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Structure-specific analytical PID tuning for load disturbance rejection
Proceedings of the 3rd IFAC Conference on Proceedings of Advances in Proportional-Integral-Derivative Proceedings of the the 3rd 3rd IFAC IFAC Conference Conference on on Control Proceedings of the 3rd IFAC Conference on Control Advances in Proportional-Integral-Derivative Available online at www.sciencedirect.com Ghent, Belgium, May 9-11, 2018 Advances in Proportional-Integral-Derivative Proceedings of the 3rd IFAC Conference on Control Advances in Proportional-Integral-Derivative Control Ghent, Belgium, May 9-11, 2018 Ghent, Belgium, May 9-11, 2018 Advances in Proportional-Integral-Derivative Control Ghent, Belgium, May 9-11, 2018 Ghent, Belgium, May 9-11, 2018
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IFAC PapersOnLine 51-4 (2018) 137–142
Structure-specific analytical PID tuning Structure-specific analytical PID tuning Structure-specific analytical PID Structure-specific analytical PID tuning tuning for load disturbance rejection Structure-specific analytical PID tuning for load disturbance rejection for load disturbance rejection for load disturbance rejection ∗∗ for load disturbance rejection Alberto Leva ∗∗ Silvano Seva ∗∗
Alberto Leva ∗∗ Silvano Seva ∗∗ Alberto Alberto Leva Leva ∗ Silvano Silvano Seva Seva ∗∗ ∗ di Elettronica Informazione Leva e Silvano Seva e∗∗Bioingegneria, ∗ DipartimentoAlberto ∗ Dipartimento di Elettronica e Informazione e Bioingegneria, Dipartimento di Elettronica e Informazione ee Bioingegneria, Politecnico di Milano, Italy (e-mail: [email protected]). ∗ Dipartimento di Elettronica e Informazione Bioingegneria, di Milano, Italy (e-mail: [email protected]). ∗∗ ∗Politecnico Politecnico di Milano, Italy (e-mail: [email protected]). Graduate student at the Dipartimento di Elettronica, Informazione Dipartimento di Elettronica e Informazione e Bioingegneria, ∗∗ Politecnico di Milano, Italy (e-mail: [email protected]). ∗∗ Graduate student at the Dipartimento di Elettronica, Informazione student at the Dipartimento di Elettronica, e Bioingegneria (e-mail: [email protected]). ∗∗ Graduate Politecnico di Milano, Italy (e-mail: [email protected]). Graduate student at the Dipartimento di Elettronica, Informazione Informazione ee Bioingegneria ∗∗ Bioingegneria (e-mail: (e-mail: [email protected]). [email protected]).
Graduate student at the Dipartimento di Elettronica, Informazione e Bioingegneria (e-mail: [email protected]). e Bioingegneria (e-mail: [email protected]). Abstract This paper addresses the important and well studied problem of synthesising PID Abstract This paper addresses addresses the important and well studied problem problem of synthesising PID Abstract paper important and studied synthesising PID controllersThis for load rejection. The tuning rationale, which of some general words Abstract This paperdisturbance addresses the the important and well well studied on problem of synthesising PID controllers for load disturbance rejection. The tuning rationale, on which some general words controllers load rejection. The tuning rationale, on which some general words are spent This infor connection to literature research, to shape the disturbance-to-output frequency Abstract paperdisturbance addresses the important and well studied problem synthesising PID controllers load disturbance rejection. The is tuning rationale, on which of some general words are spent together infor connection to literature research, is to shape the disturbance-to-output frequency are spent in connection to literature research, is to shape the disturbance-to-output frequency response, with conveniently assigning the poles of the corresponding transfer function. controllers for load disturbance rejection. The tuning rationale, on which some general words are spent in connection to literature research, is to shape the disturbance-to-output frequency response, together with conveniently assigning the poles of the corresponding transfer function. response, with assigning the of the corresponding transfer function. Analytical formulæ derived, to maximise simplicity and make the presented method are spent together intuning connection to are literature research, is shape response, together with conveniently conveniently assigning thetopoles poles of the the disturbance-to-output corresponding transferfrequency function. Analytical tuning formulæ are derived, to maximise simplicity and make the presented method Analytical tuning formulæ are derived, to maximise simplicity and make the presented method applicable on any device. Simulation results support the proposal. response, together with conveniently assigning the poles of the corresponding transfer function. Analytical tuning formulæ are derived, to maximise simplicity and make the presented method applicable on any device. Simulation results support the proposal. applicable on any device. results support the proposal. Analytical tuning are derived, to maximise simplicity and make the presented method applicable on any formulæ device. Simulation Simulation support theHosting proposal. © 2018, IFAC (International Federation ofresults Automatic Control) by Elsevier Ltd. All rights reserved. applicable on any device. Simulation results support the proposal. Keywords: PID control; controller tuning; model-based tuning; load disturbance rejection. Keywords: PID control; controller tuning; model-based tuning; load disturbance rejection. Keywords: PID PID control; control; controller controller tuning; tuning; model-based model-based tuning; Keywords: tuning; load load disturbance disturbance rejection. rejection. Keywords: PID control; controller tuning; model-based disturbance 1. INTRODUCTION Vran˘ci´ctuning; et al. load (2004) adapted rejection. the magnitude optimum 1. INTRODUCTION INTRODUCTION Vran˘ ci´ i´cc et et works al. (2004) (2004) adapted the magnitude magnitude optimum 1. Vran˘ c al. adapted the optimum technique, like Liu and Gao and Leva (2005) 1. INTRODUCTION Vran˘ ci´c et works al. (2004) adapted the(2008) magnitude optimum technique, like Liu and Gao (2008) and Leva (2005) technique, works like Liu and Gao (2008) and Leva (2005) employed relay-based tuning, others such as Shamsuzzoha In process control,1.“load disturbances are often the ma- Vran˘ INTRODUCTION c i´ c et al. (2004) adapted the magnitude optimum technique, works like Liu and others Gao (2008) and Leva (2005) employed relay-based tuning, suchadditional as Shamsuzzoha Shamsuzzoha In process control, “load disturbances are often the ma˚ employed relay-based tuning, others such as In process control, “load disturbances are often the maand Lee (2008) proposed PIDs with filters, jor process consideration” A str¨ om and H¨ agglund, 2004). works like Liu and Gao with (2008) and Leva (2005) employed relay-based tuning, others suchadditional as Shamsuzzoha In control, ((“load disturbances are often theAs ma-aa technique, ˚ and many Lee (2008) (2008) proposed PIDs filters, jor consideration” Astr¨ str¨ om m andeffort H¨ agglund, gglund, 2004). As ˚ and Lee proposed PIDs with additional filters, jor consideration” ( A o and H¨ a 2004). As a adopted an IMC-like paradigm (Liu and Gao, consequence, a lot of research has been spent on relay-based tuning, others suchadditional as(Liu Shamsuzzoha In process control, disturbances are been often theAs ma˚ and Lee (2008) proposed PIDs with filters, jor consideration” (“load Astr¨ om andeffort H¨ agglund, 2004). a employed and many adopted an IMC-like paradigm and Gao, consequence, a lot of research has spent on and many adopted an IMC-like paradigm (Liu and Gao, consequence, a lot of research effort has been spent on 2010; Skogestad and Grimholt, 2012). Also, several works ˚ their effective rejection. However, if one restricts the focus and Lee (2008) proposed PIDs with additional filters, jor consideration” ( A str¨ o m and H¨ a gglund, 2004). As a many adopted an IMC-like2012). paradigm (Liu andworks Gao, consequence, arejection. lot of research effort has been the spent on and 2010; Skogestad and Grimholt, Also, several their effective However, if one restricts focus 2010; Skogestad and Grimholt, 2012). Also, several works their effective rejection. However, if one restricts the focus concentrate on “balancing” tuning forAlso, set (Liu point tracking to model-based tuning and explicit two characterismany adopted IMC-like paradigm and Gao, consequence, lot of research effort been spent on and 2010; Skogestad andan Grimholt, 2012). several works their effective arejection. However, if rules, onehas restricts the focus concentrate on “balancing” “balancing” tuning for set point point tracking to model-based tuning and explicit rules, two characterisconcentrate on tuning set to tuning and explicit rules, two characterisand forSkogestad disturbance response, see2012). e.g.for Arrieta et al.tracking (2010); ticsmodel-based that are often considered beneficial for the industrial 2010; and Grimholt, Also, several works their effective rejection. However, if one restricts the focus concentrate on “balancing” tuning for set point tracking to model-based tuning and explicit rules, two characterisand for disturbance response, see see e.g. Arrieta Arrieta et et al. (2010); (2010); tics that areofoften often considered beneficial beneficial for the the panorama industrial and disturbance response, tics that considered the industrial Alc´afor ntara eton al. “balancing” (2013) acceptance an (auto)tuning procedure, set point to model-based tuning and explicit rules,for two characterisand for disturbance response,tuning see e.g. e.g.for Arrieta et al. al.tracking (2010); tics that are areofoften considered beneficial for the industrial concentrate Alc´ a ntara et al. (2013) acceptance an (auto)tuning procedure, the panorama aafor ntara et (2013) acceptance an (auto)tuning procedure, becomes narrower. and disturbance response, see e.g. Arrieta ettuning, al. (2010); tics that quite areof considered beneficial for the the panorama industrial Alc´ Alc´ ntara et al. al. acceptance ofoften an (auto)tuning procedure, the panorama In this work we (2013) concentrate on model-based acbecomes quite narrower. becomes quite narrower. In this work we concentrate on model-based tuning, acAlc´ a ntara et al. (2013) acceptance of an (auto)tuning procedure, the panorama becomes quitewe narrower. this work we concentrate on model-based tuning, accounting for the form of the process model but at the In this paper address the type of tuning technique just In In this work we concentrate onprocess model-based tuning, accounting for the form of the model but at the In this paper we address the type of tuning technique just becomes quite narrower. forconsidering the form of ofdifferent theonprocess process model but at at the the In this address the type of technique just same possible structures for sketched, andwe with respect a major reference, hence a counting In thistime work we concentrate model-based tuning, accounting for the form the model but In this paper paper address theto type of tuning tuning technique just same time considering different possible structures for sketched, andwe with respect to a major major reference, hence a same time considering different possible structures for sketched, and with respect to a reference, hence a it, and aiming at explicit tuning formulæ. According to good representative of the state of the art, we propose counting for the form of the process model but at the In this paper we address the type of tuning technique just time considering different possible structures for sketched, and with respect to a major reference, hence a same it, and aiming at explicit tuning formulæ. According to good representative of the state of the art, we propose it, and aiming at explicit tuning formulæ. According to good representative of the state of the art, we propose Scopus, to date the most cited paper encompassing all a solution thatwith is simpler, in athat it adopts a uniform same time considering different possible structures for sketched, and respect to major reference, hence a it, and aiming at explicit tuning formulæ. According to good representative of the state of the art, we propose Scopus, to date the most cited paper encompassing all a solution that is simpler, in that it adopts a uniform to date the most cited encompassing all aa solution that is simpler, in that it adopts aaoruniform theand characteristics above, is thatpaper by Horn et al. (1996). controller structure, andtheproduces comparable better Scopus, it, aiming at explicit tuning formulæ. According to good representative of state of the art, we propose Scopus, to date the most cited paper encompassing all solution that is simpler, in that it adopts uniform the characteristics above, is that that by by Horn Horn et al. al.which (1996). controller structure, and produces comparable or better the