insensitive observers for discrete time linear systems

Automaticu, Vol. 15, pp. 641 651 Pergamon Press Ltd. 1979. Printed in Great Britain © International Federation of Automatic Control

0005*ll]t;,~ 79 1201-0641 $02.0(J 0

Insensitive Observers for Discrete Time Linear Systems*

H A J I M E A K A S H I and H I R O Y U K 1 IMAI

A nece.s,sary and suJficient condition reveals that obserL, ers jor discrete time linear multivariable systems can be designed to reconstruct linear functions of the state in spite of parameter variations.

Key Word Index .... Insensitivity; observers; geometric state-space theory; computational methods; discrete time systems; linear systems; multivariable systems: invariance: algebraic system theory.

Abstract This paper considers the problem of designing an observer insensitive to system parameter variations in discrete time linear multivariable systems. A K-insensitive observer is defined as an observer that can reconstruct a linear function of the state vector in spite of the variations of system parameters, provided the initial state of the observer is suitably chosen. A deadbeat observer is defined to be an observer that reconstructs the linear function for an arbitrary initial condition of the observer. Then, the existence of the K- insensitive observer is examined, and the class of K-insensitive observers is characterized. A necessary and sufficient condition is derived under which the K-insensitive deadbeat observer can be designed, and a simple algorithm is proposed to design the observer. The resulting observer is shown to be stable. The order of the observer is evaluated. The condition for generic solvability of the problem is also given.

1. I N T R O D U C T I O N

USUALLY, observers are used as state es t imators for de terminis t ic l inear systems. If the system pa ramete r s are k n o w n exactly, observers can be designed so as to recons t ruc t the state of the system (Luenberger , 1971). I t is often the case, however, that some of the system pa rame te r s can not be identif ied or they undergo unpred ic tab le changes. Therefore, an observer should be designed such that observer ac t ion can be preserved under pa r ame te r var iat ions. The p rob lem of designing such an observer has been app roache d in two different ways. Car ro l l and L indor f (1973) and Luders and N a r e n d r a (1973) in t roduced the not ion of an adap t ive observer that has adjust- able parameters . These pa ramete r s are upda ted to reconst ruct the state. Ano the r a p p r o a c h is that of insensit ive or robus t observers. Bongiorno

*Received March 6 1978; revised December 8 1978; revised May 7 1979. The original version of this paper was presented at the 7th 1FAC World Congress on A Link Between Science and Application of Automatic Control which was held in Helsinki, Finland during June 1978.

The published Proceedings of this IFAC Meeting may be ordered from: Pergamon Press Ltd., Headington Hill Hall, Oxford, OX3 0BW, U.K. This paper was recommended for publication in revised form by associate editor P. Dorato.

?Department of Precision Mechanics, faculty of Engineering, Kyoto University, Kyoto 606, Japan.

(1973) discussed the possibi l i ty of realizing zero sensit ivity to the system pa rame te r var ia t ion of a quadra t i c per formance index. Mira (1975) presented a design p rocedure of a zero sensit ivity observer of a l inear funct ional for s ingle-input s ingle-output systems. Bha t t a c ha ryya (1976) considered an observer , called a robus t observer , that reconstructs a l inear function of the state for a rb i t ra r i ly small pe r tu rba t ions of pa ramete r s of the observer. F u r u t a and co-workers (1976) ap- p roached this p rob lem from the reverse d i rec t ion; they de te rmined the class of p lants that is com- pat ib le with a g iven .obse rve r , and appl ied this result to the design of zero sensit ivity state observers for con t inuous t ime l inear systems.

In this paper we consider the p rob lem of designing an observer that is insensitive to system pa rame te r var ia t ions in general mul t i - input mult i - ou tpu t discrete t ime l inear systems, th rough the geometr ic a p p r o a c h deve loped by W o n h a m (1974). First , in Section 2, we define a K- insensitive observer that reconst ructs a l inear function of the state in spite of the system pa rame te r var iat ions, p rov ided the initial s tate of the observer is sui tably set up. Also, we define a deadbea t observer that reconst ructs the l inear function in a finite number of t ime steps for a rb i t ra r i ly set initial condi t ions of the observer . In Section 3, we examine the existence of a K- insensitive observer, and character ize the class of K-insensi t ive observers. In Section 4, we derive the necessary and sufficient condi t ion under which the K-insensi t ive deadbea t observer can be designed, and p ropose a s imple a lgor i thm of designing such an observer. Final ly , in Section 5, generic solvabi l i ty of the p rob lem is examined and the necessary and sufficient condi t ion is given.

641

Notations ~ " denotes an n-dimensional eucl idean space.

642 HAJIME AKASHI and HIROYUKI IMA!

Subspaces of N" are denoted by script capitals, e.g. ,~, ;'~' . . . . . The annihilator of ,ag is denoted by ,eft±. The capitals denote linear maps or their matrix representations. The image (or kernel) of a map A is written I m A (or KerA). The set of all the n x m matrices is denoted by M,.,,. If A c M ..... and ~ ~ N", we write

A - l . N £ { x ' A x e . ~ and x~N"~.

