On the set of obtainable reference trajectories using minimum variance control

On the set of obtainable reference trajectories using minimum

variance control

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Engwerda, J. C. (1987). On the set of obtainable reference trajectories using minimum variance control. (Memorandum COSOR; Vol. 8723). Technische Universiteit Eindhoven.

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EINHOVEN UNIVERSITY OF TECHNOLOGY Department of Mathematics and Computing Science

COS OR-memorandum 87-23

On the set of obtainable reference trajectories using minimum

variance control

by

J.C.

Engwerda

Eindhoven University of Technology

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

5600 MB Eindhoven The Netherlands

Eindhoven, September 1987 The Netherlands

On the set of obtainable reference trajectories using minimum variance control

ABSTRACf

In this paper we detennine the set of obtainable, i.e. asymptotically stabilizable. target paths if we use minimum variance control. We show that any reference trajectory belonging to this set must be the solution of a dynamic equation which has the same structure as the considered dynamic system.

If restrictions are imposed on the controls we show that these restrictions, if they are modelled via the cost criterium, show up directly in the dynamic evaluation of these trajectories. In case the applied control may only vary within the limits of predescribed values, application of optimal control results in a bang-bang control. The admissibility conditions become then more com-plicated.

The obtained results are illustrated by means of a simulation study. The analyzed system is assumed to be linear, finite dimensional and discrete time.

2

-L Introduction

In economics often the question is posed whether it is possible to track (exactly) a given set of economic target variables along any lime path by means of a good choice of the policy instru-ments. This subject is known as the target path controllability (fPC) problem. For time- invari-ant systems this problem has been studied in [2,4,8,13-17] and for continuous time-varying systems in [19J. In this paper the TPC-problem is viewed as a problem of optimal stabilization (see [3,14,18]). Only such reference time paths are called admissible which are asymptotically stabilizable by means of a control algorithm that results from the minimization of a special cost criterium.

We shall trace the set of admissible reference time paths for the minimum variance (MY) cost criterium. This criterium expresses that positive and negative deviations of target variables from desired levels are weighted equally and that they are increasingly costly. The major rea-sons to study in particular this criterium are:

i) Because of the uncertainty in the real-life macro-economic situation there is a constant need for short period adaptation of control with respect to new information. A regulator which is based on minimizing a short horizon makes such an adaptation possible.

The MY -controller is a typical example of a regulator which satisfies this requirement (see [1],[5],[6]).

ii) The computational ease and relatively simple formulas of the control algorithm.

However, in practice a government is subjected to limits on the amount of its expenditure. This has its impact on the admissible reference trajectory set. This set will become smaller. In order to analyze the effect of limited control possibilities they are modelled in two ways.

First we alter the cost criterium. An additional term is introduced in the MY -cost criterium which penalizes quadratically a deviation of the applied control from its reference value. The advantage of this new created problem is that it can easily be treated mathematically.

The second, more natural, way to model restrictions is obtained by assuming that every control may only vary within a certain predescribed interval. However, the disadvantage of this model-lation is that the control scheme becomes nonlinear, and that the effect on the admissibility of a reference trajectory becomes less obvious.

Throughout the paper the obtained theoretical results are illustrated by a small economic model. The description of the model itself is postponed to the last section of the paper.

3

-D. Definitions, tools and mathematical preliminaries

The base system analyzed in this paper is described by the following linear, finite dimensional, time-varying, difference equation:

y(k + l)=A(k)y(k)+B(k)u(k)+C(k)x(k); k =0,1,··· (1)

where '1 (k) is an n -dimensional output vector observed at time k which must be controlled; u (k) is an m ·dimensional input/control vector with m S n: x (k) is a p -dimensional detenninis-tic input vector, called exogenous noise and is assumed to be known at time k. The initial values of the system are yeO)

=

yeO)

and x (0)

=

x(O). It is assumed that all matrices are bounded in time. and that the matrices B (k) are all full column rank (injective).

In this paper also time-invariant systems are studied (i.e. systems for which matrices

A (k). B (k) and C (k) are constant in time). For these kind of systems we introduce some addi-tional well-known concepts and notalion. For fonnal definitions and proofs we refer the reader to standard textbooks like [10.12,20],

Consider the system y(k + 1)

=

A y(k) + B u(k); yeO)

=

yeO).

