Robust QFT-based PI controller for a feedforward control scheme⁎

Proceedings of the 3rd IFAC Conference on Proceedings of the 3rd IFAC Conference on Advances in Proportional-Integral-Derivative Proceedings of Proceedings of the the 3rd 3rd IFAC IFAC Conference Conference on on Control Advances in Proportional-Integral-Derivative Control Ghent, Belgium, May 9-11, 2018 Available online at www.sciencedirect.com Advances in Proportional-Integral-Derivative Control Advances in Proportional-Integral-Derivative Proceedings of the 3rd IFAC Conference on Control Ghent, Belgium, May 9-11, 2018 Ghent, Belgium, May 9-11, 2018 Ghent, Belgium, May 9-11, 2018 Advances in Proportional-Integral-Derivative Control Ghent, Belgium, May 9-11, 2018

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IFAC PapersOnLine 51-4 (2018) 262–267

Robust Robust QFT-based QFT-based PI PI controller controller for for a a  Robust QFT-based PI controller for a  feedforward control scheme feedforward control scheme Robust QFT-based PI controller for a  feedforward control scheme  ´ feedforward control e Carlos Moreno ∗ scheme Jos´ e Luis Guzm´ an ∗ Angeles Hoyo ∗ Jos´

∗ ´ e Carlos Moreno∗∗ Jos´ e Luis Guzm´ an ∗∗∗ Angeles Hoyo ∗∗∗ Jos´ ∗ ´ ∗ Jos´ e Carlos Moreno e Luis Guzm´ a Angeles Hoyo ´ H¨ a gglund ∗∗ Jos´ e Tore Carlos Moreno Jos´ e Luis Guzm´ an n Angeles Hoyo Jos´ Tore H¨ agglund ∗∗ ∗∗ ∗ ∗ Tore H¨ a gglund ´ H¨ aMoreno gglund Jos´ e Tore Carlos e Luis Guzm´ an ∗ Angeles Hoyo Jos´ ∗ ∗∗ CIESOL, ceIA3, 04120 Almer´ Tore H¨ aof gglund ∗ Dep. of Informatics, University Dep. of Informatics, University of Almer´ıa, ıa, CIESOL, ceIA3, 04120 ∗ ∗ Dep. of Informatics, Informatics, University of214133; Almer´ıa, ıa, CIESOL, ceIA3, ceIA3, 04120 04120 Almer´ ıa, Spain. (Tel: +34 950 e-mail:[email protected]; Dep. of University of Almer´ CIESOL, Almer´ ıa, Spain. (Tel: +34 950 214133; e-mail:[email protected]; ∗ Almer´ ıa, (Tel: +34 950 214133; e-mail:[email protected]; [email protected]; [email protected]) Almer´ ıa, Spain. Spain. (Tel: +34 950 214133; e-mail:[email protected]; Dep. of Informatics, University of Almer´ ıa, CIESOL, ceIA3, 04120 [email protected]; [email protected]) ∗∗ [email protected]; [email protected]) of Automatic Control, Lund University, Sweden, e-mail: ∗∗ Department [email protected]; [email protected]) Almer´ ıa, Spain. (Tel: +34 950 214133; e-mail:[email protected]; Department of Automatic Control, Lund University, Sweden, e-mail: ∗∗ ∗∗ Department of Automatic Control, University, [email protected] Department [email protected]; of Automatic Control, Lund Lund University, Sweden, Sweden, e-mail: e-mail: [email protected]) [email protected] ∗∗ [email protected] [email protected] Department of Automatic Control, Lund University, Sweden, e-mail: [email protected] Abstract: Feedforward Feedforward control control schemes schemes to compensate compensate for for disturbances disturbances are are very very well well known known Abstract: to Abstract: Feedforward control schemes to for are very in process control. In those control approaches, PID controllers controllers are usually usually considered in the the Abstract: Feedforward control schemes to compensate compensate for disturbances disturbances areconsidered very well well known known in process control. In those control approaches, PID are in in process control. In those control approaches, PID controllers are usually considered in the feedback loop, where nominal design for both feedback and feedforward controllers are commonly in process control. In those control approaches, PID controllers are usually considered in the Abstract: Feedforward control schemes to feedback compensate disturbances are very known feedback loop, where nominal design for both andfor feedforward controllers arewell commonly feedback where design for both controllers are performed. This paper presents robustness analysis tofeedforward study are howusually uncertainties can in affect feedback loop, where nominal design for both feedback feedback and feedforward controllers are commonly commonly in processloop, control. Innominal those control approaches, PID and controllers considered the performed. This paper presents aa robustness analysis to study how uncertainties can affect performed. This paper presents a robustness analysis to study how uncertainties can affect the classical feedforward control scheme. Afterwards, a robust PI controller is designed by performed. This paper presents a robustness analysis to study how uncertainties can affect feedback loop, where nominal design for both feedback and feedforward controllers are commonly the classical feedforward control scheme. Afterwards, a robust PI controller is designed by the classical feedforward control scheme. Afterwards, a robust PI controller is designed by using Quantitative Feedback Theory to account for these uncertainties and to fulfill robust the feedforward control Afterwards, ato robust PI designed by performed. This paper presents ascheme. robustness analysis study howcontroller uncertainties can robust affect usingclassical Quantitative Feedback Theory to account for these uncertainties and tois fulfill using Quantitative Feedback Theory toproblem. account for these these uncertainties andand toistime fulfill robust specifications for the theFeedback regulation control Results based onPI frequency domains using Quantitative Theory to account for uncertainties and to fulfill robust the classical feedforward control scheme. Afterwards, abased robust controller designed by specifications for regulation control problem. Results on frequency and time domains specifications for regulation control problem. Results based on frequency and time domains cc the  Copyright IFAC 2018. are presented. specifications for the regulation control problem. Results based on frequency and time domains using Quantitative Feedback Theory to account for these uncertainties and to fulfill robust are presented.  