A Novel Approach to Constrained Generic Model Control Based on

number of new control algorithms have been proposed based on nonlinear programming techniques. During the past 2 decades there has been a significant ...
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Ind. Eng. Chem. Res. 2000, 39, 989-998

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A Novel Approach to Constrained Generic Model Control Based on Quadratic Programming X. Q. Xie, D. H. Zhou,* and Y. H. Jin Department of Automation, Tsinghua University, Beijing 100084, P. R. China

X. D. Xu Department of Thermal Engineering, Tsinghua University, Beijing 100084, P. R. China

The ability to handle constraints is a very important feature for process control algorithms. The conventional generic model control (GMC) uses general nonlinear programming to handle the constraints, which limits its industrial implementation. In this paper, after introducing an approximate model-based predictor, we present a quadratic programming-based optimization algorithm, which has the ability to handle linear constraints of manipulated and controlled variables and their moving velocities. By combination of the proposed optimization algorithm with the generic model control scheme, a novel approach to constrained generic model control based on quadratic programming is proposed for nonlinear affine systems with relative order 1. Computer simulation results show that the proposed approach has definite robustness against process/model parameter mismatches, it can be applied in real time, and it appears to hold a considerable promise in process control. 1. Introduction In recent years, the advanced control approaches which are based on process models have been paid great attention in fields of both control theory and industrial engineering. Since there are various constraints on the manipulated and controlled variables of processes in practical situations, the successful implementation of the advanced control strategies depends greatly on the ability to handle constraints. Model predictive control (MPC) strategies have been well accepted by industrial engineers in the past few years because of their ability to handle input and output constraints and their transparent tuning capabilities. Dynamic matrix control (DMC) is one of the most popular algorithms among the model-based predictive control strategies, and it has been used extensively in industry. In this control algorithm, an objective function with constraints is minimized on-line to conpute the future manipulated variables through linear programming or quadratic programming (QP). Although linear MPC algorithms are successful in controlling processes which are linear or mildly nonlinear, performance degradation and instability often occur in the presence of strong nonlinearities. The advent of high-speed computers provided enough motivation to propose algorithms utilizing nonlinear process models in the on-line optimization. As a result, a significant number of new control algorithms have been proposed based on nonlinear programming techniques. During the past 2 decades there has been a significant increase in the number of control systems based on nonlinear systems concepts.1 An extension of DMC to handling constraints explicitly as linear inequalities was introduced by Garcia and Morshedi, which was denoted as quadratic dynamic matrix control (QDMC),2 and Garcia * Correspondingauthor.E-mail: [email protected]. Fax: +86-10-62786911. Telephone: +86-10-62785845.

gave an extension of QDMC to nonlinear processes.3 In his approach, the future manipulated variables were predicted at every sampling time, a nonlinear model was used to compute the effect of past manipulated variables and the future estimated disturbances on the predicted outputs. A linear model was obtained by linearization at each sampling time and was used to compute the effect of future manipulated variables on the predicted outputs. Then, a quadratic programming problem was solved as in the case of linear QDMC to compute the future manipulated variables. The requirement of solving only one QP on-line makes this algorithm attractive in industrial implementation. The major disadvantage of this algorithm is that it may not perform well in controlling integral processes and may lead to instabilities when applied to open-loop unstable processes.4 For open-loop unstable and integral processes, Gattu and Zafiriou4 incorporated state estimation technique into Garcia’s nonlinear version of QDMC by using a steadystate extended Kalman filter (EKF).5 Generic model control (GMC) is a nonlinear control approach capable of using the nonlinear process model directly,6 which has achieved great progress in the past decade. Unfortunately, the GMC approach uses general nonlinear programming to handle the constraints, which limits its implementation in industrial practice. Brown et al. presented a strategy, where slack variables defining the variables departure from the chosen GMC specification curves were added to the GMC control law for both manipulated and constraint variables. 7 Selecting the weighted slack variables allowed the controller to achieve the desired compromise between constraint violation and set point tracking. The solution of the problem became a nonlinear constrained optimization. However, it is still unrealistic to apply it to real industrial processes when the nonlinear optimization problem, which should be solved on-line within every control period, requires heavy computation. Zhou et al. developed an adaptive approach for GMC to incorporate