The pseudoinverse of a matrix A is written A *

2. PROBLEM STATEMENT

Consider the system described by

x(i+l)=A(O)x(i)+B(O)u(i) i=0 ,1 . . . . (2.1)

y(i)Cx(i) i=0 , 1 . . . . (2.2)

where x( i )eN" is the state, y ( i ) e W ~ the output and u(i )eN ~ the input of the system. Some entries of A(O) and B(O) depend on an uncertain parameter vector 0~ ~R ~ with nominal value 0 . It is assumed that 0 belongs to a certain class 0 containing 0 . A(O) and B(O') will be abbre- viated to A and B respectively. It is assumed that rank C = m ,

Our situation is that a vector valued linear function of the state

v( i )=Kx( i )e~" (2.3)

has to be estimated based on the input-output data of the system despite the existence of the uncertain parameters. For this purpose we use an observer of the form

z(i+ I ) = T A Sz(i)+ TA Vy(i)+ TB u(i) (2.4)

,2(i)=Sz(i)+ Vy(i) (2.5)

where 2 ( i ) ~ " is the output of the observer and z ( i )eN ~ with l<n is the state of the observer. The output multiplied by K

v(i)- :K~(i) 12.6)

is an estimate for v(i). It is assumed that S and T are of full rank.

Denote the estimation error by

e(i) a x ( i ) - 2,(i).

Then the error dynamics are found to be

e(i+ 1 ) = S T A e(i)+ { ( I - VC)[A(O):B(O)]

[ x ( i q - STEA:B'~]}Lu(i)] (2.7)

and

e(O)=( l -VC)x(O)-Sz (O) (2.8)

where I denotes the unity matrix. The form of observer defined by (2.4~(2.5) is somewhat more restrictive than usual; see for instance Luenberger (1971). The reason for choosing this particular form is to derive the error dynamics of (2.7} (2.8) which forms the basis of this paper.

We make some definitions on the error charac- teristics of observers.

Definition 1. An observer is said to be K-insensitive, if for arbitrary OeO, x(O), u(i) i=O,Z .....

provided

,)g = K e r K i = 0 , 1,. e(i)6 ~

z(O ) = Tx(O).

Definition 2. A K-insensitive observer is said to have deadbeat property, or it is called a K- insensitive deadbeat observer, if there exists an integer ~ such that

e(i)~.Y i>=~

for arbitrary x(0), z(0), u(j), j=O, 1 ..... and 0~0. Our problem is to obtain, if possible, the K-

insensitive and/or the k-insensitive deadbeat observers.

3. K-INSENSITIVE OBSERVERS

In this section the class of K-insensitive observers is characterized. Let

D(O)~[A(O)" B(O)]-[A " B ]

and define a subspace ~ according to

(3.1)

; ~ 3~ Im D(0). (3.2) 0 c 0

We note that ( I - V C ) I m D ( O ) c ~ for any 0e0 , if and only if ( I - V C ) ~ c . ~ , where # is an arbitrary subspace. We have the following auxiliary result.

Lemma 1. An observer is K-insensitive if and only if there exists a subspace M c . Y such that

( I - VC)A t ~ t (3.3)

and

I m ( I - V C - S T ) c t" (3.4)

( I - V C ) ~ c ~ . (3.5)

Insensitive observers 643

Proof Iterating (2.7) yields

i - 1

e( i )= (STA°)ie(O)+ ~ (STA°) k ,k=O

× I ( I - V C - S T ) [ A ~ ' B °]

x ( i - 1 - k) + ( l - VC)D(O)s~uii'_" i "_'l~)~. (3.6)

We note that for arbitrarily given .~ and V we can choose (S, T) such that the conditions of Lemma 1 are satisfied: namely, the existence of (S, T) is trivial. The following lemma provides the necessary and sufficient condition on ~ for the existence of V having the property (3.3)-(3.5).

Lemma 2. There exists V such that

Suppose that an observer is K-insensitive. Then by definition we claim that e ( i ) ~ r, i>O, for 0=O- and any x(0) if z(0)= Tx(O). By setting 0 = 0 ° and z(0)= Tx(O) in(3.6) and (2.8) we obtain

and

e(i) = (STA°y(I - VC - ST)x(O)

i 1

+ ~ ( S T A ~ ) k ( I - V C - S T ) k = O

× EA°:8 I [ ; , i ; - i - ; ) ]

e(O) = (I - VC - ST)x(O).

(3.7)

(3.8)

( I - V C ) A ~ ' ~ c V c X (3.3bis)

and

( I - VC)9 c .+~ (3.5 bis)

if and only if

Ker C n ( A ° / V + 9 ) c / V c 0~. (3.10)

Proof Clearly (3.3) and (3.5) imply that

( I - VC)(A°~V + 9 )~ ~V.

Therefore

( I - V C)[-Ker C n (A '~/V + 9 )] = Ker Cc~ (A °/V + 9 ) c ~+~c ~ .

