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- 1. Research Article A novel auto-tuning method for fractional order PI/PD controllers Robin De Keyser a , Cristina I. Muresan b,n , Clara M. Ionescu a a Ghent University, Department of Electrical Energy, Systems and Automation, Technologiepark 914, B9052 Zwijnaarde, Belgium b Technical University of Cluj-Napoca, Department of Automation, Gh. Baritiu, No. 26-28, Cluj-Napoca, Romania a r t i c l e i n f o Article history: Received 5 September 2015 Received in revised form 19 December 2015 Accepted 29 January 2016 Available online 20 February 2016 This paper was recommended for publica- tion by Prof. Y. Chen Keywords: Auto-tuning Fractional order controller Robustness Validation a b s t r a c t Fractional order PID controllers beneﬁt from an increasing amount of interest from the research com- munity due to their proven advantages. The classical tuning approach for these controllers is based on specifying a certain gain crossover frequency, a phase margin and a robustness to gain variations. To tune the fractional order controllers, the modulus, phase and phase slope of the process at the imposed gain crossover frequency are required. Usually these values are obtained from a mathematical model of the process, e.g. a transfer function. In the absence of such model, an auto-tuning method that is able to estimate these values is a valuable alternative. Auto-tuning methods are among the least discussed design methods for fractional order PID controllers. This paper proposes a novel approach for the auto- tuning of fractional order controllers. The method is based on a simple experiment that is able to determine the modulus, phase and phase slope of the process required in the computation of the con- troller parameters. The proposed design technique is simple and efﬁcient in ensuring the robustness of the closed loop system. Several simulation examples are presented, including the control of processes exhibiting integer and fractional order dynamics. & 2016 ISA. Published by Elsevier Ltd. All rights reserved. 1. Introduction The proportional-integrative-derivative (PID) controller is still the most widely used control strategy [1]. In numerous processes, the PI(D) controllers account for more than 90% of the control loops [2]. Sometimes, system modeling may be difﬁcult, especially in large industrial plants with sub-system interaction. Probably the most popular method to tune the controllers in this particular situation, where the process model is unavailable, is that devel- oped by Ziegler and Nichols [3]. The design proposed by Ziegler and Nichols is based on the measurement of the critical gain and critical frequency of the plant and determining the PI/PID con- troller parameters according to these critical values. The problem with the design is that it leads to some closed loop systems with poor robustness [1]. Several methods to avoid this problem and improvements of the original tuning procedures of Ziegler and Nichols have been proposed throughout the years. Åström–Häg- glund used the same general ideas as presented by Ziegler and Nichols, but combined these with the use of robust loop shaping for control design, allowing for a clear tradeoff between robustness and performance [1]. An automatic tuning method based on a simple relay feedback test which uses the describing function analysis to give the critical gain and the critical frequency of the system has also been proposed as an alternative [4]. A modiﬁcation of this initial auto-tuning method based on a relay with hysteresis for noise immunity has also been proposed, with the new approaches based on an artiﬁcial time delay within the relay closed-loop system to change the oscillation frequency in relay feedback tests [5]. For the tuning of a PI controller based on the relay feedback test, two equations for phase and amplitude assignment are used. For the PID, in the modiﬁed Ziegler–Nichols method [6], a parameter α¼4 is used as the ratio between the integral and derivative time constants. However, since research has shown that the control performance is heavily inﬂuenced by the choice of the parameter α [7], improved methods for the auto- tuning of the PID controller have been proposed, such as for example the addition of a third condition to make the system robust to open loop gain variations [2]. The generalization of the classical PID controller to the frac- tional order one is due to Podlubny [8] and it is based on the use of a fractional integrator of order μ and a fractional differentiator of order λ, instead of the classical integer order integrator and dif- ferentiator. Researchers have shown that this generalization allows for a better shaping of the closed loop responses, mainly due to the two supplementary tuning parameters involved, the fractional Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/isatrans ISA Transactions http://dx.doi.org/10.1016/j.isatra.2016.01.021 0019-0578/& 2016 ISA. Published by Elsevier Ltd. All rights reserved. n Corresponding author. E-mail addresses: Robain.DeKeyser@ugent.be (R. De Keyser), Cristina.Pop@aut.utcluj.ro (C.I. Muresan), Claramihaela.Ionescu@ugent.be (C.M. Ionescu). ISA Transactions 62 (2016) 268–275
