Non-standard singularly perturbed control systems and differential-algebraic equations

We consider a class of linear control systems represented by equations depending on a small parameter but which are not in the standard singularly perturbed form. One of the challenging system theoretic problem is to obtain an equivalent representation for the control system which, if possible, is 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. The equations for the slow dynamics are 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. The fast dynamics are characterized by different equations in terms of matrices that define the original control system. Control design for the system being considered is studied using the composite control approach. The problem of contract force and position regulation in a robot with its end effector in contact with a stiff surface is considered as an example