Ind. Eng. Chem. Res. 2002, 41, 801-816 801 Mode Predictive Contro of Interconnected Linear and Noninear Processes Guang-Yan Zhu and Michae A. Henson* Department of Chemica Engineering, Louisiana State University, Baton Rouge, Louisiana 70803 A pant-wide contro strategy based on integration of inear mode predictive contro (LMPC) and noninear mode predictive contro (NMPC) is proposed. The design method is appicabe to pants that can be decomposed according to the noninearity properties of the individua unit operations. The basic idea is to appy LMPC and NMPC controers to the inear and noninear subsystems, respectivey. A systematic procedure for performing the pant decomposition given noninearity information is presented. Because the subsystems are couped via materia and energy fows, a sequentia soution procedure that aims to minimize the amount of unknown information in the MPC designs is deveoped. The pant decomposition and sequentia MPC soution agorithms are appied to a arge-scae styrene production fowsheet. Three controer coordination strategies are deveoped to hande the information-exchange probems caused by sequentia MPC soution. The methods are shown to be nominay stabiizing for noninear pants with a certain trianguar structure. A muti-rate extension for pants with time-scae separations is presented. A reaction/separation process with recyce is used to compare the different hybrid MPC approaches. 1. Introduction * To whom correspondence shoud be addressed. Phone: 225-578-3690. Fax: 225-578-1476. E-mai: henson@che.su.edu. Current address: Advanced Process Contro, Goba Suppy System, Praxair Inc., Tonawanda, NY 14150-0044. The synthesis of pant-wide contro structures is one of the most important probems in process contro theory and practice. 1 The pant-wide contro probem invoves seection of controed variabes, manipuated variabes, and measured variabes; formuation of the contro structure connecting the variabes; and specification of the controer type. 2,3 The probem originay was studied by Buckey. 4 Considerabe research has focused on decomposition of the pant into simper subsystems based on functiona and/or time-scae differences of the unit operations. In a series of papers by Stephanopouos and co-workers, 2,5-7 the contro structure formuation probem was posed as an optimization probem. The resuting structure was decomposed verticay on the basis of disturbance dynamics and horizontay on the basis of functiona groups to yied a moduar feedback optimizing contro system. The same authors addressed the synthesis of reguatory oops, seection of secondary measurements, estimation of states, and synthesis of contro structures for two representative chemica processes. Price and Georgakis 8 proposed a tiered framework for soution of the pant-wide contro probem. Contro oops were grouped into mutipe tiers on the basis of the reative importance of the associated contro objectives. Zheng and co-workers 9 proposed a hierarchica procedure for formuating contro structures on the basis of the minimization of economic penaties. Ng and Stephanopouos 10 deveoped a hierarchica procedure that successivey increases the resoution of the pant-wide contro structure. Beginning with the optimizing feedback approach, Skogestad 3 deveoped a procedure for identifying controed variabes that aows near-optima reguatory performance. Luyben et a. 11 proposed a heuristic design procedure for generating a decentraized contro structure. These pant-wide contro techniques are based on the use of decentraized contro structures. An exception is the moduar mutivariabe contro structure 12 used in ref 10. In principe, mode predictive contro (MPC) can be appied to very arge pant-wide contro probems. The mutivariabe and constraint-handing capabiities of MPC are very appeaing compared to decentraized contro. Linear mode predictive contro (LMPC) has been appied successfuy to industria processes with hundreds of input and output variabes. 13 Process noninearities remain one of the most difficut probems associated with pant-wide MPC appications. 10 When strong noninearities precude the successfu appication of LMPC, noninear mode predictive contro (NMPC) is required. Because NPMC utiizes a noninear dynamic mode for prediction, a noninear programming probem must be soved at each samping period to cacuate the optima input moves. Despite this difficuty, a NMPC controer has been deveoped for the Tennessee Eastman Chaenge Process by the judicious use of modeing and controer simpifications. 14 Motivated by the observation that highy noninear dynamica behavior is typicay associated with a sma number of unit operations, we have deveoped a hybrid mode predictive contro strategy for pant-wide contro appications. 15 The advantage of the proposed approach is that NMPC is utiized ony where necessary and LMPC is appied to the remaining unit operations. The pant is decomposed into a inear subsystem (the distiation coumn) and a noninear subsystem (the reactor) on the basis of the degree of noninearity. LMPC is appied to the coumn, and NMPC is appied to the 10.1021/ie001038n CCC: $22.00 2002 American Chemica Society Pubished on Web 01/18/2002
802 Ind. Eng. Chem. Res., Vo. 41, No. 4, 2002 reactor. Two controer coordination strategies are proposed to compensate for interconnections between the two subsystems. Simuation resuts for a reaction/ separation process with recyce show that the hybrid MPC controer performance is comparabe to that of a fu NMPC controer and superior to that of a conventiona LMPC controer. On the other hand, the computationa time of the hybrid controer is ess than 8% of the NMPC controer execution time. Athough our initia resuts are very promising, there remain severa unresoved probems associated with the appication of the hybrid LMPC-NMPC strategy to compex pants. A systematic procedure is needed to seect the optima pant decomposition from the arge number of possibe aternatives. The subsystem MPC controers must be soved sequentiay to achieve significant reductions in computationa cost. This requires some type of approximation because the soution of a particuar MPC subsystem probem requires information that is avaiabe ony after the other MPC subsystem probems are soved. A systematic procedure is needed to determine the soution sequence of the MPC subsystem probems such that the amount of unavaiabe information is minimized. With the exception of certain trianguar pant structures discussed beow, coupings between the subsystems precude exact soution of the MPC probems. Controer coordination strategies are needed to achieve acceptabe cosed-oop performance in the presence of such coupings. The cass of noninear pants that can be stabiized with the hybrid MPC controer needs to be