Improved Regulatory Control of Industrial Gas-Phase Ethylene

Apr 27, 1999 - Burdett, I. D. The Union Carbide UNIPOL process: Polymerization of olefins in a gas-phase fluidized bed. AIChE Annual Meeting, Washingt...
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Ind. Eng. Chem. Res. 1999, 38, 2383-2390

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Improved Regulatory Control of Industrial Gas-Phase Ethylene Polymerization Reactors E. M. Ali,* A. E. Abasaeed, and S. M. Al-Zahrani Chemical Engineering Department, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia

This paper investigates the application of two techniques to improve the proportional-integral (PI) control performance when implemented to stabilize the operation of gas-phase polyethylene reactors around an optimal operating point which is open-loop unstable. The first technique deals with improving the performance of a single input-single output control loop through employing an on-line adaptive tuning strategy. The purpose of the tuning strategy is to force the closed-loop response to fit inside a desired time-domain specification envelope by automatically adapting the PI setting values. The second technique deals with improving the multiple input-multiple output control performance by simple selection of the proper control structure design. The objective of the proposed control design is to avoid the procedure of screening various control structure candidates. Moreover, because there are more inputs than outputs, a splitrange configuration is used to utilize all available manipulated variables. This configuration allows for tight control. Simulation results demonstrated the success of the proposed methods to provide a better regulatory control performance when compared to those that have been reported in earlier work. 1. Introduction Control of polymerization reactors has long been known to be a difficult task. This is due to the high nonlinearity of the reaction and the strong interaction of the reactor variables. It has been pointed out that gas-phase polyethylene (PE) reactors are prone to instability and the reactor temperature may exhibit serious runaway.1-3 The latter must be avoided because it may lead to catalyst deactivation and serious changes in product properties. Also, most industrial gas-phase fluidized-bed polyethylene reactors are operated in a narrow temperature range between 75 and 130 °C.4 For these reasons, stabilization for polyethylene reactors is a challenging problem and needs to be addressed through good control. Dadebo et al.1 demonstrated the stabilization of the polyethylene reactor temperature through a single input-single output (SISO) control loop using the coolant temperature to an external heat exchanger as the manipulated variable. This was illustrated via dynamic simulations for set-point changes. Regulatory control was only demonstrated via bifurcation analysis and not through dynamic simulations. In addition, practically, the reactor feed temperature is constrained between physical bounds, which in turn impose a limitation on the allowable range for the coolant temperature or equivalently for the amount of heat that can be removed by the cooler. Because of this limitation, perfect regulator control over a wide range of operating conditions may not be possible. Our previous work5 investigated the effect of physical constraints imposed on the feed temperature, which in turn represent constraints on coolant temperature, and on the regulatory control of the polyethylene reactor. Furthermore, improvement of the regulatory control through a different control structure and/or implemen* Corresponding author. E-mail: [email protected]. Fax: 00-966-1-4678770.

tation of an advanced control strategy such as nonlinear model predictive control (NLMPC) was addressed. The study revealed the superiority of NLMPC over a proportional-integral (PI) controller because of its capability to handle multivariable control systems efficiently. Implementation of a PI controller suffered from two shortcomings. For the SISO control problem, it was found that the PI settings had to be retuned from case to case to ensure a good performance. For the multiple input-multiple output (MIMO) case, it was found that the control structure selection affects the closed-loop performance of the process. In fact, the study indicated that the performance of an arbitrary control structure outperforms that found by rigorous published methods such as relative gain array (RGA) and singular value decomposition (SVD). This paper, thus, deals with improving the PI regulatory performance applied to the gas-phase polyethylene reactor by proper handling of the two shortcomings encountered in the previous work. First, the SISO tuning problem will be handled by incorporating an online tuning technique developed by Ali.6 The proposed tuning method is very suitable for this case because it has full automation and adaptive features. The other published methods, which are discussed elsewhere,6 are either off-line, computationally demanding, or at least nonadaptive techniques. In the latter case, the obtained tuning parameters are constant with time; thus, it does not account for the process variation with time. Second, improvement of the MIMO control problem will be tackled through implementation of a simple control structure design that depends on the steady-state gain array of the process along with the split-range control structure. The idea is to avoid the tedious screening of various control structure candidates and to provide more degrees of freedom. The latter will allow for a tighter control performance similar to that obtained by multivariable NLMPC.

