Improving the Performance of Generalized Predictive Control for

Ind. Eng. Chem. Res. 2010, 49, 4809–4816

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Improving the Performance of Generalized Predictive Control for Nonlinear Processes Ma’moun Abu-Ayyad*,† and Rickey Dubay*,‡ School of Science, Engineering and Technology, Penn State-Harrisburg, Middletown, PennsylVania 17057, and Department of Mechanical Engineering, UniVersity of New Brunswick, Fredericton, NB, Canada, E3B 5A3

This paper presents a unique method for improving the performance of the generalized predictive control (GPC) algorithm for controlling nonlinear systems which can be extended to other forms of predictive controllers. This method is termed adaptive generalized predictive control (AGPC) which uses a multidimensional workspace of the nonlinear plant to recalculate the controller parameters every sampling instant. This results in a more accurate process prediction and improved closed-loop performance over the original GPC algorithm. The AGPC controller was tested in simulation, and its control performance was compared to GPC on several nonlinear plants with different degrees of nonlinearity. Practical testing and comparisons were performed on a steel cylinder temperature control system. Simulation and experimental results show that the adaptive generalized predictive controller provided improved closed-loop performance over GPC. The formulation of the multidimensional workspace can be readily applied to other advanced control strategies making the methodology generic. Introduction Normally proportional-integral derivative (PID) controllers are difficult to control when applied to systems with significant time-delays. To overcome the problem of delayed feedback, an advanced controller such as the Smith-Predictor control is needed to provide reasonable control. If there is no time-delay, this controller usually collapses to general conventional PID form. Another advanced form of control is model predictive control (MPC) which minimizes the squared weighted difference between the desired and actual process outputs and has been widely accepted in industry. In 1997 an industrial survey of more than 2200 applications of MPC schemes was performed by Qin et al.1 In their study, 67% of the applications are in petrochemicals and chemical industries which are generally nonlinear processes. Other applications of MPC include pulp and paper, food processing, furnaces, and automotive industries. MPC is an optimization-based control methodology that explicitly utilizes a dynamic mathematical model of a process to obtain a control signal by minimizing an objective function. It is popular in industry because of its ability to handle multiinput multi-output (MIMO) as well as single-input single-output (SISO) systems, process interactions, process constraints, and its intrinsic compensation for dead times.2 The various MPC algorithms differ mainly in the type of model used to represent the process and its disturbances, as well as the cost functions to be minimized, with or without constraints. One of the robust MPC algorithms is known as GPC, proposed by Clarke et al.,3,4 using GMV and the pole-placement approaches to be able to handle unstable plants. Clarke et al.3 showed that the CARIMA model reduces the number of parameters that can be used to represent the model in comparison to step response models. The GPC algorithm has gained increasing attention during the past decade which is essentially due to the improvements in modeling and identification techniques and digital computers. * To whom correspondence should be addressed. Tel.: +1 717 948 6786. Fax: +1 717 948 6619. E-mail: [email protected]. Tel.: +1 506 458 7770. Fax: +1 506 453 5025. E-mail: [email protected]. † Penn State-Harrisburg. ‡ University of New Brunswick.

In conventional industrial MPC, the objective of the modeling process is to determine a model that can be numerically evaluated quickly and that adequately describes the process dynamics in a neighborhood of some desired steady-state operating point. MPC based on these models has been successful in industry.5 In practice, most of stable and unstable processes can be described by using autoregressive models such as ARX, ARIMAX, or CARIMA model that is used in GPC.3,4 While these modeling approaches work well, in many instances they are specific to the process and cannot be used generically to control a wide range of nonlinear industrial systems. In many of these industrial systems, the nonlinearities are directly related to varying process gain and time constants as the system moves from one state to another. An adaptive GPC controller was proposed by Huang et al.6 to control the melt temperature of a molding machine. In this method, an online estimation of the controller parameters using the gradient of the performance index was used to update the GPC control law. Al-Ghazzawi et al.7 proposed an online adaptive strategy for constrained linear MPC which uses the closed-loop response gradient to determine the controller’s parameters. The shortcoming of this method is that it cannot be extended to nonlinear systems because of the difficulty of deriving the analytical expression for the nonlinear gradient closed-loop response. As a development on the previous method, Ali8 used the fuzzy logic technique instead of the gradient of the closed-loop response to calculate the MPC tuning parameters. In this method, the control horizon was kept constant while others (prediction horizon and weighting matrices) were adjusted using fuzzy logic techniques. However, this approach may not lead to the optimal solution of the control moves for the nonlinear workspace. This paper investigates the implementation of an adaptive GPC algorithm (AGPC) that recalculates the controller variables using process parameters that are determined from a formulated nonlinear workspace of the process or controlled variable. These recalculations occur every time step as the manipulated (control action) and controlled variables move from one state to another. The paper also highlights the key details for developing this multidimensional workspace. The adaptive and original GPC

