A Recursive Elimination Method for Optimal Control of Discrete-Time Polynomial Systems

In this paper, a solution method for finite-horizon optimal control problems of discrete-time polynomial systems is proposed. Instead of finding an optimal solution over the horizon directly, a sequence of algebraic equations for the optimal control input at each time step is constructed backward by the recursive elimination of variables in the optimality conditions starting from the terminal condition, which can be viewed as a generalization of the classical backward sweep method for finite-horizon linear quadratic control to obtain the Riccati difference equation. An example of optimal control and model predictive control using the proposed method is presented.