characterized. Finay, differences between subsystem time-scae properties can be expoited to further reduce on-ine computationa effort. A of these probems are investigated in this paper. The remainder of the paper is organized as foows. In section 2, the pant decomposition agorithm is presented and appied to a styrene pant fowsheet. The hybrid MPC contro strategy is deveoped in section 3. Simpe pants, each comprising a singe inear subsystem and a singe noninear subsystem, are used to iustrate the deveopment of controer coordination strategies. The remainder of the section is focused on more compex pants with mutipe inear and/or noninear subsystems. A specific cass of trianguar noninear systems is shown to be asymptoticay stabiized by the hybrid MPC controer. The MPC sequence seection agorithm is presented and appied to the decomposed styrene pant fowsheet. A muti-time-scae extension aso is proposed. In section 4, the hybrid MPC contro methods are appied to the reaction/separation process considered in ref 15. A summary and concusion are given in section 5. 2. Pant Decomposition The objective is to partition the pant into inear and noninear subsystems according to the noninearity properties of the individua unit operations. This approach requires a methodoogy for measuring the degree of process noninearity. Athough rigorous noninearity measures have been proposed, 16,17 the requirement of a compete noninear pant mode makes these techniques very difficut to appy to compex fowsheets. Data-driven methods such as coherence anaysis 18,19 appear to represent a more promising framework for noninearity quantification. The deveopment of such noninearity measures is beyond the scope of this paper. We beieve that the reative noninearity of typica unit operations often can be determined a priori using process knowedge. This admittedy heuristic approach is pursued in this paper. The decomposition agorithm presented in section 2.1 is designed to produce the smaest number of subsystems. The agorithm is appied to a styrene pant fowsheet in section 2.2. 2.1. Pant Decomposition Agorithm. The required input data for the pant decomposition agorithm is a vector containing the noninearity properties of a unit operations and a matrix describing the unit connections, that is, unit operation noninearity vector n such that n i ) 1ifu i is noninear and n i ) 0ifu i is inear and unit operation connection matrix C such that C i,j ) 1if u i affects u j and C i,j ) 0 otherwise, where u i denotes the ith unit operation. If the tota number of unit operations is denoted N, then n is an N 1 vector, and C is an N N matrix. Soution of the decomposition probem yieds (i) a decomposition matrix X that represent the resuting subsystems and their member unit operations, (ii) a subsystem noninearity vector Y that characterizes the noninearity property of each subsystem, and (iii) a subsystem connection matrix Γ that detais the connections between the derived subsystems. These quantities are defined formay as foows: decomposition matrix X X i,j ) 1ifu i is contained in g j ; X i,j ) 0 otherwise subsystem noninearity vector Y Y i ) 1ifg i is noninear; Y i ) 0 otherwise subsystem connection matrix Γ Γ i,j ) 1ifg i affects g j ; Γ i,j ) 0 otherwise where g j denotes the jth subsystem. The pant decomposition must satisfy the foowing constraints: (1) Every unit operation beongs to one and ony one subsystem. (2) A unit operations in a given subsystem must have the same noninearity property. (3) Every unit in a given subsystem must be connected to at east one other unit in the same subsystem. The decomposition probem can be formuated as an optimization probem in which the tota number of subsystems is minimized. As shown in Appendix A, the optimization probem can be formuated as a binary poynomia programming probem. This probem can be transformed into a binary inear programming (LP) probem 20 with Z decision variabes and 3N constraints, where N Z ) M)0 For the styrene pant fowsheet considered in the next section, N ) 16, and the optimization probem invoves 1 048 576 decision variabes and 48 constraints. Beow, we present an iterative agorithm that is more suitabe for the pant-wide contro probems typicay encountered. N M(N - M)! (1)
Ind. Eng. Chem. Res., Vo. 41, No. 4, 2002 803 Figure 1. Schematic representation of the iterative pant decomposition agorithm. A schematic representation of the decomposition agorithm is shown in Figure 1. Here, k is the subsystem index, and b is an N 1 vector whose ith eement is set to unity when unit i and a connected units with the same noninearity property as unit i are assigned to the same subsystem. The decomposition agorithm is designed to construct the decomposition matrix X such that the tota number of subsystems M is minimized. The agorithm invoves three oops. The inner oop assigns to the subsystem k a units j that have the same noninearity property as and are connected directy to a given unit i aready in subsystem k. The midde oop simpy repeats the first oop unti no additiona units can be added to the current subsystem. The outer oop creates additiona subsystems if any units remain unassigned. The subsystem connection matrix Γ is computed as foows: Γ i,j ) { 1 if X T i CX j g 1 0 otherwise where X i denotes the ith coumn of X corresponding to the ith subsystem. The scaar X T i CX j is a positive integer if any units in subsystem i affect any units in subsystem j. (2) 2.2. Appication to a Styrene Pant Fowsheet. The styrene pant fowsheet shown in Figure 2 was chosen to iustrate the decomposition agorithm because it contains a arge number of unit operations and severa recyce streams. Ethyene and benzene are fed to the first reactor, where ethybenzene (EB) is formed. The effuent from the first reactor is fed to a benzene recovery section from which both benzene and poyethybenzene (PEB) are recyced. High-purity EB is produced from one of the four coumns and fed to a second reactor, where styrene is formed. The products from the second reactor are fed to a series of coumns to produce high-purity styrene and EB for recyce back to the styrene reactor. Operationa experience 21,22 suggests that the EB reactor, the high-purity EB coumn (overhead EB purity of 99.6%), and the EB/styrene spitter (overhead styrene purity of 99.9%) are sufficienty noninear to warrant NMPC. The remainder of the 16 unit operations are considered to be approximatey inear. Therefore, the nonzero eements of the unit operation noninearity vector n are n 4, n 6, and n 13. The nonzero eements of the connection matrix C are C 1,2, C 2,1, C 2,3, C 2,5, C 3,4, C 4,2, C 5,2, C 5,6, C 6,7, C 6,9, C 7,3, C 8,9, C 9,10, C 10,11, C 11,12, C 11,13, C 12,11, C 13,14, C 13,15, C 14,9, and C 14,16.
