On the compensator (Part II): Corrections and extensions

Tilburg University

On the compensator (Part II)

Merbis, M.D.

Publication date:

1983

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Citation for published version (APA):

Merbis, M. D. (1983). On the compensator (Part II): Corrections and extensions . (pp. 1-20). (Ter Discussie

FEW). Faculteit der Economische Wetenschappen.

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No. 83.09 januari 1983 On the compensator

Part II, Corrections and Extensions

1

-1. Introduction

In part I we introduced the matrix minimum principle (MMP) as a technique to solve the control problem, the filter problem, the LQG-problem, and the two-DM Nash problem. A formulation of the b1MP as a theorem can be found in appendix A. It appears that some results of part I need to be stated more precisely to over-come the difficulties in solving the Nash problem. Therefore, the filter problem will be reconsidered; the control problem which is in fact similar by duality is omitted, but can be treated along the same lines. Also the LQG-result can be ex-tended (section 3) for other representations. These results are needed for a better understanding of the sepa.ration property, which plays a crucial role in LQG-models.

2

-2. The filter problem revisited

For completeness, the model and the problem formulation will be restated briefly.

Model: _{xt~l -~t } M~t'} xp E G(m,E) yt - Cxt f Nvt, vt E G(O,V) xt n-dimensional state

y k-dimensional output t

vt p-dimensional white noise

t E T-{O,l,...,t1} time index set

(1)

Compensator: _{ztfl - Ftzt } Gtyt}

Assumptions: zt is an n-dimensional process,z~ - m

Filter problem. determine Ft, Gt, t E T such that zt is an unbiased, minimum variance estimator of xt.

nxn From the unbiasedness, we infer Ft - A-GtC. Now let et :- xt-zt and Et: T--}R be the variance of et, then it remains to determine Gt, t E T with respect to

a suitable criterion, implied by the minimum variance condition.

The resulting problem will be formulated as follows:

Given the matrix recursion for the variance of the error

~ttl - (A-GtC)Et(A-GtC)T t (M-GtN)V(M-GtN)T, ~G

(2)

and the terminal costs

E[ J ]- E[eT M e ]- trace M F , M ~ 0 ( 3)

t1 t1 t1 t1 t1 t1 t1

determine Gt, t E T to minimize E[Jt ] 1

This problem will be solved with the aid of the matrix minimum principle. App1i-cation of the MMP yiel.ds:

Proposition

3

-Pt - (A-GtC)T Pttl(A-GtC) P - M ~ 0

tl tl

and for the first-order condition (by differentiation of the hamiltonian H)

aG - Pttl [A Et CT t MVNT - Gt (C Et CT f NVNT) ]- Onxk t

Proof. Appendix A, AS, A6 and A7 and cf. I 3.2.10 and I 3.2.11.t

Remarks. For gr.~ater clearity zero matrices will be subscripted with their dimensions.

From (4): Pt - Pt, t E T.

Here we pause a moment to discuss the result. It is not straightforward to

con-clude immediately from (5) the desired result

Gt -(A Et CT f M~INT) (C Et CT t NVNT) -1 4 Gt

since (Gt-Gt) is premultiplied with Pr}~.

~ ~

From

Pttl(Gt-Gt) - Onxk we can only deduce Gt - Gt if Pt}1 is nonsingular. This is always the case in the continuous-time setting, but not in discrete-time. If,

for example, the state matrix A is singular, it may happen that (A-GtC) is singular and by (4) Pt, t ~ tl becomes singular. This will be illustrated in the

following example.

(4)

(5)

(6)

Example

Define the Gaussian syster~ _{'xtfl -} Axt f B _w1t

yt - Cxt f _w2t

(7)

where wlt and w2t are independent white noises.

Apply a transformation to bring (7) into standard form (1)

xtfl - Axt t[ B 0] (W1) xtfl - Axt f Mvt

' 2 or yt - Cxt f Nvt

yt - Cxt f[0 I] (W11 vt E G(O,V) 2

Let xt :- E[xtlFt-1] where Ft-1 ~ a{ys, 0 ~ s ~ t-1}, then the Kalman filter obeys

4 -xttl - Axt f Kt [yt - Cxt ] Kt -(A Et CT t MV:~IT) (C Et CT t NVNT) -1 ~tfl - A Et AT f MVMT - Kt(C Et AT t NVMT) , ~0

Now take n-2, k-1, P-3, B-V-E~-I, m0 -(~), C- (1 0)

MVNT - (~)

-a, E R, a~ 0 then MVMT - NVNT - 1 and

~ C 1 0

Note that the pair (A,C) is observable since _{CA - 1 a}

has rank two if a~ 0.

