Non-interacting control by measurement feedback for"outputs complete" systemsWoude, van der, J.W.

Published: 01/01/1987

Document VersionPublisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Please check the document version of this publication:

• A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differencesbetween the submitted version and the official published version of record. People interested in the research are advised to contact theauthor for the final version of the publication, or visit the DOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and page numbers.

Link to publication

General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal ?

Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.

Download date: 22. Aug. 2018

EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of Mathematics and Computing Science

COSOR memorandum 87-09

NON INTERACTING CONTROL

BY MEASUREMENT FEEDBACK

FOR "OUTPUTS COMPLETE" SYSTEMS

by

l.W. van der Woude

Eindhoven University of Technology, Department of Mathematics and Computing Science, PO Box 513, 5600 MB Eindhoven, The Netherlands.

Eindhoven, October 1986 The Netherlands

NON INTERACTING CONTROL BY MEASUREMENT FEEDBACK FOR "OUTPUTS COMPLETE tI SYSTEMS

by J.W. van der Woude

ABSTRACf

In this paper we shall consider systems that in addition to a control input and a measurement output have two exogenous inputs and two exogenous outputs. Furthermore, we assume that the two exogenous outputs of these systems are complete, i.e. we assume that the state of these systems can be written as a linear combination of the two exogenous outputs. For this kind of systems we shall solve the problem of non interacting control by measurement feedback. That is, we shall derive necessary and sufficient conditions for the existence of a measurement feedback compensator such that the transfer matrix of the resulting closed loop system. when partitioned according to exogenous inputs and exogenous outputs, has off-diagonal blocks equal to zero.

Keywords & Phrases Non interacting control, measurement feedback, outputs complete. (A.B )-compatibility, (C.A)­

compatibility.

1. Introduction

Consider a system, in state space representation, that in addition to a control input and a meas­

urement output has two exogenous inputs and two exogenous outputs. Controlling such a sys­

tem by means of a measurement feedback compensator results in a closed loop system that has

two exogenous inputs and two exogenous outputs. Therefore, the transfer matrix of the closed

loop system can be partitioned according to the dimensions of the exogenous inputs and out­

puts as a two by two block matrix. The problem that will be addressed in this paper can then

be fomlUlated as follows.

Given a system as described above, does there exist a measurement feedback compensator such

that the transfer matrix of the closed loop system has off-diagonal blocks equal to zero?

And if so, how can this compensator be computed?

The above problem formulation falls within the framework. of non interacting control. The

approach towards non interacting control as described in this paper is initiated in Willems [5]

and is developed in Trentelman & Van der Woude [4]. In the latter paper also the distinction is

made clear between the present approach and the point of view towards non interacting control

as exposed in Morse & Wonham [2] and Hautus & Heyman [1]. The main contribution of this

paper is that unlike Willems [5] and Trentelman & Van der Woude [4], where (dynamic) state

feedback is required in the solution of the problem formulated above, in this paper we allow

the problem to be solved by (dynamic) measurement feedback. In this context we also refer to

Van der Woude [7] where measurement feedback is used to solve the almost version of the

problem formulated above. In the present paper we shall be able to solve the problem formu­

lated above for systems of which the two exogenous outputs are complete.

The outline of the paper is as follows.

In Section 2 we shall give a mathematical formulation of the main problem of this paper.

Furthennore we shall recall some well-known results coming from the geometric approach

towards control theory. In Section 3 we shall derive some preliminary results. In fact, the main

result of Section 3 consists of sufficient conditions for the solvability of the main problem of

this paper even in the most general case that the two exogenous outputs are not complete.

Necessary and sufficient conditions for the solvability of the main problem of this paper for

systems with two exogenous outputs that are complete will be derived in Section 4. In Section

5 we shall state some remarks and conclusions. Furthermore, in Section 5 we shall give a con­

ceptual algorithm, that, if it exists. provides a compensator that achieves non interaction.

- 3 -

2. Problem Formulation

Consider the finite-dimensional linear time-invariant system E given by

i(l) = A X(I) + B u(t) + G1 Vl(/) + G2V2(1) •

yet) = C X(I) •

(la)

(lb)

(Ic)

Here X(I) e JR" denotes the state of the system. U(/) e Rift the control input. Vl(t) e Rq

\

vit) e IRq'}. the two exogenous inputs, yet) e JRP the measurement output and ZI(1) e /l(\

z 2(1) eRr'}. the two exogenous outputs. A. B. C. G it G 2,. HI and H 2 are real matrices of

appropriate dimensions.

