A graphical technique for limit cycle analysis of uncertain nonlinear control systems

Abstract

A graphical technique is proposed to synthesize robust controllers to suppress the amplitude of the limit cycle persisting in uncertain nonlinear control systems using the stability equation method. First, the describing function analysis method is employed to approximate the behavior of the nonlinearity and the celebrated Kharitonov theorem is utilized to characterize the plant variations. Accordingly, a family of Kharitonov polynomials is obtained for limit cycle analysis. Further, decomposing each of the vertex Kharitonov polynomials into real part and imaginary parts results in two related stability equations. By solving the two stability equations, families of constant limit cycle amplitude loci are plotted. These loci isolate the parameter plane into several limit cycle regions. Therefore, an admissible specification‐oriented parameter region is found directly on the controller coefficient plane. Furthermore, the overlapped region of the admissible parameter region for each Kharitonov polynomial is called the Kharitonov region. The Kharitonov region constitutes all of the feasible controller parameter sets to achieve robust limit cycle amplitude suppression for the entire uncertain nonlinear control system. Hence, the controller can be designed more flexibly. Additionally, a way to tune the robust controller gains is suggested. Finally, the distinguished advantages of the proposed algorithm as compared with other proposed methods are demonstrated by two examples proposed in the literature for comparison.