Abstract
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 coprime. The derivative of is denoted by . The set of Hurwitz stable polynomials are
The 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. This 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 define the signum function by
Lemma 1 (see [8]). Let a nonzero polynomial have the even-odd components . Suppose and is coprime. Then, if and only if at the real negative roots of odd multiplicities of the following holds:
The following result determines the number of real negative roots of a real polynomial.
Lemma 2 (see [6]). A nonzero polynomial , such that , has real negative roots without counting the multiplicities if and only if the signature of the polynomial is . All roots of are real, negative, and distinct if and only if .
We now describe a slight extension of the constant stabilizing gain algorithm of [8]. Given a plant where are coprime with less than or equal to , the set is the set of all real such that has signature equal to .
Let and be the even-odd components of and , respectively, so that
Let be the even-odd components of . Also let . By a simple computation, it follows that (we replace by ):
With this setting, we have
If and if they exist, let the real negative roots with odd multiplicities of be with the ordering , with and for notational convenience.
The following algorithm determines whether is empty or not and outputs its elements when it is not empty [6].
Algorithm 3. (1) Consider all the sequences of signums
where for .
(2) Choose all the sequences that satisfy
(3) For each sequence of signums that satisfy step (2), let
The set is nonempty if and only if for at least one signum sequence satisfying step (2), holds.
(4) is equal to the union of intervals for each sequence of signums that satisfy step (3).
The algorithm above is easily specialized to determine all stabilizing proportional controllers for the plant . This is achieved by replacing in step (3) of the algorithm by , the degree of .
Remark 4. By step (2) of Algorithm 3, a necessary condition for the existence of a is that the odd part of has at least real negative roots with odd multiplicities. When solving a constant stabilization problem, this lower bound is .
3. 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 find 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 find stabilizing values of . The closed-loop characteristic polynomial is given by where
Multiplying by , we obtain
Note here that , therefore the odd part of must have all its roots real and negative. The idea behind this method which determines the set of stabilizing parameters is to divide the problem into two subproblems: first we find all values of for which the polynomial has all its roots real and negative. Next, by sweeping over all values of found in the first step and using Algorithm 3, we determine stabilizing values of . In the first step, we construct a new polynomial using Lemma 2, finding values of such that has all its roots real and negative is equivalent to stabilizing the new constructed polynomial . Hence our method consists of applying Algorithm 3 to two specially constructed plants:
Remark 5. For a fixed 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 satisfies step (2) in Algorithm 3 and therefore there is only one interval as a solution. This conclusion still holds if we use higher-order Padé approximations.
4. Robust Stabilizing PI Controllers
We now consider a second-order uncertain plant with an interval type uncertainty for the coefficients
Our aim is to find robust stabilizing PI controllers. Replacing the time delay by a second-order Padé approximation, we get the following rational transfer function
Let and , be the four Kharitonov polynomials corresponding to and , respectively. Let , be the four Kharitonov segments of , that is, where . The four Kharitonov segments , of can be defined similarly. Let denote the family of 32 segment plants
It is well known [9] that the family is stabilized by a particular controller, if and only if the 32 segment plants are stabilized by the same controller. Let denote the family of 16 segment plants
It is shown in [10] that “the entire family is stabilized by a particular PID controller, if and only if each segment plant is stabilized by that same PID controller.” In reaching this result the structure of the PID controller was used to reduce the 32 segment plants to only 16. Since we are working with PI controllers, the numerator and denominator of the controller are convex directions [9]. Therefore, stabilizing the interval plant by a PI controller is equivalent to stabilizing 16 vertex plants, namely,
The stabilizing controller, if any, can be determined by first calculating which is the intersection of ’s found for the 16 plants mentioned above. We can then sweep over the values of and apply Algorithm 3 to the 16 vertex plants to find .
5. 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. This type of process is found in many industrial systems such as furnaces, air conditioning, and so forth. The PT-326 Thermal 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. The 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 flow 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 flow rate either by(i) controlling the speed of the circulating fan or,(ii) restricting the actual flow channel itself using a controlled Vane mounted in the flow path.
Two platinum resistance thermometers ( and ) 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.
The physical principle which governs the behavior of the thermal process in the PT-326 apparatus is the balance of heat energy. The rate at which heat accumulates in a fixed 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 . The 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. Thus, 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 specified 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. The transfer function is described by the following expression:
5.1. Regions of All Stabilizing Values of
Approximating the time delay by a second-order Padé approximation we get
The closed-loop characteristic polynomial is given by
Using Lemma 2, we solve subproblem 1: find all values of such that is Hurwitz stable. Applying Algorithm 3, we get . The complete solution is given in Figure 5.
5.2. 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 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 figure a choice of couple () belonging the domain of stabilizing parameters can stabilize the studied system.
6. Conclusions
In this paper, we determine the set of all stabilizing PI controllers for first-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 fixed. 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.