characteristics above, is et (1996). controller structure, and produces comparable or better The authors adopt an IMC-centred approach, is results. It has to be noted that our technique – like its Scopus, to date the most cited paper encompassing all a solution that is simpler, in that it adopts a uniform the characteristics above, is that by Horn et al.which (1996). controller structure, produces or better The authors adopt an IMC-centred approach, is results. Itcounterpart has to to be be and noted that delay-free ourcomparable technique – like like its The authors adopt an IMC-centred approach, which is results. It has noted that our technique – its very well suited to achieve model-based explicit tuning reference – considers processes: this the characteristics above, is that by Horn et al. (1996). controller structure, and produces comparable or better The authors adopt an IMC-centred approach, which is results. Itcounterpart has to be noted that delay-free our technique – likethis its very well suited to achieve model-based explicit tuning reference – considers processes: well to achieve model-based explicit tuning reference –limitation, considers delay-free processes: this for various model and determine a controller may or may nottobebea noted in Leva The authors adopt an IMC-centred approach, which is results. Itcounterpart has that as ourdiscussed technique – likeand its very very well suited suited tostructures, achieve model-based explicit tuning reference counterpart considers delay-free processes: this for various model structures, and determine afilter controller may or may may notanyway, be aa –limitation, limitation, as discussed in Leva Leva and for various model structures, and determine a controller may or not be as discussed in and in the form of an ideal PID cascaded to a up to Maggio (2010); addressing the delay (dominated) very well suited to achieve model-based explicit tuning reference counterpart – considers delay-free processes: this for various model structures, and determine a controller may or may not be a limitation, as discussed in Leva and in the form of of anInideal ideal PID cascaded cascaded to a filter filter up to Maggio (2010); anyway, addressing the delay delay (dominated) in form PID to up Maggio (2010); addressing the thethe second the quoted paper, rules obtained case will be the subject of future as research. various model structures, determine afilter controller may or may notanyway, be a limitation, discussed in Leva and for in the formorder. of an anIn ideal PID and cascaded to aa are up to to Maggio (2010); anyway, addressing the delay (dominated) (dominated) the second order. the quoted quoted paper, rules are obtained case will will be the the subject of future research. research. the second order. In the paper, rules are obtained case be subject of future for five model structures by changing the structure of in the form ofstructures anInideal PID cascaded tostructure a are filter upthe to Maggio (2010); anyway, addressing the delay 2(dominated) the second order. the quoted paper,the rules obtained case be istheorganised subject of research. for five model by changing of the The will paper asfuture follows. Section provides a for five model structures by changing the structure of the filter, interpreted in IMC terms. This results in controllers the second order. In the quoted paper, rules are obtained The paper is organised as follows. Section 2 provides a case will be the subject of future research. for five model structures by changing the structure of the The paper is as follows. 22 provides filter, interpreted in IMC terms. This This results in controllers controllers minimal review of related andSection motivates the par-aa filter, in terms. in The paper is organised organised as work, follows. Section provides of different structure themselves, as results some parameters of fiveinterpreted model structures by changing the structure of the minimal review of related related work, and motivates the parpar- for filter, interpreted in IMC IMC terms. This results in controllers minimal review of work, and motivates the of different structure themselves, as some parameters of ticular research here presented. In Section 3 the proposed The paper is organised as follows. Section 2 provides a of different structure themselves, as some parameters of minimal review here of related work,Inand motivates the par- filter, the general form (PID plus filter), for some models are interpreted in IMC terms. This results in controllers ticular research presented. Section 3 the proposed of different some parameters of ticular research presented. In Section 33 the the general structure form (PID (PIDthemselves, plus filter), filter),as for for some models are are tuning rules are here derived, andwork, in Section 4 they areproposed applied minimal review of related and motivates the parthe general form plus some models ticular research here presented. In Section the proposed structurally zero. of different some parameters of tuning rules are derived, derived, andthat in Section Section theyaccording are applied applied the general structure form (PIDthemselves, plus filter),as for some models are tuning are and in 444 they are structurally zero. to somerules process structures areSection relevant to structurally ticular research presented. In 3 the zero. tuning rules are here derived, andthat in Section they areproposed applied the general form (PID plus filter), for some models are to some process structures are relevant according to structurally zero. to process structures that are according to The method we propose aims at obtaining the same (or thesome literature. Sections 5 and 6inrespectively report benchtuning rules are derived, andthat Section 4 they areaaapplied to some process structures are relevant relevant according to structurally The method we propose propose aims at obtaining the same (or zero. the literature. Sections and respectively report benchThe method we aims obtaining the (or the literature. Sections 555 and 666 respectively report aa benchbetter) result a uniform and completely standard mark simulation campaign and aarelaboratory experiment, to some process structures that relevant according to The method wewith propose aims at at obtaining the same same (or the literature. Sections and respectively report benchbetter) result with a uniform and completely standard mark simulation campaign and a laboratory experiment, result with uniform and completely standard mark simulation campaign and aa laboratory experiment, controller structure, i.e., aaims real PID (more precisely, a PID to evidence the achieved results and advantages, while better) The method wewith propose at obtaining the same (or the literature. Sections 5 and 6 respectively report a benchbetter) result aa uniform and completely standard mark simulation campaign and laboratory experiment, controller structure, i.e., a real real PID PID (more precisely, acould PID to evidence the achieved results and advantages, advantages, while controller structure, i.e., a (more precisely, a PID to evidence the achieved results and while with filtered derivative action). The closest work we Section 7 concludes the paper by drawing some conclusions better) result with a uniform and completely standard mark simulation campaign and a laboratory experiment, controller structure, i.e., a real PID (more precisely, a PID to evidence the achieved results and advantages, while with filtered derivative action). The closest work we could Section concludes the paper by by drawing some some conclusions with filtered derivative The closest work we Section 777 concludes the paper find to ourstructure, approach isaction). that Jin(more and Liu (2014), that andevidence outlining future work. i.e., a realby precisely, PID to the achieved results and advantages, while controller with filtered derivative The work we acould could Section concludes the paper by drawing drawing some conclusions conclusions find to our approach isaction). that byPID Jin closest and Liu Liu (2014), that and outlining outlining future work. find to our approach is that by Jin and (2014), that and future work. however does not encompass all the characteristics above, filtered derivative action). The closest work we could Section 7 concludes paper by drawing some conclusions with find to our approach is that by Jin and Liu (2014), that and outlining futurethe work. however does not not encompass encompass all the theshape characteristics above, however does all characteristics above, and in directly the disturbancefind to particular our approach isnot that by the Jinshape and Liu (2014), that and outlining future work. 2. BRIEF LITERATURE REVIEW however does not does encompass all characteristics above, and in particular does not directly the disturbance2. BRIEF LITERATURE REVIEW and in particular does not directly shape the disturbanceto-output frequency response magnitude the way we do. As 2. BRIEF LITERATURE REVIEW however does not encompass all the characteristics above, and in particular does not directly shapethe theway disturbance2. BRIEF LITERATURE REVIEW to-output frequency response magnitude we do. do. As As frequency response magnitude the way we for the tracking/rejection tradeoff, here we concentrate on and in particular does not directly shape the disturbanceMany approaches wereLITERATURE proposed to PI/PID tuning for load to-output 2. BRIEF REVIEW to-output frequency response magnitude theconcentrate way we do. As for the tracking/rejection tradeoff, here we on Many approaches werehistorically proposed to toinPI/PID PI/PID tuning for load for tracking/rejection here we concentrate on the the latter: if trackingresponse is of tradeoff, interest as well, one can recover to-output frequency magnitude the way we do. As Many approaches were proposed tuning for load disturbance rejection, an attempt to cure an for the tracking/rejection tradeoff, here we concentrate on the latter: if tracking is of interest as well, one can recover Many approaches were proposed to PI/PID tuning for load disturbance rejection, historically in an an attempt attempt to cure cure an the latter: if is interest as well, one can it for example with method in Leva and Bascetta (2006), for the tracking/rejection here we concentrate on disturbance rejection, historically to an “overemphasis on were the set point response” (Shinskey, 2002), the latter: if tracking tracking is of of tradeoff, interest as well, one can recover recover Many approaches proposed toin tuning load it for example with method in Leva Leva and Bascetta (2006), disturbance rejection, historically inPI/PID an attempt to for cure an it “overemphasis on the set point response” (Shinskey, 2002), for example with method in and Bascetta (2006), as that technique is independent of how the feedback part the latter: if tracking is of interest as well, one can recover “overemphasis on the set point response” (Shinskey, 2002), but basicallyrejection, driven the necessities of(Shinskey, process for with method in Leva andthe Bascetta (2006), disturbance historically in an attempt to control: cure an it as thatexample technique is independent independent of how feedback part “overemphasis on theby set point response” 2002), but basically driven by the necessities ofdirect process control: as that technique of feedback part of for the controller isis it with method in Leva andthe Bascetta (2006), but driven by the necessities of process control: Chenbasically and Seborg (2002) resorted toof synthesis, as thatexample technique isdesigned. independent of how how the feedback part “overemphasis on the set point response” (Shinskey, 2002), of the controller is designed. but basically driven by the necessities process control: Chen and and Seborg (2002) (2002) resorted resorted to to direct synthesis, synthesis, of thatcontroller techniqueis independent of how the feedback part Chen of the the controller isisdesigned. designed. but driven (2002) by the resorted necessitiestoofdirect process control: as Chenbasically and Seborg Seborg direct synthesis, of the controller is designed. Chen and Seborg (2002) resorted to direct synthesis, Copyright 137 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © © 2018 2018, IFAC IFAC (International Federation of Automatic Control) Copyright © 2018 137 Copyright © under 2018 IFAC IFAC 137 Control. Peer review responsibility of International Federation of Automatic Copyright © 2018 IFAC 137 10.1016/j.ifacol.2018.06.115 Copyright © 2018 IFAC 137
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3. THE PROPOSED TUNING METHOD The block diagram for the addressed control scheme is shown in Figure 1, where P (s) and C(s) are respectively the transfer functions of the process and the controller, w(t) is the set point, y(t) the controlled variable, u(t) the control signal, and d(t) a load disturbance to be rejected.
