Stability of predictive control in job shop system with reconfigurable machine tools for capacity adjustment

Zhang, Qiang; Freitag, Michael; Pannek, Jürgen Article Stability of predictive control in job shop system with reconfigurable machine tools for capacity adjustment Logistics Research Provided in Cooperation with: Bundesvereinigung Logistik (BVL) e.V., Bremen Suggested Citation: Zhang, Qiang; Freitag, Michael; Pannek, Jürgen (2020) : Stability of predictive control in job shop system with reconfigurable machine tools for capacity adjustment, Logistics Research, ISSN 1865-0368, Bundesvereinigung Logistik (BVL), Bremen, Vol. 12, Iss. 1, pp. 1-12, https://doi.org/10.23773/2019_3 This Version is available at: https://hdl.handle.net/10419/297170 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Received:06 June 2018 /Accepted:08 March2019 /Publishedonline:08 April2019 ©TheAuthor(s)2018 This articleis publishedwith Open Access at www.bvl.de/lore Stabilityof Predictive Controlin JobShop System with ReconfigurableMachineToolsforCapacity Adjustment Q. Zhang, M. Freitag, J. Pannek ABSTRACT Dueto changesin individual demand, manufacturing processeshave become more complexanddynamic. To cope with respective fluctuationsas well as machine breakdowns,capacity adjustment is one of themajor effective measures.Insteadof labor-oriented methods, we proposeamachinery-basedapproach utilizing the new type of reconfigurablemachinetoolsfor adjustingcapacities within a jobshopsystem.To economically maintain desiredwork in processlevels forallworkstations,we impose a modelpredictive controlscheme. Forthis method we show stabilityof theclosed-loop foranyfeasible initial state of thejob shop system using a terminal conditionargument.For apracticalapplication, thisreduces thecomputation of asuitable prediction horizonto controllabilityof theinitial state. To illustrate theeffectivenessand plug-and-playavailability of theproposed method,we analyzeanumericalsimulation of afour workstation jobshopsystem andcompareit to astate-of-the-art method. This articleis theextension of aconference paper entitled ”PredictiveControlof aJob Shop System with RMTs using EquilibriumTerminal Constraints” presented at the6thInternational Conference on Dynamicsin Logistics(LDIC2018). Keywords: Reconfigurable machinetool · Capacit y adjust men t ·Modelpredictivecontrol · Stabi lit y 1INTRODUCTION Nowadays,consumersdemand individualized products in smallquantities with short delivery time. Consequently,manufacturersareconfronted withthechallengeto reactto demand andmarket fluctuationsquickly, efficiently andeffectively. This tendencyrendersmanufacturingprocessesto be more complexanddynamic. In general, such processes aresubject to externaldisturbances,e.g. rush orders, andinternalfactors, e.g. machinebreakdown or intendedadjustmentsby system design.Thecurrent manufacturingparadigms, whichaim at producing products at lowcost andhigh qualities, cannot deal with this requirementin asatisfactory manner.To deal with theresultingperformancedegradationandto achieve agood shopfloor performance, capacity adjustment is one of themajoreffectivetools[1]. Here, even small modifications during ahigh load period mayimprove performance significantly [2]. Typically, capacity adjustment is done by purchasing new equipment, employingtemporary workers, extending workingtimes andso forth. Theseoptions offerflexibility,but are not long-termsustainableand expensiveespeciallyin thewesterncountrieswherethe laborcost is high [3]. Intheperspectiveof sustainability, reconfigurablemachinesystems(RMS)mayfill this gap. Similarto the mentioned alternatives, theseare flexibleby construction butalso allowto adaptcapacity andfunctionalitywithin acertainrange. Thekey LogisticsResearch (2019) 12:3 DOI_10.23773/2019_3 QiangZhang Universityof Bremen, International Graduate School forDynamics in Logistics, Faculty of Production Engineering, Badgasteiner