10.1021/ie100133k  2010 American Chemical Society Published on Web 04/13/2010

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Ind. Eng. Chem. Res., Vol. 49, No. 10, 2010

controllers are tested and compared in simulation on several plants with different degrees of nonlinearity. The algorithms are also tested on a practical temperature control setup. Adaptive Generalized Predictive Control In this section, the AGPC strategy will be formulated as well as the procedure for obtaining the multidimensional workspace of the nonlinear systems. Theoretical formulations of the AGPC scheme will then be presented in the following sections. Initial Considerations. In GPC, Clarke et al.3 showed that the CARIMA model is more appropriate for many industrial processes in which disturbances are nonstationary. This model is expressed as A(z-1)y(t) ) B(z-1)z-du(t - 1) + C(z-1)

ε(t) ∆

(1)

where Figure 1. PRBS open-loop test for process 1.

A(z-1) ) 1 + a1z-1 + a2z-2 + · · · + anaz-na B(z-1) ) 1 + b1z-1 + b2z-2 + · · · + bnbz-nb ∆ ) 1 - z-1 For simplicity C polynomial is chosen to be 1. The variable y(t) is the model output, the polynomial A(z-1) contains coefficients associated with the process output or controlled variable, and B(z-1) contains coefficients associated with the manipulated variable. In many industrial control applications, the plant can be approximated as a first order plus dead time (FOPDT) model.10 The general discrete FOPDT model as in eq 1 can be expressed as3 Θ(z-1) )

B(z-1) -d bz-1 -d z ) z -1 A(z ) 1 - az-1

(2)

In this investigation, the assumption is that this FOPDT structure can be used to model the open-loop dynamic behavior of a nonlinear plant from one time step to another. Also, the plant is piecewise linear during the sampling interval or time step ∆t. The new strategy incorporates the manipulated and controlled variables (u,ym) such that the plant or process parameters are functions of(u,ym). Therefore, the nonlinear plant parameters can be expressed as Kp ) f(u, ym) τp ) f(u, ym)

(3)

Furthermore, the FOPDT model variables (a,b,d) are directly related to variables (u,ym). It follows that using eq 3, the model variables (a,b,d) can now be expressed as a(u, ym) ) e-∆t/τp

b(u, ym) ) Kp[1 - e-∆t/τp] ) Kp[1 - a] Td d) ∆t

(4)

The parameter Td is assumed to be a constant in this investigation. Equations 3 and 4 are the premise for developing the AGPC methodology. Consequently, the model variables (a,b) are calculated every ∆t from nonlinear functions in eq 4, which are to be determined.

The usefulness and applicability of eq 4 can be better understood as a three-dimensional (3D) workspace plot. The 3D workspace can provide information on plant nonlinearities such as saturation, unstable and unbounded regions, discontinuities, and the nonlinear variations in the process parameters (Kp,τp). Subsequently, the modified CARIMA model (for AGPC) which is recalculated every ∆t is now rewritten using eq 4 as [1 - a(u, ym)z-1]y(t) ) b(u, ym)z-du(t - 1) +

ε(t) ∆

(5)

