Ind. Eng. Chem. Res. 1994,33, 3063-3069
3063
Effect of Process Nonlinearity on the Performance of Linear Model Predictive Controllers for the Environmentally Safe Operation of a Fluid Catalytic Cracking Unit Lokesh Kalra and Christos Georgakis' Chemical Process Modeling and Control Research Center and Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015
One possible approach to developing processes that are environmentally friendly and economically competitive at the same time involves the effective use of modern control technology to enable the operation of the process below the environmental constraints but close to them. However, often due to nonlinearities in the process, the performance of linear control strategies degrades as the system is operated close to constraints. Addressing these issues in the context of the specific process of fluidized bed catalytic cracking is the motivation behind the present paper. Such a study can help in the understanding of how process nonlinearity might restrict the effectiveness of linear control strategies and thus provide motivation for nonlinear strategies. 1. Introduction
The Fluid Catalytic Cracking Unit (FCCU) and its profitable operation are vital to a refinery. A major challenge in FCCU operation is in meeting the environmental restrictions on stack gases, mainly carbon monoxide. The combustion reaction to burn carbonaceous deposits from the spent catalyst produces carbon dioxide and carbon monoxide. One way of preventing the violation of stack emission constraints is by maintaining an abundant supply of air during this reaction. However air supply is limited by blower capacity and the pressure in the unit. Hence, to reduce stack gas carbon monoxide, the total feed to the FCCU may have to be reduced, affecting the profitability of the process. It is precisely these apparently conflicting issuesprofitability and operation within environmental constraints-that the present research is looking to bring to concurrence. Developments in the area of nonlinear control and the availability of increasingly powerful computer control hardware at falling prices should facilitate this process. At present however, a majority of catalytic cracking units are controlled using linear control strategies. With this in mind, it is instructive to study the nonlinearities in the FCCU and how they affect control system performance. It is expected that this investigation will yield an insight into improved controller and/or process design leading to better, safer, and environmentally friendly control of the process. For setting the appropriate background a brief description of the FCCU and its control is provided in section 2. Section 3 outlines our experiments in openloop nonlinearity assessment of the process. A closedloop look at process nonlinearities is provided through implementation of linear Model Predictive Control (MPC) on the process. Details of the algorithm used and tuning procedures are presented in section 4, and section 5 contains the results of our closed-loop simulations. Concluding remarks are the topic of section 6. 2. The Process: Fluid Catalytic Cracking Unit
The Model IV FCCU consists of two main sections: the reador, where the catalytic cracking reaction occurs, * T o whom all correspondence should be addressed. Email: cgOWlehigh.edu.
and the regenerator, where hydrocarbon deposits that are products of the cracking reaction are burned off the catalysts (Figure 1). Because of the high degree of interaction between the two sections, governed largely by the pressure differential between them, the Model IV is a challenging problem from the point of view of process identification, control, and optimization. To address the need for a detailed and realistic simulation for such research a generic but realistic model motivated by an industrial process-the Amoco FCCU-was developed through a cooperative effort between Lehigh University and the Amoco Corporation (McFarlane et al., 1993). This section contains relevant details of the model. Subsection 2.1 outlines a few salient features of the model itself and the resulting dynamic simulator. The control objectives for the process, as well as our definition of the multivariable control problem as we perceived it are defined in subsection 2.2. 2.1. The Model and Simulation. The mechanistic model of the Amoco FCCU was developed as a test bed for advanced strategies in process control (McFarlane et al., 1993). For this reason it strives to capture the major dynamics of the model IV reactorhegenerator sections, including nonlinearities and interactions as well as all equipment and operating constraints imposed on the process due to economic, environmental, and safety considerations. The authors of the model also provide an ACSL (Advanced Continuous Simulation Language) program for integrating the model. With provision for incorporating FORTRAN code for carrying out on-line control and optimization, the ACSL program yields a powerful simulation of the Model IV FCCU. For a detailed description of the FCCU and its open-loop dynamic characteristics, the reader is referred to McFarlane et al. (1993). 2.2. Control Objectives. To derive the maximum cost advantage the system needs to be operated at high throughputs. The ability t o reject disturbances that could otherwise lead to a violation of constraints, particularly at high throughputs, is thus a major objective of FCCU control. To examine control system performance, a number of operating points with progressively larger throughputs were designed. This was accomplished by increasing the feed to the open-loop
