Design of Observer for Linear Descriptor Systems

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DESIGN OF OBSERVER FOR LINEAR DESCRIPTOR SYSTEMS S. Kawaji /)p/mrlllll'lll

of Elnlriral Ellginferillg & Complller ScienCf, KWllalllolo C'lIiwrsit\" KWllamolo 860, japan

Abstract. This paper is concerned with the problem of designing an observer for linear descriptor systems Ex = Ax + Bu, y = Cx. At first, an observer is presupposed to have the same structure as Luenberger Observer, and the fundamental equations that the plant and the observer must satify are derived, Based on these equations, it is shown that an observer exists if the system is observable in the sense of Rosenbrock. And the design methods of an identity observer and a minimal-order observer are presented by utilizing the generalized matrix inverse, Secondly, the realizability condition is relaxed by eliminationg the purely static modes of the system, and it is shown that the observer can be realized if the system is observable in the sense of Verghese. Keywords,

descriptor system; observability; observers; purely static modes; system

order reduction

In system (1), there exist exponential modes, impulsive modes, and the purely static modes. Consequently, two kinds of observability are defined as follows (Cobb 1984) (i) System (1) is R-observabZe (observable in the sense of Rosenbrock) iff

INTRODUCTION Linear descriptor system is a mixture of dynamic and algebraic equations,

and can be served as a

useful method in analysis of many real and practical situations.

Due to its extensive applications,

the descriptor system has attracted much attention in recent years (See the survey paper by Lewis, 198

(2)

rank [:] = n

6). For descriptor systems, the basic design method is the feedback of the descriptor variables, and an "observer" is indispensable for the implementation of the control law via incomplete state observations. A considerable amount of research has been devoted to the design of observers (El-Tohami,1983; Shafai, 1987; Kobayasi, 1987; Kawaji, 1988). The traditional approach to such an observer is to separate the dynamic equations from the algebraic equations, and to estimate the unobservable parts from the measured outputs.

rank [A EC-A

1=

n,

VAEI[

(3)

(ii) Use an allowed transformation to bring the modal observability matrix of system (1) to the form

[

Hl - Al A2] Cl C with El of full column rank 2

Then system (1) is V-observabZe (observable in the sense of Verghese) iff

In this paper, contrary to this, an observer is presupposed to have the samt structure as Luenberger Observer (Luenberger, 1971). Firstly, the fundamental equations that the system and the observer must satisfy are derived. Based on these equations, sufficient conditions are given for the existence of observer, and methods of designing an identity

rank [El A2] = n o C2 rank [H -CA

J = n,

(4)

"'A El[

(3)

Next consider the related system driven by u(t) and yet) of system (1) and giving the output wet)

observer and a minimal order observer are presented

by utilizing the generalized matrix inverse. Secondly, the realizability condition is relaxed by eliminating the purely static parts of the system. Finally, an example is offered to illustrate the procedure.

Z = Az + By + Ju w = Cz

(5)

+ Dy

q where zER , and A, B, J,

C, 6

are matrices of proper

dimensions.

DEFINITIONS AND FUNDAHENTAL EQUATIONS

Definition

Consider the linear descriptor system described by Ex

Ax

Y

Cx n

+ Bu

If, for a specified matrix K,

lim [w ( t ) t-*

(1)

Kx ( t ) 1

0,

If x ( 0- ), z (0), u ( .)

( 6)

00

holds, then system (5) is said to be a Kx-observer for system (1). If K = In' system (5) is called a

m where xER is the descriptor variables, u ER is the input vector, YERP is the output vector, E, A, B, C are constant maticies of appropriate dimensions, and rank C = p. E is not necessarily nonsingular and rank E = r (;; n). The invertibility of AE - A is assumed for the solution of (1) to be unique.

state observer.

The following theorem containes the fundamental results of observer theory Theorem 1

241

System (5) is an observer for system (1)

from which we have n-p '; rank U,; n. It follows that minimal order of Luenberger-type observer is n-p. In the sequel, an identity observer and a minimal order observer constructed by utilizing eqn. (10) •

if

Re 1..0,)

0,

<

l.

i

= 1,···, q

(7a)

and if there exists a matrix U£R qxn such that AUE + BC = UA

Ob)

= UB

J

(7c)

CUE + DC (Proof)

K

[I] Identity observer

(7d) For system (1), choosing an any (EH, CH)£r and letting D = cll U C (13)

Let

~ ~ z - UEx then

is governed by

~

~

=

A~ + (AUE + BC - UA)x +

(J -

we get UB)u

CUE + DC = EHE + C#C = I

which by involking (7b), (7c) reduces to

t

=

A~

It follows from (7a) that

w

=

~

(t) ... 0 as t ... ~.

