# Robustness of Model Predictive Control for Ill-Conditioned Distillation Process.

код для вставкиСкачатьDev. Chem. Eng. Mineral Process. 13(3/4), pp. 31 1-316, 2005. Robustness of Model Predictive Control for 111-ConditionedDistillation Process Vu Trieu Minh and Nitin Afzulpurkar’ Mechatronics, School of Advanced Technologies, Asian Institute of Technology (Mi”), Thailand This paper briefly reviews controlling an ill-conditioned distillation process and some basic computational algorithms of model predictive control (MPC). A modified MPC for ill-conditioned distillation process with output regions is developed to improve the robustness of the controller for handling input and output constraints and rejecting disturbances. Compared with the traditional method of MPC for an ill-conditioned process that deletes some controlled variables, the modijied method with output regions proves its superiority because it still maintains penalty on the output deviation JLom the desired region into its controlled objective jimction. Introduction Model predictive control (MPC) refers to a class of algorithms that compute a sequence of manipulated variable adjustments in order to optimize the hture behavior of a plant. MPC technology can now be found in a wide variety of application areas [l]. It differs from other control in that the optimal control problem is solved on-line for the current state of the plant, rather than determined off-line as a feedback policy. MPC has been broadly applied in industry because of its ability to handle input and output constraints in the optimal control problem The success of MPC control is highly dependent on the accuracy of the open-loop predictions, which in turn depend on the accuracy ofthe process models [2]. However MPC is not designed to explicitly handle plant-model uncertainty, and the system might perform very poorly when implemented on a physical system which is not exactly described by the model [3]. Several issues about MPC still remain open and are of interest to researchers due to the lack of a theoretical basis such as nonlinear MPC, nominal stability, offset-free properties and robustness of MPC. In this paper we study the ability to increase the robustness of MPC for an ill-conditioned distillation process. Once the process is illconditioned, i.e. it has a large process condition number (defined as the ratio between the maximum and the minimum singular value of the gain matrix), it is difficult to control because the process gain is strongly dependent on the input direction. For this reason, conventional MPC controllers cannot provide an adequate performance with set point tracking and disturbance rejection. Control of 111-Conditioned Distillation Process Distillation is one of the major operations in process industries. The requirement for the control system is to achieve the desired product quality even though there may be variation in feed composition, flow rate and other uncertainty disturbances. *Authorfor correspondence (nitinaait.ac.th). 311 Vu Trieu Minh and Nitin Afiulpurkar Condenser + I Feed Flow F Bottom Flow B Reboiler Figure I . Ill-conditioned distillation column. A typical example for an ill-conditioned process is the control of top and bottom compositions of a distillation column through reflux flow rate L and boilup flow rate V (see Figure 1). The objective of the distillation is to separate the feed flow F, a mixture of a light and a heavy component, into a distillate product D and a bottom product B. A variation of each of the inputs, with the other one kept at a constant value, leads to the improvement in one product purity with a decrease in the other. Therefore, an improvement in the purity of both products is difficult. The control problem is similar to control of two different outputs while acting on only one input. An ill-conditioned problem is mathematically described by the singular value decomposition (SVD) of the system [5]. An ill-conditioned process is the one having a large condition number (ratio between the maximum and the minimum singular value). Such processes require an advanced inverse based controller in order for the control system to be effective in the presence of disturbances and set-point variations. It has been shown [4, 61 that if the process is structurally ill-conditioned, then inverse-based controllers show large sensitivity, both to input uncertainties and to uncertainties in the individual elements of the transfer function matrix. A number of techniques can be found for designing multivariable controllers characterized by robustness in case of structural ill-condition. For such processes, the use of decentralized controllers, or at most one way decouplers, is usually suggested [4]. A shortcut method is proposed in [7] for the design of a robust decoupler. The traditional method for controlling an ill-conditioned process with the same number of manipulated inputs as controlled outputs is to delete one or more controlled variables from the control objective [ 1, 81. However, this will fail to achieve a steadystate offset of the controlled outputs. This problem is investigated in this paper. MPC Controller The hndamental idea of MPC is the minimization of a performance objective h c t i o n , with respect to future input moves, over a finite time horizon. The standard MPC regulator is based on the minimization of the following open-loop quadratic objective hnction discussed in [9, 101. Consider the problem of regulating to the origin the discrete-time linear invariant system: 312 Robustness of Model Predictive Controlfor Ill-Conditioned Distillation Process ...