Let R be the set of states reach-able from 1(0)

=

O. Then R equals the linear subspace 1mB + A 1mB + '" + A I I -11mB. This

subspace is usually abbreviated by <A I B >. It can then be proved that the set of all zero con-trollable initial states consists of {y (0) I A 11-1y (0) E <A I B >}. We will abbreviate this set by

<A I B >0'

furthennore, the set of initial states '1(0) for which the solution of y(k + 1)

=

A y(k) converges to zero when k tends to infinity is called the stable subspace and is denoted by X-(A).

The following theorem for time-invariant systems is well-known: y (0) is zero stabilizable (i.e. there exists a control function u ( • ) such that y (k) -+ 0, when k -+ oo) if and only if '1 (0)

belongs to the subspace X-(A) + <A I B >0' Now return to base system (1).

Assume that a cost criterium J is given which has to be minimized.

Then. the difference between output and reference vector y (k) - y. (k). when the optimal con· trol (w.r.t. J) is applied to the system. is called the closed-loop (CL) or control error and denoted by e (k).

Definition

A reference trajectory is called admissible with respect to J and y (0) if there exists a control function u (. ). minimizing J. such that the corresponding error function e ( • ) converges to zero when k tends to infinity.

-4-Remark

From now on we shall omit J and y(O) when we talk about the admissibility of a reference

tra-jectory if it is clear which cost criterium and initial state are meant.

In this section we delennine the set of admissible reference trajectories for the following cost criterium:

J

=

eT(Ie + 1) Q(k) e(k + 1) + (u (k)-u*(k»TR (k) (u(k)-u(k» (2)

where u"'(Ie) is the desired level of control. Q(k) is a positive definite matrix. and R(k) is a semi-positive definite matrix.

The reason for studying this cost criterium is that both the MY-cost criterium and its restricted version result easily from this one.

Before we state the result we introduce some notation.

A T will denote the transpose of matrix A. K (A) an injective matrix from which the image represents the kernel of matrix A. and

Lemma 1:

Minimization of J w.r.t. system (1) results in the following CL-error equation:

e(1e + 1) ;:;: M (1e)A (k )e(k) + M(k)[A (k)y"'(k) + B (k)u*(k) + C(k )x(k) - ,·(k + 1)l

here M (k) = M (k )(Q (k ),R (k

».

Moreover, if a reference trajectory ,·(k) is admissible then there exist vector sequences u (. ) and v(· ). where v(k) converges to zero when Ie tends to infinity. such that:

, . (Ie + 1) = A (Ie )y. (k) + B (k )u· (k) + C (k )x (k) + K (M (k »u (k) + M (k )v (Ie )

Proof:

5

-ID. The admissible reference trajectories

In this section we characterize the admissible reference trajectories if MV -control is used for the regulation of the system (1).

Fonnally, the MV -cost criterium is obtained by taking R (k) and u

*

(k) equal to zero in (2). As a consequence K (M (k» then becomes B (k).

Theorem 1:

A reference trajectory is admissible w.r.t. MV-control if and only if there exist u(· ) and v(. )

such that the following two conditions are met:

i)

ii)

Proof:

e(O) := y(O) - y*(O) is stabilized by means of v ( • ) in the following linear system:

e (k + I)

=

M (k}A (k)e (k) - M (k)v (k); e (0) = e(O) . y*(k + I)

=

A (k)y*(k) + B(k)u(k) + C(k)x(k) + M(k)v(k)

.. ~" Substitution of the above mentioned parameters into lemma 1 yields the necessity of con-dition ii).

Substitution of ii) into the corresponding control error equation (see lemma 1 again) yields that:

e(k + 1)

=

M(k)A(k)e(k) - M(k)v(k), where v(k) ~ 0 when k ~ 00

e(O)

=

e(O) .

So, we have that e(O) is stabilized, which completes this part of the proof.

"e=" Equation ii) implies that M(k){A(k)y*(k) + C(k)x(k) - y*(k+l)} equals -M(k)v(k) at any

time k.

So, the error equation can be rewritten as:

e(k + 1)

=

M(k)A(k)e(k) - M(k)v(k) e(O) = e(O) .

Since. by assumption i), this vector sequence v (. ) also stabilizes e(O) it follows that a refer-ence trajectory satisfying assumption i) and ii) will be admissible.

