基于拟线性化方法的非线性系统闭环反馈控制保辛算法

提出了一种求解非线性系统闭环反馈控制问题的保辛算法．首先，通过拟线性化方法将非线性系统最优控制问题转化为线性非齐次Hamilton系统两端边值问题的迭代格式求解．然后，通过作用量变分原理与生成函数构造了保辛的数值算法，且该算法保持了原Hamilton系统的辛几何性质．最后，通过时间步的递进完成状态与控制变量的更新，进而达到闭环控制的目的．数值算例表明:保辛算法具有较高的计算精度和较快的收敛速度．此外，将闭环反馈控制与开环控制分别应用于驱动小车上的倒立摆控制系统中，结果表明:在存在初始偏差的情况下,开环控制会导致稳定控制任务的失败,而闭环反馈控制能够在一段时间后消除初始偏差的影响,并使系统达到稳定状态．

Symplectic Conservative Approach for Solving Nonlinear Closed-Loop Feedback Control Problems Based on Quasilinearization Method

A symplectic approach was proposed to solve the nonlinear closed-loop feedback control problems. First, the optimal control problems of the nonlinear system were transformed into the iteration form of linear Hamilton system’s two-point boundary value problems. Second, a symplectic numerical approach was deduced based on dual variable principle and generating function. This method can keep the symplectic geometry structure of the Hamilton system. Last, with the state vector updated and input controlted by the forwarding of time steps, the goal of closed-loop control was achieved. The numerical simulation shows that the proposed symplectic method has high precision and fast iteration speed. In addition, the closed-loop feedback control and open-loop control were used separately to analyze the inverted pendulum control system. The results show that in the case of the presence of initial errors, open-loop control will result in the failure of the stability control tasks, while closed-loop feedback control will eliminate the initial errors after a certain period of time and lead the system to a stable state.