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- 1. Research Article On the fragility of fractional-order PID controllers for FOPDT processes Fabrizio Padula a , Antonio Visioli b,n a Dipartimento di Ingegneria dell'Informazione, University of Brescia - Italy, Italy b Dipartimento di Ingegneria Meccanica e Industriale, University of Brescia - Italy, Via Branze 38, I-25123 Brescia, Italy a r t i c l e i n f o Article history: Received 2 December 2014 Received in revised form 26 August 2015 Accepted 9 November 2015 Available online 27 November 2015 This paper was recommended for publica- tion by Dr. Y. Chen Keywords: Fractional-order controllers PID control Tuning Fragility a b s t r a c t This paper analyzes the fragility issue of fractional-order proportional-integral-derivative controllers applied to integer ﬁrst-order plus-dead-time processes. In particular, the effects of the variations of the controller parameters on the achieved control system robustness and performance are investigated. Results show that this kind of controllers is more fragile with respect to the standard proportional-integral- derivative controllers and therefore a signiﬁcant attention should be paid by the user in their tuning. & 2015 ISA. Published by Elsevier Ltd. All rights reserved. 1. Introduction It is well known that a properly designed control system must provide an effective trade-off between performance and robust- ness. However, it has also been recognized that another important issue to be addressed is the fragility of the control system to the variation of the controller parameters, that is, the sensitivity of the robustness and/or performance of the control system to changes in the controller parameters. This issue has been raised in the literature in some papers (see, for example, [1]) and, in particular, in [2] where it has been stressed that design techniques based on the minimization of the H2, H1 and l1 norms can yield to high-order robust, optimal but also extremely fragile controllers, namely, a very small variation of the controller coefﬁcients can result in an unstable system. How- ever, in [3,4] it has been pointed out that this problem can be solved by using a suitable controller parametrization. As integer-order proportional-integral-derivative (IOPID) controllers are the most used controllers in industry, the fragility of such a kind for controllers has been speciﬁcally addressed in [5,6]. Therein, authors suggest to tune the IOPID controller in order to maximize the l2 norm of the controller parameter vector in the stabilizing region for a given plant. However, the typical industrial performance measures (related to the set-point following and/or to the load disturbance rejection task) are not taken into account. Further, it has been shown in [7] that this kind of approach applied to ﬁrst-order-plus-dead-time (FOPDT) and integrator-plus-dead-time (IPDT) processes yields a tuning similar to that obtained by using the Ziegler–Nichols step response method [8] which is known to be improvable under many points of view [9]. Thus, it has been recognized in the literature that one of the main reasons to investigate the fragility of IOPID controllers is to give to the user an idea of how a ﬁne tuning of the controller can be done [10–12]. In other words, as the IOPID parameters have a clear physical meaning, the operator can modify them in order to change the control system performance. In this context, it is useful to evaluate the sensitivity of the robustness/performance behavior with respect to (small) changes of the parameters. For this pur- pose, a graphic tool called fragility rings providing a visual aid for evaluation of the controller robustness/fragility has been proposed in [13]. In the recent years, there has also been a signiﬁcant interest from the academic and industrial communities for fractional- order-proportional-integral-derivative (FOPID) controllers because they are capable to provide (as there are ﬁve parameters to tune) more ﬂexibility in the control system design (see, for example, [14–17]). Many different tuning rules have been proposed in the literature to facilitate their use (see, for example, [18–23]). In this context, while the problem of stabilizing a (possibly fractional) dynamic system using FOPID controllers has been already addressed in the literature (see, for example, [24–26]), for such a Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/isatrans ISA Transactions http://dx.doi.org/10.1016/j.isatra.2015.11.010 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved. n Corresponding author. Tel.: þ39 030 3715460; fax: þ39 030 380014. E-mail addresses: fabrizio.padula@unibs.it (F. Padula), antonio.visioli@unibs.it (A. Visioli). ISA Transactions 60 (2016) 228–243
