Non interacting control by measurement feedback

Non interacting control by measurement feedback

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Woude, van der, J. W. (1987). Non interacting control by measurement feedback. (Memorandum COSOR; Vol. 8711). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1987

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EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of Mathematics and Computing Science

COSOR memorandum 87-11

NON INTERACTING CONTROL BY MEASUREMENT FEEDBACK

by

J.W. van der Woude

Eindhoven University of Technology,

Department of Mathematics and Computing Science, PO Box 513,

5600 MB Eindhoven, The Netherlands.

Eindhoven, August 1987 The Netherlands

by

J.W. van der Woude

ABSTRACT

In this paper we shall solve the problem of non interacting control by ment feedback for systems that in addition to a control input and a measure-ment output have two exogenous inputs and two exogenous outputs.

That is, we shall derive necessary and sufficient conditions that can actually be verified 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 outputs, has off-diagonal blocks equal to zero.

Keywords & Phrases

Non interacting control, measurement feedback, (A, B)-compatibility, (C. A)-compatibility, linear matrix equations.

2

-1. Introduction

Consider a system in state space fonn that, in addition to a control input and a measurement output, has two exogenous inputs and two exogenous outputs. Controlling such a system 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 outputs as a two by two block matrix. The problem that will be addressed in this paper can then be fonnu-lated 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 fonnulation falls within the framework of non interacting control. The approach towards non interacting control as described in this paper was initiated in Willems [5] and was developed in Trentelman & Van der Woude [4]. In the latter paper also the distinction was 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 was required in the solution of the problem fonnulated 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 also measurement feedback is used to solve the

almost version of the problem fonnulated above. The outline of the paper is as follows.

In Section 2 we shall give a mathematical fonnulation 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 our main problem.

Necessary and sufficient conditions for the solvability of our main problem will be derived in Section 4. In Section 5 we shall state some remarks and conclusions. Furthennore, in Section 5 we shall give a conceptual algorithm, that, if it exists, provides a compensator that achieves non interaction.

2. Problem Formulation

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

yet)

=

C x(t) ,

(I a) (lb) (Ie) Here xCt) E Rl! denotes the state of the system, u(t) E Rm the control input, vIet) E Rq1,

V2(t) E Rqz the two exogenous inputs, yet) E RP the measurement output and ZI(t) E 1/\

Z 2( t) ERr z the two exogenous outputs. A, B, C, G l' G 2' H I and H 2 are real matrices of

appropriate dimensions.

Assume that the system ~ is controlled by means of a measurement feedback compensator ~c described by

w(t)=Kw(t)+Ly(t) ,

u(t)

=

M wet) + N yet) ,

(2a) (2b) with w (t) E R k the state of the compensator and K, L. M and N real matrices of appropriate dimensions.

Interconnection of the system ~ with the compensator ~c results in a closed loop system ~cl with two exogenous inputs VI(t), v2(t) and two exogenous outputs z1(t), Z2(t). The closed loop system ~cl is described by

Xe(t)

=

Ae xe(t)

+

GI,e VI(t)

+

G₂,e V2(t) , z l(t) = H I,e Xe (t), Z 2(t)

=

H 2,e Xe (t) •

where we have denoted

Hi,e

=

[Hi • 0] (i =1,2) .

(i = 1,2) ,

(3 a) (3b)

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

[

T l1(S) T 12(S)

1

T(s) = T

21(S) T22(S)

where Ti/s) denotes the rj x qj transfer matrix between the j -th exogenous input and the i-th exogenous output. It is clear that

Tij(s)

=

Hi,e(s[ -Ae

r

1 Gj,e . (4) We are now able to give the following problem formulation.

4

-Definition 2.1.

Let L be given. The non interacting control problem by measurement feedback (NICPM) con-sists of finding a measurement feedback compensator Lc such that in the closed loop system T 12(S)

=

0 and T 21 (s)

=

O.

If a measurement feedback compensator Lc is such that it solves (NICPM), then it is said that Lc achieves non interaction.

In Section 4 we shall derive necessary and sufficient conditions for the solvability of (NICPM). The conditions obtained will be stated in geometric terms. To that extent we recall some well-known concepts Originating from the geometric approach towards control theory (cf. Wonham

[6J, Schumacher [3]).

Consider the dynamical system described by

i(t)

=

A x(t)

+

B u(t), yet)

=

C x(t)

with state space Rn, control input space Rm, measurement output space RP and matrices

A E RnXll, B E RnXm and C E RPXIl. With respect to this system we now introduce the following.