10.1021/ie990395z CCC: $19.00 © 2000 American Chemical Society Published on Web 03/08/2000

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the effect of constraints by adjusting the response time τ during the control law calculation,8 but this procedure was somewhat ad hoc in nature. Moreover, it assumed that there always existed a manipulated variable u(t) to make controlled variables tracking the prespecific trajectories unbiased, which was impossible in some cases.9 Similar to those in both QDMC and nonlinear QDMC (NLQDMC),4 an approximate nonlinear model-based predictor is used in this paper to compute the predictive values of outputs in the future at each sampling time. The multistep output prediction is computed from the effect of the past and the future manipulated variables as well as the disturbance prediction in the future, which accounts for unmodeled dynamics and external process disturbances. After the approximate predictor is introduced, a quadratic programming (QP) based optimization algorithm is proposed, which can handle linear constraints of the manipulated and controlled variables and their moving velocities. Then this optimization algorithm is combined with the GMC scheme, and a novel constrained generic model control based on QP is further proposed for a class of nonlinear affine systems with relative order 1. Since the computational burden of QP is much less than that of the general nonlinear programming, the proposed approach can be implemented in real time and has the potential for industrial implementation. The paper is organized as follows: Section 2 is devoted to an outline of GMC approach. After introducing a nonlinear predictor, Section 3 presents the main idea of QP-based constrained GMC, and gives the detailed presentation of constraint handling. In section 4, two simulation results are given to show the effectiveness of the proposed approach. Concluding remarks are provided in the last section. 2. Outline of Generic Model Control The nonlinear process I being studied is described as follows6

dx ) f(x) + g(x)‚u(t) dt

(1)

y(t) ) h(x)

(2)

where the state vector x(t) ∈ RNx, the input vector u(t) ∈ RNu, and the output vector y(t) ∈ RNy. f(‚), g(‚), h(‚) are nonlinear functions of x respectively, which have sufficient order of continuous partial derivative with regard to x in the region R0 of x space. In practice, we assume that the relative order of process I is a fixed constant when x ∈ R0. The physical meaning of the relative order is very clear, it is the number of integral states between process inputs and outputs. In this paper, we assume that the relative order of the process I is 1. By letting the process output to track a desired trajectory yr, Lee and Sullivan (1988) proposed a socalled generic model control (GMC) scheme as follows:6

min u

∫0t [a(x,u)T‚W‚a(x,u)] dt

(3)

x3 ) f(x) + g(x)‚u(t)

(4)

f

s.t.

|u| e R

(5)

a(x,u) } y3 (t) - y3 r(t) ) Lfh(x) + Lgh(x)‚u(t) - Kl

∫0t (y* - y) dt

(y* - y) - K2

f

(6)

where y* represents the process set points; for details about R, K1, and K2, see Lee and Sullivan (1988).6 If the control law is feasible with respect to the constraints, the dimensions of Nu and Ny are the same, and at least one element of u(t) appears in each of the Nu equations represented by eq 6, the above optimization problem will be simplified to the solution of Nu equations with Nu unknowns:

Lfh(x) + Lgh(x)‚u(t) - K1(y* - y) K2

∫0t (y* - y) dt ) 0 f

(7)

