Our focus in this chapter and the rest of the book is on nonlinear control of multiple time scale systems. From singular perturbation theory concepts detailed in Chapter 2 it is understood that the response of a two time scale system represented in the standard singularly perturbed form can be approximated by the reduced slow system provided the reduced fast system is uniformly asymptotically stabilizing. This model reduction technique has allowed engineers to develop control designs for two time scale systems using only the lower-order reduced slow system models. In the process of the control design, engineers often ignore the fast occurring phenomena captured by the reduced fast system. Let us elucidate this process and its shortcomings through the following simple system:

where *u* denotes the desired control variable to ensure the slow state *x* stabilizes about the origin and the fast state *z* of (3.1) always remains bounded. The singularly perturbed system in (3.1) is in standard form, and the algebraic equation 0 = −*z*^{2} + *uz* obtained by setting ∊ = 0 has two isolated manifold solutions *z*_{0} = 0 and *z*_{0} = *u*. Substituting these isolated roots in (3.1) and setting ∊ = 0 results in the following two well-defined reduced slow systems:

corresponding to *z*_{0} = 0 and

corresponding to *z*_{0} = *u*. Notice from a control standpoint only the reduced slow system given in (3.3) is of interest. In order to determine whether control of (3.3) alone guarantees stabilization of (3.1) let us examine whether *z*_{0} = *u* is an asymptotically stabilizing equilibrium of the reduced fast system (developed by writing (3.1) in the stretched time scale τ and setting ∊ = 0)