Robust controller design : recent emerging concepts for control of mechatronic systems

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JournaloftheFranklinInstitute357(2020)7818–7844

www.elsevier.com/locate/jfranklin

Robust controller design: Recent emerging concepts for control of mechatronic systems

Clara M. Ionescu

^a^,^b^,^c^,^∗

, Eva H. Dulf

^a

, Maria Ghita

^b^,^c

, Cristina I. Muresan

^a

aDepartmentofAutomation,TechnicalUniversityofClujNapoca,MemorandumuluiStreet28,Cluj400114, Romania

bDepartmentofElectromechanicalSystemsandmetalEngineering,ResearchgrouponDynamicalSystemsand Control(DySC),GhentUniversity,TechLaneSciencePark125,Ghent9052,Belgium

cEEDTcoregrouponDecisionandControlinFlandersMakeconsortium,TechLaneSciencePark131,Ghent 9052,Belgium

Received27March2019;receivedinrevisedform17March2020;accepted26May2020 Availableonline4June2020

Abstract

The recent industrialrevolution puts competitive requirements on most manufacturing and mechatronic processes.Someof theseareeconomicdriven, butmostofthemhave anintrinsic projectionon the loop performanceachieved in most of closed loops acrossthe various process layers. It turnsout that successfuloperationin a globalizationcontext canonlybeensured byrobust tuning ofcontroller parameterasaneffectivewaytodealwithcontinuouslychangingend-userspecsandrawproductprop- erties.Still,ease ofcommunicationinnon-specialisedprocessengineering vocabularymustbeensured at all times and ease of implementation on already existing platforms is preferred. Specifications as settling time, overshootand robustnesshave a directmeaning in termsof process output and remain mostpopularamongstprocessengineers.Anintuitivetuningprocedureforrobustnessisbasedonlinear systemtoolssuchasfrequencyresponseandbandlimitedspecificationsthereof.Loopshapingremainsa matureand easytousemethodology,althoughitstoolssuchasHinfremain intheshadowofclassical PIDcontrolforindustrialapplications.Recently,nexttothesepopularloopshapingmethods,newtools have emerged, i.e. fractional order controller tuning rules. The key feature of the latter group is an intrinsicrobustnessto variationsin thegain, timedelayand timeconstantvalues, hence ideallysuited for loop shaping purpose. In this paper, both methods are sketched and discussed in terms of their advantagesanddisadvantages.Areallifecontrolapplicationusedinmechatronicapplicationsillustrates

∗Corresponding authorat: Department of Automation, Technical University ofCluj Napoca, Memorandumului Street28,Cluj400114,Romania.

E-mailaddress:claramihaela.ionescu@ugent.be(C.M.Ionescu).

https://doi.org/10.1016/j.jfranklin.2020.05.046

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the proposedclaims.Theresultssupport theclaimthatfractionalorder controllersoutperforminterms of versatilitythe Hinf control, withoutlosing the generalityof conclusions.Thepaper pleads towards the useoftheemergingtoolsas theyare nowreadyforbroaderuse,whileprovidingthereaderwitha goodperspectiveof theirpotential.

Thisis anopen accessarticleunderthe CCBYlicense.(http://creativecommons.org/licenses/by/4.0/)

1. Introduction

With the recent industrial revolutionathand,agreat burdenis placedon theperformance of the global manufacturing process, due to continuously varying user-defined specification of the end-product. The implications are mainlyeconomic, but the driving force consists of better processoperation inthelowerlevel loops [1]. Thatis,agood lowerloopperformance facilitates high-end performance of supervisory loops where the economic cost is most rel- evant. Recent surveys have outlined emerging tools in the field of process control with the major reason for their success being the intrinsicrobustness they offerat ahigher degreeof freedom tooperate andtunecontroller parameters[2–4].

Inarecentsummaryontheindustrialrelevanceofvariouscontrolalgorithms[5],itisover- whelming the general agreement on top-three most prevalent inpractice: PID (proportional- integral-derivative) control,MPC(model predictive control) andsystemidentification.Atthe bottom of the same list are robust control mature algorithms such as Hinf and LQR (linear quadratic regulator), whichare based onloopshaping tuning methods.

At the core of the robust designin control systems are the frequency based loopshaping methods[6].Thisisofparticularinterestinsmartmanufacturing,whereasmechatronicappli- cations of human-robot collaborative systems play akey role [7,8].Hinf control is amature methodology andhas an established tradition incontrol theory [9].Robustness isa key feature in the optimizationprocedureused tofind the controller parameters. Givenuser defined specifications, translatedinfrequency responsedomain,thecontroller canbeoptimally tuned infrequencydomainwithresultingpolesandzeros.Othernotableapplicationsinmechatronic system controls are based on backstepping strategies [10–14], sliding mode [15] or combi- nations thereof withinformationtechnology tools[16].In miscellaneousapplications ofsuch basic elements of control systems as motors, the essential features are robustness and real time implementability[17–20].

To fulfil a high degree of robustness over a specified bandwidth in single controllers designed for process operation, the control design uses specifications interms of open loop

Table1

RealresponseperformancescomparisonbetweenPID,FO(fractionalorder)PIDandHinfcontrollers.

Brake Rise-Time Overshoot Settling-time5%

Mismatch Hinf FO PID Hinf FO PID Hinf FO PID

50% 1.1 s 1s 1.2s 13% 12.2% 37.4% 2s 2.1s 1.95s

25% 0.9s 0.9s 1.6s 10% 9.8% 43% 2s 2s 2.1s

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Fig.1. Illustrationoftheband-limitedrobustnessinopenloopcontrolledsystemsinfrequencydomain.

frequency response. Some of these specifications are illustrated in Fig. 1, introducing the principle of band-limited robustness(Ro) propertiesinthe Nyquist plane.

