Disturbance decoupling and output stabilization by measurement feedback : a combined approach

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Woude, van der, J. W. (1986). Disturbance decoupling and output stabilization by measurement feedback : a combined approach. (Memorandum COSOR; Vol. 8607). Technische Universiteit Eindhoven.

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EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of Mathematics and Computing Science

Memorandum COSOR 86-07

Disturbance decoupling and output stabilization by measurement feedback

a combined approach

by

J.W. van der Woude

Eindhoven, The Netherlands Mei 1986

DISTURBANCE DECOUPLING AND OUTPUT STABILIZATION BY MEASUREMENT

Abstract.

FEEDBACK : A COMBINED APPROACH

by

J.W. van der Woude

Eindhoven University of Technology

Department of Mathematics and Computing Science P.O. Box 513, 5600 MB Eindhoven, The Netherlands

In this paper we shall solve a very general control problem in the context of the geometric approach to linear multivariable control theory.

The problem to be considered will be general in the sense that it contains as a special case not only the well-known disturbance decoupling problem by measurement feedback (cf. Schumacher [4J) and the problem of stabilizing the output with respect to the disturbances by means of a state feedback

(cf. Hautus [3J), but for instance also the extension of the latter problem to the situation in which only the use of measurement feedback is allowed.

Furthermore, in this paper connections with and extensions of recent work of Trentelman [5J will be given.

In the paper a fruitful use will be made of a merge of concepts arising from the geometric approach to control theory on the one hand and the frequency domain approach to control theory on the other hand.

2

-1. Introduction.

In this paper we shall consider an extension of the disturbance decoupling problem with measurement feedback (DDPM).

The latter problem deals with the situation in which by means of a measure-ment feedback compensator we want to achieve decoupling between exogeneous disturbances entering the system and external, or to-be-controlled, outputs leaving the system. This problem has been studied thoroughly and is now well-understood (cf. Willems & Commault [6]. Schumacher [4], Akashi & Imai [1]). In the present paper we are concerned with situations in which the above-mentioned problem may not be solvable. That is, there may not exist a measure-ment feedback compensator such that the exogeneous disturbances are decoupled from the external outputs. In such a case a natural question to ask is the following: does there exists a part of the exogeneous disturbances that can be decoupled from the external outputs by means of an suitable measurement feedback compensator, while the remaining part of the exogeneous disturbances only influences the to-be-controlled outputs in a stable sense.

In the latter context, it is also possible to consider the dual problem in which the to-be-controlled outputs are assumed to be decomposed into two parts. The question that arises in that case is whether it is possible to find a

measurement feedback compensator such that the first part of the to-be-controlled outputs remains unaffected by the exogeneous disturbances entering the system, while the second part of the controlled outputs only depends on the incoming disturbances in a stable fashion.

In this paper we shall consider a combination of the two above mentioned con-trol problems. That is, we shall assume that both the exogeneous disturbance as well as the external outputs are decomposed into two parts.

The maLn problem of this paper can then be formulated as follows :

is it possible to find a measurement feedback compensator such that the two control problems mentioned above are solved, i.e. such that the first part of the exogeneous disturbances is decoupled from the external outputs, the first part of the external outputs is decoupled from the exogeneous distur-bances, the second part of the exogeneous disturbances affects the external outputs in a stable fashion and the second part of the external outputs only depends in a stable manner on the exogeneous disturbances.

It may be clear that the disturbance decoupling problem with measurement feed-back can be considered as a special case of our problem. Moreover, in the sequel it will become clear that our problem also generalizes the problem of stabilizing the output with respect to disturbances, (OSDP), as considered in Hautus [3J, and the disturbance decoupling problem with output stabilization,

(DDPOS), as considered in Trentelman [5J. We note that in both of these pro-blems only state feedback was allowed.

The outline of this paper will be as follows.

In section 2 the above motivated problem is stated Ln a mathematically more rigorous formulation. Furthermore, section 2 will contain the notation that we shall use in this paper. In section 3 and 4 some important special cases of our problem are considered. Section 5 will contain some preliminary results that will be needed for the solution of our problem. In section 6 necessary and sufficient conditions for the solvability of our problem will be given. In section 7 we shall consider the extensions towards measurement feedback of the problems (OSDP) and (DDPOS) as mentioned above.

In section 8 we reconsider the main problem of this paper with the additional requirement of internal stability.

4

-2. Mathematical problem formulation.

In this paper we consider the linear time invariant system given by

.

x y A x + B u + G 1d1 + G2d2 C x

withxinX;:: IRn, the state, u ~n

U

;:: IR , the contro m I ' ~nput,

( 1 a) (1b) (1 c) . q1 d 1 ~n V 1 ;:: IR and q2 d

2 in V2 ;:: IR , the disturbance inputs, y in

Y

=

IR

P,

r

1

the measurement, zl ~n 21 - IR and z2 in 22 ~

r Z

IR , the to-be-controlled outputs.

A,B,C,G₁,G₂,H₁ and HZ are real matrices of appropriate dimensions. We shall denote the system described by (la), (lb) and (lc) as

I

(A,B,C,G₁,G₂,H₁,H₂)·

Throughout this paper we shall assume that

Further on, we shall argue that ~n the context of the problem treated in

this paper assumption (2) is no restriction. and therefore can be made without harming the problem formulation.

The reason for making assumption (2) is to keep formulations in the remainder of this paper compact.

See also Trentelman [5], where a similar assumption is made.

