METHOD OF UNCERTAIN COEFFICIENTS IN PROBLEMS OF OPTIMAL STABILIZATION OF TECHNOLOGICAL PROCESSES
Keywords:Technological process, linear-quadratic optimization task, AKOR method, modal synthesis, method of uncertain coefficients, choice and correction of roots spectrum, R. Bass’s method.
Context. The equivalent transformation method is examined in the given article. Its essence lies in changing of a certain class of non-stationary systems with the stationary ones, for which optimization methods are well processed. Urgency of the method is determined by the fact that in most optimal control methods, developed for continuous systems, tasks are considered in the temporary space using the states space and the matrix theory. All real control objects are known to be non-linear and non-stationary in one way or another. Analysis and synthesis of control systems for such objects is a complex mathematical issue, and its solution is received for some separate occasions for now. As a result of using the suggested method, when the variable coefficients matrix is known, the task of the non-stationary system optimal control is reduced to the task of the equivalent stationary system optimal control for which solution methods are well-known and well processed.
Objective. Reducing energy intensity and improving the quality of products of various technological processes is an urgent task of the national economy of Ukraine.
Methods. To achieve this goal, we propose a method of modal synthesis of optimal stabilization laws using the method of uncertain coefficients, developed by the authors
Results. Algorithm of synthesis of the optimal controller in the absence and presence of delay in the control loop is developed. The method of selection and correction of the desired spectrum of roots is proposed. To eliminate self-oscillations in the presence of a delay in the control circuit, the R. Bass method is used.
Conclusions. The modal synthesis of optimal laws of stabilization of technological processes is proposed on the basis of the original method of uncertain coefficients. The complexity of the choice of the desired eigenvalues is overcome by the proposed procedure of construction and correction of the spectrum of roots in a closed system of optimal control. To eliminate the occurrence of stable self-oscillations (in the presence of a delay) in the stabilization process near a given trajectory, the Bass’s method is proposed to be used. The simulation results confirm the correctness and effectiveness of the results.
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