Application of IHMPC to an unstable reactor system: study of feasibility and performance

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1 Proceedngs of the 9th Internatonal Symposum on Dynamcs and Control of Process Systems (DYCOPS 21), Leuven, Belgum, July 5-7, 21 Mayuresh Kothare, Moses Tade, Alan Vande Wouwer, Ilse Smets (Eds.) MoAT2.2 Applcaton of IHMPC to an uable reactor system: study of feasblty and performance Alejandro H. González Eduardo J. Adam Maran G. Marcoveccho Darc Odloak Itute of Technologcal Development for the Chemcal Industry (INTEC), CONICET-UNL. Gemes 345, (3) Santa Fe - Argentna. alejgon,eadam@santafe-concet.gov.ar INGAR- Ituto de Desarrollo y Dseo, Avellaneda 3657, (3) Santa Fe - Argentna. marangm@santafe-concet.gov.ar Department of Chemcal Engneerng, Unversty of São Paulo, Av. Prof. Lucano Gualberto, trv 3 38, São Paulo, Brazl. odloak@usp.br Abstract: Almost all the theoretcal aspects of Model Predctve Control (MPC), such as stablty, recursve feasblty and even the optmalty are now well establshed for both, the nomnal and the robust case. The stablty and recursve feasblty are usually guaranteed by means of addtonal termnal corants, whle the optmalty s acheved consderng closed-loop predctons. However, these sgnfcant mprovements are not always applcable to real processes. An nterestng case s the control of open-loop uable reactor systems. There, the tradtonal nfnte horzon MPC (IHMPC), whch cotutes the smplest strategy ensurng stablty, needs to nclude an addtonal termnal corant to cancel the uable modes, producng n ths way feasblty problems. The termnal corant could be an equalty or an ncluson corant, dependng on the local controller assumed for predctons. In both cases, however, a prohbtve length of the control horzon s necessary to produce a reasonable doman of attracton for real applcatons. In ths work, we propose an IHMPC formulaton that has maxmal doman of attracton (.e. the doman of attracton s determned by the system and the corants, and not by the controller) and s sutable for real applcatons n the sense that t accounts for the case of output trackng, t s offset free f the output target s reachable, and mnmzes the offset f some of the corants become actve at steady-state. Keywords: model predctve control, doman of attracton, uable reactor. 1. INTRODUCTION When a coraned open-loop uable system, as an uable reactor system, s attempted to be controlled the guarantee of recursve feasblty and coraned stablty s a hghly desrable controller property. Frst, the maxmal stablzable sets assocated to the system equlbrum should be carefully determned snce, opposte to what happens wth stable systems, nput corants could make mpossble the rejecton of large dsturbances, ndependently of the controller. Then, a controller wth guaranteed stablty that explctly takes nto account these lmtatons should be desgned. In ths context, MPC appears to be the most sutable opton. In fact, the stablty, feasblty and even optmalty of MPC s now well establshed n the theoretcal aspects (Mayne et al. (2), Rawlngs and Mayne (29)). Standard approaches use the dual-mode predcton paradgm (Scokaert and Rawlngs (1998)) n conjuncton wth an nfnte horzon. Wthn ths paradgm t s assumed that a fxed uncoraned feedback K (local controller) proceeds for predctons beyond the control horzon. stablzng n ths way the uable modes. In ths context, a major obstacle s to establsh a trade-off between the desrable volume of the doman of attracton (the set of states for whch the controller can generate a feasble nput), the overall complexty (computatonal cost), and the achevable performance for a gven control horzon (degree of optmalty). Assumng that the control horzon s chosen small for computatonal reasons, the doman of attracton s domnated by the aggressveness of the fxed uncoraned feedback, snce addtonal (termnal) corants are needed to assure the feasblty of the local control law. For stable systems, the null local controller K = (Rawlngs and