- 2. kind of controllers, a fragility analysis has been only partially exploited until now. In particular, in [27,28], the tuning of the FOPID controllers is performed by considering the centroids of the admissible regions in the parameter space so that a non-fragile controller results. However, one of the main purposes for evalu- ating the fragility of the controller is in evaluating the sensitivity of the robustness/performance indexes to the (possibly ﬁne) tuning of the parameters. Indeed, in order to foster a widespread use of FOPID controllers in industrial plants, in addition to well-established tuning rules, clear guidelines on how to modify the controller parameters should be given to the operator in order for him/her to be con- ﬁdent with them. Thus, the aim of this paper is to provide a fra- gility analysis for FOPID controllers and to make a comparison with IOPID controllers in order to understand the differences that should be taken into account in the adjustment of the parameters starting from a given tuning. For this purpose, the tuning rules proposed in [23,29], which aim at minimizing the integrated absolute error subject to constraints on the maximum sensitivity, are used, both for FOPID and IOPID controllers. Both the tuning rules for the set-point following and the load disturbance rejection tasks are considered. They also have the signiﬁcant feature of providing a control action that is invariant when the time unit is changed. These tuning rules are therefore suitable to perform a fragility analysis with respect to both robustness and performance. It is worth stressing that the calculated fragility depends on the nominal parameters of the control system and for this reason, in order to obtain a fair comparison, we select tuning rules that solve the same optimization problem, so that the possible additional complexity of adjusting the parameters of a FOPID controller, with respect to a IOPID one, starting from a given tuning is clearly addressed. The fragility is evaluated by changing all the parameters at the same time or just one of them by keeping the other ones ﬁxed. The latter case is performed in order to investigate which parameter has more inﬂuence on the controller fragility. The paper is organized as follows. The basic deﬁnitions employed for the fragility evaluation are reviewed in Section 2, in addition to the description of the tuning rules used for both integer- order and fractional-order PID controllers. The fragility analysis related to the robustness is presented in Section 3 while that related to the performance is presented in Section 4. A discussion is made in Section 5, while conclusions are drawn in Section 6. 2. Fragility indices The fragility indices proposed in [10–12] are brieﬂy reviewed in this section for the sake of clarity and in order to introduce the notation used in presenting the results. Consider a unity feedback control system (see Fig. 1) where the process (which is assumed to be self-regulating) is denoted as P and the controller as C. In this paper, the controller is a FOPID controller, which can be expressed either in series form, i.e., CðsÞ ¼ Kp Tisλ þ1 Tisλ Tdsμ þ1 Tf sþ1 ð1Þ or in parallel (ideal) form, i.e., CðsÞ ¼ Kp 1þ 1 Tisλ þTdsμ 1 Tf sþ1 : ð2Þ In both expression, Kp is the proportional gain, Ti is the integral time constant, Td is the derivative time constant and λ and μ are the noninteger orders of the integral and derivative terms respectively. Note that it is important to consider both forms (1) and (2) because it is not possible to transform (2) into an equivalent form (1) and vice versa unless Ti Z4Td and λ ¼ μ [29]. In order to implement the fractional-order controller, the well-known Ous- taloup continuous integer-order approximation [30] has been employed