Copyright IFAC 2018. cc the are  Copyright IFAC 2018. are presented. presented.for  Copyright IFAC 2018.problem. Results based on frequency and time domains specifications regulation control © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. c Copyright are presented.  IFAC 2018. Keywords: feedforward, PID, disturbances, robustness Keywords: feedforward, PID, disturbances, robustness Keywords: feedforward, feedforward, PID, robustness Keywords: PID, disturbances, disturbances, robustness Keywords: feedforward, PID, disturbances, robustness 1. INTRODUCTION INTRODUCTION In 1. In all all these these works, works, nominal nominal models models were were used used and and the the 1. INTRODUCTION INTRODUCTION In all all these thesecase works, nominal models were used and the robustness was not studied. Notice that when uncer1. In works, nominal models were used the robustness case was not studied. Notice that whenand uncerrobustness case was not studied. Notice that when uncertainties are considered, the feedforward control scheme is robustness case was not studied. Notice that when uncer1. INTRODUCTION In all these works, nominal models were used and In process process control, control, feedforward feedforward compensation compensation helps helps the the tainties are considered, the feedforward control schemethe is In tainties are considered, the feedforward control scheme is deteriorated even for the perfect cancellation case. In that tainties are considered, the feedforward control scheme is robustness case was not studied. Notice that when uncerIn process control, feedforward compensation helps the feedback control to attenuate the effects of measurable deteriorated even for the perfect cancellation case. In that In process control, feedforward compensation helps the feedback control to attenuate the effects of measurable case, deteriorated even for the perfect cancellation case. In that the cancellation of the feedforward is not perfect and deteriorated even for the perfect cancellation case. In that tainties are considered, the feedforward control scheme is feedback control to attenuate the effects of measurable disturbances on the process. When the disturbance enters case, the cancellation of the feedforward is not perfect and feedback to process. attenuate the effects of measurable In processcontrol control, feedforward compensation helps the case, the cancellation of the feedforward is not perfect and disturbances on the When the disturbance enters the closed-loop may not be fulfilled case, the cancellation of the feedforward is not perfect and deteriorated evenspecifications for the perfect cancellation case. Inwhen that disturbances on the the process. When the disturbance enters into the process, the feedforward controller helps the clasthe closed-loop specifications may not be fulfilled when disturbances on process. When the disturbance enters feedback control to attenuate the effects of measurable into the process, the feedforward controller helps the clas- the closed-loop specifications may be fulfilled when deviates from the nominal conditions (Guzm´ a the system closed-loop specifications may not not benot fulfilled when case, the cancellation of the is perfect and into the process, process, theprocess. feedforward controller helpsaction, theenters classic feedback feedback controller, which only has by system deviates from thefeedforward nominal conditions (Guzm´ an n into the the feedforward helps the clasdisturbances on the the reactive disturbance sic controller, whichWhen onlycontroller has reactive action, by the the system deviates from the nominal conditions (Guzm´ a and H¨ a gglund, 2011). Thus, it is interesting to analyze this system deviates from the nominal conditions (Guzm´ an n the closed-loop specifications may not be fulfilled when sic feedback controller, which only has reactive action, by supplying an extra corrective signal before the disturbance and H¨ a gglund, 2011). Thus, it is interesting to analyze this sic feedback controller, which only has reactive action, by into the process, the feedforward controller helps the classupplying an extra corrective signal before the disturbance situation and H¨ aagglund, 2011). Thus, it is interesting to analyze this and to propose robust solutions for this problem. and H¨ gglund, 2011). Thus, it is interesting to analyze this the system deviates from the nominal conditions (Guzm´ a n supplying an extra corrective signal before the disturbance leads the system away from its setpoint (Guzm´ a n et al., situation and to propose robust solutions for this problem. supplying an extra corrective signal before the disturbance sic feedback controller, only has reactive action, by situation and to propose robust solutions for this problem. leads the system away which from its setpoint (Guzm´ an et al., situation and to2011). propose robust solutions fortothis problem. H¨aare gglund, Thus, it is interesting analyze this leads the system system away from its its setpoint (Guzm´ an n et et al., al., and 2015). leads the away from setpoint (Guzm´ a supplying an extra corrective signal before the disturbance There only a few works in literature where the robust2015). There areand onlytoapropose few works in literature where theproblem. robustrobust solutions for this 2015).the system away from its setpoint (Guzm´an et al., situation There are only aa few in where the robust2015). leads the feedforward control scheme has Therefor only few works works in literature literature thestudied. robustThe feedback feedback with with feedforward feedforward control control strategy strategy is is comcom- ness ness forare the feedforward control scheme where has been been studied. The ness for the feedforward control scheme has been studied. 