Muske (1993)), whch represents the poorest tuned local controller, produces the maxmal doman of attracton. Ths s the case of the classcal nfnte horzon MPC (IHMPC). However, for the general case of uable systems, a termnal corant that cancels the uable modes s needed (as they cannot be steered to the orgn by the proposed local null controller), producng agan a severe reducton of the resultng doman of attracton. Nagrath et al. (22) presented an nterestng example of a open-loop uable jacketed chemcal reactor (CSTR) controlled by MPC. The objectve s to keep the Copyrght held by the Internatonal Federaton of Automatc Control 27

2 temperature of the reactng mxture coant at a desred value, whle the dsturbances to the system nclude the feed temperature and the jacket feed temperature and the only manpulated varable s the jacket flow rate. For ths knd of system, t s usual to adopt a cascade control structure: an nner-loop for the control of the jacket temperature and an outer-loop for the control of the reactor temperature. The authors compared three dfferent controllers: a classcal cascade controller, a fnte horzon MPC and an IHMPC. The output performance s clearly better for the MPC strateges, whle only the IHMPC assures stablty. As the authors declared, the problem wth the IHMPC s that to mplement approprately the cancellaton of the uable modes (termnal corant) a prohbtve large control horzon s needed. Agan, the problem of a reduced doman of attracton for small control horzons arses. More recently, González and Odloak (29) reformulates the orgnal IHMPC to allow an augmented doman of attracton for both, stable and uable systems. The man dea was to nclude an approprate set of slack varables nto the MPC optmzaton problem that, together wth some model formulaton propertes, allows a maxmal closed-loop doman of attracton (.e., the doman of attracton s determned by the system and the corants, and not by the controller). Ths paper proposes the study of the applcaton of IHMPC to an uable reactor system n terms of the doman of attracton and the output performance. Furthermore, a generalzaton of the IHMPC proposed n González and Odloak (29) s presented, and some steady state optmalty propertes, related to the capablty of the controller to account for unreachable output references, are dscussed. Notaton: vector (a, b) denotes a T b T T ; for a gven λ, λx = {λx : x X}. For a symmetrc postve defnte matrx P, x P = x T P x denotes the weghted Eucldean norm. Matrx I n R n n denotes the dentty matrx. n,m R n m denotes the null matrx. Consder a R na, b R nb and a set Γ R na+nb, then { the projecton operaton s defned as P roj a (Γ) = a R na : b R nb, (a, b) Γ }. Gven two sets S 1, S 2, the set S 1 \S 2 s defned as S 1 \S 2 = {x : x S 1 and x / S 2 }. 2. REACTOR SYSTEM Consder a contnuous strred-tank reactor (CSTR), n whch an exothermc adabtc rreversble frst-order reacton (A B) s descrbed by the followng non-lnear state equatons (Nagrath et al. (22), Russo and Bequette (1996)): ẋ 1 = q + φk(x 2 )x 1 + qx 1f (1) ẋ 2 = βφk(x 2 ) x 1 (q + δ)x 2 + δx 3 + qx 2f (2) ẋ 3 = δ 1 δδ 2 x 2 δ 1 (q c + δδ 2 )x 3 + δ 1 q c x 3f (3) In these equatons x 1 s the dmensonless concentraton of reactant A, x 2 s the dmensonless reactor temperature, x 3 s the dmensonless jacket temperature, x 1f s the dmensonless feed concentraton of reactant A to the reactor, x 2f s the dmensonless feed temperature, x 3f s the dmensonless jacket feed temperature and q c s the jacket flow rate. The state vector s defned as x := x 1 x 2 x 3 T, whle x 1f, x 2f and x 3f are the possble dsturbances to the reactor. In ths case, x 1f s consdered coant, and then the dsturbance vector wll be gven by l := x 2f x 3f T. Clearly, the set of equatons presented before s nonlnear. Eq. (1) represents the dynamc materal balance for the reactant A, Eq. (2) represents the dynamc energy balance nsde the CSTR, and Eq. (3) represents the dynamc energy balance around the coolng jacket. For the three