Nonlinear Dynamics 38: 305–321, 2004. C 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Tuning of PID Controllers Based on Bode’s Ideal Transfer Function RAMIRO S. BARBOSA1,∗ , J. A. TENREIRO MACHADO1 , and ISABEL M. FERREIRA2 1 Department of Electrotechnical Engineering, Institute of Engineering of Porto, Rua Dr. Ant´ onio Bernardino de Almeida, 4200-072 Porto, Portugal; 2 Department of Electrotechnical Engineering, Faculty of Engineering of Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal; ∗ Author for correspondence (e-mail:
[email protected]; fax: +351-22-8321159)
(Received: 3 December 2003; accepted: 15 March 2004)
Abstract. This paper presents a new strategy for tuning PID controllers based on a fractional reference model. The model is represented as an ideal closed-loop system whose open-loop is given by the Bode’s ideal transfer function. The PID controller parameters are determined by the minimization of the integral square error (ISE) between the time responses of the desired fractional reference model and of the system with the PID controller. The resulting closed-loop system (with the PID controller) has the desirable feature of being robust to gain variations with step responses exhibiting an iso-damping property. Several examples are presented that demonstrate the effectiveness and validity of the proposed methodology. Key words: Bode’s ideal transfer function, fractional calculus, fractional-order systems, ISE optimization, PID tuning
1. Introduction In recent years we observe an increasing number of studies related with the application of the fractional calculus (FC) theory in many areas of science and engineering [1–4]. This fact is due to a better understanding of the FC potentialities revealed by many phenomena such as viscoelasticity and damping, transmission lines, diffusion and wave propagation, electromagnetism, dielectric polarization, heat transfer, percolation and irreversibility. These are examples exhibiting a fractional-order dynamics, where the application of the FC tool is adequate and, in some cases, essential for a complete characterization of the phenomena involved. In what concerns the area of automatic control systems [5–9] the application of the FC concepts is still scarce and only in the last two decades appeared the first applications. The PID controllers are the most commonly used control algorithms in industry [10]. So, there are good reasons to look for better design methods or alternative controllers because of the widespread use of these algorithms. Oustaloup [5] introduced the fractional-order algorithms for the control of dynamic systems and demonstrated the superior performance of the CRONE (French abbreviation for Commande Robuste d’Ordre Non Entier) method over the PID controller. More recently, Podlubny [3, 12] proposed a generalization of the PID controller, namely the PIλ Dµ controller, involving an integrator of order λ and differentiator of order µ (the orders λ and µ may assume real noninteger values). Podlubny also demonstrated the better response of this type of controller, in comparison with the classical PID controller, when used for the control of fractional-order systems. Along the last decades were developed many tuning techniques for the determination of the PID parameters. Among them, the most well known are the Ziegler–Nichols tuning rules [10]. However, these heuristic rules often do not produce satisfactory results giving very poor damping, typically ¨ ζ ≈ 0.2 (see Astrom and H¨agglund [10]). Therefore, other methods were developed such as root-locus based techniques [10] and methods based on optimization strategies [17].
