# Steady-state target calculation for constrained predictive control systems based on goal programming.

код для вставкиСкачатьASIA-PACIFIC JOURNAL OF CHEMICAL ENGINEERING Asia-Pac. J. Chem. Eng. 2008; 3: 648–655 Published online 17 October 2008 in Wiley InterScience (www.interscience.wiley.com) DOI:10.1002/apj.200 Special Theme Research Article Steady-state target calculation for constrained predictive control systems based on goal programming Shaoyuan Li,* Yi Zheng and Baiping Wang Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, China Received 23 July 2008; Accepted 23 July 2008 ABSTRACT: A new method of steady-state target calculation for constrained model predictive control (MPC) using goal programming has been developed to solve the problem that the result is not satisfactory when the optimization problem is infeasible or the feasible region is away from the desired working point owing to system dynamics. In this model, soft constraints adjustment and target relaxation have been adopted simultaneously to coordinate with the result. The goal priority factors are introduced to describe the priority of constraints and targets; thereby, the steady-state target calculation is transformed into a goal-programming problem with a standard linear programming form and is solved with some calculation in real time. Simulation is processed with the example of the Shell heavy oil fractionators’ benchmark problem, and the result shows the validity of the proposed algorithm. 2008 Curtin University of Technology and John Wiley & Sons, Ltd. KEYWORDS: predictive control; constraint priority; target priority; goal programming; soft constraint INTRODUCTION Currently, the optimization and control of complex systems have become very important issues as good optimization results can bring tremendous economic benefits and good dynamic behavior.[1 – 3] In the modern process industry, simple control methods cannot satisfy the control requirements of large enterprises along with the scale of commercial enterprises expanding continuously; therefore, many large enterprises use a control system, which is structured hierarchically into several layers, and each layer operates on a different timescale. Typically, layers include global steady-state calculation (everyday or week), local steady-state calculation (every 30 min to 6 h), model predictive control (MPC) (every 1–2 min) and basic dynamic control (every second). As one part of a multilevel hierarchy of control functions, MPC plays an important role in the complex control industry. It can move the plant from one constrained steady state to another while minimizing constraint violations along the way.[4] Recently, the separation of the MPC algorithm into a steady-state and dynamic calculation has been a common part of industrial MPC technology.[4 – 7] Here, the goal of the steady-state target calculation is to recalculate the targets from the local optimizer every time the MPC controller executes. This *Correspondence to: Shaoyuan Li, Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, China. E-mail: syli@sjtu.edu.cn 2008 Curtin University of Technology and John Wiley & Sons, Ltd. must be done because disturbances entering the system or new input information from the operator may change the location of the optimal steady state.[4] MPC has become an industry standard to solve complex constrained multivariable control problems in the process industry with its wide usage in refinery and petrochemical processes.[8] Currently, constraint MPC is being successfully used in the control of various dynamic systems, and its use is becoming more widespread.[9 – 11] However, chemical plants in modern industries have become increasingly complex, which leads to excessive targets and constraints. The study of constraint MPC is also a major problem.[12 – 14] In Olaru and Dumur,[13] the methods to avoid constraints redundancy in MPC are discussed. In Wang et al .,[12] a hybrid approach using a mixed logical dynamical framework to handle infeasibility and constraint prioritization issues in MPC based on dynamic model is introduced. As an important part of a two-stage constraint MPC algorithm, steady-state target calculation faces many problems and challenges. There are many manipulated variables and controlled variables in the process of MPC dynamic optimization. The steady-state target must satisfy the requirements of system safety, energy, and technical conditions. In addition, these requirements have different priority requests according to their extent of importance. In essence, these problems are multiobjective and multidegree of the freedom proposed in Ref. [15] In the classical constraint programming, the constraint cannot be changed. Once the intersection of Asia-Pacific Journal of Chemical Engineering these ‘hand constraints’ does not exist, or constraints are violated due to disturbances, operator intervention, model mismatch, and plant failure, it might result in infeasibility of the MPC optimal program. However, in satisfactory optimization control, soft constraints can be used to reduce the probability of infeasibility.