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
Highlights
The search for optimal control of specific technical applications requires an informed choice of the methods that would be most suitable for these objects
Dealing with management of industrial objects and corresponding technological processes, it is necessary to take into account that such a choice of optimal control should help cope with a number of uncertainties
In [30], one of the main problems of optimal control of the process of non-stationary thermoelectric cooling is formulated to consist in determining the optimum dependence of the supply current on time, which ensures a minimum of the cooling temperature
Summary
The search for optimal control of specific technical applications requires an informed choice of the methods that would be most suitable for these objects. The presence of an obvious, sometimes “competing”, dualism in this issue requires additional research on a number of fundamental aspects for successful determination of optimal management In this connection, it is necessary to note such key areas as the selection of the criterion of management optimality [1], the structure of the mathematical description of the control object [2,3,4], the substantiation of the effectiveness of the individual operation estimation for the implementation of the selected management [5,6,7], and the uncertainty in the estimation of state variables [8], which generates the need for their fuzzy description [9,10,11]. All the aforementioned arguments substantiate the relevance of research subjects devoted to improving the methods of searching for optimal control, both in the applied aspect and in terms of developing the mathematical theory of control
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