Synthesis of optimal control of technological processes based on a multialternative parametric description of the final state

Abstract

A method is proposed to establish control of technological processes that would be optimal in terms of speed and final state on the basis of analyzing the solution of a system of stochastic differential equations (SDEs), which is a mathematical model of a controlled process. The results of the numerical modeling have proved that, being sufficiently simple, the proposed method helps obtain solutions that are completely consistent with the results obtained using the Pontryagin maximum principle for the speed problem. It has been shown that such an approach to the search for optimal control of technological processes opens up additional opportunities in solving the task of retaining the parameters of the technological process within a given area. Two alternatives of the control implementation are proposed and justified, differing in the principle of selecting control switching times. It has been shown that the determining factor for the choice of optimal control is the initial state of the system, described by the position of the phase space point characterizing the actual initial state relative to the final state line. If the final state is described by the equation of the straight line, it is proposed to reduce it to its normal form and to calculate the corresponding deviation of the point of the preceding state from this straight line, which uniquely determines the sign of control. It has been proved that the problem of finding the optimal control of technological processes must be preceded by the problem of obtaining a mathematical description of the final state, based on the construction of regression equations in which the output variable can be the quality of the finished technological product. It is proposed to obtain a multialternative parametric description of the final state for the search for optimal control of the technological process using a ridge analysis. It has been shown that each of the alternatives represents a set of suboptimal values of the output variable, which provides optimal values of the output variable describing the quality of the finished technological product in the chosen sense. Due to this approach, it is possible to synthesize the optimal control in terms of the speed and final state of technological processes in conditions of a multialternative description of the final state of the technological system.