0

Theorem 1 leads to the, intuitively very appealing, resulL that any admissible reference trajec-tory must satisfy a recurrence equation which corresponds to the system. The only difference with the system is that the reference trajectory may posses an additional disturbation, which

6

-converges to zero when k tends to infinity. Note that, if the nonn of matrix M(k)A(k) is smaller than one, this condition is also sufficient to conclude admissibility of a reference path. Moreover. for time-invariant systems it is clear from section II that an e(O) stabilizing vector sequence v(. ) exists if and only if e(O) is an element of the subspace X-(MA) + <MA I M >0. For the Kendrick model (see section V), where m

=

n

=

2, this condition is trivially satisfied since M

=

0 and therefore X-(MA)

=

R2 and <MA I M >0:: O. A closer analysis shows that in

this case any trajectory is admissible. In fact this is due to Tinbergen's rank condition, Le. rank B (k) ~ n. In the next corollary we show that this rank condition is easily reobtained as the condition for solvability of the TPC problem.

Corollary 1:

In system (1) is every reference path admissible if and only if the number of control variables is greater than or equal to the number of target variables.

From theorem 1 it is clear that in case the number of inputs is equal to the number of outputs, every reference path may be admissible.

Straightforward calculation shows that matrix M (k) equals zero.

Therefore, we conclude from the error equation that the control error equals zero for all k. So, any reference path is admissible in this case.

On the other hand, when the number of inputs is smaller than the number of outputs, easily a reference trajectory can be constructed which is not admissible. This completes the proof.

0

A last remark w.r.t. the theorem concerns the influence of the weighting matrix Q on the admissibility of a reference trajectory.

From the second condition of theorem 1 it is clear that this matrix has no direct influence on the admissibility. However, indirectly it has its impact because it does influence the stability of the CL-error equation. Engwerda and Olter studied this aspect of the MY -controller in more detail in [9J.

IV. The influence of control restrictions

In this section the innucncc of comrol restrictions on the admissibility propcny of a reference trajectory is investigated.

First. we analyze the effect via a modified MV·cosl criLeriwn.

We assume that the cost criterium is given by (2), in which R (k) now is taken positive definite. As a reSUlt, M (k )(Q (k). R (k» is then positive definite too, which implies that K (M (Ie» is zero . . These considerations lead to theorem 2. The proof of it is analogous to that of theorem 1.

Theorem 2:

A reference trajectory is admissible if and only if there exists a vector sequence v( • ) such that the following two conditions are met:

i) e(O) := ,(0) - ,·(0), is zero slabilizable by means of v(· ) in the following linear system:

e(Ic + 1) = M (k)A (k)e(k) - M (k}v(k); e(O)

=

e(O)

ii) ,·(k + I)

=

A (k)y"'(Ie) + B(Ie)u·(k) + C(k)x(k) + M(Ic)v(k)

fJ

Note that the requirement that the applied control should not deviate too much from its set point trajectory, u.*(k), shows up nicely in the admissibility condition. In contrast with section

III. where an admissible reference trajectory which was disrupted by any control input

sequence remained admissible, we see now that only those reference trajectories from which

the dynamic evaluation is in correspondence with the set point trajectory u*(k)

are

admissible. For time-invariant systems the same propeny w.r.t. the existence of an F(O) stabilizing sequence v(. ) as in the previous section holds.

Straightforward calculation shows thai in the Kendrick model the eigenvalues of MA are

0.582 + 0.195i. So. they are situaled inside the unit circle which implies that X-(MA)

=

JR2

again. Note that also <MA I M>o equals JR2 in this case. Therefore a reference trajectory is admissible in this model jf and only if it is genenHcd as follows;

,*(Ie + 1)

=

Ay·(k) + Bu"'(k) + Cx(k) + v(Ie); ,(0)

=

YeO) and v(·) -+ 0

Now assume thai the restrictions arc modelled as follows:

minimize eT <k + 1) Q (Ie) e <k + 1), given

d.

(k) ~ uj(k):; d.(k)

"(k)

where u.(Ic)

=

(Ul(.t),· .. • u",(Ic)} and d.(k) and d,(Ie) are given constants i = l •..• m.

Then. if the control possibilities arc restricted for I consecutive time steps. the effect on the admissibility of a reference trajectory can be characterized exactly.

8

-Then, if the control possibilities are restricted for I consecutive time steps. the effect on the admissibility of a reference trajectory can be characterized exactly.

Before the theorem concerning this subject is discussed, we introduce some notation.

From now on uG (k) will denote the applied control at time step k and uapl(k) the optimal con-trol at time step k if there exist no bounds on the permitted extent of control at this time step. Furthermore. will 6 u (k) denote the difference between uG(k) and u"l"'(k).