A linear subspace V in R n is called an (A, B)-invariant subspace if A V !:; V

+

im B. It is well known, that this subspace inclusion is equivalent to the existence of a matrix F E RmXll

such that (A +BF)V !:; V.

Following Wonham [6] we call two (A ,B )-invariant subspaces VIand V 2 compatible with respect to the pair (A,B) (or simply: (A,B)-compatible) if there exists a matrix F E Rmxn

such that both (A

+

BF)V 1 !:; VIand (A

+

BF)V 2 !:; V 2' It can be proved (see Wonham [6J, Ex. 9.1) that two (A ,B)-invariant subspaces VI and V₂ are (A ,B)-compatible if and only if their intersection VI ( l V₂ is an (A ,B)-invariant subspace. If K is a linear subspace in Rn,

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 [6J, 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

n

kerC)!:; S. This subspace inclusion is equivalent to the existence of a matrix J E RnXP such that

(A

+

JC)S !:; S. Furthermore, two (C ,A )-invariant subspaces S 1 and S 2 are said to be

compa-tible with respect to the pair (C ,A) (or simply (C, A )-compatible) if there exists a matrix

that two (C ,A)-invariant subspaces S 1 and S2 are (C ,A)-compatible if and only if their sum S 1

+

S2 is a (C ,A)-invariant subspace. If L is a linear subspace in Rn, then S*(L) will

denote the smallest (C ,A )-invariant subspace containing L. An algoritlun to calculate S*(L)

can be found in Schumacher [3]. The latter algoritlun is in fact the dual of the algoritlun men-tioned previously for the determination of V*(K) with respect to a given linear subspace K.

~

6

-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 provides sufficient conditions for the solvability of (NICPM). 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 Ai E RSjXV, Bi E RW'XJi, C_{i E} RSj'XJj (i=I,2) be given matrices. There exists a matrix

X E RVXW such that A) X B 1

=

C) and AzX B2 = C 2 if and only if imAi ;;:? imCi (i

=

1,2),

ker Bi S;; kerC_i (i

=

1,2) and

[~I ~Jer[BI.Bzl

<: im

Proof. See Van der Woude [7].

o

For the second result that we need here, we refer to the linear system x(t)

=

A x(t)

+

B u(t),

yet)

=

C x(t) as described in the previous section. Theorem 3.2.

Let S). S2 be (C ,A)~invariant subspaces and VI. V₂ be (A .B)-invariant subspaces in RII such

that SIS;; V 1 and S 2 S;; V 2. Then there exists a matrix N E Rm xp such that

(A +BNC)S) S;; VI and (A +BNC)S2 S;; V₂ if and only if

[~

..:] «S I

e

SiJ

n

ker[C.C]) <: (VI

e

ViJ

+

im [: ].

Here we have adopted the following notation. If Ll is a linear subspace in R" and L

_z

is a linear subspace in Rtz then L)

e

L2 denotes the linear subspace in Rt,+lz defined as

Proof. Let X l' X 2' T 1 and T 2 be matrices such that imX;

=

Si (i

=

1,2) and kerTi

=

Vi (i

=

1,2). Then there exists a matrix N E RmXP such that (A +BNC)Si S;; Vi (i

=

1,2) if and only if there exists a matrix N E RmXP such that Ti A Xi

+

Ti BNC Xi

=

0 (i

=

1,2).

By Theorem 3.1 the latter is equivalent to im Ti B ;;:? im Ti A Xj (i

=

1,2), kerC Xi S;; kerTi A Xi (i

=

1,2) and

[TI~

XI

-T'~

xJer[C

X

I'

C X,I <: im [;::] .

and

The proof can now be completed using the observation that the conditions A Sj !:. Vj

+

im B

(i=1,2) and A (SJI kerCj)!:. Vj (i=1,2) are fulfilled trivially since SI' S2 are

(C,A)-invariant subspaces, V I> V 2 are (A, B)-invariant subspaces and S 1 !:. VI' S 2 !:. V 2,

0

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

Theorem 3.3.

Let the system 1: be given. Let S l' S 2 be (C ,A )-invariant subspaces and let VI. V 2 be (A

,B)-invariant subspaces such that the following conditions are satisfied. (a) imG1 6 SI 6 VI 6 kerB2 • imG2!:. S2!:. V 2 !:. kerB},

(b) VI (J V 2 is an (A t B)-invariant subspace,

(c) SI

+

S2 is a (C .A)-invariant subspace and

(d)

[~_~

] «S I @ S,) 1"'1 ker[C ,CD <;; (VI @ V,)

+

im [:].

Then there exists a measurement feedback compensator 1:_c such that in the closed loop system T 12(s) = 0 and T 21 (s)

=

O.