306 R. S. Barbosa et al. In this paper we propose the design of PID controllers that minimizes an ISE performance index [15]. The adopted strategy is known as reference model tuning and consists on the ISE minimization between the step responses of an ideal closed-loop system, with Bode’s ideal transfer function, and the PID controlled system. We apply the proposed strategy to a variety of plant transfer functions and, in particular, for the commonly used first-order plus dead-time (FOPDT) model. Several techniques are known for approximating plant step responses by this type of transfer function [10]. The proposed methodology makes closed-loop systems robust to gain variations and step responses exhibiting an iso-damping property. It also proves that although the closed-loop system with the PID controller is treated as integer-order it may be analysed as fractional-order and, therefore, we can (and we should) take advantage of that. Bearing these ideas in mind, the paper is organized as follows. Section 2 reviews the fundamental aspects of the FC theory. Section 3 proposes an ideal closed-loop system with Bode’s ideal transfer function. The fundamental known characteristics of the time and frequency responses are also presented. Motivated by the results, Section 4 establishes a strategy for the tuning of PID controllers based on the ISE minimization between the ideal and actual time responses. We apply the method to several plants that reveal good results and demonstrate its applicability. Finally, Section 5 draws the main conclusions and addresses perspectives towards future developments. 2. Fundamental Aspects of Fractional Calculus Fractional calculus deals with derivatives and integrals to an arbitrary order (i.e., rational, irrational or even complex). Since its foundation this area of mathematics has been the subject of several approaches leading to several definitions of fractional derivatives and integrals [1–3]. However, from a control point of view some definitions seem to be more appropriate than others, particularly for a discrete-time implementation. According to authors Oldham and Spanier [1], the most fundamental definition of a fractional derivative and integral of order γ is given by Gr¨unwald–Letnikov definition [1, 3]: ∞ 1 γ γ k D f (t) = lim f (t − kh) , γ ∈ R (1) (−1) h→0 h γ k k=0 (γ + 1) γ = (2) k (k + 1)(γ − k + 1) where (z) is the Gamma function, h is the time increment and f (t) is the applied function. Oldham and Spanier called (1) differintegral operator since it unifies on a single operator the notions of integral and derivative. This definition reveals that while integer-order derivatives imply a finite series, the fractionalorder derivatives require an infinite number of terms. This means that integer-order derivatives are “local” operators in opposition with the fractional-order derivatives that are “global” operators having a memory of all past function values. In the analysis and synthesis of automatic control systems we use the Laplace transform method. Fortunately, the Laplace transform of the differintegral operator D γ f (t), under null initial conditions, is given by the expected form: L{D γ f (t)} = s γ L{ f (t)},
γ ∈R
(3)
Expression (3) shows the straightforward extension of the classical frequency based methods to fractional-order control systems.
Tuning of PID Controllers Based On Bode’s Ideal Transfer Function 307 In the spite of the work that has been done the application of fractional derivatives and integrals has been scarce until recently. In the last years, the area of fractional calculus becomes an active field of research in almost all areas of science and engineering [3–5]. However, this work is still giving its first steps and, consequently, many aspects remain to be investigated. 3. Bode’s Ideal Transfer Function In his study on design of feedback amplifiers Bode [13] has suggested an ideal shape of the open-loop transfer function of the form: L(s) =
ωc s
γ
, γ ∈R
(4)
where ωc is the gain crossover frequency, that is, |L( jωc )| = 1. The parameter γ is the slope of the magnitude curve, on a log-log scale, and may assume integer as well noninteger values. In fact, the transfer function L(s) is a fractional-order differentiator for γ < 0 and a fractional-order integrator for γ > 0. The Bode diagrams of L(s) (1 < γ < 2) are very simple (Figure 1). The amplitude curve is a straight line of constant slope −20γ dB/dec, and the phase curve is a horizontal line at −γ π/2 rad. The Nyquist curve consists, simply, on a straight line through the origin with arg L( jω) = −γ π/2 rad. Let us now consider the unit feedback system represented in Figure 2 with Bode’s ideal transfer function L(s) inserted in the forward path. This choice of L(s) gives a closed-loop system with the desirable property of being insensitive to gain changes. If the gain changes the crossover frequency ωc will vary but the phase margin of the system remains PM = π(1 − γ /2) rad, independently of the value of the gain (Figure 1).
Figure 1. Bode diagrams of amplitude and phase of L(s) for 1 < γ < 2.
308 R. S. Barbosa et al.
Figure 2. Fractional-order control system with Bode’s ideal transfer function L(s).