Theorem 3:

If the control possibilities are restricted for I consecutive time steps. then a reference trajectory that is admissible in the sense of section III remains admissible if and only if

B Au(ko + 1- 1) + MAB 6u(KO + I - 2) + ... + _{(MA)I-IB Au(ko)} belongs to the stable space of matrix MA.

Here ko denotes the time at which the bounds first became effective.

frrulf:

9

-V. The influence of white noise

In this section we shall briefly comment on what happens with the admissibility conditions when the system (1) is disturbed by white noise. In fact we will show that these conditions remain the same. To prove this. we shall reconsider some fonnulas. The problem to be solved is now 10 minimize the expected quadratic cost functional E (J 1. see section II. subject 10 the constraint:

y(k + 1) = A(k)y(k) + B(k)u(k) + C(k)x(k) + w(k) •

where w(k) is a white-noise vector with cov.(w(k)wT(I)} = L",Oal' Here Oal.the kronecker delta, equals 1 if k = I and 0 otherwise.

Due 10 the fact that the system noise is white. we know that the optimal control minimizing

E (J) is equal to the optimal control obtained in lemma 1. So the CL-error equation now becomes:

e(k + 1) = M(k)A(k)e(k) + M(k)(A(k)y*(k) + B(k)u*(k) + C(k)x(k) - y*(k + I)}

+ w(k)

From this we obtain that the expectcd CL-error is givcn by the recurrence equation:

E (e(k + I)}

=

M{k)A(k) E (e(k») + M(k)(A(k)y*(k) + B(k)u*(k) + C(k)x(k)

- y*(k + I)} .

So, the results of theorems 1 and 2 remain valid. in the sense that they give necessary and sufficient conditions for convergence of the expected CL-error 10 zero.

However, in case the closed-loop system matrix MA is not asymptotically stable the error covariance grows 10 infinity. So, in general for practical situations these theorems make no sense. To have a more valuable analogue for time-invariant systems of theorem I, we state the following corollary:

Corollary 2:

A reference trajectory is admissible (in the sense that the expected CL-error converges to zero) if there exist u ( • ) and v (. ) such thal:

i) y*(k + I)

=

Ay*(k) + Bu*(A) + Cx(k) + K(M)u(k) + Mv(k). where v(k) -+ 0

ii) matrix MA is stable .

10

-The other results mentioned in the previous sec lions can be generalized in the same way. We do not go into any further detail about this subject.

11

-VI. A simulation study

The simulation study is perfonned on a macro-economic model estimated by Kendrick for the U.S. economy in [11].

He considered the following reduced fonn model:

where

C (k)

=

Private Consumption ;

1 (k)

=

Bruto Privale invesunent

U I (k)

=

Governmental Expendilures

u2(k)

=

Money Supply ;

x (k)

=

Exogenous Noise variable

yT(k) = (vf(k) vI(k» is a white noise vector with cov[V(k) yT(s)} = l:vliu

All quantities are measured in billions of dollars, in quaner k .

The parameters he obtained are respectively:

_ rO.914 -0.016]. _ [0.305 0.424] rC1] [ -59.437]

A - lO.097 0.424' B - -0.101 1.459 ; lC2 x(k)

=

-184.766

[

3.73

0]

l:v = 0 8.58' with initial values C (0) = 387.9 and 1 (0)

=

85.3 The reference trajectories are given by the following recurrence equations:

rc*(k)]_ [1.0075 0 ] rc*(k-I)] [Ut(k)]_ [1.0075 0 ] rut(k-I)]

l/*(k) - 0 1.0075 lr(k -I) and ui (k) - 0 1.0075 lui (k -I)

with initial values C*(O)

=

387.9; 1*(0)

=

85.3; u}" (0)

=

110.4 and ui (0)

=

157.3. The considered weighting matrices are:

rO.0625

0]

[I 0]

Q =

l

O l a n d R = 0 0.444 .

Note that the reference trajectories are growLh paths, and that matrix B is invertible.

From corollary 1 we know that in case the number of inputs equals the number of outputs any reference trajectory is admissible when MY -control is used to regulate the system. This is

12

-illustrated in fig. 1. In this experiment we took in the above mentioned model matrix R equal to zero, what corresponds with applying MV-control. We see that indeed the consumption- and investment reference trajectory are tracked exact When restrictions are imposed to the applied control sequence, things change drastically.

This is illustrated in figs. 2-4. Here it is assumed that the control should not deviate too much from a predescribed trajectory. This is modelled by the alternative cost criterium (see section IV).