2015). In (Guzm´ a n and H¨ a gglund, 2011) the robustness of ness for the feedforward control scheme has been studied. There are only a few works in literature where the robustThe feedback with feedforward feedforward control strategy is is commonly used in process industry. It is implemented in most In (Guzm´ a n and H¨ a gglund, 2011) the robustness of the the The feedback with control strategy commonly used in process industry. It is implemented in most In (Guzm´ aan and H¨ aagglund, 2011) the robustness of the feedback control with feedforward compensator was In (Guzm´ n and H¨ gglund, 2011) the robustness ofanathe ness for the feedforward control scheme has been studied. monly used in process industry. It is implemented in most distributed control systems to improve the control perforfeedback control with feedforward compensator was anamonly used in process industry. It is implemented in most The feedback with feedforward control strategy is comdistributed control systems to improve the control perfor- lyzed feedback control with feedforward compensator was anawith respect to uncertainties in the process gain and feedback control with feedforward compensator was anaIn (Guzm´ a n and H¨ a gglund, 2011) the robustness of the distributed control systems to improve the control performance in applications such as distillation columns (Nisenwith respect to uncertainties in the process gain and distributed control systems to distillation improve thecolumns controlin performonly in process industry. It is implemented most lyzed mance used in applications such as (Nisenlyzed withcontrol respect to uncertainties uncertainties in thewas process gainanaand approximated high-order dynamics. It demonstrated lyzed with respect to in the process gain and feedback with feedforward compensator was mance in Miyasaki, applications such power as distillation columns (Nisenfeld and 1973), plants (Weng and Ray, approximated high-order dynamics. It was demonstrated mance in applications such as distillation columns (Nisendistributed control systems to improve the control perforfeld and Miyasaki, 1973), power plants (Weng and Ray, lyzed approximated high-order dynamics. Itthewas was demonstrated that variability of the model parameters affects high-order dynamics. demonstrated with respect to in It process gain and feld and Miyasaki, 1973), power plants (Weng and Ray, approximated 1997)and or microalgae cultures among many(Weng other and examples that the the variability of uncertainties the process process model parameters affects feld Miyasaki, 1973), plants Ray, mance inmicroalgae applications such power asamong distillation columns (Nisen1997) or cultures many other examples that the variability of the process model parameters affects the response of the PID controller with the feedforward that the variability of the process model parameters affects approximated high-order dynamics. It was demonstrated 1997) or microalgae cultures among many other examples (Adam and Marchetti, 2004). response of the PID controller with the feedforward 1997) orand microalgae cultures among many(Weng other and examples feld and Miyasaki, 1973), power plants Ray, the (Adam Marchetti, 2004). the response of controller with the the response of the theofPID PID controller withparameters the feedforward feedforward that the variability the process model affects (Adamorand and Marchetti, 2004).among many other examples compensation. compensation. (Adam Marchetti, 2004). 1997) microalgae cultures The common feedforward compensator is formed as the compensation. compensation. response of the(Adam PID controller with the feedforward The common feedforward compensator is formed as the the (Adam and Marchetti, 2004). On the other hand, and Marchetti, 2004) presented The common feedforward compensator isand formed as the the compensation. dynamics between the load loadcompensator disturbance is the process process On the other hand, (Adam and Marchetti, 2004) presented The common feedforward formed as dynamics between the disturbance and the the hand, (Adam and Marchetti, 2004) aOn design solution for feedforward when the other other hand, (Adam Marchetti,controllers 2004) presented presented dynamics between the load disturbance and the process output, divided by the dynamic between the control signal a robust robust design solution forand feedforward controllers when dynamics between the dynamic loadcompensator disturbance the process The common feedforward isand formed as the On output, divided by the between the control signal a robust design solution for feedforward controllers when the available dynamic models include estimated limits a robust design solution for feedforward controllers when On the other hand, (Adam and Marchetti, 2004) presented output, divided byoutput the dynamic between the control signal and the process with reverse sign. However, the the available dynamic models include estimated limits output, divided by the dynamic between the control signal dynamics between the load disturbance and the process and the process output with reverse sign. However, the the available dynamic models include estimated limits for the uncertainties. A model-based design and tuning the available dynamic models include estimated limits a robust design solution for feedforward controllers when and the process output with reverse sign. However, the idealthe compensator maydynamic not bereverse realizable inHowever, many signal cases the uncertainties. A model-based design and tuning and processbyoutput withbe sign. the for output, divided the between thein control ideal compensator may not realizable many cases for the uncertainties. A model-based design and tuning procedure is proposed to account for model uncertainties. for the uncertainties. A model-based design and tuning the available dynamic models include estimated limits ideal compensator may not be realizable in many cases due to negative delay, having more zeros than poles, poles is proposed to account for model uncertainties. ideal compensator may not realizable inHowever, many cases and the processdelay, output withbe reverse sign. the