Straße 1, 28359Bremen,Germany E-mail:[email protected] MichaelFreitag BIBA –Bremer Institut fürProduktionund Logistik GmbHat theUniversity of Bremen, Hochschulring20,28359Bremen,Germany Universityof Bremen, Faculty of Production Engineering, Badgasteiner Straße 1, 28359Bremen,Germany Jürgen Pannek BIBA –Bremer Institut fürProduktionund Logistik GmbHat theUniversity of Bremen, Hochschulring20,28359Bremen,Germany Universityof Bremen, Research Cluster LogDynamics andFaculty of Production Engineering, Badgasteiner Straße 1, 28359Bremen,Germany 2 simulation(DES). However, this approachrequires a rather high modeling effort andis limitedin thetime frame. On theother hand, acontinuous time modeling andsimulation method mayprovide an additional research possibilityon manufacturingprocesscontrol [13, 14,15]. This method hasbeen evaluated via a state space model setting andfurther compared with DES. Theresultsindicated that therewas a subtle difference in terms of mean andvariationof WIPandlead time [16]. Independentfrom theapproach,stabilityof the closed-loopis of utmost importance.Here,process stabilityrefers to performance indicators (e.g.WIP) remainingboundedas converging to desiredvalues or an acceptable stabilityregion.In [17], a comparison regardinganalysis of stability regionswasconducted from both perspectives (macroscopic andmicroscopic), i.e. continuous modeling by mathematical theory and simulation resultsfrom DES. Theauthorsindicated that such an approximation made by amathematical modelis suitable andeffectiveforstabilityanalysis. This method allows to determine controlparametersto ensurestabilityof aproduction network faster than a repeated trial anderrorapproach. However, only steady state stabilitywasdiscussed,thetracking control problemwas not takeninto account. In this paper,we consider jobshopsystemsandfirst follow theapproachfrom [14] to directlycontrolthe WIPlevelby adapting thenumber of RMTs within theworkstations separately.To balancecapacities and loadsin thesystem,we then impose amodelpredictive control(MPC) scheme, whichis widely applied for mechanical or chemical systems[18],inventory management in supply chain[19]andhasgrown mature overthelast decades[20].As the method allows dealingwith constraintsexplicitly whilestudying an finitehorizonoptimization problemiterativelyand beinginherently robust,it is readilyapplicable to assign RMTs andachieve a good shop floor performancein thepresence of demand fluctuations. Forthis method, we showstabilityof theMPCclosed-loopsystem by imposing equilibriumterminalconditions.Moreover, we illustrate theeffectivenessof ourapproach by a numericalsimulation subject to arangeof orderrelease rates andlimited availablecapacity. Theremainderof this paper is organizedas follows: Theproblem definition is givenin Section2. Thereafter, thebasicMPCalgorithmwith equilibriumterminal conditions will be introduced in Section3. In Section 4, an illustrative exampleof ajobshopsystem with RMTs andDMTs is investigated andsimulationresults arepresented.Last,conclusion andfuture research directions arepresented in Section5. Notation: Throughout this work we denote thenatural numbersincludingzero by andthenonnegativereals by .TheEuclideannormis denoted by .Forany vector , , represents the 2-norm. The 2-normof matrix is denoted as ,whereis themaximum characteristics of RMSinclude modularity,scalability, convertibility,customization, anddiagnosability [4]. Such systemsshow significant impact on sustainable manufacturingto improve theresponsivenessto market changeswhileremaining cost-effective [5]. On the downside, theresultingplanning problemis of mixed integernature,whichcallsfor new methodsto costeffectively utilizeflexibility,capacity scalability and functionalityof RMSin thedynamicmanufacturing systems[6]. Themain essential componentrenderingRMS successfulis thereconfigurable machinetool (RMT). Thesemachinetoolsare