Completion of the squares in the finite horizon $H^\infty$ control problem by measurement feedback

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Completion of the squares in the finite horizon $H^\infty$

control problem by measurement feedback

Citation for published version (APA):

Trentelman, H. L., & Stoorvogel, A. A. (1989). Completion of the squares in the finite horizon $H^\infty$ control problem by measurement feedback. (Memorandum COSOR; Vol. 8927). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1989

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EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of Mathematics and Computing Science

Memorandum COSOR 89-27

Completion of the squares in the finite horizon Hoo control problem by measurement feedback

H.L. Trentelman & A.A. Stoorvogel

Eindhoven, October 1989

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COMPLETION OF THE SQUARES IN THE

FINITE HORIZON

HOO

CONTROL

PROBLEM BY MEASUREMENT FEEDBACK

by

H.L. Trentelman& A.A. Stoorvogel

Department of Mathematics and Computing Science P.O. Box 513

5600 MB Eindhoven, The Netherlands

ABSTRACT

In this paper we study the finite horizon version of the standardHoo control problem by measure-ment feedback. Given a finite-dimensiona1linear, time-invariant system, together with a positive real numbery, we obtain necessary and sufficient conditions for the existence of a possibly time-varying dynamic compensator such that theL₂[O,T]-induced norm of the closed loop operator is smaller thany.These conditions are expressed in terms of a pair of quadratic differential inequali-ties. generalizing the well-known Riccati differential equations that were introduced recently in the context of finite horizonHoocontrol.

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-

2-1. THE FINITE HORIZON HooCONTROL PROBLEM

Consider the linear time-invariant system x(t)

=

Ax (t)

+

Bu (t)

+

Ew (t) ,

y(t)=Ctx(t)+ D\w(t) ,

z(t)=C2x(t)+D2u(t) ,

(1.1)

where x E /Rn is the state, U E /Rm the control input, WE /Rl an unknown disturbance input,

y E JRP the measured output and zE /Rq the output to be controlled. A, B, E, Ct ,D \ ,C2 and

D2 are constant real matrices of appropriate dimensions. In addition, we assume that some fixed

time interval [0,T] is given. We shall be concerned with the existence and construction of dynamic compensators of the form

p(t)=K(t)p(t)+L(t)y(t) ,

u(t) =M(t)p(t) +N(t)y(t) , (1.2)

whereK, L, M andNare real, matrix valued, continuous functions on [0,T].The feedback inter-connection of LandLF is the linear time-varying system Lcl described by

[

X(t)] _ [A +BN(t)C\ BM(t)] [X(t)] [E+BN(t)Dt]

pet) - _{L(t)C t} K(t) pet)

+

_{L(t)D t} wet) ,

[ X

(t)]

z(t)=(C2+D2N(t)C t D2M(t) pet) +D2N(t)D t wet) .

Let us denote the matrices appearing in these equations byAe(t), Ee(t), Ce(t) andDeCt), respec-tively. Obviously, if we put

x

(0)

=

0,p(0)

=

0, then the closed loop systemLel defines a convo-lution operatorGel:L~[0,T] ~ L~[0,T] given by

I

(Gclw)(t)=z(t)=

J

Ce(t) <I>eCt, 't)Ee('t) w('t)d't+DeCt) w(t) ,

o

where<De(t,t) is the transition matrix ofAe(t).Thus, the influence of disturbancesw E L~[0,T]

on the output z can be measured by the operator norm ofGel,giveninthe usual way by IIGFwll2

IIGelll :=sup { IIwll2 10*-W E L~[0,T]} .

Here,IIxll2 denotes theL2[O, Tjnorm of the functionx. The problem that we shall discuss in this

paper is the following:

giveny> 0, find necessary and sufficient conditions for the existence of a dynamic compensator LF such that IIGelli

<

y.

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(2.1)

(2.2) -

3-The problem as posed here will be referred to as thefinite horizonHoo control problem by meas-urement feedback. This problem was studied before in [61 and [21. In the latter rererencesit is however assumed that the rollowing conditions hold: DI is surjective, D2 is injective. In the

present paper we shall extend the results obtained in [6J and [2] to the case that01 andD2 are arbitrary.