Suppose that the linear system as given 1n (la), (lb) and (lc) is con-trolled by means of a feedback compensator of the form

.

w Kw+Ly,u M w + Ny, (3)

k

with w in

W

=

IR , the state of the compensator. We shall denote a compen-sator given by (3) as Lfb(K,L,M,N). Application of this compensator results in the following closed loop system

.

x =

.

w A + BNC Bt-I LC K x + w z

= [

H2 ' 2

which we shall write compactly as

·e Ae e Ge _d 1 + Ge _{d 2 '} x = x + 1 2

Let

He e He e zl ₁ x , z2 ₂ x x

[

A + BNC BM

1 '

where x e , Ae LC K W and H7 ₁ [ H. ₁

a ] ,

i = 1,2 • (4a) (4b) (Sa) (5b)

:i

1

e _i 1,2 , G· = = 1

6

-We shall denote the system given in (Sa) and (Sb) as

We are now in position to formulate the main problem that will be considered in this paper.

Problem 1.

Given a linear system I(A,B,C,G

1,G2,H1,H2), with im G1 C 1m G2 and

ker Hl ~ ker H

2, find a measurement feedback compensator Ifb(K,L,M,N) such

. \ e e e e e

that the resultlng closed loop system Lct(A ,G₁,G₂,H

1,H2) satisfies: (i) He 1 (sl _ A e )-l _Ge 1 0 (ii) He (sl _ Ae)-l Ge

₌

0 2 1 (iii) ~ (sl _ Ae)-l Ge

₌

0 1 2

(iv) He (sl _ Ae )-l Ge has no poles outside

¢ .

2 2 g

Here

¢

denotes a given subset of the complex plane

¢

that is symmetric with g

respect to the real axis and that contains at least one point on the real aXls.

We shall now argue that the assumption made ln (2) in the context of problem lS no restriction. Therefore we assume that (2) may not hold and we consider matrices G

3 and H3 such that

(i) 1m G₁ + im G

2 = 1m G3 ' (ii) ker Hl

n

ker H2 = ker H3 •

Clearly we now have that im G₁ c im G₃ and ker H1 ~ ker H3 •

We may now state the following equivalence.

Problem 1 is solvable for the system given by L(A,B,C,G₁,G₂,H₁,H₂) if and only if

problem 1 is solvable for the system given by L(A,B,C,G_t,G₃,H"H₃).

The proof of this equivalence is straightforward using the fomulation of problem 1 as stated above.

It may now be clear that if we are considering a system such that assumption (2) .does not hold it is always possible, by a suitable redefinition of the disturbances and the to-be-controlled outputs, to obtain a system such that

(2) does hold. Therefore, in the context of problem 1, assumption (2) can be made without loss of generality.

In the remainder of this section we shall introduce some notations we use in this paper. Furthermore we shall recall some important concepts from geometric control theory together with their frequency domain descriptions.

1. Throughout this paper we shall use lower case letters for vectors, capitals for matrices and linear mappings, and scripts for linear subspaces and vector spaces. Linear mappings will be identified with their matrix representation.

The kernel of a mapping M shall be denoted by ker M, its image by ~m M and

its spectrum by a(M).

8

-If

V

is a linear subspace of the linear space X satisfying

MV

c

V,

where M is a mapping from X into itself, then

MIV

will denote the restriction of the mapping M to

V,

and

M:X/V

will denote the mapping induced by M in the factor space

X/V.

If V is a subspace of X then

V~

will denote the orthogonal complement of V in X.

2. Let X be a real n-dimensional space and let K be a subspace of X. We will denote the set of all n-vectors whose components are proper (respectively strictly proper) rational functions with real coefficients as X(s) (respec-tively X+(s». We will denote by K(s) (respec(respec-tively K+(s» the space of all elements ~(s) of X(s) (respectively X+(s» with the property that ~(s) E K for all s.

As mentioned before, in this paper

¢

will denote a given subset of the

g

complex plane

¢

that is symmetric with respect to the real aX1S and that

contains at least one point on the real axis. We shall call a rational function stable if the function has no poles outside

¢ .

g

3. We shall now recall some basic facts from geometric control theory to-gether with some frequency domain concepts.

Consider the dynamical system given by

I

x

=

A x + B u , Y C x , with state space X, control space U and measurement space

Y.

(i) A linear subspace V in X is called a controlled invariant or (A,B)-invariant subspace if AV c V+ im B. It is well-kno\vu that the latter sub-space inclusion is equivalent to the existence of a mapping F: X + U, defining a state feedback u

=

F x for

I,

such that (A + BF)

V

c

V.

A subspace

V

in X is called a stabilizability subspace if there exists a mapping F: X ~ U such that (A + BF)

V

c

V

and a(A + BF I

V)

c

¢ .

g

*

If K is a linear subspace of

X

then V (K) will denote the largest controlled invariant subspace in

K.

The largest stabilizability subspace in

K

will be

*

denoted by

V (K).

g Clearly

V*(K)

c

V*(K)

c

K

and if g

K1

c

K2

then

V*(K

₁

)

c

V*(K

2

)

and

K1 and K2 are linear subspaces in

X

such that

*

*

V g (K

1) c V g (K2) •

(ii) A linear subspace S in

X

is called a conditioned invariant or (C,A)-invariant subspace if A(S

n

ker C) c

S.

This is equivalent to the existence of a linear mapping T:

Y

~

X,

defining an output injection for

I.

such that

(A + TC)

S

c

S.

A linear subspace

S

is called a detectability subspace if there exists a linear mapping T:

Y

~

X

such that (A + TC)

S

c Sand

a(A + TC :

X I S)

c

¢ .

g

*

If L is a linear subspace in X then S (L) will denote the smallest conditioned invariant subspace that contains

L,

and

S*CL)

will denote the smallest

detecta-g

bility subspace containing L.

It is clear that

L c

S*(L)

c

S*CL)

g

*

*

and

SgCL

₁) c

SgCL2).