This is to say, in circumstances where the dimensions of Nu and Ny are equal and the relative order of the process is 1, explicit solution of the problem in (3) is possible. Because there are large amount of chemical processes having the characteristics of relative degree 1, the GMC approach has been first applied in chemical process control field. 3. Constrained Generic Model Control 3.1. Nonlinear Predictor. To handle the constraints of the controlled variables, we introduce a nonlinear model-based multistep predictor.4 The nonlinear predictor consists of three parts: The first part is the contribution of the past manipulated variables on the predicted outputs, which is defined as the output values when there are no input signals in the future. This is obtained by integrating the model differential equations from the current state over the prediction horizon with constant inputs. The second part is the contribution of the future manipulated variables to the predicted outputs, which is represented with the use of a step response model. A linear model obtained by linearization of the nonlinear model at each sampling time is used to compute the step response coefficients. The third part is the effect of future disturbance on the predicted outputs. The unmodeled effect at the current sampling time is computed as the difference between the process measurements and the model outputs, which is evaluated from an extended Kalman filter (EKF). In the absence of any information on unknown disturbance in the future, it is assumed that the future predicted value of disturbance is equal to the current value. However, the approximation made on the superposition of past and future effect and constant step response coefficients for each of the future moves makes the solution suboptimal,4 which is necessary in order for the on-line optimization to be a single QP at each sampling point. At the sampling instant k, the known variables are as follows: y(k), the plant measurements; xˆ (k|k - 1), the state estimates5 by EKF at k based on the information at k - 1; K(k), the gain matrix of EKF at k, and u(k - 1), the manipulated variable at k - 1. (1) Effect of Past Manipulated Variables. In the calculation of the contribution of the past manipulated variables u(k - 1) to the predicted outputs, the input values in the future are kept constant and equal to u(k - 1). For i ) 1, 2, ..., P, we successively integrate eq 1 over one sampling time from xˆ (k + i - 1|k - 1), with

Ind. Eng. Chem. Res., Vol. 39, No. 4, 2000 991

u(k + i - 1) ) u(k - 1), and then add K(k)‚γ(k|k) to obtain xˆ (k + i|k - 1). The formulas are as follows:

∫tt+∆t(f(xˆ (k + i - 1|k - 1)) +

∆xˆ (k + i - 1) )

g(xˆ (k + i - 1|k - 1))‚u(t))‚dt|u(t))u(k-1) (8) xˆ (k + i|k - 1) ) xˆ (k + i - 1|k - 1) + ∆xˆ (k + i - 1) + K(k)‚γ(k|k) (9)

turbance. To overcome these problems, we introduce dynamic optimization here, which is often involved in the predictive control. For such purpose, the objective function in (3) is approximately transformed into the form of formula 16. The physical meaning is to choose a value of ∆u(k) at the time instant k, such that in the predictive horizon P, the sum of the difference between the first-order derivative of process outputs and reference trajectories is made to be as small as possible.

J1(k) ) min (A(k)T‚W‚A(k)) ∆u(k)

yˆ past(k + i) ) h(xˆ (k + i|k - 1)), i ) 1, 2, ..., P (10) Here P is the prediction horizon and ∆t is the sampling period. γ(k|k) ) y(k) - h(xˆ (k|k - 1)) is the innovation of EKF. (2) Effect of Future Manipulated Variables. After linearizing and discretizing the nonlinear process I at xˆ (k|k - 1) and u(k - 1), we obtain the discretized linear model at k:

{

xk+1 ) Φkxk + Γkuk yk ) Ckxk

(11)

The step response coefficients Si,k(i ) 1, 2, ..., P) are then computed from the above discretized linear model, which is used to represent the effect of future manipulated variables. i

Si,k )

CkΦkj-1Γk ∑ j)1

(12)

i

yˆ future(k + i) )

∑ l)1

Sl,k∆u(k + i - l) i ) 1, 2, ..., P

where

A(k) ) [aT(k + 1) aT(k + 2) ... aT(k + P)]T (17) a(k + i) ) Lfh(xˆ (k + i|k - 1)) + Lgh(xˆ (k + i|k - 1))‚(u(k - 1) + ∆u(k)) - y3 r(k + i) i ) 1, 2, ..., P (18) In eq 18, u(k) ) (u(k - 1) + ∆u(k)) is adopted in every a(k + i). This is because there is only u(k) being searched at the time instant k. u(k) is used in the process and the predictor and is assumed to keep constant during the whole prediction horizon. To simplify the computation, the first-order derivative of the reference trajectory after the instant k is also assumed to be the same as that of the present. The reference trajectory in eq 6 can also be expressed in the following discretized form:

y3 r(k + i) ) y3 r(k) ) K1(y*(k) - y(k)) + k

K2

(13)

where ∆u is the change of the manipulated variables, being defined as

∆u(k + i - l) ) u(k + i - l) - u(k + i - l - 1) (14) (3) Output Prediction. The predicted outputs are computed as the sum of the effect of past and future manipulated variables and the future predicted disturbances