w(t) + −
C(s)
d(t) u(t) + +
P(s)
y(t)
Figure 1. Control loop with load disturbance. We consider as the controller to tune a real PID with possibly complex zeroes, that we write as bC0 + bC1 s + bC2 s2 C(s) = . (1) aC1 s + aC2 s2 The tuning goal is to constrain the disturbance-to-output transfer function Q(s) = Y (s)/D(s) to take the form QN (s) , QD (s) = (1 + sτQ )nQ , (2) Qo (s) = QD (s) with QN (0) = 0 – as inherent to the structure of (1), incidentally – to guarantee asymptotic disturbance rejection, τQ > 0 for stability, and nQ large enough to ensure high-frequency roll-off (if not for a particular case briefly discussed in Section 4.3). In fact, nQ is dictated by the structure of the considered process models, as there must be as tuning equations as controller parameters—whence also the apparent redundancy in the parametrisation of C(s), as will become clear in Sections 4.1 through 4.5. On the contrary, τQ is a tuning parameter (the only one) for the method, interpreted as a time constant connected to the desired disturbance step response convergence time—i.e., in some sense, pretty much lie the desired closed-loop dominant time constant in lambda or IMC tuning. For completeness, in the (dominantly frequent) case of a controller with real zeroes, the controller(1) is immediately converted to the standard ISA PID form 1 sTd C(s) = K 1 + (3) + sTi 1 + sTd /N by aC1 bC1 − aC2 bC0 aC1 bC1 − aC2 bC0 , Ti = , K= 2 aC1 aC1 bC0 Td =
a2C2 bC0 − aC1 aC2 bC1 + a2C1 bC2 , aC1 (aC1 bC1 − aC2 bC0 )
N=
a2C2 bC0 − aC1 aC2 bC1 + a2C1 bC2 . aC2 (aC1 bC1 − aC2 bC0 )
(4)
Coming to the process model, we assume the general form bP 0 ∓ b P 1 s ∓ b P 2 s 2 P (s) = (5) aP 0 + aP 1 s + aP 2 s 2 138
with all parameters nonnegative; notice that we are assuming positive gain as well, but obviously this does not impair generality. The rationale behind the tuning approach we adopt, is to prevent |Q(jω)| from exhibiting a plateau, thereby minimising the width of the band where the disturbance is rejected to the least extent (i.e., where |Q(jω)| is near to its maximum), while allowing some control on the maximum value of |Q(jω)| and of the high-frequency control sensitivity. The structure chosen for Qo (s), see (2), allows to easily obtain an estimate on its ∞-norm by taking the magnitude of its frequency response at ω = 1/τQ , where the coincident poles are. For example, if Qo (s) has four poles (and one zero in the origin, remember) then the only possible mutual locations of zeroes and poles are shown in Figure 2, and clearly the estimate just introduced is sensible; in the figure, asymptotic Bode plots are sketched for simplicity. Z Z P4
Z P4 Z ω
P4 Z Z ω
ω
Figure 2. Possible frequency positions of zeroes (Z) and poles (P ) of Q(s) in the fourth order case. As such, in the following we further denote the estimated ∞-norm of Qo (s) and the high-frequency value of the controller (i.e., of the control sensitivity) magnitude, respectively, as max = max |Q(jω)|, C∞ = lim |C(jω)|. Q (6) ω
ω→∞
max underlines that this is an where the hat sign in Q estimate; the exact value depends on the detailed shape of the frequency response, but the estimate is representative in any case. The nice fact in all this reasoning is that once the systems of equations that provide the controller parameters are analytically solved, as done in the following sections, both Qmax and C∞ – that respectively quantify the disturbance rejection and the high-frequency control activity – are functions of the tuning parameter τQ , and can be used as the basis for a (possibly) automatic selection of that parameter, although in this paper we do not discuss the matter. Summarising, the main advantage of our rationale is that the disturbance-to-output frequency response is shaped in a very direct and straightforward manner, and for several process structures – like those addressed below – the resulting tuning relationships are explicit. 4. APPLICATION TO RELEVANT PROCESS STRUCTURES The quoted paper by Horn et al. (1996) reports IMC controllers for five process structures, optimised for “open
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loop dynamics slower then the [desired] closed-loop dynamics”, i.e., for what is sometimes called “strong feedback”, i.e., in turn, essentially for disturbance rejection. With the notation of that reference, that we report here for the reader’s convenience, the five structures treated are kp kp (−s + z) PA (s) = , PB (s) = , τs + 1 (τ1 s + 1)(τ2 s + 1) PC (s) =
kp (−s + z) kp , PD (s) = , τs + 1 (τ1 s + 1)(τ2 s + 1) PE (s) =
(7)
kp s(τ s + 1)
4.1 Process A In the form (5), process A in (7) reads bP 0 PA (s) = , (8) 1 + aP 1 s and with controller (1), this structurally yields a thirdorder Q(s), i.e., nQ = 3. Setting QD (s) = (1 + sτQ )3 we then get 3 τQ 1 aC1 = 1, aC2 = , bC0 = , aP 1 bP 0 (9) 2 3 a (3τ − a ) − τ 3τQ − 1 P1 P1 Q Q bC1 = , bC2 = bP 0 a P 1 bP 0 and 4 + a2 bP 0 τQ P1 (10) Qmax = 3/2 2 a P 1 τQ aP 1 (aP 1 − 3τ 2 ) + τ 3 Q Q (11) C∞ = 3 bP 0 τQ 4.2 Process B
In the form (5), process B in (7) reads. bP 0 − bP 1 s PB (s) = 1 + a P 1 s + aP 2 s 2 and in this case we have to set nq = 4, whence
aC1
bC1
In the form (5), process C in (7) reads bP 0 − b P 1 s PC (s) = . (14) 1 + aP 1 s We include this case for completeness, but we have to notice that a first-order, non strictly proper process – and to top, nonminimum-phase – seems a bit unrealistic indeed in practice. Joined to a real PID, in addition, this gives rise to an open loop transfer function with zero relative degree, which has very well known drawbacks—and in the end, adopting a closed-loop pole placement approach like ours is probably among the best ways to handle such a situation. In any case, we apply the technique to show that formally it works. To do this we have to set nq = 3, whence aP 1 b2P 0 bP 1 + bP 0 b2P 1 + b3P 1 2 3 + 3bP 0 b2P 1 τQ + 3b2P 0 bP 1 τQ + b3P 0 τQ aC1 = 1, aC2 = − , b2P 0 bP 1 + aP 1 b3P 0 bC0 = −
1 bP 1 + bP 0 (3τQ + 1) , bC1 = − bP 0 b2P 0
−a2P 1 b2P 0 − aP 1 bP 0 bP 1 − aP 1 b2P 1 2 3 − 3aP 1 bP 0 bP 1 τQ − 3aP 1 b2P 0 τQ + b2P 0 τQ = b2P 0 bP 1 + aP 1 b3P 0
bC2
(15) while Qmax and C∞ are again too long to be shown. In this work we do not further concentrate on this case, as probably one should adopt a different controller structure, or simply admit that every process eventually rolls off at high frequency, and resort e.g. to case B.
In the form (5), process D in (7) reads bP 0 PD (s) = 1 + a P 1 s + aP 2 s 2 and we have to set nq = 4, whence
bC2 =
3 4 + aP 1 τQ 4a2P 2 τQ − 4aP 2 τQ , a2P 2 bP 0
139
C∞
(17)
2 3 4 6a2P 2 τQ − 4aP 1 aP 2 τQ + (a2P 1 − aP 2 )τQ a2P 2 bP 0
and
(13)
(16)
3 4 4 − aP 1 τQ 4aP 2 τQ τQ 1 , a = , bC0 = , C2 2 aP 2 aP 2 bP 0
bC1 =
aP 1 aP 2 b2P 1 + a2P 2 bP 0 bP 1 + (4a2P 2 b2P 0 2 +4aP 1 aP 2 bP 0 bP 1 )τQ − 6aP 2 bP 0 bP 1 τQ 3 4 − 4aP 2 b2P 0 τQ + (aP 1 b2P 0 + bP 0 bP 1 )τQ = , 2 2 2 3 a P 2 b P 0 bP 1 + a P 1 a P 2 b P 0 b P 1 + a P 2 b P 0
bC2
4.3 Process C
aC1 =
4 τQ 1 , bC0 = , aP 2 bP 0
2 a2P 2 b2P 1 + 4a2P 2 bP 0 bP 1 τQ + 6a2P 2 b2P 0 τQ 2 3 −(4aP 1 aP 2 bP 0 + 4aP 2 bP 0 bP 1 )τQ 4 + ((a2P 1 − aP 2 )b2P 0 + aP 1 bP 0 bP 1 )τQ = 2 2 2 3 a P 2 bP 0 b P 1 + a P 1 a P 2 b P 0 bP 1 + a P 2 b P 0
max and C∞ are too long to be while the expressions of Q reported here.
