Singular Perturbation Margin assessment of Linear Time-Invariant systems via the Bauer-Fike theorems

Singular Perturbation Margin (SPM) is a stability margin for Nonlinear systems established from the view of the singular perturbation (time-scale separation) parameter epsilon. Using the Bauer-Fike, the generalized Bauer-Fike theorems and the corollaries, Theorem 1 in this paper provides an assessment method for the SPM of a Linear Time-Invariant system, with an error on the order of a fractional power p/q of epsilon, where p is the order of truncation and q is the size of the largest Jordan block of the LTI system. As a consequence, Corollary 1 reveals the relationship between the condition number of the modal matrix of the nominal system (when epsilon approaches zero) and the SPM estimate. Through the iterative solution of matrix algebraic Riccati equation, the concepts of the pth-order approximated Chang transformation and the corresponding pth-order approximated slow and fast systems are established, and it is shown that the nominal system is singularly perturbed by way of a multivariate matrix polynomial with the powers of epsilon as the coefficients. The approach developed here will be useful in subsequent investigation on estimate of the SPM for Linear Time-Varying systems and Nonlinear Time-Varying systems, which may serve as a theoretical as well as practical stability metric.