Нелинейное квазиоптимальное управление ленточным конвейером комплекса напольного хранения зерна

Abstract

NON-LINEAR QUASI-OPTIMAL CONTROL OF THE BELT

CONVEYOR OF THE GRAIN STORAGE COMPLEX

I. Golodnyi, A. Toropov, L. Toropova

The aim of the research is developing of the optimal control system for the productivity of the belt conveyor, which ensures a reduction in the energy consumption of the complex.

One of the most popular approaches for synthesizing optimal regulators for continuous technological processes for systems with pure delay is the analytical design procedure. During this procedure the quality functional for an object described by a system of differential equations is minimized. It should be noted that analytical design of regulators is widely used for linear systems and quadratic functional of quality. In the case of the existence of nonlinear uncompensated dependencies in the model of the object, the problem of synthesis of optimal control becomes much more complicated.

To use the above approach, mathematical model of the object with sufficient accuracy of linearization was obtained, and also the general form of the minimized quality functional was chosen.  In this case, the Bellman-Lyapunov method to obtain the control algorithm using the concept of the immersion method was used.

Conveyor productivity regulation loop by two nonlinear equations with “product” and “saturation” links is described. «Product» link from the pure delay link model using Taylor expansion procedure was obtained earlier.

There were used linearization procedures for these nonlinearities, namely «in-point linearization» procedure, and «secant» method. According to concept of the immersion method, the initial synthesis problem for a nonlinear object in the form of a family of linear tasks of analytical design is represented. Therefore, optimal controls “in small” and “in large”, which have a form of linear dependence of state variables, were obtained.  At the end of the regulator design procedure, "sewing" of the solutions on the basis of the law of optimal change of the feedback coefficients was carried out. During this procedure A.A. Krasovsky functional of generalized work criterion was minimized. Using of such criterion allows obtaining arithmetical equations system of Riccati and, therefore, simplified form of Hamilton-Jacobi-Bellman equation. A solution of this equation in the quadratic form of state variables could be found.

As a result, for the conveyor productivity stabilizing loop, obtained regulator by a nonlinear law on the state variables is described. It has linear and cubical summands, which allows making significant influences at large deviations. This form of control could be simply realized at programmable logical controller that regulates all processes at grain storage complex. Thus, realization of such control system does not need of large investments.

Conclusions and perspectives

A comparing analysis between PID – control and synthesized nonlinear control was made. The adjustment of the PID controller using a block «Signal constraint» was carried out. The advantage of using a synthesized regulator by the results of a numerical simulation of a grain conveyor is confirmed. Further improvement of the dynamic performance of the system is possible using of more complete mathematical model of the conveyor and the synthesis of the regulator taking into account the elasticity of the belt.