Proof. Because of (a), (d) and Theorem 3.2 there exists a matrix N E RmXP such that

(A

+

BNC)Sj 6 Vi (i = 1,2). By (b) and (c) it follows that there exist matrices F E Rmxn and

J E R"XP such that (A +BF)Vj 6 Vi (i = 1,2) and (A +JC)Sj 6 Sj (i

=

1,2). Let W l,e and

W 2,e be linear subspaces in R 2n defined by

Wi, = {

[~

]

+ [:

lIs e

Si'

v e

Vi } and let Ae E R2nx2n be a matrix defined as

[ A +BNC Ae = BNC-JC BF-BNC ] A+BF+JC-BNC . (i

=

1,2) ,

The matrix A, can considered to be obtained by the interconnection of the system 1: and the compensator 1:c where K

=

A +BF +JC -BNC. L

=

BN -J and M

=

F -NC.

For every xe E W l,e there exist vectors S E S 1 and v E V 1 such that

[s

1

[v

1

[(A +JC)s

1

[CA

+JC)s

1

[(A +BNC)s

1

[(A +BF)v

1

Ae Xe

=

Ae ( 0 + v )

=

0 - (A +JC)s + (A +BNC)s + (A +BF)v .

Since (A +B)Vl !:. VI> (A +JC)SI !:. SI and (A +BNC)Sl !:. VI it is immediate that

8

-A" W2,e ~ W2,,,.

By (a) it is clear that im G I,,, ~ W 1,1' ~ ker H 2,1' and im G 2,1' ~ W 2,1' ~ ker H I,e.

Now there follows that HI,,, A"Ic_{G 2},e

=

0 and H 2,e A:G 1,1' = 0 for all k ~ 0 from which it is

4. Main result

In this section we shall derive the main result of the present paper. The result establishes necessary and sufficient conditions for the solvability of (NICPM) in state space terms that can actually be verified. However, before stating this result we have to introduce the following. If ZI' Z2 and Z are linear subspaces in RII such that Z 1 +Z2 !:; Z then we define the set

(1)(Z I. Z 2' Z) as follows.

(1)(2_1,22,Z) = {(M I,M z) E RIIXII X R IIXII I (M l-/)Z1 = {O}, (M 2-/)Z2 = {a} and

(M₁+M_{2-/)Z=2 1 (l22 }·}

Note that the set (1)(21,Z2'Z) is not empty.

Indeed, let [Lo,LI,L2.L3.L4] E R IIXII be a square invertible matrix such that imLo=21 ( l 2_2, im[Lo,Ltl =2_1, im[L o,L2] =2₂ and im[Lo,L₁,L2.L3J =Z. Then any pair of matrices (MhM 2) E R IIXII x RIIXII such that MdLo,LltL2,L3,L4]

=

=

[L o,L1>O,M13,M 14J and M2[Lo,L}oL2,L3,L4]

=

[Lo,O,L2.L3-M13,M24] is an element of

(1)(Z 1,22, Z). In the latter M 13' M 14 and M 24 are arbitrary matrices of appropriate dimensions.

Given the system 1:. we shall adopt the following notation

Sr

=

S*(imG_1). Sf

=

S*(imGz), S*

=

S*(im[G1>G_{2]) ,}

vt

= V*(kerH,).

vt

= V*(kerH ,). V* =

v'

(ker [::

l) .

Since SI* ,- S* S* _{>= ' 2 - '} c S* V*,- v'" and V* _{- 1} c _{- 2} V* there holds

S r

+

Sf!:; S... and V*!:; v r ( l V f . Now the main result of the present paper reads as follows.

Theorem 4.1.

Let the system 1:. be given.

Then (NICPM) is solvable if and only if there exist pairs of matrices

(D1,Dz)E (1)(st,sf,S*) and (E[,EI)E (1)(vtl,vfl,V*J.) such that D1S*!:;Vt, DzS*!:; Vf, E1St !:; V*, E2Sf !:; V*. (D1+El-I)S*!:; V* and

(AD 1

+

E lA - A )(S * ( l ker C) !:; V*

+

im B .

In the above T stands for matrix transposition and 1 denotes the orthogonal complement with respect to the euclidian innerproduct.

Before giving the proof of Theorem 4.1, we have to make some remarks.

Remark 4.2.

In the proof of Theorem 4.1 we shall frequently make use of specific non unique representa-tions of the linear subspaces sr, Sf, S*, vr, Vf and V*. More concretely, we let

10

-[X o,X I,X_Z.X_3] be an injective (full column rank) matrix such that imX o = S i" ()

si,

To

T}

im[XO,X1]=Si", im[Xo,Xz]=Si _{and im[Xo,X I,Xz,X3]=S*. Dually, we let} T

z be a

T3

suljective (full row rank) matrix such that ker To = V

t

+

V:t • ker

[~;

1

= V

t .