10.1021/ie980636n CCC: $18.00 © 1999 American Chemical Society Published on Web 04/27/1999

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The paper is organized as follows: first, the dynamic model of the process will be presented. Next, a summary of the polyethylene reactor control objective and design will be presented. This will include formulation of the proposed on-line tuning strategy and the MIMO control structure design. Finally, open-loop and closed-loop simulations will be presented to investigate the effectiveness of the improved control algorithm. 2. Dynamic Model of the Reactor Figure 1 shows a schematic diagram of a polymerization fluidized-bed reactor, the UNIPOL process of Union Carbide.7,8 The major components of the process are shown in the figure. A simplified dynamic model, which was developed previously9 to describe the polymerization reaction of ethylene in a fluidized-bed reactor, is used in this investigation. The model is based on the two-phase (emulsion or dense and bubble phases) theory for fluidized beds. For brevity, only the final form of the resulting model equations in the dense phase is given here. A detailed derivation of model equations, assumptions employed, and numerical values of the various parameters is given in previous papers.5,9

dX1 ) a3(1 - X1) - R4(X1 - 1)(1 - e-R1) dτ R6e-R5/X3X2X12 - R6e-R5/X3X2X1 (1) X1 + R7 -R5/X3

X22X1

dX2 R6e ) R8Qc dτ X1 + R7

(2)

X3 - 1 dX1 γ5X1(Tf - X3) dX3 )+ dτ X1 + γ1 dτ X1 + γ1 R6e-R5/X3X2X1(X3 - 1) R6γ3e-R5/X3X2X1 + X1 + R7 X1 + γ1 γ4(X3 - Tf)(1 - e-R2) 4γ2(X3 - Tw) (3) X1 + γ1 X1 + γ1 where X1 is the dimensionless monomer concentration, X2 is the dimensionless catalyst concentration, X3 is the emulsion phase dimensionless temperature, Qc is the catalyst feed rate, and τ is the dimensionless time. The dimensionless superficial velocity φ is embedded in the model parameters. 3. Polyethylene Reactor Control Objective and Design Basically, industrial PE reactors possess several control objectives. The bed height is maintained at a desired value by manipulating the product flow rate. The reactor total pressure is perfectly controlled via a bleed stream. The reactor feed temperature is usually controlled by manipulating the cooling water flow rate of an upstream heat exchanger which is used mainly to remove the heat produced by the exothermic reaction. The product quality is achieved through maintaining the ratio of the gas partial pressures within desired targets. The latter is achieved by manipulating the gas fresh feeds. If reliable measurements of the product properties are available, they can also be used to minimize the off-spec products during grade transition. Here we are concerned with controlling the monomer

Figure 1. Schematic diagram of the fluidized-bed reactor for polymerization reactions. Table 1. Optimum Operating Condition of the Reactor φ

Qc (g/h)

Tf

X1

X2

X3

Tw

6.001 969.5 1.1 (330 K) 0.7524 1.202 × 10-4 1.33 1.03 (309 K)

Table 2. Manipulated Variable Constraints variable

upper value

lower value

Qc φ Tf

10.6 (×100 g/h) 10 1.2 (360 K)

1 (×100 g/h) 2 1 (300 K)

concentration and reactor temperature that corresponds to maximum yield, assuming that all other basic control loops are taken care of by other means. The optimum steady-state operating conditions, which were obtained previously,5 are listed in Table 1. The importance of our proposed control objective stems from the fact that the desired optimum set point is unstable, and thus any disturbances may drift the process variables away from the desired point. In addition, tight control of the reactor temperature is required to keep it above the dew point of the gases and below the softening temperature of the polymer. The available manipulated variables for this case are the catalyst feed rate (Qc), the ratio of the superficial velocity to the minimum fluidization velocity (φ), and the set point of the feed temperature (Tf). Physical constraints are imposed on these variables as listed in Table 2. The common disturbances to the process are the wall temperature, the feed temperature or coolant temperature, the inert gas concentrations, and the catalyst feed rate. Practically, disturbances in Qc occur because of malfunctioning of the catalyst feeder, which is very common to such processes. For brevity, we will consider the objective of rejecting the influence of disturbances in the reactor wall tem-