yˆ predict(k + i) ) yˆ past(k + i) +yˆ future(k + i) + γ(k + i|k) (15) where γ(k + i|k) ) γ(k|k), i ) 1, 2, ..., P. Even though the EKF innovation γ(k|k) as in eq 9 has been used to compensate for the unmodeled effect in state prediction, the future predicted disturbance γ(k + i|k) is further added to the effect of past and future manipulated variables in eq 15 to compute the future predicted outputs. This is necessary in order to reduce the steady-state offset.10 3.2. Transformation of the NLP Problem into the QP Problem. The optimization objective function of GMC (eq 3) is the minimization of integration from 0 to instant tf of the difference between the first-order derivative of process outputs and reference trajectories. It is very difficult to calculate the trajectories of the manipulated variables from the initial state to the final state in one sampling time because of the following two reasons: the first is the heavy on-line computational load; the second is the poor predictive ability during the process implementation because of the unknown dis-

(16)

∑ (y*(j) - y(j))‚∆t

(19)

j)0

After some simple deduction, it is easy to transform eq 16 into the following standard quadratic programming (QP) form

1 J2(k) ) min ∆UT(k)‚H‚∆U(k) + GT‚∆U(k) 2 ∆U(k)

(

)

(20)

∆U(k) ) [∆uT(k) ∆uT(k) ... ∆uT(k)]TNu×P

(21)

H ) diag(H1, H2, ... HP)

(22)

where

Hi ) 2(Lgh(xˆ (k + i|k - 1)))T‚W‚(Lgh(xˆ (k + i|k - 1))) (23) G ) [G1T G2T ... GPT]T

(24)

Gi ) 2(Lgh(xˆ (k + i|k - 1)))‚W‚(Lfh(xˆ (k + i|k - 1)) yˆ r(k)) + Hi‚u(k - 1) (25) 3.3. Constraints Handling of Inputs and Outputs. (1) Constraints of Moving Velocities of Inputs.

∆ulowbound e ∆u(k) e ∆uupbound

(26)

(2) Constraints of Inputs.

ulowbound e u(k - 1) + ∆u(k) e uupbound

(27)

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i.e.

{

∆u(k) e uupbound - u(k - 1) - ∆u(k) e u(k - 1) - ulowbound

(28)

(3) Constraints of Outputs. The inputs and input moving velocities’ constraints defined above are hard constraints. In real processes, there exists a certain degree of process/model parameter mismatches due to the assumptions made in deriving the process model and the effect of the unmodeled process disturbances. Therefore, it is recommended that the constraints involving the process output variables be treated as soft constraints. Hence slack variables λ+y g 0, λ-y g 0 are introduced here:

ylowbound - λ-y e yˆ predict(k + i) e yupbound + λ+y, i ) 1, 2, ..., P (29)

{

Figure 1. Forced circulation single-stage evaporator.

Combined with eqs 13 and 15, it leads to i

∑ Sl,k∆u(k + i - l) e yupbound

l)1

+ λ+y - yˆ past(k + i) - γ(k + i|k) i

-

∑

l)1

(30)

Sl,k∆u(k + i - l) e yˆ past(k + i)

+ γ(k + i|k) - ylowbound + λ-y

where ∆u(k + i - l) ) ∆u(k), ∀i ) 1, 2, ..., P. Therefore, an additional term should be added into the objective function J2(k) in order to minimize the weighted sum of the slack variables, and it yields

1 J3(k) ) min ∆UeT(k)‚He‚∆Ue(k) + GeT‚∆Ue(k) ∆Ue(k) 2 (31)

(

)

Figure 2. Exit concentration.

with ∆Ue(k) ) [∆UT(k) λ+yT λ-yT]T

He )

[ ] H 0 0 0

,

(Nu+2×Ny)×(Nu+2×Ny)

[]

G Ge ) w 1 w2

(32)

