The disturbance decoupling problem with measurement feedback and stability for systems with direct feedthrough matrices

The disturbance decoupling problem with measurement

feedback and stability for systems with direct feedthrough

matrices

Citation for published version (APA):

Stoorvogel, A. A., & Woude, van der, J. W. (1990). The disturbance decoupling problem with measurement feedback and stability for systems with direct feedthrough matrices. (Memorandum COSOR; Vol. 9046). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1990

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EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of Mathematics and Computing Scie"cc

r

CaSOR-Memorandum 90-46

The disturbance decoupling problem with measurement feedback and stability for systems with direct feedthrough matrices

by

I

A.A. Stoorvogel J.W. van der Woude

The disturbance decoupling problem \vith Ineasurement

feedback and stability for systems \vith direct feedthrough

Inatrices

A.A. Stoorvogel* J.W. van der \Voude§ November 30, 1990

Abstract

In this paper the known results on the disturbance decoupling problem with measure-ment feedback and internal stability (DDPMS) are extended to include non-zero direct feedthrough matrices. Necessary and sufficient for the solvability of the DDPMS are expressed in three subspace inclusions.

Keywords: disturbance decoupling, geometric approach, measurement feedback.

1

Introduction

The so-called disturbance decoupling problems have been investigated extensively in the last two decades. It was the starting point for the development of a geometric approach to systems theory. The problem is to find a compensator such that the closed loop transfer matrix from disturbance to output is equal to O. Using the concept of(A,B)-invariance, the disturbance decoupling problem with state feedback (DDP) was solved in [2, 3, 18]. The problem of disturbance decoupling with state feedback and the extra requirement of internal stability (DDPS), was solved in [6, 18]. An excellent reference for the above mentioned problems is also [19]. Approximately 10 years later, the above mentioned problems were solved for the case of measurement feedback. The disturbance decoupling problem with measurement feedback (DDPM) was solved in [1, 7]. Finally, the disturbance decoupling problem with measurement feedback and internal stability was solved in [5, 15].

All the above problems have also been extended to the so-called almost disturbance decoupling problems where one investigates under which conditions we can make the closed loop transfer matrix arbitrarily small in a suitable norm. Excellent references for these extensions are [12, 16, 17].

In all of the above references the direct feed through matrices of the system are assumed to be equal to zero. In the state space approach to Hooand LQG control however disturbance

decoupling problems for systems with direct feed through matrices do playa role (see [10, 11)). Therefore, this paper will extend the results on the disturbance decoupling problems men-tioned above to include direct feedthrough matrices. We will solve the problem of disturbance

"Dept. of Mathematics and Computing Science, Eindhoven University of Technology, The Netherlands §Dept. of Technical Mathematics and Informatics, Delft University of Technology, The Netherlands

decoupling with measurement feedback and internal stability (DDP11S) with respect to some arbitrary stability set. A result for the DDPM follows as a special case from our main theo-rem. Note that an extension of the results on almost disturbance decoupling to incorporate direct feedthrough matrices has already been done in [20J.

We will use a geometric approach in this paper similar to the techniques used in the above mentioned references. We find that solvability of the DDPMS is equivalent to the requirement that three subspace inclusions hold. We will derive our results for continuous time systems but the related results for discrete time systems are identical.

In this paper we will present our main result in section 2. In section 3 we will prove our main result. In section 4 we will discuss some extensions. Finally we conclude with section 5 which contains some concluding remarks.

2

Problem formulation and results

Consider the finite-dimensionallillear time-invariant system ~ given by: Eu

+

Ed,

DId, (2.1)

(2.2) where x E Rn is the state of the system, u E Rm the control input, d E Rq the distur-bance input, y E

RP

the measurement output and z E

Rr

the output of~ to be controlled. A,E,E,GI,G2,DI and D2 are real matrices of the appropriate dimensions.