(iii) We shall denote the reachable subspace in X, i.e.

im B + A im B + ... + An-1 im B , by

<

A I im B > • and the unobservable

n-1

subspace ker C

n

ker CA

n ... n

ker CA ,by

<

ker C

I

A> •

(iv) Let pCs) := det (A-sI) be decomposed as pes)

=

Pg(s) Pb(s) where Pg(s) has only zeros in ¢g and Pb(s) has no zeros in ¢g'

- 10

-We shall denote Xg(A)

:=

ker Pg(A) and Xb(A) := ker Pb(A) , and we note that

X

_g (A), respectively Xb(A),is the span of the generalized eigenvectors corre-_. sponding to the eigenvalues in

¢ ,

respectively in the complement of

¢

in

¢.

g g

(v) In this paper we shall use the frequency domain concept of (~,oo) pairs as introduced in Hautus [3].

A given vector x in

X

is said to have a (~,w) respresentation if there exist strictly proper rational functions ~(s) in X+(s) and oo(s) in U+(s) such that

x

=

(sI-A) ~(s) - Boo(s).

3. Disturbance decoupling with output stabilization.

In this section, we shall consider a special case of our main problem.

The results of this section will be needed to derive necessary and sufficient conditions for the solvability of problem 1.

Consider the following linear system :

x

=

A x + B u + G d , zl

with state space

X

and control space U.

As explained in section 2 we shall assume that ker Hl ~ ker H2 ' 1.e! we consider z2 to be an 'enlargement' of zl'

The disturbance decoupling problem with output stabilization (DDPOS), (cf. Trentelman [5]), consists of finding a mapping F:

X

~ U, defining a state feedback u

=

F x, such that

H (sI-(A + BF»-l G is a stable rational function.

It turns out that the following subspace is crucial to the solvability of (DDPOS) (cf. Trentelman [5J).

Definition 3.1. Vg(ker H1 ' ker H₂) denotes the set of points in

X

for which there exists a (~,w) representation such that H1 ~(s)

=

0 (~(s) E ker H1 for all s), and H2 ~(s) is stable.

The following results now hold.

~,

*

Theorem 3.2. Vg(ker H1 ' ker H

2) = Vg(ker H1) + V (ker H2). Theorem 3.3. There exists a mapping F:

X

+ U such that

(i) (A + BF) Vg(ker Hl ' ker H₂) c Vg(ker Hl • ker HZ) , (ii) (A + BF)

V

*

(ker H

*

2) C

V

(ker H2) ,

(iii) _a«A + BF): Vg(ker H1 • ker HZ) / V (ker HZ»

*

C ¢g'

For the proofs of these theorems we refer to'Trentelman [5J. We may now prove the next theorem.

Theorem The following statements are equivalent.

(i) (DDPOS) is solvable.

(ii) im G C Vg(ker Hl • ker HZ)'

(iii) There exist strictly proper rational matrices Xes) and D(s) such that (sI-A) Xes) - B D(s)

=

G,

Hl xes)

=

0 and HZ xes) is stable.

(iv) There exists a strictly proper rational matrix W(s) such that

-1 -1

H

1(sI-A) B W(S) - H1(sI-A) G = 0 •

-1 1

H₂(sI-A) B W(s) - HZ(SI-A)- G is stable. Proof.

(i)

_<=>

_{(ii) See Trentelman [5J.} (ii)

<=>

(iii) By definition 3.1.

(i)

=>

(iv) Define W(s) := F(sI-(A + BF»-l G•

(iv)

=>

(iii) Define D(s) := -W(s) and

1Z

-Remark 3.5. From theorem 3.3 (i) it follows that Vg(ker Hl ' ker HZ) is an (A,B)-invariant subspace.

*

Remark 3.6. Note that

V

(ker HZ) c Vg(ker Hl • ker H₂) c ker Hl and that

*

V

(ker H₂) c ker H2 c ker Hl •

Take the mapping F: X ~ U as defined in theorem 3.3, and decompose the state

Xl i X

_z

=

Vg(ker Hl ' ker H

2). Furthermore, choose a basis in the state space adapted to this decomposition. With respect to this basis the matrices

(A + BF), Hl and HZ have the following form.

A11 A12 A₁₃

1

(A + BF) 0 _A22 A 23 ' H1 [0 0 H13

J ,

HZ

J

0 0 _A33

Note that due to theorem 3.3 (iii), o(A

22) C ¢g'

Therefore, with respect to this basis we have the following.

H 1(sI-(A + BF»-l H 2(sI-(A + BF»-l [0 [0

o

[0 H22 _H23]·

where P3(s) and Q3(s) are strictly proper rational matrices and Q2(s) is a stable strictly proper rational matrix.

4. A dual problem.

The objective of this section 1S to solve the dual of (DDPOS) as studied 1n the previous section. The results of this section may be obtained simply by dualizing the results of section 3.

It is possible to give an interpretation ~n terms of observers of the

problem considered in this section. However, since this subject is beyond the scope of this paper it is omitted.

The system considered in this section is given by

.

x

c

x , z H x ,

with state space X and measurement space

Y.

As explained in section 2 we shall assume that im G₁ c: im G₂•

The problem considered in this section consists of finding a mapping T:

Y

~ X, defining an output injection for

I,

such that

H(sI-(A + _{TC»-l G2} 1S stable.

The next definition is the result of dualizing definition 3.1. Definition 4.1. Sg(im G₁ ' im _GZ)

=

(Vg(ker

Vg(ker

G~

, ker

G~)

is computed relative AT

GT _{1 ' ker G2» , where} T .i and CT.

Note that Sg(im G₁ ' im G₂) depends only on A,C,G_{1 and G2 •}

The following theorems may now be obtained from theorem 3.Z, 3.3 and 3.4 by pure dualization.

*

* .