Q888-5885/94l2633-3Q63$04.5OJQ 0 1994 American Chemical Society
3064 Ind. Eng. Chem. Res., Vol. 33, No. 12, 1994 Stack
Downstream Separators
Preheat/ Furnace Combustion Air
Blower
Gas Oil
Spent catalyst Sdrry Recycle
Diesel Wash Oil
Figure 1. Schematic of the Model IV FCCU. Table 1. Operating Points operating points
OP-I OP-II OP-III OP-lv
feed flow rate, lb/s 126.0 127.3 128.9 131.0
stack CO concn, ppm 25 100 150 205
system and allowing it to settle a t a new steady state. Table 1 highlights the important details of these operating points. We decided on a set of manipulated and controlled variables, following to an extent Amoco’s Multivariable Predictive Control (AMPC) on the FCCU (McFarlane and Reineman, 1990). As mentioned before, the objective of FCCU control is to be able to operate the process without violating environmental and equipment constraints. Environmental restrictions can be adhered to by maintaining conditions of excess oxygen in the regenerator and by ensuring a proper regenerator bed temperature. As a component of the total air flow into the regenerator, lift air supply has an immediate effect on oxygen quantities available for burning coke off the catalyst. Thus lift air flow rate is an important manipulated variable. Total combustion operation of the regenerator can also be ensured by reducing the total feed and/or the fraction of heavy feed into the FCCU system. Thus fresh feed flow rate and the amount of the heavy slurry recycle making up the total fresh feed are also important inputs to the system. Finally, reactor-regenerator differential pressure is an input that can change catalyst recirculation and can have an important effect on the entire FCCU system. Thus, it is an important manipulated variable, keeping in mind that excessive variation should be avoided in order not t o upset the system. Regenerator temperature is best controlled by manipulating the fresh feed and slurry recycle flow rates. For total combustion operation it is desirable to keep bed temperatures above 1265 OF. The wet gas compressor suction valve is a pointer to reactor pressure behavior. A low value indicates that reactor pressure is low, and with the reactor-regenerator differential pressure controller active, the pressure in the regenerator is low. Consequently more air can be pumped into the regenerator and the throughput of the FCCU can simultaneously be raised. Typically, the wet gas com-
pressor suction valve would be set at 95% open so that it can react to disturbances in the reactor pressure. Reactor riser temperature is also an important variable to control because it has an effect on the yield of wet gases produced. Thus the inputs and outputs used in our MIMO predictive control strategy are controlled variables: ( 1 ) stack gas carbon monoxide concentration (CO,) (2) reactor riser temperature (T,) (3) regenerator bed temperature ( T r e g ) (4) wet gas compressor suction valve position (Vll) with the following manipulated variables: ( 1 ) lift air flow rate (F9set) (2) feed flow rate ( F 3 d ( 3 ) slurry recycle flow rate (F4set) (4) reactor-regenerator differential pressure (Dpst) Additional control structures, in particular, one where the riser temperature T, and differential pressure Dp,t are excluded from the multivariable MPC and from a §IS0 PI loop, were also tried out. However in this paper we will be presenting only the results of simulation experiments that were carried out with the above 4 x 4 multivariable control scheme. It is amply recognized that catalytic cracking is a nonlinear process (McFarlane et al., 1993). For a variety of reasons summarized nicely in the review paper by Bequette (19911, linear control strategies such as MPC that have not been explicitly designed to take process nonlinearity into consideration lead to operational problems. These problems are aggravated when the process encompasses a large operating range, as the FCCU does a t the three operating points. In the three sections that follow the nonlinearities in the FCCU are highlighted and their effects on control of the process close to the environmental constraints are discussed. 3. Nonlinearity Assessment
A simple and informative method of assessing process nonlinearity is a step test. With lift air flow rate as the manipulated variable, three step tests of different magnitudes were conducted at each of the operating points OP-I and OP-IV. Plotted in Figure 2 are the normalized stack gas carbon monoxide concentrations to 5, 15, and 25% step increases in lift air flow rate. The normalization was done in order to bring the responses to different step magnitudes to the same
Ind. Eng. Chem. Res., Vol. 33, No. 12, 1994 3065
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Figure 2. Normalized stack gas carbon monoxide concentrations to 5, 15,and 25%changes in lift. air flow rate at OP-I:C m m a l i z e d = ( C w t u a l - Cinitial)/(Cfinal - C i n i t i a d . "*
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Figure 3. Steady-state concentration change in oxygen. The solid line indicates the concentration changes for a linear approximation of the process (based on a 2% step).