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perature by means of a standard PI control algorithm. In all simulations hereafter, the constraints in Table 2 will always be imposed, and a sampling time of 0.001 dimensionless units is used. It should also be noted that all variables, excluding Qc, shown in the figures including time are dimensionless. It is worth mentioning that the PI settings are determined by the Ziegler and Nichols10 method. 3.1. On-Line Tuning Strategy. In the following section we present a summary of the tuning strategy developed by Ali.6 Sensitivity Equations. The adaptation of the PI tuning parameters is achieved through exploitation of the sensitivity of the closed-loop response to the tuning parameters. Hence, the process closed-loop dynamic and its sensitivity to the PI settings are described by the following augmented state-space model:

dZ ) f(z,u) dt d ∂z ∂f ∂z ∂f ∂u ) + dt ∂kci ∂z ∂kci ∂u ∂kci ∂f ∂z ∂f ∂u d ∂z ) + dt ∂τIi ∂z ∂τIi ∂u ∂τIi

(4)

i ) 1, ..., ny

(5)

where z is the state vector, u is the manipulated variable vector, y is the process output vector, and ny is the number of controlled outputs. It should be noted that although a discrete PI controller will be used in the process, a continuous formulation is used in the model to facilitate the mathematical treatment for driving the sensitivity expressions. Taking y ) Cz, the closed-loop sensitivity expressions are

∂yi ∂z ) ci ∂kci ∂kci ∂yi ∂z ) ci ∂τIi ∂τIi

i ) 1, ..., ny

where ci is the ith row of matrix C. The expressions for the partial derivatives of u with respect to kci and τIi are given in Ali.6 It should also be noted that in the standard PI control framework (multiloop SISO control system), each controller loop connects one output to one manipulated variable. Thus, only the derivative of each output to its PI settings is required because the derivative of the output of one loop, and similarly the inputs, to the PI settings of another loop is zero. Closed-loop prediction of y and its gradients over a specific prediction horizon can then be obtained by numerical integration of the above augmented state equations. The initial conditions of the actual states and the manipulated variables are their corresponding initial steady-state values, while those for the gradients are zero. To reduce model-plant mismatch, the predicted output is corrected by disturbance estimates in the standard internal model control framework.11 The estimated disturbance is assumed constant over the prediction horizon and is set equal to the difference between the plant and model outputs at the current sampling point, k. PI Settings Tuning Strategy. This section presents an on-line tuning technique that adapts the PI param-

eters in order to steer the closed-loop response to satisfy desired preset time-domain specifications. Typical timedomain specifications for disturbance rejection are shown by the solid lines in the simulation figures in section 5. The on-line adaptation strategy centers around linear approximation of the relationship between the process output and the PI tuning parameters.6 The proposed algorithm is as follows: at any sampling point k given the performance envelope in the form of upper and lower bounds, i.e., yl and yu, and before computing the control action, do the following: Step 1. Predict the closed-loop response and its gradient over the prediction horizon Pw via numerically integrating eqs 4 and 5 for fixed values of the tuning parameters space, xk, and constant estimated disturbance at k. ∆xk is initially equal to xk-1. Step 2. Evaluate the predicted violation of the specifications:

{

Mi(k+l) ) yli(k+l) - yi(k+l) yi(k+l) - yui (k+l) 0 i ) 1, ..., ny,

if yli(k+l) > yi(k+l) if yui (k+l) < yi(k+l) if yui (k+l) > yi(k+l) > yli(k+l) l ) 1 + ndi, ..., Pw + ndi

}

Step 3. Determine the sampling point at which the maximum violation of the specification occurs. Let this be for output j and point k + m:

mj(k+m) ) max

max

1eieny 1+ndielePw+ndi

Mi(k+l)

Step 4. If Mj(k+m) ) 0, go to step 5; otherwise, continue. Step 4.1. Compute the deviation from the desired specification.