4.4 Process D (12)
2 aP 2 b3P 1 + 4aP 2 bP 0 b2P 1 τQ + 6aP 2 b2P 0 bP 1 τQ 3 3 3 2 4 + 4aP 2 bP 0 τQ − (aP 1 bP 0 + bP 0 bP 1 )τQ = , aP 2 bP 0 b2P 1 + aP 1 aP 2 b2P 0 bP 1 + a2P 2 b3P 0
aC2 =
139
2 bP 0 τQ
(4aP 2 − aP 1 τQ )2 + a2P 2 , 4a2P 2 2 2 2 6aP 2 − 4aP 1 aP 2 τQ + (aP 1 − aP 2 )τQ = . 2 aP 2 bP 0 τQ max = Q
(18) (19)
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4.5 Process E In the form (5), process E in (7) reads bP 0 PE s(s) = s + aP 2 s2 and we have to set nq = 4, whence aC1
3 4 4 − τQ τQ 4aP 2 τQ 1 , a = = , bC0 = , C2 2 aP 2 aP 2 bP 0
bC1 and
4 3 2 − 4aP 2 τQ + τQ 6a2P 2 τQ 4τQ = , bC2 = bP 0 a2P 2 bP 0
(4aP 2 − τQ )2 + a2P 2 max = Q , 4a2P 2 6a2 − 4aP 2 τQ + τ 2 P2 Q C∞ = . 2 aP 2 bP 0 τQ 2 bP 0 τQ
(20)
(21)
(22)
A B
√ min. for τQ = aP 1 no analytical solution
C D E
no analytical solution strictly incr. with τQ strictly incr. with τQ
pC (σ) =
1 − σθ 1 , pD (σ) = , 1+σ (1 + σ)(1 + σψ)
(24)
1 . σ(1 + σ) where 0 < ψ < 1 since the “other” pole has to be faster, while in principle θ (positive) could be arbitrarily large. Rigorously the numerator of pE should be τ , incidentally, but in this case it just plays the role of a scale factor, as the process is type 1, and therefore can be omitted in the normalised transfer function. 5.1 Test 1
(23)
pA , τq = 0.2 pE , τq = 0.2
pA , τq = 0.4 pE , τq = 0.4
pA , τq = 0.6 pE , τq = 0.6
pA , τq = 0.8 pE , τq = 0.8
20
0
−20
dB
A first thing to notice is that the five process structures considered yield very different results as for the existence max and/or C∞ , as of values for τQ that minimise Q summarised in table 1. The somewhere opposite effect of the tuning parameter convinces of the opportunity of structure-specific tuning if high rejection performance is required. Proc.
1 1 − σθ , pB (σ) = , 1+σ (1 + σ)(1 + σψ)
pE (σ) =
4.6 General remarks
max Q
pA (σ) =
−40
−60
−80 10−1
100
101
102
Normalised frequency
C∞
√ min. for τQ = aP 1 ∃ minimising τQ (complex expression) strictly incr. with τQ strictly decr. with τQ min. for τQ = 3aP 2
Figure 3. Test 1 – avoiding the magnitude plateau in the frequency response or |Q(s)|. First, we show that the proposed rule actually avoids the magnitude plateau in the frequency response or |Q(s)|, while τq controls the position of the frequency band where the disturbance is least rejected.
Table 1. Summary of the effect of parameter max and C∞ . τQ on Q
A second notable remark is that, although critical cancellations never happen (we omitted computations for brevity), in the case of nonminimum-phase processes the PID can turn out to be unstable, i.e., to have its second pole in the right half plane. In principle this can make sense as high rejection performance is required, but further study is required on this aspect, especially to properly quantify the closed-loop stability robustness, that we cannot undertake in this paper. 5. SIMULATION RESULTS In this section we present some samples of an extensive simulation campaign, that we conducted to assess the correct behaviour of the proposed technique, and to verify that the it actually exhibits the expected strengths. For compactness we introduce a normalised complex variable σ = sτ , where τ is the only or the largest time constant of the pole(s) of processes A–E not in the origin. Defining the normalised processes pA−E (σ) as PA−E (σ/τ )/µ we get 140
Figure 3 shows an example with processes A and E; the magnitude of the frequency response of Q(s) is plotted. It is interesting to note the opposite effect of τq , depending on the processes’ structure. Also, peak values above 0dB are not an issue: this example only means to show the absence of the plateau, while of course when doing a real tuning values of τq keeping the peak under 0dB will be selected. 5.2 Test 2 We now show an example of comparison between our proposal and the reference by Horn et al. (1996). The example uses two processes in class B, namely 1 − 2σ pB1 (σ) = , (1 + σ)(1 + 0.4σ) (25) 1 − 0.4σ , pB2 (σ) = (1 + σ)(1 + 0.9σ) of which the first has a remarkably undershooting step response and a second pole at significantly higher frequency wit respect to the dominant one, while the second has two
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Horn et al.
ISEprop , (27) ISEhorn ISEprop and ISEhorn being the ISE computed with the proposed and the reference controller, respectively.
Proposed
RISE =
2 pB1 , τq = 0.6
141
0 −2
pB1 , τq = 1
1
1
0
RISE
pB2 , τq = 0.6
−1
0.5
0.5
0 1
0
1 0.8
0.6
pB2 , τq = 1
τq
0.5 0.4
0.1
ψ
Figure 5. Test 3 – comparing the load disturbance step response ISE.
0.5
0
0
2
4
6
8
10
12
14
16
18
20
time [s]
Figure 4. Test 2 – comparing load disturbance step responses. almost coincident poles and a less evident overshoot (in some sense, thus, the two processes are “quite extreme” in the class). Figure 4 reports the load disturbance step responses of (4) with a “low” and a “high” value of the parameter controlling the closed-loop band, i.e., λ in the work by Horn et al. (1996) and τq in our proposal. As can be seen, both parameters act in the respective technique in the desired manner. On one hand this proves that the analogy in their interpretation is correct, but on the other hand, as expected, there is some advantage in governing the disturbance-to-output transfer function directly. For example, with pB1 the maximum error might slightly increase, the ISE is comparable but the settling is definitely shorter, while with pB2 there is an apparently less oscillatory behaviour.
Figure 5 shows the said integral index for process class D. The green surface is the relative ISE, the red line on the surface is the unity level, its projection onto the (ψ, τq ) plane is shown as well. As can be seen, there is a wide region in the (ψ, τq ) plane, where the proposed tuning approach yields an advantage. 6. AN EXPERIMENTAL TEST In this section we present an experiment in which the proposed method is applied to a laboratory apparatus, and compared to Horn et al. (1996) plus two other PID tuning techniques well known to be suitable for disturbance rejection, namely the AMIGO (˚ Astr¨om and H¨ agglund, 2004) and the SIMC (Skogestad and Grimholt, 2012). For completeness we notice that AMIGO tunes an ideal PID, that we made proper with N = 20 as suggested in the quoted reference. As for the SIMC, given the firstorder process, we used the PI rule, with the additional advantage of achieving a zero relative degree controller without choosing the second pole arbitrarily. Finally, we notice that for process class A, the most appropriate for the addressed dynamics, the controller by Horn et al. reduces to a PI. All in all, with the inevitable shortcomings of any experimental setup, we deem this a fair enough comparison.
In this test we again compare our technique to the quoted reference, but this time in terms of the load disturbance step response ISE, defined as ∞ ISE = e(t)2 dt, (26)
The used apparatus, described in Leva (2003), is a temperature control system composed of a small metal plate electrically heated by a transistor. The command to that transistor is the control signal, while the plate temperature is the controlled variable. A second transistor is connected to heat the plate, thereby apparently providing a load disturbance. Since the controlled dynamics is dominantly first-order, all the three methods just mentioned fit. The tuning results are in table 2.
where e(t) = w(t) − y(t) is the error. More precisely, we compare the relative ISE, i.e., the quantity
Figure 6 shows the response of the controlled plate temperature Tp (top plot) and of the [0, 100] command uh to the
5.3 Test 3
0
141
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Method
K
Ti
Td
N
Proposed Horn et al. AMIGO SIMC
88 41 31 33.5
15 30 30 56
0.35 0 3.5 0
0.03 n/a 20 n/a
disturbance-to-output dynamics can be considered, and analogously, the application of the same idea to other controller structures can be studied. REFERENCES
Table 2. Controller parameters in the experimental test. Set point
SIMC
AMIGO
Horn et al.
Proposed
28
Tp [◦ C]
27.5
27
26.5 100
uh [0, 100]
80
60
40
20
0
0
100
200
300
400
500
600
time [s]
Figure 6. Laboratory experiment – controlled variable and control signal response to a 40% load disturbance step. heating transistor (bottom plot) to a 40% load disturbance impressed via the second transistor. The proposed technique yields the best disturbance rejection; there is of course a cost in terms of of a more nervous control and a higher sensitivity to measurement noise, but both phenomena appear up to a reasonable extent. The technique can therefore supposed to be also viable in practical process control applications. 7. CONCLUSIONS AND FUTURE WORK We presented a technique to analytically tune a real PID in a structure-specific manner as for the process dynamics. The underlying approach is based on assigning the closed-loop poles so as to shape the disturbanceto-output frequency response to exhibit no plateau. The technique has a single parameter, interpretable in a way similar to λ in IMC-based rules, and for which “optimal” values can be sought (although this was not discussed here in depth). Future work will proceed along three main directions. The first one is to apply the same rationale to other process structures—including delay-dominated ones. The second consists of posing and solving optimisation problems (possibly not analytically, however) to automatically select the tuning parameter. The third is to integrate the proposed method – that refers to the feedback part of a PID – into the synthesis of two-degree-of-freedom controllers. As a further possibility, then, different structures for the desired 142
Alc´antara, S., Vilanova, R., and Pedret, C. (2013). PID control in terms of robustness/performance and servo/regulator trade-offs: A unifying approach to balanced autotuning. Journal of Process Control, 23(4), 527–542. Arrieta, O., Visioli, A., and Vilanova, R. (2010). PID autotuning for weighted servo/regulation control operation. J. of Process Control, 20(4), 472–480. ˚ Astr¨om, K. and H¨agglund, T. (2004). Revisiting the Ziegler-Nichols step response method for PID control. Journal of Process Control, 14(6), 635–650. Chen, D. and Seborg, D. (2002). PI/PID controller design based on direct synthesis and disturbance rejection. Ind. Eng. Chem. Process Des. Dev., 41(19), 4807–4822. Horn, C., Arulandu, J., Gombas, C., VanAntwerp, J., and Braatz, R. (1996). Improved filter design in internal model control. Industrial & Engineering Chemistry Research, 35(10), 3437–3441. Jin, Q. and Liu, Q. (2014). IMC-PID design based on model matching approach and closed-loop shaping. ISA Transactions, 53(2), 462–473. Leva, A. (2003). A hands-on experimental laboratory for undergraduate courses in automatic control. IEEE Transactions on Education, 46(2), 263–272. Leva, A. (2005). Autotuning process controller with improved load disturbance rejection. Journal of Process Control, 15(2), 223–234. Leva, A. and Bascetta, L. (2006). On the design of the feedforward compensator in two-degree-of-freedom controllers. Mechatronics, 16(9), 533–546. Leva, A. and Maggio, M. (2010). On the use of models with delay in PI(D) autotuning. In Proc. 49th IEEE Conference on Decision and Control, 3319–3324. Atlanta, GA, USA. Liu, T. and Gao, F. (2008). Relay-based autotuning of PID controller for improved load disturbance rejection. In Proc. 17th IFAC World Congress, 10933–10938. Seoul, South Korea. Liu, T. and Gao, F. (2010). New insight into internal model control filter design for load disturbance rejection. IET Control Theory and Applications, 4(3), 448–460. Shamsuzzoha, M. and Lee, M. (2008). Design of advanced PID controller for enhanced disturbance rejection of second-order processes with time delay. AIChE Journal, 54(6), 1526–1536. Shinskey, F. (2002). Process control: as taught vs as practiced. Industrial & Engineering Chemistry Research, 41(16), 3745–3750. Skogestad, S. and Grimholt, C. (2012). The SIMC method for smooth PID controller tuning. In R. Vilanova and A. Visioli (eds.), PID Control in the Third Millennium – Lessons Learned and New Approaches, 147–175. Springer, London, UK. Vran˘ci´c, D., Strm˘cnik, S., and Kocijan, J. (2004). Improving disturbance rejection of PI controllers by means of the magnitude optimum method. ISA transactions, 43(1), 73–84.