(Nu+2×Ny)

where w1 > 0, w2 > 0 are weighting coefficients of the slack variables λ+y amd λ-y. Remark 1. In the above deduction, we have adopted linear superposition technique to calculate the output prediction in eq 15. Certainly this is just an approximate approach to describe the behavior of the nonlinear process I at k. To make the output prediction much more accurate, we need to use a small enough sampling period ∆t (see eq 8), and use a small prediction horizon P in eq 13. Remark 2. In the proposed algorithm, the control horizon is chosen as 1; therefore, at time instant k, we only need to search the optimal Nu manipulated variables u(k). On the other hand, if we choose the control horizon to be l > 1, then we will have to search l‚Nu manipulated variables u(k), u(k + 1), ..., u(k + l - 1) at time instant k, which will cause the computational burden to be greatly increased. This may hinder the implementation of the algorithm in real-time control.

In the same manner as observed for predictive control,15 to adopt a longer control horizon will lead to a stronger control effect, which will cause the process outputs to reach the set points more rapidly, but the overshoot will be larger. On the contrary, the choice of the control horizon to be 1 will make the output tracking more smooth. Most chemical processes are slowly dynamic systems; therefore, when the proposed algorithm is applied to chemical processes, it is reasonable to choose the control horizon to be 1. Despite this, it should be pointed out that it is straightforward to extend the proposed algorithm to the case with control horizon larger than 1. 4. Simulation Studies 4.1. A Forced Circulation Evaporator. A forced circulation evaporator13 is shown in Figure 1. The process exhibits both nonlinear and interacting behavior. Three state variables (the same as outputs) are P2, the pressure in the evaporator; L2, the level in the separator; and X2, the outlet concentration of solute. The manipulated variables are F2, the exit flow rate; P100, the steam supply pressure; and F200, the condenser cooling water flow rate. The nominal parameters and the initial steady-state conditions are given in Table 1.

Ind. Eng. Chem. Res., Vol. 39, No. 4, 2000 993 Table 1. Nominal Parameters and Initial Conditions F1 ) 10.0 kg/min F5 ) 8.0 kg/min T2 ) 84.6 °C P2 ) 50.5 kPa M ) 20.0 kg Cp ) 0.07 kW‚min‚kg‚K

F2 ) 2.0 kg/min X1 ) 5.0% T3 ) 80.6 °C P100 ) 194.7 kPa C ) 4.0 kg/kPa ∆t ) 0.5 min

F3 ) 50.0 kg/min X2 ) 25.0% T200 ) 25.0 °C F200 ) 208.0 kg/min UA2 ) 6.84 kW/K

F4 ) 8.0 kg/min T1 ) 40.0 °C L2 ) 1.0 m FA ) 20.0 kg/m λ ) 385 kW‚min/kg

The mass balances on the process yield the following mathematical descriptions:

dL2 ) (F1 - F2 - F4)/FA dt

(33)

dX2 ) (F1X1 - F2X2)/M dt

(34)

dP2 ) (F4 - F5)/C dt

(35)

F4 )

Q100 - F1Cp(T2 - T1) λ

(36)

Q200 λ

(37)

F5 )

UA2(T3 - T200) Q200 ) UA2 1+ 2CpF200

Figure 3. Separator level.

(38)

Q100 ) 0.16(F1 + F3)(T100 - T2)

(39)

T100 ) 0.1583P100 + 90.0

(40)

T2 ) 0.5616P2 + 0.3126X2 + 48.43

(41)

T3 ) 0.507P2 + 55.0

(42)

In the case study 1 of Brown et al. (1990),7 the set point of the outlet concentration changed from 25.0% to 26.5% and then to 30.0%; the constraint of a moving velocity of one manipulated variable, the exit flow rate F2, was restricted between (0.1 kg/min, and the constraint of one output variable, the level in the separator L2, was restricted between 0.5 m e L2 e 1.5 m. Brown et al.7 handled the constraints using single-step nonlinear optimization approach, with the weighted sum of the slack variables to be their optimization objective function. The parameter setting of their algorithm was quite difficult as the result of introducing too many slack variables. Here, we use our QP-based constrained GMC approach. In our algorithm, an extended Kalman filter (EKF) is used to be a state observer, with covariance of process noise Q ) diag{10-3, 10-3, 10-3}, and covariance of measurement noise R ) diag{10-4, 10-4, 10-4}.5 In both our approach and the nonlinear programming (NLP) based approach,7 only X2 and P2 among the three outputs are controlled as set point tracking control, while L2 is controlled as range control. Hence, only two GMC SISO loops are needed to give good set point tracking performance. The GMC control law is obtained by choosing the input-output pairs as follows: F2 f X2 and F200 f P2.13 Hence, the parameters of GMC are chosen as K1 ) diag{0.3448, 0.3448}, K2 ) diag{0.0012, 0.0012} in the two SISO control loops to give the same dynamic responses as in Brown et al.7 with ξ ) 5.0, τ )