modularly designed fora customizedrangeof operationrequirements,combining theadvantages of highproductivity of dedicated machinetools(DMT)with high flexibility of flexible machinetools(FMT). Also, thesetoolsaredesigned in accordance with theconceptof sustainability,such as improving flexibility, shortingdelivery times,reducing material consumptions,andenhancing responsiveness in thepresence of demand fluctuation [7]. RMTs maybe used to balancecapacities andloads. Yet, such an adaptation requires ashort reconfiguration time to adjust capacity andfunctionality[8].Relying on production capabilityof RMTs,theauthors studied asingle productline to satisfydemand changes. Theproductioncapacity wasincreased or decreased throughadding or removing auxiliary modules for performingdifferentoperations.In [5], purchasing new RMTs wasused to increase capacitywhileminimizing reconfiguration cost andcapital investment cost.To exploitthebest configuration, a mixed integerlinear programming problem wasformulated.Twocases concerning cost management were presented to demonstrate theefficiencyof theproposed method. Taking into accountfrequent reconfigurationcost,the responsiveness of RMTwas measured and evaluatedby operationalcapabilityandmachinereconfigurability metrics[9]. This reconfigurability andflexibility can be exploitedbest within manufacturingsystemswith high productdiversity at smalllotsizes, whichis the case,e.g.,forjobshop systems[1, 10]. In [11],the recent developmentof RMTs wasstudiedandresults indicated that thecontributionswere mainly derived from configuration optimization, architecture design andsystem integrationandcontrol. However, in theaboveliterature, RMTs areonly thesource andenablerin termsof planning.To effectivelyutilizethesetools on theoperationallevel, we requirecontrol methodsincorporatingthedynamic characteristics of RMTs anddisturbances of theprocess. As jobshopsystemsofferhigh variability, researchers focused on this type of manufacturingsystem and studiedtheimpact of RMTs andrespective control methodson performance measures of such systems. Since jobshopsystemsmaysuffer from high work in process(WIP)levels andtherefore unreliable duedates andlong lead times within aproduction network[12], they have been mostly studiedusingdiscrete event Stabilityof Predictive Controlin JobShop System withReconfigurableMachineToolsforCapacity Adjustment 3 directlythrough utilizingcapacity adjustmentsto eliminate or periodically shiftbottleneckswithin the process. An overview on typical investigations on the controlof WIPis givenin Tab. 1. The mentioned workssignificantly improvedthe controlperformance. However, they mainly focused on labor-orientedapproaches.Here,we consider machinery-basedcapacity adjustment viaRMTs,i.e. we adaptthenumber of RMTs assigned to specific tasks on theshopfloor. Hence, ouraimis to design afeedback to allocate theRMTs withinthejobshop such that a certainWIPlevelis trackedandthen show thestabilityof thecontrolled system.Within[15], theauthors modeled a jobshopsystem with RMTs anddecomposed it into twooperators forthedesign of robust stabilizingcontrollers.Afterwards,they designed aPI tracking controller with respectto WIP. Thetracking performancecouldbe ensured even in thepresence of boundeduncertaintyby robust right coprime factorization. Yet, thefeedback cannot handle constraints explicitly andeffectively. Within this paper,we consider asimpleflow model of ajobshopsystem withpworkstations.Thejob shop is givenby afullyconnected graph, wheretheset of vertexes represents the workstations and theflowprobabilitymatrixbetween theworkstations. As shown in Figure 1, represent theinputrates of each workstationto workstation j,wheredenotes the orderreleaserate to workstation j.Moreover, representtherespective output ratesand theoutput rate of final productsof workstation j.We like to note that this procedure requires that knowledgeof flowof products is digitalizedand not expert knowledgeof workerson theshopfloor. eigenvalue of thematrix A, . Furthermore, we call acontinuous function of classif it is zero at zero,strictly increasing andunbounded. Similarly,a continuous function is saidto be of class if foreach it satisfies andforeach it is strictlydecreasing in itssecond argument with lim . 