Theorem 4.Z. Sg(im G₁ ' im G_Z)

=

Sg(im G₁)

n

S

(~m _G2). Theorem 4.3. There exists a mapping T:

Y

~ X such that

(i) _{(A + TC) Sg(im G}

1 • im GZ) c: Sg(im G1 ' im GZ) ,

(ii) (A + TC)

S

*

(im G

*

Z) c:

S

(im GZ) ,

(iii) _{o«A + TC): S}

*

_{(im G}

- 14

-Theorem 4.4. The following statements are equivalent.

(i) The problem of this section LS solvable.

(ii) Ker H ~ Sg(im G

1 ' im G2).

(iii) 'l'nere exist strictly proper rational matrices xes) and yes) such that Xes) (sI-A) - yes) C

=

H,

Xes) G₁

=

0 and xes) G

2 is stable.

(iv) There exists a strictly proper rational matrix W(s) such that W(s) W(s) -1 C(sl-A) G 1 -1 C(sl-A) G 2 -1 - H(sl-A) G -1 - H(sl-A) G 1 = 0, 2 is stable.

Remark 4.5. From theorem 4.3 (i) it follows that Sg(im G₁ ' im _G2) (C,A)-invariant subspace.

*

Remark 4.6. Note that im G₁ c Sg(im G₁ ' im G₂) c S (im G₂) and

*

im G

1 c im G2 c S (im G2).

is a

Take any mapping T: Y ~ X as described in theorem 4.3, and decompose the state space X as X = Xl ~ X₂ ~

_X3,

such that Xl Sg(im G₁ ' im _G2) and

*

Xl ~ X₂

=

S (im G

2). In the same way as was done in remark 3.6 we may prove

that on a basis adapted to the above state space decomposition we have

R1 (s) (i) (sl-(A + TC»-1_G1

=

0

o

U 1 (5) (ii) (sI-(A + TC» -l_G2

=

U 2 (5)

o

J

where R₁(s) and U₁(s) are strictly proper rational matrices and U

2(s) is a stable strictly proper rational matrix.

5. Preliminaries.

The results of this section will be instrumental in the proof of problem as stated in section 2, but are also interesting for their own right. Consider the dynamical system

x

=

A x + B u + G d • y

=

C x • z

=

H x •

And assume that

I

is controlled by means of the feedback controller given by

w (A + BF + TC - Bwe) w + (BW - T) y

u = (F - We) w +

W Y ,

resulting ~n the closed loop system given by

Id,

_x 'e

₌

_Ae e e _He e x + G d , z x , where

=[

TC - Bwe]

'\ x

[A

+ Bwe BF - BWC G J e _Ae Ge

=

x

,

= w BWC - 'I'C A + BF + 0 and He =. [H 0].

- 16

-Proposition 5.1.

(i) a(Ae) = a(A + BF) U o(A + TC) where

'U'

denotes the disjoint un10n (cf. Wonham [7]).

(H) H (sI-A) e e -1 G e

=

H(sI-(A + BF» -1 G + H(sI-(A + TC» -1 G

- H(sI-(A + BF})-l(SI-(A + BWC» (sI-(A + TC})-l G.

Proof.

Note that for any regular matrix S in IR2nx2n we have

In particular with -1 and S

[ II °1]

A + BF BF - BWC we obtain

.

) o(A + BF) U o(A + TC) ,

a

A + TC

which proves part (i).

Furthermore we have that HeS- 1(sI-S AeS- 1)-1 S Ge

=

= [H 0] [ A + BF BF - BWC)

J

(sI-

-1

=

H(sI-(A + BF»

G-a

A + TC H(sI-(A + BF»-l(BF - BWC) (sI-(A + _{TC»-l G .}

Part (ii) may now be proven using the fact that

In the following proposition a connection is made between the input-output behavior of a linear dynamical system and the external behavior of the autonomous part of the system with respect to the initial conditions. For that purpose we consider the linear system

x

=

A x + B u , y C x ,

with state space

X.

Then we may prove.

Proposition 5.2.

(i) C(sI-A)-l B 0 if and only if for every vector p in

<

A I im B

>

C(sI-A)-l p

=

o.

(ii) C(sI-A)-l B 1S stable if and only if for every vector p in

<

A

I

im B

>:

C(SI-A)-l p is stable.

Proof.

Since

<

A I im B

>

1m B + A im B + ••••••• + A n-1 im B the (if)-part of (i) as well as of (ii) is obvious.

(Only if).

1 ~ CAi -1B

Note that C(sI-A)- B

I

i

from which it is immediate that i=l s

s C(sI-A)-l B - CB = C(sI-A)-lAB and that C(sI-A)-l B = 0 if and only if for

i-1

all i ~ 1 : CA B = O.

-1 -1

Therefore, we may conclude that if C(sI-A) B

=

0 then C(sI-A) AB

=

O. It is also easy to see that if C(SI-A)-l B is stable then C(SI-A)-l AB is stable. Repeated use of the previous arguments yields that

if

C(sI-A)-l B = 0 then for all k ~ 0 : C(sI-A)-l AkB = O. and if C(sI-A)-l Bis stable then for all k ~ 0 : C(sI-A)-l AkB is stable.

- 18

-By the definition of

<

A

I

im B

>

and the fact that the sum of stable rational matrices is stableJ the proof of the (only if)-part is

completed.

Again as 1n proposition 5.2 we consider the linear dynamical system

\

.

l x

=

A x + B u , Y

=

C x ,

with state space

X,

control space

U

and measurement space

Y.

We assume that Sl and S2 are (C,A)-invariant subspaces in

X,

and that

V

1 and V2 are (A,B)-invariant subspaces in X satisfying S, c S2 c V

2 and S1 c V1 c V2 • We may now state the following.

