a very nice form since all cross-coupling terms and
all nonlinear terms are multiplied by a small parameters..
This suggests that a fixed point algorithm can be efficient

for their solution. Let's propose the following algorithm
(1+1) T (1+1) - (1)

E3 123 + D333 - H3 (2.38)
(1+1) (1+1) T (1+1) - (1)

E2 23 + El Q21 + 22233 - 31 (2.39)

. T . T . .
El(1+1)Dl + D1E1(1+1) -..-. DTHI(1)+ 31(i)n + DTH3(1)D

31(0) = E2‘0’ = 33(0) = o; i = o,1,2,....

29

then the following thereom indicates its features.
THEOREM 2.2: Under the conditions stated in Theorem 2.1,
algorithm (2.38) - (2.40) converges to the exact solution B

with the rate of convergence of 0(a), i.e.
llE-E(i+l)ll =0(€)HE-E(i)!l 1=o,1,2,...
or equivalently

(18 - E‘i’li = 0(ei)

Proof: As a starting point, we need to show the existence
of a bounded solution of E1, 82 and E3 in the neighborhood
of €i= 0. To prove that by the Implicit Function Theorem,
it is enough to show that the corresponding Jacobian is

nonsingular atE: = 0. The Jacobian is given by

1‘21 P22 I’23 (2.41)

 

 

For its nonsingularity P11, F22, and P33 have to be
nonsingular. Using a Kronecker product representation,

they have the form

T T

1‘11 ‘ In]x 21 + 21 x In1

F a T I

22 D3 x n2 (2.42)
T T

3O

Matrix D3 is a closed loop matrix of the fast subsystem
(2.2) and thus stable matrix by well-known properties of
the solution of Riccati equations [7]. Matrix 21, can be
easily shown to be the closed loop matrix of the reduced
system and it is stable, too [7]. Then matrices F11, T22,
and F33 are nonsingular [8], so does the Jacobian.

For i = 0,(2.33) and (2.38) imply
T
(B3 - 33(1))23 + 23(33 - 23(1)) =s133(32,33;:) (2.43)

which by stability of Q3 and the existence of bounded

solution of £2 andTE3 gives
[:23 - 23‘1’Il = 0(a) (2.44)
Similarly, from (2.3)) and (2.40) we have

H31 - 21ml! = 0(a) (2.45)
and from (2.32) and (2.39)

[[32 - 22(1) I! = 0(a) (2.46)

Continuing the same procedure it can be easily shown that

(131 - Bl‘i'l’l! = 0(61)
1132 - az‘i’l’ii a 0(ei) 1 = 1,2,... (2.47)
H 33 _ 53(i-l)i{ = 0(ei)

which by stability of D1 and D3 and boundness of El' 22,

and E3 imply

31

Hal-21‘“); =0(si)

H82 - 132(1)]! = 0(81) (2.48)
[IE3 - 23(1’H = 0(ei)

or

HE - E‘i’l! = 0(ei) (2.49)

which proves the theorem.

Thus, the overall solution of n1 + n2 dimensional
Riccati equation can be found by solving two lower-order
Riccati and two lower-order Lyapunov equations which can
bring considerable savings in the size of required
computations, especially when systems of high order are
considered.

It is interesting to point out that this algorithm
gets better as 6 gets smaller (usually converges in only 1
or 2 iterations). On the other hand, standard methods get
worse and worse with decreasing value of s. The complete

algorithm has the following form

ALGORITHM
STEP 1.
Solve (2.25), (2.26), (2.28), i.e. find 21, £2, £3

STEP 2.

Find D3: D1: D1 2211 D22

STEP 3.

i = 0; El(0) _.._ 0; 82(0) = 0; 33(0) _._._ 0
STEP 4.

Pl(1) = 21 + €31(1)

P2(1) 3 £2 + £E2(1)

P3‘(1) = 23 + 883(1)
STEP 5.

Find 31(1), H2(1), 33(1)
STEP 6.

Solve

(1+1) T (1+1) (1)
Ea p-3 + 12333 ‘ H3

. T : . . T .
31(1+1)_12l + 9131(11-1) = DTH3(1)D + 31(1)12 + DTBIU)

+ €82(1)

132(1+1) , 'E1(i+l)Dz1D§1 _ D§233(1+1)D;1 - 31(1)D§1
STEP 7.

Check required accuracy.

If not;i a i+l, go back to Step 4.

If yes, stop.

In order to illustrate efficiency of proposed
algorithm, letfs consider the magnetic type control system

example [6].

33

EXAMPLE 2.1.

 

 

 

 

_ 0 0.4 0 0 ' "'0'
0 0 0.345 0 0
A(9) 2 8(8) 3
0 -24122 -9425: Q2222 0
C E E
0 0 0 ‘l l
. a J -EJ
R]: diag {1,011.10} R2: 1 C = 0.1.

With accuracy up to 6 decimal digits we got convergence to

 

 

 

the exact solution in 4 iterations. Componentwise, results
are given in Table 2.).

E=P(°) P(1) P(2) P(3) 13(4)g

(exact)

P12 54224152 6.155215 6.170524 6.170445 6.170447
P13 0. 0.405222 0.405345 0.405342 0.405342
P14 0.100000 0.100000 0.100000 0.100000 0.100000
P22 7.15152A 7.452224 7.467122 7.467275 7.467278
P23 0.312112, 0.394525 0.395104 0.395100 0.395100
P24 0.085122 0.089245 0.089202 0.089202 0.089202
P33 0.154222 0.122121 0.130434 0.130441 0.130441
P44 0.004512 0.006181 0.006200 0.006200 0.006200
Table 25f Solution of Riccati equation for magnetic type

control.

Using different values of e in the same example, we
can note very good convergence rates even with relatively
large values of perturbation parameter. These results are

displayed in Table 2.2.

 

No. of iterations
for convergence

0.01 1
0.1 4
0.5 . 7
0.9 10

 

Table 2.2 Dependence of number of
iterations on a.

Filter Riccati equation (2.8b) can be solved by the

same algorithm taking into account the following analogies.

T T T T T T

T T T T T
0191*G1wclr 9192*G1W‘32' 0292* G2 ”‘32; R2"V

Thus, the remaining problem is to find a corresponding

procedure for solving (2.15).
II.2.2 FIXED POINT SOLUTION OF L AND M EQUATIONS

Solutions of L, M equations (2.15) that are needed for
our LQG optimal control can be sought through the iterative

form proposed by Kokotovic [9].
(1+1) _ (N) -1 (N)

(N) -1 (i) (N) (N) -l (i)
(2.50)

35
(i+l) .. (N) _, (N) ’1 _ (i) (N) _ (N)

(N) (N) (N) (i) . (N) -1
-[(A1 - BlFl ) - (A2 - BlF2 )L ]M } (A4 ’ Bze )

i = 1'2'000'N-l (2.51)
with initial values
(0) (o) -l (o)
(o) (o) (o) -1

Note that in (2.50) 5‘1”” and Fzm) have already been
determined using (2:7) and the solution of the Riccati
equation.(2.8) which can be obtained using the algorithm of

the previous section. The same holds for L(N)

in equation
(2.51).

Equations (2.50) and (2.51) take the form of (2.38) -
(2.40) and their original forn:(2.15) can be put in form

similar to (2.35) - (2.37), i.e.,
D31.) = Y4 +534(L,€) (2.54)
MD3 = )5 +635(M,L,€)' (2.55)

where D3 is a stable matrix, ‘(4 and Y5 known constants and
.383 and 3’5 nonlinear functions. Then the following theorem

holds for L and M.

36

THEOREM 2.3: Under the conditions of Theorem 2.1,
algorithm (2.50) - (2.51) converges to the exact solution L

and M with the rate of convergence of 0(8), i.e.
HL-L‘imli = 0(1) HL-L‘i’i!
HM-M‘i”1’l| = 0(8) HM-MmH

or equivalently

HL-L(i)H =0(ei)

llM-M‘i’l! =0(ei)

Proof of this theorem uses the same arguments as the proof

of Theorem 2.2 and thus it is omitted.

II.2.3 THE ITERATIVE SOLUTION OF LQG

We developed so far the iterative procedures that
generate in a very efficient way all coefficients for the
approximate solution of LOG, (2.16) - (2.17). Those
coefficients for an 0(ek) accuracy of- the approximation of
LQG are obtained by doing k-l iterations on the same set of
equations. Contrary to the power series expansions, where
we have to solve k-different set of equations, our
iterative scheme is very useful in the case where high
order of accuracy is required. Instead of defining and
solving a new set of equations, we have to perform just one

additional iteration on the already existing set of

37

equations. On the other hand in both cases we are faced
with low-order numerical problems (in fact they are of the

same order).
Equations (2.16) - (2.17) can be written in the

following composite forms

(k) (k)*(k) (k)~(k)

uopt = ‘fl “1 - f2 ”2 (2.56)
:(k) (k)‘(k) (k) (k)

n1 = 31 01 + 91 v. (2.57)
Egék): aék)fiék) + gék) v(k) (2.58)
where v“” =:y - ifk)afk) - Eék)0ék) , with obvious
expressions for fiu‘), aifl‘), giU‘), €i(k)' i a 1,2. Then

the suboptimal criterion

 

 

(MT (k)
t x x
3“” = 1m --1-- a [I 1 R1 1
t 1-.. t1-t° t x (k) x (k)
0 2 2
t -> ..
1
( T (
+ u k) Rzu k’ dt (2.59)
is given by
(k) (k) (HT (k) (k)
(k) 1 (k)
where (k)= var x1 and (k): var n1
qll x2(k) ‘32? 520:)

(k)

Quantities qll and q22(k) can be obtained by studying

the variance equation of the following system

38

 

 

 

 

 

 

”.(kf ' (k) (k) l ' (kf
x1 A1 A2 "Blfl ’Blfz x1
-(k) . (k) (k) (k)
x2 3 A3 A4 ’Bzfl 'Bzfz x2
:(k) (k (k (k) (k)£(k) (k),(k) .(k)
n1 91 1 91 2 a1 '91 1 ’91 *2 ”1
Egm (k (k (k)r(k) (k) (k) (k) .(k)

2 92 1 92 2 '92 *1 a2 '92 52 n2
_ A h d- d
'61 o ‘ ' '
62 0 W
+ (k, (2.61)
0 91 V
(k)
° 92. L-

 

 

 

 

or in a composite form

The variance of z‘k) denoted by q(k) is given by well-known

Lyapunov equation

T - T
“((k)q(k) + q(kbg(k) + G(k)wG(k) g 0 (2.53)

where q(k) is partioned as

 

 

' (k) (k)
q(k) = q11 912
(k)T (k, (2.64)
q12 g q22

This procedure is demonstrated by a numerical example,
showing the required convergence properties
fi(k) + fiopt

ai(k) + aiopt

39

—-

gi(k) _, giopt
, (k) - o t
J(k) '* JOpt

k = 0,1,..., and i = 1,2.

EXAMPLE 2.2:

An F-8 LQG Controller

In order to demonstrate the numerical behavior of the

near-optimum design of singularly perturbed LQG regulators,

we present results for an LQG controller of an F-8 aircraft

which was considered in [3].

 

 

 

 

 

”£1” FL1.357 x 10"2
$2 1.2 x 10’4
£3 - 41.212 x 10'4

Lé4. b5.7 x 10"4

"-0.433 )
0.1394
(1+
-o.1394
b-o.1577‘
y: [o o o 1] rail“
1 o o 0 x2
£3
bX4d

 

The system model is given by

-32.2
0
O

F'
-4603

1.214

-l.214

 

 

L -9.01

 

-46.3
1.214

-1.214

-9.01

 

 

 

o . '

o

1

-0.6696
J

where the white noise processes w and v are independent and

have intensities W a 3.15 x 10"4 and V = diag[6.859 x

10", 401.

The performance criterion is

40

t - ~2 ~2
J = lim ----- E “ff[0.01x2 + 3260(x + x + u2)]dt.

tf+-w
The reader is reffered to [3] for discussion of the
modeling aspects and the choice of J.

The open-loop eigenvalues are -0.94 i j2.98 and
-0.007S i,j0.076 which shows clearly the two-time-scale
property of the system. The choice of state variables
adopted in [3] led nicely to a formulation in which the
first two variables are slow variables. A logical choice
of the parameter c is e a 0.025 which is roughly the ratio
of the magnitude of the slow eigenvalues to the magnitude
of the fast eigenvalues. The singularly perturbed nature
of this system becomes more evident [13] by using a state

transformation x = Tx where

1 1618 133.92 200‘
o 500 40.8 61
T a
o ‘o 600 o
o o o 200
" A

 

 

Introducing a artificially by multiplying the left-hand
sides by 0.025 the system takes the singularly perturbed
form of (2.1) - (2.4) with

[0.278386 -0.965256] {-0.074210 0.016017]
A = A =
1

0.089833 -0.290700 0.012815 -0.001398

41

-0.001815 0.005873] -0.030344 0.075024
A = A :2
3 0.002850 -0.009223 4 -0.075092 -0.016777
174.907714 -2.091000
B1 = 32 =
54.392760 -0.780500
T 0.010000 -0.032360
p10.1 =
-0.023260 0.104717
T -0.000032 -0.000130
“1‘2 3
0.000102 0.000421
T 0.009056 0.000000
0252 =
0.000000 0.081502
R2 = 3260 w = 0.000315 v = diag{0.000686, 40}
0 0 0 0.005000
C1 = C2 =3
1 ~3.236000 -0.003152 0.013020
-46.626960 -18.210002
G1 a 62 =
7.858776 -45.049998

Corresponding results are shown in Table ZJL

IIQB WELL-POSEDNESS 6F MODEL ORDER REDUCTION FOR

SINGULARLY PERTURBBD LINEAR STOCHASTIC SYSTEMS

Singular perturbation method can serve as a tool for
model order reductions. An interesting question that

arises in that respect is: should we do the order

42

.m-m acmcucwm soc was to :o_uzpom :o_uoswxocgam m>_mmwou=m < m.~ opncp

 

 

 

 

 

 

 

 

mwmooo.o mmmooo.o cemooo.o mmmpoo.o mmw—oo.o
memmm.n ommomm.m Nmomcm.~ eemmmm.s P—ov_w.N P
= vmpvoo.o m—Fvoo.o Nmpmoo.o vmwooo.o vmcooo.o a
mmmomm.mm mmmmmm.mm mpmmmm.mm mmvmmv.mw mmomo¢.mm
ammopo.o- mmmopo.o( ammo—o.o( ommmpo.ou mommpo.o
upmmko.o: m—mmmo.oa Npmmmo.Ou upmmmo.ou w~¢w~o.o( N
: mmmcno.o ommvxo.o ommvxo.o mvmvmo.o mmmq~0.o a
vpmpmo.o- vpm~mo.o: m—mpmo.ou oompmo.0a mmmpmo.o-
mepomv.0u mcpom¢.o( mmpom¢.o- wmmmmv.o cmmmmc.o
oqupp.o ow¢¢p_.o Nm¢¢~_.o mmvv—_.o mm¢¢pp.o _
= vwmopv.—u ewmop¢.—u movo_¢.~a Nmmmo¢.pu Nowmov.—u a
mvmmmm.o mvmmmm.o mmwmmm.o Nmnmmm.o Nmnmmm.o
mmompo.o mmompo.o moompo.o —mmmpo.o omen—0.9
mcpppo.o m¢_P_o.o mp—p—o.o wmmopo.o Nmpmoo.ou N
: mOOmoo.o moomoo.o aoomoo.o aoomoo.o scamoo.o w
cooooo.o eooooo.o vooooo.o vooooo.o .o
omemN.mn cpmmm~.m- Npmmmm.mn pmmmmm.mn pummmm.mu
cpmmmm.o e.mmmm.o v—mamm.o mmmmmm.o mmmmmm.o p
= wwwooo.On waooo.o( Fwwooo.Oc mowoco.ou momooo.On u
mmmooo.o mmmooo.o ammoco.o ammooo.o ommooo.o
mooooo.o( mooooo.ou mcoooo.ou pooooo.ou mmoooo.On N
= omeooo.¢: mmqooo.o- mmvooo.o( mm¢ooo.Ou emeooo.o( w
wmmmoo.o cmmmoo.o «mmmoo.o memoo.o Pmmmoo.o _
= mwcooo.oa mvvooo.0u mcvooo.ou w«¢ooo.o: mvcooo.o( w
.EszE 2528 9:3. 92.8mm 5m: omHmSn—Em

 

43

.As.s=ouv m.~ 6.88c

 

 

 

Nmmmoo.mN Nmmmoo.mN mmmmoo.mN wowoco.mN Nemoco.mN mNommo.mN a
mNm_oo.o mumpoo.o cum—oo.o mmwpoo.o NNm—oo.o
MNmmw¢.NN vcmmw¢.NN mcoow¢.NN mommm¢.NN NommNm.NN N
= —Nmpoo.o NNm—oo.o campoo.o mmmpoo.o wmmpoo.o m
oommo~.m mmmmo_.m «Np¢o~.m mNNemo.m mvaop.m
4<zhhdo :pmaom ommzh ozoumm hmmHu owfiuabmzmm.

