含有广义饱和函数的全局渐近稳定非线性PID控制

为解决传统饱和函数条件比较苛刻的问题,本文提出了一种饱和函数,并将其应用于线性PD(比例-微分)+非线性PI (比例-积分)控制律.应用李亚普诺夫稳定性定理和拉萨尔不变原理推导了非线性PID控制律全局渐近稳定条件.为提高非线性PID控制精度,以位置跟踪误差绝对值时间积分和关节力矩输出误差绝对值时间积分为目标函数,以全局渐近稳定性条件以及关节额定驱动力矩为约束条件,应用多目标遗传算法NSGA-II(non-dominated sorted genetic algorithm-II)进行非线性PID控制律参数整定.选择优化后位置跟踪误差绝对值时间积分最小的饱和函数,并研究含有该饱和函数的非线性PID控制律对模型不确定性、输入干扰力矩和噪声的鲁棒性.与传统PID控制律和含有传统饱和函数的非线性PID控制律相比,本文方法的位置跟踪精度分别提高了近2个数量级和1个数量级.提出的饱和函数在靠近平衡点处具有较强反作用,使误差快速收敛于平衡点,有助于提高非线性PID控制律的位置跟踪精度以及鲁棒性.

Globally Asymptotically Stable Nonlinear PID Controlwith a Generalized Saturation Function

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.