Proposition 5.3. There exists a mapping W:

Y

~ U, defining a static measure-ment feedback u = W Y for

I,

such that (A + BWC) S1 c V

1 and

A + BWC) S2 c V 2 •

Proof.

See Schumacher [4J, lemma 3.6.

6. The ma1n problem.

We are now in position to state the main result of this paper.

The result provides necessary and sufficient conditions for the solvability of problem 1 as stated in section 2.

Theorem 6.1.

Problem 1 is solvable

if and only if _{S (im G2)}

*

c Vg(ker H, , ker H₂) and Sg(im G₁ ' im G₂) c V*(ker H

Proof. (Only if).

Assume that there are matrices K,L,M and N defining a measurement feedback compensator which solves problem 1. Therefore we have that

o ,

o ,

(6)

Define the subspace Ve in

X

$ W as Ve :=

<

Ae

I

im G~

> .

From proposition 5.2 and (6) it follows that for every vector xe 1n Ve

(7)

Define subspaces S and V in

X

as follows.

S is the set of vectors 1n

X

such that

[ xo

1

is a vector in

V

e•

V

1S the set of vectors x in

X

for which there exists a vector w

in (.J such that

: 1

_{i, a vector ,n}

v

e_•

Clearly

S

c

V.

It may be proven that

S

is a (C,A)-invariant subspace and that

V

is an (A,B)-invariant subspace (cf. Schumacher [4J).

Since Ge

2 it is clear that 1m G2 e S c

V.

And therefore we have

*

*

im G

2 c

S

(im G2) c

V,

for

S

(im G2) 1S the smallest (C,A)-subspace containing im G

20

-Take a vector x in

V.

Then according to the definition of

V

there exists

a vector w in

W

such that

[ _Wx

1

Define the strictly proper rational vectors ~(s) and A(s) by

Then we have x = (sl-A) ~(s) - B(NC~(s) + MA(s», and since H~ = [Hi

OJ ,

i

=

1,2, by (7) it follows that H1 ~(s) = 0 and H2 ~(s) is stable.

By definition 3.1 we may conclude that x is 1n Vg(ker H1 ' ker H₂).

*

Consequently we have S (im G₂) e S c V c Vg(ker H1 ' ker H₂).

*

By dual reasoning we may derive Sg(im G

1 ' im G2) c

V

(ker H2).

(if) During this proof we denote = V*(ker H₂), sg

=

Sg(im G 1 ' im

vg

=

_{Vg(ker H1 ' ker H2}}

*

*

G 2) and S = S (im G2).

V*

=

,

Therefore, we now have that sg c

V*

and S* c vg, from which we shall derive a measurement feedback compensator that solves problem 1.

. g

*

*

g

SJ.nce Sand S are (C ,A)-invariant subspaces and V and V are (A,B)-invariant subspaces satisfying sg c

v*

c Vg and sg c S* c

V

g by proposition 5.3 there exists a matrix W such that (A + BWC) sg c

V*

and (A + BWC) S* c

vg.

Take the matrix F as indicated in theorem 3.3 and the matrix T as indicated

in theorem 4.3, and define

K := A + BF + TC

-

BWC

,

L := BW

-

T M := F

-

\JC

,

We now claim that the matrices {K,L,M,N} constitute a measurement feedback compensator Ifb(K,L,M,N) that solves problem 1.

\

Therefore we remark the following.

1. Note that with Lfb(K,L,M,N) as defined above the closed loop system becomes

1 [

Xw'

]+ [

GOl ] A + BF + TC - BWC BF - BWC

• Zz -

[HZ OJ [ : ]

By proposition 5.2 (ii) we now know that in the closed loop system the transfer function between disturbances d. and to-be-controlled output z. equals

J 1

H.(sI-(A + BF»-l G. + H.(sI-(A + TC»-l G.

1 J 1 J

-H.(sI-(A + BF»-l(sI-(A + BWC»(sI-(A + TC»-l G. where i,j

=

1,2.

1 J

2. Note that we have the following subspace inclusions

(i) V

*

c Vg c _{ker Hl} (ii) _V

*

_{c ker H2 c ker Hl} (iii) im G

1 c sg c

s*

(iv) im G

1 c 1m G

.

2 c

s*

Consider the following state space decomposition of X X

=

X 1 $ X2 $ X3 $ X₄ $ X5 $ X6 where Xl

=

sg

t Xl $ X 2 $ X3 =

V*

,

Xl $ X 2 $ X4 '"

s*

22

-By remark 3.6 it follows that

Hl (sI-(A + BF»-l :: [0 0 0 0 0 P 6 (s) ]

H

2(sI-(A + BF»-l =[0 0 0 Q4(s) Q5(s) Q6 (s) ]

where P6(s) and Q6(s) are strictly proper rational matrices and Q4(s) and Q5(s) are stable strictly proper rational matrices.

Analogous, by remark 4.6 it follows that on a basis adapted to the chosen state space decomposition

Rl (s)

o

o

o

o

o

U 1 (s) U 2(s)

o

U 4(s)

o

o

where R₁(S) and U

1(s) are strictly proper rational matrices, and U₂(s) and U

Furthermore, with respect to the chosen decomposition the matrices Gl ' G2 ' Hl and H2 may be written as

G_l1 G₁₂ 0 G_{22 Hl}

=

[0 0 0 0 0 H 16] 0 0 G 1 ' G2

=

0 G 42 0 0 _H2 [0 0 0 _{H24 H25} H 26] 0 0

The matrix A + BWC with respect to this decomposition becomes

All A12

*

A14

* *

A21 A22

*

A24

* *

A31 A32

*

A34

* *

0 _A42

_*

_A44

_{* *}

a

_A52

_*

_AS4

_*

_*

a

0

_*

0

_*

_*

where

'*'

denotes an unknown matrix block.