2. QUADRATIC DIFFERENTIAL INEQUALITIES

A central role in our study of the problem posed is played by what we shall call the qua-dratic differential inequality. let 'Y>

°

be given. For any differentiable matrix function P:[0,T] ~ /Rnxn,defineFy(P):[0,T] ~ JR(n+m)x(n+m) by

[

P+A'P+PA+C2'C2+

~PEETP

PB+C2'D2]

Fy(P):= _{B'P+D 2'C 2} D 2 ' D 2 ·

IfF iP)(t)~

°

for all tE [0,T], then we shall say thatP satisfies the quadratic differential ine-quality (at y). Also a dual version of (2.1) will be important to us: for any differentiable

Q:[0,T] ~ /Rnxndefine Gy(Q): [0,T] ~ JR(n+p)x(n+p) by:

[

-Q

+AQ +QA' +EE'

+

~

QC 2'C 2Q QC 1' +ED1']

Gy(Q):= _{C 1Q +D}₁E' _{D1 D}1"

IfGy(Q)(t)~

°

forall t E [0,T] then we shall say thatQsatisfies the dual quadratic differential

inequality (aty). In the sequel let

G(s):=C 2(ls-Ar1B+D2 ,

H(s):=C1(ls -A)-IE +D1

denotc the open loop transfer matrices from uto z andw to y,respectively. Furthermore, denote by normrank(G) and normrank(H) the ranks of these transfer matriccs considered as matrices with entries in the field of real rational functions. We are now ready to state our main result:

THEOREM 2.1. Let'Y

>

0. The following two statements are equivalent: (i) There exists a time-varying dynamic compensatorLFsuch that IIGcllI

<

y,

(ii) There exist differentiable matrix functionsP andQ: [0,T] ~ JRnxn such that

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-4-(b) rankF y(P)(t)=nonnrank (G) \;ItE [0,T] , (2.4)

(c) Gy(Q)(t)~O\;ItE [O,T]andQ(O)=O, (2.5)

(d) rankG..,(Q)(t)=nonnrank(H) \;ItE [0,T] , (2.6)

(e)

Yl-

Q(t)P(t)is invertible \;ItE [0,T]. (2.7)

The aim of this paper is to outline the main steps and ideas involved in a proof of the latter theorem. For a more detailed discussion we would like to refer to[5].

It can be shown that, in general, ifF ..,(P)~

°

on[0,T] then

rankF..,cP)(t)~nonnrank(G) on [0,T]

and, likewise,ifG..,(Q)~ Oon[0,T] then

rankGy(Q)(t)~nonnrank(H)on lO,TJ

This means that the pair of conditions (2.3), (2.4) can be refonnulated as:Pis a rank-minimizing solution of the quadratic differential inequality at y, satisfying the end-condition P (T)

=

0. A similar restatement is valid for the conditions (2.5), (2.6).Itcan also beshown that ifP satisfies (2.3) and (2.4) then it is unique. Also, this unique solution turns out to be symmetric for all t E lO,T].The same holds forQsatisfying (2.5) and (2.6).

We will now show that for the special case thatD1andDz are assumed to be surjective and injective, respectively, our Theorem 2.1 specializes to the results obtained before in [6] and [2]. Indeed, ifD

_z

is injective then of course

nonnrank(G)

=

rankDz

=

m

Denote

Ry(P)

:=P

+A'P

+

PA

+

_{C z' C z}

+

~

PEE'P - (PB

+

_{C z' Dz)(D z' D z}

r

1(B'p +Dz'C z)

Clearly,R ..,(P) is the Schur-complement of D

_z'

D

_z

inF ..,(P). Therefore we have

rankF ..,(P)(t)

=

m

+

rankR..,(P)(t)

for all t E lO,T]. This implies that the pair of conditions (2.3), (2.4) is equivalent to the condi-tion: P is the solution of the Riccati differential equation Ry(P)

=

°

with terminal condition P (T)

=

0. A similar statement holds forQsatisfying (2.5) and (2.6). Thus we obtain

COROLLARY 2.2. Let y> 0. Assume D1is surjective and D

_z

is injective. Then the following statements are equivalent:

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5

-(i) There exists a time-varying dynamic compensator}:.F such that IIGe/1i

<

y.

