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

In process control, many systems are represented by �rstorder 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 kkpp, kkii, and kkdd of a PID controller by replacing the time delay by a �rstorder Padé approximation. In fact, extensions of theHermiteBiehler 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 �rst-order system with time delay, where the time delay is replaced by a second-order Padé approximation. We show that for a �xed value of the proportional gain kkpp, the set of stabilizing kkii gains is a single interval. is conclusion still holds for higher-order Padé approximations. Next, we consider uncertain second-order systemswith time delay and robust stabilizing PI controllers are determined. e 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. ese results are used in Section 3 to determine stabilizing PI controllers for �rst-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.


Introduction
In process control, many systems are represented by �rstorder 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 �rstorder 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 �rst-order system with time delay, where the time delay is replaced by a second-order Padé approximation. We show that for a �xed value of the proportional gain , the set of stabilizing gains is a single interval. is 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. e 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. ese results are used in Section 3 to determine stabilizing PI controllers for �rst-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.

Proportional Controllers
In this section, an algorithm that determines the set of all proportional controllers [6] is reviewed. Let us �rst �x the notation used in this paper. Let denote the set of real numbers and denote the set of complex numbers and let − , 0 , + denote the points in the open le-half, -axis, and the open right-half of the complex plane, respectively. Given a set of polynomials 1 , … , ∈ not all zero and 1, their greatest common divisor is unique and it is denoted by gcd{ 1 , … , }. If gcd{ 1 , … , } = 1, then we say ( 1 , … , ) is coprime. e derivative of is denoted by ′ . e set ℋ of Hurwitz stable polynomials are e signature ( ) of a polynomial ∈ is the difference between the number of its − roots and + roots. Given ∈ , the even-odd components ( , ) of ( ) are the unique polynomials , ∈ such that It is possible to state a necessary and sufficient condition for the Hurwitz stability of ( ) in terms of its even-odd components ( , ). Stability is characterized by the interlacing property of the real, negative, and distinct roots of the even and odd parts. is result is known as the Hermite-Biehler theorem. Below is a generalization of the Hermite-Biehler theorem applicable to not necessarily Hurwitz stable polynomials. Let us de�ne the signum function Lemma 1 (see [8] e following result determines the number of real negative roots of a real polynomial.
Lemma 2 (see [6] We now describe a slight extension of the constant stabilizing gain algorithm of [8]. Given a plant where , are coprime with deg less than or equal to deg , the set is the set of all real such that ( , has signature equal to . Let (ℎ, and ( , be the even-odd components of ( and ( , respectively, so that ( ℎ 2 + 2 , Let ( , be the even-odd components of ( (− . Also let ( 2 ( (− . By a simple computation, it follows that (we replace 2 by ): With this setting, we have If ̸ ≡ 0 and if they exist, let the real negative roots with odd multiplicities of ( be { 1 , … , } with the ordering 1 2 ⋯ , with 0 0 and +1 −∞ for notational convenience.
e following algorithm determines whether ( , is empty or not and outputs its elements when it is not empty [6].
e set ( , is nonempty if and only if for at least one signum sequence ℐ satisfying step (2), max min holds. (4) ( , is equal to the union of intervals ( max , min for each sequence of signums ℐ that satisfy step (3). e algorithm above is easily specialized to determine all stabilizing proportional controllers ( for the plant ( . is is achieved by replacing in step (3) of the algorithm by , the degree of ( , .

Stabilizing with PI Controllers
In this section, we consider PI controllers applied to a plant transfer function where is the time constant, is a constant whose sign and value determines the open-loop stability and steady-state gain, respectively, and represents the time delay. Our aim is to �nd all values of ( , ) such that the closed-loop system is stable. We replace the time delay by a second-order Padé approximation In what follows, we show how to �nd stabilizing values of ( , ). e closed-loop characteristic polynomial is given by where ( ) = ( ) 2 2 6 12 , ( ) = 2 2 − 6 12.
Note here that (− ) , therefore the odd part ( ) ( ) of ( ) must have all its roots real and negative. e idea behind this method which determines the set of stabilizing parameters ( , ) is to divide the problem into two subproblems: �rst we �nd all values of for which the polynomial ( ) ( ) has all its roots real and negative. Next, by sweeping over all values of found in the �rst step and using Algorithm 3, we determine stabilizing values of . In the �rst step, we construct a new polynomial using Lemma 2, �nding values of such that ( ) ( ) has all its roots real and negative is equivalent to stabilizing the new constructed polynomial 1 ( , ). Hence our method consists of applying Algorithm 3 to two specially constructed plants: Remark 5. �or a �xed value of , the set of stabilizing values of , if they exist, are a single interval. In fact, using Padé approximation (− ) is always Hurwitz stable. By the interlacing property of the roots of the even and odd parts of a Hurwitz stable polynomial, it follows that only one sequence of signums satis�es step (2) in Algorithm 3 and therefore there is only one interval as a solution. is conclusion still holds if we use higher-order Padé approximations.