In fig. 2 the simulation results arc given for the model described above. We observe that the obtained closed-loop system is now unstable, though its system matrix MA is aymptotically stable. By altering the consumption- and investment reference trajectory conform theorem 2 we see that the system becomes stable again (fig. 3). That it is essential in this case that in the generation of these trajectories the desired trajectory for the input variables is considered, is shown in fig. 4. Here we see that when the chosen exogenous input sequence used in the gen-erated consumption- and investment reference trajectory differs from the desired input trajec-tory, the system becomes unstable again. The exogenous input in this experiment was taken identically zero, and the desired input trajectory similar to experiment 3.

In order to show that in case the number of inputs is smaller than the number of controlled variables there are still a lot of consumption- and investment trajectories that can be tracked, an experiment is performed with one input and two outputs. The chosen input is the money sup-ply, and MV-control is applied to regulate the system. The model parameters are taken as above. As a result the closed-loop system matrix MA is asymptotically slable again. To satisfy the conditions of theorem 1 the reference trajectories of consumption and investment are gen-erated with a growth matrix equal to A, an arbitrary exogenous input sequence and the exo-genous noise component of the system. The initial reference values for consumption and investment were respectively 300 and 170. The results are shown in fig. 5.

At last the effect of bounding the control absolute is simulated (see section IV). In this experi-ment we assume that the input reference trajectory is generated like in experiexperi-ment 2. The per-mitted deviation of the applied control from this setpoint trajectory is assumed to be at any time at most ten percent. The simulation results are shown in fig. 6. From fig. 6iii we see that the money supply brings on the instability of the closed-loop system. Furthermore we see that during the first sixty quarters the tracking properties are somewhat better than in experiment 2 and that the control exhibits a bang-bang behaviour. When both control bounds become effective we see that the destabilization effects are much greater than those in experiment 2. This in spite of the fact that the total amount of control applied to the system is greater (as well for the money supply as for the government expenditures).

-13-Conclusions

In this paper we showed that an admissible reference trajectory for the MV -controller is gen-erated confonn the system dynamics. In general this dynamic evaluation condition is not enough 10 conclude admissibility. An additional necessary and sufficient condition for admissi-bility was that components of noise appearing outside the image of matrix B and which do not show up in the system are stabilized by the closed-loop system matrix.

When we extend the cost criterium with a component in which the amount of control is penal-ized we saw that the admissibility freedom w.r.l. image B is lost.

An admissible reference trajectory is now generated according to the system dynamics and the desired setpoint sequence of the input. Any disturbance of it must be stabilized by the closed-loop system matrix.

From the simulations it was seen that the more natural way of modelling bounded control pos-sibilities (see section IV-B) resulted in the shon run to a bener tracking of the target trajec-tories at the expense of a bang-bang control. The consequences of this policy in the long run were disastrous. Compared with the cost criterium modellation of the problem we saw that more control effon was needed to oblain a much worse tracking result. So as well from a mathematical point of view as from a pmclical point of view, analyzing limited control possi-bilities via an alternative cost criterium seems to be preferable.

An indirectly rcoblained result in this paper is Tinbergen's counting rule for the TPC-problem. We showed that a discrete time-varying system, with matrix B injective, is TPC if and only if the number of inputs is greater or equal than the number of outputs.

-14-Appendix I

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-17-Appendix II

Proof of lemma 1:

By straighuOIward differentiation of cost criterium J it is seen that the optimal control for sys-tem (1) minimizing J is:

u(k)

=

_(BT (k)Q (k)B (k) + R (k )r1B T (k)Q (k )(A (k )y(k) + C(k )x(k) - y*(k +

1»

+ (BT(k)Q(k)B(k) + R{k)rIR(k)u·(k) .

Substitution of this control into base system (1) leads to the following control error equation:

e(k + 1)

=

M(k)A(k)y(k) + M(k)C(k)x(k) - M(k)y·(k+l) + + B (k )(B T (k )Q (k )B (k) + R (k )

r

I R (k )u • (k) •

where M(k)

=

M(k)(Q(k),R(k» is as defined in section II.

Some matrix manipulation shows that

B (k)(B T(k)Q (k)B (k) + R {k)rIR (k )u"'(k) equals M (k)B (k)u·(k).

So the above error equation can be rcwriLl.cn as:

e(A: + 1) =M(k)A(k)e(k)+M(k){A(k)y"'(k) + C(k)x{k)+B(k)u*{k)-y·(k + 1» •

which proves the first pan of the lemma.

