EIGENVALUE ASSIGNMENT FOR A SINGULARLY PERTURBED MINIMUM PHASE SYSTEM Othman Alsmadi Adnan Al-Smadi e-mail: othmanmk@ju.edu.jo e-mail: smadi98@yahoo.com Department of Electrical Engineering, University of Jordan, Jordan Department of Electronics Engineering, Hijjawi Faculty, Yarmouk University, Jordan Key words: Minimum phase, Eigenvalue, singular perturbation ABSTRACT In this paper, the eigenvalue assignment of a SISO singularly perturbed problem is considered for a minimum phase system. We propose an approach for the eigenvalue assignment of a singularly perturbed minimum phase system using state feedback which leads to a simple eigenvalue reassignment procedure. The singularly perturbed system, using singular perturbation, can be divided into two subsystems independent of each other. Based on the slow subsystem only, we can obtain a feedback gain that reassigns the eigenvalues to provide a desired system response. I. INTRODUCTION Nature offers many situations of systems where more than one event occurs at different time scales. For example, an electrically driven robot manipulator can have slower mechanical dynamics and faster electrical dynamics. In such cases, we can divide the systems into two subsystems one corresponding to faster dynamics and the other corresponding to slower dynamics. Then, controllers for each one of them can be designed separately. It is then common practice to consider those events occurring at the faster scale as being instantaneous with respect to the slower ones. These results in a lesser number of variables or parameters needed to describe the evolution of the system. Several techniques have been developed in relation with such events. That is, reduction and estimation of the discrepancy between the complete system and the systems arising from the reduction. The best known methods are the averaging methods, the singular perturbation methods, and the aggregation methods [1]. Singular perturbation method has been widely used in engineering and technology problems. Singular perturbation is a mathematical operation which can be used on class of linear/nonlinear problems where two dynamics operating on different time scales is present. In the singular perturbation method both slow and fast modes are retained, but analysis and design problems are solved in two stages [2]. Applications of singular perturbation method are found in physics, chemistry, mechanics, industrial process, and engineering [3, 4]. For example, Arino et al. [1] studied a model of age- structured population with two time scales. The first one is slow and corresponds to the demographic process. The second one is fast and describes the migration process between different spatial patches. In control systems, it is always desired to enhance the performance of a given system. Knowing the relation between the closed-loop poles and the system performance, the system can be designed effectively by specifying the locations of these poles [5]. The problem of eigenvalue assignment which arises from singular system control has been studied in literature extensively [6-8]. In this paper, we consider the problem of eigenvalue assignment of a single input/ single output (SISO) singularly perturbed method for a minimum phase system. State feedback approach is considered to obtain a simple suboptimal control eigenvalue reassignment procedure. The suboptimal control is based on the slow dynamics of the system. The organization of the paper is as follows. Section 2 presents the general model and the problem formulation. In section 3, simulation examples are considered. Section 4 presents our concluding remarks. II. PROBLEM FORMULATION The relationship between the input and output of an N th order linear time-invariant (LTI) system is described by the differential equation ∑ ∑ = = = M i i i i N i i i i t u dt d b t y dt d a 0 0 ) ( ) ( (1) The system transfer function of Eq. (1) is