基于反步法非线性纯反馈系统的控制器设计

Design of controller based on backstepping nonlinear pure feedback system

针对由于非线性纯反馈系统存在非仿射性结构使得用以往的坐标变换难以设计出控制器的问题,提出了一种新的坐标变换,并引入了一阶控制输入的辅助系统来处理非线性纯反馈系统。首先,结合新提出的坐标变换,计算出新状态方程;然后,基于反步法在每一步中设计出正定的Lyapunov函数;最后,通过设计虚拟控制器和实际的辅助控制器使得Lyapunov的导数负定,这样从理论上解决了非线性纯反馈系统的跟踪问题。仿真实验表明所设计的辅助控制器能使得纯反馈闭环系统所有状态信号有界,控制输出能跟踪到给定信号,跟踪误差渐近地趋于稳定,从而达到要求。

To solve the problem that it is difficult to design a controller by previous coordinate transformation because there is nonaffine structure in nonlinear pure feedback system, a new coordinate transformation was proposed and a first-order control input auxiliary system was introduced to deal with the nonlinear pure feedback systems. Firstly, a new state equation was calculated by combining the new coordinate transformation. Secondly, positive definite Lyapunov function was designed for each step based on the backstepping method. Finally, the derivatives of Lyapunov functions were made negative by designing virtual controllers and auxiliary controller, so the tracking problem of nonlinear pure feedback systems was theoretically solved. The experimental results show that the designed auxiliary controller is able to make the state of the nonlinear system bounded globally, and the control output can track the given signal, and the tracking error becomes asymptotically stable.