The closed-loop system of Figure 2 will be used (in Section 4) as reference model for tuning PID controllers. In the following sub-sections we present the fundamental characteristics of the time and frequency responses of the fractional-order control system represented in Figure 2. Motivated by these results, in Section 4 we develop a strategy for tuning PID controllers based on this fractional reference model. 3.1. FREQUENCY CHARACTERISTICS The closed-loop transfer function of fractional-order system of Figure 2, T (s) = Y (s)/R(s), is given by: T (s) =
L(s) 1 = , 1 + L(s) (s/ωc )γ + 1
γ ∈ R+
(5)
where ωc denotes the crossover frequency. In Sub-section 4.3 we shall consider the introduction of a time delay in this model. The Bode diagrams of amplitude and phase of transfer function T (s), (|T ( jω)|dB and arg [T ( jω)]), are given by: |T ( jω)|dB = 20 log10
1
(ω/ωc )2γ + 2(ω/ωc )γ cos(γ π/2) + 1 sin(γ π/2) arg[T ( jω)] = − arctan cos(γ π/2) + (ωc /ω)γ
(6)
(7)
Considering the asymptotic behaviour, as ω → +∞, of expressions (6) and (7), we have (γ > 0): ω |T ( jω)|dB ≈ −20γ log10 , arg[T ( jω)] ≈ −γ π/2 (8) ωc Hence, at high frequencies the asymptotes of magnitude and phase are given by straight lines of −20γ dB/dec and −γ π/2 rad, respectively. The low frequency behaviour approaches asymptotically the horizontal straight lines 0 dB and 0 rad, correspondingly to the magnitude and phase shift. These results are illustrated in Figures 3 and 4 which show the normalized Bode diagrams of amplitude (or gain) and phase of T (s) for 1 < γ < 2. The resonance peak Mr and the frequency ωr at which it occurs are given by the formulae: Mr =
1 , sin(γ π/2)
ωr = ωc |cos(γ π/2)|1/γ
(9)
Tuning of PID Controllers Based On Bode’s Ideal Transfer Function 309
Figure 3. Gain vs. normalized frequency of T (s) for 1 < γ < 2.
Figure 4. Phase shift vs. normalized frequency of T (s) for 1 < γ < 2.
3.2. TIME CHARACTERISTICS We start by obtaining the unit step response of fractional-order transfer function T (s). The output y(t) = L −1 [T (s)R(s)], when the input is a unit step R(s) = 1/s, has the solution: y(t) = L
−1
γ
ωc γ
γ s s + ωc
=1−
∞ [−(ωc t)γ ]n n=0
(1 + γ n)
= 1 − E γ [−(ωc t)γ ]
(10)
310 R. S. Barbosa et al. where E γ (x) is the one-parameter Mittag–Leffler function, given by the power series expansion [2, 14]: E γ (x) =
∞ n=0
xn , (1 + γ n)
γ >0
(11)
This function is a generalization of the common exponential function since for γ = 1 we have E 1 (x) = e x . Considering the asymptotic behaviour of E γ [−(ωc t)γ ], when t → +∞ and t → 0+ , as [14]: (ωc t)γ , ω c t → 0+ 1 − (γ + 1) γ E γ [−(ωc t) ] ≈ (ωc t)−γ , ωc t → +∞ (1 − γ )
(12)
we arrive to the final and initial values of the step response, y(t → +∞) and y(t → 0+ ), respectively: y(∞) = lim y(t) = 1, t→+∞
y(0+ ) = lim y(t) = 0 t→0+
(13)
Specifications for a control system design often involve certain requirements associated with the system time response. Next, we derive some useful formulae to characterize the time response of the fractional-order transfer function T (s). Like in the case of the underdamped second-order systems we develop expressions for the overshoot Mp , peak time Tp , rise time Tr , time constant Tc , and settling time Ts . Figure 5 shows the normalized step responses of T (s) while Figure 6 illustrates the percent overshoot Mp (%) versus γ , both for 1 < γ < 2. The peak time Tp is the time at which the overshoot occurs; the rise time Tr is the time for the response to evolve from 0.1 up to 0.9 of its final value;
Figure 5. Unit step responses for several values of order 1 < γ < 2.
Tuning of PID Controllers Based On Bode’s Ideal Transfer Function 311
Figure 6. Percent overshoot Mp (%) vs. order 1 < γ < 2.
the time constant Tc is the time required for the response to rise up to 63% of its final value; and the settling time Ts is the time required for the response to settle within a small fraction of its steady state value and to stay there. These variables can be approximated numerically leading to the following expressions: – The overshoot Mp : ymax − y(∞) Mp = (14) , Mp ≈ 0.8(γ − 1)(γ − 0.75), 1 < γ < 2 y(∞) – The peak time Tp (error smaller than 1%): 1.106(γ − 0.255)2 , 1