Assume that system (2.1) is controlled by means of a measurement feedback compensator ~F

described by:

~F:

{

p

= ](p

+

Ly,

u=Mp+Ny,

with pERk the state of the compensator and K, L,M and N real matrices of appropriate dimensions. Interconnection of ~ and ~F results in a closed loop system l:cl = ~ X ~F

described by

where we have denoted:

(2.3) A_e = (A+ ENGI LGI Ge = (G2+D2NGI EM) ](

,

D

2

M) ,

B = (E+ENDI₎ e LD I '

LetCgbe a subset of the complex plane C which is symmetric (.\ ECg {:}

>.

ECg) and with at least one point on the real axis (Cg

n

R :f= 0). The interconnection of a system ~ and ~F as

given in (2.3) is called Cg-stable if and only if the matrix Ae is Cg-stable, i.e. all eigenvalues

ofAe are contained inCg.

Problem: Given~ determine~Fsuch that the interconnection 2:x~F is Cg-stable and such that the closed loop transfer matrix is equal to zero.

This problem is often called the disturbance decoupling problem with measurement feedback and internal stability (DDPMS). The problem has been completely solved in case the direct feedthrough matrices DI and Dz are equal to 0 (see [4, 8, 1.5]). This paper extends these

results to include direct feed through matrices.

For the formulation of our main result we need a number of definitions:

Definition 2.1 : We define the detectable strongly controllable subspace~(A,B,C,D) as the

smallest subspace T ofRn for which there exists a linear mapping G such that the following

subspace inclusions are satisfied:

(A

+

GC)T ~ T,

1m (B

+

GD) C T,

(2.4) (2.5)

and such thatA

+

GC

I

Rn

IT

is Cg-stable. We also define the stabilizable weakly unobservable

subspace Vg( A, B,C,D) as the lm'gest subspace V for which there exists a mapping F such

that the following subspace inclusions are satisfied:

(A

+

BF)V C V,

(C'

+

DF)V {O},

and such that A

+

BF

IV is Cg-stable.

(2.6) (2.7)

o

The subspaces Vg(A,B,C',D) and ~(A,B,C',D)can be computed by means of well-known algorithms (see e.g. [9]). We also note that if(A, B)is Cg-stabilizable then forVg (A, B,C,D)

there exists an F suc.h that (2.6) and (2.7) are satisfied and moreover A

+

BF is Cg-stable. A similar comment can be made for ~(A,B,C',D) in case (C,A) is Cg-detectable.

We can now formulate the main result of this paper:

Theorem 2.2 : Let~ begivenoftheform (2.1). The following two conditions are equivalent:

(i) There exists a compensator of the form (2.2) such that the closed loop system is Cg-stable

and such that the closed loop transfer matrix is equal to O.

(ii) (A,B) is Cg-stabilizable, (C'llA) is Cg-detectable and

(a) 1mE ~ Vg(~ci)

+

B KerD

_z,

(b) KerC2

2

~(~di)

n

C'll1mDI, (c) ~(Ed£) ~ Vg(Eci),

o

Remarks:

• Note that we have still all freedom in our choice ofCg • For the disturbance decoupling

problem with measurement feedback (DDPM) we choose Cg = C. On the other hand

if we are interested in disturbance decoupling with internal (asymptotic) stability, then we chooseCg equal to the open left half complex plane.

• Since we mainly investigate properties of transfer matrices, the results for discrete time systems are immediate from theorem 2.2. For a discrete time system with the same parameters as E conditions (i) and (ii) are still equivalent. Only this time, for internal (asymptotic) stability, we have to chooseCg equal to the open unit disc.