804 Ind. Eng. Chem. Res., Vo. 41, No. 4, 2002 Figure 2. Styrene pant fowsheet. From this information, the decomposition agorithm generates the foowing connection matrix X, subsystem noninearity property vector Y, and subsystem connection matrix Γ: 0] 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 X )[1 T 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 Y ) [0 1 1 0 1 0] 0] [0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 Γ ) 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 The decomposed styrene pant fowsheet is shown in Figure 3. Subsystem 1 consists of a arge group of inear unit operations incuding a EB production units except for the EB reactor and EB coumn, which comprise noninear subsystems 2 and 3, respectivey. Most of the styrene production units are contained in inear subsystem 4. The EB/styrene coumn is contained in noninear subsystem 5. Because the styrene coumn is connected ony to the EB/styrene coumn, it forms a singe-unit inear subsystem 6. The subsystem connections characterized by the matrix Γ are aso shown in Figure 3. It is important to emphasize that engineering judgment can be appied to the pant decomposition to obtain the fina soution. For the styrene fowsheet, it might be desirabe to reduce the number of subsystems by combining subsystems 5 and 6 to produce a singe noninear subsystem. 3. Mode Predictive Contro Strategy The pant decomposition aows inear mode predictive contro (LMPC) to be appied to the inear subsystems and noninear mode predictive contro (NMPC) to be appied to the noninear subsystems. Litte reduction in on-ine computation time as compared to pantwide NMPC wi be reaized if a singe optimization probem is formuated for the entire pant using inear and noninear subsystem modes. A reasonabe aternative is to sove the individua MPC subsystem probems in a sequentia fashion. Sequentia soution is not straightforward when subsystems are couped by mass and energy fows. In this case, the soution of a particuar MPC subsystem probem might require the
Ind. Eng. Chem. Res., Vo. 41, No. 4, 2002 805 Figure 3. Decomposed styrene pant fowsheet. soution of other subsystem probems. The optima MPC soution sequence can be viewed as the sequence that minimizes the amount of unknown information from unsoved MPC subsystem probems. For most pants of practica interest, an idea soution sequence for which the subsystem MPC probems can be soved without the appication of certain approximations does not exist. In section 3.1, controer coordination methods that compensate for unknown subsystem information are presented for pants consisting of a singe inear subsystem and a singe noninear subsystem. In section 3.2, pants comprising mutipe inear and/or noninear subsystems are investigated. We show that noninear systems with a certain trianguar structure that makes soution of the MPC soution sequence probem trivia can be asymptoticay stabiized by hybrid MPC contro. For more genera pants, an agorithm for determining the optima MPC soution sequence is presented and appied to the decomposed styrene pant fowsheet. In section 3.3, an extension of the hybrid MPC strategy that expoits differences between subsystem time-scae properties to further reduce the on-ine computationa effort is presented. Before proceeding to the detaied deveopment, it is important to emphasize that the pant decomposition and MPC soution sequence probems are not independent. A poory conceived decomposition can ead to considerabe difficuties in determining an acceptabe soution sequence. Our approach is based on decouping these two probems to achieve a reasonaby simpe but suboptima soution. We beieve that combining this approach with sound engineering judgment wi ead to the deveopment of effective pant-wide contro structures. 3.1 Two Subsystem Probems. Pant Configurations. The simpest possibe pant decomposition consists of a singe inear subsystem and a singe noninear subsystem. The state-space mode equations for Figure 4. Two subsystem pant configurations. the two subsystems can be written as x L (k+1) ) A L x L (k) + A N x N (k) + B L u L (k) + B N u N (k) (3) y L (k) ) C L x L (k) + C N x N (k) (4) x N (k+1) ) f[x L (k), x N (k), u L (k), u N (k)] (5) y N (k) ) h[x L (k), x N (k)] (6) where the subscripts L and N denote the inear and noninear subsystems, respectivey; x L and x N are state vectors; u L and u N are input vectors; y L and y N are output vectors; A L, A N, B L, B N, C L, and C N are constant matrices; and f and h are noninear functions. Figure 4 shows the three possibe pant configurations: (1) The inear subsystem is unaffected by the noninear sub-
806 Ind. Eng. Chem. Res., Vo. 41, No. 4, 2002 system, so that A N ) B N ) C N ) 0. (2) The noninear subsystem is unaffected by the inear subsystem, so that f x L ) f u L ) h x L ) 0 (3) The inear and noninear subsystem are fuy couped. Ceary, different MPC soution sequences are appropriate for the three pant configurations. In the first two configurations, the MPC controer for the subsystem that unidirectionay affects the other subsystem is soved first, and then the MPC controer for the other subsystem is soved. The necessary information from the first subsystem can be generated even if the contro horizon of the first MPC controer is shorter than that of the second MPC controer because the first subsystem can be iterated in an open-oop fashion with constant input to yied the future state variabes needed. Brief descriptions of the LMPC and NMPC formuations used in this paper are presented in Appendices B and C, respectivey. The third configuration in Figure 4 is more chaenging, as the two subsystems are fuy couped. In this case, some type of approximation is required to deveop an impementabe sequentia soution strategy. Goba Controer Coordination Method. In our previous work, 15 we deveoped a soution approximation strategy in which a LMPC controer is designed for the entire pant by inear approximation of the noninear subsystem. The LMPC probem is soved first, and then the NMPC probem for the noninear subsystem is soved using the LMPC soution for the inear subsystem ony. We ca this the goba controer coordination method. It can be viewed as a transformation of the third configuration to the first configuration in Figure 4 because the entire pant is, by definition, unidirectionay connected to the noninear subsystem. Using the reaction/separation process introduced in section 4, we showed that the cosed-oop performance obtained with this coordination method is comparabe to that of a fu NMPC controer and superior to that of a conventiona LMPC controer. 