The controller parameters haveto be designed in such a wayas topresent iso-properties withinaspecifiedfrequencyrange,eitherforphase,eitherformagnitudeintervals.Intermsof fractionalcalculus,thisisachievedbycontinuousfractionexpansion,or,incontrolengineering terms,bypole-zerointerlacing.Thespacialdistributionofthepole-zerointerlacingwilldictate the slopeof the magnitude/phase infrequency domain[21–23].

Thegenericpropertyoftheresultedprocessandcontrollerclosedloopsystemisrobustness togainandphasevariationsintheprocessdynamics.Variousmethodsforpole-zerointerlacing exist,allbasedonfrequency domainspecifications.Acomprehensivereviewis givenin[21], with manifold applications in [24–26], including mechatronic systems [27–29]. This non- rationalcontrollerneedstobedeployedinreal-timesystemsanddiscrete-timeapproximations are described in[21,24–26] while an efficient, low-order,stable solution for discretizingany type of non-rationalfunction isgiven in[30].

Originally emergedfromfractional calculus,fractional ordercontrol(FOC)isincreasingly usedincontrolengineeringsincethegeneralizationofthePIDcontrollerusing anyrealorder wasintroducedbyPodlubny[Podlubny1999].PodlublynamedthisgeneralizedPIDcontroller asthefractionalordercontroller,PI^λD^μ whereλ∈(0,2)istheorderofintegrationandμ∈(0, 2)istheorderofdifferentiation.ThesuperiorityofthisgeneralizedPIDcontrollerincompari- sonwithitsclassicalintegerorderversionhasalsobeendemonstrated[21,Podlubny1999,32]. The design methods range from frequency domain tuning techniques[21,33,34],to methods using advanced optimization techniques [35–38] and to methods with time domain specifications [39]. Multiple applications of fractional ordercontrollers havebeen described while presenting the fractional calculus as a robust control design tool [29,40–42]. Most of the fractional order controller parameter tuning algorithms use an imposed performance criteria for the robustness of the designedcontrol system[24].

In thispaper, we provide ananswer for the reader interms of the following questions:

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•HowdoesarobustlytunedPIDcontrol,knownasthe’workinghorseforindustry’,perform against emerging fractional ordercontrol?.

•HowdoesHinfcontrol,asanexampleof’classical’robustcontrolmethod,performagainst emerging fractional ordercontrol?.

The presentpaper pleadstothe communityfor becomingaware ofthe powerfulemerging tools for robust control system design, while summarizing the main tuning methods used in practice for basis loops in mechatronic field. Emerging control design methods such as fractional order systems and controls are introduced from a practical perspective. Next, we provide ananalysis andcomparison bymeans of an ubiquitous industrial element, i.e. aDC motor withvarying loadconditions.This casestudy hasbeen selected beingoneof the core problems in control applications [43], but without losing the generality of conclusions. All designedcontrollersareexperimentallytestedonthemodularservosystemdesignedbyInteco [44,45].

The paper is structured as follows. The next section presents the commonly used tuning principlesfor PIDcontrolinindustry.Thethirdsectionprovides theprinciplesofrobustcon- trolbymeans ofHinf controldesign method,followedinthefourthsection bytheprinciples underlying fractional ordercontroldesign. Afifthsection provides the resultfor the real life setup, followedby asection containingrecommendations for the user.Aconclusion summa- rizes the mainidea of the paper.The Appendix containsessential informationfor thosewho aim embracingthe novel concepts.

2. Common PIDtuningrulesin industry

These methodspresented hereafter havebeen selected based ontheir industrial relevance.

They havebeen extensivelypublished inamanifoldof works,seefor instance[21,23,25,46–

48]. However, they have all been used in dynamic systems control with linear parameter varying conditions andthushavebeen testedagainstagreat dealof robustness.These works are suitable for the context of the paper, and they are summarized here for the sake of completeness. A comprehensivemethod overviewandcomparison canbe foundin [49]. 2.1. Autotuning methods

The indirect tuning methods are those which prior to control parameter tuning require identification of basic step response data to first order plus dead time (FOPDT) or second order plus dead time (SOPDT) [50,51]. By contrast, direct methods skip this identification step. Fig.2 provides thisconcept ingraphicalform.

Identification for the purpose of control is a demanding step in the process of model development. Nevertheless, latest reviews on their industrial relevance indicate that system identification playsan important role inpractice [5,52]. Identificationmethodsvary in terms of complexity,depending onthescopeof the targetmodel usage.Methods forsystemidenti- fication areavailable inbothtime-andfrequency-domains.Most commonlyused inindustry are thestep testresponse data,relaytestdataandsinusoidtestdata[1,53,54].Suitablemeth- ods forindustrial processcontrol areevent-basedalgorithms,where basicidentificationplays an important role before tuning controllerparameters[55–57].

Relay based methods have been some of the first used to automatic tune PID-type controllers asthey werepossibletoperformusinghydraulic andpneumaticinstrumentationcom-

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Fig.2. Schemeofdirectandindirecttuningmethodbasedonstepresponseapproximation.

monmly available in industry [47,48]. Recent revisions thereof allow a more robustness to short dataseries [58,59].

Some examples of auto-tuning methods using step response data are summarized in [21,23,46,60]. Of these, most commonly used are AMIGO (Approximate M-constrained In- tegral GainOptimization) [47]and SIMC(Skogestad Internal ModelController) [61].