equatons, the usual assumptons of coant volume, perfect mxng and coant physcal parameters are made. The tradtonal operaton of a CSTR conssts n an ndrect control of concentraton by means of the control of the reactor temperature (x 2 ), usng the coolng jacket flow rate (q c ) as manpulated varable. However, as Russo and Bequette (1996) and Russo and Bequette (1998) remark, a multplcty behavor could be found for the three-state CSTR model when q c s the manpulated varable. The exstence of the nput multplctes n the system can severely degrade the performance of controlled output and moreover, unfeasble operaton regons can appear. For ths reason, the selecton of an adequate operaton pont s a key for the correct operaton of the reactor. Several authors studed and derved condtons for steadystate multplctes for a CSTR. The parameters chosen n ths work are those shown n Russo and Bequette (1998), whch exhbt open-loop uable behavor for a range of reactor temperatures bounded by the lmt ponts: φ =.72, β = 8., δ =.3, γ = 2, q = 1., δ 1 = 1, δ 2 = 1., x 1f = 1., x 2f =. and x 3f = MODELING THE SYSTEM AND CONSTRAINTS The am of ths secton s to dscuss both, a lnear predcton model and the way the corants affect t when uable systems are attempted to be descrbed. As was shown n Nagrath et al. (22) block dagonal transton matrces, whch separate the uable from the stable modes of the systems, are desrable to represent uable systems for IHMPC. In ths context, a sutable model s as follows: x (k + 1) x un (k + 1) x st (k + 1) = I n F un x (k) x un (k) F st x st (k) y (k) = Υ Υ un Υ st x (k) x un (k) x st (k) +, B B un B st u (k) (4) where x X R n, n = max (nu, ny), represents the ntegratng modes artfcally nduced by the ncremental form of the model, x st X st R ns represents the stable modes, x un X un R nun represents the orgnal uable modes of the system, u (k) = u (k) u (k 1) R nu, y (k) R ny. Ths partcular block dagonal form of model (4) can be obtaned from the step response of the transfer functon model (González and Odloak (29), Rodrgues and Odloak (23)), or by an approprate smlarty transformaton of a gven state space model. The system has nun and ns uable and stables poles, respectvely. In addton, n s the number of ntegratng Copyrght held by the Internatonal Federaton of Automatc Control 271

3 modes (ntroduced by the ncremental form of the model) that s equal to the maxmum between the number of system nput and outputs. Matrx Υ s gven by Υ = In n,(n ny) f nu > ny, and by Υ = I n f nu ny. Matrces Υ un and Υ st account for the effect of the uable and stable states n the output. On the other hand, the nput feasble set U s defned as follows: U = { u : u max u u max and u mn u (k 1) u max }, where u (k 1) s the past value of the nput u. In addton, t s assumed that the states x = ( x, x un, x st) are coraned to belong to a set X, gven by X = X X un X st. Here, ths set s defned by the operatng wndow of the process. Set X must satsfy the nput corants, as follows: B u mn x B u max. Remark 1. Notce that the present formulaton takes a specal care of nput ncrements. Frstly, the nput ncrement s consdered as nput n the system (4), ead of the nput tself. Ths property gves an alternatve way to acheve an offset-free control, n comparson to the target calculaton strategy used n Nagrath et al. (22), that needs a separate optmzaton problem to be solved. Secondly, nput ncrement corants are ncluded together wth the nput corants. Ths cotutes an mportant feature of real processes usually dsregarded by the typcal MPC formulatons, and contrbutes to a better descrpton of the whole system, snce ths knd of corants lmts the maxmal doman of attracton of the system, as can be seen n the next sub-secton. 