procedure due to negative having more zeros than poles, poles procedure is to for model As conclusion, it that the procedure is proposed proposed to account for feedforward model uncertainties. for the uncertainties. A account model-based designuncertainties. andand tuning due tocompensator negative delay, delay, having more zeros than poles, phase poles in the the right-half plane, ornot in be general non-minimum conclusion, it is is derived derived that the the feedforward and the due to negative having more zeros than poles, poles ideal may realizable in many cases As in right-half plane, or in general non-minimum phase As conclusion, it is derived that the feedforward and the feedback controllers should be tuned simultaneously for As conclusion, it is derived that the feedforward and the procedure is proposed to account for model uncertainties. in the right-half plane, or few in general general non-minimum phase behaviours. In the last years, several works have feedback controllers should be tuned simultaneously for in the right-half plane, or in non-minimum phase due to negative delay, having more zeros than poles, poles behaviours. In the last few years, several works have feedback controllers should be tuned simultaneously for an efficient disturbance rejection. This idea has been used feedback controllers should be tuned simultaneously for As conclusion, it is derived that the feedforward and the behaviours. In the last few years, several works have appeared proposing new tuning rules to account for these an efficient disturbance rejection. This idea has been used behaviours. In the last few years, several works have in the right-half plane, or in general non-minimum phase appeared proposing new tuning rules to account for these later an efficient disturbance rejection. This idea has been used in other works. For instance, in (Vilanova et al., an efficient disturbance rejection. This idea has been used feedback controllers should be tuned simultaneously for appeared proposing new tuning rules to account for these inversion problems (Hast and H¨ a gglund, 2014),(Rodr´ ıguez in other works. For instance, in (Vilanova et al., appeared proposing new to account for ıguez these behaviours. In the(Hast last tuning few H¨ years, several works have later inversion problems and arules gglund, 2014),(Rodr´ later in other works. For instance, in (Vilanova et al., 2009) aa sequential tuning of feedforward controllers within later in other works. For instance, in (Vilanova et al., an efficient disturbance rejection. This idea has been used inversion problems (Hast and H¨ a gglund, 2014),(Rodr´ ıguez et al., 2013), (Rodr´ ıguez et al., 2014),(Guzm´ a n et al., 2009) sequential tuning of feedforward controllers within inversion problems (Hast and arules gglund, appeared proposing new to 2014),(Rodr´ accountanforetıguez these et al., 2013), (Rodr´ ıgueztuning et H¨ al., 2014),(Guzm´ al., 2009) 2009) sequential tuning ofisinstance, feedforward controllers within an control structure proposed. this aa sequential tuning controllers within in other Forof in In (Vilanova et the al., et al., 2013), 2013), (Rodr´ ıguezand et H¨ al., 2014),(Guzm´ n et etıguez al., later 2015). an IMC IMC controlworks. structure isfeedforward proposed. In this work, work, the et al., (Rodr´ ıguez et al., 2014),(Guzm´ aan al., inversion problems (Hast agglund, 2014),(Rodr´ 2015). an IMC control structure is proposed. In this work, the compensator is defined as the invertible part of the quoan IMC control structure is proposed. In this work, the 2009) a sequential tuning of feedforward controllers within 2015). is defined as the invertible part of the quo2015). et al., 2013), (Rodr´ıguez et al., 2014),(Guzm´ an et al., compensator compensator is as invertible of minimise the tient plus aa tunable filter is to compensator is defined defined as the the invertible part the quoquoan IMC structure iswhich proposed. Inpart this the tient pluscontrol tunable filter which is chosen chosen toofwork, minimise 2015). tient plus a tunable filter which is chosen to minimise the interaction between both controllers. Furthermore, tient plus a tunable filter which is chosen to minimise compensator is defined as the invertible part of the quothe interaction between both controllers. Furthermore,  This work has been partially funded by the following projects: the(Rodr´ interaction between both controllers. Furthermore,  This work has been partially funded by the following projects: in et al., aa robust methodology the interaction controllers. tient plusıguez a tunable filterboth which is design chosen Furthermore, to minimise  in (Rodr´ ıguez etbetween al., 2016) 2016) robust design methodology DPI2014-55932-C2-1-R DPI2017-84259-C2-1-R (financed by the  This work work has has been been and partially funded by by the the following following projects: in (Rodr´ ıguez et al., 2016) aa robust design methodology This partially funded projects: for simultaneously tuning both feedforward and DPI2014-55932-C2-1-R and DPI2017-84259-C2-1-R (financed by the in (Rodr´ ıguez et al., 2016) robust design methodology the interaction between both controllers. Furthermore, for simultaneously tuning both feedforward and feedback feedback Spanish Ministry of Science and Innovation and EU-(financed ERDF funds) DPI2014-55932-C2-1-R and DPI2017-84259-C2-1-R by  DPI2014-55932-C2-1-R and DPI2017-84259-C2-1-R (financed by the the for simultaneously tuning feedforward feedback This work has of been partially funded by and the EUfollowing Spanish Ministry Science and Innovation ERDFprojects: funds) for (Rodr´ simultaneously tuning both feedforward and feedback in ıguez et al., 2016)both a robust design and methodology Spanish Ministry Ministry of of Science Science and and Innovation Innovation and and EUEU- ERDF ERDF funds) funds) Spanish DPI2014-55932-C2-1-R and DPI2017-84259-C2-1-R (financed by the for simultaneously tuning both feedforward and feedback

