Synthesis of Guaranteed Stability Regions of a Nonstationary Nonlinear System with a Fuzzy Controller

The paper proposes a method for constructing guaranteed regions of stability of nonstationary nonlinear systems on the plane of parameters of a fuzzy PID controller. It is shown that this method allows to determine the full stability areas, which are significantly larger than the areas determined by classical methods (frequency circle criterion, quadratic Lyapunov functions). This improvement is achieved by using the algorithm for constructing spline Lyapunov functions. This type of Lyapunov functions is based on the necessary and sufficient conditions of stability, while the classical methods are only sufficient conditions of stability. In this regard, on the basis of the proposed method, it is possible to calculate the maximum sizes of the sectors in which the nonlinear characteristics in the channels of the fuzzy PID controller should be located. Examples of the synthesis of fuzzy P, PI, PID controllers for a nonstationary control object of the third order are given. Numerical experiments show that the expansion of the boundaries of nonlinear characteristics allows to improve the accuracy in the steady state, and also to almost double the speed of the automatic control system with a nonstationary object. The advantages over linear controllers are demonstrated. The proposed method guarantees the stability inside the calculated stability regions for any character of the change in the nonstationary parameter, as well as for any character of the change in the nonlinear characteristics in the corresponding sectors.The paper proposes a method for constructing guaranteed regions of stability of nonstationary nonlinear systems on the plane of parameters of a fuzzy PID controller. It is shown that this method allows to determine the full stability areas, which are significantly larger than the areas determined by classical methods (frequency circle criterion, quadratic Lyapunov functions). This improvement is achieved by using the algorithm for constructing spline Lyapunov functions. This type of Lyapunov functions is based on the necessary and sufficient conditions of stability, while the classical methods are only sufficient conditions of stability. In this regard, on the basis of the proposed method, it is possible to calculate the maximum sizes of the sectors in which the nonlinear characteristics in the channels of the fuzzy PID controller should be located. Examples of the synthesis of fuzzy P, PI, PID controllers for a nonstationary control object of the third order are given. Numerical experiments show that the expansion of the boundaries of nonlinear characteristics allows to improve the accuracy in the steady state, and also to almost double the speed of the automatic control system with a nonstationary object. The advantages over linear controllers are demonstrated. The proposed method guarantees the stability inside the calculated stability regions for any character of the change in the nonstationary parameter, as well as for any character of the change in the nonlinear characteristics in the corresponding sectors.