3.1 Characterzaton of the coraned steady states of the system Wthout loss of generalty t s assumed here that nu = ny = n (for non-square systems the procedure to characterze the steady state s smlar). For a gven output target (or set-pont) y sp, any steady state of the system ( ) x s, x un s, x st s, u s assocated wth ths target output should satsfy the followng condton I n I n F un I nun F st I ns Υ Υ un Υ st x s x un s x st s u s = B B un B st y sp. (5) From equaton (5), and assumng that rank ( B ) = nu, t follows that any steady state of the system are gven by ( ) x s, x un s, x st s, u s = (y sp,,, ). Ths last condton represents a useful property of the specfc model formulaton (4), as t says that any possble steady state can be condensed n a sngle state component ( xs), whle the other states are null 1. Now, snce the system s subject to corants, t should 1 Notce that the steady state s condensed n the state component that was artfcally ncluded n model (4) be steered to those steady states that satsfy the corants. The set of these admssble steady states s defned as 2 X s = { x s = ( x s, x un s, x st ) s X : x s X, x un s {} and x st s {} }. Notce that from the latter defnton, t follows that the set of feasble output steady states, y s = Cx s = x s, s drectly gven by X (ths means that any feasble ntegratng state s a steady state of the system). Remark 2. It s frequent n real applcatons that the desred output set-pont y sp s not reachable, that s, the desred steady state x s = (y sp,, ) s not n X s. Ths could be caused by a msmatch between the model used to compute the optmal steady state and the model used to compute the dynamc forecast. Despte n ths case an output offset wll necessarly appear, ths should not be a cause of ablty. 3.2 Characterzaton of the stablzable sets for the non-stable states A remarkable characterstc of the coraned uable systems s that there exsts a lmted set of non-stable states, called maxmal stablzable set, out of whch the system cannot be stablzed by any controller. It s useful at ths pont to group the ntegratng and the pure uable modes nto a sngle vector, x = ( x, x un), of non-stable states. In ths way system (4) can be rewrtten as x (k + 1) F x st = (k + 1) F st x (k) x st + (k) B B st u (k) y (k) = Υ Υ st x (k) x st, (k) where B T = B T B unt, F = dag (I n, F un ), Υ = Υ Υ un. Now we explot the steady state characterzaton presented n the last secton n order to defne some useful sets. Frst, let us consder the (equlbrum) set of feasble non-stable steady states, Xs = P roj x (X s ). Based on ths defnton, t s possble to defne the set of non-stable states that can be admssbly steered, by means of an admssble sequence of j control actons, from X = X X un to the equlbrum set X St j ( X, Xs ) = s : {x () X : for all k =,,j 1, u (k) U such that x (k) X and x (j) Xs }. Ths set (called the stablzable set for the non-stable states) s a control nvarant set for states x, for all j 1 (see Lemma 1 n González and Odloak (29)). Remark 3. As can be seen n Remark 1 of González and Odloak (29), the set St j (X, Xs ) tends to a lmted set as the number of steps j tends to. So, by ncreasng the ndex j up to N, n such a way that St N+1 (X, Xs ) St (X, Xs ), t s possble to N 2 Notce that, as was already sad, X s such that the nput corants are fulflled (.e., B u mn x B u max) Copyrght held by the Internatonal Federaton of Automatc Control 272

4 defne the largest possble doman of attracton for the noable states as Θ = St N (X, Xs ). Remark 4. If we now defne Θ = {x X : x Θ }, then ths set s the maxmal doman of attracton of any controller (largest possble doman of attracton of the system) snce t does not depend on the selected control law, but on the nature of the system and the states, as well as on the nput corants. Remark 5. The equlbrum set for the pure uable states s gven by Xs un = P roj x un (X s ) = {}. So, t s possble to defne the stablzable set for the uable states as St un j (X un, {}). Furthermore, the largest possble doman of attracton for the uable states s gven by Θ un = P roj x un (Θ ). 