Hybrid Model Predictive Control Applied to Production-Inventory Systems

Transcription

1 Preprint of paper to appear in the 18th IFAC Worl Congress, August 28 - Sept. 2, 211, Milan, Italy Hybri Moel Preictive Control Applie to Prouction-Inventory Systems Naresh N. Nanola Daniel E. Rivera Control Systems Engineering Laboratory, School for Engineering of Matter, Transport an Energy, Arizona State University, Tempe, AZ , USA ( nnanola@asu.eu an aniel.rivera@asu.eu) Abstract: Hybri prouction-inventory systems are characterize by iscrete ecisions on prouction levels an/or capacity. These systems have broa applicability to important, emerging applications of process control concepts, among them time-varying aaptive behavioral interventions an supply chain management. This paper examines the usefulness of hybri moel preictive control (HMPC) in these two novel application settings. In a hypothetical aaptive behavioral intervention inspire by Fast Track (a preventive intervention for reucing conuct isorer in at-risk chilren), HMPC is presente as a means to improve the assignment of frequency of home-base counseling visits to families with low parental function. In supply chain management, the usefulness of HMPC for assigning prouction capacity in an inventory control problem uner conitions of varying customer eman is presente. These problems are moele as mixe logical ynamical (MLD) systems, with HMPC consisting of a Mixe Integer Quaratic Program (MIQP) that employs a three-egree-of-freeom parametrization for achieving ease of tuning an facilitating robust performance uner uncertainty. Keywors: Hybri systems; Moel Preictive Control; prouction-inventory systems; aaptive behavioral interventions; supply chain management 1. INTRODUCTION Hybri systems are characterize by interactions between continuous an iscrete ynamics. The term hybri has also been applie to escribe processes that involve continuous ynamics an iscrete (logical) ecisions (Bempora an Morari, 1999; Nanola an Bhartiya, 28). Hybri systems occur in many iverse settings; these inclue manufacturing, automotive systems, an process control. In recent years, significant emphasis has been given to moeling, ientification, control an optimization of linear an nonlinear hybri systems (Nanola, 29). The review paper by Camacho et al. (21) notes that espite the consierable interest within the control engineering community for moel preictive control for hybri systems, the fiel has not been fully evelope, an many open challenges remain. One of these is the application to new areas outsie of the inustrial community, an the the nee for novel formulations that can be effectively use in noisy, uncertain environments. This paper emonstrates the application of hybri MPC to two non-traitional areas that can be conceptualize as prouction-inventory systems: aaptive interventions in behavioral health, an inventory management in supply chains. These problems are hybri in nature an isplay performance requirements that eman a flexible control formulation. We consier a novel Three-Degree-of-Freeom (3-DoF) Moel Preictive Control formulation evelope by Nanola an Rivera (21), Support for this work has been provie by the Office of Behavioral an Social Sciences Research (OBSSR) of the National Institutes of Health an the National Institute on Drug Abuse (NIDA) through grants K25 DA21173 an R21 DA Naresh N. Nanola is currently with ABB Corporate Research Center, Bangalore, Inia. [ nareshkumar.nanola@in.abb.com.] which offers performance an ease of tuning that is amenable for robustification in hybri systems. Prouction-inventory control is a classical problem in enterprise systems that has application in many problem arenas. Fig. 1 shows a iagram of a prouction-inventory system uner combine feeback-feeforwar control action. Two prouction noes are represente by two separate pipes, while the inventory component consists of flui in a tank. The goal is to manipulate the inflow to the prouction noes (i.e., starts) in orer to replenish an inventory that satisfies exogeneous eman. The eman signal can be broken own into forecaste an unforecaste components. A substantial literature exists that examines prouction-inventory systems from a control engineering stanpoint (Schwartz an Rivera, 21). Schwartz an Rivera (21) examine both Internal Moel Control (IMC) an Moel Preictive Control (MPC) for a linear prouction-inventory system with continuous inputs. The hybri prouction-inventory system, in which prouction occurs at iscrete levels (or is ecie by iscrete-event ecisions) is an important yet less stuie problem; we consier it the focus