Feedback Min-Max Model Predictive Control Based on a Quadratic Cost Function

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1 Proceedngs of the 26 Amercan Control Conference Mnneapols, Mnnesota, USA, June 14-16, 26 WeC5.5 Feedback Mn-Max Model Predctve Control Based on a Quadratc Cost Functon D. Muñoz de la Peña,T.Alamo, A. Bemporad and E.F. Camacho Abstract Feedback mn-max model predctve control based on a quadratc cost functon s addressed n ths paper. The man contrbuton s an algorthm for solvng the mn-max quadratc MPC problem wth an arbtrary degree of approxmaton. The paper also ntroduces the recourse horzon, whch allows one to obtan a trade-off between computatonal complexty and performance of the control law. The results are llustrated by means of a smulaton of a quadruple-tank process. Keywords: Predctve control for lnear systems; Robust control; Optmzaton algorthms I. INTRODUCTION Most control strateges are based on a mathematcal model of the process to be controlled. Usng ths model, the controller obtans the control nput n such a way that a gven cost crteron s mnmzed. Ths means that a controller s effcency depends greatly on how precsely the mathematcal model represents the real behavor of the system. Ths s even more crtcal n the case of model predctve control (MPC) where the control decson s taken on the base of the future predcted evoluton of the system whch s obtaned usng a model of the system assumed to be perfect n most cases [1]. One way to deal wth uncertantes n MPC s to consder a worst case approach [2]. Ths approach s denoted mnmax [3], [4], [5]. Recent works deal wth feedback MPC [6], [7]. Feedback mn-max MPC obtans a sequence of feedback control laws that mnmzes the worst case cost whle assurng robust constrant handlng. It requres the soluton of a very hgh dmensonal problem that makes ts practcal mplementaton very hard. For problems based on a cost functon that can be evaluated wth a lnear programm (LP), the explct soluton has been obtaned [8], [9], [1]. Lately, an effcent onlne algorthm was proposed by the authors n [11]. There are not any equvalent results n the lterature for quadratc cost functons although several approxmate solutons have been gven (see [5], [6], [12], [13]). In ths paper, the algorthm presented n [11] s extended to deal wth quadratc cost functons. The man contrbuton s that the modfed verson of the algorthm provdes a feasble soluton wth an arbtrary degree of approxmaton. To the best knowledge of the authors there s no equvalent result Dep. de Ingenería de Sstemas y Automátca, Unversdad de Sevlla, Span. Dp. d Ingegnera dell Informazone, Unversty of Sena, Italy. E-mal: davdmps@cartuja.us.es (D. Muñoz de la Peña), alamo@cartuja.us.es (T. Alamo), bemporad@uns.t (A. Bemporad), eduardo@cartuja.us.es (E.F. Camacho) n the lterature. The algorthm s based on the structure and the convexty propertes of the quadratc cost functon. It apples a nested decomposton procedure to solve the mn-max problems va a sequence of low order quadratc programs. Ths dea was frst ntroduced by Benders n [14] for solvng mxed nteger problems. However, the computatonal burden of the algorthm stll grows exponentally wth the length of the predcton horzon. In order to arbtrarly lmt the possble combnatoral exploson, n ths paper t s proposed to consder that the dsturbance only acts on a recourse horzon, whch can be shorter than the predcton horzon. Ths dea s motvated by the engneerng sense of not takng nto account possble model uncertantes for long predctons because the MPC controller s mplemented n a recedng horzon way. Ths dea s related to the control horzon concept, whch s generally used to keep manageable the number of free varables of the optmzaton problem. II. PROBLEM FORMULATION Consder the dscrete tme lnear system wth x(t +1)=φ(x(t),u(t),w(t)), (1) φ(x, u, w) =A(w)x + B(w)u + D(w), subject to constrants G x x(t)+g u u(t) g, (2) where x(t) R nx s the state vector, u(t) R nu s the nput vector and w(t) R nw s the uncertanty vector that s supposed to be bounded, namely w(t) Wwhere W s a closed polyhedron. The system matrces are defned by the uncertanty as A(w) =A + nw e T k wa k, B(w) =B + nw e T k wb k, (3) D(w) = nw e T k wd k, where e k s the k-th column of the dentty matrx of sze n w. Ths s a general descrpton of uncertanty for lnear systems and ncludes both parametrc and addtve uncertantes (see [5], [8]). In general the complexty of Feedback mn-max MPC [6], [7] grows n an exponental manner wth the predcton horzon. In order to mprove the relaton between computatonal complexty and performance, we propose to assume /6/$2. 26 IEEE 1575

