Real Time Control For Time Delay System

In this paper, we consider the control of time delay system by Proportional-Integral (PI) controller. By Using the Hermite-Biehler theorem, which is applicable to quasi-polynomials, we seek the stability region of the controller and the computation of its optimum parameters. We have used the genetic algorithms to lead the complexity of the optimization problem. An application of the suggested approach for real-time control on a heater PT − 326 was also considered. INTRODUCTION Systems with delays represent a class within infinite size largely used for the modelling and the analysis of transport and propagation phenomena (matter, energy or information)(Niculescu, 2001; Zhong, 2006). They naturally appear in the modelling of processes found in physics, mechanics, biology, physiology, economy, dynamics of the populations, chemistry, aeronautics and aerospace. In addition, even if the process itself does not contain delay, the sensors, the actuators and the computational time implied in the development of its control law can generate considerable delays (Niculescu, 2001) . The latter have a considerable influence on the behaviour of closed-loop system and can generate oscillations and even instability (Dambrine, 1994). PID controllers are of high interest thanks to their broad use in industrial circles (Cedric, 2002). Traditional methods of PID parameter tuning are usually used in the case of the systems without delays (Lequin et al, 2003; Liu et al, 2001). An analytical approach was developed in (Silva et al, 2005; Zhong, 2006) and allowed the characterization of the stability region of delayed systems controlled via PID. Indeed, by using the Hermit-Biehler theorem applicable to the quasipolynomials (Bhattacharyya, 1995; Silva et al, 2005), the authors have developed an analytical characterization of all values of the stabilization gains (Kp ,Ki ,Kd ) of the regulator for the case of first order delay system. The same technique is used to provide a complete characterization of all P and PI controller that stabilize a first order delay system which considered as less complicated than the PID stabilization problem (Silva et al, 2000). In order to have good performances in closed loop, it is necessary to suitably choose the parameters of the regulator in the zone stability. In this work, we look for optimum regulators under different criteria and we present the results of this approach when applied to the temperature control of a heater air stream PT − 326. This paper is structured as follows: in section 2, we present the theorem of Hermit-Biehler applicable to the quasi-polynomials. Section 3 is devoted to the problem formulation for first order delay system controlled via PI controller. In order to obtain optimal regulator in the zone of stability, a description of the genetic algorithms is presented in section 4. Section 5 is reserved for real time example to test the control method. PRELIMINARY RESULTS FOR ANALYZING TIME DELAY SYSTEM Several problems in process control engineering are related to the presence of delays. These delays intervene in dynamic models whose characteristic equations are of the following form (Silva et al, 2002, 2005): δ(s) = d(s) + en1(s) + e n2(s) +... + enm(s) (1) Where: d(s) and n(s) are polynomials with real coefficients and Li represent time delays. These characteristic equations are recognized as quasi-polynomials. Under the following assumptions: (A1) deg(d(s)) = n and deg(ni(s)) < n for i = 1, 2, ...,m (A2) L1 < L2 < ... < Lm (2) One can consider the quasi-polynomials δ∗(s) described by : Proceedings 23rd European Conference on Modelling and Simulation ©ECMS Javier Otamendi, Andrzej Bargiela, José Luis Montes, Luis Miguel Doncel Pedrera (Editors) ISBN: 978-0-9553018-8-9 / ISBN: 978-0-9553018-9-6 (CD) δ∗(s) = emδ(s) = emd(s) + en1(s) +en2(s) + ... + nm(s) (3) The zeros of δs) are identical to those of δ∗(s) since em does not have any finite zeros in the complex plan. However, the quasi-polynomial δ∗(s) has a principal term since the coefficient of the term containing the highest powers of s and e is nonzero. If δ∗(s) does not have a principal term, then it has an infinity roots with positive real parts (Bhattacharyya, 1995; Silva et al, 2005). The stability of the system with characteristic equation (1) is equivalent to the condition that all the zeros of δ∗(s) must be in the open left half of the complex plan. We said that δ∗(s) is Hurwitz or is stable. The following theorem gives a necessary and sufficient condition for the stability of δ∗(s) (Silva et al, 2000, 2001, 2002, 2005). theorem 1 Let δ∗(s) be given by (3), and write: δ∗(jω) = δr(ω) + jδi(ω) (4) where δr(ω) and δi(ω) represent respectively the real and imaginary parts of δ∗(jω) . Under conditions (A1) and (A2), δ ∗(s) is stable if and only if: 1: δr(ω) and δi(ω) have only simple, real roots and these interlace, 2: δ ′ i(ω0)δr(ω0) − δi(ω0)δ ′ r(ω0) > 0 for some w0 in [−∞,+∞] where δ ′ r(ω) and δ ′ i(ω) denote the first derivative with respect to w0 of δr(ω) and δi(ω), respectively. A crucial stage in the application of the precedent theorem is to verify that and have only real roots. Such a property can be checked while using the following theorem (Silva et al, 2000, 2001, 2002, 2005). theorem 2 LetM andN designate the highest powers of s and e which appear in δ∗(s) . Let η be an appropriate constant such that the coefficient of terms of highest degree in δr(ω) and δi(ω) do not vanish at ω = η . Then a necessary and sufficient condition that δr(ω) and δi(ω) have only real roots is that in each of the intervals −2lπ + η < ω < 2lπ + η, l = l0, l0 + 1, l0 + 2... δr(ω) or δi(ω) have exactly 4lN + M real roots for a sufficiently large l0 . PI CONTROL FOR FIRST ORDER DELAY SYSTEM We consider the functional diagram of figure 1, in which the transfer function of delayed system is given by (5) G(s) = K 1 + Ts e (5) WhereK , T and L represent respectively the state gain, the constant time and the time delay of the plant. These three parameters are supposed to be positive. C(s) u(t) e(t) y(t) yc (t) G(s) Figure 1: Closed-loop control of a time delay system The PI Controller is described by the following transfer function: C(s) = Kp + Ki s (6) Our objective is to analytically determine the region in the (Kp,Ki) parameter space for which the closed-loop system is stable. theorem 3 The range of Kp value, for which a solution to PI stabilization problem for a given stable open-loop plant exists, is given by (Silva et al, 2000, 2001, 2002, 2005):