Control of singularly perturbed systems

Let us now consider an augmented representation of the standard and nonstandard singularly perturbed systems discussed above, namely 

In very generic terms, controller design seeks to use the (inverse of the) model (2.83)-(2.85) or (2.86)-(2.87) to compute the inputs u as a function of the outputs y (or the states x), so as to minimize the difference between the latter and a given value, or setpoint. 

The majority of the existing literature on the control of singularly perturbed systems considers the two-time-scale, standard form (see, e.g., Kokotovic et al. 1986, Christofides and Daoutidis 1996a, 1996b). Nevertheless, the methods available for standard singularly perturbed systems can be extended to systems in nonstandard form, since these can be transformed into an equivalent standard form as mentioned above. 

Whether considering linear or nonlinear systems, and regardless of the controller design method employed, the fast component of a composite control system is typically designed to stabilize the fast dynamics (if it is unstable), while the 

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