Comparison of methods of nonlinear control systems design

The issue of designing nonlinear control systems is a complex problem. A lot of methods are known that allow us to find a suitable control for a given nonlinear object that provides asymptotic stability of the nonlinear system equilibrium and an acceptable quality of the transient process. Many of these methods are difficult to apply in practice. Thus, comparing some of the methods in terms of simplicity of use is of great interest. Two analytical methods for the synthesis of nonlinear control systems are considered. They are the algebraic polynomial-matrix method that uses a quasilinear model, and the feedback linearization method that uses the Brunovsky model in combination with special feedbacks. A comparative analysis of the algebraic polynomial-matrix method and the feedback linearization method is carried out. It is found out that the algebraic polynomial-matrix method (APM) is much simpler than the feedback linearization method (FLM). A numerical example of designing a system that is synthesized by these methods is considered. It is found out that the system synthesized by the APM method has a region of attraction of the equilibrium position twice as large as the region of attraction of the system synthesized by the FLM method. It is reasonable to use the algebraic polynomial-matrix method with the quasilinear models in case of synthesis of control systems of objects with differentiable nonlinearities.

On the practical h-stabilization of nonlinear time-varying systems: application to separately excited DC motor

Purpose The purpose of this paper is to report on the global practical uniform h-stabilization for certain classes of nonlinear time-varying systems and its application in a separately excited DC motor circuit. Design/methodology/approach Based on Lyapunov theory, the practical h-stabilization result is derived to guarantee practical h-stability and applicated in a separately excited DC motor. Findings A controller is designed and added to the nonlinear time-varying system. The practical h-stability of the nonlinear control systems is guaranteed by applying the appropriate controller based on Lyapunov second method. Another effective controller is also designed for the global practical uniform h-stability on the separately excited DC motor with load. Numerical simulations are demonstrated to verify the effectiveness of the proposed controller scheme. Originality/value The introduced approach is interesting for practical h-stabilization of nonlinear time-varying systems and its application in a separately excited DC motor. The original results generalize well-known fundamental result: practical exponential stabilization for nonlinear time-varying systems.

Numerical Design Method of Quasilinear Models for Nonlinear Objects

Design modern methods of nonlinear control systems of nonlinear objects in the majority assume transformation of initial object model to some special forms. In these cases, it is reasonable to use quasilinear models as they can be designed on condition only of differentiability of the nonlinearities of the initial objects models. These models allow to find control analytically, i.e. as a result of the solution of some equations system, if the object, naturally, meets the controllability condition. The quasilinear models are synthesized traditionally analytically, bytransformation of initial nonlinear models using operation of the taking of partial derivatives from the nonlinearities of the initial objects models and the subsequent integration of these derivatives on the auxiliary variable with application of the known formulas of differentiation and integration. However, in many cases, the objects nonlinearities have so complicated character, that the operations of the differentiation and, in particular, the integration are executed very difficult by shown way. This complexity can be overcome by application of the new numerical design method of the quasilinear models, which excludesneed of the analytical differentiation and integration, but demands considerable number of the arithmetic operations. But now it is not the big problem since the modern multiprocessor controllers can carry out all the necessary operations for a short time. The developed method allows to receive rather exact, approximate piecewise-constant quasilinear models for the objects with the complicated nonlinearities. It is convenient to apply such models at numerical control of the nonlinear objects. The efficiency of a numerical method is shown by comparison of phase portraits of piecewise constant quasilinear and nonlinear models of a simple object and also by comparison of the state variables values of these models. The offered method can be applied to nonlinear control systems design for the nonlinear, characterized by complicated characteristics objects ship, aviation, chemical, agricultural and other industries.