[16,17] Therefore, how to use the adjustable character of system constraints to get satisfying results has become one of the urgent problems in MPC steady-state target calculation. MPC steady-state target calculation comes down to constrained multiobjective multidegree of freedom optimization problem (CMMO) of complex industrial process under a dynamic uncertainty environment.[15] Reference [5] discussed the case in which the optimization problem was feasible, but it did not consider how to handle the problem in which the optimization problem was infeasible. Though Ref. [4] used soft constraints to deal with the infeasibility of the steady-state target optimization problem, it treated all constraints equally and did not consider the effects of soft constraints adjustment on control targets and the problem of multiobjectivity and priority. In Xi and Gu,[18] the problem of feasibility of CMMO is discussed. As there is an inherent relationship between the solution and the feasible sets of the optimization problem, it is necessary to consider the desired target while adjusting the constraints.[19] Reference [19] investigated the relationship between feasibility and objective coordination of the CMMO problem under flexible constraint conditions and used the mixed logic method to describe the priority of the constraints and targets. Then, the optimization problem is transformed into a mixed integer quadratic programming problem. However, a mixed integer quadratic programming problem is complex and the solution is time consuming, and it is not only used in the local steady-state calculation (every 30 min to 6 h) layer but also in the MPC (every 1–2 min) layer. Compared with the restriction of the single goal and the single optimum solution in linear programming, goal programming[20,21] permits to deal with different layers of conflicted goals. As the goal-programming model, whose form is a standard linear programming model, the goal-programming problem can be solved by the simplex method of the linear programming problem, which is fast and involves little calculation. In this study, on the basis of the structure of the MPC steady-state target calculation and dynamic control, we studied the feasibility and target coordination problems of CMMO systematically and put soft constraints adjustment and target coordination into MPC steady-state target calculation. By adopting the goal priority factor, the priority strategy is introduced into the description of system target and constraints. The priority level of soft constraints adjustment and target coordination are determined according to the requirement of system conditions. Then, the steady-state target calculation 2008 Curtin University of Technology and John Wiley & Sons, Ltd. STEADY-STATE TARGET USING GOAL PROGRAMMING is transformed into a goal programming problem with a standard linear programming form and is solved with a little calculation in real time. Through this method, the problem that the result is not satisfactory when the optimization problem is infeasible or the feasible region is away from the desired working point owing to system dynamic is systematically solved in steady-state target calculation of constrained MPC. FEASIBILITY AND TARGET COORDINATION IN STEADY-STATE TARGET OPTIMIZATION Feasibility of CMMO For the linear time-invariant multivariable system, we use the transfer function model of the plant y(s) = G(s)u(s) (1) where y ∈ R m is the output vector and u ∈ R n is the input vector, G(s) = [gij (s)]m×n , i = 1, 2, · · · , m; j = 1, 2, · · · , n. Suppose the steady-state gain matrix exists, the steady-state process can be described by ys = Gu s (2) subject to umin ≤ us ≤ umax , ymin ≤ ys ≤ ymax (3) where ys ∈ R m is the steady-state value of output, us ∈ R n the steady-state value of input, and G ∈ R m×n the steady-state gain matrix. If the pair (us , ys ) exists, which satisfies the condition (2) and (3), the CMMO problem is feasible. When CMMO problem is infeasible, some inequality constraints can be softened to guarantee the feasibility of the optimization problem. Then, the system constraints can be described by Eqn (4) y = Gu s s umin − umin ≤ us ≤ umax + umax ymin − ymin ≤ ys ≤ ymax + ymax T = [umin , umax , ymin , ymax ] ≥ 0 (4) Hard constraints correspond to i = 0 and soft constraints correspond to i ≥ 0. x1 , x2 , x3 , and x4 are auxiliary variables denoting the adjustment of the constraints, defined as x 1 x2 x3 x4 = us − umin + umin = umax − us + umax = Gu s − ymin + ymin = ymax − Gu s + ymax (5) Asia-Pac. J. Chem. Eng. 2008; 3: 648–655 DOI: 10.1002/apj 649 650 S. LI, Y. ZHENG AND B. WANG Asia-Pacific Journal of Chemical Engineering (ys∗ , us∗ ) is the steady-state us∗ = ud . However, (ys∗ , us∗ ) solution. Ideally, ys∗ = yd , may be away from (ud , yd ) because of the existence of constraints. Reference [15] introduces objective coordination algorithm based on sensitivity analysis. Moreover, define z = [x1T , x2T , x3T , x4T , T ]T umax − umin b = ymax − Gu min Gu max − ymin I I 0 0 −I = G 0 0 I −G 0 G I 0 0 −I 0 −G 0 0 −I 0 −I 0 Relationship between feasibility and target coordination of CMMO Then, the system constraints can be rewritten as z = b (6) z ≥0 The feasibility of CMMO can be judged by the following linear programming min W = c T , c T = [1, · · · , 1] s.t. (6) (7) If W = 0, CMMO is feasible, or, if W > 0, CMMO is infeasible. Reference [18] determined the soft constraints and its adjustable range based on the human–machine interface and adjusted the weight factor c on the basis of the requirement of the operators, then the optimization problem in (8) is solved min W = c T , c T = [c1 , · · · , c2(m+l ) ] s.t. (6) (8) However, in the process control, a case usually exists that a constraint is required to be softened first when the constraint region is infeasible. If the relaxation results cannot make the optimization feasible, then consider the next possibility. Obviously, it is difficult to obtain a feasible optimum by the human–machine interface. Target coordination of CMMO problem The target coordination problem in CMMO is to find an appropriate pair (us , ys ) under equality and inequality constraints. The mathematical description of the optimization problem can be formulated as min J = (us − ud 2R + ys − yd 2Q ) (9) s.t. (2), (3) where ud ∈ U ⊂ R n is the desired value of MVs and yd ∈ Y ⊂ R m is the desired value of CVs. (ud , yd ) comes from the local optimizer. In CMMO, the objective of the target calculation is to find the feasible pair (us , ys ), such that (us , ys ) is as close as possible to (ud , yd ). When CMMO problem is feasible, there exist ys and us , which satisfy the equality and inequality constraints. The optimum 2008 Curtin University of Technology and John Wiley & Sons, Ltd. As can be seen from Eqn (5), feasibility is a necessary condition for the system optimization problem. However, it is crucial to consider a satisfactory target while designing a feasible region in system optimization. Now, we discuss two aspects of the relationship between feasibility and target coordination of CMMO. First, if the feasible region is nonempty, we can get an optimum that satisfies certain priority conditions. However, the function index J becomes large when the feasible region is away from the desired working point, which means that we might not get a satisfying result, though there is a large feasible region. In this case, feasibility is not the only necessary condition for CMMO. On the one hand, if there is surplus freedom for the steady-state target, this freedom can be utilized to adjust the feasible region so that it comes close to the desired working point; on the other hand, targets may be relaxed while softening the constraints. Second, when the feasible region is empty, it must make the optimization problem feasible before coordinating the targets, which can be done by soft constraints adjustment. Generally, the polyhedral and anomalistic for the feasible region are composed of linear constraint conditions in U , Y .[19] The feasible region reconstructed by constraints adjustment may not be close to the desired target, thus leading to an unsatisfying result. Therefore, for CMMO, attention should also be given to targets along with the optimization feasibility region. In summary, it is necessary to use the adjustable character of system freedom to acquire a satisfying result whenever the optimization problem is unfeasible or the feasible region is away from the desired working point. CONSTRAINT ADJUSTMENT AND TARGET COORDINATION BASED ON GOAL PROGRAMMING In this article, target coordination is taken into account along with constraint adjustment. We use goal programming to deal with the constraint adjustment and target coordination problem in steady-state target calculation. Moreover, the goal priority factor is introduced to combine the priority of the constraints and targets. Asia-Pac. J. Chem. Eng. 2008; 3: 648–655 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering STEADY-STATE TARGET USING GOAL PROGRAMMING By introducing positive and negative error variables, Eqn (12) is transformed into equality constraints as Eqn (13) Soft constraints adjustment in steady-state target calculation Reference [18] stated the mathematical description of the