(ii) There exist differentiable matrix-functions P and Q: 10,T] ~ /Rnxn such that for all

t E [0,T]

-p

=A'P

+

PA

+

_{C z' C z}

+

~

PEE'P - (PB

+

_{C z' Dz)(D z' DZ)-l(B'P}

+

D z' Cz), P(T)

=

0 and

with, in addition,

y.l -

Q(t)P (t) invertible for allt E [0,T]

o

Itcan be shown that if the conditionsinthe statement of Theorem 2.1 (ii) indeed hold, then it is always possible to find a suitable compensator with dynamic order equal to

n,

the dynamic order of the system to be controlled.

3. COMPLETION OF THE SQUARES

In this section we shall outline the proof of the implication (i) =:> (ii) of Theorem 2.1. Con-sider the system }:.. For given

u

andw, letzu,w denote the output to be controlled, with

x

(0)= O. Our starting point is the following lemma:

LEMMA 3.1. Lety

>

O. Assume that for all 0:t:-wE L~ [0,T] we have

inf{lIzu,wllz-yllwllz I

u

E L~[O,T]}

<

0 . (3.1)

Then there exist a differentiable matrix function P: [0,T] ~ /Rnxn such that Fr<P)(t)~0 'titE [0,T], P(T) =0 andrankF y(P)(t) =normrank(G)'tit E [0,T].

PROOF. A proof of this can be given by combining the result of [2, Theorem 2.3] with ideas

used in the proof of [4, Theorem 5.4].

0

Now, assume that the condition (i) in the statement of Theorem 2.1 holds, Le. assume there exists a dynamic compensator }:.,.. such that liCe/II <y. Then condition (3.1) holds: let

w E L~[0,T] and w :t:-0 and let zbe the closed loop output with x(0)

=

0 andp(0)

=

O. Then

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-

6-and hence IIzu.wll - yIIwllz <O. Then also the infimum in (3.1) is less than O. We may then con-clude that, indeed, a differentiable matrix functionPexists that satisfies(2.3)and(2.4).

The fact that also (2.5)and(2.6)hold can be proven by the following dualization argument. Consider the dual system

~=A'l;+CI'V+ Cz'd ,

r:

T\=B'l;

+

Dz'd ,

C=E'l;+D1'v ,

and apply to

r:

the time-varying compensator

q

=K'(T - t)q +M'(T - t)T\ , v=L'(T-t)q+N'(T-t)T\ .

It can be shown that if we denote by

Gel

the closed loop operator of

r:

xLF' (with

C(O)

=

0,q(0)

=

0), and it G~l denotes the adjoint operator ofGel then the following equality holds:

(3.2) where R denotes the time-reversal operator (Rx)(t):= x(T - t). Now, if IIGelll <y then also IIG~lll

<

Yand therefore, by 3.2,IIGelII

<

y. We can therefore conclude that the quadratic differen-tial inequality associated with

r:

has an appropriate solution, say P(t), on [0,T]. by defining

Q(t):=P(T - t)we obtain a functionQthat satisfies(2.5)and(2.6).

Finally, we have to show that condition(2.7)holds. We shall need the following lemma: LEMMA 3.2. Assume that there exists P:[0,T] ~ JRnxn such that F y(P)(t)'2:.0, "iItE [0,T], and rankFy(P)(t)

=

normrank(G),"iIt E [0,T].Then there exist continuous matrix functions Cz.p andDp such that for allt

[

c

Z

P'(t)]

FiP)(t)=

Dp~(t)

(Cz.p(t) Dp(t)) .

AssumeFiP)is factorized as in(3.3). Introduce a new system, sayLp,by

(3.3)

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-

7-Xp

=

(A

+

~

EE'P)xp

+

Bup

+

Ewp ,

yp =(C}

+

~D}E'P)XP

+

D}wp, Zp= C2,pXp

+

Dpup .

(3.4)

We stress that ~p is a time-varying system with continuous coefficient matrices. If ~F is a dynamic compensator of the form (1.2).letGp,cI denote the operator from wp tozp obtained by interconnecting~pand~F'

The crucial observation now is thatIIGclll

<

Yif and only ifIIGp,clll

<

Y.that is.a compensator~F

"works" for ~

if

and only

if

it "works" for ~p! A proof of this can be based on the following "completion of the squares" argument

LEMMA 3.3. Assume thatP satisfies (2.3) and (2.4). Assumexp(O)=X(0)=0, up(t)=u(t) for all t E [0,T] and suppose that Wp and ware related by wp(t)

=

wet) - y-2E'P (t)x (t) for all t E [0,T].Then for alltE lO,T] we have

liz(t)1I2- yllw (t)1I2

=

.!L.(x'(t)P (t)x (t))

+

IIzp (t)1I2 - yllwp(t)1I 2

dt Consequently:

IIzll~ - yllwll~

=

IIzpll~- yllwpll~ (3.5)

PROOF: This canbeproven by straightforward calculation, using the factorization (3.3).

