Conjugate gradient method with pseudospectral collocation scheme for optimal rocket landing guidance

Abstract This paper aims at proposing a pseudospectral collocation conjugate gradient method (PCCG) for solving the nonlinear optimal control problems with terminal constraints and to apply it to rocket landing guidance. Firstly, the projection technique and conjugate gradient method are combined to obtain the control that can satisfy the terminal constraints. In addition, a new step-size adjustment strategy is applied to make the algorithm converge faster. Secondly, to improve the computational efficiency, the nonlinear adjoint and auxiliary differential equations are transformed into a set of linear algebraic equations by using the Gauss pseudospectral collocation scheme. Therefore, the values of adjoint and auxiliary functions at LG nodes can be obtained by solving the system of linear equations, and the special integrals needed in the control update process can be solved more effectively by using the Gauss quadrature method. Finally, PCCG algorithm is applied to the rocket landing guidance problem. A comparison with conjugate gradient method and GPOPS-II is also provided. Simulation results show that the algorithm has high convergence speed and computational efficiency, and the solution is optimal.