Eigenvector approach for optimal control of singularly perturbed and weakly coupled linear systems

We show how to exactly decompose both the singularly perturbed (SP) and the weakly coupled (WC) algebraic Riccati equation and the corresponding linear-quadratic optimal control problem at steady state in terms of reduced-order subproblems by using the eigenvector approach. The proposed algorithms may be applied to standard and nonstandard SP systems. The eigenvector approach should be used for decomposition of the SP (WC) control systems in the cases when the perturbation (coupling) parameter ε is not sufficiently small. By using duality between the linear-quadratic optimal control and the Kalman filtering, the results reported can also be applied to the reduced-order Kalman filtering problems of both singularly perturbed and regular linear stochastic systems. In addition, the eigenvector approach provides new tools and insight into the nature of the decomposition problem and finds all required solutions without solving the corresponding subsystem Riccati equations