From these we can verify in turn

Im [ ( S T A y ( I - V C - ST)] c J~ff i=0 ,1 . . . .

o r n - 1

I m ( I - V C - S T ) c / ¢ ~ a= n (STA°) - k ~ = j { ' . k = O

(3.9)

This verifies (3.4). Also, it follows from (3.9) that ( I - V C - S T ) A ° ~ U , c U,, and since .+', is an STA°-invariant subspace in .YC, the statement (3.3) follows at once. Finally, by using the fact that ~ ' , is an STA~-invariant subspace of a'ff, we can easily verify from (3.6)-(3.9) that

Im [(STA°)~(I - VC)D(O)] c ~ i=0, 1 . . . .

for any 0E0, and this verifies (3.5). For the converse, note from (3.4)-(3.5) that

For the converse, write

A° /V + 9 = Ker Cn(A° N" + 9 )GIm U,

with a suitable matrix U, and let Va=U(CU) +. It is then easy to see that

Im U [ I - (CU)+CU] c K e r C n l m u =0. (3.11)

Hence by (3.10) we obtain

(I - VC)[A ~ /V + 9]

= [ I - U ( C U ) + C ]

= [Ker C ~ (A°/V + 9 ) G I m U]

= K e r Cn(A~: /V+9)

+ I m U[I - (CU) + CU]

= K e r C n ( A ° J V + 9 ) c X ~ . f

Im {(I - VC - ST)[A ~ " B ] + (I - VC)D(O)} c Y"

for any 0e0 , and from (3.3)-(3.4) that J¢,~ is an STA°-invariant subspace of Jcd. Therefore, it is easy to see from (3.6) that e ( i ) e j~ ' coU , i=0, 1 . . . . . for any 0e0 , x(0), and u(i), i=0, 1 . . . . . as required. []

Our problem is, now, to find an observer, if it exists, such that all the conditions of Lemma 1 are satisfied. Before solving this problem, we examine the existence of such an observer: namely, the existence of a K-insensitive observer

and clearly this verifies (3.3) and (3.5). [] Next, we examine the existence of #" having

the property (3.10). For this let us introduce a class ,U of subspaces defined by

~t / ' a={J¥:KerCn(A~/V+9)c ~ } (3.12)

and the sequence ~V& {/¢'o,/V'I . . . . }, where

# 'o = 0 (3.13a)

~i+1 =KerC~(A°~Vi+-@) • (3.13b)

644 HAJIME AKASHI and HIROYUK1 [MAI

The algor i thm (3.13) is known as the controlla- bility subspace a lgor i thm (Wonham, 1974). It is known also that . J ' i s nondecreasing and there exists an integer k < n such that = 'i'k, V i > k. It is easy to see that the subspace

4 j'* % t ,

is the least member of . , i " (Wonham, 1974). Then we obtain

Theorem 1. There exists a k-insensitive observer if and only if f"* c p f .

The p roof of Theorem 1 follows immediately from Lemmas 1 and 2.

The main result of this section is given by the following theorem.

Theorem 2. Suppose A ~'* c ~:~f. Then (V, S, T) is a K-insensitive observer if and only if

V=P.i~U(CU)+ + Y-P4.±YCU(CU) + (3.14)

and

ST= p , ~(I - VC) + P ,.W (3.15a)

=P , . I ( I - Y C ) ( I - U(CU)+ C)+ P., W

(3.15b)

for some

g ' e . V(,~ff) A= { A/. :. +~ e . ~ ' and ,,t,' c .X~ ~

(3.16)

and

U ~ U ( 4 )

a_ ~U:A' f ' + @ = K e r C c 3 ( A i + ~ ) ® I m U } (3.17)

where P~-(P~-0 is the or thogonal projector on t ' (,.t±), and Y e M .... and W~M,.n are

arbitrary.

Proof L e m m a s 1 and 2 state that (V, S, T) is a K-insensitive observer if and only if there exists 4 "~ , i "(,X/) such that

and

P , I ( I - V C ) ( A f + ~ ) = 0 (3.18)

P 41( l - Vc)=p+ LST. (3.19)

Since Ker Cc~(A < J f + ~ ) c 1, it is easy to check that (3.18) is equivalent to

P ~ L(I- VC)U = 0 (3.20)

for some U e U(.V). The solvability of equat ion (3.20) for V is guaranteed by (3.11), and it is readily verified that V solves (3.20) if and only if V is given by (3.14). Also, it is obvious that (3.19) is equivalent to (3.15a), and (3.15b) follows by substi tuting (3.14) into (3.15a). [ ]

We note that for obtaining a K-insensitive observer we first have to choose a subspace

t ~ . t'(,,g.). But when we are concerned with only K-insensitivity, we may use I '* the least member of . t ; as computed by (3.13~, provided t,* c ,# .

4. K-INSENSITIVE DEADBEAT OBSERVER

In this section an algori thm is given of designing a K-insensitive observer with deadbeat pro- petty. We begin by proving the following auxiliary result.