2 that the dsturbance only acts for a fnte tme N r N denoted recourse horzon, where N s the overall predcton horzon. Ths choce has two dfferent motvatons. Frst, from an engneerng vewpont t makes sense not to take nto account possble model uncertantes after a certan predcton tme, as the control law s mplemented n a recedng horzon scheme. Second, by fxng N r <Nthe complexty of the problem (n terms of optmzaton varables and constrants) wll only grow lnearly wth N, whchs clearly advantageous from a computatonal vewpont. Note that ths dea s equvalent to usng as a termnal cost, the optmal cost functon of a nomnal MPC controller. The feedback mn-max optmal control problem wth a recourse horzon s defned by the followng problem whch optmum value s denoted J (x), mn {L(u(),x()) + max { mn {L(u(1),x(1))+ u(),x() w() u(1),x(1) max {...{ mn {L(u(N r 1),x(N r 1))+ w(1) u(n r 1),x(N r 1) max { mn { N 1 L(x(t),u(t)) + F (x(n))}...} w(n r 1) u(t),x(t),t N r t=n r (4) subject to x() = x, x(t +1)=φ(x(t),u(t),w(t)), t=,...,n r 1, x(t +1)=φ(x(t),u(t), ), t= N r,...,n 1, G x x(t)+g u u(t) g, t =,...,N, w(t) W, where L(x, u) and F (x) are the stage and termnal cost functons respectvely and are gven by L(x, u) =x T Qx + u T Ru, F (x) =x T Px, wth Q, R and P postve defnte matrces. The control law s appled n a recedng horzon scheme. At each samplng tme the problem s solved for the current state x and J (x) s obtaned. The controller apples the optmal control nput for the frst tme step whch s denoted u(). Note that ths optmzaton problem s of very hgh complexty. A. Worst Case Scenaro Tree In order to solve the mn-max problem, not all possble values of the uncertanty (whch leads to an nfnte dmensonal problem) have to be taken nto account, but only the extreme realzatons (.e. the vertces of W). Ths s a well known result (see [7], [11]). In ths way, n order to keep the sequence of decson-uncertanty realzaton, the extreme realzatons of the uncertanty can be represented n a scenaro tree as n [7]. Ths tree s used to solve the mn-max problem as a fnte dmensonal determnstc problem. The root node of the tree represents the ntal tme step j =and each new level of the tree stands for a new tme step so each node has q chldren, one for each vertex of W. Each node s then defned by an uncertanty vector w vert(w) whch characterzes the uncertanty realzaton from the parent node. Note that the uncertanty s only taken nto account n the recourse horzon. Ths means, that the w 1 =1 w 2 = 1 w 3 =1 1 w 4 = 1 w 5 =1 2 w 6 = 1 Fg. 1. Scenaro tree wth N r =2and q =2 scenaro tree s made of N r levels, and that each leaf node, has the nformaton of the predctons along the rest of the predcton horzon. All the nodes of the tree are numbered, startng from the root node (node ) to the leaf nodes, stage by stage (so the enumeraton of the nodes of a gven stage s lower than ther chldren nodes). M s the total number of nodes. Each node has a set of chldren I() and a parent node p(). Thesetof chldren s empty for the leafs nodes and the root node has no parent. Fgure 1 shows an example scenaro tree wth N r =2and two possble uncertantes realzatons (w { 1, 1}). The chldren sets I and the the parent nodes p() are gven by: I() = {1, 2} I(1) = {3, 4} I(2) = {5, 6}, p(1) = p(2) = p(3) = p(4) = 1 p(5) = p(6) = 2 The scenaro tree s used to defne an optmzaton problem that s equvalent to the mn-max problem proposed n the prevous secton. To each node of the tree s assgned a set of varables and a cost to go functon defned by the followng optmzaton problem ˆV (x p(),u p() ) = mnl(x,u )+ max ˆV j (x,u ) x,u j I() s.t. x = φ(x p(),u p(),w ), G x x + G u u g. (5) The ndex of the varables denotes node enumeraton. The cost functon ˆV depends on the prevous decson varables,.e. the varables of the father node x p(),u p() and on the cost functon of ts chldren. To obtan the control nput, the cost functon of the root node s mnmzed for a gven ntal state x,.e. the followng. 1576