ScienceDirect
IFAC PapersOnLine 51-4 (2018) 137–142
Structure-specific analytical PID tuning Structure-specific analytical PID tuning Structure-specific analytical PID Structure-specific analytical PID tuning tuning for load disturbance rejection Structure-specific analytical PID tuning for load disturbance rejection for load disturbance rejection for load disturbance rejection ∗∗ for load disturbance rejection Alberto Leva ∗∗ Silvano Seva ∗∗
Alberto Leva ∗∗ Silvano Seva ∗∗ Alberto Alberto Leva Leva ∗ Silvano Silvano Seva Seva ∗∗ ∗ di Elettronica Informazione Leva e Silvano Seva e∗∗Bioingegneria, ∗ DipartimentoAlberto ∗ Dipartimento di Elettronica e Informazione e Bioingegneria, Dipartimento di Elettronica e Informazione ee Bioingegneria, Politecnico di Milano, Italy (e-mail: [email protected]). ∗ Dipartimento di Elettronica e Informazione Bioingegneria, di Milano, Italy (e-mail: [email protected]). ∗∗ ∗Politecnico Politecnico di Milano, Italy (e-mail: [email protected]). Graduate student at the Dipartimento di Elettronica, Informazione Dipartimento di Elettronica e Informazione e Bioingegneria, ∗∗ Politecnico di Milano, Italy (e-mail: [email protected]). ∗∗ Graduate student at the Dipartimento di Elettronica, Informazione student at the Dipartimento di Elettronica, e Bioingegneria (e-mail: [email protected]). ∗∗ Graduate Politecnico di Milano, Italy (e-mail: [email protected]). Graduate student at the Dipartimento di Elettronica, Informazione Informazione ee Bioingegneria ∗∗ Bioingegneria (e-mail: (e-mail: [email protected]). [email protected]).
Graduate student at the Dipartimento di Elettronica, Informazione e Bioingegneria (e-mail: [email protected]). e Bioingegneria (e-mail: [email protected]). Abstract This paper addresses the important and well studied problem of synthesising PID Abstract This paper addresses addresses the important and well studied problem problem of synthesising PID Abstract paper important and studied synthesising PID controllersThis for load rejection. The tuning rationale, which of some general words Abstract This paperdisturbance addresses the the important and well well studied on problem of synthesising PID controllers for load disturbance rejection. The tuning rationale, on which some general words controllers load rejection. The tuning rationale, on which some general words are spent This infor connection to literature research, to shape the disturbance-to-output frequency Abstract paperdisturbance addresses the important and well studied problem synthesising PID controllers load disturbance rejection. The is tuning rationale, on which of some general words are spent together infor connection to literature research, is to shape the disturbance-to-output frequency are spent in connection to literature research, is to shape the disturbance-to-output frequency response, with conveniently assigning the poles of the corresponding transfer function. controllers for load disturbance rejection. The tuning rationale, on which some general words are spent in connection to literature research, is to shape the disturbance-to-output frequency response, together with conveniently assigning the poles of the corresponding transfer function. response, with assigning the of the corresponding transfer function. Analytical formulæ derived, to maximise simplicity and make the presented method are spent together intuning connection to are literature research, is shape response, together with conveniently conveniently assigning thetopoles poles of the the disturbance-to-output corresponding transferfrequency function. Analytical tuning formulæ are derived, to maximise simplicity and make the presented method Analytical tuning formulæ are derived, to maximise simplicity and make the presented method applicable on any device. Simulation results support the proposal. response, together with conveniently assigning the poles of the corresponding transfer function. Analytical tuning formulæ are derived, to maximise simplicity and make the presented method applicable on any device. Simulation results support the proposal. applicable on any device. results support the proposal. Analytical tuning are derived, to maximise simplicity and make the presented method applicable on any formulæ device. Simulation Simulation support theHosting proposal. © 2018, IFAC (International Federation ofresults Automatic Control) by Elsevier Ltd. All rights reserved. applicable on any device. Simulation results support the proposal. Keywords: PID control; controller tuning; model-based tuning; load disturbance rejection. Keywords: PID control; controller tuning; model-based tuning; load disturbance rejection. Keywords: PID PID control; control; controller controller tuning; tuning; model-based model-based tuning; Keywords: tuning; load load disturbance disturbance rejection. rejection. Keywords: PID control; controller tuning; model-based disturbance 1. INTRODUCTION Vran˘ci´ctuning; et al. load (2004) adapted rejection. the magnitude optimum 1. INTRODUCTION INTRODUCTION Vran˘ ci´ i´cc et et works al. (2004) (2004) adapted the magnitude magnitude optimum 1. Vran˘ c al. adapted the optimum technique, like Liu and Gao and Leva (2005) 1. INTRODUCTION Vran˘ ci´c et works al. (2004) adapted the(2008) magnitude optimum technique, like Liu and Gao (2008) and Leva (2005) technique, works like Liu and Gao (2008) and Leva (2005) employed relay-based tuning, others such as Shamsuzzoha In process control,1.“load disturbances are often the ma- Vran˘ INTRODUCTION c i´ c et al. (2004) adapted the magnitude optimum technique, works like Liu and others Gao (2008) and Leva (2005) employed relay-based tuning, suchadditional as Shamsuzzoha Shamsuzzoha In process control, “load disturbances are often the ma˚ employed relay-based tuning, others such as In process control, “load disturbances are often the maand Lee (2008) proposed PIDs with filters, jor process consideration” A str¨ om and H¨ agglund, 2004). works like Liu and Gao with (2008) and Leva (2005) employed relay-based tuning, others suchadditional as Shamsuzzoha In control, ((“load disturbances are often theAs ma-aa technique, ˚ and many Lee (2008) (2008) proposed PIDs filters, jor consideration” Astr¨ str¨ om m andeffort H¨ agglund, gglund, 2004). As ˚ and Lee proposed PIDs with additional filters, jor consideration” ( A o and H¨ a 2004). As a adopted an IMC-like paradigm (Liu and Gao, consequence, a lot of research has been spent on relay-based tuning, others suchadditional as(Liu Shamsuzzoha In process control, disturbances are been often theAs ma˚ and Lee (2008) proposed PIDs with filters, jor consideration” (“load Astr¨ om andeffort H¨ agglund, 2004). a employed and many adopted an IMC-like paradigm and Gao, consequence, a lot of research has spent on and many adopted an IMC-like paradigm (Liu and Gao, consequence, a lot of research effort has been spent on 2010; Skogestad and Grimholt, 2012). Also, several works ˚ their effective rejection. However, if one restricts the focus and Lee (2008) proposed PIDs with additional filters, jor consideration” ( A str¨ o m and H¨ a gglund, 2004). As a many adopted an IMC-like2012). paradigm (Liu andworks Gao, consequence, arejection. lot of research effort has been the spent on and 2010; Skogestad and Grimholt, Also, several their effective However, if one restricts focus 2010; Skogestad and Grimholt, 2012). Also, several works their effective rejection. However, if one restricts the focus concentrate on “balancing” tuning forAlso, set (Liu point tracking to model-based tuning and explicit two characterismany adopted IMC-like paradigm and Gao, consequence, lot of research effort been spent on and 2010; Skogestad andan Grimholt, 2012). several works their effective arejection. However, if rules, onehas restricts the focus concentrate on “balancing” “balancing” tuning for set point point tracking to model-based tuning and explicit rules, two characterisconcentrate on tuning set to tuning and explicit rules, two characterisand forSkogestad disturbance response, see2012). e.g.for Arrieta et al.tracking (2010); ticsmodel-based that are often considered beneficial for the industrial 2010; and Grimholt, Also, several works their effective rejection. However, if one restricts the focus concentrate on “balancing” tuning for set point tracking to model-based tuning and explicit rules, two characterisand for disturbance response, see see e.g. Arrieta Arrieta et et al. (2010); (2010); tics that areofoften often considered beneficial beneficial for the the panorama industrial and disturbance response, tics that considered the industrial Alc´afor ntara eton al. “balancing” (2013) acceptance an (auto)tuning procedure, set point to model-based tuning and explicit rules,for two characterisand for disturbance response,tuning see e.g. e.g.for Arrieta et al. al.tracking (2010); tics that are areofoften considered beneficial for the industrial concentrate Alc´ a ntara et al. (2013) acceptance an (auto)tuning procedure, the panorama aafor ntara et (2013) acceptance an (auto)tuning procedure, becomes narrower. and disturbance response, see e.g. Arrieta ettuning, al. (2010); tics that quite areof considered beneficial for the the panorama industrial Alc´ Alc´ ntara et al. al. acceptance ofoften an (auto)tuning procedure, the panorama In this work we (2013) concentrate on model-based acbecomes quite narrower. becomes quite narrower. In this work we concentrate on model-based tuning, acAlc´ a ntara et al. (2013) acceptance of an (auto)tuning procedure, the panorama becomes quitewe narrower. this work we concentrate on model-based tuning, accounting for the form of the process model but at the In this paper address the type of tuning technique just In In this work we concentrate onprocess model-based tuning, accounting for the form of the model but at the In this paper we address the type of tuning technique just becomes quite narrower. forconsidering the form of ofdifferent theonprocess process model but at at the the In this address the type of technique just same possible structures for sketched, andwe with respect a major reference, hence a counting In thistime work we concentrate model-based tuning, accounting for the form the model but In this paper paper address theto type of tuning tuning technique just same time considering different possible structures for sketched, andwe with respect to a major major reference, hence a same time considering different possible structures for sketched, and with respect to a reference, hence a it, and aiming at explicit tuning formulæ. According to good representative of the state of the art, we propose counting for the form of the process model but at the In this paper we address the type of tuning technique just time considering different possible structures for sketched, and with respect to a major reference, hence a same it, and aiming at explicit tuning formulæ. According to good representative of the state of the art, we propose it, and aiming at explicit tuning formulæ. According to good representative of the state of the art, we propose Scopus, to date the most cited paper encompassing all a solution thatwith is simpler, in athat it adopts a uniform same time considering different possible structures for sketched, and respect to major reference, hence a it, and aiming at explicit tuning formulæ. According to good representative of the state of the art, we propose Scopus, to date the most cited paper encompassing all a solution that is simpler, in that it adopts a uniform to date the most cited encompassing all aa solution that is simpler, in that it adopts aaoruniform theand characteristics above, is thatpaper by Horn et al. (1996). controller structure, andtheproduces comparable better Scopus, it, aiming at explicit tuning formulæ. According to good representative of state of the art, we propose Scopus, to date the most cited paper encompassing all solution that is simpler, in that it adopts uniform the characteristics above, is that that by by Horn Horn et al. al.which (1996). controller structure, and produces comparable or better the characteristics above, is et (1996). controller structure, and produces comparable or better The authors adopt an IMC-centred approach, is results. It has to be noted that our technique – like its Scopus, to date the most cited paper encompassing all a solution that is simpler, in that it adopts a uniform the characteristics above, is that by Horn et al.which (1996). controller structure, produces or better The