Assume that the system E is controlled by means of a measurement feedback compensator 4 described by

wet) = K wet) + L y(t) •

U(I) = M wet) + N yet) ,

(2a)

(2b)

with wet) e JRA: the state of the compensator and K. L. M and N real matrices of appropriate

dimensions.

Interconnection of the system E with the compensator 4 results in a closed loop system Ed

with two exogenous inputs Vl(t), vit) and two exogenous outputs ZI(t), Z2,(t). The closed loop

system Eel is described by

i.(t) = A. x,,(t) + G 1,. Vl(t) + G2,;I V2(t).

Z 1 (t) = H I.e x. (t). Z 2(t) = H 2,. X. (t) ,

where we have denoted

[X(I) 1 [A +BNC BM 1 [Gi 1

x«(t)= wet) • A. = LC K' Gi •• = 0

(i = 1,2) .

(i = 1,2) •

(3a)

(3b)

Let T(s) be the transfer matrix of the closed loop system 41' Then T(s) can be partitioned as

[Tu(S) Tds) 1

T(s) = TZ,l(S) T 22(S)

where Tjj (s) denotes the rj x qj transfer matrix between the j -th exogenous input and the i-th

exogenous output It is clear that

(4)

We are now able to give the following problem fonnulation.

-4-

Definition 2.1.

Let 1: be given. The non interacting control problem by measurement feedback (NICPM) con­

sists of finding a measurement feedback compensator l:c such that in the closed loop system

T 11(S) = 0 and T2.1(S) = o.

If a measurement feedback compensator l:c is such that it solves (NICPM), then it is said that

I:c achieves non interaction. As announced in the introduction, throughout this paper we shall

assume that the two exogenous outputs of 1: are complete, i.e. ker HI i'I ker H 1 = [O}. (See also

Wonham [6]). Oearly, this means that the state x(1) can be written as a constant linear combi­

nation of the two exogenous outputs %1(1) and %1(1).

In Section 4 we shall derive necessary and sufficient conditions for the solvability of (NICPM)

under the assumption of outputs complete. The conditions obtained will be stated in geometric

tenns. To that end we shall now recall some well-known concepts originating from the

geometric approach towards control theory (cf. Wonham [6], Schumacher [3]).

Consider the dynamical system described by

i(/) = A X(/) + B U(/), y(t) = C X(/)

with state space JR", control input space R"'. measurement output space JRP and matrices

A e JR"""'. B e R"- and C e JRP""'. With respect to this system we now introduce the fol­

lowing.

A linear subspace V in /R" is called an (A ,B)-invariant subspace if A V ~ V + imB. It is well

known, that this subspace inclusion is equivalent to the existence of a matrix F e R tIS"'" such

that (A +BF)V ~ V.

Following Wonham (6] we call two (A,B)-invariant subspaces V 1 and V1 compatible with

respect to the pair (A ,B) (or simply: (A ,B )-compatible) if there exists a matrix F e JR"'''''' such

that both (A +BF)V1 ~ Vt and (A +BF)Vl ~ V2.' It can be proved (see Wonham [6], Ex. 9.1)

that two (A ,B)-invariant subspaces V1 and V2. are (A ,B)-compatible if and only if their inter­

section V 1 i'I V 2. is an (A ,B )-invariant subspace. If K is a linear subspace in JR". then V* (K)

will denote the largest (A,B)-invariant subspace contained in K. V*(K) can be calculated by

means of the algorithm given in Wonham [6], Chapter 4.

Dualizing the concepts introduced above we obtain the following (cf. Schumacher [3]).

A linear subspace S in R" is called a (C ,A )-invariant subspace if A (S i'I kerC) ~ S. This sub­

space inclusion is equivalent to the existence of a matrix J e JRltXP such that (A +JC)S ~ S.