( 1) x ( t + l ) =Ax(t) + Bu(t),y(t)= C(t) subject to the output and input constraints: ~ m i n5 ~ ( t2)Y- 9 Umin ...(2) 5 u(t>5 urn, Then, the optimization problem: subject to: fork =O,l, ...Ny -1; ]u, for k =O,l,. ..Nu-1, u ( t + k ) = O , k > Nu, yr+klrE~-.Y-] U(+k E [u-, x,,, = x ( 0 , Xf+&+llf = 4+,l, + &+kY ...(3) Yf+&l,= Cxr+kI,3 is solved at each time t, where x , + ~denotes ~, the predicted state vector at time t + k , obtained by applying the input sequence {u,, . . . , u , + ~ to ~ - model ~} Equation (1) starting from the state x ( t ) . Q = Q'2 0,R = R'2 0 are the weighting matrices for predicted state and input, respectively. P is the solution of the Lyapunov equation P = A'PA + Q , The MPC regulator computes the optimal solution u'(t)= {t(,...,u:+N-l} in Equation (3), then applies u ( t ) = u; as input to the system in (l), and repeats the optimization in (3) at time t + 1, based on the new calculated state ofx(t + 1). However, most processes operate with target values or output set points. In some operations, the desired values might change over time. Therefore, target tracking is an important part of any control theory. The objective function for target calculation is to determine the feasible steady-state values to which the regulator converges, that minimize their deviation from the target values [ll]. Target tracking can be formulated as an optimization MPC problem that uses a quadratic objective function to minimize the deviation of the steady-state output and input from their target values: where Y , + ~and ~ , r are the predicted outputs and the output set-points, respectively. = u , + ~-~u , + ~ - ~Q , , .and R are penalty matrices. In the target tracking MPC regulator, the steady-state values of the process will be equal to the target set points if there is no constraint and disturbance. The formulation (4) is the one we consider for the remainder of this paper to verify the robustness of MPC for ill-condition process. 313 Vu Trieu Minh and Nitin Afiulpurkar MPC Methods for Ill-conditioned Process An ill-conditioned process is difficult to control because some manipulated variables have similar effect on some controlled variables. Furthermore, when there are too many input and output constraints, the process may not be able to meet all the set points. Since MPC regulator is designed for an on-line implementation, any infeasible solution of the optimization problem cannot be tolerated. To guarantee the system stability, the traditional MPC method for controlling an ill-condition process is to delete one or more of the controlled variables from its control objective function [l, 81. This is because if some output set points are deleted, the system becomes looser and the probability that MPC can find a solution will increase. Deletion of some output set points can be done by choosing the tuning matrix Q in (4). For example, one MPC controller has two outputs y = bl y21T,if we select the 2 by 2 weighting matrix Q = diag { 1,1}, implying that both outputs are required to reach set points. However, if we want to delete the output set point for y2 or it is required that only the first output yl reach the set point, we can choose the controller tuning matrix Q = diag { 1,O). In other words, the controlled variables now become y = yl . The robustness of MPC can also increase if we can relax some set points into regions. We propose a modified MPC algorithm, which changes from the set points into regions for processes with outputs that have ranges of desired values instead of specific values. When the system has output regions, the MPC formulation needs to be changed slightly because the reference set points r in (4) now become regions. An output region is defined by the minimum and maximum values of a desired range. The minimum value is the lower limit, and the maximum value is the upper limit y, Iy,+&,,Iy, . Then the modified objective for MPC with output regions is: zt+tlr = Y/ower - ~ t + k l t for < K, ; ~t+tlt ~t+klt = 0 for low I~ t + k l r IY- As long as the outputs still lie inside the desired regions, no control actions are taken because none of the control objectives have been violated ( z ~ +=~0~)., But when the outputs violate the desired regions, the control objective in the MPC regulator will activate and push them back to the desired