authors adopt an IMC-centred approach, is results. Itcounterpart has to to be be and noted that delay-free ourcomparable technique – like like its The authors adopt an IMC-centred approach, which is results. It has noted that our technique – its very well suited to achieve model-based explicit tuning reference – considers processes: this the characteristics above, is that by Horn et al. (1996). controller structure, and produces comparable or better The authors adopt an IMC-centred approach, which is results. Itcounterpart has to be noted that delay-free our technique – likethis its very well suited to achieve model-based explicit tuning reference – considers processes: well to achieve model-based explicit tuning reference –limitation, considers delay-free processes: this for various model and determine a controller may or may nottobebea noted in Leva The authors adopt an IMC-centred approach, which is results. Itcounterpart has that as ourdiscussed technique – likeand its very very well suited suited tostructures, achieve model-based explicit tuning reference counterpart considers delay-free processes: this for various model structures, and determine afilter controller may or may may notanyway, be aa –limitation, limitation, as discussed in Leva Leva and for various model structures, and determine a controller may or not be as discussed in and in the form of an ideal PID cascaded to a up to Maggio (2010); addressing the delay (dominated) very well suited to achieve model-based explicit tuning reference counterpart – considers delay-free processes: this for various model structures, and determine a controller may or may not be a limitation, as discussed in Leva and in the form of of anInideal ideal PID cascaded cascaded to a filter filter up to Maggio (2010); anyway, addressing the delay delay (dominated) in form PID to up Maggio (2010); addressing the thethe second the quoted paper, rules obtained case will be the subject of future as research. various model structures, determine afilter controller may or may notanyway, be a limitation, discussed in Leva and for in the formorder. of an anIn ideal PID and cascaded to aa are up to to Maggio (2010); anyway, addressing the delay (dominated) (dominated) the second order. the quoted quoted paper, rules are obtained case will will be the the subject of future research. research. the second order. In the paper, rules are obtained case be subject of future for five model structures by changing the structure of in the form ofstructures anInideal PID cascaded tostructure a are filter upthe to Maggio (2010); anyway, addressing the delay 2(dominated) the second order. the quoted paper,the rules obtained case be istheorganised subject of research. for five model by changing of the The will paper asfuture follows. Section provides a for five model structures by changing the structure of the filter, interpreted in IMC terms. This results in controllers the second order. In the quoted paper, rules are obtained The paper is organised as follows. Section 2 provides a case will be the subject of future research. for five model structures by changing the structure of the The paper is as follows. 22 provides filter, interpreted in IMC terms. This This results in controllers controllers minimal review of related andSection motivates the par-aa filter, in terms. in The paper is organised organised as work, follows. Section provides of different structure themselves, as results some parameters of fiveinterpreted model structures by changing the structure of the minimal review of related related work, and motivates the parpar- for filter, interpreted in IMC IMC terms. This results in controllers minimal review of work, and motivates the of different structure themselves, as some parameters of ticular research here presented. In Section 3 the proposed The paper is organised as follows. Section 2 provides a of different structure themselves, as some parameters of minimal review here of related work,Inand motivates the par- filter, the general form (PID plus filter), for some models are interpreted in IMC terms. This results in controllers ticular research presented. Section 3 the proposed of different some parameters of ticular research presented. In Section 33 the the general structure form (PID (PIDthemselves, plus filter), filter),as for for some models are are tuning rules are here derived, andwork, in Section 4 they areproposed applied minimal review of related and motivates the parthe general form plus some models ticular research here presented. In Section the proposed structurally zero. of different some parameters of tuning rules are derived, derived, andthat in Section Section theyaccording are applied applied the general structure form (PIDthemselves, plus filter),as for some models are tuning are and in 444 they are structurally zero. to somerules process structures areSection relevant to structurally ticular research presented. In 3 the zero. tuning rules are here derived, andthat in Section they areproposed applied the general form (PID plus filter), for some models are to some process structures are relevant according to structurally zero. to process structures that are according to The method we propose aims at obtaining the same (or thesome literature. Sections 5 and 6inrespectively report benchtuning rules are derived, andthat Section 4 they areaaapplied to some process structures are relevant relevant according to structurally The method we propose propose aims at obtaining the same (or zero. the literature. Sections and respectively report benchThe method we aims obtaining the (or the literature. Sections 555 and 666 respectively report aa benchbetter) result a uniform and completely standard mark simulation campaign and aarelaboratory experiment, to some process structures that relevant according to The method wewith propose aims at at obtaining the same same (or the literature. Sections and respectively report benchbetter) result with a uniform and completely standard mark simulation campaign and a laboratory experiment, result with uniform and completely standard mark simulation campaign and aa laboratory experiment, controller structure, i.e., aaims real PID (more precisely, a PID to evidence the achieved results and advantages, while better) The method wewith propose at obtaining the same (or the literature. Sections 5 and 6 respectively report a benchbetter) result aa uniform and completely standard mark simulation campaign and laboratory experiment, controller structure, i.e., a real real PID PID (more precisely, acould PID to evidence the achieved results and advantages, advantages, while controller structure, i.e., a (more precisely, a PID to evidence the achieved results and while with filtered derivative action). The closest work we Section 7 concludes the paper by drawing some conclusions better) result with a uniform and completely standard mark simulation campaign and a laboratory experiment, controller structure, i.e., a real PID (more precisely, a PID to evidence the achieved results and advantages, while with filtered derivative action). The closest work we could Section concludes the paper by by drawing some some conclusions with filtered derivative The closest work we Section 777 concludes the paper find to ourstructure, approach isaction). that Jin(more and Liu (2014), that andevidence outlining future work. i.e., a realby precisely, PID to the achieved results and advantages, while controller with filtered derivative The work we acould could Section concludes the paper by drawing drawing some conclusions conclusions find to our approach isaction). that byPID Jin closest and Liu Liu (2014), that and outlining outlining future work. find to our approach is that by Jin and (2014), that and future work. however does not encompass all the characteristics above, filtered derivative action). The closest work we could Section 7 concludes paper by drawing some conclusions with find to our approach is that by Jin and Liu (2014), that and outlining futurethe work. however does not not encompass encompass all the theshape characteristics above, however does all characteristics above, and in directly the disturbancefind to particular our approach isnot that by the Jinshape and Liu (2014), that and outlining future work. 2. BRIEF LITERATURE REVIEW however does not does encompass all characteristics above, and in particular does not directly the disturbance2. BRIEF LITERATURE REVIEW and in particular does not directly shape the disturbanceto-output frequency response magnitude the way we do. As 2. BRIEF LITERATURE REVIEW however does not encompass all the characteristics above, and in particular does not directly shapethe theway disturbance2. BRIEF LITERATURE REVIEW to-output frequency response magnitude we do. do. As As frequency response magnitude the way we for the tracking/rejection tradeoff, here we concentrate on and in particular does not directly shape the disturbanceMany approaches wereLITERATURE proposed to PI/PID tuning for load to-output 2. BRIEF REVIEW to-output frequency response magnitude theconcentrate way we do. As for the tracking/rejection tradeoff, here we on Many approaches werehistorically proposed to toinPI/PID PI/PID tuning for load for tracking/rejection here we concentrate on the the latter: if trackingresponse is of tradeoff, interest as well, one can recover to-output frequency magnitude the way we do. As Many approaches were proposed tuning for load disturbance rejection, an attempt to cure an for the tracking/rejection tradeoff, here we concentrate on the latter: if tracking is of interest as well, one can recover Many approaches were proposed to PI/PID tuning for load disturbance rejection, historically in an an attempt attempt to cure cure an the latter: if is interest as well, one can it for example with method in Leva and Bascetta (2006), for the tracking/rejection here we concentrate on disturbance rejection, historically to an “overemphasis on were the set point response” (Shinskey, 2002), the latter: if tracking tracking is of of tradeoff, interest as well, one can recover recover Many approaches proposed toin tuning load it for example with method in Leva Leva and Bascetta (2006), disturbance rejection, historically inPI/PID an attempt to for cure an it “overemphasis on the set point response” (Shinskey, 2002), for example with method in and Bascetta (2006), as that technique is independent of how the feedback part the latter: if tracking is of interest as well, one can recover “overemphasis on the set point response” (Shinskey, 2002), but basicallyrejection, driven the necessities of(Shinskey, process for with method in Leva andthe Bascetta (2006), disturbance historically in an attempt to control: cure an it as thatexample technique is independent independent of how feedback part “overemphasis on theby set point response” 2002), but basically driven by the necessities ofdirect process control: as that technique of feedback part of for the controller isis it with method in Leva andthe Bascetta (2006), but driven by the necessities of process control: Chenbasically and Seborg (2002) resorted toof synthesis, as thatexample technique isdesigned. independent of how how the feedback part “overemphasis on the set point response” (Shinskey, 2002), of the controller is designed. but basically driven by the necessities process control: Chen and and Seborg (2002) (2002) resorted resorted to to direct synthesis, synthesis, of thatcontroller techniqueis independent of how the feedback part Chen of the the controller isisdesigned. designed. but driven (2002) by the resorted necessitiestoofdirect process control: as Chenbasically and Seborg Seborg direct synthesis, of the controller is designed. Chen and Seborg (2002) resorted to direct synthesis, Copyright 137 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © © 2018 2018, IFAC IFAC (International Federation of Automatic Control) Copyright © 2018 137 Copyright © under 2018 IFAC IFAC 137 Control. Peer review responsibility of International Federation of Automatic Copyright © 2018 IFAC 137 10.1016/j.ifacol.2018.06.115 Copyright © 2018 IFAC 137
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3. THE PROPOSED TUNING METHOD The block diagram for the addressed control scheme is shown in Figure 1, where P (s) and C(s) are respectively the transfer functions of the process and the controller, w(t) is the set point, y(t) the controlled variable, u(t) the control signal, and d(t) a load disturbance to be rejected.