It can be easily checked that condition (c) implies conditions (a) and (b) in case the direct feedthrough matrices D1 and D2 are equal to O. The following lemma gives the possibility of

actually calculating a suitable compensator if it exists:

Corollary 2.3: Let E of the form (2.1) be given. Conditions (a)-(c) of theorem 2.2 are

equivalent to the following conditions: ~(Edi) ~Vg(Eci) and there is a matrix N such that

(2.8)

In addition to such an N, let F and G be the matrices satisfying the conditions of definition

2.1 for Vg(Eci) and~(Edi)respectively and such that A+BF and A+GC1 are both Cg-stable.

Then a compensatorEp making the closed loop system Cg-stable and the closed loop transfer

matrix equal to 0 is given by:

Ep: {

P

=

Ap

+

Bu

+

G(C1P - y),

u = Fp - N(C1P - y). (2.9)

o

Remark: In case Cg = Rn, Le. if we consider disturbance decoupling without stability

requirements, and if the conditions (b) of theorem 2.2 are satisfied then it can be shown that there exists a compensator achieving disturbance decoupling of dynamic order

This result was already known (see [8]) in case the direct feedthrough matrices are equal to

o

and can be extended to the more general system (2.1) investigated in this paper.

We have not been able to derive a similar result in case of stability requirements (Cg

I:-

Rn).

Note that we can easily rewrite (2.9) in the form (2.2). However in (2.9) the structure of the controller is more visible. We can also investigate when there exists a strictly proper compensator which solves the DDPMS:

Corollary 2.4 Let E be given of the form (2.1). The following two conditioH8 arE equiva-lent:

(i) There exists a compensator of the form (2.2) with N

=

a

such that the closed loop system is Cg-stable and such that the closed loop transfer matriJ' is Equal to O.

(ii) (A,B) isCg-stabilizable, (Gl,A) isCg-detectable and (a) ImE~Vg(~ci),

(b) Ker G2

:2

1;(~dd,

(c) A1;(~di) ~ Vg(~cd,

where ~ci

=

(A,B,G2,D2 ) and ~di

=

(A,E,Gl,D l ). 0

In case part (ii) is satisfied a compensator satisfying (i) is given by (2.9) with N = 0 and

F,G as described in corollary 2.3. 0

In the next section the above results will be proven.

3

Proofs of the results obtained

The following characterization (see e.g. [4]) of the subspaceVg(~)turns out to be very useful:

Lemma 3.1 : Define by R sps (s) the set of strictly proper Cg-stable real rational vectors, i. e. vectors whose elements are strictly proper rational functions with all poles in Cg. The subspace

Vg(A, B, G, D) is equal to the set of allXo ERn for which there exist €,w ERsps(s) such that:

Xo

o

(sf - A)€(s) - Bw(s),

G€(s)

+

Dw(s).

_o

\Ve will also need the following lemma which is in essence well-known:

Lemma 3.2 : Let U and W be linear subspaces, and let P,

Q

and R be matrices such that:

PU C W+ImQ,

P(U

n

KerR) C W.

Then there is a matrix X such that

(P

+

QXR)U ~ W.

_o

Proof: Let U = 1mU and W = Ker~V. The two conditions of our lemma imply the existence of matrices Xl and X2such that:

111 PU

+

WQXl 0,

WPU

+

X2RU

=

O.

WPU

+

WQXRU = O.

This X satisfies the conditions of our lemma.

•

(3.2) (3.1 )

Proof of theorem 2.2 (ii) ::;. (i). Because (A, B) and (Cl ,A) are Cg-stabilizable and Cg

-detectable respectively it follows from the definitions of Vg(~ci) and

1;

(~dd that there are matrices

F

and

G

satisfying the conditions of definition 2.1 and such that

A+BF

and

A+GC

l

are both Cg-stable.

Now observe that Vg(~ci)and ~(~di)are such that:

[~J Vg(~c;) ~ (Vg(~ci)

E&{O})

+

1m

[:J,

[A E]

((~(~d;)

E&Rq)

n

Ker

[e

l Dt ] )

~ ~(~di)'

Furthermore, it can be easily shown that condition (a) is equivalent to:

and that condition (b) is equivalent to

(~(~di)

E& Rq)

n

Ker

[C

t

D

t ]

~

Ker

[C

2

0].