For t- 0 we have: K- 1(1)0 2 0 ' A-K C0

F.1 -For t - 1: K1 aZ t 3~2 0 0 2a2f3 2 2a2t5 2a2t5 0 1 A-K1C i

and by induction it is easily seen that A-KtC is singular

for all t E T.

T

Hence Pt - ( A-KtC) Pt}1(A-KtC) will be singular for all t ~ tl.

Remark. At several occasions in the literature it is observed that in the dis-crete-time case the system matrix A is in fact the state transition matrix, and, by analogy of the continuous-time case, should be assumed nonsingular. By the same analogy Pt is assumed to be positive-definite.

-1 This enables to deal with Pt .

Now return to the first-order condition (5) and write it as

~

Pttl(Gt-Gt) - Onxk

-If we not assume that Pt, t E T is positive-definite, conclusions about Gt can only be made on the space spanned by the independent columns of Pt; on the

5

-space of Pt no conclusions can be made. It is still an open question whether there exists an alternative procedure to obtain an explicit expression for Gt, e.g. some recursive argument. Hoorever, the minimum principle only provides necessary conditions. Candidates for the optimal solution must be submitted to the sufficient conditions (i.e. the Hamilton-Jacobi-Bellman partial differential equation), which may in addition provide useful facts as uniqueness, global pro-perties, regularity conditions.

Evidently, a reasonable candidate is indeed Gt - Gt. And, as is well-known, from HJB-theory it can be proven that under certain conditions Gt - Gt is in-deed the unique solution for this problem.

The dual problem, i.e. the control problem, has been analyzed extensively in the vector-case. Then a full theory is availàble: HJB-theory provides sufficient conditions and the convexity of the Hamiltonian can be verified immediately,

2

since a2 can be calculated at once. au

3. The LQG- problem

3.1. The combined filter and control problem is dealt with in I 3.3. Just as in the filter case no constructive method can be found to obtain explicit expressions for the candidates of the optimal solution. But still reasonable candidates can be suggested, see the verification lemma below. The LQG-problem

is also discussed for the other representations of I 3.3.8, and since this ana-lysis is important, it will be presented in some detail.

First we regard the (é)-representation. Consult I 3.3 if necessary for details. To be complete, we repeat the equations for the system and the corresponding compensator.

P :

C .

(8)

(9)

In addition to the definitions of (1), ut is here the m-dimensional input or control process. Notice that the structure of the compensator is not as general as possible, but already appropriate

Let et :- xt - zt, then

E(2) .

for the problem to be discussed here.

A t BLt - BLt xt M t 0 A- KtC e t M- KtN tl-1

~~ o

[

t~-1 - Azt f But t Kt [Yt - Czt ] ut - Ltzt

i

(xTeT) i

` - LTRL

2nx2n

Let Et : T-~ R be the variance of (e) t, then:

-~-Now we can formulate the LQG-problem:

Given the "state-equation" ( 10), find Kt, Lt, t E T to minimize E[J].

Proposition. For (10) and (11), we obtain for the costate Pt: T-~R

~ T Pt - A Pttl A} Qt Qt 0 P - 1 tl 0 0

and for the two first-order conditions

[ BTP A f (BTP B t R) L](E - ET ) f

aH T 11 T11 t il 12

áLt - [ B P11BLt t B P12 (A-KtC) f RLt ](E22 - E12) - Omxn

áH P12 [(AfBLt) E12CT - BLtE22CT f MVNT] f aKt

-P22 [AE22CT f MVNT - Kt (CE22CT f NVNT) ]- Onxk

Proof. The costate equation follows from differentiating

H(Et' Ptfl' Kt, Lt) - tr{A Et A T Ptfl } MVM T Ptfl } Qt ~t}

2nx2n

(12)

(13)

(14)

with respect to Et; the first-order conditions follow from elaborating H in terms of ~T1 ~12 -: Et and pTl P12 -, pt}1 and

differentia-12 22 t 12 22

tfl ting with respect to Kt and Lt.