- 2. orders μ and λ [9,10]. Due to these beneﬁts, many researchers have focused on the design problem of fractional order controllers [9–14]. The majority of the tuning techniques for these fractional order controllers are based on some frequency domain speciﬁca- tions that most frequently refer to a certain gain crossover fre- quency, a certain phase margin of the open loop system, an iso- damping property, output disturbance rejection and high fre- quency noise rejection. The auto-tuning of fractional order controllers has seen less attention from the research community, with some notable papers that discuss the design of a classical integer order PID combined with a fractional order integrator or differentiator s α , with αϵ (À1,1), denoted as the phase shaper [15]. The basic idea in the design resides on the iso-damping property, also present for classical PID controllers and for general fractional order PI μ D λ s [16]. Here, the relay test is used in the auto-tuning procedure, with the exact approach being based on an extension of the classical approach used in the auto-tuning of integer order PID controllers. The ﬁnal controller is computed by designing separately a frac- tional order PI controller (FO-PI) and a fractional order PD (FO-PD) controller with a ﬁlter. The actual tuning of the FO-PI and FO-PD controller parameters is based on ensuring a ﬂat phase around the frequency of interest (the iso-damping property), a gain crossover frequency, and phase margin. The procedure is lengthy and is based on maximizing the robustness to plant gain variations. A different approach for the auto-tuning of fractional order PID controllers has been also proposed [17], which is inspired from both the classical Zigler–Nichols and Åström–Hägglund tuning methods. The Ziegler–Nichols tuning procedure is ﬁrstly used to determine the proportional and integrative gains of the controller, while the initial value of derivative gain is obtained using Åström– Hägglund method. The same frequency domain speciﬁcations are used here as well. Based on the critical frequency and critical gain obtained according to the Åström–Hägglund method, two non- linear equations are obtained and used to achieve a speciﬁed phase margin. A ﬁne tuning of the derivative gain is required to achieve the best numerical solutions of these two equations. The fractional orders μ and λ are obtained from these equations using an optimization technique. In case of a better step response of the closed loop system, an optimization model, developed using Simulink MATLAB, is used, which employs the previously com- puted controller parameters as initial values and produces new values for the controller parameters. In the current study, a novel auto-tuning of fractional order PI and PD controllers is proposed for stable, minimum phase, integer order or fractional order processes. The tuning equations are based on the widely used nonlinear system of equations obtained by specifying a certain phase margin, gain crossover frequency and iso-damping property. To determine the FO-PI/FO-PD controller parameters, apart from the performance speciﬁcations, the mag- nitude, phase and phase slope of the process at the gain crossover frequency are necessary. It is assumed that no process model (e.g. transfer function) is available. The magnitude and phase can be obtained directly using frequency domain data, via a sine test applied to the process. However, the novelty of the approach is that the phase slope can also be computed, via ﬁltering techniques applied to the same sine test data. The efﬁciency of the proposed auto-tuning method is demonstrated using several simulation examples. An example that considers a known process transfer function is also given, where the auto-tuning results are compared to the analytical method to validate the proposed technique. Additionally, the method is suitable and demonstrated for pro- cesses which are described by both integer order model structures, as well as generalized fractional order models. The paper is structured as follows. Section 2 describes the clas- sical tuning technique for FO-PI/FO-PD controllers, for the case of a known process model. Section 3 details the basis for the auto- tuning procedure in the case of no process model; it explains the technique for estimating the process phase slope. Section 4 includes the pseudo-algorithm for auto-tuning of FO-PI/FO-PD controllers, while Section 5 presents the simulation results that prove the efﬁciency of the proposed auto-tuning technique. The ﬁnal section contains the concluding remarks. 