0

THEOREM 3.4. Let P satisfy (2.3) and (2.4). Let ~F be a dynamic compensator of the form (1.2).Then

PROOF. AssumeIIGp •_c/1i

<

yand consider the interconnection of~and~F' p Up .::-'.:- w

~p

c..--p

~F

y

..::-

'1/1

~

-y 1

~F

Let

°

:tow E L~ [0,T], let x be the corresponding staLe trajectory of ~ and define

Wp :=w -

y2

E'px. Then clearlyyp=y, xp=x and thereforeUp=u.This implies that the equal-ity (3.4) holds. Also, we clearly have

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8

-IlzpJI~- .yzllwpll~S (IIGp.c/112 - .yz~lwpll~ (3.6) Next, note that the mapping

lIJ>

H _wp+y-2 E'Pxpdcfines a bounded operator from

Li

[0.T) to

Li[o.

T). Hence there exists a constant ~

>

0 such that IIwpll~ >~lwll~. Define

S>

0 by 52:=

r

-IIGp,clI12. Combining (3.4) and (3.5) then yields

IIzll~- .yznwll~S-52~lwll~ .

Obviously. this implies thatllGc/1I2

sf -

()2~

<

f.

IJ

We will now prove that(2.7)holds. Again assume that:Ep yields 1IGclll

<

y. By applying a version

of Lemma

3.1

for time-varying systemsitcan then be proven that thedualquadratic differential inequality associated with:Ep :

_

[-1'

+ApY

+

YAp' +EE'

+

_~

YC2,P'C2,pY

G (Y)'- T

. , . - C_I,PY +D E'_I

has a solutionyet)on [0.T].satisfyingYeO)

=

0 and

_

.

[IS -

Ap(t) -E]

rankG.,(Y)(t)

=

nonnrank CI.P(t) D

I

-n

(3.7)

for allt e [0,T). Here, we have denotedA p =A +y-2EE'Pand CI,P

=

CI +y-2DIE'P. Further-more. it can be shown that Yis

unique

on each interval [0,tI](tI S

n.

Ontheother hand. it can

be proven that on each interval [0,

ttl

on which 1-QP is invertible, the function

Y:=

(1-QPrlQ satisfies

Gy(Y)(t)~

0,

YeO)

=0 and the rank condition (3.7). Thus on any such interval [0, td we must haveY(t)=yet). aeady, since Q(O)=O, theree.xistsO

<

tlS Tsuch that

I - QP is invertible on [0,tI). assume now that tI

>

0 is the smallest real number such that

1- Q(tl)P(tl)is not invertible. Then on [0,tl)we have

Q(t)=(/- Q(t)P(t)) yet)

and hence, by continuity

-Q(tI)=(/-Q(tl)P(td)Y(tI) .

(3.8)

There exists x

*

0 such that x'(I - Q(tI)P(tI»

=

O. By (3.8) this yields x'Q(t

t>

=

0 whence

x'

=

0, which is a contradiction. We must conclude that I - Q (t)P (t) is invertible for all t e [0,T].

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- 9-4. EXISTENCE OF COMPENSATORS

In the present section we will sketch the main ideas of our proof of the implication (ii) ~

(i) of Theorem 2.1. The main idea is as follows: starting from the original system L we shall define a ncw system,Lp,Q'which has the following important properties:

(1) Let Lp be any compensator. The closed loop operator Gel of the interconnection LX Lp

satisfies IIGclll

<

Y

if

and only

if

the closed loop operator of Lp,Q

x

Lp, say Gp,Q,ch satisfies

IIGp,Q,clll

<

y.

(2) The system LP,Q is almost disturbance decoupable by dynamic measurement feedback, i.e. forallE

>

0there existsLpsuch thatIIGp,Q,clll

<

E.

Property(1)states that a compensatorLF"works" forL.if and only if it "works" forLp,Q' On the other hand, property(2)states that,indeed, there exists a compensatorLF that "works" forL.P,Q:

take any£$ yand take a compensatorLFsuch thatIIGp,Q,clll

<

E. Then by, property(1),IIGclll

<

y

soL.F works forL..This would clearly establish a proof of the implication(ii) ~ (i) in Theorem 2.1.