to approximate the fractional differintegrator. In this paper 16 poles and zeros have been used in order to approximate the fractional differintegrator in a frequency range ½ωl; ωh, where ωl and ωh have been selected as 0:0001ωc and 10000ωc respectively, with ωc being the gain crossover frequency. It is worth noting that the used number of poles and zeros leads to a computationally demanding controller and, actually, the frac- tional controller could be approximated with a lower order integer one. Nevertheless, considering that the purpose of this paper is the fragility analysis of the fractional controller, a higher computational cost is accepted in order to achieve an improved approximation. The approximated and the ideal open loop transfer function in this way are virtually indistinguishable at those frequencies that have an appreciable impact on the closed- C r ye P d Fig. 1. The considered control scheme. Table 1 The controller parameters for the considered example with L=T ¼ 0:5 and for the different control tasks, set-point (SP) following and load disturbance (LD) rejection with a maximum sensitivity of 1.4 and 2.0 respectively. Controller Kp Ti Td λ μ FOPID series SP 1.4 1.1060 0.9839 0.1554 1 1.2 SP 2.0 1.6698 1.0281 0.1975 1 1.1 LD 1.4 0.7818 0.4683 0.2617 1 1.1 LD 2.0 1.1182 0.4236 0.3105 1 1.1 FOPID parallel SP 1.4 1.3307 1.1765 0.1384 1 1.1578 SP 2.0 2.0850 1.2507 0.1757 1 1.1351 LD 1.4 1.2786 0.8824 0.1686 1 1.1351 LD 2.0 2.3611 0.9079 0.1440 1 1.1525 IOPID series SP 1.4 0.8676 0.8127 0.2074 – – SP 2.0 1.4708 0.9568 0.2347 – – LD 1.4 0.6369 1.0081 0.3031 – – LD 2.0 1.007 0.4106 0.3304 – – F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243 229
- 3. loop system dynamics. It can be also noted that an additional ﬁrst-order ﬁlter has been employed in both (1) and (2) in order to make the controller proper. The selection of the time constant Tf is done in such a way that the high-frequency noise is ﬁltered without inﬂuencing the dynamics of the controller signiﬁcantly [29,31,32]. Eventually, considering the Oustaloup approximation and that only the fractional part μ (μÀ1 if μ41) of the derivative action is approximated, an integer ﬁlter is enough to guarantee the properness of the controller. Then, it has also to be noted that by selecting λ ¼ μ ¼ 1, an IOPID controller is obtained. In this paper, just for the sake of comparison (as the analysis will be focused on FOPID controllers), we consider the IOPID controller in series form, i.e., CðsÞ ¼ Kp Tisþ1 Tis Tdsþ1 Tf sþ1 ð3Þ (note that with the employed tuning rules described below it results in Ti 44Td and therefore an equivalent IOPID controller in ideal form can always be considered, i.e., the optimal IOPID controller is unique). The typical control speciﬁcation requires that a predeﬁned performance is obtained in the set-point fol- lowing and load disturbance rejection task. In both cases, a typical performance index related to the step responses is the integrated absolute error [33], which yields, in general, a small overshoot and a small settling time at the same time and is deﬁned as Je ¼ Z 1 0 jeðtÞj dt ¼ Z 1 0 jrðtÞÀyðtÞj dt; ð4Þ where r is the set-point signal and y is the process variable. From another point of view, it is often essential that the control system is also robust to changes in the process dynam- ics. A commonly employed measure of the robustness of the system is the maximum sensitivity, which represents the inverse of the minimum distance of the loop transfer function from the critical point (À1, 0) in the Nyquist plot and it is deﬁned as Ms ¼ max ωA½0;þ1Þ 1 1þCðsÞPðsÞ : ð5Þ For this reason, speciﬁc tuning rules have been devised for each controller (1)–(3) in order to minimize the Je value subject to constraints on the maximum sensitivity [23,29]. In parti- cular, both the set-point following and the load disturbance Fig. 2. Resulting values of RFIΔε for FOPID controller in series form (‘○’), for FOPID controller in parallel form (‘□’) and for IOPID controller (‘▵’). Top left: tuning for set-point with Ms ¼1.4. Top right: tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms ¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0. F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243230