soft constraints adjustment in steady state and, in this description, both the inputs and outputs were considered as adjustable variables, whereas, in practice, constraints of manipulated variables are generally hard and cannot be violated at all. According to the principle of goal programming, the constraints in steady-state target calculation are divided into three types by introducing positive and negative error variables. Hard constraints composed by equality and input constraints Constraints of manipulated variables are usually derived from the physical property and equipment limitation, such as valve, which has limited active scope. Thereby, constraints of manipulated variables are hard ys = Gu s , umin ≤ us ≤ umax (10) Soft constraints Constraints of controlled variables can be usually violated if safety is not a concern. Therefore, steady-state output constraints can be considered as soft constraints. By introducing positive and negative error variables, the steady-state output constraints in Eqn (3) can be rewritten as ysi + d1i− − d1i+ = ymax ysi + d2i− − d2i+ = ymin i = 1, 2, · · · , m (11) where, for ‘≤’ constraints, each positive error variable d1i+ (i = 1, 2, · · · , m) is minimized and, for ‘≥’ constraints, each negative error variable d2i− (i = 1, 2, · · · , m) is minimized. Goal constraints The difference between performance and constraints is diminishing as far as the optimization problem in the industry is concerned, and can converge into each other under some conditions. The minimization of the performance index may be comprehended as ‘soft’ constraints, whose values are zero, while various constraints can be regarded as targets inversely.[19] In this way, the quadratic performance index in Eqn (5), whose objective is to minimize the differences between us and ud and the differences between ys and yd , can be transformed into inequality constraints as in Eqn (12) |us − ud | ≤ 0 (12) |ys − yd | ≤ 0 2008 Curtin University of Technology and John Wiley & Sons, Ltd. usi + d3i− − d3i+ = ud ysj + d4j− − d4j+ = yd i = 1, 2, · · · , n j = 1, 2, · · · , m (13) where only if all the positive and negative error variables d3i+ , d3i− (i = 1, 2, · · · , n) are equal to zero, then us is equal to ud . Minimizing d3i+ , d3i− (i = 1, 2, · · · , n) at the same time implies that us is as close as possible to the desired ud . Similarly, only if all the positive and negative error variables d4j+ , d4j− (j = 1, 2, · · · , m) are equal to zero, then ys is equal to yd . Minimizing d4j+ , d4j− (j = 1, 2, · · · , m) at the same time implies that ys is as close as possible to the desired yd . Priority of constraint adjustment and target coordination The core in CMMO is based on the human–machine interface to determine the requirement of control, to emphasize on people’s request, and to make the result satisfy people’s requirement.[19] The priority in soft constraint adjustment in the industrial process was mentioned in Ref. [18]. The idea can be stated as follows: when there is no feasible region or the solution of the optimization problem is unsatisfying, some constraints should be softened. If the relaxation results cannot make the optimization feasible or the optimum of the problem is still not satisfactory, the next should be considered. In the same way, soft constraint adjustment in steadystate target calculation also has a priority request. According to the operator’s desire, soft constraints are set to different priorities. The highest priority is designated to the constraint, which is preferred to be violated first, then the second and so on. At the same time, when coordinating the targets, we may want to reach some targets first and then another. So the target coordination in CMMO has additional priority requests. Targets with higher priorities are met first, and then those with lower priorities are satisfied as much as possible. Generally, priorities of output targets are higher than those of input targets. In this article, we focus on soft constraint adjustment and target coordination in steady-state target calculation when the optimization problem is infeasible or the feasible region is away from the desired working point. According to the system requirement, we can determine the priority of each target and constraint, use the goal priority factor to describe the priority of the constraints and targets and transform the target coordination problem into a goal constraint satisfactory problem. Finally, steady-state target calculation can be realized by solving a goal-programming problem. Asia-Pac. J. Chem. Eng. 2008; 3: 648–655 DOI: 10.1002/apj 651 652 S. LI, Y. ZHENG AND B. WANG Asia-Pacific Journal of Chemical Engineering Mathematical description of steady-state target calculation The goal function in goal programming is constituted