 

44

reduction from the very beginning (in order to reduce
computational requirements) or at the very end (after all
results are already obtained)? When do the results of those
two approaches coincide? In the case when they coincide,

we will be talking about well-posed model order reduction,
otherwise we have an ill-posed reduction problem. This is

a critical issue for the concept of multimodeling control

to be developed later on. We will study this problem in

the context of state estimation and LQG control problems.
11.3.1 STATE ESTIMATION

Consider singularly perturbed linear stochastic

system
*1 = Alxl + A2X2 + le (2.65)
5&2 = A3x1 + A4x2 + 62W (2.66)

with corresponding measurements

y = Clxl + C2x2 + v (2.67)

Its optimal estimates are given by the well-known Kalman

filter [12]

X1 3 Alxl + Azfiz + Kl(y ’ Clxl ’ C2X2) (2.58)

O
A

5X2 = A3£1 + A4§2 + K2(Y - C1§l ’ C2§2) (2.69)

Defining the estimation errors as el = x1 - 3:1 and e2 = x2

- §2 we have the following augmented system

45

 

 

 

 

 

 

 

 

r 7. ‘ [- 7 FA " f ‘1
LEézd -0 0 A3'K2Cl A4'K2C2J hezm1 _Gz ’de

2.70)
If we now reduce the order by formally neglecting fast
dynamics setting 5 = 0, we get the remaining slow portion

as

f; A KC 3; KB K w

[.1] g [o 00 ] [1]+[oo o] [](2.71)
£1 0 Ao-KOCo £1 Go-KOHo -Ko V

where

1

- 1
=Cl-C2A4A31 H

‘C2A4 G2

Co

-1 ‘1

0n the other hand, if the order reduction has been
performed as a starting point in our original model (2.65)

- (2.67), i.e. for

A

M.
II

x0 + Gow (2.72).

0

Cox

‘<
0
II

o + How + v . (2.73)
then corresponding Kalman filter is

x0 = ono + Ko(yo - Coxo) (2.74)

and by defining the estimation error as ea = x — SE

0 owe

have the augmented slow-filter and slow-error system

46

:2 A KC )1 RH K w
[.0] = [ ° ° ° ] [ O] + [ ° ° 0] [‘12.75)
e0 0 AO-KOCO eo GO—KOHO -Ko Y}

Comparing (2.71) and (2.75) we see that (in m.s.s.)

A A

31 = x0 and £1 = eo (2.76)

Thus, we can conclude that for an estimation problem the

order of performing model reduction is irrelevant.
II.3.2 LQG CONTROL

For the optimal control problem defined by (ZJJ -

(2.4) the optimal filter-error equations are of the known

 

 

 

 

 

 

 

 

form [12]
x1 Al-BlFl Az-BTF2 K1C1 chz‘ fxl‘
é 0 0 A —K c A -K c e
f 21 _ 3 2 1 4 2 2‘ _ zj
(0 K1 '
0 K2 .W
+ (2.77)
6]. "Kl V
G2 “K2
where the optimal control given by
L1 = -Fl;l " F2£2 (2.78)

has been eliminated from (2.77).

47

Setting e:= 0 into (2:77) we get the reduced solution

as
31 [(Ao-BCFO) (Ko+6)Co 21 [(KO+6)HO KO+5 w
= +
£1 0 (AG-KOCO) g1 GO-KOHO -Ko v
(2.79)
where

1 -1

- - ‘ _ -l .
6 - (Bl A2A4 32) [I + F2(A4 B2132) 321132.114 K2

[I + c2014 - K2C2’-1K2] (2.80)

On the other hand, by doing reduction first, plugging s = 0
in (2.2) and eliminating x2 from (2.1), (2.3), and (2.4) we

get a reduced problem

x = ono + Bouo + Gow (2.81)

yo = Cox + N u

0 o o + How + v (2.82)

. -1
WIth NO =’C2A4 Bz
and
- 1 t1 T T T '1'
J a 11m €---- E ./ (xo p0 poxo - 2xO r pouo +
tow-co l-to to
t1+ on.
T (2.83)
uoRouo)dt 4' J1 = J0 + J1

where J1 contains terms not influenced by control.
Optimizing Jo with respect to (2.81) - (2.82)

implies the optimal control

48

and the optimal Kalman filter

x0 = (A0 - KOCo)xo + (Bo - KoHo)uo + Koy ' (2.85)

Our slow-filter slow-error system has the form

x0 Ao-BOFo KOCO x0 KOHO KO w
,e0 0 Ao-KOCO eo Go-KOHo "K0 v
(2.86)

Comparing the elements of (2.79) and (2.86) we conclude

that in general (5 + 0), (in m.s.s.)

£1 + x0 and £1 = eo (2.87)

A A

Note that the result x1 + x0 was previously observed in [4]

and [11] without any detailed analysis.

If we write down corresponding Lyapunov equations for
(2.79) and (2.86) in order to find their variances [12],

exploring the orthogonality principle, we can easily get

T T
(A0 - BOFO)Ql + 91(AO - BOFO) + KOVOKO (2 88)
+5TV05 + 5V0K0T + KovoaT = 0
and
T T _
(Ao - BOFO)QTO+ ()1ng - BOFO) + KOVOKO — 0 (2.89)

where 91 = var(i1) and 010: var(8o). Obviously, in the

general case ( 6* 0) the variances are different too, i.e.

var(i1) # var(§o) (2.90)

Thus, the cause of the trouble is the quantity 5, and

49

from given equations we have

21 = x0 and var(xl) = var(xo) (2.91)
36:0 35:0-

Having 6 given by (2.80) we can discuss cases when it takes

zero value. This can happen in the following special

cases:

a) F2 = 0; which can happen when B2 = 0 or R22 = 0.

b) K2 = 0; when C2 = 0 or Gz = 0.

Thus, not penalizing fast in the criteria (R22 = 0), not

influencing fast variables by control (82 s 0), or by noise

(62 = 0), or not measuring fast (C2 2 0) comprise special

cases where the model reduction is well-defined.

Otherwise, we have an ill-defined problem. This is going

to be a crucial issue in the context of multimodeling.

III. MULTIMODEL CONTROL OF LINEAR STOCHASTIC SYSTEMS

The concept of multimodel control has been brought to
attention after the paper [14] by Khalil and Kokotovic.
According to that concept, a system may be controlled by
several independent agents using several simplified models
of the system. This procedure is common in multi-area
power practice and interacting economics (c.f. [22], [23]).
In general, models of large scale systems usually contain
weak coupling and fast dynamics among same sets of state
variables. It is a common engineering practice to neglect
weak connections and fast dynamics in order to reduce the
order and complexity of the model. In the case of systems
controlled by several control agents, such model
simplifications can be done in many ways so that each
control agent may have a different model of the system. In
fact, each of them may employ a detailed model of his area
and only a dynamical equivalent of the remainder of the
system. Deterministic version of this problem was studied
in [14], assuming linear systems, quadratic indeces and
using the theory of singular perturbation as a tool for its
analysis.

The purpose of this chapter is to consider the same
problem for the case of stochastic systems and partial

imperfect observations. The presence of white noise input

51

driving fast variables and output feedbacks makes the main
difference between the two problems disallowing straight
forward extention of deterministic results. Even though
the multimodel structure underlying in the deterministic
problem is kept as a candidate for stochastic
multimodeling, a new set of conditions for well—posedness
of the problem is required. Well posedness is sought in
the sense of asymptotic closeness of performance indeces
and trajectories at steady state compared with the
corresponding ones under perfect knowledge of the model.

It is shown that well-posedness can not be achieved without
communications of measurements and decisions. The main
reason for this is the phenomenon observed in Section II43,
that in stochastic control problems the slow dynamics of
the closed-loop systems are dependent on the fast control;
a situation which is different from deterministic problems.
Therefore, a local model in which fast dynamics of other
areas are neglected cannot be used to calculate the
decisions of other controllers. To capture the information
lost through order reduction, we allow communication of
decisions among the controllers. It is shown analytically
that such a strategy, called multimodel strategy, is well-
posed. .A numerical example illustrates closeness in
criteria and slow trajectories at steady state compared
with the optimal solution assuming exact knowledge of the

model.

At the end of this chapter, we consider the special
case of multimodeling when all decision makers agree to use
the singular perturbation technique to solve their own
problems. In this case, communication in measurements and
fast decisions only is required for the well—posedness of

the problem.
III.1 PREVIOUS RESULTS IN MULTIMODEL CONTROL

The concept of multimodel control was introduced in
1978 by Khalil and Kokotovic [14]. They have been studied
linear deterministic systems with strongly coupled slow
subsystems and weakly coupled fast subsystems. For the
case of two subsystems and two control agents, their model

is represented by

xo 3 Aooxo + Aolxl + A02x2 + Bolul + Bozuz (3.1a)
E:lxl a Aloxo + Allxl + €3A12X2 + Bllul (3.1b)
c2’E2 3 A20% + €4A2131 + A22x2 " Bzzuz (3-16’

where £1 and 62 are small positive singular perturbation
parameters whereas 63 and 84 are small regular
perturbation parameters of arbitrary sign. In addition,
the ratio of 61 and a is bounded by some positive
constants

e1
o<m$E‘SM<co
2,

53

11 n1 1'12

0 . . -
vector xosR represents slow var1aoles, xrzR and xzeR

ml m2
are fast variables, whereas ulaR and u2€R are control

vectors associated with first and second subsystems.
Multimodeling comes as a consequence of different
order reductions of the original model (BJJ. The first
controller is neglecting 82 and 83 parameters, thus weak
coupling and fast dynamics of the second subsystem, ime.,

its model now becomes

-(1) (1) (1) (l) (1)

x0 = A0 x0 + Aolxl + Bolul + 802 u2 (3.2a)
eTi{1’= Aloxél) + A11x{1)+ Bllul (3.2b)
where

(l) -1 (l) -1
Ao = Aoo ' A02A22 A20 r 562 = Boz ’ A02A22B22

On the other hand, the second controller is neglecting £1

and 64 parameters and its model is given by

~(2)= (2) (2) (2) (2)

0 A0 x0 + Aozxz + 801 ul + Bozu2 (3.3a)
.(2 (2 2
8232 )3 Azoxo ) + A223; )+ 8221.12 (3.31))
where
(2) -l (2) -1
A0 2 A00 - AolAllAlo ' Bol = Bol ' AolAllBll

Obviously in order to be able to get these two models, All

and A22 have to be nonsingular matrices.

54

With each subsystem the following cost functional is

associated

Ji = Ji(xo'xi'ui) i = 1,2 (3.4a)

Since in the j-th model, j = 1,2, variable xi, 1 9r j does
not appear, it is assumed that j-th controller knows i-th
functional (i 4 j) in the form 31 (x0, ui) which is

obtained from Ji(xo, x ui) consistently with the model

it
reduction procedure, i.eu, using equations (3.1) after

neglecting corresponding parameters. The design objective
for each controller is to optimize a linear combination of

the form

71:31 + Yij 1’]. = 1,2 1 * j Yi+Yj = l (3.4b)

This situation corresponds to a Pareto deterministic
multimodeling and it has been proved in [14] , that it is a
well-posed problem. In this chapter, we will study the
same problem in the context of stochastic multimodeling.
The basic contribution of [14] is establishing a set
of conditions under which a deterministic multimodel
strategy is well-posed. The adopted underlying structure
(fast dynamical subsystems connected through slow
variables) is suggestive and covers a wide class of large
scale systems. On the other hand, the singular
perturbation methodology used to represent and analyze the
multimodel situation is a powerful mathematical tool; and

no apriori structure (e4L, decentralized or hierarchical)

55

is imposed on the control scheme. That is why we are going
to develop a stochastic multimodeling proceeding along the
lines of [14].

Recently, a few papers bearing the features of
multimodeling were published. As a natural extention of
[14] a deterministic Nash multimodeling was studied in [15],
whereas [16] addressed to the decentralized deterministic
Nash multimodel strategies. Stochastic multimodel problems
for Nash [l7] and team theory [18) were studied for a
special case where noise inputs in subsystems are properly
scaled so that the ideas developed for deterministic
multimodeling can be extended to the stochastic one. In
fact, as it was shown in [37], considering noise as /3; WT,
i = 1,2 where w- is a white Gaussian noise affecting

1

subsystem i, and $1 is the i-th subsystem singular
perturbation parameter, causes fast stochastic subsystem

to behave as a deterministic one when::- + 0. For the

1
reason of having stochastic quantitieS' present all the
way, even when Ei'+ O, we will exclude this situation from
consideration.

Concluding this review of so far known results in the
multimodeling literature, we want to mention the following.
Attempts to identify the class of admissible strategies
which allow multimodel solutions in the case of a linear

deterministic systems which are not singularly perturbed,

were nicely treated in [20]. Even though papers [19] and

56
[21] do not intend to be called multimodeling, they can be
viewed as special cases of multimodeling. In general, many
decoupling and decentralized procedures might bear features

of multimodeling.

III.2 PROBLEM STATEMENT AND FORMULATION OF STOCHASTIC

MULTI MODEL STRATEG I ES

A linear stochastic interconnected system consisting of
connected slow dynamics and isolated fast dynamics can be

represented as

2

x0 = Aooxo + jEl (AojxJ + Bojuj + Gojwj) (3.5a)
Eixi 3 Aioxo + Aiixi + Biiui + Giiwi, 1 = 1,2 (3.5b)
y1 = Cioxo + Ciix1 + v1, i = 1,2 (3.5c)

no n1 n2 m]
where xoeR , xleR , xzeR are state variables ulsR and
In2 S1 s2

uzaR are control inputs, yleR and ysz are measured

r r2 51 52
outputs, wleR , wzeR , vleR and vzeR are independent

Gaussian white noise processes with intensities W1 > 0,
W2 > 0, V1 > 0 and V2 > 0 respectively. The small singular
perturbation parameters 61 and 62 are strictly positive.
It is assumed that 81 ands:2 are of the same order, i.e.
E1

0 < kl S -:E- S k2 (m

57

All matrices in CLS) are constant matrices and of
compatible dimensions. Note that the form of Gij and Cji'
i = 0,1,2; j = 1,2 is consistent with the underlying
structure of the system. For the sake of simplicity we
consider the case of two control agents but the results of
this chapter can be easily extended to the case of N

control agents.

With each subsystem we associate the performance criterion

t
1 1 T T
Ji = 11m €-”fE- E I (2121 + uiRiui)dt I R1 > 0
to+-oo 1 - o to
tI++m (3.63)
where
21 = Dioxo + Diixi' i 3 1'2 (3.6b)

The optimal control problem is to minimize a weighted sum

Of J1 and J2: i.e.
J = YiJl + Y2J2' Y1 + Y2 = l (3.7)

which corresponds to a Pareto-optimal cooperative strategy.
In order to concentrate attention on the effect of
multimodeling, which is a nonclassical off-line information
scheme, we assume classical on-line information, iJL all
measurements are known to all controllers. With this
assumption optimal control is the solution of a standard

LQG optimal control problem of the form

2 = Ax + Bu + Gw (3.8a)

 

58

y = Cx + v (3.8b)
. 1 t1 T T T
J = 11m ------- E j‘ (x 0 Dx + u Ru)dt (3.8c)
to+-“’ tl ' to to
t1++ at

where corresponding matrices and vectors are given in
Appendix 3.1. The appendix includes also all newly defined
matrices unless they are explicitly given in the chapter.
Assuming that the triples (A,B,D) and (A,G,C) are

stabilizable-detectable the well-known solution of ELB) is

given by
u = -F§ = -R-1BTPx (3.9)
PA + ATP - PBR-lBTP + DTD = 0 (3.10)

where x is the output of the optimal Kalman filter at

steady state‘
x = Aé + Bu + K(y - cfi); K = och"1 (3.11)

QAT + AQ - och’1 co + GWGT = 0 (3.12)

Due to the singularly perturbed nature of the system the
overall stabilizability-detectability conditions can be
replaced by stabilizability-detectability conditions on the
slow and fast subsystems. For the regulator part of the
problem that was done in [14] and is stated in the

following lemma.