2PROBLEMDEFINITION Jobshopmanufacturing systemsprovide high flexibility in conjunctionwith cross-linkinformation andmulti-directional flow, whichis indispensable for high customizationwith lowrepetition rates. These properties arebeneficial foroften changing products butmaylead to bottlenecksin one or multiple machines or workstations.Becauseit maycontainreentrantlines to complete productsin theprocess, the orders may return to thesame machinemany times to perform differentstepsof theprocess.Meanwhile, some machines or workstations maylayidle. Theresulting bottlenecks, in turn,will causehigh work in process (WIP), long lead time, lowmachineutilization, andlow duedate reliabilityfortheoverall system[1]. Generally, thedecisionsforplanningandcontrolcanbe classified into threecategories:strategic, tactical andoperational. Here,we specificallyfocus on theoperationallayer, whichconsidersshorttermdecisionsandis relatedto optimallycontrollingthemanufacturingprocess. In particular,we assume that thesequence of orders to be processedis fixed. In orderto shorten lead time andimprovethe reliabilityof delivery time, one couldrelease orders earlier, whichis intendedto increase outputrates. Doingso maydestabilize thesystem,causeunbounded growth of WIP, additional inventory cost,requirement of largestoragespaceand even loss of consumers[21]. Since theWIPlevelis essential forallkeyperformance indicators [22],we propose to controltheWIPlevel Publications Contributions Methodology J.-H. Kim et.al. [14] Presentadynamic multi-workstationmodelwith closed-loop capacitycontrol include disturbance Transferfunction and proportional controller N. Duffieet.al. [16] Build up adiscrete model of production network with local capacitycontrol and compared with DES State space and proportionalcontroller B. Scholz-Reiter et.al. [23] Analyze dynamic behavior and performance of capacitycontrol of production network via Vensim DSS software Bio-inspired H.R. Karimi et.al [24] Investigate aclass of productionnetwork for capacitychanges with time-delayand showthe stability H∞control J.K. Sagawa and M.S. Nagano [25] Presentamodel of multi-productjob shop systemtomaintain adesired WIP level Bond graph proportionalcontroller Table 1: WIPcontrolby means of controltheory 4 externally and must thereforebe consideredas disturbances UtilizingAssumptions 1 and 2 allows us to simplify WIPjto (1) As we want to include RMTs into theworkstations, we link system (1)to number ofmachinetools operatingwithin theworkstations.Note that from an economic pointof view it only makessenseto buy new machineryif thecurrentcapacity is insufficientto deal with all orders.This typicallyleadsto high WIPlevels, whichallowus to rewritetheoutput as (2) At lowlevels or other extremeoperatingconditions, however, differentcapacity adjustmentsmaybe required [26]. Ouridea,whichwe followin this paper, is to ensurefidelityof afeedback,i.e. to guaranteethat all workstations operate close to predefined WIPlevels. If this propertycanbe shownfor a feedback at hand, then theassumption of ahigh WIPlevelcanbe shown to hold. More formally,thelatter assumption reads: Assumption 3 (HighWIP level) At anytime instantn, theWIPlevels in all workstations areat leastas high as thetotalmachinecapacity, i.e. (2)holds. Onewayto prove that Assumption 3 holdsis to show that (1)is asymptotically stable for a feedback at hand.To this end, let representtheWIP levelof allworkstations and denote thevector of RMTs assigned to allworkstations.Here,thesets and allowus to incorporate possibly wanted constraintson theWIP levelandthetotalnumber