The arguments ttl and t of Pttl and Et resp. are omitted for brevity. ~ Verification lemma

The pair

Lt - - [BTP11B f R] -1 _BTP11A

Kt -(AE22CT f MVNT) (CE22CT t NVNT)-1

satisfies the first-order conditions (13) and (14).

Proof. From (10) and (12) we have by simply expanding

~22(tfl) - (A-KC)E22(A-KC)T f (M-KN)V(M-KN)T pll (t) -(AfBL)T pll (t}1) (AfBL)

t LTRL f Q P12 (t) -(Ai-BL) T P12 (A-KC) -(AfBL) T P11BL - LTRL

p22 (t) - LTBTpi 1BL -(A-KC) T p12 BL - LTBT P12 (A-KC) f(A-KC) T P22 (A-KC) f LT RL

Hence

(~22 - ~12) tfl - (A}BL) (E22 - ~12) (A-KC) T

- Kt [C E22 AT f NVMT -(C E22 CT t NVNT) KT ] , ~nxn

P12 (t) -- Lt [BT pl l A }(BT pl l B t R) Lt ] t(A-KC) T P12 (tfl )( AfBL) , Onxn and the above choice of Kt and Lt sets:

for all t E T (~22 - ~12) t - ~nxn

T - p for all t E T p12(t) _nxn

Now the two terms in (13) and (14) both vanish.

3.2. Along similar lines an equivalent result can be obtained for the (Z)-repre-sentation. The same notation is used as in 3.1.

xtf 1 ,,ztfl KtC BLt xt M f vt A t BLt - KtC zt KtN 'pt p tl-1 O E[J] - tr 1 Et f E tr ~ 0 0 1 t-0 ~tfl - A Et A~ f MVM , 1 LT ~Q 0~ costate : Pt - A T Pt} 1 A f Qt , t 1 0 0 first-order conditions: 1.0_,~ (15) (16) atí - _{[BT(p11tP12)A t BT(p12tP22)KtC] ~12 }} (17) aLt _{[RLt f BT} (P11 } P12 } p12 f P22) BLt f BT (p12 t P22) (A-KtC) ]~22 - Omxn

óH p12[A(E11 - E12)CT f BLt(E12 - E22)CT f MVNT ] t

(18) óKt

Now the verification lemma is slightly more involved.

Verification lemma

Lt --[ BT (P11 ~- P12) B f R]-1 BT (P11 ~- P12) A The pair

Kt - IA(E11 - E12)CT t MVNT ] IC(E11 - E12)CT t NVNT ]-1

satisfies the first-order conditions (17) and (18).

Proof. From (15) and (16) we have

E 1 1( tf 1) - A E 1 1 AT t A E 12 LT BT f BL E 12 AT t BL E 2 2 LT BT t MVMT

E 12 ( tf 1) - A E 11 CT KT f A E 12 ( AfBL-KC )-T -t- BL E i 2 CT KT f BL E 2 2( AfBL-KC ) T t f MVNT KT

E22 (ttl) - K C E11 CT KT t K C E12 (AfBL-KC) T f (A~-BL-KC) E12 CT KT t t (AfBL-KC)E22 (AfBL-KC)T f KNVNT KT

P11(t) - ATP11AfCTKTP12AtATP12 KC t CTKTP22KCfQ

P12 (t) - AT P11 BL t CT KT P12 BL t AT P12 (AfBL-KC) f CT KT P22 (A-1-BL-KC)

Y221~1 - LT DT P11 liL ~(AIBL-KC)T P12 BL ~ LTBT P12(AtBL-KC) ~ f(AtBL-KC) T P22 (AfBL-KC) -F- LT RL

and therefore

(ET - F. ) - (AfBL-KC) (ET - E ) (A-KC) T f

12 22 ttl 12 22

Kt [C ( E 1 1- E 12 ) AT t NVMT -{NVNT ~- C( E 11 - E 12 ) CT }KT] , ~nxn (P12 f P22) t-(AfBL-KC) T(P12 f P22) (AfBL)

-1-Lt [BT (P11 f P12) (AtB-1-Lt) f R-1-Lt ]' ~nxn

Now the above choice of Kt and Lt makes E12 -~22 - Onxn and Pi2 } P22 - Onxn for all t E T and the two terms in (17) and in (18) both vanish. ~

Remark. Lt and Kt are driven by (P11 } P12)ttl and (E11- ~12)t resn. Using the expressions for Lt and Kt as given in the verification lemma, these recursions obey:

(P11 _{} P12) t- AT (P11 } P12) ttl} (AfBLt) f Q

lU

-3.3. Summary

Since the notation E12, P22,... has been used throughout the different repre-sentations, the mnemonic Exz' ~ex' Pxx'" ' should be used now to compare the results of 3.1. and 3.2. Together with I 3.3.17 we have for the three different representations of the LQG-problem.