3 optmzaton problem s solved: ˆV (x) = mn L(x,u )+ max ˆV j (x,u ) x,u j I() s.t. x = x, G x x + G u u g. The defnton of ˆV (x) takes nto account that the state of the root node s gven by the measured state of the system. The leafs node (nodes such that I() s empty) are defned by the cost functon of a nomnal MPC control law. The cost to go ˆV (x p(),u p() ) of each leaf node s gven by the followng problem N 1 mn L(x (t),u (t)) + F (x (N)) u (t),x (t),t N r t=n r subject to x (N r )=φ(x p(),u p(),w ), x (t +1)=φ(x (t),u (t), ), t= N r,...,n 1, G x x (t)+g u u (t) g, t = N r,...,n. Ths problem optmzes the cost of the nomnal trajectory from N r to N as n standard MPC. The most mportant dfference between problems (4) and (5), s that n (5) all the parameters are determnstc. That s, each node has a correspondng known realzaton of the uncertanty and the maxmzaton s done over the correspondng cost functons of the chldren nodes (whch s a fnte set). It s a mult-stage mn-max quadratc program. In the followng sectons ths knd of problems are ntroduced and an algorthm that explots the problem structure s presented. III. MULTI STAGE MIN-MAX QUADRATIC PROGRAMMING In ths secton the mult-stage mn-max quadratc program n standard form s presented. Problem (5) can be formulated as a mult-stage mn-max quadratc program. To pose t n standard form, auxlary and slack varables have to be ntroduced as n lnear programmng (see [15]). The mult-stage mn-max quadratc problem n standard form s defned as (z p()) = mnz z T H z +max j ) j I() (6a) s.t. W z = h A z p(), (6b) z. (6c) V wth H >. Ths knd of problems are defned by a scenaro tree as the one ntroduced n the prevous secton. Each node has a set of chldren I() and a parent node p(). Note that ths set s empty for the leafs nodes and the root node has no parent. Each node s defned by matrces and vectors H,W,h and A. All these parameters are determnstc and can be dfferent for each node. In the feedback mn-max MPC case, the matrces and vectors H,W,h and A depend on the system, the cost functon, the constrants and on the value of the uncertanty from the parent node to the node (what n the prevous secton was defned as w ). The ntal state of the system defnes the constrants n the root node whch does not depends on any prevous decson, namely h depends on x. The objectve s to mnmze V, the cost functon n the root node. The boundary condtons are gven by the problem solved at each leaf node. The set of varables z correspondng to each node ncludes the state, the nput, and the slack varables needed to represent the feedback mn-max problem n standard form. Note that because of the recourse horzon, the varables z of the leafs nodes are of a hgher dmenson than the rest of the nodes. Note also, that as the set I s empty for these nodes, the cost s defned as a determnstc quadratc programmng (QP) problem. IV. NESTED DECOMPOSITION ALGORITHM The general dea of decomposton algorthms was frst ntroduced by Benders n [14] for solvng mxed nteger problems and has been successfully appled to stochastc programmng [16], [17]. In ths secton the algorthm for solvng mult-stage mn-max lnear programs based on Benders decomposton presented by the authors n [11] s extended to deal wth the quadratc case. At each step, the algorthm obtans a feasble set of varables for the orgnal problem. These varables mght not be optmal, but a bound on the error s gven. Ths bound decreases at each teraton. In order to obtan the feasble set of varables, at each step m, a subproblem s solved for each node. These subproblems have the same constrants on z, but addtonal varables and lnear constrants are added to evaluate the maxmzaton and approxmate the functons V j (z ) by an outer lnearzaton,.e. a lower bound that can