OPTIMAL CONTROL OF TEMPERATURE MODES OF THE INSTRUMENTAL CONSTRUCTIONS OF AUTONOMOUS OBJECTS

Problem of the automatic thermogradient stabilization of the constructive elements sizes is discussed. Dimensional stability of instrumental constructions improves the validity of the indicated values of instruments located on these constructions. The optimal control problem is formulated. The optimal algorithm is proposed for variation of the intensity of the distributed controlled heat sources.


Introduction
Actual problem of the stabilization of temperature modes of load-bearing constructions of autonomous objects, on which the heat generating instruments of data measuring systems are located, was formulated and proved in [1].Dimensional stability of the supporting constructions improves the validity of the indicated values of instruments located on these constructions; but thermal deformation due to heat generation of operating devices and the influence of external heat fluxes significantly increase the error of the measurements.

Solution
As it was found in [1], the measurement errors reduction can be achieved by implementation of a system for automatic thermogradient stabilization.The structural mathematical model of the controlled object is developed using the method of Green's functions [1][2][3].Let us consider the linear formulation of the problem for the constructive element of a rectangular prism form with the sizes ) -some basis value of temperature, for example, a required constant temperature, λ -coefficient of thermal conductivity, C -specific heat capacity, max P -maximum volume density of heat sources.When the heat generating device is located on the surface of the construction, the standardizing function allows to reduce the initial inhomogeneous boundary problem to the standard form of (1) -( 3) that can be written as follows with the help of application of the Green's function [2,3]: Let us consider the possible ways of development of an automatic stabilization system.The dependence of thermal deformation of the structure on the temperature can be described by the known linear relations [5].This fact does not allow using a quadratic deviation, depending on dest 7 to estimate the control quality of management of the resulting temperature field because this criterion is not sensitive to local thermal gradient, and thus to thermal deformation.Therefore, the Chebyshev norm is applied as optimization criterion in the optimal control problem: Thus, we can formulate the optimal control problem as follows.It is required to determine the optimal law of variation of the heat flows ) , , (    y,  x,  Τ from the desired value dest Τ that can be reached in the class of i-intervals control functions.It should be noted that there is a limit inf H of achievable deviations and thus the limiting number J of intervals ) (i G ' , δ=1,2,..,i; i=1,2,..J of optimal control; and the following condition holds true: element can be described by inhomogeneous differential equation of heat transfer in a standard form [2,3]: a Corresponding author: entcom@samgtu.ruDOI: 10.1051/ C Owned by the authors, published by EDP Sciences,

---
flow of heat exchange between the surface of α-th side of the prism (α=͞ 1͞ ,͞ 6) and the corresponding surface of the environment with the temperature heat flow of j-th heat generating data measuring device from the set of all D N devices located on a rectangular surface, bounded by closed intervals distributed within α-th verge heat flux (α=͞ 1͞ ,͞ 6) of the controlled heat sources of thermogradient stabilization system; 1(*) -Heaviside function (unit step function), δ(*)delta function.According to the authors opinion, the most effective way is the formulation of optimal control problem for optimization of temperature field regarding typical optimization criterion.
time-optimal problem (β=2)).If the admissible control functions in (4) can be represented in the following multiplicative forms Thermophysical Basis of Energy Technologies 2015 01036-p.3EPJ Web of Conferences

.
In the case of autonomous control of each j-th,