Now using condition (c), the combination of (3.1) and (3.3) implies that

[A E]

(~(~di)

E&Rq)

~ (Vg(~c;)

E&{O})

+

1m

[B],

C2 0 D2

while the combination of (3.2) and (3.4) with condition (c) implies that:

[;2

~] ((~(~di)

EB

Rq)

n

Ker

[C

t Dt ])

~ (Vg(~c;)

E&{O}).

(3.3)

(3.4)

The above two equations imply by lemma 3.2 the existence of a matrix N satisfying (2.8), or equivalently such that:

(A

+

BNCd~(~d;) ~ Vg(~ci), 1m (E

+

BN Dt ) ~ Vg(~ci), (C2

+

D2NCt ) ~(~di)

=

{O}, D2NDt

=

O.

(3.5)

Let such N be the feedthrough matrix of the compensator (2.2), and define the other com-pensator matrices as follows:

J( :=A

+

BF

+

GCt - BNCl l L := BN - G, M := F - NCt .

It is easy to see that the closed loop system now obtained can equivalently be described as follows: z ~cI :

(x

~

p)

=

(A +oBF

B:~t~~F)

C:

p)

+

(EE+:::

t)

d,

=

(C2

+

D2 F D2 NCt - D 2F)

C:

p)

+

D 2N DId. (3.6)

An easy calculation shows that the closed loop transfer matrix is giYen b~':

(C2

+

_{D2F)(sf - A - BF)-l(E}

+

_{BN Dl )}

-(C2

+

D_{2F)(sf - A - BF)-l(sf - A - BNCd(sf - ..}1 - Gel)-1(E

+

GDd

+(C2

+

D2NCd(sf - A - GCd-l(E

+

GD1 )

+

D2ND1 •

From (3.6) it is clear that the closed loop system isCg stable, since A

+

BF and A

+

GCl are

Cg-stable. Moreover observe that F and G are such that for all sEC:

Ker (C₂

+

D2F) (sf - A - BF)-l

2

Vg(~cd,

1m (sf - A - GC1)-1 (E

+

_{GDl )} ~ Tg(~di)'

(3.7) (3.8) Using (3.5),(3.7) and (3.8) together with condition (c) of theorem 2.2 itis straightforward to show that the closed loop transfer matrix is equal to O.

(i) :::} (ii). Let a compensator ~F be given, satisfying the conditions of our lemma. Let the closed loop system ~cl be described by (2.3). We know that all eigenvalues of Ae are in Cg

and

By the fact that Ae is Cg-stable, it is immediate that (A, B) must be Cg-stabilizable and

(C1,A)Cg-detectable. Because the closed loop transfer matrix is zero, it follows that De

=

O.

\Ve defineVe

:=<

Ael 1mBe

>,

i.e. Veis the smallest Ae-invariant subspace containing 1mBe.

Since Ce(sf - Ae)-1Be

=

0, the definition ofVeimplies that Ve~ KerCeo Define:

Clearly

T

~

V.

Moreover, it follows that:

(3.9) (3.10) (A

+

BNCl)T ~ V, (C2

+

D2NCl )T

=

{O}, 1m (E+BNDl)~V, D2NDl

=

O. (3.11)

Take any x EV. By definition ofV there exists pERk such that (x T pT)T EVe. Define

(

~(S))

:= (sf _ Ae)-l

(x) .

w(s) p

Because Aeis Cg-stable we know that ~,w E 'Rsps(s) and hence ~,(NCl~

+

Mw) E Rsps(s).

Moreover, because Ve is Ae-invariantitfollows that for all s:

(

~(S))

EVe~ Ker Ceo

Combining the above, we find that:

x (s1 - A)~(s)- B (NCI~(S)

+

Mw(s)) ,

o

C2~(S)

+

D2(.NCI~(S)

+

Mw(s)).