Lemma 3. Suppose that an observer is K- insensitive. Then the observer has deadbeat property if and only if there exists an integer i such that

K [ ( I - V C ) A ]~(I-VC)=O. (4.1)

Proof Since the observer is K-insensitive, it is easy to see from L e m m a 1 that

K [ ( I - VC)A ] ~ I ( I - VC)[A(O):B(O)]

- ST[A " B']I =O i=0 ,1 . . . .

for any 0 ~ 0, and that

(4.2)

K [ ( I - VC)A ] i=K(STA ')~ i=0 , l . . . . . (4.3)

Thus, it follows from (2.7) and (2.8) that

K e ( i ) = K [ ( l - VC)A ]'e(O)

= K [ ( I - V C ) A ]~[(I VC)x(O)-Sz(O)].

t4.4t

Therefore, if the observer has deadbeat property, (4.1) follows immediately from (4.4). For the converse, we first note from L e m m a 1 that

K [ ( I - VC)A ]i(l - VC}

= K [ ( I - VC)A l iST V i > 0 .

Therefore, since T is of full rank, (4.1) implies

K [ ( I - V C ) A ] k S = O Vk>i.

Hence by (4.4) we conclude that K e ( k ) = 0 , Vk > i, for any x(0) and z(0), as required. []

Substi tut ing (3.14) into (4.1) and noting

K [ ( I - VCjA':]~P,=O i=0 , 1 . . . . [4.5)


we find that (4.1) is equivalent to

K [ ( I - YC)QA~>]~(I- YC)Q = 0 (4.6)

provided the observer is K-insensitive, where

Q a= I - U(CU)+ C. (4.7)

Our problem is now to find X ~ , + ' ( J { ' ) and Y e M . . . . if possible, such that (4.6) holds for some i.

To solve this problem we introduce a sequence .Sea__ 1,5~o, ,Sel . . . . . 5e.} where

5% = 9~" (4.8a)

~9~i+ 1 =A~( Ker C n 5el) + N. (4.8b)

Clearly 5" is nonincreasing, so that we can define

q a--min {k : O°~ =Sek Vi>k}. (4.9)

We write

,90 a__ 5pq.

It is easy to verify that there exists a sequence ~ { ~ o , ~ , . . . . . ~q} such that

It is noted that for arbitrary U e U ( J - ) we can find ~ such that ~e~=ImU for all i, O<i<n. The main result of this paper is given by the following theorem.

Theorem 3. Let ¢- be defined by (4.13). Then a K-insensitive deadbeat observer can be constructed if and only if

Under this condition, a K-insensitive deadbeat observer is given by (3.14) and (3.15) with ,Y- in place of M, U E U ( J ) , and

Y = P s ~ Q Z ( C Q Z ) + (4.16)

where Q is defined by (4.7), and Z e Z with Im Z ~ I m U. For the proof of Theorem 3 we need some auxiliary results.

Lemma 4. Let Q be defined by (4.7) with UeU(,Y-). Then Q is a projector on I m Q ( = K e r C) along ImU. Let Z e Z with I m Z = I m U . Then

R ~ [ I - Q Z ( C Q Z ) + C]Q (4.17)

, cT i=KerCnSe i@~ i O<_i<q (4.10)

~ o ~ 1 ~ . . . ~ q . (4.11)

We note that ~ 0 is a complementary subspace to Ker C, i.e.

~ o @ K e r C = N". (4.12)

In general, such a ~ 0 is not unique. We denote by "~ the set of all such ~eo'S as having the properties (4.10)-(4.11), and define a set of matrices Z according to

Z ~ [ Z : I m Z = ~ o and ~ o ~ } .

It is noted that if Z (aZ) is of full rank, CZ is invertible. We write

,y_ a__ Ker C n ,5 P (4.13)

and note that (/'= A':',Y--+ @ (4.14)

and ,Y- = Ker C n (A o~-- + ~ )

i.e. ,Y-e A( Thus the already defined notation U(,Y-) with :+; in place of ¢- by (3.17) can be used. In view of (4.14), U(J-) can be written in the form

U (,Y--) = { U : Y = Ker Cn09a@Im U}. (4.15)

is the projector on Ker C along im Z.

Proof. Clearly, Q is an idempotent, i.e., a projector. Let x ~ K e r C . Then Qx=x. This verifies Im Q ~ Ker C. To show that

Ker Q = I m U

let xCKerQ. Then x - U ( C U ) + C x = Q x = O , so that x = U(CU) + Cx ~Im U, proving Ker Q c ImU. The reverse inclusion is obvious from (3.11). Next, it is easy to see that R is also an idempotent. It will be shown that Im R = K e r C. For this let x ~ Ker C. Then, Rx = [I - Q Z ( C Q Z ) + C ] x = x , proving Ker C c I m R. For the reverse inclusion recall from (4.12) that Ker C®Im Z = ~". Thus

CR~R" = C [ I - QZ(CQZ) + C]Q[Ker C@Im Z]

= [ I - CQZ(CQZ)+]CQ[Ker c @ I m z ]

=0 .

This verifies Ker C ~ I m R . Finally, we show that K e r R = I m Z . Since Q is a projector on ImQ ( = K e r C ) along I m U as shown above and I m Z =Ira U by assumption, we can apply LemmaA, given m the Appendix, to obtain

Ker Cc~Im QZ = Ker C n l m Qn (Im Z + Im U)

= K e r C n l m Z = 0 .

