Design Method of Quasilinear Nonaffine Nonlinear Control Systems of General Structure

The paper is devoted to present a new original design method of nonaffine nonlinear control systems of general structure. The originality of the method lies in the application to a nonlinear nonaffine plant of the general structure of an algebraic polynomial matrix design method using quasi-linear models. This method allows finding control by solving a linear algebraic system and an additional nonlinear algebraic equation. In many cases, control can be found analytically, which ensures high efficiency of the method. This research considers nonlinear plants that are nonaffine in control and disturbance. The analytical method for creation of the quasilinear models from the plant differential equations in the Cauchy form based on the theorem of the integral independence on the integration path is given in the article in brief. The design problem of the nonaffine nonlinear system has solution by this method, if the quasilinear model of the plant is controllable on the state and has a controlled output. The article presents the controllability criterion on the state and controllability criterion of the output. The functional parameters of the nonlinear control are defined by solving a system of linear algebraic equations and an additional nonlinear algebraic equation. In the system with this control, the tracking error is zero, if the reference input is stepwise, and it is sufficiently small if the reference input is periodic. The effectiveness of the suggested method is illustrated by three numerical examples of the design of nonaffine nonlinear control systems of the second and third order. Note to Practitioners— This paper focuses on creating nonlinear control systems for nonaffine nonlinear plants, which are common in actual practice. The known methods of designing nonaffine nonlinear systems are highly complex, since their application requires nonlinear transformations of the initial equations. Generally, these transformations are rather difficult to find. The new method presented in this article is based on quasilinear models and algebraic polynomial matrix method for designing nonlinear Hurwitz control systems, and is very simple and easy to implement in practice using modern technical means of automation. It is applied in conditions when the nonlinearities of the plant are differentiable, the state vector is measurable, and the plant’s quasilinear model satisfies the controllability criterion on the state and the controllability criterion of the output. The advantages of the proposed method are that the coefficients of the quasilinear model are easily found from the initial nonlinear equations of the plant. According to this model, a number of polynomials are found, and a system of linear algebraic equations is compiled using the coefficients of these polynomials. The parameters of the control law are determined as the solution of this algebraic system and of a single nonlinear algebraic equation. The three numerical examples of designing control systems for nonaffine nonlinear plants presented in the paper demonstrate the simplicity of the proposed method and the high quality of the control or tracking processes. This method can be applied to create highly efficient nonaffine nonlinear control systems for various industries: shipbuilding, aircraft, automobile, machine tools, agriculture and many others.