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.