646 HAJIME AKASHI and HIROYUKI IMAI

Therefore

Im RZ = I m QZ[I - (CQZ) ~ CQZ]

Ker CcMm QZ = 0 (4.18)

and this verifies Ker R = I m Z . For the reverse inclusion let x ~ Ker R, then

0 = Rx = [I - U (CU)+ C][I - Z (CQZ) + CQ]x

= x - U (CU )+ C[ I - Z (CQZ) + CQ]x

- Z ( C Q Z ) + C Q x

or x ~ l m U + l m Z = I m Z , which implies Ker R = I m Z . []

Lemma5. Let ~,'~ . ~(#{) and Q be defined by (4.7) with U ~ U ( I ' ) . Let ~"denote the largest (I

- YC)QA-invar iant subspace in ~ . Define

~ o ~ [ ( I - YC)Q] '~

~ 8 ~ [ A ( I - Y C ) Q ] ~.~ ~

(4.19a)

= [ A ( I - Y C ) Q ] - ' [ ( I - YC)Q] ~ (4.19b)

and suppose that . ~ = ~". Then

:~i -~ .SQ ~ 0 _< i _< k (4.20)

where 5f k_ i t ~9 ° is defined by (4.8).

Proof We show first that

(4.21)

for every i>0. It is easy to see from (3.12), (3.16) and (3.17) and Lemma 4 that

( I - Y C ) Q ~ < ( I - Y C ) Q [ A t ' + @ ] ~ i , ~ t .

This verifies (4.21) for i=0 . The last inclusion follows by the assumption that t i s the largest (I - Y C ) Q A - i n v a r i a n t subspace in ,Y{. Also, by this assumption we obtain

~ ( A ) 1 [ (1 - Y C ) Q ] - I ¢ = ( A )-1,~0.

Thus ~8o C'~I' and clearly ,~_ l = ; ~ implies , ~ ,8~+~. Therefore by induction we have shown (4.21) for every i>0. Now, if ~ k = N ", by (4.19)

:~k=[A ( 1 - Y C ) Q ] "~k , = ~ " "

It follows from Lemma 4 that

"Sk 1 ~ A ' ( I - - YC)Q~" A ' ( I - Y C ) K e r C = A KerC

and since "Sk- ~ ~ @ as shown above, we have

~ k - ~ A K e r C + ~ = # ~ .

Suppose Mk-i=,~i . Then by (4.19b)

~k (i~11DA (I-- YC)@~ k i ~ A ( I - YC)Q//~i

,4 ( I - YC)Q(Ker Cc~.~fi)

= A (Ker Ccv,(/~i)

and since ~Nk , +~ ~ ~ , we have

"~k (i . l )~ A (KerCr~//"i)+~=.!/ ' i+ l"

Thus the Lemma follows by induction. []

Proof of Theorem 3. (Only if) If a K-insensitive deadbeat observer can be constructed, by (4.6) there exists Y ~ M,,,,, and an integer k such that

.1{ -~ d k ~ (I - YC )Q[A (1 - YC )Q]k~.R~ (4.22)

for some l e , t ( J ' f ) and U e U ( ¢ ' ) . Let t be the largest ( I - YC)QA -invariant subspace of .;V. Then obviously ;~~dk , and hence it follows from (4.22) that

:~k=[A ( I - YC)Q] k[(I-- YC)Q] 1'1 ~)~,,.

Therefore, by Lemma 5, , ( f k C . 8 o = [ ( l - YC)Q] ;~and, since l ~ , ; f

.~ _~ ~ ~ ( I - YC)QJ'k ~ (1 -- YC)Q[Ker C~,(/'k]

= Ker Cm,9~k ~ Ker Cm,!/' = ,Y-

as claimed. (If) If J - c . ; f , obviously ,Y-c. ¢(X). Therefore

it is enough to show that Y as given by (4.16) solves the equation

K [ ( I - Y(" }QA ]k ( l_ YC)Q=O (4.23)

for some integer k and U eU(,Y). Substituting (4.16) into (4.23) and noting

[ ( I - YC)QA ](Y- c,£- c Y

or equivalently

K [ ( I - YC)QA ]'P.z = 0

we find that (4.23) is equivalent to

K ( R A ) k R = 0 (4.24)

where R is defined by (4.17). We write

Z) i ~ Im (RA '~ )iR.

It will be shown that '3~cKerCr~// '~ for every i _>0. For this we recall from Lemma4 that R is the projector on KerC along Im Z. Then it is clear that

~o = Ker C = Ker C~,!/;o

and if ~d~ ~KerCc~ ,7 '~ ~, then by Lemma A


and (4.10)

~i=RA°°2i- i =KerCn[A°221 1 + I m Z ]

c Ker C n [A '~ (Ker C n S ~ _ ~ ) + Im Z + ~ ]

= Ker Cn[5~i + Im Z]

= Ker C n [ K e r C n S e i O I m Z]

= Ker Cc~5~.

Thus ~ q c y c ~ f f , or K(RA)qR=O, which has to be proved. []

We summarize below the procedure of designing a K-insensitive deadbeat observer.

(1) Compute 5~ according to (4.8) and check ;y~ c iF, and if this is so,

(2) choose U e U(d-), and then Z e Z such that I m Z = I m U.