59

LEMMA.3.l: If Aii' i = 1,2 are nonsingular and the triples

(Aos'Bos'Dos) and (Aii'Bii'Dii)’ i = 1,2 are stabilizable-

detectable then, for sufficiently small 61 and 82, (3.10)
has a unique stabilizing solution.
In order to avoid unboundedness when small parameters

tend to zero, this solution has been sought in the form

 

 

Poo El P01 82 P02
- T E
P ‘ E1Poi 1 P11 '6152 P12 (3°13)
T f‘“" T
L E:21’62 E1 E2 P12 E:2 P22T

By using duality between the regulator and filter
Riccati equations we obtain, by analogy, the following

lemma.

LEMMA.3.2: If Aii' i = 1,2 are nonsingular and the triples

(Aos'Gos'Cos) and (Aii'Gii'cii)' i = 1,2, are stabilizable-

detectable then for sufficiently small 61 and 82 CLJZ) has
a unique stabilizing solution.

This solution is properly scaled in the form

" '!

Qoo Q01 002
1 l
T
Q 3 001 E; 011 -:%:ۤ 012 (3.14)
T 1 T 1
062 ‘jEFE= Q12 “ Q22
1 2 S2
.. .J

 

 

60

It is interesting to point out that in the case when
control agents are acting independently and having complete
knowledge of the optimization problem.(3.8)(each of them
will solve (3.9) - (3.12) and compute both ul and uz but
will implement only his own component of the control.

Also, each of them will implement locally the Kalman filter
(3.11) driven by locally computed ul and “2° It is
important to stress that the only on-line information
communicated between the two agents is the measurements Y1
and Y2° Even though the locally implemented Kalman filters
have to be driven by both ul and u2 there is no need to
communicate the decisions ul and u2 between the two agents
since each one of them is capable of computing those
decisions locally.

Now we are ready to formulate a multimodeling problem.
Let's assume that no one control agent has complete
knowledge of the global model, but they do have local
models available. These local models are obtained by
neglecting fast dynamics of the other subsystem, but
retaining the exact model of oneJS own subsystem. In model
(3.5) this simplification is equivalent to the assumption
that $2 is set equal to zero for the first control agent,
and 81 is set equal to zero for the second one. Thus we
have two simplified models of the interconnected system.
Assuming All and A22 are nonsingular, the simplified model

of control agent no. i, i = 1,2, is given by

.(i) (i) ('1) (i) — (i)
X0 = A0 X0 + Aoixi + BOiui + 80:} u]
+ GOiwi + GO§1&j (3.15a)
Y1 = Cioxél) + C1955) 7“ V1 (3-156)
(1) (i)
Yj - Cjo X0 + Njuj + Hjo + ij
3= 1,2. j + i, (3.15d)

Since the simplified model of each control agent lacks
information about the fast dynamics of the other agent’s
subsystem it is assumed, for consistency, that agent i
knows a version of Jj, called Jj(si) in which the effect of
those fast dynamics is eliminated, i.en

T

9 t o n
(1) l 1 (1) (1) T
J- = lim ------- E (z- z- + u- R-u- )dt
)3 t67'm tl — to [1T0 35 35 J 3 1
where
(i) (i) (i) . . . .
zjs 8 jo x0 + Ejuj 1,3 = 1,2 3 # 1 (3.16b)

is the slow component of 21.
The criterion to be optimized by the control agent i
is

(i) _ (i)

62

Thus instead of solving the overall optimal control problem
(3.5) — (3.7) (or equivalently (3.8)) using perfect model
information, each control agent will solve an optimal

control problem defined by CLJS) using his own simplified

model
i‘i’ = A(1)x(i) + B‘i)u + C(i)w (3.18a)
y = c‘i’x‘i’ + N(i)u + H(i)w + v (3.18b)
. 1 t1 . T .T . . .T .
J(1) = lim _______ ELL (x(1)D(1)D(1)x(1) + 2x(1)p(1)u
t ?-<” t - t
t§++‘” l o o
+ uTR(i)u)dt] , 1 = 1,2 (3.18c)

It is assumed here that each control agent has access to
the measurements Y1 and y2, which is consistent with the
on-line classical information structure underwhich the
optimal solution (3.11) - (3.16) was computed and
implemented. It is seen that the control problem for each
agent is an LQG optimal control problem with singularly
perturbed dynamics. It is natural to employ singular
perturbation decompositions (eug. [2] and [4]) to
simplify the solution of this probleun However, to
concentrate attention on the question of well-posedness of
multimodeling control, we postpone employing singular
perturbation decompositions till a later point of this
chapter. At this point it is assumed that each control

agent will seek an optimal solution for his control problem

63

and well-posedness of multimodeling will be studied for
this situation of local optimization.

The optimal solution of problenl(3.18) exists under
conditions stated in the following lemma [2}, [4].
LEMMA 3.3: If A11 is nonsingular and the triples
(Aos'Bos'Dos)' (Aos'Gos'Cos)' (Aii'Bii'Dii)' (Aii'Gii'Cii)
are stabilizable-detectable then, for sufficiently small

Si, (3.18) has an unique stabilizing solution given by

-1

u(i) a _F(i)§(i) = _R(i)(P(i)B(i) + T(i))T§(i) (3.19a,

“(1) = A(1),;(1) + B(i)u(i) + Km,y _ c‘i’i‘i’ _ N(1)

x u)
(3.19b)
T T 7
8‘1) = (o‘l’c‘l’ + c‘l’wu‘l’)(v + H‘l’wn‘l’)’1 (3.19c)

where P(i) and 0(1) are solutions of the corresponding

regulator and filter Riccati equations,

are properly scaled in the form

 

 

 

 

respectively, and

' (i) (1)-1
P00 8iPoi
p‘i) = = 1,2 (3.20)
T
(i) (i)
iPoi eiPii
' (i) (1) 7
O00 O01
0‘1) = = 1,2 (3.21)
T
(i) 1 (1)
Qoi ET Qii
. 1 d

64

Note that Lemma 3.3 did not require any new conditions over
those required by Lemmas 3.1 and 3.2.

The control strategy (3.19) can be computed by agent i
using his own simplified model. Moreover, since both Y1
and y2 are available to both agents, each agent can

implement the Kalman filter (3.19b) locally using his

locally computed control u§1)and uél). But, the actual

strategy applied to the system is ul = nil) and “2 = uéZ),

i.e. each agent computes his own version of both components
of the control strategy but he implements only his own

component. The control

P

(1)'
u
strategy “mm: 1 is a multimodel strategy. Is this

(2)
_u2 1
multimodel strategy well-posed? That.is thebquestion

 

 

addressed in this chapter. Let us first define what we
mean by well-posedness in this context. A multimodel
strategy is well posed if the performance of the system
under the multimodel strategy is close to its performance
had the control agents used a perfect global model to
compute their control strategies. In view of the
parameterization of model information by means of
perturbation parameters 81 and 82, we say that the
multimodel strategy umm is well-posed if application of
this strategy to the system will result in values of

performance criteria and state trajectories at steady-state

From

65
which approach asymtotically, as 61 and €2‘+ 0, their
corresponding quantities under the optimal control strategy
(3.9) - (3.12).

Since the well-posedness question under consideration
here is an extension of the one studied for the
deterministic full state problem in [14], we can make use
of the previous results to partially answer the question.
Towards that end we start by investigating the well-
posedness of the off-line computations. We compare the
regulator and filter gains (3.9), (3.11), of the optimal
strategy with the regulator and filter gains (3.19a),
(3.19c) of the multimodel strategy. We rewrite the

regulator gains F, F”), and 3(2)

 

 

 

 

as
F61 F11 F21“
F = [F0 F1 521 =
.Foz F12 F224
' (1) (1)‘
Fo1 F11
Fm . mg“ Pg”; = (3.22,
(1) (1)
_F62 F12 j
' (2) (2)“
Fol F21
(2) (2) (2)
F = [F0 F2 1 =
(2) (2)
LFoz F22 .

 

 

[l4] and Appendix 3.2, we have

66

(i) (i)

* A
FOi = Poi + 0(6), Fil = Fii + 0(:)
(i) -
Fij "' 0(5), Fji = 0(‘) (3.23)
F(i) ’ F - F (A - B F )-1(A - B F ) + 0(5)
,03 01 11 11 j: 13 30 01 03

By the duality between the regulator and filter Riccati
equations similar conclusions can be obtained for the

filter gains. Rewriting K, K‘l) and K‘2) as

 

 

 

 

 

 

 

 

. - T T
Ko K01 K02
(
K - El - 511 512
81 *81 E1
E3 531 532
8'2 E2 82
L ‘ ' ‘ (3.24)
' (1)) ' (1) (1)”
K0 K01 K02
K(1) g g
(1) (1) (1)
51-- 511- E32-
81 . L E1 E1 -
' (2) ' (2) (2)“
Ko i Kol Ko2
K(2) g g
(2) (2) (2)
52-- 521- 522-
L 82 j ..82 E2 .

 

 

 

 

the following expressions hold

 

E
* 0(a) denotes 0(l! 1 H) for any norm {E'
E H

2

 

 

(i) = (1)

K01 K01 + 0(8), K11 = K11 + 0(8)
(i) C _ :

K(i) = K - - (A - K C- )(A-- - K--C--)71K~- + 0(5)
03 03 oi 03 J) 11 31 11 11 '

Equati'ons (3.23) and (3.25) show that the regulator and
filter gains for the optimal control can be reconstructed
up to accuracy of 0(a) from the two different lower order
simplified models. In particular, the locally designed
feedback gains and Kalman filters are, asymptotically,
consistent with the optimal feedback gains and Kalman
filters. The consistency of the feedback gains follows
from (3.23) since, by definition of the multimodel
strategy, each agent applies only his control component.
To see the consistency of the filters let us consider the
case of agent no. 1 (the other case is similar). The
optimal Kalman filter (3.11) is given, in a partitioned

form, by

O
'8

x0 7 (A00 7 Kolclo 7 K02020)x0 7 (A01 7 K01C11)x1

+ (A02 ’ K02C22)x2 + Bolul + Bozuz + Ko1Y1 + KozYz
(3.26a)

2‘ A A
E‘1"1 = (A10 ’ K11‘316 ' K12C20)x0 + (A11 ‘ K11C11’x1

' K12C22X2 + B11u1 + K11Y1 + K12Y2 (3°25b’

68

E2‘2 = (A20 ' K21C16 ' K22C20)x0 ‘ K21Cllxl
+ (A22 - K22C22)x2 + Bzzu2 + K21yl + K22y2 (3.26c)

while the locally designed Kalman filter is given by

:(l) (l) (1) (l) (1) “(1) (l) “(1)
xo 7 (A0 7 K01 Clo 7 Ko2 C20 )xo 7 (A01 7 K01 Cll)x1
(1) (1) (l) (1)
+ Bolul + (302 ' K02 Nz’uz + K61 Y1 + K02 y2
(3.27a)

;,1, (1) (1) (1) .(1) (1)' (1) «
81x1 3 (A10 ' K11 C10 ‘ K12 C20 ’Xo + (A11 ’ K11 C11’x1

(1) (1) (l) (1)
+ Bllul - K12 N2u2 + K11 yl + K12 YZ (3.27b)

Viewing both filters as systems driven by u and y we
compare their coefficients. First we compare equations
(3.26b) and (3.27b) for estimating the fast variable x1.
Using (3.25) it is seen that the coefficients of the two
equations are 0(5) apart. Next we compare equations
(3.26a) and (3.278) for estimating the slow variable x0. A
direct comparison of the coefficients is not feasible this
time because the £2 term on the R.H.S. of (3.26a) has an

0(1) coefficient. So we apply the transformation

where M2 is chosen to satisfy

69
M2‘A22 ‘ Kzzczz’ ’ (A02 ’ K02C22) ‘ E2‘Aoo ' Kolclo ‘

K02C20)M2 + €2M2(A20 - KZICIO - K22C20)M2 = 0 (3.29)

The use of (3.28) transforms the optimal filter (3.26) into

p = ((AOO - MzAzo) - (Kol + 0(a))clo - (K02 -_M2K22)C2019

(1) “
+ [A01 ’ K01 C11 + 0(5)]xl + Bolul

(l)
+ (B02 ‘ M2822)UZ + (K01 + 0(5))yl + (K02 ' M2K22)Y2

(3.30a)
81x1 = [(Alo - K11C10)+ 0(8)]p + (All - K11C1)x1 + 0(e)x2
+ Bllul + Kllyl + 0(8)Y2 (3.30b)

Széz = [(A20 - K22C20) + 0(8)]p - 0(E)§l

+ [(A22 - K22C22) + 0(5)]X2 + Bzzuz + K21yl + K22y2
(3.30C)

Transformation (3.28) was introduced in such a way as
to eliminate the fast variable £2 from the slow filter

(3.26a). ZNote that an 0(8) approximation of M2 is given

by
M2 ‘ (A02 ' K02C22)(A22 ‘ K22C22’71' ‘3-31’
i.e.,

M2 3 M2 + 0(6)

70

Using (3.25) and (3.31) the following expressions can be shown

1 \
A(1) (1) ,(1) ( 7

o 7 K01 Clo 7 K02 C26 7 (A00 7 MZAZO) 7 Kolclo
" (K02 - M2K22)C20 ‘1' 0(5) (3.323)
(1) _ 8
K02 - K02 " szzz + 0( ) (3.32b)
(1) (
802 "' Kai-)Nz = 302 ’ M2322 + 0(8) (3.32C)

A comparison of the coefficients of (3.27a) and
(3.30a) using (3.25) and (3.32) confirms that they are 0(a)
apart, which establishes the well-posedness of off-line
computations.

We turn now to investigating the well—posedness of the
on-line implementations. We suggested before that the
local Kalman filter GL19b) be implemented with the localy
computed strategies u§i)and ugi)as the deriving control
terms and the locally available measurements Y1 and y2 as
the deriving output terms. This suggestion looks
reasonable especially in view of the closeness of the
regulator and filter gains which has already been
established. Interestingly enough, it turns out that this
implementation of the multimodel strategy is not well-
posed. The fact that the strategy is not near optimal will
be shown later by a counter example (Example 3.1). Let us

here explain the source of the trouble. It is known, [2],

[4], that LQG controllers for singularly perturbed systems

71

decompose asymptotically into slow and fast filters which
provide estimates of slow and fast variables that are used,
respectively, to compute the slow and fast components of
the control. The slow filter is driven by both the slow
and fast components of the control, or in other words it is
driven by the same control applied to the system. Failure
to have the fast component of the control as an input to
the slow filter results in strategies which are not near-
optimal. In our multimodel problem, since each control
agent does not have complete knowledge of the fast dynamics
of the other subsystem, his local optimization cannot
produce the fast component of the other agent's control.
Therefore, using his locally computed control to derive his
local Kalman filter will lead to a situation where the
local Kalman filter is driven by a control different from
the control applied to the system. 'Thus we conclude that
there is no way to achieve well-posedness of the multimodel
strategy without each agent having access to the fast
component of the control of the other agent. Since such a
component cannot be computed locally it has to be obtained
through communication. 'This can be achieved by changing

the on-line information structure to the following one

Control agents communicate their measurements

as well as their decisions (3.33)

72

Under this assumption the ith control agent will implement
his local Kalman filter with Y1 and y2 as driving output
terms and uii) and uj(j) as driving input terms where uii)
is computed locally while ugj) is communicated to him by
the other agent. It will be shown in the next section that
the above implementation of the multimodel strategy is
well-posed.

The following counter example demonstrates that the

failure to communicate decisions will cause the multimodel

strategy not to be near-optimal.

EXAMPLE 3.1:

 

 

 

 

 

 

-1 l l l l
A ._. -.1. - -.1. o , B ._. - 9:3 0 ,
E1 51 $1
...2. 0 - ...2. o .. -.1.
62 E52 92
I- «I L d
’ l l ‘
1 -0.5 0
G = - 77-77 0 p C =
l 0 -0.5
L 0 " :l .4
'52

73

T - .
[D10 011] [010 911] - I 1 o o
D = o 0.5 o
T
[n20 n22] [920 9221 - I _o o 0.5J

 

 

Forming a family of problems by taking limits 81,82 + 0 we
can compare the steady-state root mean square of the slow
variable xo under optimal strategy, multimodel strategy
with communications and multimodel strategy without
communications of decisionsw These results are given in
Table 3.1, where it can be seen that in the case without
communication the performance of the system does not tend
to the optimal performance.

