A computationally efficient numerical algorithm for solving cross-coupled algebraic Riccati equation and its application to multimodeling systems

In this paper, a new algorithm for solving cross-coupled algebraic Riccati equation (CARE) is proposed. Since the new algorithm is based on the fixed point algorithm, the solutions can be obtained independently as the solution of the algebraic Lyapunov equation. As a result, the convergence and the positive semidefiniteness of the obtained solutions are guaranteed. In order to show the validity of the proposed algorithm, the linear quadratic infinite horizon Nash game for general multiparameter singularly perturbed systems (GMSPS) is applied. The local uniqueness and the asymptotic structure of the solutions to the cross-coupled multiparameter algebraic Riccati equation (CMARE) is newly established by means of implicit function theorem. Moreover, a new formulation related to the reduced-order CARE is derived. Utilizing the new formulation and the asymptotic structure of the solutions to CMARE, the approximate Nash strategy is constructed