Towards Design of Quasilinear Gurvits Control Systems

Abstract

The design problem of control systems for nonlinear plants with differentiated nonlinearity is considered. The urgency of this problem is caused by the big difficulties of practical design of nonlinear control systems with the help of the majority of known methods. In many cases, even provision by these methods of just stability of equilibrium point of a designing system represents a big challenge. Distinctive feature of the method of nonlinear control systems design considered below is the use of the nonlinear plants models represented in a quasilinear form. This form of the nonlinear differential equations exists, if nonlinearities in their right parts are differentiated across all arguments. The quasilinear model of the controlled plant allows reducing the design problem to the solution of an algebraic equations system, which has the unique solution if the plant is controlled according to the controllability condition provided in the article. This condition is similar to the controllability condition of the Kalman’s criterion. Procedure of the nonlinear control systems design on a basis of the plant’s quasilinear models is very simple. Practically, it is close to the known polynomial method of the linear control systems design. The equations of the nonlinear systems designed with application of the plant’s quasilinear models also can be represented in the quasilinear form. The basic result of this article is the proof of the theorem and the corollary from it about conditions of the asymptotical stability at whole of the equilibrium point of the nonlinear control systems designed on a basis of the plant’s quasilinear models. For the proof of the theorem and consequence, the properties of simple matrixes and known theorems of stability of the indignant systems of the differential equations are used. A way of the stability research of the equilibrium point of the quasilinear control systems based on the proved theorem is illustrated by numerical examples. Computer simulation of these systems verifies correctness of the hypoyhesis of the proved theorem. Obtained results allow applying the method of nonlinear systems design on a basis of the quasilinear models for creation of various control systems for plants in power, aviation, space, robotechnical and other industries.