To

ker

[~:

1

= Vi' and ker

~:

= V',

T3

To

°

o Tl 0 TI

In the sequel we denote X = _{[Xo,X I'XZ'X}3], X = [O,X 1,XZ,X3], T

=

_T

_z

and T=

_Tz

T3 T3

o 0

and we let Qx and QT be square matrices such that X

=

X Qx and T = QT T .

Then using these specific representations, the subspace inclusions in the "unknown" matrices D It D 2. E I and E z appearing in Theorem 4.1 can be reformulated

as

linear matrix equations

in the "unknown" matrices D 1. D

_z,

Eland E

_z.

Next. the linear matrix equations obtained can, by means of Kronecker products (see Lancaster [8]), be transformed into linear equations whose solvability can be checked using standard techniques.

Hence, the conditions of Theorem 4.1 can actually be verified.

Proof of Theorem 4.1 .. (if)

Let S i" ' S i , S*. Vi", Viand V* have a representation as indicated in Remark 4.2.

Because (D 1,D0 E <J>(S i" ,S i ,S*) there follows that with respect to the chosen representation

Dl [X o,XltXZ.X3] = [Xo,Xlt0,D I3] and D z [Xo,Xt>XZ,X3] = [X o,O,XZ,D23J with

D13+Dz3

=

X3·

Indeed, because (D I -I)S i" = {O) DdXo,X 1,X2,X3]

=

[Xo,X I,D lz,D 13]

and (D

_z

-I)Si

=

{a} it is clear and D2[Xo,X I,X2,X3]

=

[XO,DZl,Xz,D23]'

that and because CD 1

+

D 2 - / )S*

=

S i" () S i there follows that D 12

=

0, D ZI

=

°

and D 13

+

D 23

=

X 3·

Dually, because (E [ , E

r)

E <J>(V i"1. , V i1. , V*L there follows that with respect to the chosen

To To To _To

T} Tl Tl 0

T2 E 1

=

0 and T2 E 2

=

T2 with E31 +E32

=

T3 .

T3 E31 T3 E32

With X and T as introduced in Remark 4.2 the subspace inclusions of Theorem 4.1 imply that

[~:

]D1X

=

0,

[~:]

D,x

=

0,

TEdXo,X,J

=

0,

TE,

[Xo,X,]

=

0,

T

CD

1

+

E 1 - J)X

=

0 and T (AD 1

+

E lA - A )X ker CX !;; im TB .

Decompose -E 32D 23 = U 3Y 3 with U 3 a surjective matrix and Y 3 an injective matrix. (It is

always possible to find such a decomposition; for instance, U 3 = -[E 32'/ ,0] and

y 3 =

[D~231.)

Let

g denote the number of rows of Y

3 and

define matrices

R ,P

r

ell' '"'

as

To ₀ T} ₀ R

[X

_O

,X

_l

,X

₂

.X

_3] = [O,O,0,Y₃

J.

T2

P=

0 T3 U3 Denote R(II+g)x(n+g)

B - [B

0]

E R(n+g)x(m+g) , g - 0 I '

G [Gi]

R(n+g)Xt1i i,g

=

0 E (i

=

1,2), C,

=

[~ ~]

e R(p+')x( .... ') and H i,g

=

[H 0] i, E In ",rjx(lI+g) (i

=

1,2).

Now let S l,g , S 2,g' V l,g and V 2,g be linear subspaces in R n+g defined as

[D1X]

[DiX]

S l,g

=

im RX ,S2,g = im -RX ' V l,g

=

ker[TEl • TP] and V2,g

=

ker[TE2.-TP] .

In order to complete the proof of the (H)-part of Theorem 4.1 it suffices to show that the sys-tem characterized by the matrices Ag, Bg , Cg , G l,g' G2,g' H 1.g and H 2.g together with the

linear subspaces S l,g' S 2,g' V l,g and V 2,g satisfy the conditions of Theorem 3.3. Indeed, if

that is the case, then by Theorem 3.3 there exist matrices Kg, Lg, Mg and Ng such that for

12

-where we have decomposed L,

=

[L I. L,]. M,

= [::]

and N,

= [::: :::].

From the

latter we can conclude that the compensator given by

•

[

w2(t) -

Wl(t)]_

[N22 N2] L2 Kg

[WI(t)]

w2(t)

+

[N21] Ll y(t).

achieves non interaction and therefore solves (NlCPM).

