Algebraic approach to nonlinear finite-horizon optimal control problems of discrete-time systems with terminal constraints

Abstract

This paper proposes a method to solve nonlinear finite-horizon optimal control problems of discrete-time polynomial systems with polynomial terminal constraints. Algebraic equations with all variables at each time step, which are independent of variables at other time steps, are derived from the necessary conditions for optimality by eliminating variables recursively. The candidates of the optimal solution are obtained by solving these equations, and algorithms to find all of these candidates are also proposed. Because of its structure, the proposed method is suitable for nonlinear model predictive control that needs only the initial optimal control law. A simple example to illustrate the methodology and a practical example with the nonlinear model predictive control framework are provided.

Algebraic Approach to Nonlinear Optimal Control Problems with Terminal Constraints: Sufficient Conditions for Existence of Algebraic Solutions

“…To apply the recursive elimination method, we have to recast the ELEs into a set of algebraic equations. Equations (6)- (8) are recast in [7], and (9) can be readily recast in the same manner as described in [7], because ∇ x φ and ∂ψ/∂x consist of algebraic functions. Therefore, only (10) remains to recast.…”

“…For FHOCPs with terminal constraints, the recursive elimination method in [8] is modified into Algorithm 1, which yields Theorem 1 and Corollary 1.…”

“…In this paper, first, we introduce a recursive elimination method for solving FHOCPs with terminal constraints where all nonlinear functions are rational or algebraic functions; note that this method has been presented in the preliminary form in conference proceedings [8], [9]. Next, we provide sufficient conditions for optimality of the problems.…”