by goal priority factors: positive and negative error variables of each goal. When each target is determined, the objective of the optimization problem is as close as possible to the desired target. Priority is determined according real requirement. If there are s priorities, then goal priority factors Pi (i = 1, 2, · · · , s) satisfy P1 P2 · · · Ps , where ‘’ denotes priority. The priority of goal constraints is higher than the priority of output constraints and the priority of output constraints is higher than the priority of input constraints. In practice, we can set the priority of constraints and targets according to the system requirement correspondingly. Assuming that d + , d − denote all the positive and negative error variables, the steady-state target calculation problem is achieved by the goal-programming problem as follows: min P1 m − − + + (ω4j d4j + ω4j d4j ) + P2 − − + + (ω3i d3i + ω3i d3i ) + P3 i =1 + m m + + ω1i d1i i =1 − − ω2i d2i (14) i =1 s.t. usi + d3i− − d3i+ = ud ysj + d4j− − d4j+ = yd ysi + ysi + d1i− d2i− − − d1i+ d2i+ Operators can determine the priority and judge whether the optimization solution is satisfying or not according to their desire, and the adjusting process would not finish until the optimization result is satisfactory. If all the adjustable constraints change into hard constraints, the original optimization problem is infeasible. This indicates that there is not enough freedom of the system and it cannot satisfy the control target; therefore, the freedom must be added. SIMULATION STUDIES j =1 n Step 2: If ys = yd and us = ud , then the algorithm ends, otherwise go to step 3. Step 3: Set the priority of constraints adjustment and target coordination according to the desire of the operator. Solve goal-programming problem (14), get the solution. Step 4: Judge the positive and negative error variables from (14); if they are satisfying, then the algorithm ends. Otherwise, give the positive and negative error variables that are receivable, then go to step 3. i = 1, 2, · · · , n (15) j = 1, 2, · · · , m (16) = ymax i = 1, 2, · · · , m (17) = ymin i = 1, 2, · · · , m (18) ys = Gu s (19) umin ≤ us ≤ umax (20) d +, d − ≥ 0 (21) Heavy oil fractionators shown in Fig. 1 can be regarded as the benchmark processes for the Shell standard control problem. The three inputs of the process represent the product draw rate from the top of the column (u1 ), the product draw rate from the side of the column (u2 ), and the reflux heat duty from the bottom of the column (u3 ). The three outputs of the process include the draw composition (y1 ) from the top of the column, the draw composition (y2 ) from the side of the column, and the reflux temperature at the bottom of the column (y3 ). The two disturbances are the reflux heat duty for the intermediate section of the column (d1 ) and the reflux heat duty for the top of the column (d2 ). where ω with super- and subscript denote the weighting coefficients and can distinguish the relative level of importance of each constraint or goal in each priority. Using the simplex method in goal programming to solve this goal-programming problem, we can get the steadystate target. In fact, the soft constraint adjustment and target coordination algorithm based on goal programming consists of the following steps: Step 1: Solve optimization problem (7); if Wmin = 0, then CMMO is feasible, go to step 2; otherwise, go to step 3. 2008 Curtin University of Technology and John Wiley & Sons, Ltd. Figure 1. problem. Heavy oil fractionator-Shell column control Asia-Pac. J. Chem. Eng. 2008; 3: 648–655 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering STEADY-STATE TARGET USING GOAL PROGRAMMING Prett and Morari presented the following model[22] : 4.05e−27s 1.77e−28s 5.88e−27s 1 60s + 1 50s + 1 50s + −18s 5.72e−14s 6.90e−15s G(s) = 5.39e 60s + 1 40s + 1 50s + 1 4.38e−20s 4.42e−22s 7.20 33s + 1 44s + 1 19s + 1 1.20e−27s 1.44e−27s 40s + 1 45s + 1 −15s 1.83e−15s Gd (s) = 1.52e 25s + 1 20s + 1 1.26 1.14 27s + 1 32s + 1 steady-state equation, and the optimization problem is feasible. We employ MPC steady-state target calculation and dynamic control. Figure 2 shows the corresponding result. Another example gives the control targets as y1 = y2 = 0.3 and y3 = −0.3, and an output target changes with no disturbance. We can see that the feasible region is away from the desired working point. In order to get a satisfactory result, it is necessary to adjust constraints and coordinate targets. According to the technical character of oil fractionators, assume that the priority order of the control targets is The inputs and outputs are constrained between −0.5 and 0.5. The inputs velocity constraints are 0.20. Both models are sampled with a period of 4 min. Its steadystate equation is 4.05 1.77 5.88 1.20 1.44 ys = 5.39 5.72 6.90 us + 1.52 1.83 d 4.38 4.42 7.20 1.14 1.26 1)|y1 − 0.3| ≤ 0 In fact, the heavy oil fractionators are a subsystem of the complex chemical process; thus, the desired working point comes from the local optimizer. Usually, the global optimizer determines optimal steady-state settings for each unit in the plant. These may be sent to local optimizers, which run more frequently than the global optimizer. The local optimizer computes an optimal economic steady state and passes this information to the MPC algorithm for implementation.