Furthennore, two (C ,A )-invariant subspaces S 1 and S 2. are said to be compatible with respect to

the pair (C ,A) (or simply (C ,A)-compatible) if there exists a matrix J e JRIlXP such that both

(A +JC)Sl ~ Sl and (A +JC)S2.!:: S2.' By the previous it is clear that two (C .A)-invariant

- 5 -

subspaces S 1 and S 2 are (C.A )-compatible if and only if their sum S 1 + S 2 is a (C ,A )-invariant

subspace. If L is a linear subspace in /R". then S"'(L) will denote the smallest (C ,A)-invariant

subspace containing L. An algorithm to calculate S"'(L) can be found in Schumacher [3]. The

latter algorithm is in fact the dual of the algorithm mentioned previously for the detennination

of V"'(K) with respect to a given linear subspace K.

3. Sufficient Conditions

In this section we shall derive a preliminary result that we shall need in the proof of our main

result The result of this section provides sufficient conditions for the solvability of (N1CPM).

In order to establish these sufficient conditions, we shall make use of the following two results.

The first result that we need is very general and is concerned with the existence of a common

solution to a pair of linear matrix equations.

Theorem 3.1.

Let A JR 'jXll B JRWlCt; C JR';lCtj (. 1 ") be . j e ,i e ,i e ,= ~ gIVen.

The following statements are equivalent

(1) There exists X e JR.xw such that A1X Bl = C 1 and AzX Bl = Cz.

(2) ImAi ~ im Cj (i = 1;2), kerB; t: kerCi (i = 1;2) and

[~' ;,] ker[B .. B~I:Un [~:l. frQQf. See Van der Woude [7]. (]

For the second result that we need here, we refer to the linear system ;i(t) = A X(I) + B U(I).

y(/) = C X(I) as described in the previous section.

Theorem 3.2.

Let SI. Sl be (C.A)-invariant subspaces and VI. V l be (A.B)-invariant subspaces in JR" such

that SI t: VI and Sl t: Vl . Then there exists N e RIllXP such that (A +BNC)Sl t: VI and

(A +BNC)Sl t: V l if and only if

[~ ~] «S, e s,) 1"\ ter[C ,CD I: (V, e v,) + Un [: ].

Here ED denotes the external direct sum, (cf. Wonham [6]).

frQQf. Let Sb Sl. WI and W 2 be matrices such that im ft = Sj (i = 1.2) and ker Wi = Vj (i = 1;2),

Then there exists N e JRIllXP such that (A +BNC)Sj t: Vi (i = 1,2) if and only if there exists

N e JR"'XP such that Wi Aft + Wi BNC S; = 0 (i = 1;2).

By Theorem 3.1 the latter is equivalent to:

im Wj B ~ im Wi Aft (i = 1;2). kerC S; t: kerWi Aft (i = 1,2)

and

In its tum this is equivalent to:

- 7 -

A Sj t: Vj + imB (i =1,2), A(S; i'I kerC) t: Vj (i = 1,2)

and

[~ -1] «($, II> SiJ n ker[C .Cl) <: (V, II> ViJ + im [:].

The proof can now be completed using the obselVation that the conditions A Si t: Vj + imB

(i=I,2) and A(Sji'l kerCj)t:Vj (i=I,2) are fulfilled trivially since Sl and Sz are (C,A)­

invariant subspaces, VI and Vz are (A ,B)-invariant subspaces and Sl t: VI and Sz t: Vz. [J

Then the following theorem is the main result of this section.

Theorem 3.3.

Let the system ~ be given. Let S h S z be (C ,A )-invariant subspaces and let V it V z be (A ,B)­

invariant subspaces such that

(a) imG1 t: Sl t: Vi t: kerH')., imG'). t: Sz t: Vz!:: kerHl ,

(b) VI i'I Vz is an (A ,B)-invariant subspace,

(c) S I + S'). is a (C ,A )-invariant subspace and

(d) [~1] «S, II> SiJ n ker[C .CD <: (V, II> ViJ + im ~]. Then there exists a measurement feedback compensator It such that in the closed loop system

T1Z(s) = 0 and TzI(s) = o. EmQf. Because of (a), (d) and Theorem 3.2 there exists N e /R"''Xp such that (A +BNC)Sj t: Vj

(i = 1,2). By (b) and (c) it follows that there exist F e /R"'XIt and I e /RA'Xp such that

(A+BF)Vj t: Vi (;=1.2) and (A+IC)S; t:Sj (i=I,2). Let WI and Wz be linear subspaces in /R 2A defined by

By (a) it is clear that

(5)

Now define the matrix A" e lR 2Ax2A by

[A +BNC BF -BNC 1

A" = BNC -IC A +BF +IC -BNC .