Now. since M{A:)A(k) is bounded. a necessary condition for convergence of e(k) to zero is that

M(A:)(A(k)y*{k) + B(k)u*(k) + C(k)x(k) - y"'(k + 1) converges to zero.

Since the kernel of M(k) is in general unequal to zero, we can rewrite this last condition as:

y"(A: + J)

=

A (k)y*(k) + B(k)u*(k) + C(k)x{k) + M(k)v(k) + K{M (k»u (k)

for some u (A:) and v (A:). where v (k) converges to zero when" lends to infinity. []

Proof of theorem 3:

Let e(q)(ko + k) denote the comrol error at Lime ko + k. when from lime ko to ko + q application of optimal control is not possible.

Then. it follows from the idcmity y(i + 1) - y"'(i + 1)

=

Ay(;) + Bu(i) + Cx{k) - y*(; + 1) that the equations

-18-e(j){ko + i)

=

Ay(ko + i - I ) + _{Bu"(k o} + i - I ) + Cx(k o + i - I ) - ,*(.1:0 + i)

and

e(i-I)(k o + i) = Ay(ko + i - I ) + BUOPI(k_o + i - I ) + _{Cx(k o} + i -1 ) - y*(k o + i) hold Subtracting one equation from thc other then yiclds:

(a)

Note that the admissible trajectories with respect to the cost criterium considered in this sub-section are a subset of those considered in sub-section Ill.

So, when after ; - 1 time steps the bounds on the ex.tent of control disappear, we have from theorem 2 that _e(i-I)(ko + i) is equal to

(b) _{MAe{i-l)(k o} + i - I ) - Mv(ko + i - I )

Substitution now of (b) into (a) gives:

e(l)(k o + I)

=

B .1u(ko + 1- 1) + MAE .1u(ko + I - 2) + ... + _{(MA)I-lB .1u(ko)} + e(ko + I) • where e(ko + I) = _{e(O)(k o} + I).

Since no restrictions exist anymore from lime step lon, il is easily shown by induction that for

k greater than zero

e(l)(ko + 1+.1:) - e(ko + I + k) equals _{MA(e(I)(k o} + I + k - 1) - e(ko + 1+ k -

1»,

-19-REFERENCES

1. Aalders L., Engwerda J.C., and Olter P.W. (1983), Selftuning control of

a

macroeconomic system

Proceedings of lhe 4lh IFAC/lFORS/llASA on Economic Dynamics and Control Washington D.C .• U.S.A., 1984

2. Aoki M. and, Canzoneri M. (1979),

Sufficient conditions for control of larget variables and assignment of instruments in dynamic macroeconomic models

International Economic Review vo1.20, no.3, pp.60S-616

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Notes and Comments: On Sufficient Conditions for Optimal Stabilization Policies Review of Economic Studies 40, pp.131-138

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6.

On a generalization of Tinbcrgen's condition in the lheory of policy to dynamic models Review of Economic Studies vol.42, pp.293-296

o

Astrom K.J. (1983),

Theory and applications of adaplive control -

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Astrom K.J., and Wiuenmark B. (1984), Computer Controlled S ystcms

Prentice Hall

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Applied Matrix Algebra in the Stalis(ica1 Sciences, pp.163 Nonh Holland. New York

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-20-9. Engwerda J.C., and Otter P.W. (1986).

The reference stability of a macroeconomic system with a recursive minimum variance control equation

Submitted for publication

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Caution and probing in a macroeconomic model

Journal of Economic Dynamics and Control noA. pp.149-170

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Stochastic Models, Estimation, and Control

Vo1.l41-3 in the series Mathematics in Science and Engineering. chapter 14.10 Academic Press. Inc.(London) Ltd.

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A Paradox in the Theory of Optimal Stabilization Review of economic Studies 39, pp.423-432

15. Preston AJ. (1974),

A dynamic generalization of Tinbcrgcn's theory of policy Review of Economic Studies vol.4I. pp.65-74

16. Preston A.J .• Pagon A.R. (19X2),

The Theory of Economic Policy. pp.237,262-264.314,334 Cambridge University Press. New York

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On the Theory of Economic Policy North Holland, Amsterdam

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Optimal Stabilization Policies for Dclcnninislic and Stochaslic Linear Economic Systems Review of Economic Studies 40. pp.79-95

19. Wohltmann H.W. (1985),

Target path controllability of linear lime-varying dynamical systems I.E.E.E. AC-30. pp.84-87

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Linear Multivariable Control: a Geometric Approach. pp.36,37,54 Springer Verlag. Berlin