So. it remains to show that the system described by the matrices Ag • Bg • Cg • Gl,B' G2,g'

HI,B and H 2,g together with the linear subspaces Sl,g' S2,g' VI,g and V2,g satisfy the

condi-tions of Theorem 3.3.

To that extent note that imG

I.,

=

imG

I

ED (O) <;; S

t

ED (O)

=

im

[~I

]

[X .. X

I]

<;; S

I.,.

Analogously it can be shown that imG₂,g !;; S2,B' Vl,g !;; kerH_{2•g and V2.g} !;; kerHJ,g'

Next we claim that S J.g !;; V l,g and S 2.g !;; V 2,g .

It is clear that proving this claim is equivalent to proving that T (E ID I

+

PR)X = 0 and

T(E 2D2+PR)X

=

O. Therefore. observe that

TOXo 0 TOX2 0 0 0 T2XO 0 T 2X 2 ToD23

o

T_{2D 23} E32XO 0 _{E32X2 E 32D}23+U 3Y3

Since [;:] D,X

=

O. TE, [X a.X,]

=

0 and E"D,,+ U ,Y,

=

0 there 0 follows th,:

T(E 2D2+PR)X

=

O. The latter implies also that Qr T(E_{2D 2+PR)X Qx}

=

_{T(E 2D2+PR)X}

o 0

where we used T. X, Qr and Qx as introduced in Remark 4.2.

o 0 0 0 0

Note that RX=RX. TP =TP, DIX+D2X=DIX+D~ =X and TE 1+TE2=

o 0 0

= TEl

+

TE2 = T. Then there follows 0 = _{T(E 2D 2+PR)X =}

= T(l-DJ-EJ)X

+

T(E1D1+PR)X. Since T(D₁+E_I-I)X

=

0 we obtain that also

T(E}Dl+PR)X

=

O. So Sl,g !;; VI,g and S2,g !;; V 2.g.

and

A,(SI., n kerC,) =A,(im

[:0 :1

~:3]

n (kerC til {O}»

=

=Ag«im[XO,X!1fl kerC)Ef:) {O})=(A(StflkerC»e {O}!;;Si e {O}!;;Sl,g

[ XO Xl D13 Xo X2 D23 ] Ag«Sl,g+S2,g)flkerCg)=Ag(im 0 0 Y 3 0 0 -Y3 flkerCg)= [ Xo Xl X

2

X3

Dl3]

= Ag (im 0 0 0 0 Y 3 fl (ker C

e

{O}))

=

=

Ag«im [Xo,X _{l'X 2,X3]} fl kerC) e {O}) =

Hence, Sl,g' SZ,g and Sl,g +SZ,g are (Cg,Ag)-invariant subspaces in Rn+g. Dually we can prove that V l,g' V Z,g and V l,g fl V 2,g are (A_{g ,} B g)-invariant subspaces in R n+g .

Finally we have to prove that

To that extent denote

-

[A,

0

1

Ao = 0 -Ag , Eo=

[::

].

Co = [Cg.Cgl • D1X 0 s= RX 0

1'=

[TEl TP

0 0

1

0 DzX and

o

0 TE2 -TP . 0 -RX

Then the above inclusion reads

A

o(im

S

fl ker C 0) !;; ker

l'

+ im

E

o. It is easy to see that the latter subspace inclusion is equivalent to

A

1 ker C 1 !;; im

B

1 where

[

TE1AD1X 0

1

[TEtB TP

1

A

I =

TAoS

= 0

-TE~zX'

B

=

TEo

= TE2B -TP and

14

-[

CDtX CD2Xl

Ct=CJ=

RX -RX

-Now let _U -₁

₌

[I _Qx

01

_{1 and V}

-

_I

₌

[I QT _{0 1}

1

-A

2 ker C 2 ~ im

B

2 where

A

2

=

V)A

I

U

I.

B

2 =

VI B

J and C 2

=

C 1

U

1-o 0 0 0

A straightforward calculation using TP

=

TP, RX

=

RX. D IX

+

D 2X

=

X and TE 1

+

TE 2

=

T

shows that

and

Recall that

[~l

= [:.

~

: '

~:

1

with the matrix Y 3 injective and note that any vector in kerC 2 consists of 8 "components".

