Global Asymptotic Nonlinear PID Control With a New Generalized Saturation Function
Guojun Niu, Cuicui Qu
Abstract
In order to solve the problem of the strict condition of traditional saturation function, a new generalized saturation function was proposed and applied in nonlinear PID (Proportion-Integration-Differentiation) control laws, which consisted of linear D + nonlinear PI and linear PD + nonlinear PI. The new generalized saturation function has powerful reaction near the equilibrium point, and has the capability to make the control converge to the equilibrium point swiftly. The global asymptotic stability condition of nonlinear PID control laws were derived by employing Lyapunov's method and LaSalle's invariance principle. In order to improve the accuracy of nonlinear PID control laws, time integration of the absolute value of position tracking error and time integration of the absolute value of torque error were chosen as the objective functions. Global asymptotic stability conditions and rated driving torque of each motor were set as the constraint conditions. Nonlinear PID controller parameters were tuned by employing multi-objective genetic algorithm, non-dominated sorted genetic algorithm-II (NSGA-II). Compared with the optimization results of nonlinear PID with traditional saturation function, the accuracy of position tracking using the proposed method was improved by nearly one order of magnitude. The new generalized saturation functions with minimum time integration of position tracking error were selected to study the robustness of the nonlinear PID controller in modeling uncertainty, input torque disturbance, and noise. The position tracking accuracy of the proposed method compared to those of the traditional PID controller and nonlinear PID controller with traditional saturation function was improved by nearly two orders of magnitude and one order of magnitude, respectively. The introduced saturation function significantly improves position tracking accuracy and robustness of the nonlinear PID controller.