This implies by lemma 3.1 that x E Vg(~ci)' Hence V ~ Vg(2:ci

l.

By dual reasoning it can be derived that ~(~di) ~ T. From the above it is clear that condition (c) of (ii) is satisfied. From (3.11) it is clear that:

1m(E

+

GN DI ) ~ Vg(~ci), Ker(C2

+

D2NCI )

2

~(~di), D2NDI = O. (3.12) (3.13) (3.14) Now (3.12) and (3.14) together imply condition (a) of (ii). Indeed take any x ERq and denote

u

=

NDIx. Then u E KerD2 and there is a v E Vg(~ci) such that Ex

=

v - B2u. Dually it

can be shown that (3.13), (3.14) together imply condition (b) of (ii). • Corollaries 2.3 and 2.4 immediately follow from the proof of theorem 2.2.

4

A more general case

In section 2 we discussed a system ~ of the form (2.1). However, in the most general linear, time-invariant case, there are two more direct feedthrough matrices unequal to 0 in the system. In this section we discuss this more general case and we assume that our system is of the form: { X= Ax

+

Eu

+

Ed, ~ : y = CIx

+

D3u

+

DId, z = C2x

+

D2u

+

D4d. (4.1)

For this more general case we have to discuss the admissibility of controllers of the form (2.2) in more detail (see e.g. [13]). Consider the following interconnection where

t

and ~F are described by (4.1) and (2.2) respectively:

w z

-VI U

~

-+ ~F Y + V2 -(4.2)

The closed loop system is well-defined and "internally proper" if the closed loop transfer matrices from w,Vb V2 to z, u,yare well-defined and proper. In that case we call the

in-terconnection well-posed. It can be shown that this is equivalent to the requirement that

1 - N D3 is invertible or, equivalently, that I - D3N is invertible. Therefore we require that

our controller ~F is such that1- N D3 is invertible. Moreover, if1 - N D3 is invertible, the

closed loop system ~ x ~F can be written in the form (2.3) where

AI' := ( A

+

BN(I - D3N)-IGI B(l - N D3)-1J1 ) . L(I - D31V)-IGI J{

+

L(I - D3N)-1 D3J1

\Ve require that the interconnection is Cg-stable, Le. the matrix At has all eigenvalues in

Cg • If the realizations for

:t

and ~F are both Cg-detectable and Cg-stabilizable, then this

is equivalent to the requirement that in the interconnection (4.2) the closed loop transfer matrices from w,VI, V2 to u,y,z are all Cg-stable, i.e. the rational matrices have all poles in

Cg •

\Ve can derive the following theorem:

Theorem 4.1 : Let ~ be given of the form (4.1). There exists a compensator of the form

(2.2) such that the closed loop system is well-posed and Cg-stable and such that the closed loop

transfer matrix is equal to 0 if and only if the following conditions are satisfied:

(i) (A,B) is Cg-stabilizable, (GilA) is Cg-detectable, (ii) J;(~di)S;;;; Vg(~ci),

(iii) there exists a matrix

N

such that:

and such that I

+

N

D3 is invertible.

A controller solving the DDPMS is then described by:

~

. { p=Ap+Bu+ G(G1P+D3u-y),

~F .

-u

=

Fp - N(G1P

+

D3u - y).

(4.4)

o

Remark: Note that (4.4) describes a compensator of the form (2.2) because I

+

N

D3 is

invertible. The reason for defining the compensator in this implicit way is to show the relationship with the compensator we found in (2.9).

We can again express solvability of (4.3) in terms of subspace inclusions. However the well-posedness constraint (I

+

N

D3 is invertible) we can notexpress in subspace inclusions.