∆y )

{

ylj(k+m) - yj(k+m) if ylj(k+m) > yj(k+m) yj(k+m) - yuj (k+m) if yuj (k+m) < yj(k+m)

}

Step 4.2. Define a ) ∇xyjT(k+m) and scale it by postmultiplying with ψ, which is a matrix that has the current values of the settings on the diagonal and zero elsewhere. Step 4.3. If |a|∞ e β (a user-defined number equal to 10-5 in this paper), go to step 4.4; otherwise, solve

min |a∆xk - ∆y|2 ∆xk

subject to ∆xl e ∆xk e ∆xu where ∆xl ) xl - xk, ∆xu ) xu - xk, and xl and xu are the lower and upper bounds on x, respectively. Go to step 4.5. Step 4.4. Set ∆xk ) 0. Step 4.5. Set xk ) xk + ∆xkψ. Step 5. Compute and implement the control action. Shift to the next sampling time; set k ) k + 1. Go to step 1. The optimization problem is solved by MATLAB software. To implement the above algorithm, the user should provide a nominal performance envelope with a specific window size, threshold values within which the output variation is considered tolerable, and a value for the prediction horizon, Pw. The adjustment of the

2386 Ind. Eng. Chem. Res., Vol. 38, No. 6, 1999 Table 3. Steady-State Gains between Manipulated and Controlled Variables X1 X3

Qc

φ

Tf

0.2412 -0.2795

-0.3781 0.4227

0.009 -0.0099

gradient scaling matrix ψ is also sometimes useful. The effect of these variables is discussed in section 5. Activation of the Algorithm. Process Operating at Steady State. The closed-loop response predictions of the outputs over Pw will be monitored every sampling time. The adaptation algorithm will then be triggered automatically only when any of the prediction outputs violate their threshold value because of the influence of disturbance. The algorithm will be turned off again when the time exceeds the window size of the specs. The tuning algorithm will scale the nominal performance specs to properly suit the actual behavior of the process under disturbance. To achieve this, a scaling factor will be computed on-line at the triggering point based on an estimate of the actual effect of the disturbance on the output. The nominal envelope is then multiplied by the scaling factor for adequate adjustment.6 To avoid ambiguous values, the computed scaling factor should also be constrained between lower and upper limits which should be defined by the user. Selection of the upper and lower limits depends on the degree of uncertainty of the nominal specs. The algorithm triggering and envelope adjustment for the case when the process is operating at changing set point are discussed elsewhere.6 3.2. PI Control Structure Design. When the PI controller is implemented to a nonsquare MIMO system, the best control structure should be determined first, followed by determining the best control pairing. The best structure is the square subsystem which is expected to have the best closed-loop performance among all of the other sets of square subsystems. There are many approaches reported in the literature that deal with this situation, among which is the nonsquare relative gain array (NRGA) method.12 Cao and Rossiter13 utilized this tool for input selection. Cao and Biss14 used another method to select the control structure. They divided the nonsquare steady-state gain matrix into a set of square subsystems and computed the SVD for each set. The set with the largest SVD is taken as the most effective set. These two methods were used in our previous work5 to select the best two manipulated variables for our MIMO control problem. Then the RGA15 was used to obtain the best input-output pairing. First, the nonlinear model is linearized around the optimal operating point from which the steady-state gain matrix (G), for two-output and three-input system, is obtained as listed in Table 3. This matrix is used with the NRGA and SVD methods, which in turn suggested that X1 (monomer concentration) should be controlled by φ and X3 (reactor temperature) by Qc. However, our previous closed-loop simulations showed a poor performance for this control structure.5 In fact, another arbitrary control structure which relates X1 to φ and X3 to Tf had a superior performance.5 Therefore, based on the above finding and to avoid the procedure for best control structure selection and pairing, a simpler procedure is adopted here. The procedure will utilize all available manipulated variables to regulate the process controlled variables.