~~1) : (Z) representation -P - P xz zz ~~2) : (é) representation ~~3) : (é) representation EZe - 0 P - 0 ex P - P zz ze

11

-4. A remarkable formula for the LQG-case

Here we specialize to the (é)-representation. Analogous calculations can be made for the other two representatíons.

We repeat the model, costate equation and first-order conditions, see I 3.3.12-3.3.15. Let A M -A-KtC `S`t ut ~ K N' t K N t

then the state and costate equation respectively are

~tfl - AEtAT f MVMT

~ T ~

Pt - A Ptf 1 A } Qt '

The first-order conditions are:

( `1

`1 ~

Qtl

Qtl ~

3H ~(BTP11BfR)Lt t BTP11A]~11 }

iiLt -~BT pl l KtC f BT P12(A-KtC) _{]~12 - OmXn}

aH (P11 - P12) [(AfBLt) E12 CT f Kt(C E22 CT f NVNT)] ~ aKt -(p12 - p22) ~A ~22 _{CT f MVNT - Kt(C E22 CT f NVNT)]} - Onxk

12

-[(AtBLt) E12 CT t Kt(C E22 CT t NVNT) ] Kt t ~11(ttl)

-(AtBLt) ~ il(AtBLt) T t Kt C E 1 2(Ai-BLt) T~~T T [A E22 CT t MVNT - Kt(C E22 CT t NVNT)] Kt t E (tfl)

-12

(A-Kt C) E12 (AtBLt) T. ~nxn

of Lt and Kt

The E- and P-blocks at the RHS of these expressions should be evaluated at t and ttl respectively.

Now we see the remarkable fact that the coefficients show up in the first-order conditions (21) and (22).

nx2n 2nxn

of IIt : T-~ R and St : T-~ R such that

~t '- [ (P11 - P12) (P12 - P22)] t

(25)

(26)

in (23)-(26), This leads to the definition

If we postmultiply (23) and (24) by E11(t) and E12(t) resp. and premultiply (25) and (26) by _{(P11 - P12) ttl} and (P12 - P22) ttl resp. , after substitutíng the first-order conditions (21) and (22) and arranging the different terms, we arrive at

T[t St -(A-Kt C) T~tt1 A St ~ttl Stfl - ~tt1 A St ( A-F-BLt)

T

(27)

(28)

(27) can be seen as a backward recursion in Iit, (28) can be seen as a forward recursion in St.

Note at the dominant role of the product lIt}1 A St, which will show up again later in the Nash-compensator case and also in an alternative derivation of the LQG-result in section 5.

Also notice that from the optimal LQG-model we have P11 - P12 - 0 and E12 - 0 for all t E T, implying that T[t St - 0 for all t E T.

However, there seems to be no constructive way to obtain this or a símilar result (e,g. _{Ilttl A St - 0) directly} from the first-order conditions combined with (23)-(26). It is conjectured that this result (]?tSt - 0 or ~ttl A St - 0) is intimately related with the separation property.

Therefore the separation property must be understood in the LQG-case and to that aim a more concise and insightful derivation of the LQG-result will be presented

14

-5. The separation property for the LQG-model

In this section we use the model of section four, i.e. the (é)-representation, under the additional assumption that system- and measurement noise are uncorre-lated (MVNT - 0). This is no serious restriction, but will elucidate the duality between the control and the filter part more clearly.

The presented derivation here will be different from the one in section 3. Ex-panding the state and costate equations is rather tedious, especially in the

two-DM case with dimensions 3nx 3n and nine blocks for Pt and Et. This can be E R2nX2n

such that avoíded by introducing E1, E2

Ei ;- (ó) E2 :- (~) where I is the n x n unity matrix.