We shall now describe how the new system L.P,Q is defined. Assume that there existPand

Qsatisfying(2.3)to(2.7). Apply Lemma3.2 to obtain a continuous factorization(3.3)ofFyep) and let the system L.pbe defined by(3.4).Next, consider the dual quadratic differential inequality GlY)2 0 associated with the systemLp,together with the conditions YeO)

=

0 and the rank con-dition (3.7). As was already noted in the previous section, the conditions (2.5), (2.6) and (2.7) assure that there exists a unique solution Yon [O,T]. (In fact, yet)=

(1'

I-Q(t)p(or1Q(t).) Now, it can be shown that there exists a factorization

withEp,Q andDp,Q continuous on[O,T].Denote

Ap,Q(t):=Ap(t)+yet) C2,P'(t)C2,P(t) , Bp,Q(t) :=B

+

Y(t) C 2,P'(t) Dp(t).

Then, introduce the new systemLp,Q by:

Xp,Q

=

Ap,Q xp,Q

+

Bp,Q up,Q

+

Ep,Q wp,Q '

YP,Q

=

Cl,pXp,Q

+

Dp,Q wp,Q '

zp,Q

=

C 2,pxp,Q +Dp up,Q.

Again,L.P,Q is a time-varying system with continuous coefficient matrices. We note that L.P,Q is in fact obtained by first transforming L into Lp and by subsequently applying the dual of this transformation toL.p. We shall now first show that property (1)above holds. IfL p is a dynamic compensator, then letGp,Q,cIbe the closed loop operator fromwp,Q tozp,Q in the interconnection

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-10-P,Q Up,Q 1.:.£ _{f -}1/1

"

~1',Q Q

'B

F z YP,

-:::-

1/1 f

-"

~ y U

"

~F

Recall thatGel denotes the closed loop operator from wto z in the interconnection of LandLF'

We have the following: THEOREM4,1.

PROOF. Assume LF yields IIGelll

<

y. By Theorem 3.4 then alsoIlGp,elll

<

y, Le.• LF intercon-nected withLp (given by 3.4) also yields a closed loop operator with norm less than y.Itis easily seen that the dual compensator

"£/

(see section 3), interconnected with the dual ofLp:

~=Ap'(T -t)~

+

Cl,p'(T -t)v

+

C2,p'(T -t)d

Lp' T]=B'~+ DP'(T-t)d

~ =E'~+OI'V

yields a closed loop operator

6

P,el(fromdto~) with

116

P,elll

<

y. Now, the quadratic differential inequality associated withLp' is the transposed. time-reversed version of the inequality G-yCY);:: 0 and therefore has a unique solution

yet)

=

yeT -

t)such that

yeT)

=0 and the corresponding rank condition (3.7) holds. By applying Theorem 3.4 to the system LP' we may then conclude that the interconnection of L/ with the dual LP,Q' of LP,Q yields a closed loop operator with norm less thany. Again by dualization we then conclude that IIGp,Q,clll <y. The converse implication is

pro-ven analogously.

0

Property (2) is stated formally in the following theorem:

THEOREM 4.2. For all e>0 there exists a time-varying dynamic compensator LF such that

IIGp ,Q,clll

<

y.

0

Due to space limitations, For a proof of the latter theorem we refer to [5]. By combining theorems 4.1 and 4.2 we immediately obtain a proof of the implication (ii) ~ (i) in Theorem 4.2.

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11 -5. CONCLUDING REMARKS

In this paper we have studied the finite horizonHoo control problem by dynamic measure-ment feedback. We have noted that the results obtained can be specialized to re-obtain results that were obtained before [6] and [2]. The development of our theory runs analogously to the theory developed in [4] and [3] around the standardHoo control problem (theinfinite horizon version of the problem studied in the present paper). In the latter references the main tools are the so-called quadratic matrix inequalities, the algebraic versions of the differential inequalities used inthe present paper. For the special case that D1 is surjective and D2 is injective these quadratic matrix inequalities reduce to the algebraic Riccati equations that were also obtained in [6] and [ll.

REFERENCES

[1] 1. Doyle,K. Glover, P.P. Khargonekar, B.A. Francis, "State space solutions to standard H2

andH00 control problems", IEEE Trans. Aut. Contr., Vol. 34, No.8, 1989, pp. 831-847.

