Symplectic Pseudospectral Methods For Nonlinear Optimal Control Problems And Their Applications

Complicated factors,such as strong nonlinearity,constraints and time-delays are overwhelmingly involved in optimal control problems existed in practical engineering.Hence,it becomes impossible to solve such complex optimal control problems by traditional analytical methods.When constructing numerical methods for solving optimal control problems,most researches merely focus on reducing the truncation error of the numerical solution to the real solution,while the inherent mathematical structure of optimal control problems is rarely utilized.In fact,an optimal control problem can be transformed into a Hamiltonian system by the Pontryagin's Maximum Principle.And symplectic numerical methods can solve Hamiltonian systems efficiently and accurately.In addition,pseudospectral methods,which are developed under the framework of direct methods,have become the most popular numerical methods for solving nonlinear optimal control problem due to their excellent numerical precision.However,a pseudospectral method itself is essentially a universal approximation scheme.Hence,pseudospectral methods should not be limited to constructing numerical methods under the framework of direct methods.Based on the above facts,the dissertation devotes to develop symplectic numerical methods for solving nonlinear optimal control problems under the framework of indirect methods while utilizing the outstanding accuracy of pseudospectral methods.Consequently,the following researches are carried out in the dissertation.1.A multi-interval symplectic pseudospectral method for solving general nonlinear optimal control problems without constraints is proposed.Numerical simulations demonstrate that the proposed method has obvious advantages on both accuracy and efficiency when comparing to the symplectic method which utilizes uniform Lagrange interpolation scheme.In addition,to avoid blind mesh refinement operations for the purpose of improving numerical precision,an adaptive hp mesh refinement technique based on the relative curvature of state variables is developed.2.A multi-interval symplectic pseudospectral method for solving nonlinear optimal control problems with inequality constraints is proposed by combining the sequential convexification technique.Three kinds of inequality constraint,i.e.,pure-state,pure control and mixed state-control are treated under the uniform framework and strictly satisfied by using the Lagrange multiplier method.Numerical simulations demonstrate that the proposed method has obvious advantages on both accuracy and efficiency when comparing to the classical and the adaptive hp pseudospectral methods.3.A multi-interval symplectic pseudospectral method for solving nonlinear optimal control problems with state-delays is proposed by combining the sequential convexification technique.To the best known of the author,this is the first symplectic numerical methods developed for optimal control problems with time-delay.Numerical simulations demonstrate that the proposed method has obvious advantages on both accuracy and efficiency when comparing to pseudospectral methods and the homotopy shooting method.4.To meet the requirement of closed-loop control,based on the symplectic pseudospectral method developed for solving nonlinear optimal control problems with inequality constraints,meanwhile utilizing the concept of receding horizon optimization,a symplectic pseudospectral model predictive control method and a symplectic pseudospectral moving horizon method where constraints can be taken into consideration are constructed.The effectiveness of the two developed methods are validated by a trajectory tracking problem of overhead crane systems and a state estimation problem of a spacecraft,respectively.The series of multi-interval symplectic pseudospectral methods developed in this dissertation have abundant convergence features,i.e.,the developed methods should exhibit linear convergent rate and exponent convergent rate by tuning the number of sub-intervals and the degree of pseudospectral interpolation approximation,respectively.The core matrices involved in the developed method are inherently sparse and symmetric due to the benefit of the utilization of least action principle.Meanwhile,the developed methods have inherent convenience for the implementation of parallel computation since multi-interval strategy is used.The above three advantages provide the developed multi-interval symplectic pseudospectral with potential abilities to accurately and efficiently solve large-scale nonlinear optimal control problems.In addition,for practical trajectory optimization problems,the offline trajectory planning together with the online trajectory tracking and state estimation can be solved by an identical symplectic pseudospectral method,providing extreme convenience for the integration of control algorithms on hard devices.