Now, cf. (19) and (20) the augmented system is characterized through:

A - E1 A E1 t E2 A E2 f E1 BL E1 t(E1-E2) KC E2

,~, T (E1-E2)KNVNT KT(Ei E2) T t E2 MVMT E2 f

Q MVM -E2 MVNT KT (E1--E2) T t(E1--E2) T KNVMT -E2 - (E1fE2)Q(E1fE2)T t E1 LT RLE1 H - tr {A Et AT_{Pttl } MVM T Pt}1 } Qt ~t}} aH - BT ET P _{A E E -f- RL ET E E - 0} (29) aLt 1 ttl t 1 t 1 t 1 mxn aH _{-(E -E ) T p A E E CT f( E -E ) T P (E -E ) K NVNT - 0} (30) aK 1 2 tfl t 2 1 2 tfl 1 2 t nxk t

By expanding (29) and (30) we obtain the earlier derived expressions (21) and (22). Notice the duality between (29) and (30). Obviously the factor Pt}lA ~t is invariant under duality: Now from this viewpoint we will investigate the se-paration property. To obtain a separation result, we need a solution of (29) and

(30) in the form:

K - K(E)

15 -T

So from ( 29) it is seen that ElEt E1 must cancel,

from (30) it is sec:n that _{(E1-E2)T Pttl ( E1} -E2) must cancel.

This is indeed the case, as will be verified now.

We know that for the optimal solution of the LQG-model we have

L

T

E12 - 0

T

P11-P12 - 0

Using this condition, the first-order conditions reduce to:

AfBL K C ~E i) _{BTE1Ptf1A~tE1 (31)}

BT(p11P12)

t t 11

0 A-Kt C ~0

-(p12-P22) (A-KC) E22 C -(E1-E2)_Ptfl E2 (A-KC) E2 Et E2 C AtBL K C 0 ii) (E1-E2) T ptfl A~t E2 CT (31) (0 P12 p22) t t CT 0 A-Kt C E2` T - T T So what happens?

If we insert (31) in the áL - 0 equation, A"reduces" to t E1 (AfBLt) E1 -A for all t E T AfBLt BT P11 (AfBLt) E11 - BT E1 Pttl E1 (AtBLt) E1 Et E1

and in the áK - 0 equation t

` 0 0

"reduces" to EZ(A-Kt C) E2 - 1 ~ So, from (29) ,(30) and (31) we conclude

BT E1 pttl E1 (AfBLt) E1 Et E1 f RLt Ei Et E1 - Omxn

(E1-E2)

T Ptfl (E1E2) (AKt C) E2 Et E2 CT

-(E1-E2)T _pttl (E1-E2) Kt NVNT - Onxk

(31)

Assuming Ei Et E1 and

16

-BT E1 Ptfl E1 (AfBLt) f RLt - Omxn

17

-Appendix A: The matrix minimum principle

Theorem

Given:

(A1) state equation Xt}1- Xt - F(t,Xt,Ut), X~ tl-1

(A2) costs J- K(Xt ) f E L(t,Xt,Ut)

1 t-0

(A3) Hamiltonian H(Xt'Pttl'Ut) ~ L(t,Xt,Ut) f tr [F (t,Xt,Ut) Ptfl~ where X : T ~ Rnlxn2 mlxm2 U : T ; R nixn2 mlxm2 nlxn2 F: T x R x R -~ R nlxn2 K : R -~ R nlxn2 mixm2 L: T x R x R -~ R P : T x Rnlxn2 nlxn2 nlxn2 mlxm2 H: R x R x R ~ R

T - {O,l,...,ti} time index set.

If Ut is the optimal unconstrained control and Xt the corresponding state trajec-tory, then there exists a costate matrix Pt, t E T such that

18

-(A7 ) aH

_aut

x mlxm2

19

-Appendix B: Reduced order compensators

A still partially unsolved problem is the reduced-order compensator. xtf i- A xt t B ut t r~ vt

Let the system be:

yt - C xt f N vt

and let the compensator have the most general structure zttl - Ft zt t Gt ut f Kt yt

ut - Lt zt

where _{xt is the n-dimensional state process,}

zt is the nl _{(~n) dimensional compensator and} Ft, Gt, Kt, Lt are unknown matrices.

(B1)

(B2)

These four unknowns cannot be determined in a straightforward way if nl ~ n; compare part I, section 4.

Cohen [1] _{gives only necessarily conditions for this problem, derived for a}

COT1tir1Lni~c-tim2 modcl ir. an approach compïeteiy parallel as presented here.

Sims [4] _{assumes that the matrix F is known by some device and applies the MMP}

to determine Kt (no control process present).

Due to these problems we will restríct to n-dimensional compensators.