- 4. rejection tasks have been considered separately and, for each task, the values of Ms ¼1.4 and Ms ¼2.0 have been selected (note that tuning rules related to integral and unstable pro- cesses have been proposed in [34]). In general, it has been shown that the FOPID controller provides a better performance than the IOPID controller and the improvement is achieved by using an integer-order integrator and a fractional derivative order μ41 [23]. The fragility of the controller can be evaluated with respect to either the robustness or the performance. By denoting as θ 0 c the vector of the controller parameters (that is, θ 0 c ¼ ½Kp; Ti; Td; λ; μ for the FOPID controller and θ 0 c ¼ ½Kp; Ti; Td for the IOPID controller), the loss of robustness of the control system when the controller parameters are perturbed can be expressed by the so-called Delta- Epsilon-Robustness-Fragility Index which is deﬁned as RFIΔε ¼ Mm sΔε M0 s À1 ¼ maxfMsðð17δεÞθ 0 c Þg Msðθ 0 c Þ À1; ð6Þ where Mm sΔε is the extreme maximum sensitivity, that is, the highest loss of robustness of the control system that occurs when all the parameters of the controller can vary of the same δε quantity with respect to their nominal values θ 0 c , considering all the possible combinations of the perturbed parameters. On the contrary, Msðθ 0 c Þ is the nominal sensitivity, that is, the sensitivity obtained with the nominal controller. It appears that, an index RFIΔε ¼ 0 implies that the controller is absolutely robustness-non-fragile. It is however recognized that a reasonable variation of the parameters is up to 20%. For this reason, a controller is considered to be robustness resilient if its delta 20 robustness fragility index is less than 0.10 (that is, RFIΔ20 o0:10), robustness non-fragile if RFIΔ20 r0:50 and robustness fragile if RFIΔ20 40:50. It is also important to evaluate the relative inﬂuence of a single parameter on the robustness fragility. In order to do that, the Parametric-Delta-Epsilon-Robustness-Fragility Index has been deﬁned as RFI pi δε ¼ M pi sδε M0 s À1 ¼ maxfMsðð17δεÞpi; θ 0 c Þg Msðθ 0 c Þ À1: ð7Þ Similar to the previous case, the loss of performance of the control system when the controller parameters are per- turbed (note again that the tuning rules minimize the int- Fig. 3. Resulting values of RFI Kp δε for FOPID controller in series form (‘○’), for FOPID controller in parallel form (‘□’) and for IOPID controller (‘▵’). Top left: tuning for set-point with Ms ¼1.4. Top right: tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0. F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243 231
- 5. egrated absolute error) can be expressed by the so-called Delta-Epsilon-Performance-Fragility Index which is deﬁned as PFIΔε ¼ Jm eΔε J0 e À1 ¼ maxfJeðð17δεÞθ 0 c Þg Jeðθ 0 c Þ À1; ð8Þ where Jm eΔε is the extreme performance and Je 0 is the nominal performance. The relative inﬂuence of a δε variation of a single controller parameter pi on the performance fragility of the control system can be expressed by the following Parametric-Delta-Epsi- lon-Performance-Fragility Index: PFI pi δε ¼ J pi eδε J0 e À1 ¼ maxfJeðð17δεÞpi; θ 0 c Þg Jeðθ 0 c Þ À1: ð9Þ Similarly again to the robustness case, by assuming a reasonable threshold of 20%, a controller is considered to be performance resilient if its delta 20 performance fragility index is less than 0.10 (that is, PFIΔ20 o0:10), performance non-fragile if RFIΔ20 r0:50 and performance fragile if RFIΔ20 40:50. Remark 1. It is worth stressing that the fragility indices obviously depend on the tuning of the IOPID or FOPID parameters. For this reason, in order to provide meaningful results, it is important to compare the FOPID and IOPID con- trollers with parameters selected in order to optimize the same performance index. 3. Robustness fragility The robustness fragility have been evaluated for the FOPID controllers in both series and parallel form and the results are compared with the IOPID controller. In particular, FOPDT pro- cesses have been considered. They are described by the transfer function PðsÞ ¼ K Tsþ1 eÀ Ls : ð10Þ Then, for the normalized gain K¼1 and for different values of the normalized dead time L=T in the interval ½0:1; 1, the tuning rules for the minimization of integrated absolute error of the set-point step response and of the load disturbance step response have been applied. In both cases, the values of the target maximum sensitivity are set to either Ms ¼1.4 or Ms ¼2.0. Fig. 4. Resulting values of RFITi δε for FOPID controller in series form (‘○’), for FOPID controller in parallel form (‘□’) and for IOPID controller (‘▵’). Top left: tuning for set-point with Ms ¼1.4. Top right: tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms ¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0. F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243232