3D Nonlinear Workspace. Initially, open loop tests are performed on the plant using several pseudorandom binary sequences (PRBS) (inputs) in order to define the stable envelope of the process. From these tests, the process parameters (Kp,τp) are determined for each PBRS test. It is to be noted that each PBRS test commences from a state ym which is zero initially and nonzero throughout. The PBRS algorithm was tested on the following processes. Process 1. This process is a nonlinear dynamic process formulated in a discrete form as11 y(k) ) y(k - 1)3 - 0.2|y(k - 1)|u(k - 1) + 0.08u(k - 1)2 (6) The PRBS test signal for testing process 1 is shown in Figure 1. From this test (Kp,τp) were evaluated and shown in Figure 2 highlighting the discontinuity region of (Kp,τp). Also, the nonfilled region in Figure 2 represents the steady state regions where (Kp,τp) f 0. The nonlinear model variables (a,b) are now evaluated using eq 4. These functions are used to generate a nonlinear 3D workspace for the process as shown in Figure 3. It should be noted that the workspace for a(u,ym) is almost flat over the range of u ∈ [0,2] which can be explained as τp is constant over this range while Kp is not. This observation demonstrates the importance of generating the nonlinear workspace which defines the stable operating envelope of the process. The degree of nonlinearity for each process can be readily shown if 3D plots of (Kp,τp) vs (u,ym) were performed, but are not added due to emphasis on evaluating (a,b). As a comparative test of GPC and APGC, the highly nonlinear discontinuity region will be used in closed-loop control of process 1. Note that all the above details on the nonlinear workspace will not be stated for other processes.


Figure 2. The process parameters (Kp,τp) for process 1.

Figure 4. The 3D workspace system for process 2.


Figure 5. The 3D workspace system process 3.

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Process 2.12 y(k - 1)y(k - 2)[y(k - 1) + 2.5] + u(k) 1 + [y(k - 1)]2 + [y(k - 2)]2

y(k) )

(7)

Process 3.13 y(k) ) y(k - 1) + u(k - 1)e-3|y(k-1)|

(8)

14

Process 4. y(k) )

2.5y(k - 1)y(k - 2) + [1 + y(k - 1)2 + y(k - 2)2 + y(k - 3)2] 0.4y(k - 4) + u(k) + 1.1u(k - 1) (9)

Figures 4, 5, and 6 show the 3D workspaces of the model variables (a,b) for processes 2, 3, and 4, respectively. It can be noted that process 2 has a higher degree of nonlinearity than process 1 since (a and b) variables are varying with (u,ym) more severely, indicating the importance of the 3D nonlinear workspace. The workspace plot of process 3 shows the variables (a,b) are varying significantly over the range of ym ∈ [-3,3] where the nonlinearity in the process variables (a,b) are varying rapidly over this region. Real-Time Applications of SISO AGPC In this study a real-time application of AGPC was conducted to control the temperature of a steel cylinder which represents


a slow reacting nonlinear SISO system. The steel cylinder is encased circumferentially by individual electrical heater band (400 W) as shown in Figure 7. The dimensions of the cylinder are 25 mm in radius and 150 mm in length. The energy that heats the steel cylinder is manipulated using an electronic solid state relay (SSR) that is pulse-width modulated using a digital output which can change state (high or low). The duty cycle

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Figure 7. Temperature control system.

Figure 9. The 3D workspace for the steel cylinder.

the heater is at its maximum (100%) and minimum (5%) inputs. This is because the process is very slow reacting and the calculation of a as in eq 4 results in relatively small values. The region to the right the discontinuity of the workspace represents positive changes in u (heating). In the absence of an analytical process model as in many practical systems, it is apparent that the formulation of the workspace is critical in the evaluation of (a,b) every ∆t for the AGPC algorithm. Adaptive Generalized Predictive Control Formulation. The formulation of AGPC commences with the prediction of the process (following GPC3) each ∆t using the manipulated and controlled variables (u,ym) without the noise term evaluated as Figure 8. Several open-loop tests of the steel cylinder.

used has a fixed duration of 5 s, that is, the sum of the high and low state. The temperature is measured by an ungrounded E-type thermocouple. Therefore, in control, the AGPC controller would calculate the duration of the control signal u of the high logic state. Practical experiments using the proposed algorithm were conducted and compared to the original GPC method. More details on the temperature control process can be found in ref 15. Several open-loop tests were conducted on the steel cylinder using different input signals to demonstrate the nonlinear dynamic behavior of the system as shown in Figure 8. The input signal here u is the percentage of the high state of the duty cycle (the heater is on). The value of u is in between 0 and 100%. The normalized open-loop tests (not shown) of the steel cylinder are obtained as follows:

(

)