15 As compared to the aternative strategies discussed beow, shortcomings of this method incude an increase in the LMPC probem size and approximation of the noninear subsystem by a inear mode in the LMPC probem. Loca Steady-State Controer Coordination Method. Another controer coordination method briefy discussed in ref 15 utiizes the MPC probem soutions at the previous time step to sove the MPC probems at the current time step. This method eiminates the probem of unknown subsystem information and aows the LMPC and NMPC probems to be formuated separatey for each subsystem. We pursue here a simpified version of this method in which the current state and input variabes from the first subsystem are assumed to remain constant in the future for soution of the second MPC probem. This is caed the oca steady-state controer coordination method. Consider the case where the LMPC subsystem probem is soved first. The noninear subsystem variabes are treated as constant disturbances in the LMPC steady-state target cacuation min x s L (k),u s L (k) ) [u ref L (k) - u s L (k)] T R s [u ref L (k) - u s L (k)] (7) subject to x L s (k) ) A L x L s (k) + B L u L s (k) + A N x N (k k) + B N u N (k k-1) y L ref (k) ) C L x L s (k) + C N x N (k k) + dˆl(k) u L,min e u L s (k) e u L,max y L,min e C L x L s (k) + C N x N (k k) + dˆl(k) e y L,max where x L s (k) and u L s (k) are the steady-state and input targets, respectivey, for the inear subsystem and dˆ L- (k) is an estimated step output disturbance. The targets are cacuated by minimizing the difference between the desired input, u L ref, and the input target subject to steady-state equaity and inequaity constraints invoving the desired output, y L ref. If the NMPC subsystem probem is soved first, the NMPC steady-state targets can be cacuated anaogousy using x L (k k) and u L (k k- 1) from the previous LMPC soution. Loca Dynamic Controer Coordination Method. A more sophisticated controer coordination strategy proposed in ref 15 invoves the use of predicted variabe trajectories from previousy soved MPC probems for soution of the current MPC probem. The technique differs from the oca steady-state coordination method in that variabes from the first subsystem are aowed to vary over the contro horizon of the second MPC controer. This method is caed the oca dynamic controer coordination method because dynamic information from each subsystem is used in the MPC cacuations. Consider the case where the LMPC subsystem probem is soved first. At time k, estimates of the future noninear state and input variabes are avaiabe from the NMPC soution at time k - 1 X N (k-1) ) [x N T (k k-1) x N T (k+p N -1 k-1) x N T (k+n L -1 k-1)] T U N (k-1) ) [u N T (k k-1) u N T (k+n N -2 k-1) x N T (k+n L -1 k-1)] T where it has been assumed that the LMPC contro horizon (N L ) is arger than the NMPC contro horizon (N N ) and the NMPC prediction horizon (P N ). In this case, the ast N L - P N eements of X N (k-1) can be obtained by open-oop simuation of the noninear subsystem with constant input u N (k+j k-1) ) u N (k+n N -2 k-1) j g N N - 1. A simper approach is to assume that x N (k+j k- 1) ) x N (k+p N -1 k-1) j g P N. The quadratic program (QP) formuation of the LMPC probem must be modified from that given in ref 23 to incorporate X N (k-1) and U N (k-1). The modified QP matrices are presented in Appendix D. The steady-state target cacuation in
Ind. Eng. Chem. Res., Vo. 41, No. 4, 2002 807 eq 10 is modified by repacing x N (k) and u N (k k-1) with x N (k+n L -1 k-1) and u N (k+n L -1 k-1), respectivey. The foowing information is avaiabe to the NMPC controer after the LMPC probem is soved at time k: X L (k) ) [x L T (k k) x L T (k+1 k) x L T (k+n L k)] T U L (k) ) [u L T (k k) u L T (k+1 k) u L T (k+n L -1 k)] T These predicted vaues can be incorporated directy into the equaity constraints representing the discretized noninear mode equations in the NMPC probem. Extension of these controer coordination methods to pant decompositions with mutipe inear and/or noninear subsystems is tedious but conceptuay straightforward. 3.2 Mutipe Subsystem Probems. Trianguar Pant Decompositions. As discussed beow, the determination of an appropriate MPC soution sequence for pant decompositions with mutipe inear and/or noninear subsystems can be difficut. The soution sequence probem is soved triviay for the foowing cass of pant decompositions x 1 (k+1) ) f 1 [x 1 (k), u 1 (k)] x 2 (k+1) ) f 1 [x 1 (k), x 2 (k), u 1 (k), u 2 (k)] x M-1 (k+1) ) f M-1 [x 1 (k),..., x M-1 (k), u 1 (k),..., u M-1 (k)] x M (k+1) ) f M-1 [x 1 (k),..., x M (k), u 1 (k),..., u M (k)] (8) where M is the tota number of inear and noninear subsystems; x i (k) and u i (k) are the state and input vectors of the ith subsystem, respectivey; and f i ( ) are (possiby) noninear functions. This is caed a trianguar decomposition because the ith subsystem depends ony on variabes from the first i subsystems. Because of the trianguar structure, the optima MPC soution sequence is {g 1, g 2,..., g n-1, g n } where g i represents the ith subsystem. In this case, each MPC probem can be soved exacty without any approximations about the other subsystems. Such trianguar decompositions represent the cass of noninear systems for which we are abe to verify that hybrid LMPC-NMPC contro is stabiizing. The proof of the foowing resut, which is based on the theorems in ref 24, is omitted for the sake of brevity. Assume that each MPC probem is feasibe and that the resuting noninear feedback contro aws are represented by u i (k+j k) ) h j i [x 1 (k),..., x i (k)] j [0, N i - 1], i [1, M] (9) where N i is the contro horizon of the ith MPC controer and h j i ( ) are noninear functions. If the assumptions (1) f i (x 1,..., x i, u 1,..., u i ) is Lipschitz in its arguments i [1, M] and (2) h j i (x 1,..., x i ) is Lipschitz in its arguments j [0, N i - 1], i [1, M] hod, then x(k) ) [x 1 T (k) x M T (k)] T ) 0 is a ocay asymptoticay stabe fixed point of the cosed-oop system x i (k+1) ) f i [x 1 (k),..., x i (k), h 0 1 [x 1 (k)],..., h 0 i [x 1 (k),..., x i (k)]], i [1, M] Note that the functions f i ( ) and h j i ( ) are guaranteed to be Lipschitz if the ith subsystem is inear. 25 Athough hybrid LMPC-NMPC contro is stabiizing for this cass of noninear systems, sequentia soution of subsystem MPC probems typicay wi resut in a oss of performance compared to that obtained via pantwide NMPC contro. The motivation for decomposing the pant into subsystems is that an intractabe pant-wide NMPC probem can be transformed into a set of smaer LMPC and NMPC probems. It is expected that performance wi degrade as the number of subsystems increases. Consequenty, pant decomposition shoud be pursued ony to the extent necessary to produce a computationay tractabe set of MPC probems. MPC Soution Sequence Agorithm. The determination of an appropriate MPC soution sequence is more difficut for compex decompositions with highy interconnected subsystems. The styrene pant fowsheet shown in Figure 3 is an exampe of such a nontrivia decomposition. Ceary, the amount of unknown information from other subsystems required by each MPC controer depends on the soution sequence. The optimization probem presented in Appendix E yieds a MPC soution sequence with the east amount of information required from unsoved subsystems. Athough it might be possibe to sove the optimization probem using methods