- 6. Thus, for each process and for each of the three controllers, four cases have been considered and the RFIΔε index has been calculated for different values of δε, by iteratively considering all the possible variations of the parameters. For the sake of brevity, only the results related to the process with normalized dead time L=T ¼ 0:5 are shown. In this case, assuming K¼1, T¼1 and L¼0.5, we obtain, for the different con- trol speciﬁcations, the controller parameters shown in Table 1. Actually, the results related to the other processes are very similar to them. Results are shown in Fig. 2 where it has to be stressed that when the data is missing it means that the overall control system is unstable. Thus, it can be easily noted that the FOPID controller (both in series and parallel form) is much more robustness fragile (thus, the ﬁne tuning is more critical) than the IOPID controller. In order to evaluate better the inﬂuence of the single con- troller parameters, the Parametric-Delta-Epsilon-Robustness- Fragility Index RFI pi δε has also been computed for the different controller parameters. Results are shown in Figs. 3–7 (note that the IOPID controller does not include the λ and μ parameters). It appears that the robustness fragility of the FOPID controllers is less critical if one parameter at a time is ﬁne tuned and, in any case, the fractional order of the derivative term is the most dangerous parameter, especially when the employed tuning rule aims at achieving a more aggressive controller (namely, the target Ms ¼2.0 has been selected) and when a FOPID con- troller in parallel form is used. A discussion about this issue will be done in Section 5. 4. Performance fragility The same analysis done for the robustness fragility has been performed also for the performance fragility, that is, for each FOPDT process and for each of the three controllers with the four different considered tuning rules, the PFIΔε index has been cal- culated for different values of δε. Results related again to the process with a normalized dead time L=T ¼ 0:5 are shown in Fig. 8. The same conclusions as for the robustness fragility can be made for the performance fragility. The FOPID controllers (especially that in parallel form) are more sensitive than the IOPID controller with respect to changes in the parameters so that their ﬁne tuning can be more critical. By evaluating the Parametric-Delta-Epsilon- Performance-Fragility Index PFI pi δε for each single parameter (see Fig. 5. Resulting values of RFITd δε for FOPID controller in series form (‘○’), for FOPID controller in parallel form (‘□’) and for IOPID controller (‘▵’). Top left: tuning for set-point with Ms ¼1.4. Top right: tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0. F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243 233
- 7. Figs. 9–13), it can be deduced that the FOPID controllers are more sensitive for changes in the fractional order of the integral and, most of all, of derivative terms than in the other parameters. 5. Discussion In the previous sections it has been pointed out that the fra- gility of the FOPID controllers is mainly motivated by the presence of the fractional-order derivative term. The reasons for this are analyzed in this section by means of an illustrative example. Consider the process PðsÞ ¼ 1 sþ1 eÀ0:5s ð11Þ and the tuning rules applied devised for the load disturbance rejection with a target maximum sensitivity of Ms ¼2.0. For the parallel FOPID controller (the series form is omitted for the sake of brevity, but results are very similar to the parallel case) these yield Kp ¼2.361, Ti ¼0.908, Td ¼0.144, λ ¼ 1, μ ¼ 1:153, while for the IOPID controller we obtain Kp ¼1.008, Ti ¼0.411, and Td ¼0.330. From the analysis of the Bode plots obtained in the nominal case, shown in Fig. 14, it is evident that the FOPID controller allows an increment of the bandwidth with respect to the IOPID controller (the gain crossover frequency is ωgc ¼ 2:01 for the FOPID controller and ωgc ¼ 1:73 for the IOPID controller), with the same level of robustness. This is achieved by exploiting the phase advance introduced by the fractional derivative of order greater than one. While this implies a better performance in the step response, it is also evident that the frequency response function monotonicity is no longer guar- anteed (see Fig. 14) because of the increased high frequency roll-up that the fractional differentiator may exhibit and this implies an incremented fragility of the controller. This can be better analyzed by considering the frequency derivative of the magnitude of the loop transfer function, which results in d CðjωÞPðjωÞ
- 8. dω ¼ d CðjωÞ
- 9. dω PðjωÞ
- 10. þ CðjωÞ
- 11. d PðjωÞ
- 12. dω ; ð12Þ where CðjωÞ
- 13. ¼ Kp
- 14. ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ N2 r ðωÞþN2 i ðωÞ D2 r ðωÞþD2 i ðωÞ s ð13Þ Fig. 6. Resulting values of RFIλ δε for FOPID controller in series form (‘○’) and for FOPID controller in parallel form (‘□’). Top left: tuning for set-point with Ms ¼1.4. Top right: tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms ¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0. F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243234
- 15. PðjωÞ
- 16. ¼ Kj j ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þðTωÞ2 q ð14Þ d PðjωÞ
- 17. dω ¼ À Kj jT2 ω 1þT2 ω2 3 2 ð15Þ and where the controller CðjωÞ has been expressed as CðjωÞ ¼ NrðωÞþjNiðωÞ DrðωÞþjDiðωÞ ð17Þ and NrðωÞ ¼ 1þTiωλ cos π 2 λ þTiTdωλþ μ cos π 2 ðλþμÞ NiðωÞ ¼ Tiωλ sin π 2 λ þTiTdωλþ μ sin π 2 ðλþμÞ DrðωÞ ¼ Tiωλ cos π 2 λ þTiTf ωλþ 1 cos π 2 ðλþ1Þ DiðωÞ ¼ Tiωλ sin π 2 λ þTiTf ωλþ1 sin π 2 ðλþ1Þ ð18Þ and dNrðωÞ dω ¼ TiλωλÀ1 cos π 2 λ þTiTdðλþμÞωλþ μÀ 1 cos π 2 ðλþμÞ dNiðωÞ dω ¼ TiλωλÀ1 sin π 2 λ þTiTdðλþμÞωλþμÀ 1 sin π 2 ðλþμÞ Fig. 7. Resulting values of RFIμ δε for FOPID controller in series form (‘○’) and for FOPID controller in parallel form (‘□’). Top left: tuning for set-point with Ms ¼1.4. Top right: tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms ¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0. d CðjωÞ
- 18. dω ¼ Kp
- 19. NrðωÞ dNrðωÞ dω þNiðωÞ dNiðωÞ dωﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ N2 r ðωÞþN2 i ðωÞ q ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ D2 r ðωÞþD2 i ðωÞ q À ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ N2 r ðωÞþN2 i ðωÞ q DrðωÞ dDrðωÞ dω þDiðωÞ dDiðωÞ dωﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ D2 r ðωÞþD2 i ðωÞ q D2 r ðωÞþD2 i ðωÞ ð16Þ F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243 235
- 20. dDrðωÞ dω ¼ TiλωλÀ1 cos π 2 λ þTiTf ðλþ1Þωλ cos π 2 ðλþ1Þ dDiðωÞ dω ¼ TiλωλÀ1 sin π 2 λ þTiTf ðλþ1Þωλ sin π 2 ðλþ1Þ : ð19Þ Actually, a deeper analysis can be performed by evaluating the frequency response function when one parameter at a time changes in the FOPID controller. Results are shown in Figs. 15–19. As expected from the results shown in Figs. 7 and 13, it appears that the (fractional) derivative action is the most critical one. In particular, the increment of the fractional order μ leads to high-frequency peaks in the sensitivity function, to a loop gain with multiple gain crossover frequencies and, eventually, to instability. Indeed, μ is the only parameter that is able to destabilize the loop in spite of variations smaller than 30%. This hap- pens because increasing μ also means an increased non-monotonic behavior of the frequency response function (see Fig. 19) as a con- sequence of the increased high frequency roll-up. Another critical parameter is the derivative time constant Td. Indeed the optimal FOPID controller has a derivative action with a derivative order μ greater than 1. This means that the optimal frequency response function is already non-monotonic and an increased derivative time constant pushes up the frequency response close to the 0 dB axes (see Fig. 17) creating again high frequency peaks in the sensitivity function with a con- sequent loss of robustness and performance. Again, this behavior is expected