2. Classical tuning procedure for fractional order PI/PD controllers The transfer functions of the fractional order PI/PD (FO-PI/FO- PD) controllers are indicated below: HFOÀ PIðsÞ ¼ kp 1þ ki sμ ð1Þ HFOÀ PDðsÞ ¼ kp 1þkdsλ ð2Þ with the controller parameters deﬁned as follows: μ; λA 0C2ð Þ the fractional orders and kp, ki and kd the proportional, integrative and derivative gains, respectively. One of the most widely used tuning procedures for fractional order PI/PD controllers starts with a set of three frequency domain performance speciﬁcations that are used to determine the FO-PI/FO-PD controller parameters [9]. These performance speciﬁcations refer to: 1. A gain crossover frequency ωgc: the gain crossover frequency is related to the settling time of the closed loop system and may be thus used as an important tuning parameter for the con- troller [18]. A large gain crossover frequency will result in a smaller closed loop settling time. In order for a system to ensure the imposed gain crossover frequency, the following condition must hold: HopenÀloopðjωgcÞ ¼ 1 ð3Þ where Hopen-loop(s) is the loop transfer function deﬁned as: Hopen-loop(s)¼P(s). HFOC(s), where P(s) is the transfer function of the process to be controlled and HFOC(s) is either the FO-PI or FO-PD controller deﬁned in (1) or (2), respectively. 2. A phase margin φm: Phase margin is an important measure of system stability and also an indicator of the closed loop over- shoot [18,19]. Usually, an interval between 45° and 65° is used for a proper phase margin selection. In order for a system to ensure a certain phase margin, the following condition must hold: ∠Hopen ÀloopðjωgcÞ ¼ Àπþϕm ð4Þ 3. Iso-damping property: This condition ensures that the system is more robust to gain changes and the overshoot of the response is almost constant within a gain range. To ensure a constant overshoot, a constant phase margin needs to be maintained around the desired gain crossover frequency, which ultimately implies that the phase of the open-loop system must be kept constant around the speciﬁed ωgc [9]. In order for a system to ensure the iso-damping property, the following condition must hold: d ∠HopenÀloopðjωÞ À Á dω ω ¼ ωgc ¼ 0 ð5Þ The complex representations in the frequency domain of the transfer functions describing the FO-PI or the FO-PD in (1) and (2) R. De Keyser et al. / ISA Transactions 62 (2016) 268–275 269
- 3. are given below: HFOÀPIðjωÞ ¼ kp 1þkiωÀ μ cos πμ 2 Àj sin πμ 2 h i ð6Þ HFOÀPDðjωÞ ¼ kp 1þkdωλ cos πλ 2 þj sin πλ 2 ! ð7Þ The corresponding modulus and phase for each of these two controllers are then computed as HFOÀPIðjωÞ ¼ kp ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þ2kiωÀ μ cos πμ 2 þki 2 ωÀ 2μ r ð8Þ ∠HFOÀPIðjωÞ ¼ Àa tan kiωÀμ sin πμ 2 1þkiωÀμ cos πμ 2 ! ð9Þ for the FO-PI controller and HFOÀPDðjωÞ ¼ kp ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þ2kdωλ cos πλ 2 þkd 2 ω2λ r ð10Þ ∠HFOÀPDðjωÞ ¼ a tan kdωλ sin πλ 2 1þkdωλ cos πλ 2 ! ð11Þ for the FO-PD controller. Then, for a FO-PI controller, the conditions in (3)–(5) may be further written as kp ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þ2kiωÀμ gc cos πμ 2 þki 2 ωÀ2μ gc r ¼ 1 PðjωgcÞ ð12Þ kiωÀ μ gc sin πμ 2 1þkiωÀμ gc cos πμ 2 ¼ tan πÀϕm þ∠PðjωgcÞ À Á ð13Þ μkiωÀ μÀ 1 gc sin πμ 2 1þ2kiωÀμ gc cos πμ 2 þk 2 i ωÀ2μ gc þ d∠PðjωÞ dω ω ¼ ωgc ¼ 0 ð14Þ To tune the FO-PI controller, the system of nonlinear Eqs. (12)– (14) needs to be solved using either optimization techniques or graphical methods [9,10,20]. For the optimization techniques, the Matlab optimization toolbox may be used, namely the fmincon function, for which the modulus condition in (12) may be used as the function intended to be minimized, while the phase condition in (13) and the robustness condition in (14) are speciﬁed as the nonlinear constraints. Initial values for the controller parameters need also to be speciﬁed, with the possibility