On the Exponential Stability of Singularly Perturbed Systems

This paper establishes some results and properties related to the exponential stability of general dynamical systems and, in particular, singularly perturbed systems. For singularly perturbed systems it is shown that if both the reduced-order system and the boundary-layer system are exponentially stable, then, provided that some further regularity conditions are satisfied, the full-order system is exponentially stable for sufficiently small values of the perturbation parameter $\mu $, and its rate of convergence approaches that of the reduced-order system $(\mu = 0)$ as $\mu $ approaches zero. Exponentially decaying norm bounds are given for the “slow” and “fast” components of the full-order system trajectories. To achieve this result, a new converse Lyapunov result for exponentially stable systems is presented.