T(t) - Ta θ) /∆u Tmax - Ta

(10)

where Ta is the ambient temperature and Tmax is the maximum steady state temperature (due to 100% on; the heater is fully on). These normalized responses were used to calculate (Kp,τp) and the model variables (a and b) as shown in Figure 9. It should note that the variation in the variable a is very small even though the time constant range varies between 17 and 45 minutes when

ˆ m(t + j) ) Ej(z-1)b(u, ym)∆u(t + j - d - 1) + Y Fj(z-1)ym(t) (11) The major improvement for AGPC is the use of a(u,ym) and b(u,ym) in determining Ej(z-1) and Fj(z-1) in comparison to GPC which uses a Diophantine equation. The polynomial Fj(z-1) can be directly formulated as a function of a(u,ym) which represents the nonlinear behavior of the τp only.

[ ] Fd+1(z-1)

Fj(z-1) ) .

Fd+2(z-1) l Fd+N(z-1)

[

(1 + a) - az-1 (1 + a + a2) - a(1 + a)z-1

2 3 2 -1 ) (1 + a + a + a ) - a(1 + a + a )z l (1 + a + a2 + · · · + aN) -

a(1 + a + a2 + · · · + aN-1)z-1

where N ) N2 - N1. In a summation form, Fj is

]

(12)

[ ] ∑a

i-1

- a(

∑a

∑a

i-1

- a(

˜ j(z-1)∆u(t + j - d - 1) + Fj(z-1)ym(t) ˆ m(t + j|t) ) G Y (18)

∑a

)z-1

i-1

i)1 3

i)1 4

∑a

i-1

- a(

i)1

∑a

(13)

-1

i-1

)z

i)1

l

N+1

[

ai-1 - a(

i)1

∑a

i-1

)z-1

i)1

Similarly, the polynomial Ej in a summation form is Ej(z ) ) 1 2

1+(

∑a

i-1

)z

-1

i)1 2

1+(

3

∑a

)z-1 + (

i-1

i)1

∑a

)z-2

i-1

i)1

l 2

1+(

3

∑a

)z-1 + (

i-1

i)1

4

∑a

)z-2 + (

i-1

i)1

∑a

)z-3 +

i-1

i)1

N+1

∑

· · · +(

ai-1)z-N

i)1

]

˜ TG ˜ + λI)∆u + 2(Y ˜ m - Ysp)TG ˜ ∆u + J˜ ) ∆uT(G ˜ ˜ m - Ysp) (19) (Ym - Ysp)T(Y After minimizing J˜, the control law of unconstrained AGPC is given as ˜ TG ˜ + λI)-1G ˜ T(Ysp - Y ˜ m) ∆u ) (G

(14)

[ ]

In a summation form, 1

i-1

i)1 2

0

i-1

i)1

∑a

i-1

···

0

which is the control move that contains the nonlinear characteristics of the process variables a(u,ym) and b(u,ym). Figure 10 shows the new AGPC architecture where the model and controller parameters are recalculated every ∆t using (u,ym). Notice that only the first element u(1) of control moves vector ∆u is applied as in Figure 10, and the procedure is repeated at the next sampling instant.

The proposed algorithm was tested on four processes with different degrees of nonlinearity. The following steps are used to control processes in simulation and practically: (1) At t ) 0, ym is at an initial value, and a small change in the manipulated variable u is sent to the nonlinear workspace to provide an initial (a,b). (2) Using (a,b), the corresponding matrices as in eq 13 and 17 are calculated. (3) The vector of control moves ∆u is determined from eq 20 and is added to u-. The first element of this result is input to the process. (4) At the next sampling instant, the closed-loop manipulated and controlled variables are injected to the 3D nonlinear workspace to reevaluate (a,b). The control algorithm repeats steps 2 and 3. The parameters for control using GPC and AGPC for each process and simulation results are provided. Note that the parameters (N, nu, ∆t, and λ) were the same for both algorithms. Process 1. The tuning parameters for this process are N ) 30, nu ) 3, ∆t ) 0.003 s, and λ ) 1.05. The closed-loop responses of both schemes are shown in Figure 11. The AGPC scheme performs better at the discontinuity region of the controlled variable where the set point is set to 0.5. The control is smoother and settles faster for all set points. Process 2. The tuning parameters that were used for both schemes are N ) 30, nu ) 3, ∆t ) 0.003 s, and λ ) 1.05.