3. It ~s now easy to see that for (i,j) = (1,1), (1,2) and (2,1)

H.(sI-(A + BF»-l G. = 0

~ J , H.(sI-(A +TC»-l G. 1 J

=

0 and

H.(sI-(A + BF»-l(sI-(A +

~ BWC»(sI-(A + TC»-l G. _J

=

0 .

For (i,j) = (2,2) we have H₂(sl-(A + H 2(sI-(A + -1 -1 BF» G 2 = Q4(s)G42, H2(sI-(A + TC» G2 -1 -1 BF» (sl-(A + BWC»(sl-(A + TC» G 2

=

Q4(s)(-A₄₂U₂(s) + (sI-A₄₄)U 4(s» + QS(S)(-A52U2(s) - AS4U4(s» ,

24

-where Q4(s), Q5(s), U

2(s) and U4(s) are stable strictly proper rational matrices.

By proposition 5.1 we now may conclude that in the closed loop system the transfer function between the disturbance d. and the to-be-controlled

J

output z. equals zero, where (i,j)

=

(1,1), (l,Z) and (2,1). Furthermore,

~

by proposition 5.1, it is easy to see that the transfer function between disturbance d

_Z

and the to-be-controlled output

Zz

is a stable strictly proper rational function.

This completes the proof of theorem 6.1.

Remark 6.2. Note, as it should be, that the two subspace inclusions

appearing in theorem 6.1 are each others dual with respect to taking ortho-gonal complements and corresponding transposition.

Similar to the theorems 3.4 and 4.4 we can extend the results of theorem 6. 1 as follows.

Theorem 6.3. The following statements are equivalent.

(i) Problem 1 is solvable.

(ii) _S

*

_{(im G}

2) c Vg(ker H1 ' ker H2) and

Sg(im G₁ ' im G

Z) C V*(ker H2) •

(iii) There exists a strictly proper matrix Xes) and a proper matrix U(s) such that for (i,j)

=

(1,1), (1,2) and (2,1) H.X(s) G.

=

0,

1 J while H₂X(s) G₂ is stable, and Xes) and U(s) satisfy

(sl-A)

=

(sI-A) X(s)(sl-A) + B U(s)

c.

(iv) There exists a proper matrix T(s) such that for (i,j)

=

(1,1), (1,2) and (2,1) -1 Hi(sl-A) B T(s) -1 H₂(sI-A) B T(s) -1 C(sI-A) G. J -1 C(sl-A) G₂ -1 + H. (sl-A) G. 1- J =

° ,

and -1 + H 2(sI-A) G2 is stable.

7. Special cases of theorem of problem 1.

In this section we shall consider some special cases of problem 1.

(I) The problem of stabilizing the output with respect to disturbances using measurement feedback, abbreviated (OSDPM).

This problem, (OSDPM), can be formulated as follows. Given the linear system

.

x A x + B u + G d , y

ex,

z H x ,

determine a feedback controller given by

.

w K w + L y , u M w + Ny,

such that in the resulting closed loop system, compactly given by

·e

x He e x ,

(8)

This problem can be seen as a generalization of the problem of stabilizing the output with respect to disturbances using state feedback. The latter problem, abbreviated as (OSDP), was introduced and solved in Hautus [3] anp is formulated as follows.

Given a system described by (8), find a feedback u

=

F x such that 1n the resulting closed loop system H (sI-(A + _{BF)-1 G is stable.}

The problem considered in this paragraph, (OSDPM), can be obtained from problem 1 by setting G

1

=

0, G2

=

G, H1

=

0 and H2 ~ H. By theorem 6.1 we may conclude.

26

-Corollary 7.1.

(OSDPM) is solvable if and only if

*

S

(im G) c

V (X,

ker H) and

g

*

S ({O}, im G) c V (ker H).

g

Now it is well-known (cf. Schumacher Hence, by theorem 3.2, we have

V (X,

g

*

[4]) that V (X) =

<

A I im B

>

+ X (A).

g g

ker H)

=

V*(ker H) +

<

A I im B

>

+ X (A). g

In this formulation it ~s easy to prove that A

V (X,

ker H) c

V (X,

ker H).

g g

*

Using the fact that im G c

S

(im G) c

<

A

I

im G

>

we have

*

S (im G) c

V (X,

ker H) if and only if

g

im G c

V (X,

ker H) •

g

By dual reasoning it follows that

S

({O},

g

S

({O},

g

*

im G) c V (ker H) if and only if im G) c ker H.

Therefore we may condlude

Corollary 7.2.

(OSDPM) is solvable if and only if im G c

V (X,

ker H) and

g

S ({O}, im G) c ker H.

g

We note that im G c

V (X,

ker H) is necessary and sufficient for the

solva-g

Remark 7.3.

The remarkable fact shows up that, due to the A-invariance of

V (X,

ker H)

g

and S ({O}, im G), the solvability of (OSDPM) is equivalent to the

solva-g

bility of (OSDP) and (OSDP)*, where (OSDP)* is the dual of (OSDP) and is formulated as follows. Given the linear system described by (9), find ~ matrix I, defining an output injection for (9), such that in the resultiing closed loop system H(sI-(A + IC»-l G is stable.

Remark 7.4.

Note that due to the A-invariance of both

V (X,

ker H) and S ({O}, im

q)

in

g g

the construction as indicated in the proof of theorem 6.1 the mapping W:

Y

7 U

can taken to be zero.

i

I

Hence, we do not need a static measurement feedback u

=

W y, and consequently the transfer function of the measurement feedback controller solving (qSDPM)

is a strictly proper rational matrix.

Remark 7.5.

From theorem 6.3 a number of equivalent statements can be derived conce~ning

the solvability of (OSDPM). Furthermore, by the fact that the solvabililty

I

of (OSDPM) is equivalent to the solvability of both (OSDP) and (OSDP)* ~he

connection to theorems 4.3 and 4.4 will yield an additional number of state-ments equivalent to the solvability of (OSDPM).