Spanish Science and Innovation and EUERDF funds) Copyright © 2018, 2018 of IFAC 262Hosting by Elsevier Ltd. All rights reserved. 2405-8963Ministry © IFAC (International Federation of Automatic Control) Copyright © 2018 IFAC 262 Copyright © 2018 IFAC 262 Peer review under responsibility of International Federation of Automatic Copyright © 2018 IFAC 262Control. 10.1016/j.ifacol.2018.06.093 Copyright © 2018 IFAC 262

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controllers is presented. Disturbance rejection performance condition is expressed as a degradation band above a desired shape. Finally, in (Elso et al., 2013), Quantitative Feedback Theory (QFT) was used to design robust feedback and feedofrward controllers. New QFT bounds were obtained for the design stage and it was the first time that the feedback was linked to the existence of a feedforward controller in QFT. The aim of this paper is to propose a QFT-based robust solution for the feedforward control scheme presented in (Guzm´ an and H¨ agglund, 2011), where PI control is combined with feedforward compensators. The idea consists in moving the uncertainties effect to bounds in the QFT specifications and then designing a robust PI controller. The main contribution of this paper consists in modifying the original boundaries of the QFT methodology for the regulation problem, and to design a robust PI controller to account for the uncertainties. Notice that the feedforward compensator is not designed from a robust point view, since its effect is moved to the QFT specifications. The paper is organized as follows. In section 2, the classic feedforward control scheme is described. Section 3 is focused on the robustness problem analysis and the proposed control design approach. In section 4, a numerical example is presented to demonstrate the contributions described in section 4. Finally, section 5 conducts the conclusion of the work.