of setting their lower and upper bounds, as well. The fmincon function will return the controller parameters, such that the modulus condition is mini- mized and the nonlinear constraints are met. An alternative pos- sibility of determining the controller parameters is based on gra- phical methods. This approach consists in the evaluation of the ki parameter as a function of the fractional order μ from (13) and (14). Then for different values of μϵ(0,2), the corresponding ki values are computed. The graphical approach then consists in plotting of the different ki values determined based on (13), as a function of the fractional order values μϵ(0,2), as well as the cor- responding ki values computed using (14). The intersection point of these two graphs gives the ﬁnal values for the ki and μ para- meters. Once these have been determined, the modulus condition in (12) may be used to compute the ﬁnal value for the kp para- meter. Similar equations may be obtained using the FO-PD con- troller, by simply following the substitutions kd ¼ki and λ¼ Àμ, in (12)–(14). Nevertheless, regardless of the approach taken to determine the controller parameters, to completely tune the FO- PI/FO-PD controllers, the modulus, phase and phase slope of the process at the gain crossover frequency have to be known, as indicated in the above Eqs. (12)–(14). 3. Proposed methodology As mentioned in the previous section, in order to tune the FO- PI/FO-PD controllers, the phase, magnitude and phase slope of the process at the imposed gain crossover frequency need to be determined. The phase and magnitude of any stable process at a speciﬁc gain crossover frequency ωgc may be easily determined by applying a sinusoidal input signal to the process as indicated below: uðtÞ ¼ Ai sin ωgct À Á ð15Þ The output signal, denoted as y(t), of the process may be represented as indicated in Fig. 1, where the input signal in (15) has also been represented. Assume in what follows that the pro- cess may be described by a transfer function P(s), considered unknown. Then, the process magnitude and phase at the test frequency may be computed according to M ¼ PðjωgcÞ ¼ Ao Ai ð16Þ ϕ ¼ ∠PðjωgcÞ ¼ ωgcτ ¼ ωgc ti Àtoð Þ ð17Þ where Ao is the output amplitude and τ¼ti Àto is the time shift between the input u(t) and output y(t) signals, M is the magnitude and φ is the phase of the process at the speciﬁc gain crossover frequency ωgc. Thus, if the imposed gain crossover frequency ωgc is used, then by applying the signal u(t) in (15) to any stable unknown process, the experimental results obtained may be further used to deter- mine the process modulus and phase at the gain crossover fre- quency, using (16) and (17). However, to further tune either the FO-PI controller according to (12)–(14) or a FO-PD controller according to a similar set of equations, the phase slope d∠PðjωÞ dω ω ¼ ωgc has to be determined as well. Assuming a signal x(t) as the output to an input signal v(t)¼t.u (t) applied to the process P(s), it follows that L tUuðtÞð ÞUPðsÞ ¼ XðsÞ ð18Þ which leads to À dUðsÞ ds UPðsÞ ¼ XðsÞ ð19Þ where the property of the Laplace transform has been used, L ÀtUuðtÞð Þ ¼ dUðsÞ ds . Consider also that if a signal u(t) is applied at the input of the process derivative dPðsÞ ds , then the corresponding output would be Fig. 1. Input and output signals of a stable process (blue line—input signal, green line—output signal); arbitrary time units on X-axis and arbitrary amplitude units on the Y-axis. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.) R. De Keyser et al. / ISA Transactions 62 (2016) 268–275270
- 4. yðtÞ, and in the Laplace domain this leads to UðsÞU dPðsÞ ds ¼ YðsÞ ð20Þ as well as UðsÞUPðsÞ ¼ YðsÞ ð21Þ where Y(s) is the Laplace transform of the output y(t), when the signal u(t) is now applied to the process P(s). Then, consider also the derivative of the process output signal Y (s): dYðsÞ ds ¼ L ÀtUyðtÞð Þ ) ÀtUyðtÞ ¼ LÀ1 dYðsÞ ds ¼ LÀ1 d PðsÞUUðsÞ½ ds ð22Þ where relation (21) has been used. However d PðsÞUUðsÞð Þ ds ¼ dPðsÞ ds UUðsÞþ dUðsÞ ds UPðsÞ ð23Þ If in this last equation, we use (20) and (19), then (23) may be rewritten as d PðsÞUUðsÞð Þ ds ¼ YðsÞÀXðsÞ ð24Þ Using (24) in (22) and assuming zero initial conditions, leads to Àt UyðtÞ ¼ LÀ1 YðsÞÀXðsÞ À Á ð25Þ which may be written in the time domain as xðtÞÀtUyðtÞ ¼ yðtÞ ð26Þ where yðtÞ is the output of the process derivative, as denoted in (20). If we now consider the sinusoidal input signal in (15), its Laplace transform is given by UðsÞ ¼ Aiωgc s2 þω2 gc ð27Þ The derivative of this input signal with respect