Because of the injectivity of the matrix Y 3 the 8-th component of any vector in ker C 2 is zero. This means that the 8-th column of

A

2 does not playa role in the description of

A

2 kerC

2-Furthermore there follows that the 6-th component of any vector in C 2 can be chosen com-pletely arbitrarily. However. the contribution of this 6-th component to the description of

A

2 ker C 2 is annihilated because the 6-th column of

A

2 is a zero column. Hence. we can delete the 6-th as well as the 8-th column in both

A

2 and C 2 and still have a description of the

sub-space

A

2kerC

2-By dual arguments it can be shown that we can delete the 6-th as well as the 8-th row in both

A

z

and

B

_z

and still we are left with a subspace inclusion equivalent to

A

2

kerC z

~ imB z. In view of this all denote

- _[TCAD1+E1A;-A)X-TEZAX'] - [TB] _ A 3 - _-T'AD

2X -T'AX" B 3

=

T'B and C 3

=

[CX, CX'] where X' = [X •• X,] and T' =

[~:].

Now there holds

A

2 ker C 2

~

im

B

2 if and only if

A

3 ker C 3

~

im

B

3- Let Qx I a[n:

~~~]

matrices of suitable dimensions such that XQx' = X' and QT'T = T' and let

U

3 = 0 I

-

[I

0]

_ _ _

and V 3 = -QT I I - Then it follows that A 3 ker C 3 ~ im B 3 if and only if

A

4kerC 4 ~ imB4 where

A4

=

V~3U3'

B4

=

Vl

l 3

and C 4

=

C

3

U

3· Again straightforward calculation shows that

where X"

=

[X o.OJ and T"

=

[:0 ].

Because of the structure of the matrices

A

4'

B

4 and

t

4 it follows that

A

4 ker

t

4 !;;;;; im

B

4 is equivalent to T (E lA + AD 1 - A )X ker CX !;;;;; im TB • ker CX !;;;;; ker T" AX and im TAX" !;;;;; im TB .

Translated into the original subspaces the latter three conditions mean that

(AD I +E IA -A)(S*

n

kerC) !;;;;; V* +imB, A (S*

n

kerC) !;;;;; Vr + vt and

A (S r n S

t )

!;;;;; V* + im B . By now we have proved that the subspace inclusion

[A~ _~,

]

«S,.,

tll

S~,)n

ker[C,.C,]),;;(V ,., tll

V~,)+im

[::] is equivalent to (ADl+ElA -A)(S*n kerC)!;;;;; V*+imB, A(S*n kerC)!;;;;; vt +vt and

A (S

t

n

S

t)

!;;;;; V* + im B .

Now of the latter three the first subspace inclusion holds by assumption while the second and third subspace inclusion hold because it can be shown that S"'!;;;;; V r + V t and Sr n Sf !;;;;; v"'. Indeed,S'"

=

(DI+D2_(Dl+D2_I)S* !;;;;;

!;;;;;D IS*+D 2S"'+(D1+D 2-l)S"'!;;;;;vt +Vf +st nSf =vt +Vf +(D1StnD 2Sf)!;;;;;

!;;;;;vt

+

_{Vi' +(D IS*nD 2S*)=Vr}

+

Vi' .

By dual arguments it can be shown that S

t

n

S

i'

!;;;;; V"'. Hence, the conditions of Theorem 3.3 are fulfilled and we have completed the proof of the (if)-part of Theorem 4.1.

(only if)

In order to prove the (only if)-part of Theorem 4.1 we assume that (NICPM) is solvable. That is, we assume that there exists a measurement feedback compensator kc such that in the closed loop system kel there holds Ht.,(sl-AerIG2.e = 0 and H2,e(sl-A,rIGI., = O.

Recall that R n+k is the state space of kel and that we denote

A,

=

[A

~NC

B:]. G

i .,

=

[~

]

(I

=

1.2) and Hi., = [Hi.oJ (I

=

1.2) .

Let WI" and W 2,e be linear subspaces in R,,+k defined as

W1,e=<A,limG_1•e> (=imG1,e+A,imG1,e+ ... +A;+k-1imGI,e) and

Then we have imG1,e !;;;;; W1,e !;;;;; kerH 2,e, imG 2,e !;;;;; W 2,e !;;;;; kerH1,e' AeWl,e !;;;;; W1,e and

AeW 2,e!;;;;;W2,e'

But also there holds Ae(W l,e + W 2,e) !;;;;; (W l,e + W 2,e) and

16

-Let S 1, S 2. S, V I, V 2 and V be linear subspaces in R n defined as

Si = (x E 1/" I

[~]

E Wi.,) • (i = 1,2) •

S = (x E 1/" I

[~]

E (WI., +W,.,)) •

Vi

=

(x E 1/" 11wEl/k: [:] e Wi.,). (i

=

1,2) and