The AMIGOmethod usesaFOPDT process approximation of the form:

Ke^−τ^s

1+Ts (1)

with K the gain, τ the time delay and T the time constant of the approximation to step response data of the process. It then uses a parallel PID configuration with the following tuning rules:

Kp= 1 K

0.2+0.45T τ

Ti =τ

0.4τ +0.8T τ+0.1T

Td =τ

0.5T 0.3τ +T

(2)

The SIMCmethod usesaSOPDT process approximation of the form:

Ke^−τ^s

(1+T₁s)(1+T₂s) (3)

with K the gain, τ the time delay and T1>T2 the time constants of the approximation to step response data of the process.It then usesaseries PID configuration withthe following tuning rules:

Kp= T1

2Kτ

(6)

Ti=min(T1,8τ )

Td =T2 (4)

2.2. Frequency responsetuning methods

Thefrequencyresponsefunction(FRF)ofadynamicalsystemisameasureofthemodulus andphaseof theoutputsignalasafunctionofaninputfrequency,relative tothe inputsignal appliedtothesystem.Toappropriatelycharacterizetheprocessdynamicsinagivenfrequency interval, the gathered information must cover the modulus, phase and their corresponding slopes withrespecttofrequency.ClassicalmethodsforestimatingFRFarebasedonavailable inputandoutput dataandapplication ofthe FastFourierTransform(FFT).Theseprocedures require multiple or persistent exciting tests with input signals of various frequencies, such that thefrequency response canbe estimated overthe required frequency range [62].

When the FRFisrequired around certainfrequency, it ispragmatic toreducethe number of necessary experiments, complexity and time-to-deliver by using an efficient and reliable algorithm.Theminimalrequiredinformationisthemodulus,thephaseandthecorresponding frequency response slopein/around aspecified frequency [43,49,63].

A largenumberof applications requirethe frequency responseslopeandseveraldetection algorithmsareavailable.Relay-basedmethodsareusedin[48],withtheidentificationmethod being automated and thus useful for autotuning applications. The slope of the frequency response modulusis estimatedatthegaincrossoverfrequency.Computationof thefrequency response slope has also found applications in the estimation of non-stationary sinusoidal parametersfor sinusoidswithlinearamplitude/frequencymodulation.An enhanced algorithm for frequency domain demodulation of spectral peaks is proposedin [64], whichdelivers an approximate maximumlikelihood estimate of the frequency slope.

In [65], Bode’s integrals are used to approximate frequency response slope of a system at a given frequency, without any model of the process. This information is then used to design aPIDcontroller forslopeadjustmentof the Nyquistdiagram andimprovethe overall closed-loop performance. The frequency response slope is also employed in the estimation of thegradient andthe Hessian of afrequency criterionin aniterative PID controller tuning method.

Emergingindustrialcontrollersof higherdegreeoffreedomarefractionalordercontrollers [3,4,24]. The phase slopeof the FRFhas been used inthe design of fractional orderPD/PI controller based on an auto-tuning method that requires knowledge of the process modulus, phase andphase slopeat animposed gaincrossoverfrequency [32].

A popular direct PID tuner is based on the common relay feedback test, with amplitude d.Thistest identifiesthe modulus atthe critical frequency,i.e. the pointof intersectionwith the negative real axis in the Nyquist plane. The period of oscillation Tc, and the amplitude of the oscillation a are thenused to tunethe PID-typecontrol parameters. The most known method is themodified Ziegler Nichols withtuning rules [47]:

Kc= 4d πa Kp=0.6Kc

Ti=0.5Tc

Td =0.125Tc (5)

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Fig.3. Theblockschemewithgainuncertainty.

Thiscommonlyusedinindustrytunerprovidesastandardrobustnessof0.5(onarange from 0 to 1) inthe Nyquist plane.

A method based on the same test was proposed in [66], with a degree of freedom to specify the desired phase margin of the loop. In thisway, the user mayalter the robustness providedby the tuner,andvaryits valueaccording totheprocess dynamics.Usingthe same relay feedback test and specified phase margin PM value (commonly selected between 40^◦ and75^◦), the tuningrules are:

Kc= 4d πa Kp=KccosPM

Ti=Tc

1+sinPM πcosPM

Td =0.25Ti (6)

NoticetherelationTi=4Td;thisisacommonlyusedchoicetosimplifythetuningprocedure;

it assumestwo identicalreal zerosinthePID controller.Someotherchoicesof thisratioand their analysis canbe found in[67].

3. Classical robustcontrol: Hinf controllerdesign

Amanifoldofmechanicalactuatorsandothermechatronicsystemsingeneralarecontrolled by robust control techniques, with prevalence of Hinf control algorithm. The reason is due to the nonlinear characteristic (staticor/and dynamic) that mechanical actuators exhibit and temperature dependent dynamics. The instrumentation is often described as being part of a linear parameter varying (LPV) system, requiring high degrees of robustness to maintain performance over wide operation range. An important aspect is then played by the model mismatchormisalignment.Thecontrolobjectiveistodesignrobustcontrollerstoensureboth stability andgoodperformances ofthe systemdespite disturbancesandmodeluncertainty. A popular solution is Hinf-optimal control, where the model used for design incorporates the possible uncertainties, illustratedhereafter.

Consider a simplified process transfer function H(s)=_T_s^K₊₁. Representing the gain uncertainty as an upper linear fractional transformation [6], results in the block scheme from Fig. 3. In this figure, K=KN(1+pkδk), Mk=

−pk 1 KN

, KN is the nominal gain value andpk, δk are thepossible gainperturbations.