of this paper. The paper highlights two application areas that map into the hybri prouction-inventory problem space. The first is aaptive interventions in behavioral health, which is a topic receiving increasing attention as a means to aress the prevention an treatment of chronic, relapsing isorers, such as rug abuse (Collins et al., 24). In an aaptive intervention, osages of intervention components (such as frequency of counseling visits or meication) are assigne to participants base on the values of tailoring variables that reflect some measure of outcome or aherence. Recent work has shown the relationship between forms of aaptive interventions an feeback control of prouction-inventory systems (Rivera et al., 27). In prac-

2 Preprint of paper to appear in the 18th IFAC Worl Congress, August 28 - Sept. 2, 211, Milan, Italy (k) K 1 (Yiel) (Throughput Time) Net Stock (Controlle) y(k) K 2 (Yiel) θ 1 θ 2 (Throughput Time) LT Actual (k) θ (Delivery Time) Controller u 2 (k) F (k) Forecast θ F Forecast Horizon Deman (Disturbance) Fig. 1. Diagrammatic representation of a prouction-inventory system consisting of two prouction noes (one primary, one seconary) an a single inventory, with combine feeback-feeforwar control. tice, these problems are hybri in nature because osages of intervention components correspon to iscrete values. These interventions have to be implemente on a participant population that may isplay significant levels of interiniviual variability. Hence a novel problem formulation is necessary that insures that the ecision policy makes appropriate ecisions for all members of a population, without emaning excessive moeling effort for each iniviual participant. Inventory management in supply chains also represents a rich application area for hybri prouction-inventory systems. Consier a scenario where an enterprise is require to make the ecisions on startup / shutown of an auxiliary manufacturing facility in orer to achieve operational goals uner changing market eman. These problems are hybri in nature, an the ynamics of these systems can be highly uncertain (Wang an Rivera, 28). This necessitates a novel problem formulation to insure efficient use of resources an an optimal operating policy for the supply chain management problem, while taking account of the changing market eman. The paper is organize as follows: Section 2 summarize the MPC formulation with the multi-egree-of-freeom tuning for MLD systems. Two case stuies, namely a hypothetical aaptive intervention base on Fast Track program, an the supply chain management problem escribe previously are iscusse in Section 3. Summary, conclusions, an irections for future research are presente in Section MODEL PREDICTIVE CONTROL FOR HYBRID SYSTEMS Moel preictive control (MPC) is wiely accepte in the process inustries ue to its ability to systematically inclue constraints, the capability to hanle plants with multiple inputs an outputs, the flexibility given to the user to efine a cost function, an its isturbance rejection properties. 2.1 Controller Moel The MPC controller presente in this paper relies on the following moel framework (Nanola an Rivera, 21), x(k + 1) = Ax(k) + B 1 u(k) + B 2 δ(k) + B 3 z(k) + B (k) (1) y(k + 1) = Cx(k + 1) + (k + 1) + ν(k + 1) (2) E 5 E 2 δ(k) + E 3 z(k) E 4 y(k) E 1 u(k) + E (k)(3) where x an u represent states (both iscrete an continuous) an inputs (both iscrete an continuous) of the system. y is a vector of outputs, an, an ν represent measure isturbances, unmeasure isturbances an measurement noise signals, respectively. δ an z are iscrete an continuous auxiliary variables that are introuce in orer to convert logical/iscrete ecisions into their equivalent linear inequality constraints summarize in (3) (for etails, see Bempora an Morari (1999) ). The framework permits the user to inclue an prioritize constraints, an incorporate heuristic rules in the escription of the moel. Because isturbances are an inherent part of any process, it is necessary to incorporate these in the controller moel that efines the control system. Equations (1)-(3) are the MLD framework shown in (Bempora an Morari, 1999), which is moifie by incorporating measure an unmeasure isturbances. The moel lumps the effect of all unmeasure isturbances on the outputs only, which is a common practice in the process control literature (Wang an Rivera, 28; Lee an Yu, 1994). We consier, the unmeasure isturbance, as a stochastic signal, escribe as follows, x w (k + 1) = A w x w (k) + B w w(k), (k + 1) = C w x w (k + 1)(4) where A w has all eigenvalues insie the unit circle an w(k) is a vector of integrate white noise. Here, it is assume that the isturbance effect is uncorrelate. Thus, B w = C w = I an A w = iag{α 1, α 1,, α ny } where n y is number of outputs. In orer to take avantage of well unerstoo properties of white noise signal consiering ifference form of isturbance an system moels an augmenting them as follows, X(k + 1) = A X(k) + B 1 u(k) + B 2 δ(k) + B 3 z(k) +B (k) + B w w(k) (5) y(k + 1) = C X(k + 1) + ν(k + 1) (6) Here X(k) = [ x T (k) x T w(k) y T (k)] T, (k) = (k) (k 1) an w(k) is white noise sequence. Augmente matrices A, B i an C are given in Nanola an Rivera (21). 