of RMTs.Utilizing Assumption 3, then from (1)we obtain (3) seealso[27] forfurther modellingdetails. Usingthe latter-notation, we candefine theconceptof asymptotic stabilityformally: Definition 1 Suppose asystem (3), apredefined reference valuex* and a controlu(·)to be givensuch thereexists aforwardinvariantset .If there existsfunction such that jth workstation (j=1,2,3...p) i 1j(n) i 2j(n) . . . i pj (n) oj1(n ) oj2(n ) . . . ojp(n ) i 0j (n) oj0(n) Fig. 1: jth workstationin multi-workstation production system To avoid order-machinespecialties – e.g. certain manufacturingstepsfor a productcanbe executed on one machine only –we suppose that each of the workstations features identicalDMTs,whichmay operate with production rate .Thenumber of RMTs is controlled by ourinputvariable uandeach RMTmayoperate with production rate .The work in processlevelWIPjis defined as thenumber of orders waitingto be processed at workstation j, hence itsrate of change is givenby thedifferencebetween rates of inputandoutput orders IandO.Basedon Fig. 1, we obtain Therefore, time dependentdynamics of WIPjcanbe computed viaitspreviousvalue, theinputs from other workstations,thedifferencebetween self-loop input with output of itself,andtheexternal input, i.e. To linkoutputsto theinputs, we impose thefollowing: Assumption 1 (Flowconservation) Thejob shop systemis mass conservative, that is forgiven flow probability matrix Pwe have foreach workstationj. We like to note that Assumption 1is appropriate here fortworeasons: First, loss of products within thejob shopsystem canbe included by modifyingtheflow probabilitiesbetween theworkstations.Andsecondly, theassumption rulesout dissipation, whichsimilar to friction in a mechanical system easesthetask of stabilizing a system.Hence, Assumption 1represents themoredifficult case andallfollowing resultsalso applyto thecase with dissipation. Here,we additionally focuson theoperationallayer only.As aconsequence, we cannot determine the order releaserates to anyof theworkstations. Assumption 2 (Operational layer) Theorderrelease rates to each workstation jaredetermined Stabilityof Predictive Controlin JobShop System withReconfigurableMachineToolsforCapacity Adjustment 5 thesolution of theinfinite horizonoptimalcontrol problemwith keyperformanceindex (6) subject to thedynamics (3)andconstraints, .Note that thedirect integrationof both thekey performanceindexandof theconstraintspresents a majordifference to aPIDcontroller.Whilethelatter needsto be tuned by an expert to adhere theconstraints andperform well givenan externalindex, no further action is required fortheMPCcontroller. Before we specifytheMPCproblemto oursetting, we introducethegeneralbackground of the method. Followingliterature[20],we impose thefollowing standard assumptions: Assumption 4 Forthereexists such that . Assumption 5 Thestage cost satisfies andforall if . Theoptimalvaluefunction correspondingto (6) is givenby andbased on dynamicprogrammingprinciples we obtain andcanderive an optimalfeedback controllaw by usingBellmans optimalityprinciple. Sincethis optimal control problem is typically computationally intractable, MPCapproximates therespective solution via a threestep procedure: Afterobtaining thecurrent state of thesystem, a truncatedoptimalproblem with finiteprediction horizonis solved to obtaina correspondingoptimalcontrolsequence.Then, only thefirst elementof this sequence is appliedandthe prediction horizonis shifted, whichrendersthe method to be iterativelyapplicable.Then computationally complexpart is thesolution of thetruncatedproblems min(7) subject to required in thesecond step of Algorithm1. For simplicity of exposition we assume that a minimizer argmin of (7)is unique. Combined,thesestepsreveal thefollowing algorithm: (4) holdsforallandall,then thecontrol u(·)asymptotically stabilizes x*. Fig. 2: Illustrate example of definition 1 Nowwe