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The corresponding state space equations are:

⎧⎪

⎨

⎪⎩

˙ x=−1

Tx− pk

T uk+K Tu yk=−pkuk+Ku y=x

(7)

resulting the packed matrix [6]: P=

⎡

⎣

⎡

⎣ A B₁ B₂ C1 D11 D12

C2 D21 D22

⎤

⎦

⎤

⎦=

⎡

⎣

⎡

⎣ −_T¹ −^p_T^k ^K_T 0 −pk K

1 0 0

⎤

⎦

⎤

⎦ (8)

ThecontrollerCisdesignedtoachieverobuststability,meaningthattheclosed-loopsystem remains internallystablefor allpossibleplantmodelsPandsatisfythe performancecriterion [6]:

Wp(I+PC)⁻¹ WuC(I+PC)⁻¹

∞<1 (9)

where Wp, Wu are weighting functions chosen to represent the frequency characteristics of performance andcontrol-effort constraints.

To design robust controllers is possible using computer aided design tools and dedicated control systems software such asMatlab [6].Dedicatedautomatictuning functionsare available within the Robust ControlToolbox inMatlab and examples are manifold. Despite this, industryrelevanceofrobustcontrolandHinfcontrolinparticularislowerthanofPIDcontrol, as reportedin[5].

4. Emerging robustcontrol: fractional ordercontroller design 4.1. Design principles

ThefractionalorderPI^λD^μ controllermaybedefinedasageneralizedclassicalPID,having an integrator and a differentiator of order λ and μ, respectively, where generally λ, μ∈(0, 2). The transferfunction of thisfractional ordercontroller is givenas:

lHFO−PID(s)=kp

1+ ki

s^λ +kds^μ

(10)

The tuning of the fractional order PID controller in Eq. (10) implies determining the five controller parameters, the proportional,integrative andderivative gains, kp,ki andkd,as well as the fractional orders of integration anddifferentiation, λ andμ. To determine these parameters, fivenonlinearequations areusedresultingfromfive performancecriteria[21,68]. The performance criteriausually refer tothe following:

•a gaincrossoverfrequency, ωgc,that ensures acertainclosed loopsettling time.The gain crossoverfrequencyrequirement istranslatedintothe modulusconditionasindicatednext:

Hol(jωgc)=1 (11) where Hol(s) isthe open looptransferfunction.

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•AphasemarginPMthatlimitsthe closedloopovershoot.Thisistranslatedintothe phase condition as given below:

∠Hol(jωgc)=π+PM (12)

•The iso-damping property, which ensures a constant overshoot value of the closed loop systemdespite gainvariations.Thisisspecifiedintermsoftheopenloopphasederivative:

d(∠Hol(jω)) dω

ω=ωgc

=0 (13)

•Goodoutput disturbancerejection.Thisisspecifiedas aconstraintonthe sensitivityfunc- tion as follows:

S(jω)= 1

1+HFO−PID(jω)HDC(jω)

≤B dB (14)

for all frequenciesω≤ωS rad/s.

•High frequency noise rejection. This is specified as a constraint on the complementary sensitivity functionas:

T(jω)= HFO−PID(jω)HDC(jω) 1+HFO−PID(jω)HDC(jω)

≤A dB (15) for all frequenciesω≥ωT rad/s.

The complex frequency domainform of the fractional orderPI^λD^μ controller is givenas:

HFO−PID(jω)=kp

1+kiω^−λ

cos^πλ₂ − jsin^πλ₂ +kdω^μ

cos^πμ₂ + jsin^πμ₂ (16)

The tuning of the fractional orderPI^λD^μ controller is now defined as the solution of the systemconsisting of complex nonlinear equations givenbyEqs. (11)–(15).

4.2. Optimization

To solve Eqs. (11)–(15) and determine the controller parameter values, the Matlab optimization toolboxisconsidered, using the fmincon() function.Thisfunctionattemptstosolve problems of the form: minXF(X) subject to some linear constraints specified as inequali- tiesA^∗X≤B andequalities Aeq∗X =Beq, nonlinear constraintsspecified also as inequalities C(X)≤0 and equalities Ceq(X)=0. The solution is bounded in the interval LB≤X≤UB, where X =[kpkikd λ μ]isavectorcontainingthefivecontrollerparametersandLBandUB are the minimum andmaximumadmissible valuesfor the controller parameters.

The modulus condition in Eq. (11) isfurther considered as the main function tobe min- imized, whereas the phase margin condition in Eq. (12) and the robustness condition in Eq. (13) are defined as nonlinear equality constraints. The complementary sensitivity and sensitivity functions in Eqs. (14) and (15), respectively, are defined as nonlinear inequality constraints. Since there are no linear constraints, the A, B, Aeq and Beq parameters in the fmincon() functionare not defined.

For the fmincon() function to be successful, a proper set of initial values X0= [0.01;3;0.5;0.3;0.5] needs to be specified. The fmincon() function uses the interior-point algorithm, which can handle large, sparse problems, as well as small, dense problems. The

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Fig.4. Themodularservosystemusedasacasestudy.

advantage of this algorithm is that it satisfies bounds at all iterations, and can also recover from Nota Numberor Infinity results.

5. Abasic element in mechatronic industry: theDCmotor

LPV mechatronicsystems are aclass of (non)linear systems whose properties depend on a set of time-varying parameters that are not known in advance. A wayto deal with model uncertainty is to probethe system dynamics invariousconditions to extract aset of models characterizing these possible situations. This is a tedious but necessary task when accurate performance isrequired[69].Ifperformance isnotthe primarygoal,butrathercompensation of unknowndisturbances,orunknowndynamics,thenthecontrol designparametersaretuned for slower control actions,with highrobustness.

