Globally Asymptotically Stable Nonlinear PID Controlwith a Generalized Saturation Function
NIU Guojun1, QU Cuicui2, PAN Bo3, FU Yili3
1. School of Mechanical Engineering and Automation, Zhejiang Sci-Tech University, Hangzhou 310018, China; 2. Hangzhou SIASUN Robot&Automation CO. LTD., Hangzhou 311200, China; 3. State Key Laboratory of iRobotics and System, Harbin Institute of Technology, Harbin 150001, China
Abstract：In order to solve the problem of the strict conditions of traditional saturation function, a saturation function is proposed and applied to the linear PD (proportional-differential) + nonlinear PI (proportional-integral) control law. The globally asymptotic stability conditions of nonlinear PID control law are derived using Lyapunov's stability theorem and LaSalle's invariance principle. In order to improve the accuracy of nonlinear PID control, the tuning of nonlinear PID control parameters is accomplished by the multi-objective genetic algorithm NSGA-II (non-dominated sorted genetic algorithm-II), taking both the time integral of the absolute value of position tracking error and the time integral of the absolute value of input torque error as the objective functions, regarding the globally asymptotic stability conditions and the rated driving torque of each motor as the constraint conditions. The saturation function with minimum time integral of position tracking error is selected, and then the robustness of the nonlinear PID control law with the saturation function to model uncertainty, input disturbance, and noise is studied. Compared with the traditional PID control law and the nonlinear PID control law with the traditional saturation function, the position tracking accuracy of the proposed method is improved by nearly two orders of magnitude and one order of magnitude, respectively. The proposed saturation function shows strong reaction near the equilibrium point, which makes the errors converge to the equilibrium point quickly. And it is helpful to improve the position tracking accuracy and the robustness of nonlinear PID control law.
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