(II) The second problem we shall consider in this section can be seen a~ a

I

generalization of the above-mentioned problem, (OSDPM), but also as an I

extens.ion of (DDPOS) as described in section 3.

For that reason we consider the following linear system

I

x

=

A x + B u + G d y

=

C x

,

_zl

=

H

1x , z2 H2x , where we assume that ker Hl => _{ker H2}

.

28

-The problem of stabilizing the output with respect to the disturbances by means of a measurement feedback (DDPOSM) now consists of finding a feedback compensator given by

.

w K w + L y , u

=

M w + N y

such that the resulting closed loop system, which may be written compactly by

satisfies

(DDPOSM) may be obtained from problem 1 by setting G₁ ; 0 and G₂ G. From theorem 6.1 we may conclude

Coro llary 7.6.

(DDPOSH) is solvablejif and only if

*

S (im G) c Vg(ker H1 ' ker H

2) and

*

Sg({O}, im G) C

V

(ker H

2).

As we showed in the previous paragraph we also have the following equivalence.

(DDPOSM) is solvable if and only if

*

S (im G) c V (ker H • ker H) and

g

*

But, since ~n general Vg(ker Hl ' ker H₂) is not A-invariant no further simplification is possible. Moreover, ~n constructing a measurement

feedback compensator as described in the proof of theorem 6.1, in general, we do need the mapping W:

Y

~ U, such that

*

(A + BWC)

S

(im G) c Vg(ker Hl ' ker HZ) and

*

(A + BWC) Sg{{O}, im G) c V (ker HZ) •

This implies that the measurement feedback controller solving (DDPOSM), ~n general, has a proper transfer function.

Remark 7.7.

Again, it is possible, by application of theorem 6.3, to obtain a number of statements equivalent to the solvability of (DDPOSM).

(III) We will conclude this section by considering the well-known disturbance decoupling problem by meaSurement feedback (DDPM) (cf. Willems

&

Commault [6], Schumacher [4J, Akashi

&

Imai [1]).

The latter problem may be obtained from problem 1 by setting G₁

=

G_Z

=

G and H1

=

H2 = H in the formulation of problem 1. Moreover, if we set in the proof of theorem 6.1 G₁ = G

2

=

G and H1

=

HZ

=

H, we obtain a new, straightforward and elegant proof of the following well-known equivalence :

Corollary 7.S.

*

'*

(DDPM) is solvable if and only if S (im G) c V (ker H).

Although (DDPM) is solved in a nice way nothing can be said with respect to internal stabilization or pole placement. Therefore, in the following section we shall discuss the extension of problem 1 to the case that in addition to problem 1 internal stability is required.

30

-8. Problem 1 with internal stabilization.

In this section we shall reconsider problem 1 as described in section 2. In addition to the mentioned requirements involving disturbance decoupling and output stabilization with respect to disturbances, we now also require

internal stabilization.

For that reason we make the following decomposition of the complex plane ¢, ¢f C ¢s C ¢. Where ¢f and ¢s are symmetric with respect to the real axis

and contain at least one point on the real axis. With respect to such a decomposition of the complex plane we call a rational function f-stable,

(resp. s-stable) if the rational function has no poles outside ¢f (resp. ¢s)'

The problem considered ~n this section now consists of the following.

Problem 2.

Given a linear system I{A,B,c,G

1,G2,H1,H2), as described in section 2, with

im G

1 C im G2 and ker Hl C ker H2, find matrices K,L,M,N defining a

measure-ment feedback compensator Lfb(K,L,M,N), as defined in section 2, such that

h 1 · 1 d 1 \' (e e e e e) . f'

t e resu t~ng c ose oop system Lct A ,G₁,G₂,H₁,H₂ sat~s ~es:

(i)

H:(sI-Ae)-l_G:

~ J

o

for (i,j) ( 1 , 1 ) , ( 1 ,2), and ( 2, 1) ,

(H) H;(Sl-Ae)-l G; 18 f-stable

(iii) (sI-Ae )-l is s-stable.

The solution of problem 2 closely resembles the solution of problem 1, as given in section,6, and will therefore not be given in detail.

Some elementary results, however, will be mentioned. Before doing this, some additional notation shall be introduced.

Consider the linear system

I

i

= A x + B u , Y C x, with state space

X, control space

U

and measurement space

Y.

Throughout this section we shall call the pair (A,B) s-stabilizable if for every A in ¢ / ¢s (the complement of ¢s in ¢) rank [A-Al,B]

=

n

(cf. Hautus [2J).

It is well-known that this rank condition is equivalent to the existence of a mapping F: X ~ U such that o(A + BF) C ¢ •

s

We shall call the pair (C,A) s-detectable if the pair (AT,CT) is s-stabilizable.

*

*

Given a linear subspace

K

in X, we call

Vf(K)

(resp.

Vs(K»

the largest sub-space

V

in

K

for which there exists a mapping F:

X

~

U

such that

(A + BF) V C V and a(A + BFIV) is in ¢f (resp. ¢s)' Given a linear subspace

.

*

*

L ~n

X,

Sf(L) (resp. Ss(L» denotes the smallest subspace S containing L for which there exists a mapping T:

Y

~

X

such that (A + TC) S c Sand

a(A + TC:X/S) is in ¢f (resp. ¢s)'

*

Another way to characterize V (K) is the following (cf. Hautus [3J). s

*

V (K) is the subspace of points in

X

for which there exists as-stable «(,w) s

representation with (s) in

K.

Here we note that a «(.w) representation is called s-stable if both (s) and w(s) are strictly proper s-stable rational functions.