263

The models of the process Pu (s) and the disturbance Pd (s) are described by first-order transfer functions without time delay Pu (s) =

ku τu s + 1

(2)

Pd (s) =

kd τd s + 1

(3)

where the gains and the time constants in both cases, ku , τu for the process and kd , τd for the disturbance, are the parameters that bring the uncertainty to the system as described below. Notice that in this work, the freedelay case is considered as a first approach to the robust problem. However, the solution can easily be extended to the time delay case. From Figure 1, the closed loop transfer function between the output y and the disturbance signal d, Tdy (s), is given by the following equation Tdy (s) =

−F F (s)Pu (s) + Pd (s) 1 + C(s)Pu (s)

The proposed feedforward compensator is given by the following transfer function F F (s) = −Kf f

2. PRELIMINARIES

(4)

Tz s + 1 Tp s + 1

(5)

where Tz = τu , Tp = τd and Kf f is set as follows: The feedforward control scheme used in this work is shown in Figure 1. It consists of a feedback controller C(s), a process Pu (s), a setpoint signal r, a control signal u, and a process output y. The disturbance d, which is measurable, influences the feedback loop as shown in the figure. The transfer function between the load d and the output y is Pd (s). The feedforward compensator F F (s) feeds the disturbance d, and its output is added to the feedback control signal.

Kf f =

kd ku

(6)

in order to cancel the effect of the disturbance effect as suggested in (Guzm´an and H¨agglund, 2011). 3. ROBUSTNESS ANALYSIS AND DESIGN Such as described in the previous section, the feedforward element is commonly used as a lead-lag compensator described by (5). Usually, it is calculated from models in Eq. (2) and (3), assuming that there is no uncertainty (Guzm´an and H¨agglund, 2011). Notice that for the nominal case and if there are no inversion problems, the disturbance effect can be totally cancelled. In this paper, we assume uncertainties in gain and time constant in the process transfer functions Pu and Pd to perform a robustness analysis of the classical feedforward control design. Thus, now we have a set of plants given by the following equations Pu = {Pu : ku ∈ [ku,low , ku,high ], τu ∈ [τu,low , τu,high ]} (7)

Fig. 1. Classical feedforward control scheme. A PI controller is considered as feedback regulator C(s) such as used in (Guzm´ an and H¨ agglund, 2011):   1 C(s) = Kp 1 + Ti s

(1) 263

Pd = {Pd : kd ∈ [kd,low , kd,high ], τd ∈ [τd,low , τd,high ]} (8) Pu0 (s) =

ku0 +1

τu0 s

(9)

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kd0 τd0 s + 1

(10)

where Pu0 (s) ∈ Pu and Pd0 (s) ∈ Pd are the nominal plants.

The first issue to be analyzed is to observe how the use of the feedfoward control scheme and the presence of uncertainties affect the system specifications in the regulation control problem. Lets rewrite equation (4) as follows Tdy (s) =

Pdg (s) 1+C(s)Pu (s)

where Pdg is given by the following equation Pdg (s) = −F F (s)Pu (s) + Pd (s)

(11)

Notice that if F F = 0 we have the classical regulation problem. When Pu and Pd are uncertain and QFT is used to design the feedback regulator, the specification for the robust regulation problem is given as

|Tdy (jω)|dB

    Pdg (jω)   = 1 + C(jω)Pu (jω) 

dB

≤ δdB

(12)

or equivalently as

    1    1 + C(jω)Pu (jω) 

dB

≤ δdB − |Pdg (jω)|dB = γ(ω) (13)

for all plants Pu ∈ Pu , Pd ∈ Pd and where γ is the new specification bound. Therefore, it can be observed how the feedforward compensator and the uncertainties affect the bound in the specification problem. Thus, according to (12) and (13), there are two different ways to account for the robust control problem: • The first one would be to use (13) and follow the classical robust design with QFT. In this case, the specification bound γ depends on the uncertainties and the feedforward compensator presented in Pdg (jω). Then, the minimum value of γ for all plants Pu ∈ Pu and Pd ∈ Pd , and for all evaluated frequencies, must be considered as specification. Once this minimum bound is calculated, classical QFT is used to obtain the resulting controller. However, a very conservative solution will be obtained since the specification bound is computed for the worst possible case. • On the other hand, a second solution to this problem is to consider the specification (12) explicitly and to modify the boundary calculation in QFT. So, new boundaries are obtained considering the presence of the feedforward compensator and then a robust controller is obtained based on these new limits. This is the new solution proposed in this paper and presented in the following. First, the nominal plants Pu0 (s) and Pd0 (s) are selected, and the nominal feedforward compensator is calculated using 264

