The Design of Hybrid Control System for Nonaffine Plants

Abstract

In the theory of automatic control, an urgent problem is the development of methods to design nonaffine control systems. In such systems, the control affects the input of the object nonlinearly, so it affects the state variables in a non-additive way. The purpose of this article is to develop a design method that ensures the stability of the zero-equilibrium position of a closed nonaffine control system in a certain area. The objects described by a nonlinear system of differential equations with one control and one output are considered. A constraint is introduced, which consists in the differentiability of the right-hand sides of differential equations with respect to all state variables. The task of designing control in the form of a function of the setting action, a vector of state variables and control values at previous points in time is set. This problem is solved using a quasilinear model of the control object. Such model allows you to preserve all the features of the nonlinear equations of objects without simplifying them. In the quasilinear model, matrices and vectors are functions of the variables of the state of the control object. The control is found using an algebraic polynomial matrix method. This method allows you to find control under the condition of controllability of the object in the form of inequality. This article presents the calculation ratios for calculating the control in accordance with the polynomial-matrix method. Based on the given coefficients of the desired polynomial, as a result of solving an algebraic system of equations, coefficients are found that are a function of control and state variables. At the same time, the fulfillment of the controllability condition guarantees the existence of a solution of the specified algebraic system. An expression has been found that allows calculating the control by the coefficients found. The article also presents a condition for the possibility of providing a non-zero value of the output-controlled quantity of a nonlinear Hurwitz system in a steady-state mode. Under this condition, a zero value of the static error for the setting effect can also be provided. Further, the transformation of the obtained continuous control into a discrete one is proposed, which is implemented in a digital computer. The article also provides a numerical example of designing a control system for a nonaffine object of the second order, and the results of modeling a closed nonaffine system. The given example confirms the theoretical results obtained. Thus, the proposed approach makes it possible to design stable Hurwitz control systems for nonaffine plants using the algebraic polynomial matrix method with sufficiently small sampling periods of variables of the control plant and small modules of the roots of the characteristic polynomial of the matrix of a closed system in its quasilinear model.