regions. We illustrate the robustness improvements of the modified MPC controller with the following example. Example Consider an ill-conditioned model of a distillation column as discussed in [ 121: Y&)p [L(s) V ( s ) p 1 + h(s) G,(s) = 0.045 -0.069 .. where y~ and y2 are the changes of the top and the bottom compositions (%); L and V are the changes of the reflux and the boilup rate (in m3/h). The steady-state operating 314 Robustness of Model Predictive Controlfor Ill-Conditioned Distillation Process v conditions are: 7, = 96.54, 7, = 3.75, i = 7.3916 and = 6.4677 . Use of MPC controller will take the system from these initial conditions to the target regions 98 I y, I 100 and 0 I y2 I 2 . It is assumed that the actual plant is governed by: G, (s) = IYl(S) Y2WI’ [L(s) V(s)]’ - -0.068 0.045 1+ 1 .. 47) The condition numbers for the model and the plant are 68 and 63, respectively. So the process can be considered as an ill-conditioned one. The constraints on the inputs are: 0 I L 5 14.78 0 I V 213.78 and -0.03 5 Au 5 0.03 . ...(8) The discrete state-space matrices of the model are (sampling time of one unit): . 0.2241 4.34361 Bp= 0.221 4.33861 , c =[I O] 0.9502 0 A 0 0.9502 B’=[-0.2191 0.3586 4.2W 0.3586 0 1 Three MPC controllers are compared without constraints and disturbances (Figure 2): MPCl: Conventional MPC with two output set-points at r = [99, 1IT,the prediction horizonN,,=Nu=5,thetuningmatricesQ=diag{2,2}andR=diag{l,l}. MPC2: MPC with only one set-point imposed on the output y, = 99, the output y2 is now changed into an output region 0 s y , s 2 as in Equation (9,the prediction horizonN, = Nu= 5 , the tuning matrices Q = diag{2,2} and R = diag{ l,l}. MPC3: MPC with only one output required to reach the set-point y, = 99; the set-pointy2 is deleted by choosing Q = diag{2,0} and R = diag{ l y l } ;Ny= Nu= 5. When there are no input and output constraints, all three MPC controllers take the system from initial output condition 7, = 96.54, y2 = 3.75 to the output targets. But when the control objective for y 2 is changed from set-point into region, the penalty yf+L,f - r in Equation (4) will be greater than the penalty z , + ~ ,in~ ( 5 ) for output region. Therefore, both outputs in MPC2 approach the targets slower than in MPC1. A similar result is also found in MPC3 when we control only one set point foryl. In the next simulation, we impose the system with input and output constraints as in (8). At t = 250, a step disturbance of -0.03 enters y2.The MPCl outputs jump out of the output desired regions due to the plant-model mismatch, while both modified MPCZ and MPC3 outputs are returned back to the output desired regions (Figure 3). It is observed that only yI violates the desired region in MPC2; while both yI and y2 for MPC3 violate the desired regions for a considerably longer time. Thus MPC2 is more attractive due to its better ability to maintain outputs in the desired regions. =[ 1’ 3 [ Conclusions A modified MPC with output regions for an ill-conditioned distillation process has shown its ability to reject disturbance and maintain closed loop stability. Compared to the traditional method that deletes some controlled variables from the controlled objective, the modified MPC with output regions shows its superiority because it always maintains penalty on any output violation from the desired regions. If outputs violate the desired regions, MPC regulator with output regions will rapidly push the violated outputs back to the desired regions. 315 Vu Trieu Minh and Nitin Afiulpurkar i im, 5 - MPCl 1 I Figure 2. Outputs of three MPC methods without constraints and disturbances. Figure 3. Outputs of three MPC methods with constraints and step disturbance. Even though the example shows that the modified MPC is successful in controlling ill-conditioned process, model uncertainty and model mismatch that affect the closed loop stability are still open issues. Further analysis is needed for the effectiveness of the modified MPC with output regions regarding model mismatch. References 1. Qin, S.J., and Badgwell, T.A. 1997. An Overview of Industrial Model Predictive Control Technology. Fij?h Int. Con$ on Chem. Process Control, CACHE and AIChE, 316(93), 232-256. 2. Berber, R. 1995. Methods of Model Based Process Control. NATO AS1 Series. Series E, Applied Sciences. 3. Zheng, Q., and Morari, M. 1993. Robust Stability of Constrained Model Predictive Control. Proc. Amer. Control Con$, California, 379-383. 4. Morari, M., and Zafiriou, E. 1989. Robust Process Control. Prentice Hall, USA. 5 . Klema, V.C., and h u b , A.J. 1980. The Singular Value Decomposition: its Computation and Some Application. IEEE Trans. Auto. Control., (25), 164-176. 6 . 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Condensate Processing Plant Project - Process Description, Document No. 82036-02BM-01. PetroVietnam. Received: 28 November 2003; Accepted afer revision: 12 May 2004. 316

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