We conclude this section by commenting on the system
structure that has been adopted in the problem statement.
That system structure has several special features. First,
no connections between the fast subystems. Second, special
structures for the matrices B,(L C and D. Third,
statistical independence of wl, wz, v1 and v2. Most of
these features are essential for the well-posedness of the
multimodel strategy. The only feature that was adopted for

convenience is the independence of w- and v- for each i.

i 1
This one can be removed at the expense of more complicated
algebraic expressions. However, (w1,vl) on one hand and
(w2,v2) on the other hand must be independent of each
other. Counter examples can be constructed to show that

violation of the noise independence or the structure of the

matrices AWB,G,C and D will lead to ill-posed multimodel

.mcofimwoop cw mcowumoficsesoo nu“: can usosuwa
xmmumbum Hmcoefiuass now mosam> mumsvm cams uoo~ manmwump 30am 21m wanna

 

74

 

No.0 mmooow.o hm.mm mammam.o Nmmmmm.o voco.o
no.0 mammww.o mm.vm mNHHom.o mommm©.o Hco.o
hd.o mNmlo.o mo.Hm mammmm.o HHHQOF.O voo.o
Hv.o mmmwmh.o em.mm mmhvmm.o hOthh.o Hc.c
Nm.~ wmthH.H Hm.HH HmomMN.H hmHHHH.H Vo.c
mm.H whhdhw.d hm.N fimommood wmmmmw.H H.c
mm.m NHmhmH.m mm.m ameMH.m havmmm.m v.6
mcoflumowcss mcoHumowcss
w 1500 :ufi3 a 1500 usonuw3 Houucou «undo
Hmooefiuasz Hmnoefluasz Hmefiumo
0H 0H
11.... n N< onvéédnmu ...ila u as onvéééuau onvwéduou

oulmu ounuu

 

75

strategies, some of them are given in Appendix 3.3. The
only possible relaxation of those requirements is to

replace zero terms by 0(9) terms, e.g., A can be taken as

o ( 1) o (1) o < 1)

A g Bill Bill 915.).
El 81

99. 9&3). 92’.
‘32 8:2

 

 

It is interesting to mention that in the multimodel
solution of the deterministic Nash problem studied in [16)
it was shown that the weak connection assumption could be
violated as long as the closed-loop system is
asymptotically stable. This is not the case in the
stochastic problem where the contribution of the fast is
very important. A counter example to verify this can be

easily constructed too.
III.3 PROOF OF WELL-POSEDNESS OF THE MULTIMODEL STRATEGY

In this section we want to show that the performance
of the system under the multimodel strategy approaches the
performance of the system under the optimal strategy,
asymptotically, as €1,t:2 +-~0. The performance of the
system is characterized by values of performance criteria
and state trajectories at steady-state. The idea of the
proof is to represent the system under feedback control in

each case as a system driven by external white noise. By

76

appropriately transforming the equations into new
coordinates it will be shown that the system under
multimodel control is a regular perturbation of the system
under optimal control, ine., they differ only in right-hand
side coefficients which are 0(6) apart. Then, lemmas
generalized from [4] to the multiparameter case will
establish the effect of such regular perturbations on
solutions of the stochastic differential equations and on
the performance criteria. Finally by applying those
lemmas we will conclude the proof.

Let us start by considering the system under the
multimodel strategy. The multimodel strategy has been

defined as

F . . - -
(l) (l) (l) ~(l)
u1 Fol P11 0 0 x0
mm
(2) (2) (2) “(1)
u2 0 0 F02 F22 x1

- - - d

 

 

 

 

“(2)
x0

~ (2)
"2
(3:34)

 

 

where (iél’, kil’) are the estimates provided by the local

Kalman filter (3.19b) which is derived by the measured
output y and the multimodel control umnv Writing down the
equations for the closed-loop system under the multimodel

strategy and using (3.5a) and (3.34) to eliminate y and

77

“mm' respectively, yields the following set of equations:

2
_ ' (j)*(j ) (j)~(j)
_ A x + .: [A03xjm - BOj(FOj XOJ + Fi lj .j )

+ Gijj] (3.356)

' i)‘(i) (1)‘(1)
i im7 ioxom ii im7 ii F‘oi xo +1fliix )7Giiw i'

i ='l,21(3.35b)

:(i) (i) (i) (i) (i) “(i) (i) ‘(i)
X = (A0 ‘ K01 CiO ‘ KOj Cjo )Xo + (A01 ' K01 Cii)xi

(1)~(1) (1)*(1) ) _ (B(1) _ K(1)N j)(F(.j)'~(j)

7 Boimoi xo 7 ii x1 03 0303 o

(j)*(j) (i)
jj Xj ) + K01 (CioXOm + C

(i)

‘+ Koj (Cjoxofll-t ijxjfll.+ V

+ v-)

+ F iixim 1

j), i,j = 1,2, 3 f i (3.35c)

i) (i) (i) A(i) (i) “(1)
io ' Kii Cio ‘ Kij Cjo)xo + (A11 ‘ K11 Cii)xi

(1)«(1) + F(1);(1)) + K(1)N (F(j)“(j) + F(j)‘(j)

(Foi x0 11 1 13 j 03 x0 33 xj

)

(i) (i)
ii (Ciox + C-.x- + Vi ) + Ki (C- x + C -x- + v-),

7 K 11 1m 13 3o om 33 3m 3

1,3 = 1,2, j + i (3.35d)

where (xom, xlm' x2m) denotes the state of the system under

multimodel strategy. Setting

om

 

om

“(1)

“(2)
x0

 

d

im

 

78

im

“(i)

 

' i = 1,2,

equation (3.35)'can be rewritten as

i im

where the matrices 513'

way.

3.25)

om

= S

ioS

S" = 0(8),

1]

om

ii

5

it follows that

im

Siijm

It is pointed out that using Ki]

0(8),

7 soosom 7 Solslm 7 S0252m 7 Tow

+ TiW,

(i)

j t i

W

ilj = 1:2:

" q

w1
w2
v1

 

 

LVZJ

(3.36a)

j k

(

(3.36b)
Ti can be obtained in an obvious

= can, 3 # 1 (see

(3.37)

where W a Block diag[W1,W2,V1,V2] is the intensity matrix

of w.

Expressions (3.37) are crucial because they

guarantee that the fast variables of the closed-loop are

weakly connected which is needed to carry out the proof.

Using the asymptotic properties of the solution of the
regulator and filter Riccati equation of multimodel

solution of Lemma 3.3 (see e.g. [2] , [4]), it can be shown

that

Re A(S

ii I-
81

=82: O

<

0

510 ' 80252

1

252°)! )

(3.38a)

(3.38b)

i

79
Hence, for sufficiently small 51 and 62, the closed-100p
system is asymptotically stable.

Let us now consider the system under the optimal
strategy. As we explained in section III.2, a comparison
of the coefficients of the optimal and multimodel
strategies will not be useful if the optimal Kalman filter
is expressed in its original coordinates per (3.26). The
state information.(3.28) should be used to facilitate such
comparison. Since the multimodel strategy contains two

different estimates of the slow variable x we use two

0'
transformations

Pi = i0 - eijfj: i.j = 1.2. j + i (3.39)
where Mj satisfies

Mj(Ajj - ijij) - (Aoj - Kojij) ejiAoo KOiCio

7 K0jCjo)Mj 7 eij(Ajo 7 Kjicio ' ijcjo)Mj = 0,

(3.40)

ya
u-
U

which coincides with (3.29) when 3 = l, i = 2. The
transformations (3439) introduce a redundant vector in the

description of the optimal control since x is replaced by

0

both p1 and p2. Using (3.26) and (3.39) we obtain

80

E3i 7 [A00 7 Kolclo 7 Kozczo 7 M3(Ajo 7 KjiCio 7 ij cjoHpi
+ [(Aoi - KoiCii) + ijjici 1x1
+ Boiui + (Boj - Mijj)uj (3.41)
+ (K0i - ijjiwi + (Koj - Mj ij)yj, i,j = 1,2 3 # i

where (3.40) has been used to eliminate the ij term on the
R.H.S. of (3.41) Using (3.25) expressions (3.32) for M2 and
analogous expressions for M1, it can be shown that

(i) (i) (i)

(A. C '- KI - C- -+ 0( d))pi

pi 7 o 7 oi io 03 30
(i) (i) (i)
+ (A01 - Koi Cii + 0(3))xi + Boiui + (Boj " Koj Nj
(i) (i)
+ 0(8))uj + (KO + 0(a))yi + (xoj + 0(5))yj,
1,3 = 1,2 j + i (3.42)

Moreover, using (3.39) and (3423) the optimal control can

be expressed as

U' a ‘ [F + FiiXi + (F- )X 1

oipi lj 7 6i FoiMj j
(3.43)

- [(F(i) + 0(9))pi + (F(i) + 0(8))xi + 0(€)xj]

Defining
. x07 x
i .
so 2 p1 , Si = . , 1 = 1,2,
xi
92

 

 

81

and using (3.42), (3.43), it is straight-forward to show
that the closed-loop system under the optimal strategy can

be represented by

éo = 50050 + 50151 + 80252 + Tow (3.44a)
7151 = 81080 + Siis1 + 51353 + Tiw,
i,j = 1,2, j + 1 (3.44b)
where
éij - sij = 0(5), i,j = 0,1,2 (3.45a)
ii - Ti = 0(5), 1 = 0,1,2 (3.45s)

The comparison of the performance of the closed-100p system
under the multimodel and optimal strategies has been
reduced to a comparison of the solutions of CLBG) and
(3.44). The following two lemmas carry out the latter
comparison.

LEMMA 3.4: Under the above conditions, and for
sufficiently small 81 andiiz, the following expressions

hold at steady-state (as t‘*“’).

E([[so - somllz} 0(62)

0(5), 1 = 1,2

B£!1si - Simllz}

Proof: See Appendix 3.4.
To be able to compare the value of the performance

criterion under the multimodel strategy, denoted by Jm,

82

with its optimal value, denoted by J, we express Jm and J

in terms of the variables of (3.36) and (3.44)

 

 

 

 

 

 

 

 

respectively.
.. .T
som (soml
- - T
Jm - 1:11“; )3 51m (LO,L1,L2) (LO,L1,L2) slm (3.46)
LSZm ‘ _52m
. . T . .
so 8o
J 2 lim a s1 (io,i1,£2)T(£o,Ll,i2) sl (3.47)
t-.-)- 00
S2 82
Using (3.43) it can be verified that
ii - Li = 0(5), i = 0,1,2 (3.48)

LEMMA 3.5: Under the above conditions*, and for
sufficiently small 51 and :2, the following expression

holds

Proof: See Appendix 3.5.
With Lemmas 3.4 and 3.5, we have completed the proof

of the following theorem.

 

* To exclude trivial cases it is assumed throughout the

chapter that J +'0, as $1, 82" 0.

83
THEOREM 3.1: Suppose that the condition of Lemmas 3.1 and
3.2 are satisfied, then, for sufficiently small El and 82,

the multimodel strategy u defined by CL34) is near-

mm'
optimal in the sense that
xom = x0 + 0(a)
1/2 .
xim=Xi+0(€ )7 1:1(2
(Jm - J)/J = 0(5)

III.4 FURTHER DECOMPOSITIONS OF MULTIMODEL STRATEGIES

As we indicated earlier, the local design problem for
each agent is a singularly perturbed LQG optimal control
problem. Well-documented decomposition techniques (e4;,
[2], [4]) can be used to simplify the local strategies. As
an example we consider a slow-fast decomposition technique
due to Haddad and Kokotovic [2]. Instead of solving his
local singularly perturbed LQG problem, each agent defines
slow and fast LQG control problems which are extracted from
the singularly perturbed one. For agent 1, the slow
control problem is defined by, formally, setting 61 = 0 in
his local model. IBut this local model has been obtained by
setting 62 = 0 in the overall model of the system. Thus
the slow control problem of agent 1 is obtained from the
overall problem by setting 62 = 0 followed by 51 = 0.
Therefore, the control agents end up with the same slow

problem which is given by

84

x8 = AsxS + Bsus + Gsw (3.49a)
y = CsxS + Nsus + st + v (3.49b)
t \
- . l l T T T T T
“s = 11m --------- E‘ I [xSDSDSxs + 2xSDSESuS + uSRsusldt
t-+-m (t1 - t0) to

The newly defined matrices are given in Appendix 3.1.

The optimal solution of this problem is given by [2]

u F 1
us = 15 = - [13 x5 (3.50)
u25 F2s

where £5 is provided, for any control input u, by the slow

Kalman filter

x3 = AsxS + Bsu + Ks(y - CsxS - Nsu) (3.51)

The fast control problem for agent i,i = 1,2, is given by

Eixif = Aiixif + Biiuif + Giiwi (3.52a>
Yif = C1131: + V1 (3.52b)
1 t
. 1 T T T
Jif 7 11m """" E ]. [xifDiiDiixif + “ifRiuidet
toérmvtl - to to
t11.. (3.52c)

and its optimal solution is given by [2]
uif = -Fif£if (3.53)

where §if is provided by the fast Kalman filter

85

1 . 5(1) , ~
1‘15 = Aiixif + Biiuif + K1f‘Y1 ’ Cisxs ' N1uis ‘ Ciixif)'

(3.54)

E

“(1)

In (3.54), (Yi - Cisxs ) is the realization of the

“ Niuis
postulated fast output yif' where xéi) is the output of the
locally implemented slow filter. Expressions for Fis' Ks,
Fif and Kif can be found in [2] or [4]. The important
thing for us here is that these matrices are related to the

optimal solution regulator and filter gains as shown per

(3.55).
- F (A B F )71(A B F ) 0( )
F13 - oi ‘ P11 11 ‘ 11 11 io ' 11 1o + E
K - x c 71 K — K15
is - oi ‘ (A01 ' Koicii)(Aii ' K11 11’ + 0‘8)? 5 *
K25

A multimodel strategy can be formed as before. Agents
communicate their measurements as well as their decisions.
The local slow filters are driven by the actual
measurements and by the same control inputs applied to the
system. The local fast filters are driven by the actual
measurements and by the locally computed fast control
inputs. Finally, each agent applies his locally computed
component of the control strategy which is formed as the

sum of the slow and fast controls, i.e., u- = u

1 is 7 uif'

Thus, the multimodel strategy is given by

86

,
Si9(1))

“(2)
5 x8

u2 0 F23 0 F21 x11

 

(“25

 

13.56)
The near-optimality of the multimodel strategy (3.56) is
established per the following theorem.

