A New Parameter Estimate in Singular Perturbations.

Abstract

A new upper bound is obtained for the singular perturbation parameter of an asymptotically stable singularly perturbed system. General time invariant systems with a single small parameter are considered. The paper employs a Riccati equation whose solution is known to facilitate the exact decoupling of fast and slow dynamics. An application of the Brouwer fixed point theorem to the Riccati equation and of Liapunov's direct method to the fast and slow subsystems results in the desired upper bound. Computation of the estimate requires only the solution of two Liapunov matrix equations. For the case of scalar fast and slow variables, an expression for the estimate is evaluated and compared with the upper bound of Zien (L. Zien, J. Franklin Institute, 295, pp. 373-381, 1973).