DUAL CONTROL OF THE EXTREMAL MULTIDIMENSIONAL REGRESSION OBJECT

. The statement of the problem of the dual control of the regression object with multidimensional-matrix input and output variables and dynamic programming functional equations for its solution are given. The problem of the dual control of the extremal regression object, i.e. object response function of which has an extremum, is considered. The purpose of control is reaching the extremum of the output variable by sequential control actions in production operation mode. In order to solve the problem, the regression function of the object is supposed to be quadratic in input variables, and the inner noise is supposed to be Gaussian. The sequential solution of the functional dynamic programming equations is performed. As a result, the optimal control action at the last control step is obtained. It is shoved also that the optimal control actions obtaining at the other control steps is connected with big difficulties and impossible both analytically and numerically. The control action obtained at the last control step is proposed to be used at the arbitrary control step. This control action is called the control action with passive information accumulation. The dual control algorithm with passive information accumulation was programmed for numerical calculations and tested for a number of objects. It showed acceptable results for the practice.


Introduction
The problem of the dual control of the multidimensional regression object is formulated as follows [1][2][3][4][5]. The control system with controlled object O, controller C, feedback path and driving action s g is considered (Fig. 1).
is minimal, is called optimal system. There The task consists of determining the strategies of the controller C, i. e. sequence of the conditional probability density functions , for which the total average risk R (1) is minimal.
As it is known [2][3][4][5], the optimal strategies of the controller C are not randomized, i.e. the control actions s U are not random and will be denoted s u . In this conditions the controller C will be described by conditional probability density function where s u is the fixed value of the variable s U . We will use the following simplified notation: The optimal control algorithm, i.e. the sequence of the control actions 0 1 ,..., , u u u n n  is determined in pointed inverse order from the following functional equations: in which and * 1  m n u is optimal control action for the instant of time

Dual control of the extremal regression object
Let us consider the case of dual control when the controlled object has an extremal characteristic, and the task consist of the search and support this extremal state. The task is concretized in this case as follows.
The controlled object is described at the instant of time s by the gaussian probability density function: ), c is a some set of the parameters (generalized parameter of the object). Let us note that we natation now the generalized paramener as c instead of  in expressions (5), (6). We suppose too that the regression function is quadratic: are the substitutions of transpose of the type "back" and "onward" respectively [6]. Provided the regrassion function (8), the probability density function of the object (7) take the following form: For the task of the dual search of the minimum of the regression function we should to choose the loss function in the form by the Bayes formula (6). We will consider the right hand part of the equality (8) with the parameters and will suppose the general case, when the output variable y Let us agree to use below the following notations: ,... , 2 1 Let the random cell , has the Gaussian priory probability denity function described by the following expresion [7]: n is the number of the scalar elements of the cell t C . Then the posterior probability density function ) (c f n (6) is defined by the following expression [7]: in which ) ,..., , ( The substitutions of transpose j i T , in (11) and i T in (12) have the following forms: It is of interest in dual control to use the single measurements for updating the estimations (10)-(14).
We will have for this the following expressions:  (5), where ) ( t n c f is determined by the formula (10). We will use for this the following theorem from [7]: Theorem (total probability formula for the joint Gaussian distribution of the multidimensional random matrices). Let the conditional probability density function ) / (  y f has the following form where i h is a ) ( then the integral (6) (the total probability formula) is defined by the following expression: where The matrices  The control action  1 n u is defined by the following expression (formula (3)): where ) ( (α( ))) (α( ( ) )) (α( )) Taking into account the expression (21) for the Y  , we get: The search of the optimal control action 

Computer simulation
The algorithms of the optimal dual control with passive information storage (22), (27) were realized programmatically, utilized at a number of objects and showed results acceptable for practice. For instant, the object with Booth function as the regression function was simulated: This function has minimum at the point

Conclusion
To sum up, the general solution to the problem of the dual control with passive information storage of the extremal multidimensional regression object in the Gaussian case was obtained for the first time. This solution can be applied to control various technological processes, but each of them requires separate consideration. One of them is the allowance distribution problem [8].