Figure 4. Evaporator pressure.

29.0. The weighting matrix W is selected to be the identity matrix, the slack weighting coefficients are chosen to be w1 ) [1, 1], w2 ) [1, 1], and the prediction horizon P ) 5. The simulation results of the proposed algorithm shown in Figures 2-7 are satisfactory, and have a performance similar to that of the NLP-based approach in Brown et al.7 For comparison purposes, with an IBM 586-166 compatible PC, we simulated the process operating for 100 min; it cost 45.91 s for our approach and 271.66 s for the NLP-based GMC approach.7 The software used for simulation is Matlab5.0. The QP and NLP are realized using subroutine “QP.m” and “LEASTSQ.m” in “Optimization Toolbox”, respectively. In order to test the robustness of the proposed QPbased constrained generic model control approach and the conventional NLP-based approach,7 we made some simulations by adopting the following mismatched model parameters: FA ) 23.0 kg/m, C ) 3.8 kg/kPa

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GMC approach have definite robustness against process/ model parameter mismatches. 4.2. CSTR. This benchmark is adopted from Li and Biegler,11 which describes a kinetic model of a catalytic reaction in a CSTR with multiequilibrium points at steady-state. For the reaction

A+BfP where A is in excess, the reaction rate equation can be written as

rB )

Figure 5. Exit flow rate.

k1CB (1 + k2CB)2.0

where k1 ) 1.0, k2 ) 1.0, and CB is the concentration of component B in the reactor. The reaction occurs in an ideal continuously stirred tank.11 The concentrations of B in the two inlet flows are assumed to be fixed: CB1 ) 24.9%; CB2 ) 0.1%. Both inlet flows contain an excess amount of A. The tank is well stirred with a liquid outflow rate determined by the liquid height h in the tank; i.e., F0 ) F(h) ) 0.2h0.5. The cross-sectional area of the tank is 1. The sampling time of this problem is set to 1.0 min. After simplification, the process model becomes4

dx1 ) u1 + u2-0.2x10.5 dt

(43)

dx2 ) dt u1 u2 k1x2 + (CB2 - x2) (44) x1 x1 (1 + k x )2.0 2 2

(CB1 - x2)

Figure 6. Cooling water flow rate.

Figure 7. Steam supply pressure.

(their nominal values are given by Table 1). The simulation results are shown in Figures 8-13. The simulation results demonstrated that both our proposed approach and the conventional constrained

y 1 ) x1 ) h

(45)

y2 ) x2 ) CB

(46)

where u1 ) F1 is the inlet flow rate with condensed B, u2 ) F2 is the inlet flow rate with dilute B, x1 ) h is the liquid level in the tank, and x2 ) CB is the concentration of B in the reactor. The model has two stable equilibrium points and one unstable equilibrium point. At the unstable equilibrium point (x1 ) 100m, x2 ) 2.787%), the linearized model has two eigenvalues at -0.01 and 0.01286.4 The model is marginally unstable with respect to the concentration. Here, we choose this unstable equilibrium point as the set point to demonstrate the performance of our proposed algorithm for this openloop unstable systems.11 The constraints of this nonlinear control problem are: 0 e u1 e 5 kg/min, 0 e u2 e 5 kg/min, -1.5 kg/min e ∆u1 e 1.5 kg/min, -1.5 kg/min e ∆u2 e 1.5 kg/min, 50 m e y1 e 101 m, 0.5% e y2 e 3.5%. Figures 14-17 illustrate the response of the reactor controlled by our approach and the results of NLQDMC4 for comparison purpose. The set point changes from an initial condition (x1 ) 40.0m, x2 ) 0.1%) to the unstable equilibrium point with values (x1 ) 100.0m, x2 ) 2.787%), and then to the stable equilibrium point at (x1 ) 100.0m, x2 ) 0.6327%). The time span for simulation is 100 min. In our algorithm, an extended Kalman filter (EKF) is used to be a state observer, with covariance of process noise Q ) diag{0.01, 0.001} and covariance of measurement noise R ) diag{0.01, 0.001}.5 The parameters of