2.2 MPC Problem In this work, we use a quaratic cost function of the form, min {[u(k+i)] m 1 i=, [δ(k+i)]p 1 i=, [z(k+i)]p 1 m 1 + i= p 1 + i= i= } J = Q u ( u(k + i)) 2 m i= Q (δ(k + i) δ r ) 2 p p i=1 Q y (y(k + i) y r ) 2 2 Q u (u(k + i) u r ) 2 2 i= Q z (z(k + i) z r ) 2 2 (7) subjecte to mixe integer constraints of (3) an various process an safety constraints such as boun constraints on inputs, outputs an move. Here p is the preiction horizon, m is the control horizon, ( ) r stans for reference trajectory an 2 is for 2-norm. Q y, Q u, Q u, Q, an Q z are penalty weights on the control error, move size, control signal, auxiliary binary variables an auxiliary continuous variables, respectively. The aforementione MPC problem is governe by both binary an continuous ecision variables hence it is a mixe integer quaratic program (miqp). Moreover, it requires future preictions of the outputs an the mixe integer constraints in (3),

3 Preprint of paper to appear in the 18th IFAC Worl Congress, August 28 - Sept. 2, 211, Milan, Italy which can be obtain by propagating (3), (5) an (6) for p steps in future. These multi-step preictions are then use to convert aforementione MPC problem to a stanar miqp (for etails, see Nanola an Rivera (21)). This problem can be solve using any miqp solver available in the market. In this work, we have use the Tomlab-CPLEX solver. It shoul be note that the algorithm also requires externally generate reference trajectories, an externally generate forecast of the measure isturbance an estimate of (isturbance free) initial states X(k) that influence the robust performance of the MPC. The output reference trajectory is generate using an asymptotically step (a Type-I filter per Morari an Zafiriou (1989)) as: y r (k + i) = (1 α r j )q y target q αr j, 1 j n y, 1 i p (8) The setpoint tracking spee can be ajuste by choosing αr j between [,1) for each output. The smaller the value, the faster the response for particular setpoint tracking. Thus, setpoint tracking spee can be ajuste for each output iniviually. The spee require to reject each measure isturbance can be ajuste inepenently by using a filter, f (q,α j ), 1 j n ist, for each measure isturbance. Here n ist is the number of measure isturbances an α j is a tuning parameter between [,1), for the jth measure isturbance. Smaller the α j, faster the spee of particular isturbance rejection. Thus, filtere signal can be use as an anticipate unmeasure isturbance. The transfer function f (q,α j ) can be assume as an asymptotically step or an asymptotically ramp (i.e. Type-I or Type-II signal as per Morari an Zafiriou (1989)), as per the nature of the system ynamics. Type-I filter structure is given in (8) an Type-II filter structure can be given as f (q,α j ) = (β + β 1 q β ω q ω ) (1 α j )q q α j (9) 6kα j β k = (1 α j 1 k ω )ω(ω + 1)(2ω + 1), (1) β = 1 (β β ω ) (11) The states of the system can be estimate from the current measurements, y(k) while rejecting the unmeasure isturbance using a Kalman filter as follows: X(k k 1) = A X(k 1 k 1) + B 1 u(k 1) + B 2 δ(k 1) +B 3 z(k 1) + B (k 1) (12) X(k k) = X(k k 1) + K f (y(k) C X(k k 1)) (13) Here K f is the filter gain, an optimal value of which can be foun by solving an algebraic Riccati equation. We use the parametrization of filter gain (Lee an Yu, 1994) as follows, K f = [ F b F a ] T (14) Here F a = iag{( f a ) 1,,( f a ) ny }, F b = iag{( f b ) 1,,( f b ) ny }, ( f b ) j = (( f a ) j 2 )\(1 + α j α j ( f a ) j ), 1 j n y an ( f a ) j is a tuning parameter between an 1. While the unmeasure isturbances are rejecte using the state observer presente in (12)-(14), the spee of rejection is proportional to the tuning parameter ( f a ) j. As ( f a ) j approaches zero, the state estimator increasingly ignores the preiction error correction, an the control solution is mainly etermine by the eterministic moel, (12). On the other han, the state estimator tries to compensate for all preiction error as ( f a ) j approaches to 1, with a corresponing increase in the aggressiveness of the control action. In practice, the juicious selection of ( f a ) j requires making the proper traeoff between performance an robustness. 