A + (B - AC#)C = lA

Then

C~ + (CUE + DC)x

A

A Q.E.D.

J=

z

In the preceding section, a sufficient condition that a Luenberger-type system (5) can qualify as an observer for the descriptor system (1) was derived. The observer design problem is to find a matrix U such that the conditions (7a)-(7d) are satisfied when E, A, B, C, and observer's order q are given. In this section, an identity observer and a minimal order observer will be eonstructed.

(16)

(17) Therefore, we obtained an identity ob-

C

= n,

I1 I1 [AIn ~ EH AJ [:11 A~I'J [H ~ A]

(19)

.

(20)

P

and det EH = n. rank

Therefore we have

[H -C A]

= n,

(21)

which .implies that the exponential modes of system (1) must be observable (Cobb, 1984).

which implies that the impulse modes and the purely static modes of system (1) must be observable (Cobb, 1984). I t is noted from (9) that r ~ n - p. When the condition (9) is satisfied, there exist two nxn matricies EH£R and CH£R nxp such that In

EHA]

AIn [

is necessary. Notl.cing (E , C ) £ r

(9)

rank [:] = n

Thus we come to the theorem giving sufficient condition for the existence of an identity observer. Theorem 2 An identity observer for system (1) can be realized if (9) and (21) holds, i.e. system (1) is R-observable.

(10)

CH] is a generalized matrix inverse of

[11] Minimal order observer

Here we define a class

r ~ {(EH, CH)

As is well known, the utilization of all the output information results a reduced-order observer. We will construct a minimal order observer since rank C = p.

det EH 1 0, EI'E + CI'c = I } n

(ll)

Then we have the following (Kawaji, 1988). Lemma

H

(E#A - KC)z + (E#AC# + K - KCCH)y + E#Bu (18)

rank

Firstly, it is required from the condition (7d) that

CT]T.

KCC

For this, in (15), (C, E#A) must be observable (the detectability is sufficient, strictly speaking. But an observer with freedom to choose all eigenvalues is frequently important.). That is,

DESIGN OF OBSERVERS

[ET

K -

The remaining problem is the determination of the parameter K in such a way that a stable observer results.

In the following, only the case K = In' that is, a state observer is investigated.

i.e., [EH

+

x = z + CHy

where U ~ UE. It is well known in linear system that (8) is the necessary and sufficient condition.

=

l1

E#B

from (7c). (8)

[~]

(15)

lAC

K =

server as

CU + DC = K

CI']

KC

+

Furthermore,

UE-lA

UE-lB

[EH

E#A ll

Ac

B

Remark: Note that Theorem 1 provides a sufficient condition. While, if E is nonsingular, (7b)-(7d) reduce to AU + BC

(14)

H

A

Letting K = B - AC , we get

and hence, from (7d), w(t) ... Kx(t) as t ... ~.

J=

n

Condition (7d) is thus satisfied, and rank U = n. Substituting (13) into (7b) and use of (10) yields

There exists a matrix UII ER (n-p)x n such that

r is not empty. T-l = [UI'J }n-p

(The proof is omitted). An algorithm to find an element of r is given in Appendix.

C

is nonsingular. Also the condition (7d) indicates

T

=

[H

(22)

}p

Denote

D]

(23)

then we have

(12)

HUll + DC = I

242

n

(24)

(l.

Vsing any

HV"E"E

ell) £ rand (24). we get

We have thus obtained a minimal order observer as

(I -De)(I -elle) n

I

n

- (e"+D(I -ee")}e p

n

or

(39) (25)

So. taking account of rank V = n-p. set

e

H

D

ell + D(I - cell)

/l

V

Theorem 3 A minimal order observer for system (1) can be realized if (9) and (21) holds. i.e. system (1) is R-observable.