[4] An example of output target change with disturbance d = [0.6 0.5]T is given below. The control objective is y1 = y2 = 0.3 and y3 = −0.3, which satisfies the system 2)|y2 − 0.3| ≤ 0 3)|y3 + 0.3| ≤ 0 Suppose the combinational priority order of control target and constraint is 1)|y1 − 0.3| ≤ 0 2)|y2 − 0.3| ≤ 0 3)y1 4)y2 5)y3 6)|y3 + 0.3| ≤ 0 then the mathematical description of this goalprogramming problem is given below min P1 (d1− + d1+ ) + P2 (d2− + d2+ ) + P3 (d3+ + d4− ) (a) (b) Figure 2. (a and b) Model predictive control result at steady state. 2008 Curtin University of Technology and John Wiley & Sons, Ltd. Asia-Pac. J. Chem. Eng. 2008; 3: 648–655 DOI: 10.1002/apj 653 654 S. LI, Y. ZHENG AND B. WANG Asia-Pacific Journal of Chemical Engineering + P4 (d5+ + d6− ) + P5 (d7+ + d8− ) + P6 (d9− + d9+ ) s.t. y1 + d1− − d1+ = 0.3 y2 + d2− − d2+ = 0.3 y1 + d3− − d3+ = 0.5 y1 + d4− − d4+ = −0.5 y2 + d5− − d5+ = 0.5 y2 + d6− − d6+ = −0.5 y3 + d7− − d7+ = 0.5 y3 + d8− − d8+ = −0.5 y3 + d9− − d9+ = −0.3 From this result, manipulating variables u to satisfy the hard constraints of the system, both y1 and y2 reach their desired value, and y3 is close to its desired value that can satisfy the requests of the operators. The corresponding simulation result is shown in Fig. 3, and the time taken by steady-state calculation is 0.109 s (CPU: 1.8 GHz, Memory: 512 Mb, simulation tool: Matlab). When the result is compared with that shown in Fig. 4, which uses the method that only constraints softening of outputs are adopted in steady-state target calculation, it is obvious that the result of steady-state target calculation obtained by the method proposed in this article is more satisfactory than that without target coordination. y1 = 4.05u1 + 1.77u2 + 5.88u3 + 1.20d1 + 1.44d2 y2 = 5.39u1 + 5.72u2 + 6.90u3 + 1.52d1 + 1.83d2 y3 = 4.38u1 + 4.42u2 + 7.20u3 + 1.14d1 + 1.26d2 −0.5 ≤ u1 ≤ 0.5 −0.5 ≤ u2 ≤ 0.5 −0.5 ≤ u3 ≤ 0.5 di+ , di− ≥ 0, i = 1, 2, · · · , 9 Then, using the simplex method in goal programming to solve this optimization problem, we get a satisfying result us = [ 0.5 −0.1018 ys = [ 0.3 0.3 −0.2627 ]T −0.15151 ]T CONCLUSIONS It can be concluded that soft constraints adjustment and target coordination were investigated according to the characteristics of constrained multiobjective and multifreedom in steady-state target calculation. When system optimization is infeasible or the feasible region is away from the desired working point, the soft constraints adjustment and objectives coordination in MPC steadystate target calculation are solved along with a little calculation by transforming steady-state target calculation into a goal-programming problem. Through constraints adjustment and objectives coordination, the system optimization becomes feasible or the feasible region comes near the desired working point. This method has strong practical significance. (a) (b) Figure 3. (a and b) Model predictive control result at steady state based on local optimization. 2008 Curtin University of Technology and John Wiley & Sons, Ltd. Asia-Pac. J. Chem. Eng. 2008; 3: 648–655 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering STEADY-STATE TARGET USING GOAL PROGRAMMING (a) (b) Figure 4. (a and b) Model predictive control result at steady state without target coordination. Acknowledgements This work was supported by the National Nature Science Foundation of P. R. China under Grant 60774015 & 60534020, the High Technology Research and Development Program of China (Grant: 2006AA04Z173), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant: 20060248001). REFERENCES [1] B. Sankararao, S.K. Gupta. Asia Pac. J. Chem. Eng., 2006; 1, 21–31. [2] R.R. Tan, D.C.Y. Foo, D.K.S. Ng, C.L. Chiang, S. Hul, V. Ku-Pineda. Asia Pac. J. Chem. Eng., 2007; 2, 566–574. [3] R. Cheng, J.F. Forbes, W.S. Yip. J. Process Control, 2007; 17, 429–438. [4] D.E. Kassmann, T.A. Badgwell. AIChE J., 2000; 46, 1007–1024. [5] C.V. Rao, J.B. Rawlings. AIChE J., 1999; 45, 1266–1278. [6] J.V. Kadam, M. Schlegel, W. Marguardt, R.L. Tousain, D.H.V. Hesssem, J.V.D. Berg, O.H. Bosgra. Proc. Eur. Symp. Comput. Aided Process Eng., 2002; 12, 511–516. [7] D.X. Huang, J.C. Wang, Y.H. Jin. Chin. J. Chem. 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