to the Laplace variable s is computed according to (27): dUðsÞ ds ¼ À 2Aiωgcs s2 þω2 gc 2 ð28Þ which means that in this case, the signal X(s) in (19) may be fur- ther computed as XðsÞ ¼ À dUðsÞ ds UPðsÞ ¼ 2Aiωgcs s2 þω2 gc 2 UPðsÞ ¼ Aiωgc s2 þω2 gc 2s s2 þω2 gc UPðsÞ ð29Þ where if we now use (27) and (21), we have the ﬁnal Laplace transform for the x(t) signal: XðsÞ ¼ 2s s2 þω2 gc UYðsÞ ð30Þ The experimental scheme in Fig. 2 may be used in the auto- tuning of FO-PI/FO-PD controllers. If a sinusoidal signal u(t) as indicated in Fig. 2 is applied at the input of the process with unknown transfer function, then the resulting experimental data for the process output y(t), as well as for the process derivative yðtÞ, are later used to determine the process magnitude, phase and phase slope at the imposed gain crossover frequency ωgc. As indicated in (26), the signal yðtÞ is computed as the difference between the signal x(t) and the term tUyðtÞ, also shown in the right part of Fig. 2. To determine the signal x(t), (30) is used and thus, as indicated in Fig. 2, the process output signal y(t) is passed through the ﬁlter 2s s2 þω2 gc . In Fig. 2, M and φ denote the modulus and phase of P(s) at a speciﬁc frequency ω. Moreover, based on (20), M and ϕ denote the modulus and phase of the process derivative dPðsÞ ds . In the frequency domain, the derivative dPðsÞ ds at the gain cross- over frequency may be computed by replacing the Laplace variable s¼jωgc: dPðjωÞ dðjωÞ ω ¼ ωgc ¼ Me j φ ) Àj d MðωÞUejφðωÞ À Á dω ω ¼ ωgc ¼ Me j φ ð31Þ Extending the left hand side of (31) leads to Àj dM dω ω ¼ ωgc ejϕ þM dejϕ dϕ dϕ dω j ω ¼ ωgc 0 @ 1 A ¼ Mejϕ or Àj dM dω ω ¼ ωgc ejϕ þMUejϕdϕ dω ω ¼ ωgc ¼ Mejϕ ð32Þ Dividing (32) by ejϕ, the next equation is obtained as Àj dM dω ω ¼ ωgc þM dϕ dω ω ¼ ωgc ¼ Mej ϕÀ ϕ À Á ¼ M cos ϕÀϕ þjM sin ϕÀϕ ð33Þ Equating the real and imaginary parts of the left and right hand sides of (33), leads to the ﬁnal equalities that represent the slopes of the process magnitude and phase at the speciﬁed gain crossover frequency ωgc: dM dω ω ¼ ωgc ¼ ÀM sin ϕÀϕ ð34Þ dϕ dω ω ¼ ωgc ¼ M M cos ϕÀϕ ð35Þ Then, if the test frequency used for the sinusoidal input signal u (t) in Fig. 2 is the gain crossover frequency, (35) may be used to determine the phase slope of the process P(s) at the gain crossover frequency: d∠PðjωÞ dω ω ¼ ωgc ¼ dϕ dω ω ¼ ωgc ¼ M M cos ϕÀϕ ð36Þ where M ¼ Ay Ai ð37Þ ϕ ¼ ωgcτy ¼ ωgc ti Àty À Á ð38Þ with Ay the amplitude of the sinusoidal signal yðtÞ, τy the time shift between the two signals u(t) and yðtÞ. Note: the theory presented in this section and the related experimental scheme in Fig. 2 describe the underlying concept of obtaining the process phase slope (as well as the magnitude slope) via a simple sine test. However, the scheme in Fig. 2 is prone to stochastic disturbances that are acting on the measured process output y(t), if these disturbances have a frequency spectrum close Fig. 2. Experimental scheme to obtain the (sine) signals y(t) and yðtÞ and compute the phase slope of the process at the gain crossover frequency. R. De Keyser et al. / ISA Transactions 62 (2016) 268–275 271
- 5. to the test frequency ωgc. The scheme can however be converted to an equivalent scheme which is very robust against such stochastic disturbances. This conversion - although important for practical implementation - is of a pure technical matter which has no impact on the fundamental idea presented in this paper. It is beyond the scope of this paper. These practical implementation issues as well as their demonstration on real-life test processes - corrupted with stochastic noise - are the subject of a subsequent paper. For that reason, the fundamental idea of this paper is illu- strated in the following section on noise-free simulated processes. 4. Illustrative simulation examples To auto-tune the FO-PI or alternatively the FO-PD controller, the following steps are necessary: 1. Given the imposed gain crossover frequency, ωgc, apply a sinusoidal input signal of the form (15) to the process to be controlled. 2. Perform the experimental test described in Fig. 2 and determine Ao and Ay, τ and τy. Compute M and ϕ using (16) and (17) as well as M and ϕ using (37) and (38). 