(26)

p

then condition (7d) is satisfied. Substituting (26) into (7b) and rearranging them. we get Av ll

+ (B- Avllell)e

= VI'EIIA

Remark: For the case E = I • (I • 0) Er. It follows that the realizability coRditi8n (9) is obvious. and (21) reduces to the observability condition of (C. A). And an identity observer (18) leads to CA - KC) z + Ky + Bu

z

(40)

x :;::: z

(27)

which was proposed by Luenberger (1971). minimal order observer (39) is written as

Also. a

(28)

(29)

x = S

Also condition (7c) gives

J=

(30) RELAXATION OF RELAIZABILITY CONDITION The condition in Theorem 1 or Theorem 2 is severe. In this section. the realizability condition for observer is relaxed by using the elimination of the purely static modes of the system.

[~] ( ~ S-l)

Vnder the regularity assumption of system (1). two nonsingular matrices P and Q exists such that

° (22) is c h osen as V II sat1of y1ng

is nonsingular. = [I

L] S-l

n-p

(31)

From (28) we have A = [I

n-p

S [::] y

which was proposed by Gopinath (1971).

vIIEI'B

The stability of Allstill rema~ns: it is necessary to find a matrix V rendering A in (28) stable. For this. e is firstly constructed such that

V"

[I~_p ] z +

(41)

L]S-lEI'AS [In_ p]

PEQ

diag{I

PAQ

diag{A • I • I } l n2 n3

• J. O}

° rixri d1ag {J rl ••••• J rs }' Jri£R is nilpotent (ri" 2). nl = deg det(AE-A). n2 = rl+··+rs. nl + n2 + n3 = n (Gantmacher 1959). Then system (1)

where J

0

(32)

All + LA2l

nl

=

is r.s.e. to

where

n-p

p

It follows that if (A 21 • All) i& observable. we can determ1ne tne parameter L so that Ahas arbitrarily assignable poles. But we have

[~-l

I0j[AI

n _/'A]s

It is noted that z3 = -B u. i.e. z3 is directly connected with the input 3and has no dynamics. We call it the purely static modes.

p AI n-p [

All

-~21

-A 12 ]

Up -I A22

By eliminatin this mode. we obtain a reduced system as

P

since (Ell. ell) £ r.

Therefore. the observability of

2l (A

• :::) [U n _ p r

All] = n _ P.

~\

€

(43)

e (38)

A21 is equivalent to (19). hence to (21).

Next we obtain is the relation between the observability of system (42) and that of system (43).

243

We know

Proposition If system (1) is V-observable, then the reduced system obtained by eliminating the purely static modes is R-observable.

0

r''''r'

rank

AJ-I

-: 1

0

If A E ~

6

z

n3 C 3

C 2

Cl

Therefore

G.

n,

= nl + n2 , (44) C Cl 2 Next, there exist two nonsingular matrices Sand T for nilpotent matrix J such that SJT

"A

0

~] ,

rIn~-s

=

=

ST

r

u

=

[C

2l

C22l

n2-s

s

E

= Fw +cy+ = Hw + Dy

J u 2

(48)

c"c

E"E

+

[F

Cl =

[H

Dl =

C

(49)

[J l J 2 l = [Inltn2-k

Td

T21

G=

J

D=

e" +

l

+ C(I k

Denote

(the proof is simple and omitted) • CT 2

(47)

where

[""'-" ,,-q

rank

n + n 2 l

Then an obsever for system (46) is constructed by using our procedure as

0 n2

I~ I=

rank

(Proof) Assume that system (1) is V-observable. By applying condition (3) to system (42),

cc")

D(I - CCII) k

It follows that a minimal order observer for system (1) is obtained as w

(50)

Then condition (4) is equivalent to 0

0

0

n2-s 0

T12 0

0

0 0

0

0

0

0

C 22

I

nl I

0 rank

x =

=n

0 I

n3 C 3

EXAMPLE Consider a following descriptor system (n = 3, p = 2,

which shows that rank C 22 will now obtained by system (42) satified by be satified

\~e

rank

=

s

r = 2)

(45)

check the R-observability of the system eliminating the purely static modes of (i.e., system (43». Condition (3) is (44). Also condition (4) is shown to by the relation

[I~l ~ 1= Cl

rank

C2

0

0 Ip

o o Cl

I

_ n2 s 0 C 2l

[~ ~ ~]

First we find (El' , c") Eras

[: :l U ,', .n:

0

E" =

C 22

and use of (45).

x

We se~ that conditions (9), (21) are satisfied, and hence this system is R-observable. It follows from Theorem 2 or Theorem 3 that an observer with arbitrarily assignable poles can be realized.