3. CASE STUDIES In this section, we present applications of MPC with 3-DoF algorithm on case stuies from two ifferent non-traitional areas escribe in the Introuction: aaptive behavioral interventions an supply chain management. 3.1 Aaptive Time-Varying Behavioral Interventions As a representative case stuy of a time-varying aaptive behavioral intervention we examine the hypothetical problem inspire by the Fast Track program (C.P.P. Res. Grp., 1992). Fast Track was a multi-year, multi-component program esigne to prevent conuct isorer in at-risk chilren. Youth showing conuct isorer are at increase risk for incarceration, injury, epression, substance abuse, an eath by homicie or suicie. In Fast Track, some intervention components were elivere universally to all participants, while other specialize components were elivere aaptively. In this paper we analyze a hypothetical aaptive intervention inspire by Fast Track for assigning home-base family counseling visits, which are provie to families on the basis of parental functioning. There are several possible levels of intensity, or oses, of family counseling. The iea is to vary the oses of family counseling epening on the nees of the family, in orer to avoi proviing an insufficient amount of counseling for very trouble families, or wasting counseling resources on families that may not nee them or be stigmatize by excessive counseling. The ecision about which ose of counseling to offer each family is base primarily on the family s level of functioning, assesse by a family functioning questionnaire complete by the parents. As escribe in Collins et al. (24), families with very poor functioning are given weekly counseling; families with poor functioning are given biweekly counseling; families with near threshol functioning are given monthly counseling; an families at or above threshol are given no counseling. Family functioning is reassesse at a review interval of three months, at which time the intervention osage may change. This goes on for three years. We consier the scenario in which the total number of home base family counseling visits is limite ue to resource constraints, an a monthly group meeting option is mae available to participant families once the maximum number of home-base family counseling visits has been reache. Following as in Rivera et al. (27), the open-loop ynamics of the intervention is moele by means of a flui analogy represente in Fig. 1. Parental function y 1 (k) is treate as flui in a tank, which is eplete by exogenous isturbances (k). The tank is replenishe by the intervention components (k) (home base counseling visits) or u 2 (k) (monthly group meeting), which are manipulate variables. The use of flui analogy enables eveloping a mathematical moel of the openloop ynamics of the intervention using the principle of conservation of mass. This moel can be escribe by following ifference equations which relates parental function y 1 (k) with the intervention components (k) an u 2 (k) as follows:

4 Preprint of paper to appear in the 18th IFAC Worl Congress, August 28 - Sept. 2, 211, Milan, Italy Counseling visits PF (%) 1 5 Weekly Bi Weekly Monthly No Visits Total visits Counseling visits PF (%) 1 5 Weekly Bi Weekly Monthly No Visits Total visits Group Meeting Yes Unlimite counseling visits Limite (48) counseling visits No Time (Month) Group Meeting Yes No Time (Month) Fig. 2. Aaptive behavioral intervention case stuy. Control performance for unlimite an limite (48) counseling visits, in the absence of the group meeting component. Step isturbance D(k) = 5, α r =, α =., f a = 1, Q y = 1, Q u =.5, Q u = Q = Q z =. y 1 (k + 1) = y 1 (k) + K 1 (k θ 1) + K 2 u 2 (k θ 2) (k) (15) K 1 =.15 an K 2 = 5 represent the intervention gain for counselor home visits an monthly group meeting, respectively, θ 1 = θ 1 1 = represents the time elay between the intervention component an its effect on parental function, θ 2 = θ 2 1 = represents the time elay between the intervention component u 2 an its effect on parental function. (k) is the unknown source of parental function epletion (i.e. unmeasure isturbance). The measure isturbance an hence feeforwar control is not consiere here in this application. Here it shoul be note that (k) has a restriction on the frequency of counselor visits such that these can be only possible either weekly, biweekly or monthly, or none at all. This problem requires imposing a restriction on