4.1 Classcal IHMPC 4. IHMPC FORMULATIONS As was establshed n Nagrath et al. (22), the man advantage of the nfnte over the fnte horzon s the elmnaton of the requrement of tunng for nomnal stablty. For uable systems, however, the non-stable modes must be canceled to acheve coraned stablzablty of the predctons (a corant s ncluded that forces ths cancellaton at the end of the control horzon m). For trackng a non-zero target, the classcal IHMPC formulaton adapted to model (4) s as follows: Problem 1 mn V k = u k subject to: m 1 j= { x (k + j k) x sp 2 Q + u (k + j k) 2 R u (k + j k) U, j =,, m 1 } + x st (k + m k) 2 P x (k + m k) y sp = (6) x un (k + m k) = (7) where x sp = (y sp,, ), x (k k) = x (k), u (k + j k) s the control move computed at tme k to be appled at tme k+j, m s the control horzon, Q and R are postve weghtng matrces of approprate dmenson, y sp s the output reference, and u k = ( u (k k),, u (k + m 1 k)). Because of the termnal corants (6) and (7), the cost of the IHMPC can be wrtten as a fnte horzon cost wth a termnal penalty term (the termnal matrx P s computed by solvng the Lyapunov equaton P = Q + F stt P F st ). Two man problems arse when ths formulaton s ntended to be appled. The doman of attracton for the non-stable modes s gven by the m-stablzable set St m (X, {}), because of corants (6) and (7). Ths set s very small, manly f the practcal case of nput ncrement corants s consdered, because t consders as target set the orgn (ead of the equlbrum set ) and because t consders only m steps to reach ths target set. On the other hand, f an unreachable set-pont s used, the cost becomes unbounded and feasblty/stablty s lost. Xs 4.2 IHMPC wth large doman of attracton A possble soluton to these problems s the ncluson of slack varables to relax the termnal corants. In González and Odloak (29) a formulaton was presented that accounts for the problem of ncludng slack varables nto the nfnte horzon MPC optmzaton problem, wth the fnal objectve of enlargng the resultng doman of attracton. The man dea of ths approach was to separate the convergence (man objectve of an IHMPC) n two steps: the frst one s devoted to steer the non-stable state x to the set St m (X, Xs ) n a fnte number of steps; whle the second one s devoted to steer the ntegratng and stable states to the desred equlbrum pont (( x, x st) (y sp, ) ). The slack varables assure the feasblty at any tme and for any non-stable state n the maxmal doman of attracton St (X, Xs ). As a result, the controller has the largest possble doman of attracton,.e. the doman of attracton s gven by Θ, whch does not depend on the control law. 4.3 A novel IHMPC wth large doman of attracton The strategy presented n González and Odloak (29) has some lmtatons, snce t does not deal wth an arbtrary number of uable states. A possble generalzaton of ths strategy conssts of usng a generalzed Mnkowsk functonal, assocated to the stablzable sets St un j (X un, {}), for m j N (where N s as n Remark 3), as the cost functon of a frst optmzaton problem of a two-stage MPC formulaton. Before presentng the novel IHMPC formulaton, we ntroduce the followng defntons: Defnton 1. Gven a convex set S X, the Mnkowsk functonal Ψ S assocated to S s defned as Ψ S (x) = nf{µ : x µs}. To see the propertes of the functon Ψ S (x), see Blanchn (1999). Consder now a sequence of -symmetrc convex sets S 1 S 2... S n. Then, a correspondng sequence of parwse dsjont sets, Υ +1 = nt(s +1 )\nt(s ), = 1,..., n 1, whch are a partton of S n, could be defned. Now, followng the dea of the Mnkowsk functonal presented above, t s possble to assocate a functon to the whole sequence, n such a way that the level surfaces of ths new functonal are the contours of the sets S : Defnton 2. Gven a sequence of convex sets S 1 S 2... S n, the generalzed Mnkowsk functonal Ψ {S1,...,S n} assocated to S s defned as { ΨνS Ψ {S1,...,S n}(x) = (x) f x Υ +1, = 1,..., n 1 Ψ ν1s 1 (x) f x S 1 where the coeffcents ν are such that ν S ν +1 S +1. Remark 6. To obtan the coeffcents ν the followng optmzaton problem must be solved: ν = mn{ν : νs ν +1 S +1 }, for = 1,..., n 1, and ν n = 1. Once the coeffcent are obtaned, the sequence of sets ν S are such that ν S ν +1 S +1, for = 1,..., n 1. Copyrght held by the Internatonal Federaton of Automatc Control 273