3. The tuning of the fractional order controller is then based on a speciﬁed phase margin φm and on an optimization routine for the system of equations given in (12)–(14), where PðjωgcÞ ¼ M ð39Þ ∠PðjωgcÞ ¼ ϕ ð40Þ d∠PðjωÞ dω ω ¼ ωgc ¼ M M cos ϕÀϕ ð41Þ The autotuning method presented above will be illustrated via some simulation examples, considering both processes that may be described by integer order or fractional order models. For the integer order case, a ﬁrst order time delay process and a higher order process are considered, as illustrative examples from the 131 benchmark processes proposed by Åström and Hägglund [21]. In the ﬁrst example, it is shown that the classical fractional order controller tuning procedure and the proposed auto-tuning method produce similar results. In the second example, the proposed auto- tuning method is detailed for a higher order process, including the closed loop step response results, demonstrating the robustness of the design procedure against gain variations. Finally, a third example is included which presents the auto-tuning of the frac- tional order PI controller for a process described by a fractional order model. This last example extends the auto-tuning procedure to general processes. The effectiveness of the proposed auto-tuning procedure in terms of closed loop results greatly depends upon the perfor- mance speciﬁcation set. This is similar to the classical tuning procedure for fractional order controllers. Several rules regarding the feasibility range of performance speciﬁcations may be found in [22] and [23]. In this paper, the purpose is to show that the clas- sical tuning procedure for fractional order controllers can be suc- cessfully extended to situations where a mathematical process of the model is not available. Therefore, in the numerical examples, rather than an optimal set of performance speciﬁcations to obtain excellent closed loop results, a set of random, but feasible, fre- quency domain performance speciﬁcations are selected, without any major concern regarding the ﬁnal closed loop results. 4.1. First order time delay process Consider the plant given by the following transfer function: P1ðsÞ ¼ 1 6sþ1 eÀs ð42Þ The performance speciﬁcations are: a gain crossover frequency ωgc ¼0.1 rad/s, a phase margin φm ¼55° and a ﬂat open loop phase around the gain crossover frequency to ensure the iso-damping property. To tune the controller, the modulus, phase and phase Table 1 Process modulus, phase and phase slope at the gain crossover frequency. P1ðjωgcÞ ∠P1ðjωgcÞ (deg) ∂∠P1 ðjωÞ ∂ω ω ¼ ωgc Analytical computation 0.8575 À36.69 À5.4118 Experimental identiﬁcation 0.8574 À37.81 À5.4551 Fig. 3. Graphical solution for the classical tuning approach. Table 2 Fractional order controller parameter values using the classical method and the auto-tuning method. Integral gain ki Proportional gain kp Fractional order μ Classical approach 0.1412 0.5065 1.245 Proposed auto-tuning method 0.1407 0.5142 1.238 0 50 100 150 200 250 300 350 400 -6 -4 -2 0 2 4 6 Time (s) Amplitude Input u(t) Output y(t) Output y bar (t) Fig. 4. Experimental results for ﬁrst order time delay process ybar tð Þ ¼ yðtÞ À Á . R. De Keyser et al. / ISA Transactions 62 (2016) 268–275272
- 6. slope of the process P1(s) at the gain crossover frequency may be determined analytically based on the process transfer function in (42). The results are given in Table 1. To tune the fractional order PI controller, using the classical approach described in Section 2, the modulus, phase and robust- ness conditions in (12)–(14) may now be expressed as kp ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þ2ki0:1À μ cos πμ 2 þki 2 0:1À 2μ r ¼ 1 0:8575 ð43Þ ki0:1À μ sin πμ 2 1þki0:1À μ cos πμ 2 ¼ tan ð1:5413Þ ð44Þ μki0:1À μÀ 1 sin πμ 2 1þ2ki0:1Àμ cos πμ 2 þk 2 i 0:1À2μ À5:4118 ¼ 0 ð45Þ Fig. 3 shows the values obtained for the ki parameter, as a function of the fractional order μ, as resulting from (44) and (45). The intersection point gives the ﬁnal controller parameter values: ki ¼0.1412 and μ¼1.245. Using now (43), the proportional gain is determined to be kp¼0.5065. A summary of the controller para- meter values is given in Table 2. The experimental auto-tuning procedure described in Section 3 is now used to obtain the modulus, phase and phase slope of the process, assuming that the process transfer function (42) is unknown. A sinusoidal signal is used with unit amplitude and test frequency taken as ωgc ¼0.1 rad/s. The experimental results are given in Fig. 4. The amplitude of the input signal at ti ¼329.8 s is Ai ¼1, as indicated in Fig. 4. The amplitude of the output signal at to¼336.4 s is Ao ¼0.8574. Using (16) and (17), the parameters M and φ are computed, with M¼0.8574 and φ ¼ 0:1U 329:8À336:4ð Þ ¼ À0:66 rad ¼ À 37:81o . Then, according to (39) and (40), P1ðjωgcÞ ¼ 0:8574 and ∠P1ðjωgcÞ ¼ À37:81o . The amplitude of yðtÞ is also determined from Fig. 3, Ay ¼ 5:1659 