i)1

l

·

l

N-1

∑a ∑a

∑a

i-1

i)1 N

i-1

··

∑a

i-1

···

i)1 N-1

(17)

∑

ai-1

∑a

i-1

i)1 N-nu

i-1

i)1

l

N-(nu+1)

N-2

i)1

0

1

∑a

˜ )b G

···

···

i)1

(20)

Simulation Results

The polynomial Ej(z-1)b(u,ym) represents the dynamic matrix ˜ Gj+1 for AGPC which contains the step response coefficients of the process model of the form Kp[1 - e- i(∆t/τp)]. Now using ˜ matrix is shown in equation 15. From eq 4, b ) eq 4, G Kp[1 - a] and eq 15 becomes eq 16.

∑a

The new cost or objective function in the AGPC algorithm required to compute the future control move vector ∆u to drive ˆ m(t + j|t) close to the reference trajectory Ysp(t + j|t) can be Y now expressed as

N

∑

-1

4813

prediction vector of AGPC is now calculated as

)z-1

i-1

i)1 2

i)1 3

Fj(z-1) )


1

2

N×nu

A major improvement of AGPC over GPC is that the matrix ˜ comprises only (a,b) which directly introduces the nonlinear G characteristics of the controlled and the manipulated variables. ˆ m being more accurate This results in the prediction of Y commencing at any closed-loop state ym of the process. The

Figure 10. AGPC structure.

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Figure 11. Closed-loop comparisons between AGPC and GPC schemes for process 1.

Figure 13. Complex set point profile comparison between AGPC and GPC schemes for process 2.

Figure 12. Closed-loop responses of AGPC and GPC schemes for process 2.


Figure 12 shows the closed-loop responses for both AGPC and GPC schemes with improved performance of AGPC having faster responses and shorter settling times. Process 2 exhibits highly nonlinear characteristics (Figure 4) demonstrating that AGPC can perform better due to a more accurate prediction of ˆ m. The algorithms were further tested where the objective was Y to track a complex trajectory as shown in Figure 13. Overall, the AGPC strategy performs better than GPC. Process 3. Comparisons were made using both approaches having the same tuning parameters N ) 25, nu ) 3, ∆t ) 0.003 s, and λ ) 1.05. The results show that the AGPC approach achieves better control response in terms of shorter settling times and minimal overshoot as shown in Figure 14. Because of the high degree of nonlinearity, AGPC performs well, reducing the severity of the closed-loop oscillations. Process 4. The tuning parameters for this simulation are N ) 30, nu ) 3, ∆t ) 0.5, and λ ) 1.05. The results show that the AGPC achieves better control response in terms of shorter settling times and minimal overshoot as shown in Figure 15. Practical ImplementationsTemperature Control of a Steel Cylinder. Practical application of AGPC was carried out on controlling the temperature of a steel cylinder. The closed-loop response of AGPC was compared to the original GPC approach as shown in Figure 16 for a single set point change. The tuning parameters N ) 20, nu ) 3, ∆t ) 5 s,


and λ ) 1.1 were used for both schemes. The results show that the AGPC settles in shorter time in comparison to the original GPC scheme. At approximately the 3 min time period, the rate of change in the manipulated variable using AGPC (Figure 17) is almost zero resulting in a smooth


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Figure 16. Closed-loop response comparisons between AGPC and GPC.

Figure 18. Closed-loop response comparisons between AGPC and GPC schemes.

Figure 17. Manipulated variable comparisons between AGPC and GPC.

Figure 19. Closed-loop response comparisons due to different operating region for both schemes.

transition to steady-state. Also, the AGPC reduces its control action at a faster rate than GPC, thereby suppressing the potential for high overshoot. This is possible since the revaluated model process parameters (a,b) during closed-loop control facilitate accurate predictions of ym which would otherwise for GPC be based on a fixed process model. The algorithms were tested on multiset point profiles as shown in Figure 18 and Figure 19. The AGPC method provides improved closed-loop responses than the original GPC approach. Conclusions A major improvement on the original GPC control strategy termed AGPC has been derived theoretically and tested in simulation and practically on several nonlinear processes. The AGPC scheme provided better control performance in all instances over the GPC method. The improved method is particularly suitable for nonlinear processes that are regulatory and can be modeled as FOPDT from time step to time step (∆t f 0). The FOPDT model parameters are revaluated every control time step from a 3D workspace that can be created from a nonlinear analytical plant model or from experimental data. Therefore the Diophantine polynomial and its solution in regular GPC is not required to formulate the controller matrices. The nonlinear workspace is particularly important for visualization of the process nonlinearities and its direct relation to the