V

=

(x e 1/" 11wel/ k : [:] e (WI., 1"\

W~,)).

Note that Si is the intersection of W_{i •}e with Rn, (i

=

1,2), S is the intersection of (W l,e + W 2.e) with Rn , Vi is the projection of W_{i •e} onto Rn along Ric, (i = 1,2) and V is the projection of (W I

.

e (l W Z _, e) onto R n along Ric, From the latter it is immediate that

(i

=

1,2) ,

(SI(l Si)

e

{OJ

=

(Sl

e

{O}) (l (Sz

e

{O}) ~ (WI,e (l WZ,e) ~ V

e

Rk and S

e

{OJ ~ (WI,e+W2,e) ~ (VI eRic.)

+

(V₂ eRic)

=

(VI

+

Vi)

e

Rk .

Clearly, there follows that Sj ~ Vi (i

=

1,2). S 1 (l Sz ~ V and S ~ V I + V

2-Because the linear subspaces WI,e' W₂,e and _{(WI,e +W2,e)} are Ae-invariant, it follows that the subspaces S 1. S 2 and S are (C , A)-invariant subspaces. Indeed. let XES 1 (l ker C. Then

[~

] e WI., and

[~]

=A,

[~]

e WI.,. Hence A(S 11"\ kerC) <; S I' Similarly. it can be shown that also Sz and S are (C .A)-invariant subspaces.

By dual reasoning it can be shown that due to the Ae -invariance of the linear subspaces W I,e'

W Z,e and (W l,e (l W 2,e) the subspaces VI. V 2 and V are (A ,B )-invariant subspaces.

Next observe that im G 1 ~ S 1, im G 2 ~ S 2. V 1 ~ ker H 2, V 2 ~ ker HI' im [G 1. G z] ~ S and V <; ker [:: ].

From the previous we may conclude that

S

t

~ V

t .

S

t

~ V

t ,

S* ~ V

t

+

V

t

and S

t

(l S

t

~ V*

where we have used the notation as given in the beginning of the present section. If, in addi-tion, we assume that the linear subspaces S

t ,

S

t ,

S ... , V

t ,

V

t

and V'" have representation as indicated in Remark 4.2. then there follows that

To 0

W •.• <; ker [;:

~].

W'., <; ker [;:

~]

and (W •••

~ W~.)

<; ker ;;

~

T3

0

So there exist matrices D 13, D 23, Y 3, E 31, E 32 and U 3 of suitable dimensions such that

and

Hence, the matrices D_{13• D 23•} Y_{3• E 3I• E32•} and U₃ are such that

.

[X~

XI 1m 0 0

with DI3+D23

=

X3 and E31 +E32

=

T 3.

Let D₁.D₂.E_I.E₂ E RnXII and R,pT E R"XII

D I [X 0' X I'X2,X3]

=

[Xo,X I.O.D 13].

R [XO,Xl>X2.X3]

=

[O.O.O'Y3]' To To To _To Tl Tl TI ₀ T2 E1= 0 T2 E 2= T2 T3 E3l T3 E32 and be matrices detennined by

D:dX o,X I.X2,X3] = _{[X o,0.K2,D 23],}

To ₀

Tl 0

and

T2 p= 0

T3 U3

It is clear that (D1-I)[Xo,Xtl

=

0, _{(D 2-I)[Xo,X2}J

=

0 and

(D 1+ D z-/) [X o,X l>X 2,X3]

=

[Xo.O,O,O] which means that (D 1-/)S

r

=

{OJ.

(D2

-/)St

=

{OJ and (D}+D 2-/)S*

=

Sr (1

st.

Hence, (DI,Dz) E

<P(st ,st

.S*).

18

-which imply T(EID1+PR)X = 0 and T(EzDz+PR)X = O.

In its tum the latter two expressions imply that for instance

o

=

[~:]

(E,D, +PR)X

=

[~:]

D,X which means that D,S' c;;

vt.

Analogously it can be shown that D~* !.: Vi, E IS

t

!.: V* and E₂Si !.: V*. Also there

fol-o 0 0

lows that Q_{TT(E zD 2+PR)X Qx}

=

T(E2D2+PR)X = O. Recall that _{D 2X}

=

X -D1X,

o 0 0 0 0 .

TE2

=

TEI-T, RX = RX and TP

=

_{TP. Then T(E 2D 2+PR)X}

=

=

T (E}D 1

+

PR)X

+

T (I - E 1-D I)X, Hence, T (D 1

+

E 1-/)X

=

0 which means that

(D }

+

E 1 -l)S

*

!.: V*.

Finally we recall that the subspaces WI" and W 2,e are A, -invariant from which it is immedi-ate that

In tum these two inclusions are equivalent to

and

[D2X] [N M] [CDz)(] [TEz,-TP]Ae -RX

=

TEzADz)(

+

[TE 2B,-TP] L K -RX

=

O.

Observe that the matrix

[~ ~

] is a common solution to two linear matrix equations. There-fore, by Theorem 3.1, it follows that

A

1 ker

C

I !.: im.8 1 where we have denoted

[

TE lAD IX 0

1

[TE 1B TP

1 - _

[CD IX CD

iX

1

Al

=

0 _-TE2AD

iX'

B1

=

TE2B -TP and C1 - RX -RX .