Robust Stabilizing PI Controllers
We now consider a second-order uncertain plant with an interval type uncertainty for the coefficients Our aim is to �nd robust stabilizing PI controllers. �eplacing the time delay by a second-order Padé approximation, we get the following rational transfer function

Application to the PT-326 Thermal Process
An application example for the theoretical approaches presented in the precedent sections can be given by the temperature control of an air stream heater (process trainer PT-326), Figure 1. is type of process is found in many industrial systems such as furnaces, air conditioning, and so forth. e PT-326 ermal Process Control models the industrial situation commonly found in such equipments as air conditioning plants where temperature control is achieved through a combination of more than one means. e process contained within the PT-326 comprises an air duct through which air may be circulated using an electrically driven variable speed fan. An electrically heated process block is mounted in the air �ow path such that it attains thermal equilibrium by balancing the heat gained through the energy supplied to it via the heater coil and the heat lost through convection and conduction. Temperature control is achieved either by (1) varying the heat energy input to the system by regulating electrical power to the heater coil or, (2) varying the heat transfer rate by regulating the air �ow rate either by (i) controlling the speed of the circulating fan or, (ii) restricting the actual �ow channel itself using a controlled �ane mounted in the �ow path.
Two platinum resistance thermometers ( and 2 ) monitor the actual temperature of the block, being in the direct thermal contact with the block and being mounted on an insulation spacer to introduce thermal inertia and additional time constant effects into the control loop. Figure  1 below shows the front panel of the apparatus.
e physical principle which governs the behavior of the thermal process in the PT-326 apparatus is the balance of heat energy. e rate at which heat accumulates in a �xed volume enclosing the heater is where is the rate at which heat is supplied by the heater, is the rate at which heat is carried into the volume by the coming air, is the rate at which heat is carried out of the volume by the outgoing air, and is the heat lost from the volume to the surroundings by radiation and conduction. Figure 2 below depicts the volume . e behavior of the PT-326 thermal process is governed by the balance of heat energy. When the air temperature inside the tube is supposed to be uniform, a linear delay system model can be obtained. us, the transfer function between the heater input voltage and the sensor output voltage can be obtained as ( ( − ( as shown in Figure 3. A typical control objective in thermal systems is to maintain the temperature of some component at a user speci�ed value called the set point. Figure 4 below depicts a closed-loop system designed to maintain the output temperature of the PT-326 apparatus at a desired set point. For the experiment,  the damper position is set to 30, and the temperature sensor is placed in the third position. e transfer function is described by the following expression: . e complete solution is given in Figure 5.

Simulation Results.
In this section, we simulated the process PT-326 using MATLAB Simulink, we tested the evolution of output of the studied system in the presence of a disturbance. For the simulation, we took a step unit. We Step also took the block (Random Number) as a disturbance where its mean equal to 0 and variance equal to 0.01 and a simple time equal to 0.5. Figure 6 shows the MATLAB Simulink schematic of our application. Figure 7 shows the evolution of the output as a function of time. Notice that from this �gure a choice of couple ( , ) belonging the domain of stabilizing parameters can stabilize the studied system.

Conclusions
In this paper, we determine the set of all stabilizing PI controllers for �rst-order systems with time delay. By using a second-order Padé approximation for the time delay, we show that the set of stabilizing values is a single interval when is �xed. Robust stabilizing PI controllers are determined for plants with uncertain parameters. Note that higher-order Padé approximations can be used without changing the analysis given in this paper, as the Hermite-Biehler framework is applicable to plants irrelevant of the order.