On the connection between nonlinear differential-algebraic equations and singularly perturbed control systems in nonstandard form

Abstract

We consider a class of control systems represented by nonlinear differential equations depending on a small parameter. The systems are not in the standard singularly perturbed form and therefore one of the challenges is to show that the control systems do represent singularly perturbed two-time-scale systems. Assumptions are introduced which guarantee that an equivalent representation for the systems can be obtained which is in the standard singularly perturbed form, thereby justifying the two-time-scale property. The equations for the slow dynamics are characterized by a set of differential-algebraic equations which have been studied recently in the literature. The fast dynamics are characterized by differential equations. Both the slow and the fast dynamics are easily derived and are defined in terms of variables that define the original control system. Control design for the class of systems being considered is studied using the composite control approach. The results of this chapter are used to establish a relationship between the dynamic models arising in compliant force and position control tasks and constrained force and position control tasks in robotics.