at ty ¼ 372:2 s. Then, using (37) and (38), M ¼ Ay Ai ¼ 5:1659 and ϕ ¼ 0:1U 329:8À372:2ð Þ ¼ À4:24 rad or ϕ ¼ À 242:93o . Finally, using (41), the process phase slope is computed as d∠P1ðjωÞ dω ω ¼ ωgc ¼ À 5:4551 s. These results are also included in Table 1, clearly showing that the experimental test in Fig. 2 produces similar results as those obtained analytically. A FO-PI controller is tuned next according to the design pro- cedure presented previously using the experimentally computed values in Table 1 and based on (12)–(14). The graphical approach yields in this case a proportional gain kp¼0.5142, an integrative gain ki ¼0.1407 and a fractional order μ¼1.238. These controller parameter values - obtained without knowledge of the process transfer function - are included in Table 2, along with the con- troller parameter values as computed using the classical tuning procedure - i.e. with knowledge of the process transfer function. The results in Table 2 indicate that the auto-tuning procedure described in Section 3 produces similar results as the classical tuning procedure described in Section 2. The transfer function of this FO-PI controller, based on the auto-tuning approach, is HFOÀ PIðsÞ ¼ 0:5142 1þ 0:1407 s1:238 ð46Þ The Bode diagram of the open loop system considering the designed FO-PI controller, as well as the process transfer function in (42) is given in Fig. 5, showing that in nominal conditions the phase margin and the gain crossover frequency meet the imposed performance speciﬁcations. Also, the phase around the gain crossover frequency is ﬂat, which implies that for 730% gain variations the phase margin remains unchanged, as indicated in the Bode diagram. The same conclusions may be drawn from the closed loop responses in Fig. 6, where the overshoot remains the same as in the nominal case, despite the 730% gain variations. Fig. 5. Bode diagram of the open loop system for example 1. 0 20 40 60 80 100 0 0.5 1 1.5 Time (s) Output 0 20 40 60 80 100 0.5 1 1.5 2 Time (s) Input nominal +30% uncertainty -30% uncertainty Fig. 6. Closed loop step responses for example 1: output and input signals. 0 50 100 150 200 250 300 -6 -4 -2 0 2 4 6 Time (s) Amplitude Input u(t) Output y(t) Output y bar (t) Fig. 7. Experimental results for higher order process ybar tð Þ ¼ yðtÞ À Á . R. De Keyser et al. / ISA Transactions 62 (2016) 268–275 273
- 7. 4.2. Higher order process As it has been shown in the previous example, the analytical and auto-tuning methods produce similar results. Consider now the plant given by the following transfer function: P2ðsÞ ¼ 1 sþ1ð Þ6 ð47Þ The performance speciﬁcations are: a gain crossover frequency ωgc ¼0.1 rad/s, a phase margin φm¼50° and a ﬂat open loop phase around the gain crossover frequency to ensure the iso-damping property. The auto-tuning of a FO-PI controller, as described in Section 3, starts with an experimental test similar to Fig. 2, where the input signal is applied of the form (15), with ωgc ¼0.1 rad/s and Ai ¼1. The experimental results are given in Fig. 7, where the amplitude Ai of the input signal is measured at ti ¼141.4 s. Simi- larly, the amplitude of the output signal is Ao¼ 0.9706 at to¼147.4 s. Using (16) and (17), the parameters M and φ are computed, with M¼0.9706 and φ ¼ 0:1U 141:4À147:4ð Þ ¼ À0:6 rad ¼ À 34:37o . Then, according to (39) and (40), P1ðjωgcÞ ¼ 0:9706 and ∠P1ðjωgcÞ ¼ À34:371. The amplitude of yðtÞ is also determined from Fig. 6, Ay ¼ 5:7932 at ty ¼ 180 s. Then, using (37) and (38), M ¼ Ay Ai ¼ 5:7932 and ϕ ¼ 0:1U 141:4À180ð Þ ¼ À3:86 rad or ϕ ¼ À 221:16o . Finally, using (41), the process phase slope is computed as d∠P1ðjωÞ dω ω ¼ ωgc ¼ À 5:927 s. With the process modulus, phase and phase slope determined experimentally, the system of equations in (12)–(14) is solved to determine the controller parameters: kp ¼0.4627, ki ¼0.1210 and μ¼1.32. The transfer function of the designed FO-PI controller is HFOÀ PIðsÞ ¼ 0:4627 1þ 0:121 s1:32 ð48Þ The closed loop simulation results considering a unit step reference are given in Fig. 8, considering both the nominal gain equal to 1, as indicated in (47), as well as 730% gain variations, with the process gain now considered 0.7 and 1.3 respectively. The results show that the designed FO-PI auto-tuner is robust to gain variations as it manages to maintain a constant overshoot, despite these modeling uncertainties. 