the intervention (k) such that it takes only four values:, u weekly, u biweekly an u monthly. In orer to capture iscreteness in the intervention, four binary auxiliary variables, δ 1, δ 2, δ 3, δ 4 an three continuous auxiliary variables, z 1, z 2, z 3 are introuce. The etaile escription of logical conitions are not presente here for the brevity of the paper. Moreover, u 2 (k) has a restriction such that it can be available if an only if intervention component of home base counseling visits is exhauste (i.e. total 48 home visits complete). Thus, an u 2 can not be given simultaneously. In orer to implement this restriction one aitional binary variable δ 5 is introuce with the following aitional relationships, y 2 (k) = y 2 (k 1) + b 1 z 1 (k) + b 2 z 2 (k) + b 3 z 3 (k) (16) δ 5 (k) = 1 48 y 2 (k) & u 2 (k) = δ 5 (k) (17) Here y 2 refers total number of counselor home visits. Equation (17) make sure that intervention component corresponing to monthly group meeting is available only in a situation where home base counselor visit is not available. Thus, the problem has inherent iscreteness along with the continuous ynamics, which can be characterize by the hybri ynamical system an can be moel using the MLD framework. Control performance for unlimite number of counseling visits in absence of group meeting (i.e. nominal case) is ocumente in Fig. 2 using soli line. From the figure, it can be seen that the parental function achieves esire goal satisfactorily using 69 in-home counseling visits uring the perio of 3 years. While Fig. 3. Aaptive behavioral intervention case stuy. Control performance for limite (48) counseling visits, with a group meeting component available. Step isturbance D(k) = 5, α r = α =., f a = 1, Q y = 1, Q u =.5, Q u = Q = Q z =. ashe-otte line in Fig. 2 represent control performance for the case where total home base counseling visits are restricte to 48 without availability of the group meeting (i.e. K 2 = ). From the figure, it is observe that the osage constraints places a funamental limit on the effects of the intervention an it is unable to achieve esire parental function goal in absence of any other aitional intervention. However, as it is clearly seen from the figure, the controller oes the best that it can an uses available resources such that it can reach as close as possible to the the esire goal. A ecline in the parental function after 24 months is notice because of the epletion of the parental function by some unknown isturbances an at the same time unavailability of the intervention osage (i.e inhome counseling visits). Fig. 3 ocuments performance for the case with similar restriction on total number of visits. However, in this case, option for the monthly group meeting is available (i.e. K 2 = 5) on completion of 48 home base counseling visits. Here it can be seen that the ecision policy assigns intervention osages (in-home counseling visits) similar to the nominal case until total number of visits reach to maximum value of 48 an then it assigns group meeting in orer to reach an maintain the esire parental function goal. Thus, the propose MPC-base ecision policy is capable of enforcing constraints on intervention osages, as well as switching between two ifferent interventions as neee. 3.2 Supply Chain Management Case Stuy In this case stuy, we consier a supply chain management problem comprise of a prouction-inventory system with one inventory an two prouction noes. The prouction noes consist of one primary factory an a seconary auxiliary one, as shown in Figure 1. Throughput time θ 1 = θ an yiel K 1 for the primary factory are 4 ays an.9, respectively; for the auxiliary factory, the throughput time θ 2 = θ an yiel K 2 are 9 ays an.8, respectively. Here we consier that the operating cost for the auxiliary factory is greater than the primary one because of the longer throughput time an lower yiel. Therefore, this auxiliary prouction noe shoul be accesse if an only if the primary noe is running at its full capacity an unable to meet future market eman; likewise, prouction from the auxiliary noe must be iscontinue if future eman

