Finding the LQR Weights to Ensure the Associated Riccati Equations Admit a Common Solution

This article addresses the problem of finding the linear quadratic regulator (LQR) weights such that the associated discrete-time algebraic Riccati equations admit a common optimal stabilizing solution. Solving such a problem is key to designing LQR controllers to stabilize discrete-time switched linear systems under arbitrary switching, or stabilize polytopic systems (e.g., Takagi–Sugeno fuzzy systems and linear parameter varying systems) in the entire operating region. To ensure problem tractability and reduce the searching space, this article proposes an efficient framework of finding only the state weights based on the given input weights. Linear matrix inequality conditions are derived to conveniently check feasibility of the problem. An iterative algorithm with quadratic convergence and low computational complexity is developed to solve the problem. Efficacy of the proposed method is illustrated through numerical simulations of systems with various sizes.