manipulated and controlled variables (u,ym). The AGPC method has significant potential for further investigations on sensitivity and robustness analyses, matrix conditionality, AGPC continuous form derivations, application to other forms of nonlinearities, and other control types. Acknowledgment The authors would like to thank the Natural Sciences and Engineering Research Council of Canada and National Scientific Foundation of the United States for the financial assistance to conduct this research investigation. Nomenclature A(z) ) denominator of process transfer function in the z-domain a1, an ) coefficients of A(z) polynomial B(z) ) numerator of process transfer function in the z-domain b1, bn ) coefficients of B(z) polynomial C(z) ) noise model in the CARIMA model e(t) ) noise term in the CARIMA model G ) process step response coefficient matrix J ) cost function of model predictive control Kp ) process gain N1, N2 ) minimum and maximum cost horizon nu ) control horizon

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na ) order of A(z) polynomial nb ) order of B(z) polynomial Td ) process dead time τp ) process time constant u(t) ) process input or control signal ym ) controlled variable z ) z-transform operator λ ) move suppression coefficient

Literature Cited (1) Qin, J. S.; Badgwell, T. A. An Overview of Industrial Model Predictive Control Technology Proceedings of the 5th International Conference on Chemical Process Control, AIChE Symposium Series 93, Tahoe City, CA, 1997, 316. (2) Garcia, C. E.; Prett, D. M.; Morari, M. Model Predictive Control: Theory and PracticesA Survey. Automatica 1989, 25, 335. (3) Clarke, D. W.; Mohtadi, C.; Tuffs, P. S. Generalized Predictive ControlsPart I. The Basic Algorithm. Automatica 1987, 23, 137. (4) Clarke, D. W.; Mohtadi, C.; Tuffs, P. S. Generalized Predictive ControlsPart II. Extensions and Interpretations. Automatica 1987, 23, 149– 160. (5) Clarke, D. W.; Mohtadi, C. Properties of Generalized Predictive Control. Automatica 1989, 25, 859. (6) Huang, S. N.; Tan, K. K.; Lee, T. H. Adaptive GPC Control of Melt Temperature in Injection Moulding. ISA Trans. 1999, 38, 361.

(7) Al-Ghazzawi, A.; Ali, E.; Zafiriou, E. On-Line Tuning Strategy for Model Predictive Controllers. J. Process Control 2001, 11, 265. (8) Ali, E. Heuristic on-Line Tuning for Nonlinear Model Predictive Controllers Using Fuzzy Logic. J. Process Control 2003, 13, 383. (9) Levin, A. U.; Narendra, K. S. Control of Nonlinear Dynamic Systems Using Neural Networks: Controllability and Stabilization. IEEE Trans. Neural Network 1993, 4, 192. (10) Dougherty, D.; Cooper, D. A Practical Multiple Model Adaptive Strategy for Single-Loop MPC. Control Eng. Pract. 2003, 11, 141. (11) Zhao, J.; Wertz, V.; Gorez, R. A Fuzzy Clustering Method for the Identification of Fuzzy Models for Dynamic Systems. Proceeding of the 9th IEEE International Symposium on Intelligent Control, 1994, Columbus, OH, 172. (12) Narendra, K. S.; Parthasarathy, K. Identification and control of dynamical systems using neural networks. IEEE Trans. Neural Networks 1990, 1, 4. (13) Roffel, B.; Betlem, B. Process Dynamics and Control; Wiley: Chichester, U.K., 2006. (14) Gao, F.; Wang, F.; Li, M. An Analytical Predictive Control Law for a Class of Nonlinear Processes. Ind. Eng. Chem. Res. 2000, 39, 2029. (15) Abu-Ayyad, M.; Dubay, R. Real-Time Comparison of a Number of Model Predictive Controllers. ISA Trans. 2007, 46, 411.

ReceiVed for reView January 20, 2010 ReVised manuscript receiVed March 10, 2010 Accepted March 22, 2010 IE100133K

Improving the Performance of Generalized Predictive Control for

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