(3) Compute Y according to (4.16). (4) Substitute Y and U, just obtained, into

(3.14) and (3.15) with Y in place of ~+" to obtain V and ST.

(5) Factor ST such that S and T are of full rank.

This procedure involves tedious computation of pseudoinverses in the steps (3) and (4). These steps can, however, be bypassed by use of the following corollary.

Corollary 1. Suppose C - c J ( ; let U e U ( ~ ) ; let Z ~ Z with full rank such that Im Z = Im U. then

and

V = P ~ Z ( C Z ) 1 (4.25)

S T = P j i ( 1 - Z ( C Z ) - ~ C ) (4.26)

give a K-insensitive deadbeat observer. Proof Substituting (4.16) into (3.14) with ~-- in

place of M gives

v=P~-±[U(CU) + + QZ(CQZ) + (I - c u ( c u ) + )].

(4.27)

Multiplying (4.27) on the right by CZ yields

VCZ =P f ~[U (CU ) + CZ + QZ(CQZ ) + CQZ].

Remark. Let A

p = m i n { k : K e r C n S P i c , ~ Vi>k} .

Then the observer obtained by Theorem 3 or Corollary 1 reconstructs the linear function Kx( ' ) in at most p steps.

Although the observer reconstructs the linear function in p steps, if the observer is unstable, difficulties may arise; for instance, signal levels in the observer may exceed allowed levels of oper- ation. So we have to examine the stability problem of the observer. The following result show that T A S is nilpotent, so that the observer is stable.

Corollary 2. Under the conditions of Corollary 1, suppose that ST is given by (4.26). Then

(TA°S)i=O

for some integer i.

Proof. By (4.26) we have

STA ° = P j ~RA' (4.28)

where R a = I - Z ( C Z ) - I C is the projector on Ker C along Im Z. We show that

Im(STA~)~cS~ s l n K e r C (4.29)

for every j > l . For this we note that ,Y- c KerCnSP s for every j. Then it follows from (4.28) and Lemma A that

Im S TA ° c P ¢ iRgt" = ,Y- ± c~ Ker C c 5P0 n Ker C.

This verifies (4.29) for j = 1, and if (4.29) holds for j = k

Im(S TA ~')k + 1 c ST A ° (5~k_ 1 n Ker C)

c Pzr ±R,C/'k

= P j JR(Ker C n 5Pk®~k)

= ~ ± n ( K e r Cn,~k) c Ker Cn,~k.

Therefore by induction we have shown (4.29) for every k > 1. Hence

By (4.18) the second term in the brace can be replaced by QZ, and we have

VCZ = P j z[U (CU ) + CZ + QZ] = P s~Z.

Im(STAO)O+ 1 c K e r C n ~q =,J'--.

Moreover, since Im S T c 3 - - ' , we obtain

Im(STA,,)q+ 1 c j - n y i = 0

Since Z has full rank, so that CZ is invertible, we see that (4.25) and (4.27) are equivalent. Then, (4.26) follows immediately by substituting (4.25) into (3.15a) and setting W = 0 . []

and hence we have (TA°S)q+2= TA'(STA~)q+IS =0. []

Reducing the order of observers is also an important problem. It will be shown that the

648 HAJIME AKASHI and HIROYUKI IMAI

order of K-insensitive deadbeat observer can be made less than the order of usual state observers, n - m, by dim ,Y-.

Corol lary 3. The minimum order, written l, of the K-insensitive deadbeat observer obtained in Theorem 3 or Corol lary 1 is

l = n - m - d i m , ~ .

Proof We write R ~ I - Z ( C Z ) ~C. Then we obtain

1 = rank S T = rank P.#.R = n - dim (Ker P~-,R)

= n - d i m R l j -

and, since R is the projector on Ker C along I m Z , we have by Lemma A and (4.13)

I = n - dim (Ker C c~ .y-@Im Z) = n - m - d i m , Y - . [ ]

Remark2. In the design procedure described above, the parameter variations need not be small. Large variations are allowed, provided I m D ( 0 ) c c ~ . It is often the case that possible variations of the parameters are rather small so that only the infinitesimal variations have to be considered. If the parameter variations are infinitesimal and D(O) is differentiable with respect to 0 at 0 , D(O) can be written as

Here

D(O)= ~ Dk(Ok--O~). k = l

=~ ?D(O)~,, Dk

~Vk Ok--Ok

where Ok(Ok) is the kth element of 0 (0) . We observe that

- 0 0

0 0

1 0 B =

0 0

0 1

_0 0

Ei 0 C = 0

0

_

0

0

0

0

1_

0 0

1 0

0 0

=_El 0 0 K 0 0 1

where O =

0

0

0 e ~ 1.

- 0 0

1 0

0 0 Ker C = lm

0

0

0 p

- 0 1

1 0

0 0 .*{ = lm

0 0

0 1

0 0

Also we have

s

Im ~'. Dk(Ok--Ok)c Y. ImDkA @O. k - I k - 1

Therefore, if the parameter variations are infinitesimal, the subspace cj can be replaced by ~ o in the arguments of this paper.

The procedure developed in this paper is illus- trated by' the following example. Example.

