%0 Journal Article
%T PI Controller Design for Time Delay Systems Using an Extension of the Hermite-Biehler Theorem
%A Sami Elmadssia
%A Karim Saadaoui
%A Mohamed Benrejeb
%J Journal of Industrial Engineering
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/813037
%X We consider stabilizing first-order systems with time delay. The set of all stabilizing proportional-integral PI controllers are determined using an extension of the Hermite-Biehler theorem. The time delay is approximated by a second-order Pad¨¦ approximation. For uncertain plants, with interval type uncertainty, robust stabilizing PI controllers are determined. 1. Introduction In process control, many systems are represented by first-order plants with time delay. Although several tuning rules are reported in the literature [1, 2], the problem of determining the entire set of stabilizing controllers of a given order, being PI or PID, for such systems is recently addressed in [3, 4]. In [5], the Hermite-Biehler theorem is used to determine analytically the set of stabilizing gains , , and of a PID controller by replacing the time delay by a first-order Pad¨¦ approximation. In fact, extensions of the Hermite-Biehler theorem were effectively used to determine the set of all stabilizing controllers of a given order and a given structure for systems without delay, see [1, 6, 7]. In this paper, we use an extension of the Hermite-Biehler theorem to determine the set of all stabilizing PI controllers for a first-order system with time delay, where the time delay is replaced by a second-order Pad¨¦ approximation. We show that for a fixed value of the proportional gain , the set of stabilizing gains is a single interval. This conclusion still holds for higher-order Pad¨¦ approximations. Next, we consider uncertain second-order systems with time delay and robust stabilizing PI controllers are determined. The paper is organized as follows. In Section 2 we present some preliminary results which can be used to determine stabilizing proportional gains for systems without delay. These results are used in Section 3 to determine stabilizing PI controllers for first-order systems with time delay. In Section 4 robust stabilizing PI controllers are determined for uncertain systems. Illustrative examples are given in Section 5. Finally Section 6 contains some concluding remarks. 2. Proportional Controllers In this section, an algorithm that determines the set of all proportional controllers [6] is reviewed. Let us first fix the notation used in this paper. Let denote the set of real numbers and denote the set of complex numbers and let ,£¿£¿ ,£¿£¿ denote the points in the open left-half, -axis, and the open right-half of the complex plane, respectively. Given a set of polynomials not all zero and , their greatest common divisor is unique and it is denoted by . If , then we say is
%U http://www.hindawi.com/journals/jie/2013/813037/