[2] D.J.N. Limebeer, B.D.O. Anderson, P.P. Khargonekar, M. Green, "A game theoretic approach toHoo control for time varying systems", preprint, 1989.

[3J A.A. Stoorvogel, "The singularH00control problem with dynamic measurement feedback",

preprint, 1989, Submitted to SIAM J. Contr.& Opt.

[41 A.A. Stoorvogel& H.L. Trentelman, "The quadratic matrix inequality in singularH00

con-trol with state feedback", preprint 1989, To appear in SIAMJ. Contr.& Opt.

r

51

A.A. Stoorvogcl & H.L. Trentelman, "The finite horizonHoo control problem with dynamic measurement feedback", in preparation.

l6] G. Tadmor,"H00 in the time domain: the standard four blocks problem", 1988, To appear in

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EINDHOVEN UNIVERSITY Of TECHNOLOGY

Depanrnent of Mathematics and Computing Science

PROBABIUTY

THEORY, STATISTICS.

OPERATIONS

RESEARCH

AND SYSTEMS

THEORY

P.O. Box 513

5600 MB Eindhoven - The Netherlands

Secretariate:

Dommelbuilding 0.03

Telephone:

040 - 47 3130

List of COSOR-memoranda - 1989

Number

Month

Author

Title

M 89-01

January

D.A. Overdljk

Conjugate profiles on mating gear teeth

M 89-02

January

A.H.W. Geerts

A priori results in linear quadratic optimal control theory

M 89-03

February

A.A.

Stoorvogel,

The quadratic matrix inequality in singular

H _

control with stall

H.L. Trentelman

feedback

M 89-04

february

E. WilJekens

Estimation of convolution tail behaviour

N. Veraverbeke

M 89-05

March

H.L. Trentelman

_The

_{totalJy singular linear quadratic problem with indefinite cost}

M

89-D6

April

B.G. Hansen

_{Self-decomposable distributions}

and

branching processes

M 89-07

April

B.G. Hansen

Narc on Urbanik's class

L. M 89-08

April

B.G. Hansen

Reversed self-decomposability

M

89-09

April

A.A.

Stoorvogel

The

singular zero-sum differential game with stabiJity using

H _

con·

trol

theory

M 89-10

April

L.J.G. Langenhoff

An

analytical theory of multi-echelon production/distribution systems

W.H.M.Zijm

(15)

-

2-Number

Momh

Author

Title

M 89-12

May

D.A. Overdijk

De geometric van de kroonwieloverbrenging

M 89-13

May

U.B.F.Adan

Analysis of the shonest queue problem

J. Wessels

W.H.M.Zijm

M 89-14

June

A.A. Stoorvogel

TIle

singular

H _

control problem with dynamic measurement

feed-back

M 89-15

June

A.H.W. Geerts

The

output-stabilizable subspace and linear optimal control

M.L.J. Hautus

M 89-16

June

p.e.

Schuur

On the asymptotic convergence of the simulated annealing algorithm

in the presence of a parameter dependent penalization

M 89-17

July

A.H.W. Geerts

A priori results in linear-quadratic optimal control theory (extended

version)

M 89-]8

July

D.A. Overdijk

The curvature of conjugate profiles in points of contact

M 89-19

August

A. Dekkers

An

approximation for the response time of an open CP-disk system

J. van der Wal

M 89-20

August

W.FJ. Verhaegh

On randomness of random number generators

M 89-21

August

P. Zwietering

Synchronously Parallel: Boltzmann Machines: a Mathematical Model

E. Aarts

M 89-22

August

U.B.F. Adan

An

asymmetric shonest queue problem

J. Wessels

W.H.M.Zijm

~

89-23

August

D.A. Overdijk

Skew-symmetric matrices in classical mechanics

~

89-24

September

F.W. Steutel

The

gamma process and

the

Poisson distribution

J.G.F. Thiemann

.t

89-25

September

A.A. Stoorvogel

The

discrete time

H _

control problem: the full-information case

I 89-26

OCtober

A.H.W. Geerts

Linear-quadratic problems and the Riccati equation

M.L.J. Hautus

I 89-27

October

_{H.L.Trentelman}

_{Completion of the squares in the finite horizon}

A.A.Stoorvogel

ROO

_{control problem by measurement feedback}