As in the (H)-part of the proof of Theorem 4.1 let

-

[I

0

1 -

[I

QT

1

U 1

=

Qx I and VI

=

0 I .

Then there holds that

A

2 ker

C

2 !.: im.8 2 where

A

z

=

VIA

16

I'

B

z

=

V

1.8 1 and

C

z

=

C

1

6

1·

[ T(AD1+E1A:A)X -TE2AD2X] A2

=

-TE~~ -TE2AD~'

__ [CX

CD2Xl C2 - 0 -RX . [ T (AD 1

+

E lA -A)X

1

Now note that -:-TE2ADiX kerCX ~

A

_2kerC\ ~

imB

2 from which it is

immedi-ate that T (AD 1

+

E lA - A )X ker CX ~ im TB . The latter means that

(AD 1

+

E lA - A )(S

*

( l ker C) ~ V*

+

im B .

- 20-S. Conclusions and Remarks

U.

In this paper we studied systems that apart from a control input and a measurement out-put have two exogenous inout-puts and two exogenous outout-puts.

For this kind of systems we derived verifiable 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.

U.

To end this section we shall give a conceptual algorithm that, if it exists, determines a compensator 1:_c that achieves non interaction.

To that extent let the system 1: be given.

Then the algorithm consists of the following steps where we use the notation as introduced in Remark 4.2.

(1) Calculate the subspaces S

t,

S! , S *, V

t,

V! and V*, for instance by means of the algorithms mentioned in Section 3, and let these subspaces have representations as indi-cated in Remark 4.2.

(2) Check whether or not there exist matrices D 1> D 2, Eland E 2 such that the conditions of

Theorem 4.1 are fulfilled. See Remark 4.2. If these matrices do not exist, then (NICPM) is not solvable. If they do exist the algorithm continues as follows.

(3) Determine matrices P and R as indicated in the proof of the (H)-part of Theorem 4.1.

(4) Let A" B" Cg , G 1.g' G2.,' H I., and H 2.g be matrices and let S 1.,' S 2". V 1.g and V 2,g be linear subspaces as described in the proof of the (if)-part of Theorem 4.1.

(5) Compute matrices Fg, Jg and Ng such that (Ag +BgFg)Vi.g ~ Vi,g (i = 1.2),

(Ag +JgCg)Si,g ~ Si,g (i

=

1,2) and (Ag +BgN,Cg)Si,g ~ Vi,g (i

=

1,2).

The computation of for instance N_g can be performed as follows. Let ~ (i

=

1,2) and f; (i

=

1,2) be matrices such that im~ = Si,g (i

=

1,2) and kerf;

=

Vi,g (i

=

1,2). Then it suffices to compute Ng such that f;Ag~

+

f;BgNgCg~ = 0 (i = 1.2), Using Kronecker products the latter two linear matrix equations can be written as one linear equation that can be solved using standard techniques. See also Van der Woude [7]. Next, rearrangement of the obtained solution provides N_{g •}

Similar remarks can be made with respect to the computation of Fg and Jg .

~ D~ne~=~+~~+~~-~~~.~=~~-~.~=~-~~ ~

partition

(7) Define

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

22

-References

[1] M.L.J. Hautus and M. Heyman, 'New result on linear feedback decoupling', Proc. 4th Intern. Conf. Anal. & Opt. Systems, Versailles Lecture Notes in Control and Inf. Sci-ences, 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] 1.M. Schumacher, 'Dynamic Feedback in Finite and Infinite Dimensional Linear Sys-tems', Math. Cent. Tracts 143, Amsterdam, 1981.

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

[5] J.e. Willems, 'Almost non interacting control design using dynamic state feedback', Proc. 4th Intern. Conf. Anal. & Opt. Systems, Versailles Lecture Notes in Control and Inf. Sci-ences, 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] 1.W. van der Woude, 'Almost non interacting control by measurement feedback', System & Control Letters, Vol. 9, pp. 7-16.

[8] P. Lancaster and M. Tismenetsky, 'The Theory of Matrices', 2nd ed., Academic Press, Orlando, 1985.