Abstract
The approach of singular perturbation is useful for the problems of analysis and design of control systems model by using the small parameter. For example they are appeared in systems which have parasitics elements or high gain feedback. Kokotovic and his co-workers studied the convergence of the solution and behavior of the poles, and proposed the pole assignment algorithm. But their methods are restricted in the meaning that the reduced systems, which is obtained by ignoring their parasitics, must be proper. But recently descriptor form, a generalized representation of state space form, are widely treated by many authors, and analysis and design tools for improper system have been proposed.
The purpose of this paper is to extend the traditional results of singular perturbation method on the base of descriptor form. The restriction that the reduced system must be proper is removed, and a new algorithm to specify the pole location of singular perturbed systems is proposed. Two associated descriptor systems, which are named as the reduced system and the boundary-layer system, and the impulse index play an important role. The main results of this paper are as follows.
1) When the impulsive index is zero, the behavior of the singular perturbed system is separated into those of the two associated descriptor systems.
2) A sufficient condition to make the impulse index of closed loop system zero is that the reduced system of the open-loop system is impulse controllable.
3) The necessary and sufficient condition of the arbitrary pole assignablility is both the reduced system and the boundary-layer system are exponential mode controllable.
The proposed algorithm is consisted in numerical oriented routines and applicable in many areas.