4.3. Fractional order process Consider now the plant given by the following transfer function: P3ðsÞ ¼ 1 5s0:5 þ1 ð49Þ The performance speciﬁcations are: a gain crossover frequency ωgc ¼1 rad/s, a phase margin φm ¼70° and a ﬂat open loop phase around the gain crossover frequency to ensure the iso-damping property. The auto-tuning of a FO-PI controller, as described in Section 3, starts with an experimental test similar to Fig. 2, where the input signal is applied of the form (15), with ωgc ¼1 rad/s and Ai ¼1. The experimental results are given in Fig. 9, where the amplitude Ai of the input signal is measured at ti ¼39.27 s. Fig. 9 also shows that the yðtÞ signal at the output of the process deri- vative will exhibit a very long transient regime, due to the half order integrator involved [9]: d P3ðsÞ½ ds ¼ À 2:5 5s0:5 þ1 À Á2 s0:5 ð50Þ The amplitude of the y(t) signal is computed in the transient regime as Ao ¼ AoðtoÞ þAoðt1Þ 2 ¼ 0:1771 þ 0:171 2 ¼ 0:1741, where to¼39.932 s and t1 ¼43.1 s. Then, using (16) and (17), the para- meters M and φ are computed, with M¼0.1741 and φ ¼ 1U 39:27À39:932ð Þ ¼ À0:662 rad ¼ À 37:93o . Then, according to (39) and (40), P1ðjωgcÞ ¼ 0:1741 and ∠P1ðjωgcÞ ¼ À37:931. The amplitude of yðtÞ is also determined from Fig. 9, similarly to how Fig. 8. Closed loop step responses for example 2: output and input signals. 0 10 20 30 40 50 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Time (s) Amplitude Input u(t) Output y(t) Output ybar (t) Fig. 9. Experimental results for fractional order process ybar tð Þ ¼ yðtÞ À Á . 0 5 10 15 0 0.5 1 1.5 Time (s) Output 0 5 10 15 0 2 4 6 Time (s) Input nominal +30% uncertainty -30% uncertainty Fig. 10. Closed loop step responses for example 3: output and input signals. R. De Keyser et al. / ISA Transactions 62 (2016) 268–275274
- 8. Ao has been determined. An analysis of the yðtÞ signal in Fig. 9, gives the following Ay ¼ AyðtyÞþ Ayðt2Þ 2 ¼ À 0:0465 þ 0:1999 2 ¼ 0:0767, with ty ¼ 44:523 s and t2 ¼47.67 s. Then, using (37) and (38), M ¼ Ay Ai ¼ 0:0767 and ϕ ¼ 1U 39:27À44:523ð Þ ¼ À5:235 rad or ϕ ¼ À 301o . Finally, using (41), the process phase slope is computed as d∠P1ðjωÞ dω ω ¼ ωgc ¼ À 0:0533 s. With the process modulus, phase and phase slope determined experimentally, the system of equations in (12)–(14) is solved to determine the controller parameters: kp¼0.38, ki ¼14.77 and μ¼0.8418. The transfer function of the designed FO-PI controller is: HFOÀPIðsÞ ¼ 0:38 1þ 14:77 s0:8418 ð51Þ The closed loop simulation results considering a unit step refer- ence are given in Fig. 10, considering both the nominal, as well as 730% gain variations. The results show that the designed FO-PI auto- tuner is robust to gain variations as it manages to maintain a con- stant overshoot, despite these modeling uncertainties. 5. Conclusions One of the most important aspects in control engineering consists in an accurate modeling of the process to be controlled. In a wide range of applications, an accurate model of the process is difﬁcult to be obtained if not impossible. In such case, the tuning of controllers becomes a tedious task. In this paper, a novel auto- tuning method for fractional order PI or PD controllers has been presented, that yields a robust controller despite the lack of an actual process model. A simple experiment is sufﬁcient in order to determine the process parameters that are required for the tuning of the fractional order controller: the process phase, the process magnitude and the slope of the process phase at a speciﬁed gain crossover frequency. The auto-tuning technique is exempliﬁed considering some benchmark processes, a ﬁrst order time delay process, a higher order process, as well as a process exhibiting some fractional order dynamics. The closed loop simulation results show that the proposed technique meets the performance criteria and ensures the robustness of the closed loop system despite the lack of an actual process model. A comparison between the tuning of a fractional order PI controller based on the classical approach (with knowledge of the process model) versus the tuning based on the proposed auto- tuning method (with lack of a process model), indicates that the two tuning procedures lead to similar results. Results are available for a future paper that will demonstrate the application of the proposed method on real-life processes, which are subject to stochastic disturbances. Acknowledgements Clara Ionescu is a post-doctoral fellow of the Flanders Research Centre (FWO), grant no. 12B345N. Cristina I. Muresan is ﬁnanced by a grant of the Romanian National Authority for Scientiﬁc Research and Innovation, CNCS – UEFISCDI, project number PN-II- RU-TE-2014-4-0598, TE 86/2015. 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