Cl C2 0 0

0

+

Y =

[I~l ~ ~] [I~l ~ 1[I~l ~] 1nl

n: :1' [J

[: : :J'

Q.E.D.

c' -

(i) identity observer Remark: Armentano (1986) showed the same results by using a geometric approach. Our derivation is based on the structure of regular pencile and is simple.

:1

Letting {-a, -B, -y} be the set of poles of A, we have, by simple calculation,

Finnaly, an observer for the V-observable system is constructed as follows: Denote y * = y + C B1 u and 3 k = rank [Cl C2l. Let R be the matrix wh1cfi picks up k independent rows from y*, and let Y be new output. Then system (43) is written as Ez y

A'i. + Bu

K

-1

(46)

Cz

where k4

where z

B=

[:~] ,

E

[:~] ,

C

= =

[I~l R[C

l

~] ,

A=

[-~ ~:] k6

= aBy,

kS

= a + B+ y

, k6

= 1 + aB + By+

Eqn.(18) gives the identity observer:

0] 01'

[AI

[:

n2

C l 2

244

ya.

Trans. of SICE, ~, 141-148 Kawaji, 5., Iwai, Z. and Inoue, A. (1988). Observers (in Japanese), Koronasha, Tokyo Kobayashi, N. and Nakamizo, T. (1987). On observers for linear descriptor systems (in Japanese), Trans. of SICE, 23, 1342-1345 Lewis, F.L. (1986) A-Survey of linear Singular systems, Circuit Syst. & Signal Process, 2, 3-36 Luenberger, D.G. (1971). An introduction to observers, IEEE Trans. Autom. Control, 16, 596-602 Rosenbrock, H.H. (1974). Structural properties of linear dynamical systems, Int. J. Control, lQ, 191-202 Shafai, B. and Carroll, R.L. (1987). Design of minimal order observer for singular systems, Int. J. Control, 45, 1075-1081 Verghese, G.C., Levy, ~C. and Kailath, T. (1981). A generalized state-space for singular systems, IEEE Trans. Autom. Control, ~, 811-831

(ii) minimal order observer Choosing as C = [0 0 1] so that

,-1 " [: : :] is nonsingular, then we have

In this case All = 0, A2l = [0

So letting -a

be the pole of observer, we get L = [Zl -a], where Zl is arbitrary. Eqn.(39) gives the m~nimal order oBserver: APPENDIX Algorithm for (E", CI' )

E

r

step 1. Find two nonsingular matrices PI and P 2 such that

x

[~l

CONCLUSION In this paper, an observer for descriptor systems is presupposed to have the same structure as one of Luenberger, and the fundamental observer equations have been derived. Based on these equations, the realizability conditions have been studied. As a result, an observer can be constructed if the system is observable in the sense of Rosenbrock. And the design method has been presented. The results are extension of those for regular systems.

l'l

h

E2 }n-r

p:][:J

Cl }n-r C }p-n+r 2

where r, rank

rank El

[:~J

step 2. Let

w Further it has been shown that the realizability conditions are relaxed by elimination of the purely static modes, and an observer can be constructed if the system is observable in the sense of Verghese,

0] }r

M

It should be mentioned that other observer problems

o

}n-r

n-r p-n+r

such as linear function observer, unknown-input ob-

II step 3. land C are given as Ell w-lp

server, etc. could be discussed based on the fundamental equations.

1

C"

ACKNOWLEDGEMENT The author would like to express his appreciation to Prof. Katsuhisa Furuta of Tokyo Institute of Technology. REFERENCES Armentano, V.A. (1986). The pencil (sE-A) and controllability- observability for generalized linear systems; A geometric approach, SIAM J. Control and Optimizization, ~, 616-63-8-----Cobb, J.D. (1984). Controllability, observability and duality in singular systems, IEEE Trans. Autom. Control, 29, 1076-1082 Dai, L. (1988). Observers for discrete singular systems, IEEE Trans. Autom. Control, 12, 187191 El-Tohami, M., Nagy, V.L. and Mukundan, R. (1983). On the design of observers for generalized state space systems using singular value decomposition, Int. J. Control, 38, 673-683 Gantmacher, F.R. (1974). The Theory of Matrices, Chelsea, New York Kawaji, S. and Narahashi, S. (1988). Design of observer for descriptor systems (in Japanese),

245

w-~P2

=

n