Research On The Symplectic Conservative Approach For Solving Nonlinear Closed-Loop Feedback Control Problems And The Applications

The numerical algorithms of optimal control has been increasingly important in the field of optimal control. As the problems are becoming more and more complex, sometimes we cannot study them by purely theoretical analysis. Thus it is necessary to design efficient numerical methods for approximate calculation. In most cases the numerical algorithms of optimal control cares only about the precision of computation, and thus ignore many intrinsic properties of the system. Symplectic conservative algorithms can keep the intrinsic properties of the system as well as meet the precision requirement. Thus this paper aims at the study of symplectic conservative numerical methods for optimal control and its applications in some hot aerospace problems. We presented a symplectic conservative numerical method for solving nonlinear optimal control problems, and extended this method into solving problems with control inequality constraints. In addition, we designed an efficient feedback control strategy, which can allow us to study some nonlinear optimal control problems when considering various disturbances from the outside environment. Then by using the novel numerical algorithms we presented, we studied the feedback control problems of the tethered satellite system and the spacecraft rendezvous problem between Halo orbits. The specific work of this paper are as follows:1. A symplectic approach was proposed to solve the nonlinear closed-loop feedback control problems in this paper. 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, update the state vector and control input by the forwarding of time steps and thus achieve the goal of closed-loop control.2. In order to solve the nonlinear optimal control problems with control inequality constraints, we transformed the original problems into the Hamiltonian two-point boundary value problems coupled with linear complementary problems by using the qusi-linearization method. In addition, we designed an efficient closed-loop feedback control strategy which can allow us to study the nonlinear optimal control problems under various disturbances under complex environment. The feedback control strategy was realized by the fast computation of open-loop control problems at each feedback point. 3. The symplectic conservative approach for solving nonlinear receding horizon control problems was applied on the closed-loop feedback control problems of the subsatellite’s deploy and retrieval process of tethered satellite system. First, the dynamic equations of two-body tethered satellite system were deduced based on Second Lagrange equations. Then we analysis and solve the problem using the symplectic conservative approach for solving nonlinear closed-loop feedback control problems. The numerical simulation showed that compared with the Legendre pseudospectral method, the symplectic approach has desirable computation and iteration speed when solving feedback control problems of tethered satellite system. Furthermore, the numerical simulations of the open-loop control and closed-loop feedback control problems of tethered satellite system showed that with the presence of initial errors, the open-loop control could not lead the system to a stable state, while the closed-loop feedback control can eliminate the initial errors within a certain period of time and the final state was still stable.4. A nonlinear closed-loop feedback control strategy for the spacecraft rendezvous problem with finite low-thrust between libration orbits in the Sun-Earth system was presented. The model of spacecraft rendezvous takes the perturbations in initial states, the actuator saturation limits, the measurement errors, and the external disturbance forces into consideration from an engineering point of view. The proposed nonlinear closed-loop feedback control strategy is not analytically explicit; rather, it is implemented by a rapid re-computation of the open-loop optimal control at each update instant. To guarantee the computational efficiency, a novel numerical algorithm for solving the open-loop optimal control is given. With the aid of the quasilinearization method, the open-loop optimal control problem is replaced successfully by a series of sparse symmetrical linear equations coupled with linear complementary problem, and the computational efficiency can be significantly increased. The numerical simulations of spacecraft rendezvous problems in the paper well demonstrate the robustness, high precision and dominant real-time merits of the proposed closed-loop feedback control strategy.