THEOREM 3.2: Suppose that the conditions of Lemmas 3.1 and
3.2 are satisfied, then, for sufficiently small 61 and 6 ,

the multimodel strategy u defined by (3.56) is near-

ml
optimal in the sense that

+0(S)' X'

1m 1’2'

X :1!

om = x1 + 0(21/2), 1

0

Outline of Proof: The proof of this theorem is almost
_identical to that of Theorem 3.1. There are only two
points of difference. First, because the locally

.(1) «(2) ~

implemented slow filters are identical, x8 = x8 = xs,

in studying the closed-loop system under the multimodel

strategy both iél’ and x;2)

are represented by the same
equation.(3.51L. Therefore, there is no need to introduce
a redundant vector in studying the optimal solution as we
did in the proof of Theorem 3.1. Second, the
transformation used to represent the optimal closed-100p

system in new coordinates, where the coefficients are 0(5)

perturbation of the coefficients of the multimodel system,

is given by

 

 

 

87

 

 

 

90 I0 ‘ e1“1“1 ‘ E2M2A2 ‘ S1M1 ’ 52“21 *0“
ql = Al 11 0 x1
92 ( A2 0 I2 , ,32.
where A1 and Mi are chosen to satisfy
(A11 ' B11F11’A1 ' (A10 ‘ B11Fo1’ ' BiiFjiAj
7 6““00 7 BoiFoi 7 BojFoj) 7 (Aoi 7 BoiFiiMi
- . - . .. x. =
(AOJ BOJFJJ)IJ] 0 (3.57)
,\
M1HA11 ‘ K11C11’ + E1“1<Aoi ‘ K01C11’] ' (A01 ’ K01‘311)
7' %I(Aoo 7 KoiCio 7 Kojcjo) 7 (Aoi 7 KoiciiHAi
- o — o o . .A.. 0
(A03 KOJCJJ) 31:11
+ EijAJ-(Aoi - Koicii) + MjKjiCii = 0 (3.58)
i,j = 1,2 1 3 j

88

APPENDIX 3.1

MATRIX DEFINITIONS

 

 

 

 

 

 

PAoo A01 A02- 7 B01 B02.
A - 512 511 o B _ 511 0
E1 81 E1
522 o 522 0 323
. 82 ' 82. . 52.
r
7
G G
01 02
G Clo Cll O
c - -ll 0 c -
_ e -
1 C20 0 c22
c
0 -32
- 62..
i T T q T T
{lDloDlo + szzonzo (1°10D11 Y2920022
T _ T , T
D D ‘ Y1911910 )1D11911
T T '
)2922920 0 Y2922022

 

 

 

xo
21 y1 h1
x = x1 , z = , y = , u = , v =
22 Y2 u2
x2
(1) -l (i) -1
A0 = Aoo ‘ AojAjjAjo' Boj = Boj ‘ AojAjijjr

 

89

(i)

 

 

 

 

- . . . . . - . .. . .A.
C30 C30 C33A33A30' D30 D30 D33A33 30
(i) -1 . . _ . é .
Gjo ‘ Goj AOjAijjj' 1:3 - 1,2, 3 . 1
- A7113 H - A716 E - A7113 1 2
N1 ' ”C11 11 11 1 C11 11 11 1 “D11 11 11 1 - ,
' (1) 1
A0 A01 (1)
A(i) = x‘i) _ 0
A10 A11 (1)
-_- --- xi
81 E1
L 1
. T
i T (1) (1) y T ‘
T QDiODio 7 Yijo Djo {iDioDii
D(1)D(1) = i,j = 1,2,
T T
_ Y1911910 Y1911911‘
(1). r (2) -
(1) 7301 '302 (2) 301 Bo2
B = , B =
B B
.%E 0 0 -23
' l .1 " 24
. (1)- ' (2) 5
G = I G =
G11 922
--_ o o -2.
.. El .1 ' 2..
V ‘ F (2) 1
c c c 0
C(1) = 10 11 ' c‘2) = 10
(1)
.C20 0 3 ,Czo C223

 

 

 

 

 

 

 

 

 

 

 

 

i h

j

N(1)

B(1)

R(l)

Q
I

O
I

90

 

 

 

 

 

 

= 0 0 ' N(2) = [Na 0
0 N2 0 OJ
0 0 H 0
l
= 0 H ' H(2) = o
0
2
L 1
Y R 0 Y (R + ETE ) 0
= l l R(2) l 1. l 1
. T '
’ T ' T I
0 7213221).) E22 71137:) El 0
= ' I.(2) _
0 0 L 0 0 i
2 -1
A00 ’ j_1 AojAjjAjo, BS = [815' 825],
B - B A A-lA — B
is ' oi oi ii 10 ‘ 01
_ -1 _ (j)
[Gls' G251' Gis ’ Goi 7 AoiAiiAio Goi
Cls (
J)
[c ] , C1s C10 C11A11A10 - C10
2s
N1 0 H1 0 fflsl o
I HS 3 ' ES 3
_ Y T _ -1 _ (j)
7 YlDlles 7 2D25D2s' Dis " D10 7 DiiAiiAio ’ Dio

91

APPENDIX 3 . 2

In [14], the following limits have been shown

(1) _,
Foi 7 Foi ' 0
(1)
F11 ' F11 ‘* 0
(i)
Fij 0
Fji "* 0
F Su-[F - - F--(A-- - B--F--)71(A- - B--F .)l + o
03 03 J] 33 33 3] 3° 33 03

by assuming the existence of a bounded solution of
algebraic Riccati equations for a multiparameter case. Now
we want to verify this assumption and show a little bit
stronger result that we stated in (3.23).

Solutions of algebraic Riccati equations (3.10) or

(3.12) are zeros of a continuous function.

E

1 ..
';(Q, 81' 62' 'é'z') = 0 (11.1)

where E113(0,€1*] and ‘EE (0,é2*] are perturbation
parameters with a bounded ratio, i.e.,

e
-1 ..
0 < kl S E5 S k2 < °° (11.2)

By taking

92

El ..
E5 = a k1 S a S k2 (11.3)

equation (11.2) can be written as
.310, €2,cx) = 0 525(0,32*] kl g czg k2 (ii.4)

It can be shown that the solution of (ii.4) or (ii.l), in

the limit, does not depend on a, i.e.,

(£100, 0, (I) = (QC, 0’ 0) 7-7 0 (11.5)
thus there-exists QO such that

00 + 00(0) (ii.6)

By the implicit function theorem [241 there exist constants

8,<31, and 02 such that

a) J = 305(00, 0,01) = 0(1)l =. 8(a) = {13‘1(a)H (11.7)
8 2:0
iue., q - Jacobian is invertible and in addition J has to
62:0

be continuous at (0°, 0, a) which ask for the existence of
b) (71(0) > 0

(ii.8)
C) (12(6) > 0
such that 5121?;de (31) and V5 26S€(0,02), Q is a unique
solution of (ii.4), i.e., there exists a continuous mapping

H satisfying Q = H82. We are going to prove that B, 01

93

and 02 can be chosen independently of a and thus the
Implicit Function Theorem can be applied uniformly in a.
It can be easily shown that the Jacobian in the limit

is given by

 

 

r q 1
100 F01 F02 0 0 0
I10 P11 0 I13 114(1) 0
J|= I‘20 0 I‘22 0 1"24(0) T25 = 3(a)l
.. I" e ..
62—0 33 0 0 2..0
0 0 1314(3) 0
(11.9)
0 0 F55
L 1
where

. _ .- _ T
1714““ - In]. x 5 (A20 I322302)

1 T
I724”” = In1 1;:(7110 " 511301)
G.

T 1 T
17.44%) In] X v/C: (A22 - Bzzfizz) 'l" --(All - Bllzll) X In

/E 2

- - T - -
I‘oo ’ In x (A00 7 Bol£ol 302302) 7 (A00 B02301

O T
302302) x InO

- .. T _ T . 3
Poi - InOX (A10 Biigoi) + (A10 BiiEoi) X Inc 1 1,2
”T-=(A--B.F..)Tx1 1:12
10 01 01-11 no I

I. .' on - 0. on T . 3
ii _ (A11 Bllfill) x Ino 1 1,2

= - T - - T
I733 I111 x (A11 311311) ”‘11 3115311) x In1
(11.10)

T _ _ T _ T
55 * I 3 (A22 322322) + (A22 B22E22) X In2

“2

94

_ _ T
F13 - Inl X (A10 Bolfiol)

1
ll

_ T
I1'12 x (A20 BOZEOZ)

where x denote Kronecker product.

Determinant of (ii.9) can be expressed as

-l -l
d6t(J(a)) ‘3 det(1‘oo " F01 T11 F10 ' 102.1"22 T20) det(r33)
l52=0
det(1"44(a)) ° det(1"55) (ii.ll)

Obviously. P33, F44(a) and F55 are nonsingular matrices.

It can be shown that

v -1 F - T
00 ' ‘olrllrlo ' *02F22120 = In x (As ' Bst) + (As '

T ..
BSFS) x In (11.12)

is a nonsingular matrix too. Thus, det(Jh))I 2? 0,
E =0
i.e., J(oz)i is nonsingular. On the other hand, acobian at
E:2.=0
€2=O is a continuous function of astkl, k2] which follows from

(11.10).

Leth observe that J'1(a) is a continuous function of a
8:0
since det J(a)‘ (as a product of eigenvalues) and all
5' =0
cofactors of J(a?|€ are continuous functions ofcz. Thus
2‘0

all elements in J'IW)IE are continuous functions of a for
=0
qukl. k2] so that a max1mum norm of J"1(0t)l can be found
, E =0
independently of a, hence the constant defined in (115n

can be chosen as

e= max [lJ‘l(aHi (11.13)
a 18230

95

thus independent of a. ‘On the other hand, since a is
limited to a compact set.(ii.3), then by a known theorem of
mathematical analysis (every continuous function on a
compact set is uniformly continuous on that set) Jacobian
is in fact uniformly continuous in a, which imply the
existence of constants defined in (ii.8) independent of a,

i.e., there exist

01 > 0 such that 01 4 01(a)
and (ii.l4)

02 > 0 such that 02 + <§(a)

Thus all constants required by the Implicit Function
Theorem can be chosen independently of a., so that given
theorem can be applied 'uniformily in.a' and guarantees the
existence of a unique and continuous (thus bounded)
solution for all Q€§O(Qo' 01) and all €2€§€(0, 02).

Note that 02 determines upper bound for E , i.e.,
32** = 02 (ii.lS)‘
and by (ii.3)
el** = k182** = kl‘E (ii.lG)

In a summary, we have proved the following
THEOREM 343: For a sufficiently small 61 and €2 there
exists a unique and bounded solution of Riccati equation

(3.10) or (3.12) represented in general form by

, E1
J(Qp El, %( f-) = 0 £1 E (0' E

'*1 i=l’2
c2 1

E

o < k1 38-1- 3 k2 < .. (11.17)
2

It is obtained for 81 E (0, ((1021 and 62 8(0, <72].
Combining results of Theorem 3.3 and [14], we can

conclude that all results proved in the limit in [14] are

in fact proved with an 0(8) closeness.

97

APPENDIX 3.3

EXAMPLE 3.2:

Using the same data as in the Example 3.1 except for the
matrix B which we are choosing now in the form that does

not possess multimodeling structure, i.e.,

' 1 1 l
B = - 2:: 9.1?-
81 81
..l _ -2.
L 52 E2 .

 

 

we get the following results for the optimal and multimodel

strategy

 

 

 

81’82 Jopt Jmm
0.4 8.111271 21.814347
0.1 14.425567 83.131649
0.04 19.161903 209.724864
0.01 34.406927 771.714490
0.004 62.758270 1596.65
0.001 203.331059 3432.47
Table 3.2

Obviously, the criterion under the optimal and under the
multimodel strategy are far apart. We can conclude that in

this case multimodeling cannot be used.

98

EXAMPLE 3 . 3 :

Let us now violate the structure of matrix A with other

matrices like in Example 3.1. Take A to be

 

 

' -1 1 1 ‘
A = —l _ ..l 0
£1 E 1
-3 - .5}. .. -3
.. 82 E42 $2 ..

In fact, we have changed only A21 element from 0 to -l/ez.
The results for the optimal and the multimodel strategies

are given in the next two tables.

 

 

8l=€2 Jopt Jmm

0.4 4.912880 12.072639
0.1 7.925279 19.540843
0.04 11.357365 24.677558
0.01 26.444723 40.754760
0.004 56.145159 70.668324
0.001 204.396474 219.066383

 

Table 3.3

99

 

 

 

)

51:32 Optimal Multimodgf A = 77-79

Control Control 0

%
0.4 1.278237 1.334436 4.40
0.1 1.437707 1.532534 6.60
0.04 1.486750 1.613071 8.50
0.01 1.514677 1.662574 9.76
0.004 1.520575 1.673268 10.04
0.001 1.523564 1.678719 10.18
Table 3.4

Even though in the Table 3.3 we note only a constant bias
in the criterion, Table 3.4 shows that slow trajectories
get further apart as E gets smaller. Thus, multimodeling
is inappropriate in this case, too.

Violating multimodel structure of weighted matrix DTD
is going to produce the same effect as in the previous two
examples. In fact since DTD is a symetric matrix, it can
be diagonalized by a nonsingular transformation (getting
rid of undesired elementsL. But this transformation will
destroy the multimodel structure of the remaining matrices
in the model.

By analogy of regulator and filter problem changing
structure of matrices G and C will produce the same effect

as in the previous examples.

100

APPENDIX 3.4

PROOF OF LEMMA 3.4

T T T T

. T
Let e. = S- - S , 1 = 0,1,2, and XT = (so, 51, 52, e0, e1,

1 1 im

T T .
e2) , then x satisfies the following system of equations

driven by white noise '

 

 

 

 

 

 

 

 

F ' ' 1
x = AX + Bw = x + w (iv.l)
A21 A22 B2
L .1 L a
where _ P _
S00 S01 S02 T0
510 S11 S12 T1
A11 = '5' ‘5‘ "2' ' Bl = 1' .
l l 1 . fl
€22 €21 523 $1
82 E2 E2 82
h ' J L .1
A21 takes the form of All with éij replaced by (Sij - éij”
A22 takes the form of All with Sij replaced by Sij and 82

~

takes the form of El with T1 replaced by Ti - Ti‘
The variance of x at steady-state is determined by

solving the algebraic Lyapunov equation
QAT + A0 + BWBT = 0 (1v.2)

where W = Block-diag [W1' W2, V1, V2] and Q is partitioned

compatibly with the partitioning of A and B as

101

Because of the special form of the matrices A1 and Bi' the

3'
solution of (iv;2) is sought in the form

FHoo H01 H02 3
l l
H = H -- H ------ H (iVo3)
10 8.1 11 ‘45—‘52 12
1
H ------ H ..l H
L 2o r——-—€l€2 21 52 22

 

 

With similar expressions for M and N. Because of the
symmetry of H and N, we have Hio = 8:1, H21 - Hi2, N10 =
Ngi and N21 = 832. Substituting the expressions (ivw3) in
(ivuZ), partitioning the Lyapunov equations and multiplying
some equations by 81 and/or 52 to remove unbounded
coefficients as 8‘1, 82 *0 we end up with twenty one

equations in H M--, Ni'

ii" 11 J
$2 either analytically or through the ratio 21782 which is

whose coefficients depend on 81.

bounded from below and above. Using the implicit function
theorem [17] it can be verified that for sufficiently small
$1 and E2 there exists a unique bounded solution (Hij' Mij'
Nij' i,j = 1,2) satisfying the above-mentioned equations.
In applying the implicit function theorem the stability
conditions C138) are employed to ensure the invertibility
of the Jacobian. With the existence of a bounded solution

established, the equations for Mi' can be written as

J

_ —m -T

SooMoo + SolMlo + SoZMZO + MooSoo + MloSol + M20502 = 0(5)
(iv.4)

102

 

_ '6' _ _ ._
S-Mo-+\/-]-'-S-M--+M Sir-+14 ST =0(€),

01 11 Ej 03 31 oo 1o 01 Sii

1 # j, i,j = 1,2 (1v.5)
-' M + _' M + T 6i §T —
Sio oo Sii io Miisoi + \/;3 Mij oj ' 0(5).

1.+ j, i,j = 1,2 (iv.6)
_ _T - . - .
S11M11 + M11511 7 0‘8) 1 - 1:2 (1v.7)
./:.§ ii “ii :7 Mii §jj = ”(6"

1 # j, i,j = 1,2 (iv.8)

_ 5 '~_'

where S.- = S To arrive at (ivu4f-

13 ijl S=0 ‘ S1j h:=0 -
(iv.8) we have used (3.37) and (3.45). Using the stability
conditions (3.38), the following series of conclusions are

obvious. First from (iv.7) and (iv.8)'

"ii = 0(5), 1 = 1,2, M12 = 0(5), M21 3 0(8) (1V.9)

Using (iv.9) in (iv.5) and (iv.6) yields

_-1 _ T -—-l—

(iv.10)

Substituting (iv.10) in (iv.4) we get

103

—— .91. _. _-1_
(500 ' Solsllslo ' S02322520)Moo

- — ——1— _ _-1.. T
+ Moo(Soo ' solsllslo ' So2522820) = 0(5)

Hence

Moo = 0(a), Moi = 0(5), Mic = 0(a), 1 = 1,2 (1v.11)

Using (iv.9)-(iv.ll) the equations for Ni' can be written

3
as
T T T T T ,2
SooNoo + SolNol + SoZNOZ + Noosoo + NolSol + NOZSOZ ‘ 0(8 )
(iv.12)
GTT
W1 T T T 2
SoiNii + \/E; SojNij + Noosio + NoiSii + eisoiNoi 3 0‘5 )'
1 # j, i,j = 1,2 (iv.13)
T T 81 T
S11N11 + N11511 + E131o o1 + 81No131o + 2; N1j51j
61 NT 2) . - . . . _ .
+ E; Sij ij = 0(e , 1 f 3, 1,3 - 1,2 (1v.14)
S N + 82 S N S N + NT ST
82 lo o2 E; 11 12 + 12 22 81 01 20
T 81 T
.j2

From (iv.14) we have Nii = 0(6), 1 = 1,2, then from (iv.15)

we get N12 = 0(8). Using that in (iv.13) yields

104

T -T .
N ' =‘N Siosii+0(€) 1-1'2,

01 00

which when substituted in (iv.12) results in

-1 -1
(500 ’ SolSllslo ' S02822520)Noo
-1 -1 T
+ Noo(soo ' S01511S1o ‘ 502522320) = 0‘5)

Thus N00 = 0(8) .-.. Noi = 0(5). Going back to (iv.14) we see
that, in view of the above orders of magnitude, (iv;l4) can

be rewritten as

w T _ 2
S N +01 811-0(8)

11 11 11

Therefore Nii = 0(62). Now (iv;15) can be rewritten as

’52 161 2
l 22
J81 11 2 V 82 12

so that N12 = 0(82). Similarly (iv.13) is rewritten as

T T _ 2
Noosio + Noisii ' 0(5 )
T -T 2 .
”N01 3 -NOOSiOSii + 0(E ) (117.16)

Substituting (iv.16) in (iv.12) yields

-1 -1
(500 ' solsllslo - 802822820)N00

105

“'1 "l T 2)

+ N (500 ' So151151o ‘ 502522320) = 0‘5

00

which shows that Noo = C(62). Thus we have shown that

Nii = 0(62), 1 = 0,1,2 (iv.17)

. T
Efllso - som H2} tr E{eoeo} tr N00 = 0(52)

2
imli}

E11151 - s tr Efeief} ;- tr Nii = 0(5) Q.E.D.