The matrix All can considered to be obtained by the interconnection of the system ~ and the compensator given by

wet) = (A +BF +JC -BNC)w(t) + (BN -J)y(t) ,

"(t) = (F -NC)w(t) + N y(t)_

For every x., e WI there exist S E S 1 and v e VI such that

[s 1 [v 1 [(A +JC)s 1 [(A +JC)s 1 [(A +BNC)s 1 [(A +BF)v 1

A., x, = A" ( 0 + v ) = . 0 - (A +JC)s + (A +BNC)s + (A +BF)v -

From this it is clear that if x" e WI also A., x, E WI- Hence, A, WI {;; WI- Analogously, we can

prove that A., W 2 {;; W 2- So we have

(6)

From (5) and (6) it is easy to see that for any k ~ 0 Hz., A:G 1,c = 0 and H t " A:Gz., = 0, from

which it is immediate that T Z1(s) = Hz.,(sl-A"r1G 1,c =0 and Ttz(s) = H1,c(sl-A.r1 Gz., =0. The choice of the matrices K, L and M that together with N describe a compensator ~ that

achieves non interaction is obvious from above. I]

- 9 -

4. Main ResuJt

In this section we shall derive the main result of this paper. The result establishes necessary

and sufficient conditions for the solvability of (NlCPM) under the assumption that

kerHt n kerH2 == (O}. However, before stating this result we have to point out the following.

Let Q 1 e RIf.XIt be a matrix representing a map whose restriction to the subspace

V*(kerH1) + V*(kerHv is equal to the projection from V*(kerH1) + V*(kerHv onto V*(kerHv

along V*(kerH 1)' Also, let --Q2 e R"XIt be a matrix representing a map whose restriction to the

subspace V*(kerH1) + V*(kerHv is equal to the projection from V*(kerH 1) + V*(kerHv onto

V*(kerH1) along V*(kerHv. Since we assume that kerHI n kerHz == {OJ, it follows that

V*(ker H 1) and V*(ker H v are linearly independent subspaces. Because of this linear indepen­

dency, it is clear that Ql - Qz represents a linear map whose restriction to the subspace

V* (ker H 1) + V*(ker H v is equal to the identity map restricted to V* (ker H 1) + V*(ker H V.

Now, let 51 and 52 be linear subspaces in R2A defined by

5, = {[ ~ ] + [Q~ b ]Ia e S'(imG,), b e S*(im [GIoG,l)} (I = 1,2) .

The following theorem is the main result of this paper.

Theorem 4.1.

Let :E be given and assume that ker H 1 n ker H 2 = (O). Then we have the following.

(NlCPM) is solvable if and only if

S*(imG 1) ~ V*(kerHv. S*(imGV!;; V*(kerH 1). S*(im [G.,Gi]) t: V*(kerH 1) + V*(kerHv and

[~ ~ --1 ~] «S, $ S,)" tez [~ ~ ~ ~ 1) !: (V*(tezH,) U*(tezH,» + 1m [: ].

Before proving this result we note that the conditions of the theorem can be checked construc­

tively.lndeed. S*(imG 1). S*(imGv. S*(im[GltGz]) and V*(kerH 1), V*(kerHv can be calculated.

using the algorithms described in Section 2. Next Qt. Q2 and 51. 52 can be detennined after

which the conditions of the theorem can be checked.

We now proceed with the proof of Theorem 4.1.

Proof (if).

Throughout the (if)-part of the proof we shall denote Sr == S*(imG 1), Sf = S*(imGv.

S* = S*(im [GJyGz]), vr = V*(kerHv and Vi = V*(kerH t). Consider the extended state space

R2A together with the linear subspaces S 1 and S 2 as defined above. Furthennore. let VIand

V 2 be linear subspaces in m. 2A defined by Vi = Vi* ED JR." (i = 1,2). Now consider the linear

system obtained from :E by adding to 1: a bank of n integrators :Eo : x.(t) = u.(t). y.(I) = X.(I)

with x. (1). Ua(1 ),Ya(l) e JR.".