w(t) + −
C(s)
d(t) u(t) + +
P(s)
y(t)
Figure 1. Control loop with load disturbance. We consider as the controller to tune a real PID with possibly complex zeroes, that we write as bC0 + bC1 s + bC2 s2 C(s) = . (1) aC1 s + aC2 s2 The tuning goal is to constrain the disturbance-to-output transfer function Q(s) = Y (s)/D(s) to take the form QN (s) , QD (s) = (1 + sτQ )nQ , (2) Qo (s) = QD (s) with QN (0) = 0 – as inherent to the structure of (1), incidentally – to guarantee asymptotic disturbance rejection, τQ > 0 for stability, and nQ large enough to ensure high-frequency roll-off (if not for a particular case briefly discussed in Section 4.3). In fact, nQ is dictated by the structure of the considered process models, as there must be as tuning equations as controller parameters—whence also the apparent redundancy in the parametrisation of C(s), as will become clear in Sections 4.1 through 4.5. On the contrary, τQ is a tuning parameter (the only one) for the method, interpreted as a time constant connected to the desired disturbance step response convergence time—i.e., in some sense, pretty much lie the desired closed-loop dominant time constant in lambda or IMC tuning. For completeness, in the (dominantly frequent) case of a controller with real zeroes, the controller(1) is immediately converted to the standard ISA PID form 1 sTd C(s) = K 1 + (3) + sTi 1 + sTd /N by aC1 bC1 − aC2 bC0 aC1 bC1 − aC2 bC0 , Ti = , K= 2 aC1 aC1 bC0 Td =
a2C2 bC0 − aC1 aC2 bC1 + a2C1 bC2 , aC1 (aC1 bC1 − aC2 bC0 )
N=
a2C2 bC0 − aC1 aC2 bC1 + a2C1 bC2 . aC2 (aC1 bC1 − aC2 bC0 )
(4)
Coming to the process model, we assume the general form bP 0 ∓ b P 1 s ∓ b P 2 s 2 P (s) = (5) aP 0 + aP 1 s + aP 2 s 2 138
with all parameters nonnegative; notice that we are assuming positive gain as well, but obviously this does not impair generality. The rationale behind the tuning approach we adopt, is to prevent |Q(jω)| from exhibiting a plateau, thereby minimising the width of the band where the disturbance is rejected to the least extent (i.e., where |Q(jω)| is near to its maximum), while allowing some control on the maximum value of |Q(jω)| and of the high-frequency control sensitivity. The structure chosen for Qo (s), see (2), allows to easily obtain an estimate on its ∞-norm by taking the magnitude of its frequency response at ω = 1/τQ , where the coincident poles are. For example, if Qo (s) has four poles (and one zero in the origin, remember) then the only possible mutual locations of zeroes and poles are shown in Figure 2, and clearly the estimate just introduced is sensible; in the figure, asymptotic Bode plots are sketched for simplicity. Z Z P4
Z P4 Z ω
P4 Z Z ω
ω
Figure 2. Possible frequency positions of zeroes (Z) and poles (P ) of Q(s) in the fourth order case. As such, in the following we further denote the estimated ∞-norm of Qo (s) and the high-frequency value of the controller (i.e., of the control sensitivity) magnitude, respectively, as max = max |Q(jω)|, C∞ = lim |C(jω)|. Q (6) ω
ω→∞
max underlines that this is an where the hat sign in Q estimate; the exact value depends on the detailed shape of the frequency response, but the estimate is representative in any case. The nice fact in all this reasoning is that once the systems of equations that provide the controller parameters are analytically solved, as done in the following sections, both Qmax and C∞ – that respectively quantify the disturbance rejection and the high-frequency control activity – are functions of the tuning parameter τQ , and can be used as the basis for a (possibly) automatic selection of that parameter, although in this paper we do not discuss the matter. Summarising, the main advantage of our rationale is that the disturbance-to-output frequency response is shaped in a very direct and straightforward manner, and for several process structures – like those addressed below – the resulting tuning relationships are explicit. 4. APPLICATION TO RELEVANT PROCESS STRUCTURES The quoted paper by Horn et al. (1996) reports IMC controllers for five process structures, optimised for “open
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loop dynamics slower then the [desired] closed-loop dynamics”, i.e., for what is sometimes called “strong feedback”, i.e., in turn, essentially for disturbance rejection. With the notation of that reference, that we report here for the reader’s convenience, the five structures treated are kp kp (−s + z) PA (s) = , PB (s) = , τs + 1 (τ1 s + 1)(τ2 s + 1) PC (s) =
kp (−s + z) kp , PD (s) = , τs + 1 (τ1 s + 1)(τ2 s + 1) PE (s) =
(7)
kp s(τ s + 1)
4.1 Process A In the form (5), process A in (7) reads bP 0 PA (s) = , (8) 1 + aP 1 s and with controller (1), this structurally yields a thirdorder Q(s), i.e., nQ = 3. Setting QD (s) = (1 + sτQ )3 we then get 3 τQ 1 aC1 = 1, aC2 = , bC0 = , aP 1 bP 0 (9) 2 3 a (3τ − a ) − τ 3τQ − 1 P1 P1 Q Q bC1 = , bC2 = bP 0 a P 1 bP 0 and 4 + a2 bP 0 τQ P1 (10) Qmax = 3/2 2 a P 1 τQ aP 1 (aP 1 − 3τ 2 ) + τ 3 Q Q (11) C∞ = 3 bP 0 τQ 4.2 Process B
In the form (5), process B in (7) reads. bP 0 − bP 1 s PB (s) = 1 + a P 1 s + aP 2 s 2 and in this case we have to set nq = 4, whence
aC1
bC1
In the form (5), process C in (7) reads bP 0 − b P 1 s PC (s) = . (14) 1 + aP 1 s We include this case for completeness, but we have to notice that a first-order, non strictly proper process – and to top, nonminimum-phase – seems a bit unrealistic indeed in practice. Joined to a real PID, in addition, this gives rise to an open loop transfer function with zero relative degree, which has very well known drawbacks—and in the end, adopting a closed-loop pole placement approach like ours is probably among the best ways to handle such a situation. In any case, we apply the technique to show that formally it works. To do this we have to set nq = 3, whence aP 1 b2P 0 bP 1 + bP 0 b2P 1 + b3P 1 2 3 + 3bP 0 b2P 1 τQ + 3b2P 0 bP 1 τQ + b3P 0 τQ aC1 = 1, aC2 = − , b2P 0 bP 1 + aP 1 b3P 0 bC0 = −
1 bP 1 + bP 0 (3τQ + 1) , bC1 = − bP 0 b2P 0
−a2P 1 b2P 0 − aP 1 bP 0 bP 1 − aP 1 b2P 1 2 3 − 3aP 1 bP 0 bP 1 τQ − 3aP 1 b2P 0 τQ + b2P 0 τQ = b2P 0 bP 1 + aP 1 b3P 0
bC2
(15) while Qmax and C∞ are again too long to be shown. In this work we do not further concentrate on this case, as probably one should adopt a different controller structure, or simply admit that every process eventually rolls off at high frequency, and resort e.g. to case B.
In the form (5), process D in (7) reads bP 0 PD (s) = 1 + a P 1 s + aP 2 s 2 and we have to set nq = 4, whence
bC2 =
3 4 + aP 1 τQ 4a2P 2 τQ − 4aP 2 τQ , a2P 2 bP 0
139
C∞
(17)
2 3 4 6a2P 2 τQ − 4aP 1 aP 2 τQ + (a2P 1 − aP 2 )τQ a2P 2 bP 0
and
(13)
(16)
3 4 4 − aP 1 τQ 4aP 2 τQ τQ 1 , a = , bC0 = , C2 2 aP 2 aP 2 bP 0
bC1 =
aP 1 aP 2 b2P 1 + a2P 2 bP 0 bP 1 + (4a2P 2 b2P 0 2 +4aP 1 aP 2 bP 0 bP 1 )τQ − 6aP 2 bP 0 bP 1 τQ 3 4 − 4aP 2 b2P 0 τQ + (aP 1 b2P 0 + bP 0 bP 1 )τQ = , 2 2 2 3 a P 2 b P 0 bP 1 + a P 1 a P 2 b P 0 b P 1 + a P 2 b P 0
bC2
4.3 Process C
aC1 =
4 τQ 1 , bC0 = , aP 2 bP 0
2 a2P 2 b2P 1 + 4a2P 2 bP 0 bP 1 τQ + 6a2P 2 b2P 0 τQ 2 3 −(4aP 1 aP 2 bP 0 + 4aP 2 bP 0 bP 1 )τQ 4 + ((a2P 1 − aP 2 )b2P 0 + aP 1 bP 0 bP 1 )τQ = 2 2 2 3 a P 2 bP 0 b P 1 + a P 1 a P 2 b P 0 bP 1 + a P 2 b P 0
max and C∞ are too long to be while the expressions of Q reported here.