deveoped for the traveing saesman probem, 26 the inherent computationa compexity motivated us to investigate aternative approaches. For the decomposed styrene pant probem, it is feasibe to enumerate and evauate a 6! possibe soution sequences. This approach might not be tractabe for more compex pant decompositions. Beow, we present an iterative agorithm for determining the optima MPC soution sequence using the subsystem connection matrix Γ and the subsystem noninearity property vector Y derived from the pant decomposition. The agorithm produces a soution sequence represented by the M M matrix Ψ with eements Ψ i,j ) 1 if the jth subsystem is the ith subsystem soved. The agorithm shown schematicay in Figure 5 consists of two oops. The outer oop simpy increments the soution sequence index, whereas the inner oop determines the next subsystem to be soved. Three vectors are introduced and evauated during every iteration of the outer oop on the basis of the updated Ψ matrix. The vector A represents the tota number of subsystems that affect a given subsystem, the vector B represents the tota number of subsystems that are affected by a given subsystem, and the vector C represents the subsystems chosen in previous iterations. The criteria for determining which subsystem to seect are ranked beow according to priority: (1) The subsystem is affected by the east number of other subsystems not yet seected. (2) The subsystem affects the greatest number of other subsystems not yet seected. (3) The subsystem is inear. (4) The subsystem has the owest number in the fowsheet. The resuts obtained by appying the MPC sequence seection agorithm to the decomposed styrene fowsheet (Figure 3) are shown in Figure 6. Both the EB coumn
808 Ind. Eng. Chem. Res., Vo. 41, No. 4, 2002 Figure 5. Iterative MPC seection sequence agorithm. Figure 6. MPC soution sequence for the decomposed styrene pant. and EB/styrene coumn are affected by one other subsystem, affect two other subsystems, and are noninear. The EB coumn is soved first because it is further upstream. The EB/styrene coumn is seected second. Both the styrene pant units (subsystem 4) and the styrene coumn (subsystem 6) are affected by zero unseected subsystems, affect zero unseected subsystems, and are inear. The styrene pant subsystem is soved third because it is further upstream. The styrene coumn is soved fourth. The EB pant units (subsystem 1) are soved before the EB reactor because subsystem 1 is inear. Athough the iterative agorithm generates a unique soution, aternative sequences that yied an equay acceptabe soution might exist. For exampe, soution of subsystems 1 and 2 before subsystems 4 and 6 yieds a soution sequence that is identica to that in Figure 6 with regard to unknown information. Therefore, engineering judgment shoud be used to determine the fina soution sequence. 3.3. Muti-Rate Probems. Pant decompositions based on noninearity properties of the unit operations can produce subsystems with significant differences in their characteristic time scaes. For exampe, a reaction/ separation process in which the reactor has much faster dynamics than the distiation coumn is considered in the next section. For such pant decompositions, computationa efficiency can be further enhanced by soving the MPC controers for the sow subsystems at a ower frequency than the MPC controers for the fast sub-
Ind. Eng. Chem. Res., Vo. 41, No. 4, 2002 809 systems. This approach is particuary beneficia if the noninear subsystems have sower dynamics as it aows ess frequent soution of the NMPC probems. Beow, we present a muti-rate formuation of the hybrid LMPC-NMPC strategy for two subsystem probems based on the oca dynamic coordination method discussed previousy. Furthermore, the deveopment is restricted to the case where the inear subsystem dynamics are sower because the soution is appicabe to the reaction/separation exampe. Extensions for other types of decompositions are conceptuay straightforward. Let the samping rate of the NMPC controer be denoted t N and that of LMPC probem be denoted t L such that t L ) n t N where n is an integer. The time index is denoted k N for the NMPC probem and k L for the LMPC probem. At time step k L, where t ) k L t L, the LMPC probem is soved using the current inear state variabes x L (k L ) and the noninear state and input variabes avaiabe from the NMPC soution at time step k N - 1 where k N ) nk L X N (k N -1) ) [x T N (k L k N -1) x T N (k L +P N k N -1) x T N (k L +N L -1 k N -1)] T U N (k N -1) ) [u T N (k L k N -1) u T N (k L +Ñ N k N -1) x T N (k L +N L -1 k N -1)] T Here, P N and Ñ N are the NMPC prediction and contro horizons, respectivey, expressed as integer mutipes of the inear samping time t L, such that P N ) int( P N - 1 n ) and Ñ N ) int( N N - 2 n ) If necessary, x N beyond the NMPC prediction horizon can be obtained via open-oop simuation of the noninear subsystem with constant u N. The LMPC soution at time k L is impemented, and it is not recacuated unti time k L + 1. The NMPC probem is soved n times at time steps k N, k N + 1,..., k N + n - 1 between the two LMPC soutions. At time step k N + j the NMPC soution is determined using the inear state and input predictions avaiabe at time k L X L (k L ) ) [x T L (k N +j k L ) x T L (k N +j+1 k L ) x T L (k N +j+nn L k L ) ] T U L (k L ) ) [u T L (k N +j k L ) u T L (k N +j+1 k L ) u T L (k N +j+n(n L -1) k L ) ] T where j [0, n - 1]. The NMPC soution is recomputed and impemented every t N time units. 4. Simuation Exampe Given the pant decomposition and MPC soution sequence shown in Figures 3 and 6, respectivey, it is possibe to deveop a hybrid LMPC-NMPC contro strategy for the styrene pant. This woud require the design of three LMPC controers and three NMPC Tabe 1. Nomina Operating Conditions for the Reaction/ Separation Process variabe vaue variabe vaue variabe vaue F 45.022 L/min E/R 8750 K X A1 0.95 F R 54.978 L/min k 0 5.14 10 10 1/min X A2 0.826 C Af 1 mo/l UA 5 10 4 J/(min K) X A3 0.709 T f 350 K T c 309.480 K X A4 0.619 V r 100 L C A 0.567 mo/l X A5 0.559 F 1000 g/l T 350 K X A6 0.506 C p 0.239 J/(g K) M 1, M 9 200 mo X A7 0.394 (- H) 5 10 4 J/mo M 2,..., M 8 50 mo X A8 0.235 F m 1 mo/l L 29.2 mo/min X A9 0.1 R 4 V 84.2 mo/min controers aong with deveopment of the associated subsystem modes. Such an effort is beyond the scope of this paper. Instead, the reaction/separation process studied in ref 15 is used to evauate the MPC controer coordination methods in section 3. This exampe offers the advantage that pant decomposition yieds a singe inear subsystem and a singe noninear subsystem. 