We may now consider the problem of section 3 with the additional requirement of internal stabilization (DDPOS)'.

(DDPOS), : Given the linear system

.

x _A x + B u + G d , zl

with state space X and control space U and with the assumption ker Hl ~ ker

HZ'

find a state feedback u

=

F x such that :

3Z

-H,(SI-(A + BF»-'G = 0 ,

HZ(sI-(A + BF»-l G 1S f-stable and

(sI-(A + BF»-l is s-stable.

We shall proceed analogously as in section 3.

Definition

8.'.

Vfs(ker Hl ' ker HZ) is the set of points in

X

for which there exists a s-stable (~,w) representation with ~(s) in ker Hl and HZ~(s) is f-stable.

The following theorems generalize the theorems 3.Z, 3.3 and 3.4.

Theorem 8.Z.

Theorem 8.3.

Assume that (A,B) 1S s-stabilizable. Then there exists a state feedback u = F x such that

(i) (A + _{BF) Vfs(ker Hl ' ker HZ)} c Vfs(ker Hl ' ker HZ) ,

(ii) (A + _{BF) V (ker HZ)}

*

c _{Vs(ker H2) ,}

*

s

(iii) o(A + BF) c ¢ , _s

Theorem 8.4.

The following statements are equivalent.

(i) (DDPOS)' is solvable .

(ii) im G c Vfs(ker Hl ' ker H₂) , and (A,B) is s-stabilizable • (iii) There exists s-stable strictly proper rational matrices

X(s) and U(s) such that (s1-A) Xes) - B U(s) = G, H

1X(s)

=

0 and H2X(s) is f-stable and (A,B) is s-stabilizable •

(iv) (And if (H

2,A) is s-detectable) .

There exists a strictly proper s-stable rational matrix T(s) such that

-1 H 1(sI-A) B T(s) -1 - H 1(s1-A) G

=

0 , -1 H

2(sI-A) B T(s) - H2(SI-A)-l G is f-stable and (A,B) is s-stabilizable.

Remark 8.5.

The necessity of the detectability assumption in (iv) may be shown by means of an example (cf. Hautus [3]).

By pure dualization of the previous the problem mentioned in section 4 with the additional requirement of internal stabilization may be solved.

The result of this dualization will be omitted. We now come up with the main result of this section.

34

-Theorem 8.6.

Problem 2 described in this section 1S solvable if and only if

*

Ss (im G2) c Vfs(ker H, • ker H_{2) •}

.

*

Sfs(im G, , 1m G₂) c _{Vs(ker H2) ,}

(A,B) is s-stabilizable and (C,A) is s-detectable •

Where Sfs(im G

1 ' im G2)

=

(Vfs(ker

G~

ker

G~»~

with

Vfs(ker

G~

• ker

G~)

computed relative AT and CT.

Remark 8.7.

Analogous, as is done in section 6 a number of results concerning the solva-bility of problem 2 stated 1n frequency domain terms can be derived.

Furthermore it is possible to derive results involving (OSDPM) and (DDPOSM), as mentioned in section 7, with the additional requirement of internal stabili-zation. Let (OSPDM)',(resp. (DDPOS.t)') be the problem obtained from problem 2 by setting G

1

=

0, Hl = 0, G2

=

G and H2

=

H, (resp. G,

=

0, G2 = G).

Note that (OSDPM), (resp. (DDPOSM)') obtained in this sense is the extension of (OSDPM) (resp. (DDPOSM» to the case of additional internal stabilization. From theorem 8.6 we can conclude.

Corollary 8.7.

(OSDPM), is solvable if and only if

*

S

_s (im G) c Vfs(X, ker H)

,

Sfs({O}, 1m G)

.

c V

*

(ker H)

_•

s

(A,B) is s-stabilizable and (C,A) is s-detectable.

Note that corollary 8.7 is the generalization of corollary 7.1. In the same way corollary 7.2 was derived from corollary 7.1 we can obtain the following.

Corollary 8.8.

(OSDPM), is solvable if and only if

im G c Vfs(X, ker H) , Sfs({O}, im G) c ker H

(A,B) is s-stabilizable and (e,A) is s-detectable •

The following result is the generalization of corollary 7.6.

Corollary 8.9.

(DDPOSM), is solvable if and only if

*

Ss (im G) c Vfs(ker H1 ' ker HZ) ,

.

*

Sfs({O}, ~m G) c V (ker HZ) , (A,B) is s-stabilizable and (C,A) is s-detectable •

36

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[1J H. Akashi & M. Imai. 'Disturbance localization and output deadbeat control through an observer in discrete time linear multivariable systems'

IEEE T~ans. Automat. Contr., vol. AC-24, pp 621-627, 1979.

[2J M.L.J. Hautus. 'Stabilization, controllability and observability of linear autonomous systems'.

Nede~l. Akad. Wetensch. Proc. Se~. A 73, pp 448-455, 1970.

[3J M.L.J. Hautus. '(A,B)-invariant and stabilizability subspaces, a frequency domain description'.

Automatica, vol. 16, pp 703-707, 1980.

[4J J.M. Schumacher. Dynamic Feedback in Finite and Infinite Dimensional

Linea~ Systems.

Mathematical Centre Tracts 143, Amsterdam, 1981.

[5] H.L. Trentelman, 'Almost Disturbance Decoupling with Bounded Peaking'. To appear in SIAM J. Cont~.

[6J J.C. Willems

&

C. Commault. 'Disturbance Decoupling by Measurement Feedback with Stability or Pole Placement'.

SIAM

J.

Cont~.vol. 19, pp 490-504, 1981.

[7J W.M. Wonham. Linea~ Multiva~iable Cont~ol: A Geomet~ic App~oach. (2-nd ed.), Springer-Verlag, New York, 1979.