Ascent trajectory optimization with singular arc using linear Gauss pseudospectral model predictive control

This paper proposes an efficient method for solving the singular optimal control problem (S-OCP) of rocket ascent trajectory, which is based on the linear Gauss pseudospectral model predictive control framework. Firstly, the singular thrust is expressed as a function solely dependent on the state variables through the calculus of variations, and the problem is reformulated as a multi-stage optimal control problem with an undetermined switching time and terminal time. Furthermore, additional optimality conditions regarding the switching time and terminal time are introduced by performing the first-order variational analysis on the augmented performance index. That establishes a complete set of necessary conditions to determine the singular and non-singular optimal control. Subsequently, using the small perturbation assumption and Gauss pseudospectral discretization, the complete necessary conditions are transformed into a system of linear algebraic equations. And, an analytical expression to eliminate the terminal error can be successfully derived within the orthogonal polynomial space. Through a closed-loop iterative mechanism, the analytical solution comes close to the optimal solution of the nonlinear multi-stage S-OCP. This method can achieve high-precision discretization for S-OCPs with few orthogonal points, and ensures high computational efficiency through analytical closed-loop iterations. Finally, various numerical simulations, comparative studies with the typical methods, and Monte Carlo simulations are conducted. The simulation results show that the proposed method can provide the singular optimal control with high accuracy. And it has high computational efficiency, rapid convergence, insensitivity to initial guesses, and strong robustness.