We will only give a sketch of our proof. \Ve will treat these extra feed through matrices in two steps. In the next subsection we show how we can reduce the disturbance decoupling problem with measurement and stabjJity(nDPMS)for (4.1) to the same problem for a different system which has a direct feed through matrix from disturbance to output which is equal to zero. In the second subsection we show how we can reduce the nDPMS for a system (4.1) to the same problem but again for another system which this time has the form (2.1). On the latter system we may apply theorem 2.2.

(4.10)

4.1 A direct feedthrough matrix from disturbance to output

We first solve DDPMS "at infinity". It is easily checked that there must exist a matrix S such that I - S D3 is invertible and

D4

+

D2(I - SD3)-1 SD1= O. (4.5)

This fact is expressed in theorem 4.1 by the condition that, in addition to I

+

N

D₃ being invertible, the matrix N has to be such that amongst others D4

+

D2 }\'D1

=

O. Clearly for

such N (4.5) can be satisfied by S:= (I

+

_ND3)-lN.

Then we apply the preliminary feedbacku = Sy+vto our system. The system we thus obtain will have a direct feedthrough matrix from dto z which will be equal to O. Clearly solvability of the disturbance decoupling problem for the original system is equivalent to solvability of the disturbance decoupling problem for the system we obtain after this preliminary feedback. Therefore we can reduce the disturbance decoupling problems for (4.1) to the same problems for a new system which has a direct feed through matrix from d to z which is equal to O.

4.2 A direct feedthrough matrix from input to measurement

By the previous subsection we may assume that we have system of the form (4.1) with D4 = O.

Assume that a compensator1;Fof the form (2.2) is given such that the interconnection

t

x1;F is well-posed, Cg-stable and has a closed loop transfer matrix which is equal to O. We define:

k

J(

+

L(I - D3Nr1 D3AI, (4.6)

1 .-

L(I - D₃N)-1, (4.7)

if .-

(I - N D_3)-1M, (4.8)

N .- N (I - D₃N)-1. (4.9)

Then it is easily checked that the following compensator satisfies condition (i) of theorem 2.2 for the system 1; given by (2.1).

'0.{i>=k

p

₊₁

_{y ,}

LJF •

-y=Mp+Ny.

On the other hand assume that we have a compensator of the form (4.10) such that I

+

D3

N

is invertible and such that condition (i) of theorem 2.2 is satisfied for 1;. In that case, the following compensator makes the interconnection

t

x

t

F well-posed, Cg-stable and yields a closed loop transfer matrix which is O.

J(

.-

_{K -}- _L(I-

+

_{D3N)- D3M,}- 1

-L

.-

1(I

+

_D3N)-1, M

.-

(I

+

_ND3)-1£J,

N

.-

(I

+

N

D3)-1

N.

We can now apply theorem 2.2 to E described by (2.1) to obtain necessary and sufficient conditions for the solvability of DDPMS for1;. We only have to do some work to incorporate the well posedness constraint (I

+

D₃

N

invertible). The results of this subsection can be used to obtain necessary and sufficient conditions for the solvability of DDPMS of

t

described by (4.1) with D4

=

O. The results of the previous subsection can then be used to obtain

necessary and sufficient conditions for the solvability of DDPMS for the general system f; of the form (4.1) without any restrictions. These conditions are given in theorem 4.1.

5

Conclusion

In this paper we have treated the most general disturbance decoupling problem: the dis-turbance decoupling problem with measurement feedback and stability for some arbitrary stability setCg • We have shown how the known results can be extended to incorporate direct

feed through matrices in the system. In our opinion this paper completes the results already available.

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[10] A.A. Stoorvogel, "The singular minimum entropy Hoocontrol problem", CaSaR

mem-orandum 90-37, Eindhoven University of Technology, 1990, Submitted for publication. [11] A.A. Stoorvogel, "The singular linear quadratic Gaussian control problem", Submitted

for publication.

[12] H.L. Trentelman, Almost invariant subspaces and high gain feedback, CWI Tracts, Vol. 29, Amsterdam, 1986.