directlyobtainthefollowing. Corollary1Considersystem(1)andapredefined referencevalue.If forthecontrol u(·)a set is forwardinvariantsuchthat (5) holds, then Assumption3holds. In practicalterms,inequality (5)ensuresthat forany chosen time instantntheWIP levelis high enough such that allmachinetoolswithin aworkstationwork at full capacity.Forwardinvariance in turn meansthecontrol uensures, that thebuffersof allworkstations will be refilledsuch that full capacity utilizationin thenext time step is guaranteed. We like to note that in practice perfecttracking (Definition1) is almost surely impossible,but needs to be extendedto practicalstability,cf.[20, Chapter 2]. Apartfrom workers, whomayinterfere with the processes, thelatter is dueto two facts: For one, any reconfiguration requirestime, whichresultsin atime delayedprocess. Andsecondly,only an integernumber of RMTs canbe assigned to a workstation, whereas typicalfeedbacksconsider convex sets. To deal with both issues in long term, we propose to consider MPC as acontrolscheme, whichis able to explicitly handle such constraints. To make thefirst step into thisdirection, in thispaper, we consider controlof themanufacturing processin the continuous optimizationcase andcomparean MPCto thestandardPIDimplementation. 3MODELPREDICTIVE CONTROL To achieve thegoal of asymptotic stability,we propose to utilizeMPC. Theidea of thelatter is to approximate n 0 β(�x0−x∗�,n) �x(n)−x ∗ � 6 forsome predefineddesiredequilibrium ( x*,u*). Then Theorem 1 holdsfor N = 2 with wher e Moreover,the stabilizing MPCfeedback is given by Thedetailsof proofaregivenin theappendix. In practice,thestagecost (11) mayrepresentany performanceindicator or a scalarizedcombination of severalindicators.Hence, it is possible to model maximization of throughput, profit or qualityas well as minimizing lead time, energy requirements or costs directly. Giventheresult from Proposition1, MPCis as readilyavailableforthejobshop system problem includingRMTs as PID. Aparticular conclusion from this result is that, upon implementation, one does nothave to worryaboutstability of theclosedloop when choosing theprediction horizonlength as stabilitycomes forfree.Hence, similarto PID, no expert knowledgeis required to controlthesystem. In contrastto stability, however, PIDrequires internal knowledge of thekeyperformanceindexesused to evaluate thefeedback as well as good commandof howto appropriatelyadaptthePIDparametersto perform well fortheseindexes. Thelatter task becomes even moredifficult if connectedPIDcontrollers, e.g., one perworkstation, need to be considered andto be adjusted simultaneously.In such aMIMOcase, one mayapplyoptimization mehtods,e.g. particle swarm optimization[29] or iterated linear matrixinequalities [30].ForMPC, no further knowledgeandnoadaptation phaseis required as KPIs canbe used as cost criterion andaretherefore optimized by design. Remark 1 Dueto theadditional terminal endpoint constraints, recursivefeasibilityis guaranteed automatically, i.e. if theinitial state of thejobshop system adheresallconstraints, then therealways exists asolution to problem(7). [20] andtheMPCprocedure canbe appliedwithout runninginto adead end. Remark 2 Whilethesolution derivedfrom optimizationtypicallyoutperformsthedecision basedonworker experience,thecomputationalcost growswith thedimensionof thesystem andmaybe intractable, i.e. thebest solution maynot foundin a reasonable time. Within ajobshopsystem,this may Applying Algorithm1, for a giveninitial value ,we obtaintheclosed-loop solution (8) Note that optimality in each iterate is notsufficient to guaranteestabilityin thesenseof Definition 1. Yet, we can utilize theoptimalvaluefunction to recapitulate thefollowing from [20, Lemma5.4, Theorem5.13]: Lemma1Considertheoptimalcontrolproblem(7) with prediction horizonand theadditional terminal conditionand supposethat Assumptions 4and 5hold.Then foreach and each we have . Theorem 1 Supposetheassumptions