LPV mechatronic systems are encountered in heavy duty machines (e.g.mining, agricul- ture, road engineering) and in automotive industry, mainly due to changing environmental conditions [70]. Space engineering applications have increased degrees of LPV dynamics, due to their size, relative position, remote location and complexity. The mechanical design of such systems may not be optimal from control point of view, but it is from a practical point of view (e.g. remote access, testing, validation protocols, etc) and it poses additional challenges forcontrol[71].Moreover,manytypesof motorssuch asbrushless, inductionand DC motorsare part of other complex systems, e.g. windmill parks, unmanned systems and parts of manufacturing andproduction systems [72].

TheDCmotorisaversatileexecutionelementwhichrequiresacertaindegreeofrobustness due to varying operation conditions, load changes and other varying variables linked to it, making it alinear parameter varying system. Wehaveexplicitly chosen thisexample dueto its simple dynamic operation. Consequently, differences among the Hinf and FOC methods will besolelyduetothe controlstrategies andtheirintrinsic properties,andnot duetosome exotic process dynamics.

The experimental unit consists inthe modularservo systemdesigned by Inteco [44]used in the particular configuration indicated in Fig. 4. The plant is composedof a tachogenera-

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tor (used to measure the rotational speed), inertia load, backlash, incremental encoder, and gearbox withoutput disk.

The mathematicalmodel of themodularservosystemhasbeen determinedusing classical experimental identificationmethod as being:

P_ω(s)= ω(s) u(s) = k

Ts+1 (17)

where ω is the angular velocity of the rotor, and k=158 and T =0.95 s are the motor nominal gain and time constant, respectively. This model is used to design the robust controller using Hinf rules and the fractional order PID controller. For sake of comparison, a classical integer order PID controller has also been tuned using computer aided tools from Matlab, providing the parameters:Kp=0.02, Ti=2 andTd =0.5 in the textbook structure.

All controllershave aspecified phase marginof PM =60^◦.

ThenextstepconsistsinthetuningoftheHinfrobustcontrollerintermsofEq.(9)forthe processdescribedinEq. (8)usingthetuningprocedurepresentedinSection3.Theconsidered gain uncertainty is k=kN(1+pkδk), with kN =158, pk=0.25 and −1≤δk≤1. Further specifications are the gain crossover frequency ωgc = 3 rad/s for settling time and a phase margin ofPM =60^◦,whilethe control effortislimited to|u(t)|≤1 innormalizedunits. The Robust ControlToolbox from Matlab [73]delivers the controller:

C(s)= 1.821s+14.22

s²+250.7s+277.8 (18)

TheBodediagramof theopenloopispresentedinFig.5,highlightingtheimposedvalues of performance measures.

ThenextstepconsistsinthetuningofafractionalorderPI^λD^μcontrollerfortheDCmotor using the tuning proceduredetailed in Section4. The imposedgaincrossover frequency and thephasemarginarethe sameas fortheHinf controller.Forthesensitivityfunction,abound B=−20 dB is considered, for all frequencies ω≤0.001 rad/s while for the complementary sensitivity function, a limit A=−20 dB is considered, for all frequencies ω≥100 rad/s.

The resulting controller parameters are kp=0.007, ki=5.5882, kd =0.078, λ=0.78 and μ=0.9,inthe form:

HFO−PID(s)=0.007

1+5.5882

s⁰^.⁷⁸ +0.078s^0.⁹

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Toimplementthe fractionalorderintegratoranddifferentiator,we usetheefficientmethod for approximation described in Appendix [30]. The approximation parameters have a low frequency bound of 10⁻² and a high frequency bound of 10² rad/s, as well as an order of approximation N =5. To test the last two remaining performance specifications, the Bode diagrams for the sensitivity function andcomplementary sensitivity function need to be an- alyzed. The sensitivity andcomplementary functions are given inFig. 6, confirmingthat all tuning conditions are met.

Forthe sakeofcomparison,aclassicalintegerorderPIDcontrollerhasalsobeendesigned usingthesamephasemarginspecification.Theresultsofthecomparisonbetweentheclassical PID control and fractional ordercontrol are depicted in the set of Figs. 7–10. The test has been performed over a wide range of the operating interval of the DC motor, for various conditions(i.e.changingthemagneticbrakeat50%andat25%,affectingbothgainandtime constantvaluesofthesystem).TheFO-PIDoutperformsthePIDcontrollerinallcases,while exhibiting similaror less control effort.

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Fig.5. TheBodediagramoftheopenloopwithHinfrobustcontroller.

ThecomparisonoftheexperimentalresultswiththetwodesignedrobustHinfandfractional order - controllers reveals the same output performances, as indicated in Fig. 11 and with similar control effort. Robustness tests have been performed at different speed values for setpoint, while maintaining the samebrake position (i.e. at25%), illustrated inFigs. 12and 13.

From theexperimentalvalidationit followsthatFO-PID controloutperformsclassicalPID control andhasasimilarperformance asthe Hinf control.The detailsaresummarized inthe following table.

6. Recommendations

Asasuggestedlist ofrecommendations forindustrial useof therobust methodspresented in thispaper, we propose the following:

•If automatictuning from industrial devices isavailable, usethe standardPID tuning rules available at hand - however, take notice to retune the PID parameters periodically for avoiding performance deterioration; thishas been addressedin [74].