Because there are more manipulated variables than controlled variables, tighter control can be achieved. In this case, each controlled output will be paired with the manipulated variable that has the corresponding maximum magnitude of steady-state gain:

um f yn

where

(n, m) ) max |Gi,j| i,j

This choice is obvious because it provides a wider range of controllability. This procedure is repeated for each controlled output, eliminating in each step the mth column from the gain matrix. The remaining manipulated variable is then connected to the last controlled variable (because it is paired to the input with the smallest static gain) or to the most important one. The last variable would then be connected to the manipulated variables in a split-range configuration.15 The common split-range configuration is to switch the error signal among the inputs in a coordinated scheme. In this paper, the error signal will be used to actuate the candidate inputs simultaneously and continuously. However, the error signal should be partitioned properly before being sent to actuators. The idea is to produce equally effective control actions. The next issue is, into what proportion should the error signal be split to activate the two manipulated variables? There are many possible choices, among which are the following: 1. To connect the same error signal equally to each input (S1). 2. To connect half of the error signal to each input (S2). 3. To connect a portion of the error signal equal to the proportion of each input static gain to the other (S3). 4. To connect a portion of the error signal equal to the inverse of the proportion of each input static gain to the other (S4). The first choice (S1) may lead to overloaded control action because twice the error signal is being taken care of each sampling time. The third choice (S3) may also lead to a poor performance because the input with the larger static gain will be given more weight. This may overly use that input, causing aggressive control action. The second (S2) and especially the fourth (S4) choices are expected to give the best performance. In S4, the error signal will be weighted appropriately such that each input will produce the same order of control action. 4. Open-Loop Simulations Effect of Changes in the Wall Temperature. To demonstrate the nonlinearity and instability of the reactor, we simulate the process for step change in the wall temperature (Tw) of magnitude of (0.002 dimensionless with (equivalent to (0.6 °C). The results of the open-loop simulations are shown in Figure 2. As shown by this figure, even for small changes in the wall temperature, the reactor temperature (X3) and consequently the monomer concentration (X1) diverge away from the desired steady state. When the wall temperature increases, the heat loss to the surrounding area decreases, leading to an increase in the reactor temperature. Because of internal instability, the reactor temperature keeps growing until it reaches another stable steady-state point. The opposite trend is true when the wall temperature decreases. Although a higher reactor temperature indicates a higher monomer conversion, it occurs at a temperature much higher than the polymer softening temperature. Therefore, in the above two

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Figure 2. Open-loop simulation for step changes in ∆Tw. Table 4. PI Controller Parameters for the X3 f Tf Scheme case

kc

τI

∆Tw ) 0.002 ∆Tw ) 0.01 ∆Tw ) 0.02

6000 1500 600

0.02 0.01 0.01

cases, the reactor behavior is unacceptable, and thus a controller is needed to stabilize the process. 5. Closed-Loop Simulations for Disturbance Rejection (a) SISO Case. First, a single-loop controller, where the feed temperature (Tf) was used as the manipulated variable and the reactor temperature was used as the controlled variable, will be examined. Usually, Tf is used to stabilize the unstable temperature of fluidized-bed reactors.1,16-18 The process closed-loop response to different values of step changes in Tw using a PI controller with Tf as the manipulated variable was examined in our earlier work.5 The feed temperature was constrained between 1 and 1.2 corresponding to 300 and 360 K to ensure normal practical reactor operation. The simulation results are omitted here for brevity. A good control performance was observed for very small perturbations in ∆Tw, e.g., 0.002 and 0.01. However, for changes in Tw as low as 0.02 dimensionless units (6 °C), the feed temperature saturates and reactor ignition occurs, leading to a poor performance. Moreover, although the controller was tuned initially using the Ziegler and Nichols method for ∆Tw ) 0.02, it had to be retuned for a good performance for each different value of ∆Tw as listed in Table 4. Therefore, using the PI settings for ∆Tw ) 0.02 as the initial guess, the proposed tuning method will be applied to this control problem for two different values of ∆Tw, e.g., 0.002 and 0.01. The purpose is to adapt the controller settings on-line such that it results in an acceptable closed-loop performance as specified by the performance envelope. A nominal disturbance envelope is designed such that it allows the controlled variable to overshoot by 1% for the first 15 sampling instants after the triggering point, to come to within 0.5% of its steady state value for the next 15 sampling instants, and finally to settle within 0.2% of its steady-state value. The shape of this envelope will be shown in the simulation figures. The threshold value is set to (0.2%