from the results shown in Fig. 5 where robustness fragility is considered. It can be appreciated that the IOPID controller always results in less fragile compared to the FOPID one. This happens because of its monotonic behavior. On the contrary, variations in the integrator order do not generate dramatic changes in the frequency response function (see Fig. 16). Indeed, the monotonicity of the frequency response function is inde- pendent from the selected value of λ, unless Ti⪢Td and a series FOPID controller is considered, but, evidently, this is not a meaningful tuning of the controller. This is inde66pc4.68 pendent from the fact that the optimal FOPID controller is obtained with λ ¼ 1. Summarizing, the relevant difference between FOPID and IOPID controllers is that the former ones are capable (indeed because of the fractional-order derivative action) of providing a reduction of the integrated absolute error but their fragility should be carefully considered when the tuning is performed and, in particular, when the fractional-order derivative term is changed, as a small varia- tion can modify the performance signiﬁcantly. Fig. 8. Resulting values of PFIΔε for FOPID controller in series form (‘○’), for FOPID controller in parallel form (‘□’) and for IOPID controller (‘▵’). Top left: tuning for set-point with Ms ¼1.4. Top right: tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms ¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0. F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243236
- 21. Fig. 9. Resulting values of PFI Kp δε for FOPID controller in series form (‘○’), for FOPID controller in parallel form (‘□’) and for IOPID controller (‘▵’). Top left: tuning for set-point with Ms ¼1.4. Top right: tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0. F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243 237
- 22. Fig. 10. Resulting values of PFITi δε for FOPID controller in series form (‘○’), for FOPID controller in parallel form (‘□’) and for IOPID controller (‘▵’). Top left: tuning for set-point with Ms ¼1.4. Top right: tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms ¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0. F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243238
- 23. Fig. 11. Resulting values of PFITd δε for FOPID controller in series form (‘○’), for FOPID controller in parallel form (‘□’) and for IOPID controller (‘▵’). Top left: tuning for set-point with Ms ¼1.4. Top right: tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0. F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243 239
- 24. Fig. 12. Resulting values of PFIλ δε for FOPID controller in series form (‘○’) and for FOPID controller in parallel form (‘□’). Top left: tuning for set-point with Ms ¼1.4. Top right: tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms ¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0. F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243240
- 25. Fig. 13. Resulting values of PFIμ δε for FOPID controller in series form (‘○’) and for FOPID controller in parallel form (‘□’). Top left: tuning for set-point with Ms ¼1.4. Top right: tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms ¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0. F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243 241
- 26. Fig. 14. Magnitude Bode plots in the nominal case for the illustrative example. Solid line: FOPID controller. Dashed line: IOPID controller. Fig. 15. Magnitude Bode plots for the illustrative example (FOPID controller) when the proportional gain Kp changes in the range 730%. Fig. 16. Magnitude Bode plots for the illustrative example (FOPID controller) when the integral time constant Ti changes in the range 730%. Fig. 17. Magnitude Bode plots for the illustrative example (FOPID controller) when the derivative time constant Td changes in the range 730%. Fig. 18. Magnitude Bode plots for the illustrative example (FOPID controller) when the fractional integral order λ changes in the range 730%. Fig. 19. Magnitude Bode plots for the illustrative example (FOPID controller) when the fractional derivative order μ changes in the range 730%. F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243242
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