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Figure 8. Exit concentration.

Figure 11. Exit flow rate.

Figure 9. Separator level.

Figure 12. Cooling water flow rate.

Figure 10. Evaporator pressure.

GMC are chosen as K1 ) diag{1, 1} and K2 ) diag{0.4, 0.4}, the weighting matrix W is selected to be the identity matrix, the slack weighting coefficients are chosen to be w1 ) [1, 10] and w2 ) [1, 1], and the prediction horizon is P ) 5. In the NLQDMC algorithm, just as in Gattu and Zafiriou (1992),4 we have selected Γ ) diag[1, 20], Λ ) 0.5, the prediction horizon P ) 5,

Figure 13. Steam supply pressure.

and the control horizon M ) 5. These parameters have been shown to give the best control performance among the various parameter choices.4

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Figure 14. Liquid level of the reactor y1.

Figure 17. u2 Inlet flow rate with dilute B.

Figure 15. Concentration of B in the reactor y2.

Figure 18. Liquid level of the reactor y1.

Also for comparison purpose, with an IBM PII-200 compatible PC, we simulated the process operating for 100 min; it cost 49.60 s for our approach and 153.08 s for the NLQDMC approach.4 Figures 18-21 illustrate the response of the two approaches with the same set point change as in Figures 14-17, but with model parameter uncertainties as

k1 ) 0.8, k2 ) 1.2, CB1 ) 22.0%, CB2 ) 0.2%

Figure 16. u1 Inlet flow rate with condensed B.

Here we have assumed that there are no process/ model mismatches. The results show that the performance of our approach is comparable with that of NLQDMC.4

(47)

In Figures 14-21, the control performance of the proposed constrained GMC is plotted as solid lines and that of NLQDMC 4 as dashed lines. The dash-dotted lines in Figures 14-15 and Figures 18-19 show the changes of the set points. Remark 3. It is well-known that the estimates by EKF are usually biased. How large the offsets in the estimated states are has not been thoroughly resolved in Kalman filtering theory;16 this problem is out of the scope of the present paper. We believe the reason our proposed algorithm has achieved “high accuracy” output tracking (see Figures 2, 14, 15, 18, and 19) in the above two simulation examples is that the EKF has been used in closed-loop control, and the state estimate bias has been efficiently corrected by the GMC controller, which

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Figure 19. Concentration of B in the reactor y2.

Figure 20. u1 Inlet flow rate with condensed B.

(1) The performance of set point tracking of the two control approaches, the proposed constrained GMC and the NLP-based approach,7 are comparable. But the computational burden of our approach is reduced to about one sixth of that of the NLP-based approach.7 See Figures 2-7. (2) Because of the excellent input-output decoupling performance of GMC,6 under the control of constrained GMC, the trajectory of y1 is not affected when the set point of y2 jumps in both conditions of exact and approximate model. On the contrary, under the control of NLQDMC, the trajectory of y1 has a distinct disturbance and violates its upper bound when the set point y2sp jumps, which may be harmful in industrial implementation. See Figure 14 and Figure 18. (3) The GMC approach requires that the process outputs track the reference trajectories. Although the two curves will not follow each other exactly after adding the constraints handling technique into the algorithm, the output trajectories of our constrained GMC algorithm are still smooth and derivable. This is because the GMC belongs to the family of input-output linearization methods, and the nonlinear characteristics of the process can be compensated for. On the contrary, the NLQDMC4 approach cannot compensate for the nonlinear characteristics of the process; the value of the manipulated variables are calculated from the minimization of an optimization objective function of prediction error. Hence, the output curves of the NLQDMC approach are not very smooth as compared to those of our proposed approach. See Figure 15 and Figure 19. Moreover, the computational burden of our approach is only about one-third of that of the NLQDMC approach. (4) Just like the conventional constrained GMC, the proposed constrained GMC approach has a definite robustness against process/model parameter mismatches; see Figures 8-13. (5) In the above two examples, our approach needs less than 1 CPU second per control step only. Therefore, the proposed approach has potential in the real-time control of nonlinear chemical processes. 5. Conclusions The simulation results show that the performance of the proposed constrained GMC approach is similar to that of the conventional NLP-based approach,7 while the computational load is greatly reduced. Furthermore, compared with the NLQDMC approach, the proposed constraint GMC can realize dynamic input-output decoupling and thus has much better set point tracking ability. Moreover, the proposed approach has definite robustness against process/model parameter mismatches and appears to hold a considerable promise in process control. Acknowledgment