scale. It can be observed that the normalized transient curves are almost identical. This implies that the dynamic characteristics change very little for these magnitudes of steps. To see the potential differences in process gain behavior, not shown in Figure 2 because of the normalization, the changes in the steady-state value of the concentrations to the same steps in lift air flow rate were plotted. For stack gas oxygen the results are shown in Figure 3. The lines show how the concentration would have changed with increasing flow rate if the process were linear. This line is based on a 2% step change in flow rate and is a good, if approximate, baseline for demonstrating nonlinearity. The circles and crosses show the actual concentration change, and it can be observed that deviations from linear behavior increase with increasing size of the step change. In this figure we present two sets of data-one for OP-I and another for OP-IV. It can be observed that the data for OP-IV deviates from linear behavior more than the one for OP-I, which is quite linear. Figure 4 shows the steady-state change in carbon monoxide concentrations to the same step increases in lift air flow rate. As with oxygen concentrations in Figure 3 it can be seen that the process deviates more from linear behavior moving from OP-I t o OP-IV. However, Figure 4 illustrates another point that possibly explains the choice, in at least one industrial
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Figure 4. Steady-state concentration change in carbon monoxide. The solid line indicates the concentration changes for a linear approximation of the process (based on a 2% step).
installation, of oxygen as a controlled variable rather than carbon monoxide itself for keeping the FCCU below the environmental constraints. Apart from the local nonlinearity, which is the deviation from linear behavior around an operating point, there is another issue of considerable interest. In Figure 4 it can be seen that the slopes of the two lines representing local linear behavior are quite different from each other. This means that a t operating point IV the magnitudes of concentration change to the same change in air flowrate are much greater than at operating point I. This has two major implications from the point of view of process control. First, while it is definitely possible to get acceptable performance at any operating point by tuning the controller for that operating point, one has also to be prepared for the eventuality that a large and unexpected disturbance might push the process outside the range for which this controller has been tuned. Hence, from this point of view, it would be useful to study the open-loop characteristics of the plant over its expected range of operation, assumed here t o be well represented by the space spanned by the operating points OP-I to OP-IV. The second implication relates t o process optimization. In processes where optimized operation offers considerable economic incentives, an on-line optimizer usually operates atop the multivariable controller guiding the process along the best path continuously adjusting controller setpoints. In the case of the FCCU such as optimizer would eventually take the process to operating point OP-IV, which is closer to the environmental constraint on carbon monoxide and which will be more profitable because of the higher feed rate. Hence, a control system should be so designed that it can take the FCCU from an initial conservative operating point to a less conservative, more profitable operating state. Because of the substantial difference in steady-state gains, as evident from the slopes of the straight lines in Figure 4, a strategy based on linear control theory that is well tuned for one operating condition may not perform satisfactorily for another operating point. It might be too conservative in its dynamic response and would thus fail to meet environmental constraints. Thus a controller that can handle large changes in process gain during operation will be ideal for this application.