106

APPENDIX 3.5

PROOF OF LEMMA 3.5.

The expressions (3.46) and (3.47) can be rewritten as

T T T T 1 T
Jm = trfLOLOHOO + ZLOLIHOI + 2L0L2H02} + tr1E; L2L2H22}

1 T 2 T T
+ tr{€; LlLlHll + ’?E%F§.L1L2le} (Vol)

and

~T~~ ~T~~T ~T~~T 1 T~
J = tr{LoLoBoo + 2L ooL18 1 + 2L ooL28 2} + tri-g LZLZBZZ}

1 T 2 "T~ ~T
+ tr{-I L1L1811 + -:%§§§ Llele (v.2)
From Appendix 3.4 we have 313' = ~Hij + 0(5) and 312 = 0(..)
so that
1 -T~ ~T~
AJ 3 Jm ' J = E; tr{(LlL1 ‘ L1L1)Hll + 2L1L1M 11
(v.3)
"T"(T + ) l L T ) TL M }
+’ Lle M12 M21 1 + E; tr{(L2 2 - L2L2 322 + 2L2 2 22 + 0(8)
Two cases can happen. The first when
tr£LiLiHii1l = 0 i = 1,2 (v.4)
€l=£é=0

which implies directly by (3.48) and Appendix 3.4 that

AJ/J 2 0(43. The second case arises when

107

} = 0 i = 1,2 (v.5)

Equations for Hii’ defined in Appendix 3.4, are given by

a ”T ~ ~T .
SiiHii + Hiisii + TiWTi = 0(5) 1 2 1’2 (V06)

Their solutions have the form

- L ~ T
°° Snt" ~T S-- t .
Hii = j; e 11 TiWTie 11 dt + 0(5) 1 = 1,2 (v.7)
Using (v.7) in (v.5) we get in the limit
Liesiitii l = O Vt, i = 1,2 (v.8)
S .8 _
1‘ 2"0
and
A
LiHiil = 0 =Liflii 1 = 1,2 (v.9)
a1""32=°

Then using the fact that Li = Li + 0(a) and Hii = Hii +
0(a) easily can be shown that in this case
1; .1...

LiHii 3 0(8) and tr{(L§Li ’ LiLi)Hii} 3 0(62) (v.10)

Similarly from (ivu?) and (ivu8), taking into account 0(a)

terms, we have

Q! ~

T
SiiMii + “115

ii + Hii0(a) + TiW0(€) = 0(22)

1 = 1,2 (v.11)

108

e o .2. C.
J 1 T c
‘/2; SiiMij + ,/23 ”ijsjj * 3110‘“)

+ Tgwo<a) = 0(22) (v.12)
i,j = 1,2 i # j

The solutions of (v.11) and (v.12) are given by

co ~ , ..T
“11 = .[ esiittfliiO(e) + Tiwo<e)1esiitdt + 0(22)
0
i = 1,2 (v.13)
s . ~ 8
/ J 1 T
w E; Siit e Sllt
_ * - 2
Mij — j;e [3110(5) + TiW0(c)]e dt + o<8 )
‘ i,j = 1.2 i f j (v.14)

By (v.8) and (v.9) the last two equations imply

~ ~ T
E: 1=€2=0 E 1:5220 Sleazzo
i = 1,2 (v.15)

Having in mind that Mij = 0(5), i,j = 0,1,2 and Li = -L-i +

0(6),(v.15)iJIfact.states that

~

_ ‘2 , “ _ 2 ” T _ -2
LiMii - 0(t. ), Lzle - 0(8 ) and L2M12 - 0(c. ) (v.16)

109

Using (v.10) and (v.16) in (v.3) we have AJ‘= 0(6). Since,

by assumption, J + 0 as el,az-+ 0, we finally get

-_ = 0(6) (v.17)

which completes the proof of Lemma 3.5.

IV. DECENTRALIZED MULTIMODEL ESTIMATION

The Optimal solution for the estimation problem with
different information sets [26] requires infinite
dimensional state estimators. From a practical point of
view this is an unacceptable result. That is why the very
first researchers in this area [261-[28] defined and solved
an approximate problem with constrained state estimators
specifying in advance dimensions of the corresponding
estimators. Recently, an alternative way of defining
practically implementable decentralized estimation problems
came from Speyer [29] and has been studied in [301-[32].
The main idea is to rearrange the optimal centralized
estimator to be implemented via a decentralized scheme.

In this chapter we will give some general improvements
over the work of [291-[32]. Scattering diagrams, already
used in [32] to clarify results of Speyer's decentralized
estimation technique, will explain easily the construction
and advantage of the new proposed scheme. In the case of
systems defined in Chapter III, presence of slow and fast
modes will lead to further reduction in computational
requirements of our method, leading to the so—called
multimodel estimation. In fact final results resemble the
multimodeling ones even though from the very beginning we
followed decentralized estimation ideas and multimodeling
results in as a by product.

Both steady state and finite time estimation problems

are considered. In order to be able to formulate the

110

lll

finite time multimodel estimation scheme the existence of a
bounded solution of differential Riccati equations for

multiparameter singular perturbation was established.

IV.1. A NEW METHOD FOR DECENTRALIZED ESTIMATION

In this section we present a new method for
decentralized estimation. Instead of solving local
estimation problems by using locally optimal estimators
(idea originated by Speyer [29], and mathematically
verified by Wilsky and his coworkers using scattering
theory, [32]) we propose an estimation technique with non-
optimal local estimators. Doing this we have the advantage
of both construction of local estimators and generation of
global estimates. Using [29], [32] in order to get local
estimators N+l Riccati equations (one global and N local,
where N is the number of estimation stations) have to be
solved. In our case only solution of one Riccati equation
is necessary. Note that all local estimators and Riccati
equations are of the same order in both cases. We
construct our global estimate by a simple addition of local
estimates. In their case the global estimate is a linear
combination of local estimates and a correction which
propagates by its own dynamics. Thus, a considerable
amount of additional computations has to be done

as a price for having local optimal estimators

112

while local Optimality is not used by any means in the
given problem and probably is unnecessary. We will briefly
summarize the main results of [32]. For the reason of
simplicity we will consider only two estimation stations.
Consider a linear system and decentralized measurements

corrupted by white Gaussian noise

i(t) = Ax(t) + w, E{x(to)} = x

o.
E{(x(to) - io)(x(to) - xo)T} = Q0 (4.1)

Y1 = Clx + v1 (4.2a)

Y2 = sz + ,2 . (4.2b)

where A, C1 and C2 are constant matrices. Noise v1 and v2

are independent. In addition the following statistics are

known

Efw} = 0, Efvi} = O i=l,2 (4.3a)
E{w(t)wT(T)} = W6(t-’t) i=l,2 (4.3b)
Eivi(t)vfcr)} =‘Vid(t~r) i=l,2 (4.3c)

where W and Vi' i=1,2 are intensity matrices.

The main idea of the decentralized scheme of [29],
[32] is to achieve the same performances with decentralized
measurements as if one has centralized measurements. The
centralized solution is given by the well known Kalman

filter

113
X = (A ‘ chl ’ K2C2)X + Klyl + K2Y2 (4.4)

T —1 .
Ki = QCiVi , 1=1,2 (4.5)

where Q is the solution of differntial Riccati equation
. T - T -
' (4.6)

Note that in order to simplify formulas we assume that w,
v1, and v2 are mutually independent.
In the case of decentralized processing of information

each local problem can be defined as

i. = Ax.

1 + w, i=1,2 (4.7a)

yi = Cixi + Vi (4.7b)

and its locally optimal estimator has the form

£1 = (A - Kgci)§i + Kgyi, i=l,2 , (4.8)

with corresponding Riccati equation

° T -1
01 = A01 + QiAT + W - Qicivi CiQir Qi(t0) = QC (4.9)
and estimator gain
L T -l .
Ki 3 Qicivi ' 1:1,2 (4.10)

114

In order to reconstruct global estimator 0L4) from the
local ones (4.8) the following relationship has been

derived in [32]

/\ /\ /\
x ' C

II
$2
H
X
[.4
+
N
N
+
J

(4.11)

where a1 and a2 are known matrices in terms of previously

defined quantities, whereas 5 is a correction vector which

has its own dynamics
5 == a6 +-t;f§3 +~1q§25 «4.12)
with a, b1 and b2 known. The block diagram of this

estimation scheme is given on Figure 4.1.

\

L Optima] l
K local a2
1 filter 1 _]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

y] b‘l
_ A I + I)?
+
I SYSTEM 1 LCorrgction Hfg
5’2 b2
L Optimal I
K2 10C31 C11 (5——
1 filter 2 I -——A.

 

 

 

 

 

Fig. 4.1. Decentralized estimation with locally
optimal estimates.

115

These results have been derived analytically in [31]:
and by using scattering theory in [32]. Since scattering
diagrams will easily clarify our approach we will present
their form for the estimation problem. Global scattering
diagram corresponding to (4.l)-(4.2) is given in Figure
4.2. The variable of interest is a wave moving to the
right, i.e., Q“). The wave moving to the left, i.e., Mt)
has no interest for the estimation problem. It is
important for the so-called smoothing problem which is not
a subject of our consideration. 15 is an infinitesimal

discretization quantity.

 

 

 

 

 

 

 

 

 

x(t)
1
I c'v c1 + czvzcz]
+
.‘______ A}
x(t) _ , A(t+;)

 

 

T -1 -1
(€1V1 y] + C2V2 Y2)A

Fig. 4.2. Global scattering diagram for the
estimation problem.

Corresponding scattering diagrams for locally optimal

estimators are given in Figure 4.3.

/

116

x t l ""' + A »
1( ) I+AA‘ X1(t+;)
- +
1 _

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

X2(t) x2(t:L)
T -1
czv2 c2 I
+ i
A2(t) - 12(t+5)
T -1
C2 2 Y2A

Fig. 4.3. Scattering diagrams for locally
optimal estimators.

In order to be able to apply superposition scattering
media have to be identical. Note that only blocks
corresponding to C matrix are different. There are two
ways to make global and local scattering media identical.
The first one is to perform neutral transformation (does
not change dynamics) on the local scattering diagram. This
approach has been taken in [32] and is represented in

Figure 4.4.

 

 

 

 

 

 

 

 

 

 

 

   

T -1 T
1

C2 2 yZA + C V1 C1x2(t)&

 

 

 

 

 

 

 

|I+-ATA I

A1(t) X1(t+A)

 

' T -1 T
C1 1 YIA + C2

vg‘c2x1(t)a

Fig. 4.4. Modified local scattering diagrams.

Applying superposition, we see from Figure 4.2 and Figure
4.4 that the correction term exists because of the presence

T -l /\

of driving inputs of the form CjVj iji, i,j=l,2, i#j, into

the local filters. ‘

The other way of making global and local media
identical, which in fact characterizes our method, is to
start from global scattering diagram and decompose it such
that no correction term is necessary. Of course we don't
expect local estimators to be optimal anymore. This can be
achieved by simply dividing inputs into global scattering

diagram into two independent parts and let these inputs

118

drive local estimators which have exactly the same
scattering media as the global estimator. This procedure

is presented in Figure 4.5.

x](t) + ;1(t+A)
={ I +A al——-————-<f; ==
+

1

 

 

 

 

T

V- C -IA

 

 

 

 

T -1
[C1V1 C1+C
+

 

 

 

 

 

 

 

 

T -1 T -1 *
[cw1 c1+c2v2 c2

 

 

 

 

A2(t+A)

T 1

C2 V2 yza

 

Fig. 4.5. Scattering diagrams for non-optimal
local estimators.

From Figure 4.5 we can conclude that the local non optimal

estimators have the form

A A \
X1 3 (A - chl " K2C2)xl + Klyl (4.1331
£3 = (A - chl - chzyfié + K2y2 (4.13b)

Then the global optimal filter is obtained by a simple

addition

119
x = £1 + £2 (4.14)

Corresponding block diagram is given on the Figure 4.6. A

similar scheme has been derived analytically in [30].

A

K‘ Fflter A]
l (4.13a)
y]
x
SYSTEM 1»
Y2 ‘
7 Filter l
(4.13bZJT .

x2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 4.6. Decentralized estimation scheme with
locally nonoptimal estimators.
Comparing these two methods represented by Figures 4.1 and
4.6 we note simplicity of the second one :which from
implementation point of view reduces considerably on-line
and off-line computations. In the next section where we
study systems with slow and fast modes we will see

additional good features of this method.

IVLZ. PROBLEM STATEMENTS AND FORMULATION

OF MULTIMODEL ESTIMATION

Let us consider the following system-measurements

structure

120

X0 = A X + Aolxl + A02X2 + GOlwl + 602W2' E{X0(to)} = X

00 0 O
(4.15a)
51x1 = Aloxo + Allxl + Gllwl' E{x1(to)} = fl
(4.15b)
E2X2 = A20xo + A22X2 + Gzzwzt " E“Mme” = x2
(4.15c)
c o 0
Y1 = 01 x + v1 (4.16a)
C10 C11 0
c o o
y2 = 02 X 4' V2 (4.16b)
C20 0 C22

‘which has been studied in Chapter III. Equations (4.15),

(4.16) can be written as

i = Ax + Gw, E(x(to)) = i, E((x(to) - x)(x(to) - x)T) = 00
(4.17)

Y1 ‘3 C1" + V1 (4.18a)

Y2 = 02x + V; (4.18b)

with obvious definition of matrices A,G, Cl and C2.

We are going to explore the impact of the slow-fast
nature of (4.15) on computing the decentralized
implementation of the optimal solution in an efficient way
by using ideas from previous section. The steady-state

optimal Kalman filter is given by

121

/\ /\ ..
x = (A - K1C1 - K2C2)x + Klyl + K2y2 , 9(0) = x (4.19)

where
T -l T -1 -
and
o = A0 + OAT + owcT - och'lco (4.21)

r)

n
<

u
z
u

A

The optimal estimate x can be represented as the sum of two

estimates 9(1) and /x\(2)

/x\=/x\(l)_+/x\(2) (4.22)

where‘§(l) is calculated using Y1 and‘Q‘Z) is calculated

using y2. We have seen in the previous section from the

infinitesimal scattering diagram for the Kalman filter that

the local filters are given by

43(1)

(A - K1C1 - K2C2)€r\(l) + Klyl (4.23a)

/x\(2)

The initial conditions of (4.22) and (4.23) will be
chosen later on in order to minimize computation

requirements. In general they have to satisfy

122

Qél’(0) + §é2)(0) - so (4.24a)
x{1)(0) + §;2)(0) = fl (4.24b)
xé1)(0) + xé2)(0) = 22 (4.24c)

In this section we will study steady state estimation,
whereas corresponding finite time problem is a subject of
consideration in the next section.