•If robustness is not an issue, but rather fast control (not accurate, but in large tolerance intervals) then classicalPIDcontrol may be used;

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Fig.6. Bodediagramsofthesensitivityfunction(left)andthecomplementarysensitivityfunction(right).

Fig.7. OutputspeedcomparisonbetweenPIDandFO-PIDwithabrakeof50%.

•If robustness isimportant, if available, use standardtools from control-related instrumentation industrial architecture.

•If the above isnot available, considerusing control specificanalytical tools, such as gain margin, phasemargin, specifications basedtuning rules.

•For accurate robustness specification in frequency domain (e.g. bandwidth, etc) consider loopshaping tools, e.g.classicalrobust control design.

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Fig.8. ControleffortcharacteristicbetweenPIDandFO-PIDwithabrakeof50%.

Fig.9. OutputspeedcomparisonbetweenPIDandFO-PIDwithabrakeof25%.

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Fig.10. ControleffortcharacteristicbetweenPIDandFO-PIDwithabrakeof25%.

Fig.11. OutputsignalsusingfractionalorderandrobustHinfcontrollerinnominalconditions(i.e.basedonnominal processmodelparameters).

•Ifthe systemexhibits significantLPV dynamics,considerfractionalordercontrolfor both optimizationinterms ofoutputperformance andcontroleffort(e.g.inloops whereenergy is costly, or the fuel consumption isrelatedto varying operating conditions).

•Ifthe LPVdynamics are extreme,considerschedulingasetof fractional ordercontrollers for stringent performance requirements.

These items are by no means exhaustive, but they offera good support for the beginner user or plantoperator interms of tuning control loops for industry.

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Fig.12. Stepresponseofthemodularservosystemconsidering100radreferenceangle.

Fig.13. Stepresponseofthemodularservosystemconsidering40radreferenceangle.

7. Conclusions

Essentially, thispaper pleadsfornewemergingconcepts inrobustcontrol design.Westart by providing a revisited summaryof the robust control methodsat hand andmost prevalent design methodsforindustrial processcontrol.Amature robust controldesignmethod- Hinf, and anewly emerging one-fractional order PIDcontrol, are described.

An example ofgreat simplicityandprevalenceinmechatronicapplicationsandbasicloop control inmanifoldofproductionsystemsisused:theDCmotor.Acomparisonisperformed, illustratingthegreatpotentialofthefractionalordercontrolinrobustdesign.Theexperimental resultssuggestthatonecanchoose thefractionalorderPI^λD^μ controllertomaintainthesame robustness of the control system as with the classical robust controllers, such as the Hinf controller. Both methods are frequency based design tools and their intrinsic complexity is similar. However, the fractional order control is a direct generalization of the classical PID control, a very popular control in industry, hence it holds the potential to become easier accepted by end-users thanHinf control.

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Acknowledgments

This research was supported by the Janos Bolyai research fellowship of the Hungarian Academy of Sciences (Dulf).

CristinaI.MuresanisfinancedbyagrantoftheRomanianNationalAuthorityforScientific ResearchandInnovation,CNCS/CCCDI-UEFISCDI,projectnumberPN-III-P1-1.1-TE-2016- 1396,TE 65/2018.”

This workis financially supported by aSpecial Research Fund of Ghent University, MI- MOPREC, BOFSTG 020-18, 01J01619(Ionescu) and doctoralgrant nr 01D15919(Ghita).

Thisworkwas in partsupported by aSTSM Grantfrom COST action15225.

Appendix A.On Fractional Order Systems andControls

In this section we have selected a basic set of information to support the readers who aim to embrace the concepts of fractional order systems and control. The information is a summaryfrom the followingtextbooks andarticles: [21,24, 25,30,Podlubny1999].

A1. Preliminaries

The fractional order system models are a generalization of the LTI system models. For instance, in the case of continuous time models, we have the transfer function givenin the form:

G(s)= Y(s)

U(s)= bms^β^m +bm−1s^β^m−1+· · · +b₀s^β⁰

ans^αⁿ+an−1s^αⁿ⁻¹+· · · +a₀s^α⁰ (A.1) Forcommensurate-order system,the continuous-time transferfunction isgiven by:

G(s)= m

k=0

bk(s^α)^k n

k=0

ak(s^α)^k

(A.2)

A2. Stability

Stabilityanalysis of fractional-ordersystems studiesthe solutionsof the differentialequa- tions [75]. Alternatively, onemay study the transferfunction of the system(A.1).Consider:

ans^αⁿ+an−1s^αⁿ⁻¹+· · · +a0s^α⁰ (A.3) with αi∈R⁺ a multi-valued function of the complex variable s whose domain can be seen as aRiemann surface of a number of sheets. This is finite onlyin the case of ∀i,αi∈Q⁺, where the principal sheetis defined by−π <arg(s)<π.

For αi∈Q⁺, i.e. α=1/q, q a positive integer, the q sheets of the Riemann surface are determined by:

s=|s|e^j^φ, (2k+1)π <φ <(2k+3)π, K=−1,0,...,q−2 (A.4)

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Fig.A1.TextbookexampleofaRiemannsurfaceforω=s¹^/³.

Fig.A2.Correspondingω-planeregionsfortheRiemannsurfaceforω=s¹^/³.