the rules proposed in (Guzm´an and H¨agglund, 2011). That is, Kf f is set as shown in (6), Tp = τd0 and Tz = τu0 Then, the algorithm proposed in (Moreno et al., 2006) to compute classical boundaries in QFT is modified in order to include a new kind of boundary that assures the satisfaction of specification in Eq. (12) when F F = 0. It is important to remember the concept of crossection defined in (Moreno et al., 2006). Fixed a phase in Nichols plane (NP) (and the frequency ω for which the boundary is being computed), the crossection is a function of the magnitude of the nominal open loop transfer function L0 = CPu0 , that provides a value of interest, the value of M ax|Tdy (jω)|dB in this paper. Obviously the location of L0 (jω) in the NP depends on the value of C(jω) because Pu0 is fixed. Figure 2 shows an example of crossection for the regulation problem for the cases with and without feedforward compensator and for ω = 1 rad/s and phase(L0 (jω)) = −100 degrees. In this figure, two different specifications are shown, for δdB = −20 and δdB = −10, respectively. It can be observed how for specifications where δdB < −20, both cases are equal for this frequency. See for instance the case where both solutions cut the specification of δdB = −20 at the value of 5.66 dB. However, for specifications with δdB ≥ −20 both solutions are different. This means that different boundaries will be obtained for both cases and thus different control design must be done. For instance, for the specification of δdB = −10, the case where F F = 0 does not cut the limit, while the case with F F = 0 cuts the specification bound at the value of -4.19 dB. Thus, a more restrictive solution is given for the case when the feedforward is included in the control scheme. This result indicates that the use of the feedforward compensator can affect the control problem negatively when modelling errors appear in the system. This fact can be better seen from Figure 3. This figure shows the boundaries for the specification of δdB = −10. As observed, the boundary is open when F F = 0 and closed when F F = 0, thus being the first one much more restrictive. However, it is interesting to see how for the zone around (-180o , 0 dB), the boundary for F F = 0 is smaller and thus less restric40

20

0

Max|Tdy (j )| (dB)

Pd0 (s) =

-20

-40

-60

-80 -60

-40

-20

0

20

40

60

|L0 (j )| (dB)

Fig. 2. Crossections for the regulation problem for F F = 0 (red) and for F F = 0 (blue) for ω = 1 rad/s and phase(L0 (jω)) = −100 degrees. Two speficiations are shown, for δdB = −20 (–) and δdB = −10 (-).

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10

265

40

20 0

0 -10

-|Pdg (j )|dB

-20

-40

dB

|L0 (j )| - dB

-20

-60

-30

-80 -40

-100 -50 -350

-300

-250

-200

-150

-100

-50

0

-120 -2 10

Phase(L0 (j )) - degrees

Fig. 3. Boundary comparisons for the regulation problem for F F = 0 (red) and for F F = 0 (blue) for ω = 1 rad/s and phase(L0 (jω)) = −100 degrees and for the specification δdB = −10.

-1

10

0

10

1

10

2

(rad/s)

Fig. 4. Right side of Eq. (13) with F F = 0 (*) and with F F = 0 (-) for nominal models Pu0 given by ku = 1 and τu = 10 and Pd0 given by kd = 3 and τd = 11.

tive. In any case, this is usually protected by the stability boundaries and thus it is not an important advantage.

40

20

0

( ) (dB)

Thus, it is concluded that QFT can be used as robust control design method when a feedforward control scheme is considered to account for the uncertainties in the process. However, the presence of the feedforward compensator affects to the calculation of the classical boundaries in QFT and new specifications must be fulfilled during the controller design stage such as shown in the following section with numerical examples.

10

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4. NUMERICAL EXAMPLE -60

This section presents a numerical example to demonstrate the contributions described in the previous section. Lets assume the following models for the process:

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10 2

(rad/s)

Pu (s) =

ku , ku ∈ [1, 10], τu ∈ [1, 10] τu s + 1

(14)

Pd (s) =

kd , kd ∈ [3, 7], τd ∈ [11, 15] τd s + 1

(15)

and

with nominal models Pu0 given by ku0 = 1 and τu0 = 10 and Pd0 given by kd0 = 3 and τd0 = 11. Such as commented in the previous section, there are two possible solutions to the problem based on considering the specifications as shown in (12) or (13). The following subsections show the analysis and results for both cases. 4.1 Classical solution One solution for the problem would be to use the specification of the problem according to (13) for the worst case and then use the classical stages for QFT with classical boundaries. Figure 4 shows different values for the γ function defined by Eq. (13) for all the plants in Pu and Pd . The curve represented by asterisks shows the case when F F = 0 265

