Recursive Reduced-Order Algorithm for Singularly Perturbed Cross Grammian Algebraic Sylvester Equation

Abstract

A new recursive algorithm is developed for solving the algebraic Sylvester equation that defines the cross Grammian of singularly perturbed linear systems. The cross Grammian matrix provides aggregate information about controllability and observability of a linear system. The solution is obtained in terms of reduced-order algebraic Sylvester equations that correspond to slow and fast subsystems of a singularly perturbed system. The rate of convergence of the proposed algorithm isOε, whereεis a small singular perturbation parameter that indicates separation of slow and fast state variables. Several real physical system examples are solved to demonstrate efficiency of the proposed algorithm.

Optimal Control of Wind Turbine Systems via Time-Scale Decomposition

“…The Chang transformation [3] has been widely used to get the exact pure-slow and pure-fast subsystems, even when the perturbation parameter is not very small. Moreover, a number of recursive algorithms [4][5][6] have been developed to avoid the problems involving ill-conditioned systems and to obtain an approximate solution to the algebraic Riccati equation (ARE), which can be decomposed after that into slow and fast parts.…”