The case of k=−1 represents the principal sheet. For the mapping ω=s^α, these sheets become regionsinthe plane ω defined by:

ω=|ω|e^j^θ, α(2k+1)π <θ <α(2k+3)π (A.5)

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The completemapping is illustratedin Figs. A.1andA.2 for the caseof ω=s¹^/³.These three sheetscorrespond to:

k=

⎧⎨

⎩

−1 for −π <arg(s)<π principalsheet 0 for π <arg(s)<3π second sheet 1 for 3π <arg(s)<5π third sheet Hence, anequation of the type:

ans^αⁿ+an−1s^αⁿ⁻¹+· · · +a0s^α⁰=0 (A.6) which in general is not a polynomial, will have an infinite number of roots, among which onlyafinitenumberof rootswillbe ontheprincipal sheetof theRiemann surface.It canbe said that the roots which are in the secondary sheets of the Riemann surface are related to solutionsthatarealwaysmonotonicallydecreasingfunctionsandonlytherootsthatareinthe principal sheetofthe Riemannsurfaceare responsiblefor differentdynamics; e.g.i)damped oscillation,ii)oscillationofconstantamplitude,oriii)oscillationofincreasingamplitudewith monotonic growth.

Afractional-ordersystemwithanirrational-ordertransferfunctionG(s)=_Q^P⁽₍^s_s⁾₎ is bounded- input bounded-output(BIBO)stable ifand onlyif thefollowing condition is fulfilled:

∃M : |G(s)|≤M, ∀s (s)≥0 (A.7)

Theprevious conditionissatisfiedifallthe rootsofQ(s)=0 inthe principalRiemannsheet havenegative real parts.

Forthecaseofcommensurate-ordersystems,whosecharacteristicequation isapolynomial of the complex variableλ=s^α,the stability condition is expressedas:

|arg(λi)|>απ

2 (A.8)

where λi are the roots of the characteristic polynomial in λ. Forthe particular case of α= 1, the well known stability condition for linear time-invariant systems of integer order is recovered:

|arg(λi)|>π

2 ∀λiQ(λi) (A.9)

Consider asystemdefined bythe transfer function:

G(s)= 1

ansⁿ^α+an−1s⁽ⁿ⁻¹⁾^α+· · · +a1s^α+a0

(A.10)

where α= ¹_q,withq,n∈Z⁺,ak∈R.Introducingthemapping λ=s^α toobtain thefunction G(λ)andapplyingthecondition(A.8),thestabilityofthesystemcanbestudiedbyevaluating the function G(λ)along the curve Γ defined inthe λ-plane:

Γ =Γ1∪Γ2∪Γ3 (A.11)

with Γ1 : λ

arg(λ) =−απ

2, |λ|∈[0,∞), Γ2 : λ= lim

R→∞R^j^φ, φ∈

−απ 2,απ

2 ,

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Fig.A3. TheΓ evaluationcontourintheλ-plane.

Γ3: λ

arg(λ) =απ

2, |λ|∈[∞,0) and illustratedinFig. A.3:

A3. Time and frequency domain response

The generalequation for theresponse ofafractional-ordersysteminthetimedomaincan be determined by using the analytical methods. The response depends on the roots of the characteristic equation, having six differentcases:

1.There are no roots in the Riemann principal sheet. In this case the response will be a monotonically decreasing function.

2.Rootsare locatedin(s)<0,(s)=0.In thiscasetheresponse willbeas previouscase.

3.Roots are located in (s)<0, (s)=0; the response will be a function with damped oscillations.

4.Roots are located in (s)=0, (s)=0; the response will be afunction withoscillations of constantamplitude.

5.Rootsare locatedin(s)>0,(s)=0;the responsewillbeafunctionwithoscillationsof increasing amplitude

6.Roots are located in (s)>0, (s)=0; the response will be a monotonically increasing function.

In theparticular caseof commensurate-ordersystems, theimpulseresponse canbewritten as follows:

L⁻¹{H(λ)}=L⁻¹

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩ m

k=0

akλ^k n k=0

bkλ^k

⎫⎪

⎪⎪

⎪⎬

⎪⎪

⎭

=L⁻¹ _n

k=0

rk

λ−λk

(A.12)

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withλ=s^α.

Taking into account the generalequation:

L⁻¹ s^α−β

s^α−λk

=t^β−1E_α,β(λkt^α) (A.13)

where Eα,β isthe MittagLeffler functionin twoparametersdefined as:

Eα,β(z)= ∞

k=0

z^k

Γ (αk+β), (α),(β)>0. (A.14)

it follows that the impulse response, g(t), can be obtained by setting α=β inthe previous equation:

g(t)= n

k=0

rkt^α−1Eα,α(λkt^α). (A.15)

The equivalent step response:

y(t)=L⁻¹ _n

k=0

rks⁻¹ s^α−λk

(A.16) can beobtained by settingα=β−1 inEq. (A.15):

y(t)= n k=0

rkt^αEα,α+1(λkt^α). (A.17)

The shape of theseresponses will be:

•monotonically decreasing if|arg(λk)|≥ ^απ₂ ;

•oscillatory withdecreasing amplitudeif ^απ₂ <|arg(λk)|<απ;

•oscillatory withconstantamplitude if |arg(λk)|= ^απ₂ ;

•oscillatory withincreasing amplitude if|arg(λk)|<^απ₂ and

|arg(λk)|=0;

•monotonically increasing if |arg(λk)|=0.

In general, the frequency response has to be obtained by the direct evaluation of the irrational-order transferfunction of the fractional-order system along the imaginary axis for s= jω, ω∈(0, ∞). However, for the commensurate order systems we can obtain the frequency response by the addition of the individual contributions of the terms of order α resulting from the factorization of the function:

G(s)= P(s^α) Q(s^α)=

m

k=0

(s^α+zk) n

k=0

(s^α+λk) ,

zk : P(zk)=0, λk : Q(λk)=0, zk=λk

(A.18)

For each of these terms, referred to as (s^α+γ )^±¹, the magnitude curve will have a slope from 0 to ±α20 dB/decfor higher frequencies,andthephase plot willgo from0 to±απ

2.