4.1. Process Mode. The process consists of continuous stirred tank reactor that is used to manufacture a product B by irreversibe reaction of a reactant A. The reactor is modeed by assuming first-order kinetics, constant-voume operation, and negigibe cooant jacket dynamics. Furthermore, the reactor feed stream, which is obtained by mixing a fresh feed stream with a recyce stream from a downstream distiation coumn, is assumed to be maintained at a constant temperature and fow rate by fast reguatory contro oops. As shown in Appendix F, the reactor mode consists of two highy noninear differentia equations for the temperature and the component concentration. The reactor contro objective is to reguate the reactor temperature by manipuation of the cooing jacket temperature. The effuent from the reactor is introduced into the fourth tray of a distiation coumn that consists of seven trays, a partia condenser, and a reboier. The overhead stream enriched in A is recyced to the reactor, whie the bottom stream enriched in B is recovered as the product. Standard assumptions such as equimoar overfow are used to derive the distiation coumn mode. 27 As shown in Appendix F, a component baance on an equiibrium stage yieds a moderatey noninear mode consisting of nine differentia equations. The coumn contro objective is to reguate the overhead and bottom compositions by manipuation of the vapor and refux rates. For this simpe process, the inear subsystem (coumn) and the noninear subsystem (reactor) are easiy determined. The combined mode can be represented as in eqs 6-9 where x L ) [X A1 X A2 X A3 X A4 X A5 X A6 X A7 X A8 X A9 ] T u L ) [L V] T, y L ) [X A1 X A9 ] T x N ) [T C A ] T, u N ) T c, y N ) T where X An and Y An are the moe fractions of component A on tray n in the iquid and vapor phases, respectivey; V and L are the moar fow rates in the coumn of the vapor and iquid streams, respectivey; C A and T are the concentration of A and the temperature of the reactor effuent stream, respectivey; and T c is the temperature of the reactor cooant stream. Nomina vaues of the mode parameters are given in Tabe 1. 4.2. Controer Design. The goba controer coordination method presented above is appied to this
810 Ind. Eng. Chem. Res., Vo. 41, No. 4, 2002 reaction/separation process in ref 15. When compared to conventiona LMPC controer, the hybrid LMPC- NMPC controer is shown to provide vasty superior performance for operation at an unstabe operating point of the reactor. The hybrid controer provides performance comparabe to that of a conventiona NMPC controer with ess than 10% of the computing effort. In this section, a three controer coordination methods and the muti-rate controer formuation discussed in section 3 are evauated using the reaction/ separation exampe. The controer tuning parameters for each coordination method are isted beow, where the samping rate t ) 10 s. Goba coordination method LMPC: N L ) 15, P L ) [0.1 0 0 4] Q ) 0 5 10 4 0 0 0 5 10 [0.01 0 0 ] R ) 0 0.01 0 0 0 0.01 [50 0 0 1] S ) 0 1 0 0 0 NMPC: N N ) 1, P N ) 4, Q ) 2, R ) 0.001, S ) 0.001 Loca steady-state coordination method LMPC: N L ) 15, P L ) Q ) [ 4] 5 104 0 0 5 10 R ) [ 0.01 0 0 0.01 ] S ) [ 1 0 0 1] NMPC: N N ) 1, P N ) 5, Q ) 2, R ) 0.001, S ) 0.001 Loca dynamic coordination approach A tuning parameters are identica to those for the oca steady-state coordination method. 4.3. Resuts. In Figure 7, the three controer coordination methods are compared for a +10-K change in the reactor temperature setpoint. A three controer rapidy bring the reactor temperature to the new setpoint whie rejecting the disturbance that propagates through the distiation coumn. The cosed-oop reactor dynamics are sighty improved for the second and third methods because of the onger prediction horizons used. Note that the input moves generated by the three controers for the noninear subsystem are virtuay identica. More significant differences are observed in coumn performance, as the goba and oca dynamic coordination methods yied much better disturbance rejection than does the oca steady-state method. This resut can be attributed to the steady-state approximation for future state and input variabes used in the steady-state method. The goba method yieds sighty better contro of the coumn compositions than does the oca dynamic method. This might be a resut of the assumption in the oca dynamic method that the noninear subsystem variabes are constant beyond the reativey short NMPC horizon. In Figure 8, the three methods are compared for an unmeasured +5-K disturbance in the reactor feed temperature. The goba method yieds the smaest overshoot in the reactor temperature. The oca dynamic method provides very simiar performance, whereas the oca steady-state method yieds reativey poor coumn performance. In Figure 9, a muti-rate hybrid MPC controer is compared to a singe-rate hybrid MPC controer when the oca dynamic method is used for controer coordination. The singe-rate controer is executed for 10 s. Because the coumn dynamics are significanty sower than the reactor dynamics, the muti-rate LMPC controer is executed at a frequency of 60 s, whie the muti-rate NMPC controer is executed every 10 s (i.e., n ) 6). As expected, the two hybrid controers yied very simiar performances for the reactor. The coumn performance obtained with the muti-rate controer is sighty degraded because of the arger samping time used for the muti-rate LMPC controer. Note that P N ) 0 and Ñ N ) 0 because both P N and N N are ess than n ) 6. Therefore, the muti-rate controer is equivaent to a oca steady-state controer except that the mutirate LMPC controer is soved at a ower frequency with a onger contro horizon because of the arger samping period. Note that the muti-rate controer outperforms the singe-rate oca steady-state controer (compare Figures 7 and 9). This might be a resut of the onger contro horizon used for the muti-rate LMPC controer. The muti-rate controer provides ony a modest improvement in computation time as compared to the singe-rate controers. More significant reductions in computationa effort are expected when NMPC controers can be executed ess frequenty than LMPC controers. 5. Summary and Concusions A systematic methodoogy for integrating inear mode predictive contro (LMPC) and noninear mode predictive contro (NMPC) for pant-wide contro appications has been presented. The method is appicabe to pants that can be decomposed into a number of inear and noninear subsystems on the basis of the noninearity properties and interconnections of the individua unit operations. An iterative decomposition agorithm was deveoped and appied to a compex styrene pant fowsheet. The pant decomposition enabes LMPC and NMPC to be appied seectivey to subsystems according to their degrees of noninearity. An iterative agorithm designed to minimize the amount of unknown information resuting from sequentia soution of the MPC probems was deveoped and appied to the decomposed styrene fowsheet. The hybrid LMPC-NMPC controer was shown to be stabiizing for a cass of trianguar noninear systems for which the MPC soution sequence probem is triviay soved. Three controer coordination strategies were presented to hande pants with more compex interconnections. An extension for muti-rate contro was presented for pants that can be decomposed into subsystems with different characteristic time scaes.