Having established the existence of a unique solution
of the algebraic Riccati equation (4.21), and using the
fact that 012 = 0(3), (see ChapterIII) our optimal filter
(4.22), i.e. its components 52‘“ and 12“” (4.23) can be

represented as

_ :(1f‘ _ T
xo

T -1 l
Aoo'(QooCol+QolCl)Vl Col

T T -
Aol'(QooCol+QolCl)Vl C1

T T -1
"(Qoocoz+Qozc2)V2 C02

 

;(1) T -1

T -l
81x1 3 (AlO-Qllclvl Col)+0(€)

.“-.----4r-—-------—---

0... ---r—

2(1) T -1
€2X2 (A20‘022C2V2 C02)+0 (8)

. . L

 

 

 

123

 

 

 

 

 

 

 

 

 

 

 

 

 

 

- . q r 1
I
' T T -l Adl) T . T -l
L .......................................................
I
' a '/\(l) T -1
E 0(5) 1 x1 + Qllclvl + 0(5) yl
L---_---_--—-—---—--——--J ------- F .......
I
: T - .AJl)
E (A22'922C2V2 C2)+°<€) X2 (5)
3 d h " ' (4.25a) ‘
° ‘ r - F ' ‘
f fuz) ,212)1 T T -1
x0 . xo (Qoocoz+QOZC2)V2
81912) = a! Ԥ{2) + 0(6) yZ
'(2) (2) T -1
85?} ’?2 022C2V2 +0(€)
'- - u d b .J L .-
(4.25b)

d4 indicates the same system matrix as in (4.25a)

Observe the nice form of the differential equations for‘QiZ)
/~(l)
and x2
‘O«2) T '1 , /«2)
81x1 2 [(A10 - Qllclvl Col) 'I’ 0(a)]xo
(4.26a)
T _
+ [(All ’ 911C1V11C1’ + 0(e)T§i2) + 0(e)yl
. T -
€2§él) = [(Azb ‘ szcnglcoz) + 0(e)1§g1)
(4.26b)

T -l /«1)

124

which, up to accuracy 0(El/2), and by choice of initial
condition in order to eliminate fast transient can be

replaced with algebraic equations

T -1 A(2) T -1 .(2)
° ‘ (A10 ’ Q11C1V1 Col’fio + (A11 ’ Q11C1V1 Cl)31
(4.27a)
T -1 .(1) T -1 .(1)
° = (A20 ’ Q22C2V2 Coz’xo + (A22 ‘ Q2202V2 Cz’xz
(4.27b)

. T -l T -1
Since (All " QllClvl Cl) and (A22 "' 022C2V2 C2) are

nonsingular matrices (4.27) has unique solutions given by

«(2) T -1 -l T -l .(2)
(4.28a)
(1) T -1 -1 T -1 .(1) '
22 (t) = “[3322 ’ Q22C2V2 C2] [A20 " 022C2V2 COZJXO (t)
(4.28b)

Obviously the initial conditions in (4.26) have to chosen -

as
£{2’(to) = 2:2)(to) (4.29a)
§él’(to) = fiél)(to) (4.29b)

Now we are able to determine initial conditions. From
(4.24a) we can arbitrarily split f6 between two filters,

let's take

0
x0 (t ) = -5 x0 (to) = -5 (4.30)

then from (4.24) and (4.28) we have

,41) _ — T -1 -1 T -1 X0
(4.3la)
r42) — T -1 -1 T "1 x0

(4.31b)

After substituting (4.28) into (4.25) our approximations

for decentralized filters become

£41) T T -1 T T -1
1‘o 3 {A00 ' (Qoocol + Qolclwl Col ' (000C02 + Q02C2)V2 C02

T -1 T -1 -1
[A02 ’ ‘Qoocoz + Qozég’vz C210322 ' Q22C2V2 C2)

T -1 /4]J T T
(A20 ' Q22C2V2 C02)}30 + {A01 - (Qoocol +'Qolcl)

, -1 t(1) T T -1 .
'VlCl} £1 + {(QOOCO]. + 001C1)V1 1’11 (4.323)
)41) T -1 ,r4l) T -1 .241)
61951 = (A10 ' Q11C1V1 Co1)3o + (A11 ’ Q11C1V1 Cl’xl
T 41

+ Qllclvl Y1? (4.32b)

126

and
/¢2) T -l T T -l
-o 3 {A00 ' (Qoocol + 001C§)V1 Col “ (Qoocoz + 002C2)V2 C02
T -l T -l -1
T --lC .r4l) T

(A10 ' 011C1V1C01)}xo + {A02 ’ (QOOCOZ

2) CT T -1
+ 002C2)V21C2}g\2( ‘I' {(000C 02 + 002C2)V2 )Yz
(4.33a)
}42) T -1C 4,42) T —1 .942)
82352 3 (A20 ' Q22C2V2C Mz’ + (A22 ‘ Q22C2V2 Cz’x 2
1 (4.33s)

+ Q22C2V2 Y2

Thus we have the parallel processing represented by the

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

_following block diagram “(1)
- x
l -4
TH (2.13;) F, 4” +4:
. —1 3%))
[ SYSTEM I 4(2)
, 51
F. ,
yz Filter .1 tit+ ‘—
xrz '
3g; (4. 33) £6 ) ‘+

 

 

 

“(2)

x0

‘1.
Fig. 4.7. Decentralized multimodel estimation.

127

With obvious expressions for £1 and £2 that come from
(4.28). Note that filter 1 and filter 2 are of reduced
dimensions. This result so far is interesting in its own ‘
right. It could be called multilevel estimation. First
reduced order local filters have been used, then Simple
static equations have been solved and finally estimates are
combinedsnufi1that they produce a good approximation for a
global estimate. This approach leads to the use of
parallel processing (decentralized estimation in this case)
from a computational point of view and it can be used

intentionally in order to speed up estimation process for

models of the given slow-fast nature.

IV.3. FINITE TIME MULTIMODEL ESTIMATION

In this section we will study a finite time estimation
problem following its formulation from Section IV.2 where a
steady state estimation problem has been considered. The
main difference between the two problems comes from the
fact that the filter gain is a function of the solution of
a differential Riccati equation rather than an algebraic
Riccati equation.(4.21%. Thus optimal filter coefficients
are time varying functions. In fact, the optimal Kalman
filter is given by
Sim = [A - xlmc1 - K2(t)C2]x(t) + Kl(t)y1(t)

+ K2(t)y2(t), £(0) = x (4.34)

128

where

T -l T -l
Kl(t) = Q(t)C1Vl K2(t) = Q(t)C2V2 (4.35)
and

Q(t) = Q(t)AT + AQ(t) + GWGT - Q(t)ch'1CQ(t) ,
0(t0) = (20 (4.36)

As in the previous section from the scattering diagrams we

have ’x\(t) = ’x‘m (t) + ’x‘m (t) where

x :.[A - Kl(t)Cl - K2(t) CZYQ“1) + Kl(t)yl (4.37a)
’x‘m - [A - Ki‘t’cl - K2(t)c2)€?‘2’ + K2(t)y2 (4.37b)

Since we are going to study a finite time estimation
problem, let us first examine the time behavior of the
solution of the Riccati equation (4.36) for t£[to,Tl. By

partitioning Q as

O00 O01 Q02
T 1 1
Q = 001 g; 011 'V'_'EIۤ 012
T 1 T 1
902 fi? Q12 g; Q22
L ..J

 

 

129

and substituting in UL36) we get the following systems of
differential equations
Q00 3

T T T
AooQoo + Qovoo + AolQol + Qolel + A02002
T

T T
+ Qoonz + Golleol + GoZWZGoz

T T T
(QoosooQoo + Qolsolooo + Q02502000 + QoosolQol

T T T
+ Q01511001 + Qoosonoz + 002522002)
° T
81001 = Q01A

(4.38a)
11 +

T
QooAlo + AolQll + aleoQol
‘ T T
+ V 0‘ A02012+ Gollell

(000501911 + Q01511011
\f— T T
+ 0‘ Q02502912 + V; 002522912)

T I T
E1(QoosooQol + Qolsolool + 002502001)

(4.38b)
- T
€1°02 = ‘QozAzz + QooAZO + A02°22 +

--- A Q
\F;' 01 12

81 T

G.

+ -- AOOQO2+ G02W2G22)G- u(QooSQZQZZ + 002822022
1
+ ---

{—- Q00501012 + "‘
a

\fi;- 001511012) " S1(000500002

T T
+ Q01501002 + 002502902)

(4.38c)

130

m

° T - T T n T
E1°11 = A11911 + Q11A11 + ‘1‘A1oQol + QolAlo) + G11"1611
s , s T e T s
‘ (911 11°11 + 4012 22°12) ’ 1‘001 01°11
T T T
+ QllsolQol + ElQolsooQol I {5 001802001
-— T T
+ ‘Va 001802012) (4.38d)

' _ T
81012 ' (A11 ' Q11511’012 + a“312“‘22 ' 922522)

F" T T 1
+ €l( a QolA20 +.;fif AloQoz)

T T 1 T
' €1‘012502002 + Q01501012 + ye? Q11501002
«—— T 81 T
+ a Q01502022 + Qfif QolsooQoz) (4'389)

- T T
E1°22 = (A22022 + Q22A22 ’ Q22522022 + Gzzwzczz)“

T T
+ E1‘Aonoz + QozAzo’ ‘ Q12511012

- T T 1 T
*' °1‘902502°22 + Q22502002 * ya; Q02301012

1 T 81 T
+‘ Qfif'ol2solooz + 7;- 002500002) (4'38f)

where

131

T -l T -l
800 = Co1V1 Co1 + Cozvz C02
T -1 T -1
501 = C01"1 C1 502 = Cozvz C2
T -1 T -1
811 = C1V1 C1 S22 = C2V2 C2
Initial conditions for (4.25) are given by
Qoo(to) = var(xo(to))
001(to) = cov(xo(to), xl(to))
002(t0) = cov(xo(to), x2(to))
Qll(t0) = 61 Var(X1(to))
012(t0) = V €182 COV(Xl(to)' XZ(to))
= -:¥ co (x (t ) x (t ))
{E— V 1 O ' 2 0
El '
( 3 ( = —--— ( )
022 to) £2 var(x2 to)) a var(x2 to )

(4.39a)
(4.39b)
(4.39c)
(4.39d)

(4.39e)

(4.39f)

where by assumption a is bounded from below and above

0 < m g a

E
=El$M<oo
2

132

In Appendix IV.l we prove the existence and uniqueness of a
bounded solution of (4.38).

Having established the existence of the unique
continuous solution on the finite time interval, which is

bounded too, we can simplify (4.38e), i.e.
€1O12(t) = (All - 011(t)sll)012(t) + 012mm22
- 022(t)522)T + 0(5)
012(to) = 0(5) (4.40)

Let @11(t,T) and @22(t;r) be the transition matrices of the

time varying systems

81;]. 3 [All " Qll(t)Slllzl(t) (4.418)
and

respectively. From the properties of the solution of the

Riccati equation (Appendix IV.1) we have

©11(t,T)‘ 3 const exp(-al(t - T)/€1) (4.42a)

 

T22(t,T)l 3 const exp(-a2(t - T)/€2) (4.42b)

Now solution of (4.40) is given by

133

.. _T
012(t) = Qll(t’tO)le(LO)Q22(t't0)

+ 1:0

t 1 T

Using the exponentially decaying bounds on $11 and $22 we

have that

012(t) = 0(a) VtéitoyT] (4.44)

Then (4.37) is given by

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P . 1 - - )- 1 - -
/\(l) /\(1)
X0 “(00(t) d401(t) d102(t) X0 3301(t)
.841) /\(1)
‘341) ‘ A ,41)
€2X2 ‘ud20(t) 0(6) a422(t) x2 0(6)
. . 1 . L . 1 .
(4.45a)
- 1 _ 1 - 1 _ 1
A(2) N2)
X0 400(t) 9401(t) 4402(t) X0 «902(t)
eixi2) a 4410(t) d411(t) 0(a) ’QiZ) + 0(a) y2
L. .1 “L — L d L d
(4.45b)

Since coefficients on the right hand side of (4.45) contain

time varying boundary layer terms, matching the initial

134

conditions for the fast filters (as we did in the previous
section) will not help us in reducing the order of filters.
In fact that approximation will not produce near
optimality. That is why we propose here the following
scheme: Transform fiIter (4u45) such that slow components
are eliminated from the fast filters. The required

transformations are given by

’n‘l a /x\{2) - 1.1932) (4.46a)
/\ A(1) A(1)

02 = x2 - L230 (4.46b)
where

81L]. 3 9411111" 0410 (40473)
The existence of L1 and L2 is guaranteed by (4.42).
Then (4.45) gets the form

 

 

 

 

 

 

 

 

F 241,1 IF (1) ' - -241; - 1
"o J «400 (t) .1401“) 14020:) xo £01m
Au) x41)

81x1 = “410‘“ 411%) 0(5) x1 + 3110;) yl
22/11} 0(5) 0(5) 422m 6‘2 0(a)
L d L d _ J L a
(4.48a)

135

q

y2

 

 

 

 

 

 

 

' f'
/\(2) (2) 242) F-
x0 4400 (t) .1401 Jozm 1 x0 302(0
21. A .- 45
-Nz) N2)
L- d L d - J -
(4.48b)
where
~4(l)(t) -.4 (t) +54 (t)L
oo ' oo 02 2
«(ZN ) - ( ) .4 ( )
00 t ‘4400 t + 01 t Ll

Observe the nice form of differential equations for £1 and

/\ .
n2 estimates

/\

elnl wvnuflfi + 0(6) , ’nyro) arbitrary (4.49a)
A

5202 =a¢22(t)4E + 0(6), ‘G2(to) arbitrary (4.49b)

Using the same arguments as for equation OLAO) and being
able to choose their initial conditions arbitrarily we

conclude

”(11%) = 0 a ’n‘fit) = 0(a) , Vteltom] (4.5021)
’n‘zuo) = o .. @(t) = 0(8) . Vtettom (4.50m

 

136

with this fact in mind and neglecting all 0(3) coefficients

on the rJLs.

filters

 

 

J

 

 

 

of'(4.48)

(l)

 

we have the following approximate

Jana)

«451(t)

d411(t)

 

PAUfl
£0

 

 

d

 

(2)
3‘

/\(2)

 

 

 

5802(t)

33

 

22(r)

1
3301(r)

Y1

 

(4.51a)

Y2

 

d

(4.51b)

Thus our decentralized estimation scheme can be represented

by block diagram,

137

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

11‘” 1
Filter ;:
81 (4.51a) I ,1
y] x51)
_ + i
LSYSTEM ' + ‘0;
~(2)
5’2 F'lt .0 ‘
1 1 er , x
82‘ (4.51b) L *1
4(2) "
.2

Fig. 4.8. Decentralized multimodel estimation scheme for
time varying problems.

Note that we have an even simpler block diagram than in

the time invariant case, but for the price of solving

additional systems of differential equations UL47),

instead of algebraic ones (4.27).

138

APPENDIX IV.l

We want to establish the existence of a unique bounded
solution of (4u38). This is a singularly perturbed system
of differential equations parameterized in a parameter a
constrained to a compact domain aelm,M]. Singularly
perturbed differential Riccati equations have been studied
in [33]. The additional features we have on our problem
are the presence of a.and the partition of the fast vector
into two weakly connected subvectors. We study the impact
of these features. Following [331 we set El = 0 in (4.25)

to get

' - T T T T T
Qoo = AooQoo + Qovoo + A01901 + Qolel + A02902 + Q02A02

T T T
+ GolWlGol + GoZWZGoz ' (QoosooQoo + QolsolQoo
T T T T
+ Q02802900 + QooSolQol + Q01511901 + Q00802902

T

T T T T
0 3 901A11 + QooAlo * A01911 + '0‘ A02912 + G01"“1‘111

’ “100501911 4‘ Q01311911 + ”7902502912

+ ”90252291? (IV.lb)

 

139

- T T 1 , T
0 = “‘QozAzz + QooAZO + A02922 + 355 A01912 + GoZW2922)

1
' “(902502922 + Q02522922 + Qfif-Qoosolg12

l

T T _
° = A11911 + Q11A11 + G11171511 ‘ Q11511911 “3912522912

(IV.ld)
T
0 3 (All ' Qllsll)912 + Ct912(A22 ' 922522) (IV.le)
- A AT 5 G w GT
0 - “( 22922 + 922 22 ‘ $222 22922 + 22 2 22’
- 012811912 (IV.lf)

Stability of All - 911511 and A22 - 922522 and (IV.le)
imply that

912 z 0 (IV.2e)

So that Qij(t), i,j=0,l,2, terms are in fact independent of

parameter a, i.e.