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A4. Implementation aspects

A stable and efficientmethod for direct approximation of fractional order systems in the form of discrete-time rational transfer functions or non-rational transfer functions (NRTF), consists of four steps, summarized from [30]. The corresponding Matlab implementation is also givenin the samereference.

Step1) Discretize the fractional-orderLaplace operator usinga suitable generating func- tion,that is equal toan interpolation betweenEuler and Tustindiscretizationrules:

sNRTF(z⁻¹)= 1+α T

1−z⁻¹

1+αz⁻¹ (A.19)

with α∈[0÷1] and T the sampling period. When α=0, we obtain the classical Euler dis- cretization rules, while when α=1 we obtain the Tustin rules.The choice of the parameter αhasaweightingeffectonthe frequencyresponse,penalizingtheerrorsonthe magnitudeor onthephase:largervaluedecreasesthephaseerrorneartheNyquistfrequency,whilealower value ensuresalowermagnitudeerror.Hence,thisparametermaybetuned foratrade-offin performance athighfrequencies amongthe magnitude andthe phase of the process.

The first stepproducesadiscrete timefractional ordersystem,G(z⁻¹),replacingsby the form in Eq. (A.19), with given α weighting parameter and maximum frequency ωh values.

The maximumsampling period is selected according tothe Nyquist sampling theorem. The rational discrete-time approximation of the general fractional ordersystem is determined in the frequency rangeω∈(0, ωh).

Step 2)Calculate the frequency responseof the discrete-time fractional ordersystem. The frequency response is computedbased on the classical relationbetween the continuoustime anddiscrete timedomains,withz=e^sT,withs= jω,where ωisavector spacedequally in the frequency interval:

ω= 2π NsT

0 1 · · · Ns

2

(A.20)

withNs the totalnumberof samples(the higherthisvalue,the betteristhe approximationat lower frequencies).

We obtain a vector of frequency response values of the fractional order discrete-time transfer function:

GNRTF(e⁻^j^ω^T) (A.21)

Step 3) Calculate the impulse responseof the discrete-time fractional order system.This step employs the inverse Fast FourierTransform (FFT)algorithm, which convertsthe previ- ously computed frequency domainresponse into atimedomainresponse, atdiscrete instants [0,T,2T,...,(NS−1)T]. The result of this step consists of a vector with NS impulse response value:

g[n]= 1 Ns

Ns−1

k=0

G[k]e^j^2π^N^s^nk n=0,1,· · ·,NS−1 (A.22)

where G[k] isthe frequency response of the original GNRTF(s).

Step4)Determinearationaldiscrete-timetransferfunctionthatproducesasimilarimpulse response as obtained from the inverse FFT. To determine the rational discrete-time transfer

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Fig.A4.IntegralWindup.

Fig.A5. Controlschemewithsaturationblock.

function we employ simple signal modelling techniques such as the Steiglitz-McBride, also available as Matlab built-infunction. The rationaldiscrete timetransferfunction:

GRTF(z⁻¹)= c₀+c₁z⁻¹+· · · +cNz⁻^N

d₀+d₁z⁻¹+· · · +dNz⁻^N (A.23) where ci anddi are thecoefficient determinedon thedesired orderNof theresulted approximation, with i=1,2,· · · ,N. The accuracy of the approximation increases with N. Lower valuesfor αimprovethe approximationofthe magnitudecurve,whilehighervaluesimprove the approximation of the phase curve.

A5. Anti-Reset windup

Integratorwindupor resetwindup,referstotheclosedloopsituation wherealargechange insetpointoccurs(say,e.g.apositivechange) andthe integralterm accumulatesasignificant errorduringtherise(windup).Often,thisresultsinovershootingandmonotonicallyincreasing dynamics as thisaccumulated error is unbound (offset by errorsin the other direction).The concept is illustratedin Fig.A.4.

Integral windup occurs in presence of physical system limitations, i.e. process input sat- uration: the input of the process being limited atthe top or bottom of its scale, makingthe errorboundtoanon-zerovalue.In thiscase,acommonanti-windupsolutionistheintegrator being turnedoff forperiods of timeuntil the responsefallsnaturally backinto an acceptable range [1].

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Fig.A6.Schemeforback-calculationanti-windup.

Fig.A7. Illustratedexampleofavoidingintegratorwind-upthankstoback-calculation.

Among the additional functionalities that a FO-PID controller should possess, the anti- windup plays a major role. In fact, it is well known that the performance of any PID-type controller canbe severely limited inpractical casesby thepresence of actuator saturationas depicted inFig.A.5.

Consider the control schemeof Fig.A.5,where u is thecontroller output, u is theactual control effort,y isthe processoutput, risthe set-point referencevalue,ande=r−yis the control error. The integrator windup mainly occurs when a step is applied to the reference set-point signalratherthantothemanipulatedvariable(i.e.for aloaddisturbance).Thereare differentanti-windup techniquesthat aretypicallyemployedforinteger-orderPIDcontrollers, which can be also applied tofractional-order controllers.In particular, the best performance is obtained withthe back-calculation technique.

Back-calculation consistsof recomputingthe integraltermonce thecontrollersaturates.In particular, the integral value is reduced by feeding back the difference of the saturated and unsaturated control signal, as shown inFig. A.6 where Tt is an additional parameter called tracking timeconstant.The valueofTt determinesthe rateatwhichtheintegral term isreset