Ind. Eng. Chem. Res., Vo. 41, No. 4, 2002 811 Figure 7. Cosed-oop simuation for +10-K change in reactor temperature setpoint. A reaction/separation process was used to evauate the various hybrid LMPC-NMPC controer design strategies. Acknowedgment Financia support from the Nationa Science Foundation (Grant CTS-9501368) and the DuPont Company is gratefuy acknowedged. Appendix A: Optimization Formuation of the Pant Decomposition Probem The pant decomposition probem can be formuated as the foowing optimization probem max N X,Ỹ k)1 N [ i)1 N (1 - X i,k )] + (1 - Ỹ k ) (10) k)1 subject to N i)1 N X i,k ) 1 i [1, N] k)1 N n i X i,k ) Ỹ k i)1 X i,k k [1, N] N i-1 i)2 (1 - C i,j )(1 - C j,i )X i,k X j,k ) 0 k [1, N] j)1 Because the number of subsystems M is unknown unti the probem is soved, it is necessary to introduce the N N matrix X and the N 1 vector Ỹ. The eements of X and Ỹ are defined identicay to the eements of the matrix X and vector Y, respectivey, introduced in section 2 except that the number of subsystems is assumed to be N instead of M. There is a tota of (N +
812 Ind. Eng. Chem. Res., Vo. 41, No. 4, 2002 Figure 8. Cosed-oop simuation for +5-K disturbance in reactor feed temperature. 1)N decision variabes corresponding to the eements of X and Ỹ. The term N i)1 (1 - X i,k ) for a given coumn k is 1 if and ony if every eement of coumn k is 0; this corresponds to an empty subsystem. Thus, the first term in the objective function represents the tota number of empty subsystems. If the subsystem k is empty, then Ỹ k can be either 0 or 1. The second term in the objective function ensures that each empty subsystem k is assigned Ỹ k ) 0. The first constraint ensures that each unit operation is aocated to one and ony one subsystem. The second constraint guarantees that a unit operations in a given subsystem have the same noninearity property (i.e., inear or noninear). The third constraint ensures that every unit operation in a given subsystem is connected to at east one other unit operation in the same subsystem. The matrices X and Y are constructed from X and Ỹ, respectivey, by eiminating zero coumns that correspond to empty subsystems. Appendix B: LMPC Controer Formuation The formuation of the LMPC controer for the case in which the inear and noninear subsystems in eqs 6-9 are uncouped is briefy outined here. The objective function is a quadratic function of the future state and input variabes given by 23 min U L (k) [x L (k+n L k) - x s L(k)] T Qh [x L (k+n L k) - x s L (k)] + N L -1 u T L (k+n L k)s u L (k+n L k) + {[x L (k+j k) - j)0 x s L (k)] T C T L QC L [x L (k+j k) - x s L (k)] + u L T (k+j k)s u L (k+j k) + [u L (k+j k) - u L s (k)] T R[u L (k+j k) - u L s (k)]} (11)
Ind. Eng. Chem. Res., Vo. 41, No. 4, 2002 813 Figure 9. Singe-rate and muti-rate contro for +10-K change in reactor temperature setpoint. subject to x L (k+j k) ) A L x L (k+j-1 k) + B L u L (k+j-1 k) j [1, N L ] u L (k+j k) ) u L (k+n L -1 k) u L,min e u L (k+j k) e u L,max j g N L u L,min e u L (k+j k) e u L,max y L,min e C L x L (k+j k) e y L,max where x L (k+j k) and u L (k+j k) are predicted state and input vectors, respectivey; x s L (k) and u s L (k) are state and input target vectors, respectivey; u L - (k+j k) ) u L (k+j k) - u L (k+j-1 k); Q, R, and S are weighting matrices; u L,min, u L,min, and y L,min are ower imits; and u L,max, u L,max, and y L,max are upper imits. An infinite prediction horizon is reaized by determining the penaty matrix Qh by soution of a Lyapunov equation. 23 The decision variabes are future vaues of the input vector over the contro horizon N L U L (k) ) [u L (k k) u L (k+1 k) u L (k+n L -1 k) ] T (12) Future vaues of the state variabes are predicted from the inear mode eqs 6 and 7. The objective function in eq 24 can be manipuated to yied a quadratic programming probem that is we-suited for rea-time soution. 23 Appendix C: NMPC Controer Formuation The formuation of the NMPC controer for the case in which the inear and noninear subsystems in eqs 6-9 are uncouped is briefy outined here. The NMPC
814 Ind. Eng. Chem. Res., Vo. 41, No. 4, 2002 optimization probem is posed as min P N {h[x N (k+j k)] - U N (k),x N (k) j)0 h[x s N (k)]} T Q{h[x N (k+j k)] - h[x s N (k)]} + N N -1 {[u N (k+j k) - u s N (k)] T R[u N (k+j k) - u s N (k)] + j)0 u N (k+j k) T S u N (k+j k)} (13) subject to x N (k+j k) ) f[x N (k+j-1 k), u N (k+j-1 k)] j [1, P N ] u N (k+j k) ) u N (k+n N -1 k) u N,min e u N (k+j k) e u N,max j g N N u N,min e u N (k+j k) e u N,max y N,min e h[x N (k+j k)] e y N,max where the noninear subsystem variabes are defined simiary to the inear subsystem variabes in the LMPC probem. Note that NMPC controer has a finite prediction horizon P N. The decision variabes are future vaues of the state and input vectors, respectivey X N (k) ) [x N (k k) x N (k+1 k) x N (k+p N k) ] T U N (k) ) [u N (k k) u N (k+1 k) u N (k+n N -1 k) ] T Because of the noninear constraints arising from the mode equations, the NMPC formuation eads to a noninear programming probem that is very difficut to sove on-ine for pants of reasonabe compexity. Appendix D: QP Formuation of LMPC for Loca Dynamic Controer Coordination Beow, we show that the LMPC probem can be formuated as a quadratic programming (QP) probem even though the couped inear system (eq 6) depends on the noninear subsystem state and input variabes. The LMPC objective function in eq 11 can be agebraicay manipuated to yied the foowing QP probem min U L (k) [U L (k)]t HU L (k) + 2[U L (k)] T [G 1 X N (k) + G 2 U N (k) + G 3 x L (k) - Fu L (k-1)] (14) where U L (k), X N (k), and U N (k) are defined as U L (k) ) [u L (k k) u L (k+1 k) u L (k+n L -1 k)] T X N (k) ) [x N (k k) x N (k+1 k) x N (k+p N k) x N (k+n L -1 k) T U N (k) ) [u N (k k) u N (k+1 k) u N (k+n N -1 k) x N (k+n L -1 k)] T The NMPC prediction horizon (P N ) and contro horizon (N N ) typicay are chosen to be shorter than the LMPC contro horizon (N L ). As discussed in section 3.1, the ast N L - P N - 1 eements of X N (k) can be obtained by iterating the noninear subsystem mode in open-oop fashion with constant input vector u N (k+n N -1 k)orby assuming that the state vector remains constant at x N - (k+p N k). The