5 The generalzed Mnkowsk functonal appled to uable stablzable sets has the followng propertes, whch are useful for the MPC formulaton: Ψ {St un m,,st un N } (x un ) s greater than zero for all non-null values of x un, Ψ {St un m,,st un N } (x un ) s null f and only f x un s null, and the contour of the stablzable sets {St un m, St un N } are level surfaces of the functonal Ψ {St un m,,st un N } (x un ). Based on the above defntons, the proposed IHMPC formulaton, whch conssts of the followng optmzaton problems that are solved sequentally: Problem 2a Problem 2b mn u b,k,δ k,δun k mn u a,k V a,k = Ψ {St un subject to: m,,st un N } (x un (k + m k)) u a (k + j k) U, j =,, m 1 V b,k = m 1 j= + u b (k + j k) 2 R subject to: u b (k + j k) U, j =,, m 1 { x (k + j k) x sp + δ (k, j) 2 Q } + x st (k + m k) 2 P + δ k 2 S x (k + m k) y sp + δ k = (8) x un (k + m k) + F unm δk un = (9) Ψ {St un m,,st un N } (x un (k + m k)) Ψ (1) where S s a postve( weghtng matrces ) of approprate dmenson, δ (k, j) = δk, F unj δk un, are slack varables, and Ψ s the optmal cost of Problem 2a. Because of the slacked termnal corants (8) and (9), the cost of ths IHMPC can be wrtten as a fnte horzon cost wth a termnal penalty term, as t s done n a classcal IHMPC. Corant (1) forces the uable states to reman n the stablzable set determned by Problem 2a. The followng algorthm produces a stablzng control law wth a doman of attracton gven by Θ: Algorthm 1. Solve Problem 2a and pass the optmal value of the cost to Problem 2b. Implement the frst control acton u b (k k). Also, the control sequence obtaned from the executon of Algorthm 1 at successve tme steps drves the output of the closed loop system asymptotcally to a pont that mnmzes δk 2 (partcularly, f the output set-pont S s reachable, the output of the closed loop system s asymptotcally steered to t wthout offset). Remark 7. The termnal corant for the uable and ntegratng states n Problem 2b can be wrtten as C y sp x (k) δ C un k u b,k = F unm (x un (k) δ un k ) where C = B B B, C un = F unm 1 B un F unm 2 B un B un. So, wth ths two-stage formulaton, the control horzon m should be, large enough to assure that matrx rank. C T C unt T s full 5. APPLICATION TO THE UNSTABLE REACTOR The am of the smulaton results s to compare the performance and feasblty of the proposed formulaton wth the classcal IHMPC, when the uable reactor of secton 2 s consdered. The objectve s to control one output varable (reactor temperature) manpulatng one nput varable (jacket flow rate). Usng Taylor seres expanson and a convenent transformaton, the followng dagonalzed dscrete tme lnear model s obtaned (T=.5): A =.9531, B = and C = The nput corants are gven by: u max = 1.32, u mn =, u max =.1. The nput ncrement bounds are chosen small to clearly show ts effect on the controllers doman of attracton. The tunng parameters of the proposed IHMPC are: Q = 5, R =.1 and S = The coeffcents ν of the generalzed Mnkowsk functonal used for Problem 2a are gven by: ν 5 = 1.8, ν 6 = 1.5, ν 7 = 1.4, ν 8 = 1.3, ν 9 = 1.15 and ν 1 = 1. The tunng parameters of the classcal IHMPC are: Q = 5,R =.1. Frst, we analyze the doman of attracton for the noable modes of both controllers. As was already sad, the doman of attracton of the IHMPC depends on the control horzon, and t s gven by St m (X, {}). Fgure 1 shows these sets for m = 1, m = 2, m = 24, m = 28 and m = 3 (dashed-lne). On the other hand, the doman of attracton of the proposed controller s the maxmal doman of attracton of the system (and so, t does not depend on the control horzon), and t s gven by St 15 (X, Xs ) (sold-lne). Notce that for m > 28 the doman of attracton of the IHMPC remans almost the same as the one obtaned wth m = 28 and furthermore, ths maxmal set s smaller than the maxmal doman of attracton of the system. Ths shows that f nput ncrement corants are