5 Preprint of paper to appear in the 18th IFAC Worl Congress, August 28 - Sept. 2, 211, Milan, Italy cannot justify its operation. The iscrete ecision of startup or shutown of the auxiliary factory is a function of the continuous variables f (k) (eman forecast) an the work-in-progress () in the primary prouction noe. Consequently, the system can be categorize as a hybri system, where the inventory ynamics are governe by continuous variables (inventory, factory starts, eman an ) an a iscrete ecision (shutown/start-up of the auxiliary factory). The ynamics of this system can be escribe using first principles moel follows: y(k + 1) = y(k) + K 1 (k θ 1) + K 2 u 2 (k θ 2) (k) (18) (k + 1) = θ 1 i= (k i) (19) where (k) = f (k θ f ) + u (k) is the total customer eman comprising forecaste f an unforecaste u components, (k) [, 2] represents the starts for the primary prouction noe, u 2 (k) [, 2] is the starts for the auxiliary prouction noe, an [, 6] represents the work-in-progress in the primary prouction noe. In orer to ensure that the auxiliary factory is activate if an only if in the primary factory is at its maximum capacity, we embe the following logical conition u 2 (k θ 2 ) (k + 1) > Cap max = 6 into the ynamical moel. The implication ( ) can be converte into linear inequality constraints using Big-M constraints (Raman an Grossmann, 1991). This conversion can be accomplishe by introucing an auxiliary binary variable (δ) an an auxiliary continuous variable (z) followe by a MLD representation of (18)-(19) as in (1)-(3). The MLD moel of (18)-(19) an linear constraint from aforementione logical conition along with boun constraints on process variables are then use to formulate an MPC problem. The operational goal of this system is to meet the customer eman (k) relying on prouction from the auxiliary factory only when necessary, while maintaining the net stock inventory level at a preefine setpoint. This can be accomplishe by manipulating factory starts (k), u 2 (k) an feeforwar compensation of forecaste eman f (k) simultaneously imposing the above escribe logical conition. Here we consier a sampling interval of T = 1 ay, θ f = p = 3 ays. In practice, it is esirable to keep the factory starts as constant as possible (i.e., avoi factory thrash ) while maintaining inventory at esire levels in the face of uncertain customer eman (Wang an Rivera, 28). In orer to examine the performance of the propose multi-egree-of-freeom MPC formulation, we consier a piecewise stochastic customer eman signal an its forecast, which are shown in Figure 4. Figure 5 emonstrates the performance of the propose formulation that uses multiegree-of-freeom tuning parameters, α r =.9, α =, f a =.1, penalty weight parameters Q y = 1, Q u = iag{ }, Q u = Q = Q z =, preiction horizon p = 3 an control horizon m = 25. From the figure, it can be seen that the propose MPC algorithm is able to satisfactorily maintain inventory at its preefine target, while proucing little variation in the starts of the factories. Moreover, it is able to ecie on the start-up / shutown of the auxiliary factory in a esirable manner (i.e., by making minimal use of the auxiliary factory). This reuces operational cost while allowing the prouction system to meet customer eman without backorers. In orer to assess the effectiveness of the propose formulation, we compare its performance with the MPC formulation that relies on a constant plant-moel mismatch over the preiction horizon. This formulation uses the move suppression weight (Q u ) in lieu of f a to reuce the variation in the manipulate Time (ay) Fig. 4. Customer eman (k) (re ashe line) an eman forecast f (k) (soli black line), SCM case stuy. y, u u Time (ay) Fig. 5. Response of net stock (y), work-in-progress (W IP), an factory starts (,u 2 ) for the propose multipleegree-freeom formulation with tuning parameters: α r =.9, α =, f a =.1 an Q y = 1, Q u = iag{ }, Q u = Q = Q z = for the eman in Fig. 4. variables (i.e., starts) of the factories. Figure 6 presents the simulation results using Q u = iag{ } while keeping all other parameters as in the previous case. The responses show evience of very poor inventory control an very high variation in the starts of the factories as compare to the propose formulation. To reuce the variation in the starts of the factories an verify the performance against the propose formulation, we apply various values of the move suppression weight Q u. Table 1 ocuments the maximum (peak) value of the inventory (y max ) an the close-loop performance metrics J e an J u using the MPC formulation relying on a constant plant-moel mismatch over the preiction horizon for five values of Q u between to 2. The last row of Table 1 ocuments these values for the three-egree-of-freeom (3-DoF) MPC formulation. The performance metrics J e an J u are measures of cumulative error (e(k) = y(k) y r ) an variation in the rate-of-change of factory starts ( u(k) = u(k) u(k 1)), respectively, which are given as J = t/t s k=1 (k)t (k), where = e, u. From the table, it can be seen that the propose 3-DoF formulation outperforms the constant plant-moel mismatch base MPC formulation in terms of maximum peak in the inventory an J e for all values of Q u. On the other han, increasing the move suppression weight Q u lowers J u, with the last two cases (i.e. Q u = iag{1 1} an Q u = iag{2 2}) yieling lower values of J u than the 3-DoF MPC formulation. f

