Control Problem under UncertaintyAuthor(s): M. A. Ali and N. SinghSource: Advances in Applied Probability, Vol. 11, No. 2 (Jun., 1979), p. 308Published by: Applied Probability TrustStable URL: http://www.jstor.org/stable/1426840 .

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308 CONFERENCE ON STOCHASTIC PROCESSES AND APPLICATIONS

IIc. Control Theory

Control problem under uncertainty

M. A. ALI* AND

N. SINGH, Monash University

Suppose that our observational vector Y'= (y,, Y2, i s, Y,) is generated by a

multiple regression process

(1.1) Y=Z+u,

where Z = {jZ}, i = 1, 2, , n; j = 1, 2, - - -, k is an n x k(n > k) input matrix of rank k on k independent variables ZI, Z2," ? ", Zk; 3' = (31, 32,

... , 3k) is a

1 x k vector of unknown-regression parameters, and u is an n x 1 unobservable random vector of errors with E(u)= 0 and E(uu') = a2I, 2( <oo) being un- known. Among k independent variables let Zk be the only one that is controllable and the others are exogenous. A control problem within the context of (1.1) is to decide on future values Zn+l,k,

Zn+2,k,''", Zn+q,k of the

controlled variable Zk (with given values Zn+1.,Z,n+2,1, * ,Zn+.; Z,+1,2,

Zn+22, - - - , Zn+q,2; ...

Zn+l.k-1, Zn+2,k-1,*. Zn+q,k-1 of the (k - 1) exogen- ous variables Z1, Z2," ?

, Zk-1) that will bring each element Yn+,

Yn+2?," Yn+ql as close as possible to some target values y*, y, ..., y.

The problem has many applications in the field of stabilization and regulations. When the control problem involves optimization over several time periods

with a stochastic model having unknown parameters which have to be esti- mated, the problem becomes more difficult. The setting of control variables in one period importantly affects the information regarding parameter values we have for the other time periods. Thus in both the single- and multi-period control problems, more precise setting of the control variables are very essential. This article sets out to develop and examine the behaviour of an alternative control rule of setting the control variable for single- and multi- period control problems.

The alternative rule suggested in this article performs better than the classical control rules studied by Anderson and Taylor (1976). For single and several time period problems this rule will be more useful.

Reference ANDERSON, T. W. AND TAYLOR, B. (1976) Some experimental results on the statistical

properties of least squares estimates in control problems. Econometrics 44, 1289-1302.

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