The system made up of :E and :Eo is described by

- 10-

i(/) = A x(t) + B "(I) + G1 Vl(t) + Gzvz(t) ,

y(n = t X(/) ,

z(t)=H1x(t). zz{t) = Hzx(t)

where we have denoted

A [A 0] ,., [BO] A= 00' B= 0/'

e = [~ ~]. H, = [H,.OJ (i = 1,2).

(i = 1,2).

(7a)

(7b)

(7c)

In order to complete the (if)-part of the proof it suffices to show that system (7) together with the linear subspaces S I. S 2. V 1 and V z satisfies the conditions of Theorem 3.3.

Indeed. if the conditions of Theorem 3.3 are fulfilled we may conclude that there exist matrices i. e /R'box:2J4, i e /R'box(p+ll), it e /RcIfI+ll)X2A and N e R(IfI+11)X(P+ll) such that for (i ,j) = (l,2}.(2.1)

A [A + B Nt B M ]-1 [G j ] 0= [Hi.O](sI - it i.) 0 =

In the latter we have partitioned

K= [~:: ~:]. i= [~~ ~:l. M= [:~ ::] and N= [~~ ~:]. From this we may conclude that the compensator given by

achieves non interaction and therefore solves (NICPM). So there remains to show that the sys­tem described by (7) together with the subspaces S 1. S 2. V 1 and V z satisfies the conditions of Theorem 3.3.

Note that im G 1 ~ 51. im G" ~ 5" and V 1 ~ ker H ". V" s:; ker HI. Furthennore, since S* s:; vr + vr it follows that QI S* s:; vr and Q"S* ~ Vr from which it is immediate that

imG 1 s:;5 1 S:;V 1 s:;kerH" and imG l s:;5"s:;V l s:;kerH l • Because kerHl n kerHz = {OJ also Vr n Vr = {OJ. Hence the subspaces vr and vr are (A,B)-compatible. Therefore there exists a matrix F E /R"'XII such that (A +BF)Vr s:; vr and (A +BF)Vr s:; Vr.

From 1his it is clear that with F = [~ ~ 1 also (). +B F)V, I: v, and (). +BF)V ,I: v,. Conse·

quently the subspaces VIand V 1 are (A ,E )-invariant and even (A .E )-compatible.

Note that A(5 1 n kerC) = A «Sr n kerC) e (OJ) = (A (Sf n kerC» e {OJ s:; sr e {OJ {;;; 51

and analogously A(51n kerC) s:; 52' Hence the subspaces 51 and 51 are (C.A)-invariant

Let i e (05,+05,) n me. Since i e $, + $, we can write i = [::] with

Xl = at + Ql bi + al + Qlbl and Xl = bi + b2 where al E Sr. a2 E sr and bl.b" E S*. But since also x E kerC it follows that C Xl = 0 and X2 = O. So b i = -b2 and

Xl = al + a2 + (QI-Qz)b 1 = at + a2 + bit where the latter equality follows from properties of

Q.. Q, and (Q, - Q,). Therefore. any i e ($, + S ,) n ker C can be written as i = [~' 1 with

" e s· n k .. C. from which it is immediate that A x = A [~] = [~] with yeS·.

Note that

which implies that A x E 51 + 52' Consequently. A «5 1 + 5 z) n ker C) s:; 51 + 52' Fmally we note that from the last condition mentioned in the theorem it follows that

A 0 0 0 o 0 0 0 o 0 -A 0 [ "" [c 0 C 0ll (S 1 e S z) n ker 0 I 0 I s:; «Vr e (OJ) e (Vr e (OJ)) + im o 0 0 0

from which it is immediate that

A 0 A It. AA A II> B [" 1 [" 1 o -A [(S 1 e S z) n ker [C, C] ] s:; (V 1 e V z) + im B .

Now the conditions of Theorem 3.3 are fulfilled and the proof of the (if)-part is completed.

(Only if)

Assume there exists a compensator l:c such that T 12(S) = 0 and T 21 (s) = O. Note that 00

Tij(s) = Hj~(sl-A"rlGj# = L (Hi .- A: Gj,,,)S-{hl), from which it is clear that for all k ~ 0 1:=0

[~ I

- 12 -

H 1 •• A:G2,e = 0 and H2o• A:GI,M = O. The latter implies that for all k ~ 0 A:imG2,e c;: kerH I ,.

and A:imG I.- c;:kerH2,e' Now define the linear subspaces WI and Wz in R,,+I; by -Wi = :E A:im Gi ,. (i = 1,2). 1=0

It is clear that im G I", c;: WI c;: ker H 2,e' im G2,e s:; W 2 s:; kef H 1.-' A. WI c;: WI and A. W 2 c;: W 2'

Let S 1, S 2. S. V 1 and V 2 be linear subspaces in R" defined by

Si = {ze R"I [~ ] e W, } (i =1,2),

S = {% e R"I [~ ] e W, + w,} ~ Vi = {ze R"llhere exists w e R' such that [: ] e Wi} (i = 1,2) .