Fig. 5. Right side of Eq. (13) with F F = 0 (*) and with F F = 0 (-) for nominal models Pu0 given by ku = 10 and τu = 1 and Pd0 given by kd = 3 and τd = 11. and the rest of the curves are all γ values when F F = 0. Thus, it is observed how for this nominal choice, there are many cases of the uncertainty where the presence of the feeforward compensator results in a more restrictive specification (a more aggressive controller will be required) since they are below the case when F F = 0. Therefore, this result indicates that in this case, the selection of the nominal plant cannot be done arbitrarily. It would be necessary to obtain the nominal plant that gives the maximum value of γ for all possible combinations. If we perform this for this example, the results presented in Figure 5 are obtained for the best case. That is, it is the selection of the nominal models that give the less restrictive γ values. This solution has been obtained for the nominal models Pu0 given by ku0 = 10 and τu0 = 1 and Pd0 given by kd0 = 3 and τd0 = 11. It can be seen that when F F = 0 all γ functions are greater than or equal to the γ function corresponding to F F = 0. Then, the function that must be used as specification in Eq. (13) is given by γ(ω) = δdB −|Pd (jω)|dB , which is the same specification as

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10 40

0

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|L0 (jω)| - dB

|P(jω)| (dB)

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when the feedforward term is not considered. If any other value for nominal used to compute F F is chosen, some γ functions will be located below the line with asterisks in the Fig. 5 as shown in the case of Figure 4. Thus, the specification would be more restrictive and a more demanding feedback controller will be necessary in order to assure the specifications. Hence, we can conclude that when the specification (13) is considered, the same specification as the case when F F = 0 can be used for the robust control problem. 4.2 New solution

Fig. 7. Nominal open-loop shaping and stability and disturbance on output rejection bounds taking the FF element into account. -40

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|T dy (jω)| (dB)

Fig. 6. Templates for ω ∈ {0.1, 1, 10, 100} rad/s

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In this case the specification (12) is considered and the new solution described in section 3 is used for this example.

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-110

Then, classical stability specifications in QFT and the new kind of disturbance rejection specifications including the feedforward element are taken into account. A phase margin greater than or equal to 45 degrees for the whole uncertainty set is used as stability specification. So, this specification on the closed loop transfer function is given by

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10 0

10 1

10 2

ω (rad/s)

Fig. 8. Tdy (jω) transfer functions and specification

4

(16)

On the other hand, a disturbance rejection specification given by δdB = −40dB. The set of design frequencies chosen is W = {0.1, 1, 10, 100} rad/s.

Figure 6 shows the templates for the set of design frequencies and for the set of plants from Eq. (14). Figure 7 shows the stability bounds, all of them are closed boundaries, and the disturbance rejection bounds, all open boundaries, for the same set of frequencies. A nominal open loop transfer function shaped fulfilling all the boundaries is drawn, given by a PI controller with Kp = 400 and Ti = 100. The parameters used for the F F element are Tz = 10, Tp = 11, and Kf f = 3. Figures 8 and 9 show that the control system satisfies all the specifications from a frequency domain point of view. 266

2

0

|L(jω)/(1+L(jω))| (dB)

   C(jω)Pu (jω)    ≤ 2.32dB |T (jω)| =  1 + C(jω)Pu (jω) 

-2

-4

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-8

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10

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10

0

10

1

ω (rad/s)

Fig. 9. T (jω) transfer functions and specification

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process output

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would result in very conservative results and the presence of the feedforward compensator would not give remarkable advantages. The second solution is based on modifying the boundaries of the regulation problem with QFT to include the presence of the feedforward controller. In this case, new boundaries were obtained and the QFT method was used to design a robust PI controller to account for uncertainties obtaining promising results.

0

REFERENCES

controller output

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Time (s)

Fig. 10. Time domain simulations for the proposed robust control design. A unitary step disturbance was included at time t = 3 seconds 0.2

process output

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controller output

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Fig. 11. Time domain simulations for nominal control design (Guzm´ an and H¨ agglund, 2011). A unitary step disturbance was included at time t = 3 seconds Finally, Figures 10 and 11 show the results in time domain for the proposed robust control approach presented in this paper and for the nominal control design presented in (Guzm´ an and H¨ agglund, 2011), respectively. It can be observed how for the proposed case, the disturbance is almost rejected beside the uncertainties with an important performance improvement with respect to the nominal case in Figure 11. Moreover, this result is achieved with very similar control effort as shown in the controller output signal of both figures. 5. CONCLUSIONS This paper has analyzed the classical feedforward control scheme for measured disturbances for the case when uncertainties are considered. QFT has been used as robust control design method. It was shown that the presence of the feedforward compensator changes the classical QFT specification for the regulation problem. This modification leads to two different solutions. The first one consists in using the same specification as the case when the feedforward is not considered, and classical QFT boundaries for the control design process are calculated. This approach 267

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