[13] M. Vi dyasagar , Control System Synthesis: A factorization approach, MIT Press, Cam-bridge, Ma., 1985.

[14] J .C. Willems, "Least squares stationary optimal control and the algebraic Riccati equa-tion", IEEE Trans. Aut. Control, Vol. 16, 1971, pp. 621-634.

[15] J.C. 'Willems, C. Commault, "Disturbance decoupling by measurement feedback with stability or pole-placement", SIAMJ. Contr. (3 Opt., Vol. 19. Ko. -L 1981. pp. 490-504. [16] J.C. Willems, "Almost invariant subspaces: an approach to high gain feed back design-Part I: almost controlled invariant subspaces", IEEE Trans. Alit. Control,Vol. 26, 1981, pp. 235-252.

[17] J.C. Willems, "Almost invariant subspaces: an approach to high gain feedback design-Part II: almost conditionally invariant subspaces", IEEE Trans. Aut. Control, Vol. 27, 1982, pp. 1071-1084.

[18] W.M. Wonham, A.S. Morse, "Decoupling and pole assignment in linear multivariable systems: a geometric approach", SIAMJ. Contr. (3 Opt., Vol. 8, No.1, 1970, pp. 1-18. [19] W.M. Wonham, Linear multivariable control: a geometric approach, 2nd ed., Springer

Verlag, New York, 1979.

[20] J.W. van der Woude, "Almost disturbance decoupling by measurement feedback: a fre-quency domain analysis", IEEE Trans. A uf. Control,Vol. 35, 1990, pp. 570-573.

EL'.,'DHOVEN UNIVERSITY OF TECHNOLOGY

D~parunent of Mathematics and Computing Science

PROBABILITY THEORY, STATISTICS, OPERATIONS RESEARCH AND SYSTE:\IS THEORY

P.O. Box 513

5600 MB Eindhoven - The Netherlands Secretariate: Dommelbuilding 0.03 Telephone: 040 - 47 3130

List of COSOR-memoranda - 1990

:'-:umber Month Author Title

'\'190-01 January !.J.B.F. Adan Analysis of the asymmetric shonest queue problem J. Wessels Part 1: Theoretical analysis

W.H.~.Zijm

.\190-02 January D.A. Overdijk Meetkundige aspeeten van de productie van kroonwielen

:"190-03 February !.J.B.F. Adan Analysis of the assymmetric shonest queue problem

1.Wessels Pan II: Numerical analysis W.H.M.Zijm

.\190-04 March P. van der Laan Statistical selection procedures for selecting the best variety L.R. Verdooren

M 90-05 March W.H.M.Zijm Scheduling a flexible machining centre E.H.L.B. Nelissen

~,,190-06 March G. Schuller The design of mechanizations: reliability, efficiency and flexibility W.H.M. Zijm

~190-07 March W.H.M.Zijm Capacity analysis of automatic transpon systems in an assembly fac·

~umber Month Author Title

M 90-08 March G.J. v.Houturn Computational procedures for stochastic multi-echelon produCti0:-.

W.H.M.Zijm systems

(Revised version)

M 90-09 March P.J.M. van Production preparation and numerical control inPCB assembly Laarnoven

W.H.M.Zijm

M 90·10 March F.A.W. Wester A hierarchical planning system versus a schedule oriented plannil;~

J.Wijngaard system W.H.M.Zijm

M 90-11 April A.Dekkers Local Area Networks

M 90·12 April P. v.d. Laan On subset selection from Logistic populations

M 90-13 April P. v.d. Laan DeVan Dantzig Prijs

M 90-14 June P. v.d. Laan Beslissen met statistische selectiemethoden

M 90-15 June F.W. Steutel Some recent characterizations of the exponential and geometric distributions

M 90-16 June J.van Geldrop Existence of general equilibria in infinite horizon economies wil:

C. Withagen exhaustible resources. (the continuous time case)

M 90-17 June P.C. Schuur Simulated annealing as a tool to obtain new resultsinplane geomeu:

M 90-18 July F.W. Steutel Applications of probability in analysis

M 90·19 July I.J.B.F. Adan Analysis of the symmetric shonest queue problem J. Wessels

W.HM.Zijm

M 90-20 July I.J.B.F. Adan Analysis of the asymmetric shortest queue problem with threshoL J. Wessels jockeying

:\urn ber ~onth .\1 90-21 July ~1 90-22 July M 90-23 July M 90-24 1uly ~1..90-2S 1uly ~1 90-26 July M90-27 August ~190-28 August M 90-29 August y190-30 August • M 90-31 August M 90-32 August Author K. vanHam F.W. Steutel A. Dekkers 1.vander Wal

A. Dekkers J.vander Wal

D.A. Overdijk 1.vanOorschot A. Dekkers 1.vanOorschot A. Dekkers D.A. Overdijk A.W.J. Kolen 1.K. Lenstra R Doornbos

M. W.I. van Kraaij W.Z. Venema J.Wessels 1. Adan A. Dekkers F.P.A. Coolen P.R. Mertens MJ.Newby Title

On a characterization of the exponential distribucon

Performance analysis ofa volume shadowing model

Mean value analysis of priority stations without preemption

Benadering van de kroonwielflank met behulp van regeloppervlakken in kroonwieloverbrengingen met grote overbrengverllouding

Cake.

a

concurrent Make CASE tool

Measuring and Simulating an 802.3 CSMNCD LAN

Skew-symmetric matrices and the Euler equations of rotational motion for rigid systems

CombinatoricsinOperations Research

Verdeling en onafhankelijkheid van kwadratensommen in de variantie-analyse

Suppon for problem solving inmanpower planning problems

Mean value approximation for closed queueing networks with multi server stations

A BayeS-Competing Risk Model for the Use of Expen Judgment in Reliability Estimation

Number. Month Author Tille

.\190·33 September B. Velrman Mwuproccssor Scheduling wlll1CJmrnwuc~on Delays B.J. ugcweg

J.K..~ua

M~()·34 S4:ptember U.B.F. Adan Flexible assembly andsnonest queue problems

J. Wessels W.H.M. Zijm

M 90·35 Sepc.ember F.P.A. Coolen AnoLC ontheuscoftheproduct of spacings in Bayesian inference

MJ.Newby

•

M90-36 September A.A.Stoorvogel Robust st.1bilization of systems with multiplicative pcnurbations

M90·37 OCLOber A.A.StoOlVOgei Thesingular minimwn entropy H_ control problem

M90·38 October Jan H.van Gcldrop Gc;neral equilibrium and international trade with natural exhaustible

Cees

A.A.M. resources Withagcn

M 90·39 OCLober U.S.F. Adan Analysis

of

1heshortest queue problem

J.Wessels (Revised version) W.H.M. Zijm

M 90-40 October M. W.P. Savelsbergh

M. Goetschalckx

An Algorithm for ll1e Vehicle Routing Problem wilh Stochastic Demands

M 90-41 November Gerard lGndervater Jan Karel Lensua Martin Savelsbergh

M 90-42 November F.W. SteuLcl

M90-43 November A.A. Stoorvogel

M 90-44 November H.L. Trentelman

J.e.

Willems

M 90-45 November A.C.M. Ran

Sequential and parallelloc:u search forthe time--consuained traveling

..

salesman problem

Theset of geometrically infmit.e1y divisible distributions

The singular linear quadra~ic Gaussian control probleo The dissipation inequality and the algebraic

Riccati equation

Linear quadratic problems with indefinite cost for H.L. Trentelman discrete time systems

M 90-46 November A.A. Stoorvogel The disturbance decoupling problem with measurement J.W. van der feedback and stability for systems with direct