be evaluated usng a lnear problem. These lower bounds are mproved at each teraton and converge to the real values of V j (z ). The lower bounds are denoted by V m defned as follows (problems P m ) (z p() ) and are mn z T H z + θ z,θ,θ,j (7a) s.t. W z = h A z p(), (7b) z, (7c) D,j k z + θ,j d k,j, j I(), k rm,j, (7d) θ θ,j, j I(), (7e) θ,θ,j, j I(). (7f) Constrants (7b) and (7c) are the constrants (6b) and (6c) of the orgnal problem. The rest of the constrants evaluate the maxmzaton and the lower bounds on Vj ). Note that these constrants do not lmt the feasble set of z. Each θ,j s a lower bound on the value of Vj ), that s, the value of the cost functon of the chldren. Ths value s obtaned evaluatng an outer lnearzaton, namely constrants (7d). In ths way, the followng nequaltes hold Vj (z ) Vj m (z ) θ,j, j I(). At the frst teraton rj 1 =for all nodes =,...M. Ths means that for the frst teraton, the value of θ,j of 1577

4 the chldren of each node s consdered to be zero (recall that θ,j ). Each tme a new optmalty cut s added (r,j m ncreases), the approxmaton of Vj ) becomes tghter. The followng theorem shows how to defne constrants (7d) n order to evaluate a lower bound of Vj ). Theorem 1 (c.f. [15]): Defne D = λ T A and d = λ T A z p() + V m (z p() ),whereλ are the dual varables of the equalty constrants (7b) for a gven z p() and V m (z p() ) s the optmal value of the cost functon. Then t holds that for all z, V m (z) d D z. (8) Proof: The dual problem D m s defned as (note that strong dualty holds): V m (z p() ) = max g(z p(),λ,s,μ k λ,s,μ k,j,μ j),j,μj s.t. 1 μ j, μ j j I() r m j μ k j, j I(), s,μ j,μ k,j. (9) where λ,s,μ k,j,μ j are the dual varables correspondng to constrants (7b), (7c), (7d) and (7e) respectvely and g(z,λ,s,μ k,j,μ j) s the dual functon for a gven value of z p() = z defned as wth 1 2 f T H 1 f +(h A z) T λ + f = W T λ + r m j j I() r m j d k j μk j j I() D kt j μk j + s. The parameter z p() does not affect the constrants of the dual problem so gven the set of optmal dual varables λ,s,μ k,j,μ j for z p(), the followng nequalty holds for all z V (z) g ( z,λ,s,μ k,j,μ j), because the set of dual varables s feasble for (9) n z. Then takng nto account ths nequalty, that g(z,λ,s,μ k,j,μ j) g(z p(),λ,s,μ k,j,μ j), s equal to λ T A z p() λ T A z, that g(z p(),λ,s,μ k,j,μ j)= V (z p() ) and the defnton of d and D,tsprovedthat(8) holds. These optmalty cuts are obtaned from feasble solutons to the dual problem of P m. Note that as the number of optmalty cuts rj m s ncreased at each step m for each chldren node j, the set of dual constrants of a prevous optmum soluton may not be optmal, but stll remans feasble f new zero varables μ k j of the new optmalty cuts are added. Ths way, although problems P m may dffer on each teraton, the lower bounds on the optmal value reman vald. Constrants (7e) evaluate the maxmzaton over all the chldren of node usng an epgraph approach. As n the prevous secton, when solvng the root node, constrants (7b) are replaced by W z = h, because the root node has no parent. For the feedback mn-max problem, h depends on the ntal state of the system x. When solvng a leaf node, varables θ,j and constrants (7d) are omtted because these nodes do not have chldren. Note that ths means that by defnton, V m (z) = V (z) for each leaf node and every algorthm teraton m. The algorthm solves problems wth relatve complete recourse [16],.e. feasblty of the root problem P 1 assures feasblty of all the problems of the nodes of the scenaro tree for all steps m. For general problems, feasblty cuts can be added to the algorthm as n stochastc lnear programmng [16], [17]. Note that feedback mn-max can be formulated to have relatvely complete recourse (see [8], [11]). Summng up, the general dea s to approxmate the cost functon of each node wth an outer lnearzaton from the leaf nodes, up to the root node. At each teraton, f the error on a node s greater than a gven value, the algorthm adds a new optmalty cut. The bound of the error s evaluated from the leaf nodes to the root node n a recursve way, takng nto account that no approxmaton