4.4 Process D (12)
2 aP 2 b3P 1 + 4aP 2 bP 0 b2P 1 τQ + 6aP 2 b2P 0 bP 1 τQ 3 3 3 2 4 + 4aP 2 bP 0 τQ − (aP 1 bP 0 + bP 0 bP 1 )τQ = , aP 2 bP 0 b2P 1 + aP 1 aP 2 b2P 0 bP 1 + a2P 2 b3P 0
aC2 =
139
2 bP 0 τQ
(4aP 2 − aP 1 τQ )2 + a2P 2 , 4a2P 2 2 2 2 6aP 2 − 4aP 1 aP 2 τQ + (aP 1 − aP 2 )τQ = . 2 aP 2 bP 0 τQ max = Q
(18) (19)
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4.5 Process E In the form (5), process E in (7) reads bP 0 PE s(s) = s + aP 2 s2 and we have to set nq = 4, whence aC1
3 4 4 − τQ τQ 4aP 2 τQ 1 , a = = , bC0 = , C2 2 aP 2 aP 2 bP 0
bC1 and
4 3 2 − 4aP 2 τQ + τQ 6a2P 2 τQ 4τQ = , bC2 = bP 0 a2P 2 bP 0
(4aP 2 − τQ )2 + a2P 2 max = Q , 4a2P 2 6a2 − 4aP 2 τQ + τ 2 P2 Q C∞ = . 2 aP 2 bP 0 τQ 2 bP 0 τQ
(20)
(21)
(22)
A B
√ min. for τQ = aP 1 no analytical solution
C D E
no analytical solution strictly incr. with τQ strictly incr. with τQ
pC (σ) =
1 − σθ 1 , pD (σ) = , 1+σ (1 + σ)(1 + σψ)
(24)
1 . σ(1 + σ) where 0 < ψ < 1 since the “other” pole has to be faster, while in principle θ (positive) could be arbitrarily large. Rigorously the numerator of pE should be τ , incidentally, but in this case it just plays the role of a scale factor, as the process is type 1, and therefore can be omitted in the normalised transfer function. 5.1 Test 1
(23)
pA , τq = 0.2 pE , τq = 0.2
pA , τq = 0.4 pE , τq = 0.4
pA , τq = 0.6 pE , τq = 0.6
pA , τq = 0.8 pE , τq = 0.8
20
0
−20
dB
A first thing to notice is that the five process structures considered yield very different results as for the existence max and/or C∞ , as of values for τQ that minimise Q summarised in table 1. The somewhere opposite effect of the tuning parameter convinces of the opportunity of structure-specific tuning if high rejection performance is required. Proc.
1 1 − σθ , pB (σ) = , 1+σ (1 + σ)(1 + σψ)
pE (σ) =
4.6 General remarks
max Q
pA (σ) =
−40
−60
−80 10−1
100
101
102
Normalised frequency
C∞
√ min. for τQ = aP 1 ∃ minimising τQ (complex expression) strictly incr. with τQ strictly decr. with τQ min. for τQ = 3aP 2
Figure 3. Test 1 – avoiding the magnitude plateau in the frequency response or |Q(s)|. First, we show that the proposed rule actually avoids the magnitude plateau in the frequency response or |Q(s)|, while τq controls the position of the frequency band where the disturbance is least rejected.
Table 1. Summary of the effect of parameter max and C∞ . τQ on Q
A second notable remark is that, although critical cancellations never happen (we omitted computations for brevity), in the case of nonminimum-phase processes the PID can turn out to be unstable, i.e., to have its second pole in the right half plane. In principle this can make sense as high rejection performance is required, but further study is required on this aspect, especially to properly quantify the closed-loop stability robustness, that we cannot undertake in this paper. 5. SIMULATION RESULTS In this section we present some samples of an extensive simulation campaign, that we conducted to assess the correct behaviour of the proposed technique, and to verify that the it actually exhibits the expected strengths. For compactness we introduce a normalised complex variable σ = sτ , where τ is the only or the largest time constant of the pole(s) of processes A–E not in the origin. Defining the normalised processes pA−E (σ) as PA−E (σ/τ )/µ we get 140
Figure 3 shows an example with processes A and E; the magnitude of the frequency response of Q(s) is plotted. It is interesting to note the opposite effect of τq , depending on the processes’ structure. Also, peak values above 0dB are not an issue: this example only means to show the absence of the plateau, while of course when doing a real tuning values of τq keeping the peak under 0dB will be selected. 5.2 Test 2 We now show an example of comparison between our proposal and the reference by Horn et al. (1996). The example uses two processes in class B, namely 1 − 2σ pB1 (σ) = , (1 + σ)(1 + 0.4σ) (25) 1 − 0.4σ , pB2 (σ) = (1 + σ)(1 + 0.9σ) of which the first has a remarkably undershooting step response and a second pole at significantly higher frequency wit respect to the dominant one, while the second has two
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Horn et al.
ISEprop , (27) ISEhorn ISEprop and ISEhorn being the ISE computed with the proposed and the reference controller, respectively.
Proposed
RISE =
2 pB1 , τq = 0.6
141
0 −2
pB1 , τq = 1
1
1
0
RISE
pB2 , τq = 0.6
−1
0.5
0.5
0 1
0
1 0.8
0.6
pB2 , τq = 1
τq
0.5 0.4
0.1
ψ
Figure 5. Test 3 – comparing the load disturbance step response ISE.
0.5
0
0
2
4
6
8
10
12
14
16
18
20
time [s]
Figure 4. Test 2 – comparing load disturbance step responses. almost coincident poles and a less evident overshoot (in some sense, thus, the two processes are “quite extreme” in the class). Figure 4 reports the load disturbance step responses of (4) with a “low” and a “high” value of the parameter controlling the closed-loop band, i.e., λ in the work by Horn et al. (1996) and τq in our proposal. As can be seen, both parameters act in the respective technique in the desired manner. On one hand this proves that the analogy in their interpretation is correct, but on the other hand, as expected, there is some advantage in governing the disturbance-to-output transfer function directly. For example, with pB1 the maximum error might slightly increase, the ISE is comparable but the settling is definitely shorter, while with pB2 there is an apparently less oscillatory behaviour.
Figure 5 shows the said integral index for process class D. The green surface is the relative ISE, the red line on the surface is the unity level, its projection onto the (ψ, τq ) plane is shown as well. As can be seen, there is a wide region in the (ψ, τq ) plane, where the proposed tuning approach yields an advantage. 6. AN EXPERIMENTAL TEST In this section we present an experiment in which the proposed method is applied to a laboratory apparatus, and compared to Horn et al. (1996) plus two other PID tuning techniques well known to be suitable for disturbance rejection, namely the AMIGO (˚ Astr¨om and H¨ agglund, 2004) and the SIMC (Skogestad and Grimholt, 2012). For completeness we notice that AMIGO tunes an ideal PID, that we made proper with N = 20 as suggested in the quoted reference. As for the SIMC, given the firstorder process, we used the PI rule, with the additional advantage of achieving a zero relative degree controller without choosing the second pole arbitrarily. Finally, we notice that for process class A, the most appropriate for the addressed dynamics, the controller by Horn et al. reduces to a PI. All in all, with the inevitable shortcomings of any experimental setup, we deem this a fair enough comparison.
In this test we again compare our technique to the quoted reference, but this time in terms of the load disturbance step response ISE, defined as ∞ ISE = e(t)2 dt, (26)
The used apparatus, described in Leva (2003), is a temperature control system composed of a small metal plate electrically heated by a transistor. The command to that transistor is the control signal, while the plate temperature is the controlled variable. A second transistor is connected to heat the plate, thereby apparently providing a load disturbance. Since the controlled dynamics is dominantly first-order, all the three methods just mentioned fit. The tuning results are in table 2.
where e(t) = w(t) − y(t) is the error. More precisely, we compare the relative ISE, i.e., the quantity
Figure 6 shows the response of the controlled plate temperature Tp (top plot) and of the [0, 100] command uh to the
5.3 Test 3
0
141
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Method
K
Ti
Td
N
Proposed Horn et al. AMIGO SIMC
88 41 31 33.5
15 30 30 56
0.35 0 3.5 0
0.03 n/a 20 n/a
disturbance-to-output dynamics can be considered, and analogously, the application of the same idea to other controller structures can be studied. REFERENCES
Table 2. Controller parameters in the experimental test. Set point
SIMC
AMIGO
Horn et al.
Proposed
28
Tp [◦ C]
27.5
27
26.5 100
uh [0, 100]
80
60
40
20
0
0
100
200
300
400
500
600
time [s]
Figure 6. Laboratory experiment – controlled variable and control signal response to a 40% load disturbance step. heating transistor (bottom plot) to a 40% load disturbance impressed via the second transistor. The proposed technique yields the best disturbance rejection; there is of course a cost in terms of of a more nervous control and a higher sensitivity to measurement noise, but both phenomena appear up to a reasonable extent. The technique can therefore supposed to be also viable in practical process control applications. 7. CONCLUSIONS AND FUTURE WORK We presented a technique to analytically tune a real PID in a structure-specific manner as for the process dynamics. The underlying approach is based on assigning the closed-loop poles so as to shape the disturbanceto-output frequency response to exhibit no plateau. The technique has a single parameter, interpretable in a way similar to λ in IMC-based rules, and for which “optimal” values can be sought (although this was not discussed here in depth). Future work will proceed along three main directions. The first one is to apply the same rationale to other process structures—including delay-dominated ones. The second consists of posing and solving optimisation problems (possibly not analytically, however) to automatically select the tuning parameter. The third is to integrate the proposed method – that refers to the feedback part of a PID – into the synthesis of two-degree-of-freedom controllers. As a further possibility, then, different structures for the desired 142
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