' T T T
0 = Q01A11 + QooAlo + A01911 + G01971611 ' Q00501911

- 901511911 (IV.2b)

 

140

T T T
0 a QOZAZZ + QOOAZO + A02922 + Gozwzc22 — 902502922

' 902522922 (IV.2C)

- AT 5 + w GT ( 2d

0 ’ A11911 + Q11 11 ’ Q11 11911 G11 1 11 IV- ’
_ + T ' T

0 - A22922 Q22A22 ' Q22522922 + G22W2G22 (IV-2f)

Using (IV.2b)-(IV.2f) in (IV.la) we can easily get

' ~ ~T T -l T ~
Qoo= AooQoo + Qovoo + Gs‘W ' WHsvs HsmGs ' QooSooQoo
900(to) = var(xo(to) (IV.2a)
where
~ 1 T -1
A00 = A5 ‘ Gsmasvs Cs
T
VS - V + HSWHS
~ T -1
S00 = Csvs Cs

The solution of the Riccati equation can be represented as

the sum of two terms; one in time scale t and the other in

time scale T [35]. Neglecting 0(a) quantities, Qij can be
represented as

- t-to
Qij(t) = Qij(t) + Pij(1) + 0(8), T = ---- (IV.3)

141

The following expressions for Pij(1) - boundary layer terms

are obtained

dP
--22 = o (IV.4a)
dT
dpll - A P P AT P s P P 5 PT ( v 4 )
“5;"- 1111+1111‘111111‘0‘122212 1°C
dP
22 - ” P + KT 3 P ) s P ( 4f)
”5;“ " 0‘ (A22 22 P22 22 ’ p22 22 22 ‘ P12 11 12 1"-
dP
01 ~T ~T ~ ,——-~ T
'5}- = PolAll + PooAlo + AolPll + “ A02P12 ‘ Poosolpll
PSP -~' (PSPT+PSPT) (IV4b)
01 ll 11 a 02 02 12 02 22 12 °
dP
02 ~T ~T ~T ~
73? = O“PozAzz + Aozpzz 1‘ PooAZO) + V 0‘ AolPlZ
-a(P-sp +PSP +--]3-PSP
oo 02 22 02 22 22 V3 oo 01 12
1 P s P ) ( 4d)
+ a: 01 11 12 IV-
dplz ~ ~T

 

142

Q22322

II

I]!
N
N
I

A22
A11 ‘ A11 ‘ Q11511

~

Alo

ll
y
1.4
O
I

Q11501
Aol = Aol ' Qoosol - Qolsll

Qoosoz ' Q02522

ll

3»
O
n)

A02

II

5
h)

I

A20 0 Q22522

Observe that the initial condition for (IV4.e) is given by

P12(0) = 012(t0) - ‘ 912(0)
El-
81
= --- cov(xl(to), x2(to) - 0 = 0
(3 £130
loeo

by stability of 311 it follows from (IV.4e) and (1v.5)
that

P12(T) = o VT 2 0 (IV.6e)

then equation (IV.4) can be simplified to

---- = O (IV.6a)

143

GP
01 ~T ~T ~T
’5;' = PolAll + P00 10 + Aolpll ‘ PoosolPll ’ P01511911
(IV.6b)
GP
02 ~T ~T ~T
'5;’ = O“(1’02‘1‘22 + Ao2P22 + PooAzo’ ' (Poosozp22
+ P02522P22)] (IVo6C)
dP
11 ~ ~T
’52- g Allpll + PllAll ' Pllsllpll (IV.66)
dP22 ~T ~
.5;- = GIP22A22 + A22P22 ’ P22822P22] (IV.6f)
with initial conditions
Poo(0) = 0 (IV.7a)
Pol(0) = 001(0)L - " 901 (IV.7b)
Icl‘o
P02(0) 3 002(0q8 -0 ‘ 902 (IV.7C)
1-
Pll(0) = 011(0)k -0 ’ 911 (IV.7a)
1-
P22(0) = 022(0)F =0 ' 922 (IV.7f)
'1

Note that all initial conditions are independent ofa .

Using (IV.7a) in (IV.6a) we have

POO(T) = 0 VT 2 0 (IV.8)

 

144

so that
dP
‘a;' = PolAll + Aolpll “ Polsllpll
dP

02 ~T ~T
'a;’ = O‘u’ozl“*22 + A021’22 ‘ P02522P22]
dPll

~ ~T
'5;‘ ‘ A11P11 + P11A11 ’ P11511P11

~T ~
‘5;’ = O“221‘22 + A22P22 ‘ P22522P22]

(IV9.b)

(IV9.c)

(IV9.d)

(IV9.f)

Observe that (IVth) and (IVKQd) are independent of the

other two equations, and that the initial conditions for

all boundary layer equations are independent of the

parameter d. Then results of [33], where boundary-layer

controllability and boundary layer observability conditions

were imposed can be used to guarantee the existence of a

unique bounded solution of (IV.9b) and (IV.9d) for all T6

[0,+w). In fact following arguments of [33] and using the

fact that boundary layer equations are of Riccati type [34]

and thus possess exponentially stable solutions we are able

to apply general results of [35] and [36] to our equations.

Consider now equations (IV.9c) and (IV.9f).

scale in these equations by

ar=o, adr=do

Rescaling time

145

we have
dP22 ~T ~
'53‘ = [P22A22 + A22P22 ‘ P22522P22] (IV°1°°’
dP
02 ~T ~T
-53- = [POZAZZ + A02P22 ‘ P02522P22] (IV-10f)

and again results from [331-[36] can be used to guarantee
the existence of unique solutions of PZZQI) and P02(°) for
all oe[0,+w).

In summary we have the following Lemma.
LEMMA IVzl: Let fast problems be locally boundary
controllable and locally boundary observable, that is, for

each fixed te[t0,T] pairs
(Cii'Aii) , 1:112, observable
(Aii' VWi) , i=l,2, controllable

then there exist a unique bounded solution of boundary
layer Riccati equations.

Given Lemma.IV;l together with conditions that guarantee
the existence of solutions for the slow part (in addition
all matrices Aij' Ck' Gk Vk' Wki,j=0,l,2; k = 1,2 have to
be continuous functions of t and e, and QO 2 0, Wk > 0, Vk
> 0, which are our starting assumptions) make the main

' result of this Appendix, i.e. establish that

146

Qij(t) = Qij(t) + Pij(r) + 0(a) i,j=0,l,2

Let us conclude by observing that the time varying

matrix (Aii ’ Qiisii)/€i has a transition matrix Q’ii(t,T)

which is exponentially decaying in t/ei, i.e.
|¢ii(t,r)l 5 const exp(-ai(t-T)/ei)

This follows from

(i) R6>\(Aii " Qiisii) < 0 VI:
(ii) IPii(T)I 5 const exp(-bir)
where ai > 0 and bi > 0, i=l,2.

V. CONCLUSIONS

The fixed point method applied for numerical solution
of singularly perturbed linear control problems produces a
very efficient scheme where a small perturbation parameter
plays the role of covergence radius.. As a consequence of
this, the accuracy of 0(Ek) can be achieved by doing only
k-l iterations on preposed algorithm. On the other hand
analicity conditions of system coefficients in 6 are not
required any more. It is enough to have continuous
coefficients on a compact set in order to be able to apply
fixed point methods. The proposed method can be extended
for numerical solution of other linear-quadratic control
problems for singularly perturbed systems: for example
solution of Lyapunov equations, Nash game problems and so
on. Hopefully it can be generalized for nonlinear sytems.
It seems that general theory for singular perturbations
can be built from the point of view of fixed point theory
instead of Taylor series expansions. These problems can
be part of future research.

In the third chapter we have shown that, under
appropriate conditions, the basic result of [14] concerning
well-posedness of multimodel strategies can be extended to
stochastic control problems with partial observations. The
analysis of this chapter has revealed a new feature of

multimodeling. When each control agent uses a simplified

147

148

model of the system that retains only the details of his
subsystem and assumes a certain "equivalent” reduced-order
model of the rest of the system, he cannot locally compute
the decisions of the other agents. When soch decisions are
needed, eqy, for deriving local Kalman filters, they have
to be provided by the other agents. Otherwise, the
multimodel strategy could be ill-posed. .Another feature
that distinguishes the stochastic problem studied here from
the deterministic problem studied in [14] is the more
important role played by the fast dynamics which manifests
itself in two ways. First, the fast component of each
local control strategy cannot be neglected even when the
open-loop fast dynamics are asymptotically stable. In
[14], neglecting fast controls will not destroy near-
optimality as long as the open-loop fast dynamics are
asymptotically stable. Second, the weak connections among
the fast dynamics are necessary and cannot be relaxed by
allowing strong connections that preserve asymptotic
stability as it might be possible in deterministic problems
[16].

The decentralized estimation scheme proposed in the
fourth chapter of this thesis is an interesting result in
its own right. Doing local estimation with non-optimal
local filters, rather than with optimal ones brings
computational advantages and speeds up the estimation

process. Based on this idea a multimodel decentralized

149

In this case the estimation stations exchange their
estimates in order to form a global optimal estimate.
Should we proceed along this line (exchange of estimates)
or should we allow the exchange of the measurements (as we
studied in chapter III) is an open question and it has to

be the subject of further consideration.

LIST OF REFERENCES

LIST OF REFERENCES

[l] A. Haddad, "Linear Filtering of Singularly Perturbed
Systems,‘ IEEE Trans. Aut. Control, AC-21, pp. 515-
519, 1976.

[2] A. Haddad and P. Kokotovic, I'Stochastic Control of \
Linear Singularly Perturbed Systems,‘ IBEE Trans. Aut.
Control, AC-22, pp. 815-821, 1977.

[3] D. Teneketzis and N. Sandell, “Linear Regulator Design
for Stochastic Systems by Multiple Time-Scale Method,‘
IEEE Trans. Aut. Control, AC-22, pp. 615-621, 1977.

[4] H. Khalil and Z. Gajic, 'Near Optimum Regulators for
Stochastic Linear Singularly Perturbed Systems,‘' IEEE
Trans. Aut. Control, AC-29, pp. 531—541, 1984.

[5] M. Jamshidi, 'An Overview on the Solutions of the
Algebraic Matrix Riccati Equation and Related
Problems,‘ J. Large Scale Systems, pp. 167-192, 1980.

[6] JQB. Chow and ruv. Kokotovic, EA Decomposition of
Near-Optimum Regulators for Systems with Slow and Fast
Modes,I IEEE Trans. Aut. Control, AC-21, October,
1976.

[7] P.V. Kokotovic and R.A. Yackel, I'Singular Perturbation
of Linear Regulators: Basic Theorems, IEEE Trans.
Aut. Control, AC-l7, 29—37, 1972.

[8] R. Belman, 'Introduction to Matrix Analysis,‘ McGraw—
Hill, 1960.

[9] P.V. Kokotovic, J.J. Allemong, J.R. Winkelman, and
J.H. Chow, "Singular Perturbation and Iterative
Separation of Time Scales,” Automatica, 16, pp. 23-33,
1980.

150

[101

[ll]

[12]

[13]

[14]

[15]

[161

[171

[18]

[19]

[20]

[21]

151

K. Chang, “Singular Perturbations of a General
Boundary Value Problem,“ SIAM J. Math. Anal., 3, pp.
520-526, 1972.

J} CPReilly, “Dynamical Feedback Control of Singularly
Perturbed Linear Systems Using a Full-Order Observer,“
Int. J. Control, Vol. 31, pp. 1-10, 1980.

H. Kwakernaak and R. Sivan, “Linear Optimal Control
Systems,“ Wiley-Interscience, 1972.

J;H. Chow, et a1., “Time Scale Modeling of Dynamic
Networks,“ Springer-Verlag, Lecture Notes in Control
and Information Sciences, p. 47, 1982.

H.K. Khalil and P.V. Kokotovic, “Control Strategies
for Decision Makers using Different Models of the Same
System,“ 1883 Trans. Aut. Control, AC-23, pp. 289-298,
April, 1978.

ELK. Khalil, “Multi-Model Design of a Nash Strategy,“
J. Optimization Theory App1., pp. 553-564, 1980.

V.R. Saksena and J.B. Cruz, Jr., “Nash Strategies in
Decentralized Control of Multiparameter Singularly
Perturbed Large Scale Systems,“ J. Large Scale
Systems, pp. 219-234, November, 1981.

V.R. Saksena and J.B. Cruz, Jr., “A Multimodel
Approach to Stochastic Nash Games,“ Automatica, Vol.
17, pp. 295-305, May, 1981.

\LR. Saksena and T. Basar, “A Multimodel Approach to
Stochastic Team Problems,“ Automatica, Vol. 18, pp.
713-720, December, 1982.

ELJ. Davison and U. Ozguner, “Synthesis of the
Decentralized Robust Servomechanism Problem using
Local Models,“ IEEE Trans. Aut. Control, AC-27, pp.
583-600, June, 1982.

V.R. Sakesena, J.B. Cruz, Jr., W.R. Perkins, and T.
Basar, “Information Induced Multimodel Solution in
Multiple Decision Maker Problems,“ IEEE Trans. Aut.
Control, AC-28, pp. 716-728, June, 1983.

Th Basar, “Decision Problems with Multiple
Probabilistic Models,“ Proc. American Control
Conference, pp. 1091-1096, 1983.

 

[22]

[23]

[24]

[3251

[26]

[27]

[28]

[29]

[30]

[31]

[32]

152

6.14. Peponides,- P.V. Kokotovic, anng.B. Chow,
“Singular Perturbations and Time Scales in Nonlinear
Models of Power Systems,“ IEEE Trans. on Circuits and
Systems, Vol. GAS-29, pp. 758-767, November, 1982.

G.M. Peponides and IhV. Kokotovic, “Weak Connections,
Time Scales, and Aggregation of Nonlinear Systems,“
IEEE Trans. Aut. Control, AC-28, pp. 729-735, June,
1983.

J1M. Ortega and WkC. Rheinboldt, “Iterative Solution
of Nonlinear Equations in Several Variables,“ Academic
Press, New York, 1970.

z. Gajic and H. Khalil, “Multimodel Strategies under
Random Disturbances and Imperfect Partial
Observations,“ invited paper at the 9th World Congress
of IFAC, Budapest, July, 1984.

I. Rhodes and D. Luenberger, “Stochastic Differential
Games with Constrained State Estimator,“ IEEE Trans.
Aut. Control, AC-14, 476-481, 1969.

C. Chong and M. Athans, “On the Stochastic Control of
Linear Systems with Different Information Sets,“ IEEE
Trans. Aut. Control, AC-16, 423-430, 1971.

D. Looze, P. Houpt, N. Sandell, and M. Athans, “On
Decentralized Estimation and Control with Application
to Freeway Ramp Metering,“ IEEE Trans. Aut. Control,
AC-23, 268-275, 1975.

J. Speyer, “Computation and Transmission Requirements
for a Decentralized Linear-Quadratic-Gaussian Control
Problem,“ IEEE Trans. Aut. Control, AC-24, 266-269,
1979.

1L Chang, comments on “Computation and Transmission
Requirements for a Decentralized Linear-Quadratic-

Guassian Control,“ IEEE Trans. Aut. Control, AC-25,
609-700, 1980.

A. Willsky, M. Bello, D. Castanon, B. Levy, and G.
Verghese, “Combining and Updating of Local Estimates
and Regional Map along Sets of One-Dimensonal Tracks,“
IEEE Trans. Aut. Control, AC-27, 799-813, 1982.

B. Levy, D. Castanon, G. Verghese, and A. Willsky, “A
Scattering Framework for Decentralized Estimation
Problems,“ Automatica, Vol. 19, pp. 373-384, 1983.

[33]

[34]

[35]

[36]

[37]

153

R.A. Yackel and P.V. Kokotovic, “A Boundary Layer
Method for the Matrix Riccati Equations,“ IEEE Trans.
Aut. Control, AC-18, (1), 17-24, 1973.

D. Vaughan, “A Negative Exponential Solution for the
Matrix Riccati Equation,“ IEEE Trans. Aut. Control,
AC-72, 1969.

ILB. Vasileva, “Asymptotic Behavior of Solutions to
Certain Problems Involving Nonlinear Differential
Equations Containing a Small Parameter Multiplying the
Highest Derivatives,“ Russian Math. Surveys, 18 (3),
13-81, 1963.

w; Wasow, “Asymptotic Expansions for Ordinary
Differential Equations,“ Interscience, New York, 1965.

H. Khalil, A. Baddad, and G. Blankenship, “Parameter
Scaling and Well-Posedness of Stochastic Singularly
Perturbed Control Systems,“ Proc. Twelfth Asilomar
Conference, Pacific Cove, California, 1978.