s done at the leaf nodes. As both problems are subject to the same constrants on z,a feasble soluton of V m, s a feasble soluton for the orgnal mult-stage mn max problem. The proposed algorthm s the followng: Algorthm 1: Proposed algorthm. m =,rj =,=1,...,M 1, j I(). f P s unfeasble Mult-stage mn-max problem s unfeasble. End of the algorthm. do (V m,z,λ,e )=solvep (,-). m = m +1. whle e >ɛ. wth solvep a recursve functon that obtans for a gven node and a gven father varable z p(), the value of the cost functon V m, the varable z, the dual varables of the equalty constrants λ and a bound on the error e for teraton m. Note that for the root node (node ), there s no parent. Ths functon s defned as follows: (V m,z,λ,e )=solvep (,z p() ) Solve P m usng z p() and obtan V m, z, θ,j and λ. for j I() (Vj m,z j,λ j,e j )=solvep (j,z ) f Vj ) θ,j >. r,j m+1 = r,j m +1. D rm+1,j,j = λ T j A j. d rm+1,j,j else r m+1,j = r m,j. end for. = λ T j A jz + Vj ). e = θ +max j I() V m j (z )+e j. 1578

5 Theorem 2: The soluton obtaned applyng Algorthm 1, denoted z, s a feasble suboptmal soluton of (6) and the followng nequaltes hold V m V V m + e. Proof: Feasblty of the soluton s assured because both sets of problems are subject to the same constrants on z (recall constrants (6b)-(6c) and (7b)-(7c)). In order to proof that the error s bounded by e,alower and an upper bound of the optmal cost functon are obtaned. Frst we derve the lower bound. Takng nto account Theorem 1, the followng nequalty holds for all teraton m and node j Vj m (z) max d k,...,r m p(j),j Dk p(j),jz. (1) p(j),j For the leafs nodes, I() s empty so t holds that V m (z) (z) because n ths case (6) and (7) are equal. Takng = V ths nto account and applyng (1) backwards from the leaf nodes, t holds for all teraton m and node V m (z) V (z). (11) The upper bound s obtaned n a recursve way. Suppose that for a gven node and a gven varable z Vj (z ) Vj m (z )+e j, (12) for all j I(). Then, the followng expressons hold V m V (z p() )=z T H z + θ, (z p()) z T H z +max j I() V j (z ), z T Hz +max V j (z ) z T H z +max V j )+e j, j I() j I() so takng nto account the defnton of e, the followng nequalty holds V (z p()) V m (z p() )+e. (13) For the leafs nodes, (12) holds for e j =. Applyng (12) backwards to the root node, t s easy to see that V V m + e. Convergence of the algorthm s not proved n ths paper. Future works wll tackle ths ssue. V. QUADRUPLE-TANK PROCESS In ths example, feedback mn-max MPC s appled to a quadruple-tank process lke the one presented n [18]. Ths plant has four connected tanks. See Fgure 2 for a layout of the plant. The plant s made of.6m 2 secton tanks wth normalzed dampng factors of e 4m 2. The 3-way valves have factors γ 1 =.3 and γ 2 =.4. (see [18] for a complete descrpton of the dynamcs) 1. The nonlnear model s lnearzed n the equlbrum pont gven by the followng heghts ([m]) and flows ([m 3 /h]): h = [ ] T, q = [ ] T. 1 Ths model belongs to a real quadruple tank process of the Unversty of Sevlle. TABLE I COMPUTATIONAL TIMES [S] FOR DIFFERENT RECOURSE HORIZONS FOR THE QUADRUPLE-TANK PROCESS. N = N r mn TABLE II COMPUTATIONAL TIMES [S] FOR DIFFERENT RECOURSE AND PREDICTION HORIZONS FOR THE QUADRUPLE-TANK PROCESS. N/N r and the followng lnear dscrete tme model wth a samplng tme 1s s obtaned x k+1 = x k.9473 (14) u k w k The state and the nput are the error from the equlbrum pont. The system has flow constrans 1.2 (q + u) 2 and state constrants h + x 1. The uncertanty s restrcted to the set W = {w : w.1}. The weghtng matrces are Q = P =1I, R = I. Note that the complexty of the feedback formulaton grows wth 4 Nr because n w =2, renderng the computaton of ths control law a very hard problem. The proposed algorthm s appled to mplement a feedback mn-max controller for ths system 2. Tables I and II show the mean computaton tme over a hundred dfferent ntal states of the proposed algorthm for dfferent predcton and recursve horzon. The bound of the error s set to ɛ =1 3. It can be seen that the computaton tme grows exponentally wth the recourse horzon but not wth the predcton horzon. As the samplng tme s 1s, a mn-max controller wth a recurson horzon of up to 5 and a predcton horzon of 1 can be appled. Note that a feasble soluton wth a bound on the error s avalable at any teraton. A smulaton s shown n fgure V. The smulaton lasts 4 tme steps.the dashed lnes represent the reference, whle the full lnes represent the smulated trajectores. Note that around tme step 16, h 1 goes beyond the reference and the control law changes. 2 The smulatons have been done wth MATLAB 6.3 n an Athlon 28 usng CPLEX 9.1 as QP solver. 1579

