PI Controller Design for Time Delay Systems Using an Extension of the Hermite-Biehler Theorem

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 Padé 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 Padé 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 Padé 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 Padé 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