6 Preprint of paper to appear in the 18th IFAC Worl Congress, August 28 - Sept. 2, 211, Milan, Italy y, u u Time (ay) Fig. 6. Response of net stock (y), Work-in-Progress (), an factory starts (,u 2 ) for the MPC formulation that relies on constant plant-moel mismatch an move suppression tuning, with Q u = iag{ } an Q y = 1, Q u = Q = Q z = for the eman profile in Fig. 4. y, u u Time (ay) Fig. 7. Response of net stock (y), Work-in-Progress (), an factory starts (,u 2 ) for the MPC formulation that relies on constant plant-moel mismatch an move suppression tuning with Q u = iag{2 2} an Q y = 1, Q u = Q = Q z = for the eman profile in Fig. 4. Table 1. J u, J e an y max using MPC relying on constant plant-moel mismatch for various Q u an a 3-DoF MPC formulation for f a =.1. Sl. No. Q u J u J e y max 1 iag{ } iag{1 1} iag{1 1} iag{1 1} iag{2 2} DoF Contrasting Fig. 5 with Fig. 7 shows that for the 3-DoF formulation, factory starts remain constant over a significant portion of the simulation, without leaing to the substantial maximum inventory peak an corresponing emans on warehouse space resulting from Q u = iag{2 2}. Thus, we observe that the 3-DoF-MPC algorithm can be useful for reucing overall operating costs an efficiently managing this class of hybri prouction-inventory systems. 4. SUMMARY Control applications of hybri systems are becoming increasingly important in many fiels, among them process control. In this work, the application of hybri moel preictive control (HMPC) to aaptive time-varying behavioral interventions an inventory management in supply chains, two problems that can be conceptualize using flui analogies, are presente. These systems are represente as mixe logical ynamical (MLD) moels, which can then be use to specify a hybri moel preictive control law relying on the three-egree-of-freeom (3-DoF) tuning formulation of Nanola an Rivera (21). The use of 3-DoF tuning enables iniviually ajusting the spee of isturbance rejection (measure an unmeasure) an setpoint tracking for each output; this has intuitive appeal for the user an facilitates achieving robust operation uner uncertainty. The effectiveness of the propose formulation is emonstrate in these applications uner iverse scenarios involving variations in constraints, isturbances, an tuning. REFERENCES Bempora, A. an Morari, M. (1999). Control of systems integrating logic, ynamics, an constraints. Automatica, 35(3), Camacho, E.F., Ramirez, D.R., Limon, D., Muñoz e la Peña, D., an Alamo, T. (21). Moel preictive control techniques for hybri systems. Ann. Rev. in Control, 34, Collins, L.M., Murphy, S.A., an Bierman, K.L. (24). A conceptual framework for aaptive preventive interventions. Prevention Science, 5(3), C.P.P. Res. Grp. (1992). A evelopmental an clinical moel for the prevention of conuct isorers: The Fast Track program. Development an Psychopathology, 4, Lee, J.H. an Yu, Z.H. (1994). Tuning of moel preictive controllers for robust performance. Comput. Chem. Eng., 18(1), Morari, M. an Zafiriou, E. (1989). Robust Process Control. Englewoo Cliffs, NJ: Prentice-Hall. Nanola, N.N. (29). A Multiple Moel Approach for Moeling, Ientification an Control of Nonlinear Hybri Systems. Ph.D. thesis, Inian Institute of Technology Bombay, Inia. Nanola, N.N. an Bhartiya, S. (28). A multiple moel approach for preictive control of nonlinear hybri systems. J. Process Control, 18(2), Nanola, N.N. an Rivera, D.E. (21). A novel moel preictive control formulation for hybri systems with application to aaptive behavioral interventions. In Amer. Cntrl. Conf., Baltimore, Marylan, USA. Raman, R. an Grossmann, I.E. (1991). Relation between milp moelling an logical inference for chemical process synthesis. Comput. Chem. Eng., 15(2), Rivera, D.E., Pew, M.D., an Collins, L.M. (27). Using engineering control principles to inform the esign of aaptive interventions a conceptual introuction. Drug an Alcohol Depenence, 88(2), S31 S4. Schwartz, J.D. an Rivera, D.E. (21). A process control approach to tactical inventory management in prouctioninventory systems. International Journal of Prouction Economics, 125(1), Wang, W. an Rivera, D.E. (28). Moel preictive control for tactical ecision-making in semiconuctor manufacturing supply chain management. IEEE Trans. Control Syst. Technol., 16(5),