-0 1

0 A=

0 0

_0

Let

0 I 0 0 0

0 - - 1 1 0 0

1 2 0 1 1 + 0

0 0 0 1 0

0 0 1 - 1 0

0 0 0 0 1

~ = I m

1

0

0

0

0

0

I

0

0

Compute ,9 ~

- 0

0

1 ,9 "~ ~ = I m

0

0

co

- 0

0

1 >f2 = Im

0

0

_0

0

1

'o °o3 It is easy to Check that

0-7

0

1

0

0

- 1

il -:1 0

I o: .oO _0

according to (4.8) to obtain -(/o = ~-)1~'

0

0

0

0

1

0

0

1

0

0

0

0

1 0 - -

0 1

0 0

0 0

0 0

- 1 0

0 -

0

0

0

1

O_


Thus

Y = I m

- 0 7

0 I

0 I

o I

_0__1

P j l =

-1 0 0 0 0 0-

0 0 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

_0 0 0 0 0 1_

We see that .Y-~ ~ , and therefore a K-insensitive deadbeat observer can be constructed. According to (4.10) and (4.11) we can choose

- 0 0 -

0 0

1 0 ~e 2 = Im

0 0

0 1

0 0_

Thus we obtain

and by Corollary 1

- 1 0 0-

0 0 0

V = 1 1 0

0 0 0

0 0 1

_ - 1 0 0 .

- 0

0

1 ,~,~e 1 =~e o = I m

0 0

0 0

_0 0

- 0 0 1-

0 0 0

1 0 0 Z =

0 0 0

0 1 0

0 0 - 1 _

B

ST=

0 1"-

0 0

0 0

0

0

- 1

0 0 0 0 0 0 -

0 0 0 0 0 0

- 1 0 0 0 0 - 1

0 0 0 1 0 0 0 " 0 0 0 0 0

1 0 0 0 0 1_

A full rank factorization of ST yields

- 0 0-

0 0

- 1 0

0 1

0 0 _ 1 0_

° ° ° ° 0 0 1 0 '

S =

z( i+ 1 ) = [ ~ ~]z( i ) + [ ~

[ ° ° , 1 u(i) + 0 0 0

+ [ i O1 -IoIY(i)

, ° 1 0 1 y(i)

The observer obtained above reconstructs linear function in at most one step.

the

5. GENERIC SOLVABILITY OF THE PROBLEM OF K-INSENSITIVE DEADBEAT OBSERVERS

In the preceding section, it was shown that the problem of K-insensitive deadbeat observers is solvable if and only if ~Jut~--. There arises naturally the question how generally or generically this condition is satisfied. The purpose of this section is to answer this question. In the following, the concepts of generic properties and related definitions are assumed to be known. Or see, for example, Fabian and Wonham (1974). Also, we use the same notations as in the paper cited: recall AeM, ' , , CEM .... K~M~.,, and define

D~Mn, d by Im D = ~ where d ~ d i m D ; we regard

pa=(A, C, K, D) as a point in ~N, where Na=n 2 +nm+nv+nd; let f~ denote an assertion; then f~ (g) implies that f~ is generically true, i.e. f~ is true at all points p in 9~ N except for points on a proper variety in 9~u; the symbols v and A are used as

n v m = m a x (n, m)

n A m =min (n, m).

We say that the problem of K-insensitive deadbeat observers is generically solvable if Y d = Y ( g ) . It is assumed in the sequel that m<n and rank K > 0 to exclude trivial cases. Then we obtain

Theorem 4. The problem of K-insensitive deadbeat observers is generically solvable if and only if

d<m.

Proof Let 6¢ be the sequence defined by (4.8) ~7- A and define ~ i = K e r C~Sai. We see by (4.8) that

the sequence .Y- so defined can be computed recursively by

As a result, we obtain the following second order observer:

3-i+1 =KerCc~(A°~-i+-~) (5.1a)

Yo = Ker C. (5.1b)

650 HAJIME AKASHI a n d H I R O Y U K I IMAI

We write ti a= dim,Y-i. It was shown by Fabian and Wonham (1974) that+

t ~ + , = 0 v [ n A ( t i + d ) - m ] (g) (5.2a)

t o = n - m (g). (5.2b)

Since .y~ is nonincreasing so that t~<n-rn, if d < m we have from (5.2a)

t i+~=Ov(ti+d-m) (g).

Thus the sequence .Y- monotonically decreases (g) until ,Y-~=0 for some i<n-m, if d<m. Therefore, we have shown ,~D,Y-=0 (g). On the other hand if d > m, by (5.2)

ti+ 1 ~rt, ,x (ti + d ) - m

>(n-m)At i (g)

> t~ (g).

(g)

(5.3)

The last inequality follows by q < n - m . Since ,~- is nonincreasing, by ( 5 . 3 ) , Y - = J - q = K e r C (g). It is readily verified that ,~¢ :~ Ker C (g) because by assumption K e r C ~ 0 and , ; 4 " ~ " . Hence ,* ~ .<(g). []

6. CONCLUSION

This paper has considered the problem of designing an observer insensitive to system parameters variations. K-insensitive and deadbeat observers are defined. Necessary and sufficient conditions are derived under which the K-insensitive and the K-insensitive deadbeat observers can be designed. A simple algorithm is proposed to design the K-insensitive deadbeat observer.