6 Fg. 2. Quadruple-tanks process. h 3 h 4 γ 1 γ 2 h 1 h 2 q 1 q 2 Fg. 3. Smulaton results for the quadruple-tank process wth a feedback mn-max MPC wth Nr =5and N =1. h 1 h 2 h 3 h 4 q 1 q Tme step (a) Levels Tme step (b) Flows. VI. CONCLUSIONS Mn-max MPC approaches are used to ncrease the robustness propertes of a controller. However, most worst case approaches, have a great computatonal burden n common, specally n the case of feedback MPC. The algorthm presented n ths paper along wth the noton of the recourse horzon makes possble to mplement for real plants an approxmated feedback mn-max control law based on a quadratc cost functon. The approxmaton error bound s computed at each step of the algorthm and can be made arbtrarly small ncreasng the number of teratons. It s mportant to remark that to the best knowledge of the authors, ths s one of the few results avalable to mplement the feedback mn-max MPC controller presented. The computatonal tmes, although grow n an exponental manner, are manageable at least for the system under consderaton. Future works nclude the proof of the algorthm convergence, and an applcaton to the real quadruple tank process. REFERENCES [1] E. Camacho and C. Bordóns, Model Predctve Control, 2nd Edton. Sprnger-Verlag, 24. [2] H. S. Wtsenhausen, A mn-max control problem for sampled lnear systems, vol. 13, no. 1, pp. 5 21, [3] P. Campo and M. Morar, Robust model predctve control, vol. 2, 1987, pp [4] J. Allwrght and G. Papavaslou, On lnear programmng and robust model-predctve control usng mpulse-responses, Systems & Control Letters, vol. 18, pp , [5] M. Kothare, V. Balakrshnan, and M. Morar, Robust constraned model predctve control usng lnear matrx nequaltes, Automatca, vol. 32, pp , [6] J. Lee and Z. Yu, Worst-case formulatons of model predctve control for systems wth bounded parameters, Automatca, vol. 33, no. 5, pp , [7] P. O. M. Scokaert and D. Q. Mayne, Mn-max feedback model predctve control for constraned lnear systems, IEEE Transactons on Automatc Control, vol. 43, no. 8, pp , [8] A. Bemporad, F. Borrell, and M. Morar, Mn-max control of constraned uncertan dscrete-tme lnear systems, IEEE Transactons on Automatc Control, vol. 48, no. 9, pp , September 23. [9] E.C.Kerrgan and J. Macejowsk, Feedback mn-max model predctve control usng a sngle lnear program: Robust stablty and the explct soluton, Internatonal Journal of Robust and Nonlnear Control, vol. 14, pp , 24. [1] M. Dehl and J. Björnberg, Robust dynamc programmng for mnmax model predctve control of constraned uncertan systems, vol. 49, no. 12, pp , 24. [11] D. Muñoz de la Peña, T. Alamo, and A. Bemporad, A decomposton algorthm for feedback mn-max model predctve control, n Proceedngs of the 44rd IEEE Conference on Decson and Control, Sevlle, Span, December 25. [12] Y. Wang and J. Rawlngs, A new robust model predctve control method : theory and computaton, Journal of Process Control, vol. 14, p , 24. [13] V. Sakzls, N. Kakals, V. Dua, J. Perkns, and E. Pstkopoulos, Desgn of robust model-based controllers va parametrc programmng, Automatca, vol. 4, pp , 24. [14] J. Benders, Parttonng procedures for solvng mxed-varables programmng problems, Numer. Math, vol. 4, pp , [15] S. Boyd and L. Vandenberghe, Convex Optmzaton. Cambrdge Unversty Press, 24. [16] J. Brge and F. Louveaux, Introducton to Stochastc Programmng. Sprnger, New York, [17], A multcut algorthm for two-stage stochastc lnear programs, European J. Operatonal Research, vol. 34, no. 3, pp , [18] K. Johansson, The quadruple-tank process, Transactons on Control Systems Technology,