Figure 3. Closed-loop simulation for step change in ∆Tw ) 0.002 using a tuning procedure with Pw ) 3.

of the steady-state value, and the envelope window size is set equal to 40 samples. The lower and upper limits of the scaling factor are 0.25 and 4, respectively. The simulation result for ∆Tw ) 0.002 is shown in Figure 3. The solid lines represent the specs envelope after scaling (reduced by 75%). The dotted curve demonstrates the regulatory response without tuning, while the dashed curve demonstrates that with adaptive tuning. It is clear that the adapted response substantially outperformed the nonadapted one. Pw ) 3 is used in this simulation. Different values would result in slightly different tuning performances, as will be explained by the last example. The tuning algorithm is triggered twice. The second triggering point occurred even when the plant controlled variable was within the bounds. This is because, at that point, the model closedloop prediction, which is corrupted by model-plant mismatch, does violate the bound. Figure 3 also shows the adaptation of Kc and τI with time. As expected, Kc increases to around 4000 as Table 4 reported a large value at ∆Tw ) 0.002, while τI decreases initially and then goes back to an acceptable value of 0.008. Figure 4 illustrates the effect of the tuning algorithm for the case of ∆Tw ) 0.01 using Pw ) 3. The figure legends are as before. In this case, the algorithm was triggered at the beginning where the envelope was reduced by 75% and was turned off after 40 samples and the envelope was set equal to the nominal threshold value. Substantial improvement in the closed-loop performance was obtained for the adapted setting case. This is achieved, as shown by the figure, at higher values for Kc and smaller values for τI. In the two above cases, the adaptation was achieved on-line and automatically starting from the same initial values for the PI settings. No user intervention was needed. In both cases, the tuning algorithm scaled down the nominal specs envelope by 75% to cope with the actual process behavior. This large reduction is an indication of an overly relaxed nominal envelope design. The tuning algorithm was not tested for ∆Tw ) 0.02, because the poor controller performance for that case was due to input saturation6 and, thus, PI settings adaptation does not help. To explore the strength of the tuning algorithm, we test its implementation to the control of a sequence of

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Figure 4. Closed-loop simulation for step change in ∆Tw ) 0.01 using a tuning procedure with Pw ) 3.

Figure 5. Closed-loop simulation for step change in ∆Tw ) 0.002 at t ) 0.01 followed by step change in ∆Tw ) 0.01 at t ) 0.1 using a tuning procedure with Pw ) 3.

two-step disturbances in the wall temperature. The simulation result is shown in Figure 5a. The dotted line represents the closed-loop response without tuning to reject a step disturbance of 0.002 in ∆Tw that occurs at t ) 0.01 followed by another one of magnitude 0.01 that occurs at t ) 0.1. Fixed PI settings equal to the last row of Table 4 were used. The solid lines in Figure 5a represent the specs envelope. In the beginning of the simulation, the solid line represents the threshold until the algorithm is triggered around the 12th sampling point because of the effect of the first step disturbance where the solid line becomes the desired specs envelope scaled down by the algorithm to 75% of its nominal value. The algorithm was then turned off after 40 samples from the triggering point where the specs envelope switched to the threshold value. At t ) 0.11 the algorithm was activated again automatically because of the effect of the second step change. The dashed line in Figure 5a represents the adapted response, which indicates dramatic performance improvement over that of the nonadapted response. It is obvious from Figure 5b,c that the PI settings were adjusted in the first 50 samples to improve the closed-loop response. This was

