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

Abstract

This paper addresses the problem of finding the linear quadratic regulator (LQR) weights such that the associated discrete-time algebraic Riccati equations admit a common optimal stabilising solution. Solving such a problem is key to designing LQR controllers to stabilise discrete-time switched linear systems under arbitrary switching, or stabilise 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 paper 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.