Note that Sj is the intersection of Wi with R" (i = 1,2), S is the intersection of WI + W 2 with

JR" and Vi is the projection of Wi onto JR" along JRl (i = 1,2).

Now it can be shown (cf. Schumacher [3]). that SI, S2 and S are (C.A)-invariant subspaces

and that Vh V2 are (A ,B)-invariant subspaces. Furthermore. it is clear that

imG t c;: SIC;: VI c;: ker H 2, im G2 c;: S2 c;: V 2 c;: ker H 1 and im {G1.GiI c;: S c;: VI + V 2 from which

it follows that S"'(imG 1) c;: V"'(kerHz), S"'(imGz) c;: V"'(kerH1) and

S"'(im[Gt.GiI) c;: V"'(kerH l ) + V"'(kerHz).

Let % e S" then [~] e W,. By 1he A. -invariance of W, it follows !bat A. [~] e W, and more

specific (A + BNC)x e V l' Therefore, (A + BNC)S 1 c;: Viand (A + BNC)S 2 c;: V 2.

Now let Q 1 and Q 2 be matrices as described in the beginning of this section and let xeS.

Since S c;: VI + V2• VI c;: kerH2• V2 c;: kerHI and kerH1 n kerH2 = {OJ the vector x can be

decomposed uniquely as x =XI-X2 with Xl e VI and X2 e V2• In particular Xl = QtX and

%, = Q,%. Because % e S we have [~] e W, + W,. Therefore, !here exist [ :: ] e W, and

[::] • W, such !bat [~] = [::] + [::] with " e V" '2. V, and blob, e R'. It is now

clear !bat " = %, = Q,%, ., = -x, = -Q,% and b, = -1>,. So ifz • S then [~] can be written

as [~] = [Q;% ] - [Q;X] with y e R' such that [Q;X]. w, and [Q;%] e W,. Because

W, and W, are A,-invariant subspaces we have [A 1':C B:] [~%]. Wi (i = 1,2) and in

particular we have «A + BNC )Qj X + BM y) e Vi (i = 1,2).

- 13 -

Let (XltX2> •••• X,} be a basis for S. By the previous it follows that for every j e {l,2 ••..• t]

there exists Yj e R" such that «A +BNC)Qtxj +BM Yj) e V l and «A +BNC)Qzxj + BM Yj) e V1. Now define matrices N' e RmXp and M' e RmXtl. by N' = N

and M' {Xt.X1 ..... x,} = {M Yt>M Y1 ••••• M y,l. Then it follows that (A +BN'C)St!:: Vh

(A +BN'C)Sl!:: V1• «A +BN'C)Ql + BM')S !:: Vt and (A +BN'C)Ql + BM')S !:: V1.

Define the linear subspaces S 1 and S 1 in R 2a by

By the previous it is clear that [A +BN'C.BM1S l !:: VI and [A +BN'C,BM1S1!:: V l •

Now let [::] eSt and [::]05, re such tltat [~ ~] [::] + [~ ~] [::] = O.

There exist Vi e Vl and V1 e Vl such that (A +BN'C)al + BM'b i = Vt and (A +BN'C)al + BM'bl = -Vl.

Denote

[::]= [~~] [::]=- [~~] [::] then it follows that

Hence,

[AO 00][- - [coco]] [B] o 0 -A 0 (S 1 f:& S ~ n ker 0 I 0 I !:: (VI f:& V V + im B .

The (only H)-part is now proved by the observation that S 1 !:: SitS 2 !:: S 2, VI!:: V*(ker H V and V 1 t: V*(ker HI)' []

- 14 -

5. Conclusions and Remarks

ll. In this paper we studied systems that apart from a control input and a measurement out­

put have two exogenous inputs and two exogenous outputs. Furthennore. we assumed that the

two exogenous outputs of the systems that we consider are complete. This assumption is

expressed in a fonnula by kerHI '" kerHz = {OJ.

