Differential-algebraic equations and nonstandard singularly perturbed control systems

A class of control systems is studied that are represented by equations which are not in the standard singularly perturbed form. Assumptions are introduced which guarantee that an equivalent representation can be obtained which is in the standard singularly perturbed form, thereby justifying the two time scale property. It is then possible to show that the equations for the slow dynamics can be characterized by a set of differential-algebraic equations which are easily derived from the original equations by setting the parameter to zero; the original assumptions guarantee the existence of solutions of the obtained differential-algebraic equations. In addition, the equations for the fast dynamics can be expressed in terms of matrices that define the original control system. Control design for the system being considered is studied using the composite control approach. As an example, the problem of contact force and position regulation in a robot with its end effector in contact with a stiff surface is considered