-20-References

[1] G. Cohen: Partage de 1'information entre systèmes stochastiques a priori indépendants, RAIRO Automatique~Systems Analysis and Control, vol. 15, pp. 45-68 (1981).

(2] M. Athans, Optimal Control, McGraw-Hill (1966). P.L. Falb:

[3] M.D. Merbis: On the compensator.

Part I: problem formulation and preliminaries, Reeks "Ter Discussie", 82.12, July 1982.

1

IN 1982 REEDS V~RSCFiENEN:

O1. W, van Groenendaal _{Building and analyzing an jan,}

econometric model with the use of a hybrid computer; part I.

02. M.D. Merbis _{System properties of the jan,}

interplay model

-03. F. Boekema _{Decentralisatie en regionaal maart}

sociaal-economisch beleid

04. P.T.W.M. Veugelers _{Een monetaristisch model voor}

maart de Nederlandse economie

O5. F. Boekema _{Morfologie van de "Wolstad".}

april Over het ontstaan en de

ont-. _{wikkeling van de ruimtelijke}

geleding en struktuur van Tilburg .

06. P. van Geel _{Over de (on)mogelijkheden mei}

van het model van Knoester. 07, J.H.M. Donaers,

F.A,M, van der Reep

08. R.M.J. Heuts

09. B.B. van der Genugten

10. J. Roemen

li. J. Roemen

12. M.D. Merbis

13. P. Slangen

14. M.D. Merbis

De betekenis van het monetaire beleid voor de Nederlandse eco-nomie, presentatie van een ana-lys~ aan de iiand van een een-voudig model

The use of non-linear

trans-formation in ARIMA-Models when the data are non-Gaussian distributed

mei

juni Asymptotic normality of least

squares estimators in auto-regressive linear regression

models . j ~i

Van koetjes en kalfjes I juli

Van koetjes en kalfjes II juli On the compensator

Part I

Problem formulation and

prelimi-naries _juli

Bepaling van de optimale beleids-parameters voor een stochastisch kasbeheersprobleem met continue

controle _aug.

Linear - Quadratic - Gaussian

2

15. P. Hinssen J. Kriens ~

J. Th. van Lieshout

Een kasbeheermodel onder

onzekerheid sept.

16. A. Hendriks en

T, van der Bij-Veenstra

17. F.W.M. Boekema A.J. Hendriks L.H.J. Verhoef 18. B. Kaper 19. P.F.P.M. Nederstigt 20. J.J.A. tioors 21. J. Plasmans H. Meersman 22. J. Plasmans H. Meersman

23. B.B. van der Genugten

24. F.A. Kense

25. R.T.P. Wiche

26. J.A.M. Oonincx

"Van Bedrijfsverzamelgebouw

naar Bedrijvencentrum" okt. Industriepolitiek, Regionaal

beleid en Innovatie okt.

Stability of a discrete-time, macroeconomic disequilibrium model.

Over de toepasbaarheid van het Amerikaanse 'Diagnosis Related Group'-systeem in Nederland

Auditing and Bayes' Estimation

An Econometric Quantity Ratio-ning Model for the Labour Market.

okt.

nov. nov.

nov. Theorieën van de

werkloos-heid. nov.

Een model ter beschrijving van de ontwikkelino van de veestapel

in Nederland. nov.

De omzet~artikel

concentratie-curve als beleidsinstrument nov. Populaire wetten~specificatieve

wetten, oftewel

IN 1983 REEDS VERSCHENEN O1. F. Boekema L. Verhoef 02. R.H. Veenstra J. Kriens 03. J. Kriens J.Th. van Lieshout J. Roemen P. Verheyen 04. P. Meys 05. H.J. Klok 06. J. Glombowski M. Kruger 07. G.J.C.TH. va-, Schijndel 08. F. Boekema L. Verhoef Enterprise Zones.

Vormen Dereauleringszones een adeauaat instrument van

regio-naal sociaal-economisch beleid? jan. Statistical Sampling in Internal Control Systems by Using the A.O.Q.L.-System.

Management Accounting and Operational Research

jan.

jan.

Het autoritair etatisme jan.

De klassieke politieke

economie geherwaardeerd _febr.

Unemployment benefits and

Goodwin's growth cycle model febr. Inkomstenbelasting in een

dynamisch model van de onder-neming

Local initiatives: local enter-prise agency~trust, business in

the community

febr.

Bibliotheek K. U. Brabant

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