of Lemma 1to hold.If thereexistfunctionsandsuch that (9) (10) holds, then is asymptotically stabilizing the closedloop (8)in the sense of Definition1. Theorem 1 provides thegeneralbackground forour task of asymptotically stabilizing a jobshopsystem with RMTs.Hence, ouraim now is to provethat the method appliesto ourcase.Theparticular difficulty with MPCis that a suitable predictionhorizonNis typically unknown, yetif Nreveals astabilizing control, then alsothefeedback with for stabilizes theclosedloop[28].Here,we showthat for N=2, thesolution of problem (7)canbe computed explicitly withouttherequirement of an optimization routine. Therefore, also allfeedbackswith asymptotically stabilizetheclosed-loop,whichrenders the method to be applicable in general. Fortechnical reasons,we require Assumption 6 Theflowprobability matrix P between theworkstations is invertible. Then we canutilizeourdynamics (3)together with Assumption 6to show thefollowing: Proposition 1 Considerproblem (7)forthejob shop system (3)together with thestage costs (11) Al gorithm1Basic modelpredictive control method Inpu t: N∈N. 1: for n=0,... do 2: Measurecurrent WIP levels x(n)and set x0:= x(n) 3: Compute controlinputs u(n)bysolving(7) 4: Apply µN(x(n))=u � (0) to all workstations 5: end for Outpu t: Feedback µN( · ) Stabilityof Predictive Controlin JobShop System withReconfigurableMachineToolsforCapacity Adjustment 7 wherethelatter include ademand fluctuation in a certainperiod,whichis modeledby asinfunction (12) Then,ourgoal wasto steertheWIPlevelof each workstationto therespective desiredvaluewhile consideringthestate andcontrolconstraints (13) We imposedthestagecost function (11),which satisfies Assumption 5. From Assumption 4, we then obtained (14) In order to increase thebasinof attraction, we chose N=16 andobtained thesimulation results sketched in Figs.4and5. As benchmark, we complemented these figures by respective graphs using a PIDcontroller. As PIDdoes notallowto includeconstraintson the totalnumber of RMTs,we additionally imposedthe truncation (15) such that PIDalways adheresto this constraint.The parameters of thePIDcontrollerwere tuned manually by extensive simulationsto reduceoscillations as be thecase if theinitial state of thejobshop system is farfrom thedesiredequilibrium. In this case, decentralized or distributedcontrolcouldbe applied forthehigh ordersystem [31, 32], whichareout of the scopeof this article. To complement andcheckourtheoreticalfindings, we next consider anumericalexampleto illustrate our results. 4CASE S TU DY Within thissection, we considered themulti workstationsystem sketched in Fig. 3. Since products canbe manufactured cost efficientlywith ahigh productivityby meansof dedicatedmachines,and diversityof customized product at a lowquantity effectivelyviareconfigurablemachines,we included both of RMTs andDMTs forallworkstations.This kind of combinationandco-existence in industrial practice canbe observed frequently [33] andnaturallyoccurs when a new type of machinetool is introduced.The respective dynamics aregivenby (3)with parameters andinitial values accordingto Tab. 2as well as flow matrixandexternal inputrates Variable Description x(0) =[40 40 40 30]�Work in process (WIP) level rDMT =3 Production rate of DMT rRMT =2 Production rate of RMT nDMT =[5452]�Number of DMTs for eachWS u=[2121] �Number of RMTs for eachWS m=6 Maximum value of RMTs x∗=[25,22,25,16]�Plannedworkinprocess (WIP) Time [hour] 010 20 30 40 50 60 70 80 WIP 1 20 40 PlannedWIP MPC PID Time [hour] 010 20 30 40 50 60 70 80 WIP 2 20 40 Time [hour] 010 20 30 40 50 60 70 80 WIP 3 20 30 40 Time [hour] 010 20 30 40 50 60 70 80 WIP 4 10 20 30 Table 2: Thevariablesdefinition in thejob shop system Fig. 3: Multiworkstationmultiproduct job shop system Fig. 4: WIPlevel forMPCand PID 8 To further testtheproposed method,we additionally simulated casesfordifferentinitial values.In the contextof MPCwith