Figure 6. Closed-loop simulation for step change in ∆Tw ) 0.002 at t ) 0.01 followed by step change in ∆Tw ) 0.01 at t ) 0.1 using a tuning procedure. (a-c) Dashed line: Pw ) 1. Solid line: Pw ) 3. Dotted line: Pw ) 5. (a) Dark solid line: bounds.

also enough to minimize the effect of the second step load change. However, the PI settings were adapted further due to the second triggering of the algorithm. To clarify the effect of the prediction horizon, Pw, on the tuning algorithm, the last test was repeated for three different values of Pw. The simulation result is demonstrated by Figure 6. For small Pw, the algorithm will trigger later; however, aggressive adaptation may result because the computed sensitivity of the closedloop response to the PI settings would be the smallest. This is clearly illustrated in the figure for the case of Pw ) 1. For larger Pw, the algorithm will trigger faster and hence provide earlier corrections. However, the computed sensitivity increases with longer horizon, leading to smaller corrections, i.e., a smoother adaptation as is clear for the case of Pw ) 3 and 5. Thus, a tradeoff exists and the choice is left to the user which also depends on the process dynamics behavior. For process dynamics faster than the gradients (sensitivity) dynamics, aggressive adaptation may be desirable. The opposite is also true. Adjustment of the adaptation speed can also be achieved through gradient scaling, as mentioned in step 4.2 of the tuning algorithm. In fact, scaling is used to properly weigh the PI settings in order to provide the same order of adaptation magnitude and consequently to relax the speed of adaptation. If the diagonal elements of the scaling matrix are equal to the current values of the PI settings, then the scaled gradient would grow with time, resulting in a progressively slowing rate of adaptation. While setting, the diagonal elements of the scaling matrix set equal to the upper limit of the PI setting would result in a very slow adaptation in the beginning and an even much slower one as time progresses. On the other hand, an identity scaling matrix would result in the fastest rate of adaptation. In fact, the desired proper rate of adaptation depends on the process dynamics. In all of the above tests, the dynamics of the gradients are faster than the process dynamics. For this reason rapid adaptation was necessary, so an identity scaling matrix was used. (b) MIMO Case. To improve the closed-loop performance, a multiloop control scheme should be used.

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Figure 7. Closed-loop simulation for step change in ∆Tw ) 0.01 using a MIMO control structure. (a and b) Solid line: S5 control structure. Dotted line: S2 control structure. Dashed line: S4 control structure. (c and d) Solid line: S1 control structure. Dashed line: S3 control structure. Table 5. PI Controller Parameter Values scheme

kc

τI

X1 f φ X3 f Qc X3 f Tf

120 300 500

0.01 0.01 0.001

Specifically, the reactor temperature and monomer concentration should be controlled by any combination of the available three manipulated variables. Our previous work5 illustrated that using Qc and φ as manipulated variables as suggested by the NRGA and SVD methods leads to a poor performance. An improved closed-loop performance was obtained when the arbitrary control structure (X1 f φ, X3 f Tf denoted as S5) was used. Therefore, selection of the best control structure is not straightforward, and even when found, it failed to provide a good performance. For this reason we seek, in the following, improved control performance using the proposed control structure design. When the steady-state gain matrix of Table 3 is inspected and the criteria discussed in section 2.2 are used, φ is connected to X1 and Qc to X3. Accordingly, the remaining variable, Tf, should be connected to X3. The PI controller settings for all loops are listed in Table 5. All of the split range options listed in section 2.2 will be tested. For S3, 77% of the error signal will be directed to actuate Qc and the remainder for Tf. While for S4, 0.23% of the error signal will be directed to actuate Qc and the remainder for Tf. Figure 7 illustrates the effect of the proposed control structure to reject the effect of step disturbance in ∆Tw of magnitude 0.01. As demonstrated by the figure, the second and fourth options (S2 and S4) had superior performances over options 1 and 3 (S1 and S3). Also, as expected, S1 has the worst feedback response because it overly uses the error signal. When compared to the performance of structure S5, which is shown by the solid line in the figure, the second and fourth options had, despite the initial overshoot, superior performances in the sense of no offset (tight control). This tight control is the result of a higher degree of freedom, i.e., three inputs to two outputs configuration. The corresponding manipulated variable responses of S2, S4, and S5 for this test are shown in Figure 8. The manipulated

Figure 8. Manipulated time response for the simulation in Figure 6. Solid line: S5 control structure. Dotted line: S2 control structure. Dashed line: S4 control structure.