Figure 21. u2 Inlet flow rate with dilute B.

was known to have a definite robustness against process/model parameter mismatches.6 From the above computer simulations, we have obtained the following experiences:

The authors gratefully acknowledge financial support from the National Natural Science Foundation, the National “863” Plan, the National Key Lab, and the National Education Ministry of China. Notation M ) hold-up of solute λ ) latent heat of vaporization F1 ) inlet feed flow rate

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F2 ) exit flow rate F4 ) vapor flow rate F5 ) condensate flow rate K1,K2 ) GMC specification diagonal matrices for control performance L2 ) level of liquid in the separator P2 ) pressure in the evaporator P100 ) steam supply pressure Q100 ) heat transferred in the evaporator Q200 ) heat transferred in the condensor F100 ) steam feed flow rate F200 ) condensor cooling water flow rate T1 ) inlet temperature T2 ) outlet temperature T3 ) vapor temperature T100 ) steam supply temperature T200 ) cooling water supply temperature X1 ) inlet concentration of solute X2 ) outlet concentration of solute

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(6) Lee, P. L.; Sullivan, G. R. Generic Model Control (GMC). Comput. Chem. Eng. 1988, 12, 573. (7) Brown, M. W.; Lee, P. L.; Sullivan, G. R.; Zhou, W. A Constrained Nonlinear Multivariable Control Algorithm. Trans. Inst. Chem. Eng. 1990, 68, 464. (8) Zhou, W.; Lee, P. L.; Sullivan, G. R.; Brown, M. W. An Adaptive Strategy for Constrained Generic Model Control. Chem. Eng. Commun. 1990, 97, 109. (9) Dunia, R. H.; Edgar, T. F. An Improved Generic Model Control Algorithm for Linear Systems. Comput. Chem. Eng. 1996, 20, 1003. (10) Ricker, N. L. Model Predictive Control with State Estimation. Ind. Eng. Chem. Res. 1990, 29, 374. (11) Li, W. C.; Biegler, L. T. Process Control Strategies for Constrained Nonlinear Systems. Ind. Eng. Chem. Res. 1988, 27, 1421. (12) Ricker, N L. Model Predictive Control with State Estimation. Ind. Eng. Chem. Res. 1990, 29, 374. (13) Lee, P. L.; Newell, R. B. Generic Model Control: A Case Study. Can. J. Chem. Eng. 1989, 67, 478. (14) Li, W. C.; Biegler, L. T. Process Control Strategies for Constrained Nonlinear Systems. Ind. Eng. Chem. Res. 1988, 27, 1421. (15) Zafiriou, E. Robust Model Predictive Control of Processes with Hard Constraints. Comput. Chem. Eng. 1990, 14, 359. (16) Boutayeb, M.; Rafaralahy, H.; Darouach, M. Convergence Analysis of the Extended Kalman Filter Used as An Observer for Nonlinear Deterministic Discrete-time Systems. IEEE Trans. Autom. Control 1997, 42, 581.

Received for review June 2, 1999 Revised manuscript received November 5, 1999 Accepted January 11, 2000 IE990395Z