Design of Discrete and Hybrid Nonlinear Control Systems

In this article the new method of discrete control systems design for nonlinear plants with differentiable nonlinearities is suggested. The increasing demands on the quality of control processes and the widespread use of computer technology provide ample opportunities for the design and implementation of digital control systems. However, discrete models of control plants are needed to solve this problem. In the case of linear plants, such models are created on the basis of z-transformation, Euler or Tustin formulas. In the case of nonlinear plants, these transformations are not applicable, so a large number of approximate discretization methods have been developed to date. Euler and Runge-Kutt transformations are used for these purposes most often, but they lead to satisfactory results only with very small period of discretization. In the case of automatic control systems, this requires the use of digital automation tools with very high speed, which is often economically impractical. Methods of discretization with a long period were most often developed on the basis of decomposition into series of the right-hand sides of the differential equations, transformed on Euler. Here, firstly, the problem of selecting the number of the series members, which to be retained arises, and secondly, already in the third or fourth order of the plant, the calculating ratios turn out to be extremely complex. The discretization method suggested below differs in that it is not the equations of nonlinear plants in the Cauchy form that are discretized, but the corresponding quasilinear model. In this case, a modified trapezoid method is used, and the discretization purpose is not the most accurate approximation of the original equations of the plant, but the stability of a closed nonlinear control system with rather big period. This system is designed using the algebraic polynomial-matrix method for designing of the nonlinear control systems. As a result, a hybrid nonlinear system with fairly simple algebraic calculation expressions is formed. The suggested approach makes it possible to create the control systems for nonlinear controlled plants using conventional computational automation tools.