Figure 9. Closed-loop simulation for step change in ∆Tw ) 0.02 using a MIMO control structure. (a and b) Solid line: S5 control structure. Dotted line: S2 control structure. Dashed line: S4 control structure. (c and d) Solid line: S1 control structure. Dashed line: S3 control structure.

variable responses for the other cases, i.e., S1 and S3, are excluded for clarity. Similarly, Figure 9 depicts the effect of the proposed control design when rejecting a step disturbance of magnitude 0.02 in ∆Tw. The same earlier observation is also obtained here. It should be noted that, among all control design options, S4 had the best performance. Moreover, S4 outperformed S2 in the sense of much less overshoot and downshoot. The reason is that even if, for the case of S2, the error signal was split into equal halves, this does not guarantee equal order of control action for the two candidate manipulated variables because the large difference between their corresponding static gain values. 6. Conclusions Our previous work5 revealed two shortcomings of the application of PI control to stabilize the polyethylene

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reactor under load step changes. This paper dealt, therefore, with improving the controller performance for the same control objective. The first problem was the need for proper tuning to improve the SISO control performance from one value of load change to another. This was handled by employing an adaptive automatic tuning strategy. The tuning strategy depends on the development and utilization of sensitivity expressions for the closed-loop response with respect to the PI tuning parameter. This requires a theoretical model for the process to exist. Simulation results revealed the success of the tuning algorithm to improve the regulatory performance when the process is under the influence of two different values of step disturbances and of two consecutive step changes in the wall temperature. The effect of the algorithm main parameter, namely, Pw, is addressed. Similarly, proper scaling of the sensitivity equations was found useful for a better tuning performance. The second problem was the proper selection of the best control structure and pairing for the MIMO control problem. Taking advantage of the process high degrees of freedom and the steady-state gain matrix, a simple and straightforward design criterion was proposed and investigated. The extra input is paired using the splitrange configuration with special signal-partitioning schemes. The simulation results demonstrated the ease of the control design and good performance in the sense of tight control (offset elimination). It was also found that splitting the signal in the same proportion as that of the inverse of the static gain for each input leads to the best closed-loop performance. This is because it results in equally effective control actions. Nomenclature a ) vector of output sensitivity C ) constant matrix d ) disturbance estimate G ) steady-state gain matrix k ) sampling instant kc ) controller proportional gain Mi ) bound violation for output i ndi ) delay time for output i ns ) envelope window size ny ) number of controlled outputs Qc ) catalyst injection rate (g/h) Pw ) closed-loop prediction horizon Si ) control structure i Tf ) dimensionless feed temperature (K) tk ) sampling time Tw ) dimensionless wall temperature (K) u ) vector of manipulated variables X1, X2 ) dimensionless monomer and catalyst concentrations in the dense phase X3 ) dimensionless dense phase temperature x ) vector of PI tuning parameters y ) vector of outputs xl, xu ) lower and upper values of PI tuning parameters yl, yu ) lower and upper values of process output yp ) vector of plant outputs z ) vector of state variables Greek Letters Ri, γi ) parameters used to nondimensionalize the model equations

β ) constant small value for the tuning algorithm ∆x ) deviation of tuning parameters from their current values ∆y ) deviation of outputs from their active bounds ∆Tw ) deviation of the wall temperature from its nominal value ψ ) scaling factor matrix for the tuning algorithm τ ) dimensionless time τΙ ) integral time

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Received for review October 5, 1998 Revised manuscript received February 24, 1999 Accepted February 25, 1999 IE980636N