terminalconditions,initial values farfrom thedesiredequilibriumrequirea largepredictionhorizon N,whichin turn maycause problems in solvingproblem(7). To testthejobshop system problem, we considered thecases1) x(0) =[30 30 30 20]T,2) x(0) =[4030 20 30]T,3) x(0) =[4040 40 30]T,and4) x(0) =[5040 50 40]T,forwhichthe resultingtrajectories aredisplayedin Fig. 6. We like to note that this cannot be regardedas acomplete test, whichwouldrequireto checkforcontrolforward invariance of asetof initial conditions.Yet, thecases alreadyassume quiteexcessive deviations from the desiredpointof operationandfrom a practicalpointof view, such occasionsarealready rather unlikely. Still, we observed that in allcasesthetrajectories converged to thedesiredvalues forallworkstations j=1, ...,4. Additionally,we like to highlight apeculiarityarising forworkstations 2and 4 at time instant n=23,when thetrajectories were almost at ,cf.themagnified sectionin Fig. 6. Here,thetrajectories showedan almost inversebehavior. In fact,MPCchose to cause a deviationfrom on purposeas it recognized that this is necessaryto steerthetrajectory exactlyto at later time instances. Remark 3We additionally like to note that the terminalconditionhasto reachablewithin thepredictionhorizon. Hence, dependingon therange of initial conditions andof theexternalinputrate d, whichtheuser wantsto allow, theprediction horizon must be chosen largeenough.If theinput rate and initial conditions arecontained in aregion forwhich afeasible solution exists, then Proposition1together with Theorem 1 guaranteethat MPCasymptotically stabilizes thejobshop system. much as possible butwithoutexternaloptimization technique.As simulationshave shown that theD component shows no further improvement,we chose a PI controller with kp=0.5, ki=0.01andkd=0. As expected,in Fig. 4we observethat theproposed method is capable of tracking thedesiredWIPvaluefor each workstation. Theallocation of RMTs is displayed in Fig. 5. From Fig. 5, we observed that forboth MPCandPID, thenumber of RMTs assigned to particular workstation is identicalfrom timeinstant n=34 onwards. Comparingthis to theWIPlevels in Fig. 4, we found that MPCis tracking thedesiredvalues perfectly forallworkstations j=1, ..., 4. PID, on theother hand, showsan offset,whichwe were unable to solvedespite extensive tuningof thecontrolparameters. Forin Fig. 5, we observed that both controllersresultin avery differentassignment of RMTs,whichhoweveris notwell reflected in standard comparisonerrors.Here,we considered integral absolute error(IAE), integral square error(ISE), integral absolute control(IAU)andintegral square control(ISU)to analyzetheresultsdisplayedin Fig. 4 and5, cf.Tab. 3fordetails. Consideringthedifferences in theassignments, time instants n= 0 and n=22 were of particular interest:At n=0, PIDchose almost identicalassignmentsof RMTs forallworkstations, whereasMPC movedRMTs to workstations 3 and 4 only.As aresult,theWIPlevels forworkstations 3 and4in theMPCcase droppedsignificantly faster than forPID, whereasWIPlevels forworkstations 1 and 2 were only slightly higher,cf.Fig. 4. At n=22, both PIDandMPCreactedto thechange of theinput rate.Thefollowing curvewasalmost identicalforboth controllersandapproximates asinfunction,whichwas to be expected giventhenature of theinputrate change. In contrastto PID, however, MPCstartedwith avery strong peakin workstations 1, 2and3. Table 3: Comparison betweenstandard PIDandMPC IAE IAUISE ISU PID WS1 199.40 77.41 1496.40 97.02 WS2 161.06 40.25 1032.42 37.80 WS3 254.92 74.51 2245.99 104.74 WS4 162.59 37.10 866.20 31.85 MPC WS1 155.24 76.83 1882.57 100.05 WS2 165.45 39.73 2163.74 33.81 WS3 83.10 73.47 622.92 103.40 WS4 62.36 36.38 400.50 39.38 Time [hour] 0102030 40 50 60 70 80 u 1 0 5 MPC PID Time [hour] 0102030 40 50 60 70 80 u 2 0 5 Time [hour] 0102030 40 50 60 70 80 u 3 0 5 Time [hour] 0102030 40 50 60 70 80 u 4 0 5 Fig. 5: Assigned RMTs by MPCand PID