consdered, the doman of attracton of the classcal IHMPC could not be the largest possble, even for very large control horzons. Now, the output performance of both controllers are compared. Frst, a state dsturbance s smulated, such that the orgnal non-stable states s n St 7 (X, Xs ). The dsturbance s gven by: x() = T, whch correspond to the followng dsturbance n the orgnal state varables: concentraton of reactant =.12, reactor temperature =.23 and jacket temperature =.25 (wrtten as devaton varables). The proposed controller steers frst the system to St m (X, Xs ), wth m = 5, n two steps, and then regulates the system to the desred equlbrum pont. Fgure 2 shows the evoluton of the non-stable states (sold-lne and crcles). On the other hand, the classcal IHMPC needs a control horzon equal Copyrght held by the Internatonal Federaton of Automatc Control 274

6 Fg. 1. Doman of attracton for both, the proposed controller and the classcal IHMPC. Fg. 3. Input, nput ncrement and output responses for a set-pont change. new IHMPC formulaton was presented that explots the propertes of a partcular model structure and exhbts the largest possble doman of attracton. As other recent formulatons, t guarantees recursve feasblty and stablty for trackng both, reachable and unreachable output set-ponts. Furthermore, n the latter case, the resultng controller steers the system to an admssble statonary output that mnmzes the dstance to the desred setpont, wthout the necessty of a target calculaton stage. Fg. 2. Non-stable state evoluton to 3 to account for ths dsturbance. Fgure 2 shows the non-stable state evoluton for the IHMPC and m = 15 (dashed-lne and crcles), where t can be seen that the corants are volated on the rght hand sde of the fgure. In a second stage, a set-pont change of.1 s smulated. A control horzon of m = 5 was selected for the proposed controller, whle a control horzon of m = 9 and m = 12 was selected for the IHMPC n order to make t feasble. Fgure 3 shows the tme responses, where t can be seen that the proposed controller has a slghtly better performance. Fnally, a set-pont change of.35 s smulated, whch s an unreachable target (t corresponds to an unreachable equlbrum state x sp = 1.77 T ). In ths case, the IHMPC cannot be used, whle the proposed controller, because of the effect of the slack varables, steers the system to a feasble pont that mnmzes the dstance from the desred set-pont. Fgure 2 shows the non-stable state evoluton (dotted-lne and crcles), together wth the unreachable equlbrum state (crcle-star). 6. CONCLUSION A study was developed of the capablty of IHMPC to account for uable reactor systems. Performance and feasblty (together wth stablty) was studed, and a REFERENCES Blanchn, F. (1999). Set nvarance n control. Automatca, 35, González, A.H. and Odloak, D. (29). Enlargng the doman of attracton of stable mpc controllers, mantanng the output performance. Automatca, 45, Mayne, D., Rawlngs, J., Rao, C., and Scokaert, P. (2). Coraned model predctve control: Stablty and optmalty. Automatca, 36, Nagrath, D., Prasad, V., and Bequette, W. (22). A model predctve formulaton for control of open-loop uable cascade systems. Chemcal Engneerng Scence, 57, Rawlngs, J. and Mayne, D.Q. (29). Model Predctve Control: Theory and Desgn. Nob Hll Publshng. Rawlngs, J. and Muske, K. (1993). The stablty of coraned recedng horzon control. IEEE Trans. on Autom. Control, 38, Rodrgues, M. and Odloak, D. (23). Mpc for stable lnear systems wth model uncertanty. Automatca, 39, Russo, L.P. and Bequette, W. (1996). Effect of process desgn on the open-loop behavor of a jacketed exothermc cstr. Computers chem. Engng., 2, Russo, L.P. and Bequette, W. (1998). Operablty of chemcal reactors: multplcty behavor of a jacketed styrene polmerzaton reactor. Chemcal Engneerng Scence, 53, Scokaert, P.O.M. and Rawlngs, J.B. (1998). Coraned lnear quadratc regulaton. IEEE Trans. on Autom. Control, 43, Copyrght held by the Internatonal Federaton of Automatc Control 275