Discrete‐Time Linear‐Quadratic Optimal Control With Indefinite Weighted Matrices

Abstract

This article addresses a linear‐quadratic (LQ) optimal control problem over the infinite horizon for discrete‐time deterministic systems, where the dynamic system contains the control and state variables. For classical deterministic LQ optimal control problems, it is commonly assumed to be regular , that is, the control and the state weighted matrices in the cost function are taken as positive‐definite and positive‐semidefinite, respectively. However, we consider a case in this article where the control and the state weighted matrices are indefinite, which results in an indefinite LQ problem. To analyze such a problem, first, we establish the equivalence between the feasibility of the employed linear matrix inequalities (LMIs) and the solvability of the discrete‐time algebraic Riccati equation (DARE). Secondly, by applying the properties of semidefinite programming (SDP) and primal‐dual optimization, we demonstrate that the maximal solution to the DARE is acquired via solving the associated SDP. Moreover, we demonstrate that the optimal solution to the SDP coincides with the unique maximal solution to the DARE. Finally, an unmanned aerial vehicle (UAV) case study is conducted to verify the correctness of this approach.