It is straightforward to modify the algorithms to the design of unknown input observers (Guidorzi and Marro, 1973; Basile and Marrow, 1973; Sundareswaran, McLane and Bayoumi, 1977) without any knowledge of the unknown disturbance input. Also, it is possible to apply the result of this paper to the design of an insensitive controller for such a system that the state vector is not available for direct measurement. Although, the problem of insensitive control has been considered by several authors (Bonivento, Guidorzi and Marrow, 1975; Karlin, Locatelli and Zanardini, 1974), it appears thal the problem has not been solved so far by using an

q-Precisely, a difference exists between the initial conditions of .~- and the corresponding sequence ~¢/2 appearing in Fabian and Wonham (1974):i.e. J o ~ K e r C here, while ;~0 =0 in the paper cited. But their proof can apply to the proof of (5.2) with a trivial change.

observer. This problem is under investigation. In this paper, we have been concerned with the

ssslem in which only maps A and B depend on the uncertain parameter. In general, also the map (" and the initial condition, x(0), may depend on the parameter. The observer designed by the algorithm of this paper is insensitive, as well, when x(0) depends on the parameter, because the observer can reconstruct the linear function for arbitrary x(0). On the other hand, if the map C depends on the parameter, the problem becomes more complicated. This problem is left for future study.

REFERENCES

Basile, G. and G. Marro (1973). On the synthesis of unknown-input observers and inverse systems by recursive algorithms. Proc. o1' 3rd IFAC Symposium on Sensitivity, • tdaptivitv and Optimality, 171 176.

Bhattacharyya, S.' P. (1976). The structure of robust observers. IEEE Trans. Aut. Control AC-21,581 588.

Bongiorno, J. J. (1973). On the design of observer tor insensitivity to plant parameter variations. Int. J. Control 18, 597-605.

Bonivento, C., R. Guidorzi and G. Marro (1975). Parametric insensitivity and controlled invariance. Automat ica ! !. 381 388.

Carroll, R. L. and D. P. Lindorff (1973). An adaptive observer for single-input single-output linear systems. IEEE Trans. Aut. Control AC-18, 428 435.

Fabian, E. and W. M. Wonham (1974). Generic solvability of the decoupling problem. SIAM J. Control 12, 688 694.

Fabian, E, and W. M. Wonham (1975). Decoupling and data sensitivity. IEEE Trans. Aut. Control AC-20, 338 344.

Furuta, K., S. Hara and S. Mori (1976). A class of systems with the same observer. IEEE Trans. Aut. Control AC-21, 572 576.

Guidorzi, R. and G. Marro (1971). On Wonham stabiliza- bility condition m the synthesis of observers for unknown- input systems. IEEE Trans. Aut. Control AC-t6, 499 500.

Karlin, A., A. Locatelli and C. Zanardini (1974). Trajectory insensitivity via feedback. Automatica 10, 517 524.

Locatelli, A. and N. Schiavoni (1976). Further results on trajectory insensitivity. Automatica 12, 285 288.

Luders, G. and K. S. Narendra (1973). An adaptive observer and identifier for a linear system. IEEE Trans. Aut. Control AC-18, 496 499.

Luenberger, D. G. (1971). An introduction to observers. IEEE Trans. Aut. Control AC-16, 596 602.

Mira, T. (1975). Design of a zero sensitive observer. Int. ,l. Control 22, 215 227.

Sundareswaran, K. K., P. J. Mclane and M. M. Bayounu 11977). Observers for linear systems with arbitrary plant disturbances. IEEE Trans. Aut. Control AC-22, 870 87l.

Wonham, W. M. (1974). Linear Multivariahle Control. Springer, Berlin.

APPENDIX: LEMMA A

Let //q ~ , :t ' be subspaces such that ~ @ f = 9{", and P~,,, denote the projector on .~ along * ". Then

P~.~ / f=;#m(.~t+~ ' ) (A.I)

( p , ~ . ~ ) I .< l _ : ~ r > , < f ~ ) 't . (A.2)


Proof ( A , 1 ) c : I f yeP~,~5 P, there exists se~9 e such tha t y = Pa, ~ s. Since P~, ~ = I - P ~ ..~

y=P:~,~ s=s-P~ , , t s~5~ + ~

and since y E ~ , we have ye~c~(SP+~) . : I f y e ~ n ( S P + ~ ) , then for some s e . 9 ~ and ve ~ , y=s

+v. No t ing tha t y e ~ and P.~,~.v=O, we have

(A.2) c : L e t ye (P~ ,~ ) i cf. Since ~ G ~ = N ", we can write

Y = Y J + Y 2 , y l c : ~ and y2~ l "

and so

P.~,, Y=P$,* Y~ +P~t,~ Y2=Yl •

Since by as sumpt ion P ~ , , y e Y , i t follows tha t y ~ e ~ n . ~ . Thus

as claimed. The reverse inclusion is obvious. [ ]

insensitive observers for discrete time linear systems

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