ast N L - N N eements of U N (k) are assumed to be equa to u N (k+n N -1 k). The matrices H, G 1, G 2, G 3, and F in eq 14 can be determined from the inear mode and tuning matrices as foows: H ) T Qh B L + R + 2S B T L A T L Qh B L - S B T L (A T L ) NL-1 Qh B L B T L Qh A L B L - S B T L Qh B L + R + 2S B T L (A T L ) NL-2 Qh B L B T L Q (A L ) NL-1 B L B T L Qh (A L ) NL-2 B L B T L Qh B L + R + 2S] [BL G 1 ) T Qh A N B T L A T L Qh A N B T L (A T L ) NL-1 Qh A N ] B T L Qh A L A N B T L Qh A N B T L (A T L ) NL-2 Qh A N B T L Qh (A L ) NL-1 A N B T L Qh (A L ) NL-2 A N B T L Qh A N [BL G 2 ) [BL T Qh B N B T L A T L Qh B N B T L (A T L ) NL-1 Qh B N ] B T L Qh A L B N B T L Qh B N B T L (A T L ) NL-2 Qh B N B T L Qh (A L ) NL-1 B N B T L Qh (A L ) NL-2 B N B T L Qh B N G 3 )[B T L Qh A L N] B T L Qh (A L ) 2 B T L Qh (A L ) ] 0 F )[S 0 where Qh is defined as foows for stabe systems 23 Qh ) [A T L ] i C T i L QC L A L i)0 (15) For unstabe systems, the QP matrices can be obtained by a simiar extension of the deveopment in ref 23. Appendix E: Optimization Formuation of the MPC Soution Sequence Probem The subsystem connection matrix Γ is required for soution of the MPC sequence probem. Reca that Γ i,j ) 1 if subsystem i affects subsystem j and Γ i,j ) 0 otherwise. We define a vector d such that d j ) M i)1 Γ i,j for j ) 1,..., M where M is the tota number of subsystems. The decision variabes are eements of the M M matrix Ψ, where Ψ k,j ) 1 if and ony if subsystem j is the kth subsystem soved. An optimization probem that yieds a MPC soution sequence with the east amount of information required from unsoved sub-
Ind. Eng. Chem. Res., Vo. 41, No. 4, 2002 815 systems is formuated as subject to min M M ψ k)1 j)1 k-1 M [d j - )1 Ψ,i Γ i,j ]Ψ k,j (16) i)1 M Ψ k,j ) 1 k)1 M Ψ k,j ) 1 j)1 The first constraint guarantees that subsystem j is soved ony once, and the second constraint aows ony one system to be soved at a given time. Appendix F: Mode Equations for the Reaction/ Separation Process Ċ A ) 1 V r [FC Af +F m F R X Ar - (F + F R )C A ] - Ṫ ) (F + F R ) V r k 0 exp ( - E RT) C A (17) (T f - T) + - H k FC 0 exp p ( - RT) E C A + UA (T VFC c - T) (18) p Ẋ A1 ) 1 M 1 (VY A2 - LX A1 -F m F R X A1 ) (19) Ẋ A2 ) 1 M 2 [L(X A1 - X A2 ) + V(Y A3 - Y A2 )] (20) Ẋ A3 ) 1 M 3 [L(X A2 - X A3 ) + V(Y A4 - Y A3 )] (21) Ẋ A4 ) 1 M 4 [L(X A3 - X A4 ) + V(Y A5 - Y A4 )] (22) X A5 ) 1 M 5 {(F + F R )C A + LX A4 - [L +F m (F + F R )]X A5 + V(Y A6 - Y A5 )} (23) Ẋ A6 ) 1 M 6 {[L +F m (F + F R )](X A5 - X A6 ) + Ẋ A7 ) 1 M 7 {[L +F m (F + F R )](X A6 - X A7 ) + Ẋ A8 ) 1 M 8 {[L +F m (F + F R )](X A7 - X A8 ) + Ẋ A9 ) 1 M 9 {[L +F m (F + F R )]X A9 - VY A9 - V(Y A7 - Y A6 )} (24) V(Y A8 - Y A7 )} (25) V(Y A9 - Y A8 )} (26) [L +F m (F + F R ) - V]X A9 } (27) where F, C Af, and T f are the voumetric fow rate, concentration of A, and temperature, respectivey, of the reactor feed stream; F R, X A1, and F m are the voumetric fow rate, moe fraction of A, and moar density, respectivey, of the recyce stream; (F + F R ), C A, and T are the voumetric fow rate, concentration of A, and temperature, respectivey, of the effuent stream; T c is the temperature of the cooant stream in the jacket surrounding the reactor; the condenser and reboier are denoted as trays 1 and 9, respectivey; X An and Y An are the moe fractions of A in the iquid and vapor phases, respectivey, on tray n; M n is the iquid moar hodup on tray n; and V and L are the moar fow rates of the vapor and iquid streams, respectivey, in the coumn. The vapor-iquid equiibrium on each tray is described by Y An ) where R is the reative voatiity. RX An 1 + (R -1)X An (28) Literature Cited (1) Stephanopouos, G.; Ng, C. Perspectives on the synthesis of pant-wide contro structures. J. Process Contro 2000, 10, 97-111. (2) Morari, M.; Arkun, Y.; Stephanopouos, G. Studies in the synthesis of contro structures for chemica processes. Part I: Formuation of the probem. Process decomposition and the cassification of the contro tasks. Anaysis of the optimizing contro structures. AIChE J. 1980, 26, 220-232. (3) Skogestad, S. Pantwide contro: The search for the sefoptimizing contro structure. J. Process Contro 2000, 10, 487-507. (4) Buckey, P. S. Techniques of Process Contro; Wiey: New York, 1964. (5) Arkun, Y.; Stephanopouos, G. Studies in the synthesis of contro structures for chemica processes. Part IV: Design of steady-state optimizing contro structures for chemica process units. AIChE J. 1980, 26, 975-991. (6) Morari, M.; Stephanopouos, G. Studies in the synthesis of contro structures for chemica processes. Part II: Structura aspects and the synthesis of aternative feasibe contro schemes. AIChE J. 1980, 26, 232-243. (7) Morari, M.; Stephanopouos, G. Studies in the synthesis of contro structures for chemica processes. Part III: Optima seection of secondary measurements within the framework of state estimation in the presence of persistent unknown disturbances. AIChE J. 1980, 26, 247-260. (8) Price, R. M.; Georgakis, C. Pantwide reguatory contro design procedure using a tiered framework. Ind. Eng. Chem. Res. 1993, 32, 2693-2705. (9) Zheng, A.; Mahajanam, R. V.; Dougas, J. M. Hierarchica procedure for pantwide contro system synthesis. AIChE J. 1999, 45, 1255-1265. (10) Ng, C.; Stephanopouos, G. Pant-wide contro structures and strategies. In Proceedings of the IFAC Symposium of Dynamics and Contro of Process Systems; Esevier Science: New York, 1998; pp 1-16. (11) Luyben, M. L.; Tyreus, B. D.; Luyben, W. L. Pantwide contro design procedure. AIChE J. 1997, 43. (12) Meadowcroft, T. A.; Stephanopouos, G.; Brosiow, C. The moduar mutivariabe controer. I: Steady-state properties. AIChE J. 1992, 38, 1254-1278. (13) Qin, S. J.; Badgwe, T. A. An overview of industria mode predictive contro technoogy. In Chemica Process Contro V; Kantor, J. C., Garcia, C. E., Carnahan, B., Eds.; CACHE: Austin, TX, 1997; pp 232-256. (14) Ricker, N. L.; Lee, J. H. Noninear mode predictive contro of the Tennessee Eastman chaenge process. Comput. Chem. Eng. 1995, 19, 961-981. (15) Zhu, G.-Y.; Henson, M. A.; Ogunnaike, B. A. A hybrid mode predictive contro strategy for noninear pant-wide contro. J. Process Contro 2000, 10, 449-458.