For this kind of systems we derived necessary and sufficient conditions for the existence of a

measurement feedback compensator such that the resulting closed loop system has off-diagonal

blocks equal to zero.

Of course analogous results may be derived for systems of which the two exogenous inputs

have a property (im G I + im G z = /R ") dual to the previous mentioned property of outputs com­

plete. An interesting problem for future research will be the above problem for systems where

no additional assumptions are put upon the two exogenous inputs or the two exogenous out­

puts. In this context we want to refer to Van der Woude [7]. In the latter paper it is argued that

the solvability of this general problem is equivalent to the existence of a proper rational matrix

X such that A1X BI = C I and AzX Bz = Cz where Ait A 2• Bit Bz• C 1 and Cz are all strictly proper rational matrices. It is clear that the latter requires a generalization of Theorem 3.1 to

matrices over the ring of proper rational functions.

U. To end this section we shall give a conceptual algorithm that, if it exists. detennines a

compensator 4: that achieves non interaction. Therefore, let the system 1: with the additional

property of outputs complete be given and assume that (NICPM) is solvable. Hence. assume

that the conditions of Theorem 4.1 are fulfilled. Then the algorithm proceeds through the fol­

lowing steps.

(1) Let A. B, C. 01• O2• iII and iI 2 be matrices and let S 1> S 2, Viand V 2 be linear sub­

spaces in R III as defined in the (H)-part of the proof of Theorem 4.1.

(2) Compute matrices F e R(II+m)x2ll, t e /RlIIx(II+P) and it e /R(II+m)x(II+P) such that

(A+BF)Vi !;;V1 0=1.2). (A+TC)Si!;;S (i=I,2) and (A+BitC)Si !;;V1 (i=I,2). The

existence of F. T and it is proved in the (if)-part of the proof of Theorem 4.1.

The computation of for instance it can be perfonned as follows. Let S; (i = 1.2) and Wi

(i = 1,2) be matrices such that im S; = S i (i = 1.2) and ker Wi = Vi (i = 1,2). Then it suffices to

compute it such that Wi A S; + "'i B it C Si = 0 (i = 1,2). Using Kronecker products the latter

two linear matrix equations can be written as one linear matrix equation which can be solved

using standard techniques. whereupon rearrangement of the obtained solution provides it . Similar remarks can be made with respect to the computation of F and t. (3) Define matrices i E RlIIxlII, i E /RlIIx(lI+p) and M e R(II+m)xlll by

i = A + B F + TC - B we, i = B W - T. M = F - WC. Now decompose

- 15 -

Then the matrices K, L, M and N constitute a measurement feedback compensator that achieves non interaction.

- 16 -

References

[1] M.L.J. Hautus and M. Heyman, 'New result on linear feedback decoupling', Proc. 4th Intem Conf. Anal. & Opt. Systems, Versailles Lecture Notes in Control and Inf. Sci­ences, Vol. 28. pp. 562-575, Springer Verlag. Berlin, 1980.

[2] A.S. Morse and W.M. Wonham, 'Status of non interacting control'. IEEE Trans. Aut Contr., Vol. AC-16, No.6, pp. 568-580, 1971.

[3] JM. Schumacher. 'Dynamic Feedback in Finite and Infinite Dimensional Linear Sys­tems', Math. Cent. Tracts 143, Amsterdam, 1981.

[4] H.L. Trentelman and l.W. van der Woude, 'Almost invariance and non interacting con­trol: a frequency domain analysis', Memorandum COSOR 86-09, Eindhoven University of Technology, 1986. (To appear in Lin. AIg. Appl.)

[5] J.C. Willems, • Almost non interacting control design using dynamic state feedback'. Proc. 4th Intern. Conf. Anal. & Opt. Systems, Versailles Lecture Notes in Control and Inf. Sci­ences, Vol. 28, pp. 555-561, Springer Verlag, Berlin, 1980.

[6] W.M. Wonham, 'Linear Multivariable Control: a geometric approach', 2nd ed., Springer Verlag, New York, 1979.

[7] l.W. van der Woude, 'Almost non interacting control by measurement feedback', Memorandum COSOR 86-16, Eindhoven University of Technology, 1986. (To appear in System & Control Letters.)