3066 Ind. Eng. Chem. Res., Vol. 33, No. 12, 1994
4. Model Predictive Control
To examine the effects of these changes in gain on the performance of the controller, model predictive control was implemented on the catalytic cracker. This study of MPC on the FCCU can also be viewed as experiments at closed-loop assessment of process nonlinearity. A brief recap of the MPC algorithm is presented in subsection 4.1 for a better understanding of the section to follow. Controller tuning is an important aspect of any control strategy implementation. While the literature abounds in studies on MPC tuning, in our opinion it was lacking in details on how one should arrive at weighting parameters that define tightness of control action. Described in subsection 4.2 are our contributions toward rectifying this deficiency. 4.1. The MPC Algorithm. For the MIMO control problem defined above, in this study the well established linear model predictive control strategy was used, mainly because of its simplicity and ability to handle multivariable processes and input, output, as well as rate constraints (Morari et al., 1993). In MPC the control program can be posed as an optimization problem where a sequence of m future control moves Au1, Aug, ..., Aum, has to be determined. These inputs have to be such that they minimize the weighted squared sum of predicted deviations of the outputs from setpoint over p future time steps, as well as the weighted squared sum of manipulated variable moves over m future time steps. In other words
where Ep is the vector of predicted errors from setpoint (E;l, ..., E;p)T t o p steps into the future, A U is the ~ vector of control moves (AUT, A U ~ ,..., A U ~ ) Tsteps into the future and
r;
ry,..., ry}, r;
= diag{rY,
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r",..., r"} (2)
are the weighting matrices of appropriate size in block diagonal form. The objective function (eq 1)can be solved to find the vector of future control moves. For unconstrained MPC an expression can be found for the controller gain KMPC off-line leading to considerable savings in real-time computation. Constrained MPC is usually solved online as a quadratic programming problem. Whatever the method, in the receding horizon approach that we used, only the first of the calculated control moves Aul is implemented and the whole vector is calculated a t the next control instant. 4.2. Tuning the Controller. The parameters whose values are required for the tuning of the MPC algorithm are (1)the sampling time (27,(2) the number of time steps to steady state after an input change (model horizon, n),(3) the number of time steps into the future over which error from setpoint has to be minimized (prediction horizon, p ) , and (4) the number of control moves into the future calculated by the controller (control horizon, m). Following the literature and our own experiments, for the present set of experiments the values used were T = 1 min, n = 60, p = 20, and m = 15. Once the basic tuning parameters are determined, the elements of the weighting matrices P and Tu need to be decided on. P adjusts the relative importance of
tightness of control on the outputs and Tu penalizes large control moves. P and Tu have significant effect on control performance, so much so that some of the other parameters such as p and m have an effect only when Tu = 0. Hence elements of these matrices need to be determined with some justification and considerable care. Not finding any guidelines in the literature, we attempted to come up with a tuning approach ourself. For convenience and due to lack of reasonable grounds to do otherwise P and Tu are chosen to be diagonal matrices. Then, say at time instant 1, the prediction errors of each of the outputs is contained in the vector Epl and the moves of the inputs are in the vector Au1. The very first term of eq 1 corresponding to this time instant is
where ny is the number of outputs, and the remaining terms are completely analogous. In order that all terms in the quadratic objective function be brought to an equal basis, it is suggested that each y is chosen equal to the inverse of a value related to the normally allowed window of change in the variable This definition makes dimensionless each of the additive terms in the performance criterion in eq 1. For predicted errors this dividing factor can be expressed as a product of the setpoint yss and the allowable percent variation. A similar procedure is followed for the manipulated variables. To allow an additional degree of freedom in order to adjust the relative movements of the variables, a parameter a is introduced into the y's. To achieve all this the y's is the weighting matrices would need to look like
y , = a,/(steady state),(allowable % deviation), = a/(allowable absolute plus or minus deviation from steady stateIi (4) Thus the smaller the allowable deviation from steady state, the greater is the weight on that variable, irrespective of whether it is an input or an output. For the a's, the choice of unity as an initial guess has been adequate for most of the cases that were tried out, more so for constrained than for unconstrained MPC. An example relevant to our 4 x 4 MIMO control objective helps in clarifying this concept. At OP-11, for the outputs, the steady-state values and the tolerable deviations from setpoint are stack gas carbon monoxide (100 f 30 ppm), riser temperature (995 f 10 OF), regenerator temperature (1265 i5 OF), wet gas suction valve position (0.95 f 0.05), giving, with the a's set to unity, the weighting matrices - ...
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Ind. Eng. Chem. Res., Vol. 33, No. 12,1994 3067
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A factor of 6 has been introduced into the weighting matrices for controller moves. Once the relative sizes of the elements of P and pl are decided on, the relative sizes of the two r matrices themselves may need t o be adjusted to fine tune the controller. The simple parameter 6 serves as a handle for this purpose, with unity being a good initial guess value. It has been our experience with a number of processes and operating conditions we have worked with that the suggested guidelines consistently yield a very good starting point in controller tuning. This is demonstrated through the following two examples where a disturbance-an upward step in the coking factor of the feed (McFarlane et al., 1993)-is introduced 5 min into the simulation. Figure 5 shows the effect of changing 6 from 1.0 to 0.1 as well as to 10. As expected, for the smaller value of 6 the controller is more aggressive and the larger 6 results in a detuned controller. The extent of movement of one variable via-a-vis that of the others can be affected by changing the corresponding a value. For example if we decided that lift air flow rate F9 was moving too much, we could reduce this movement by increasing the value of the a corresponding to lift air flow rate -am,,, say from 1 to 100. The results of a simulation doing just that are shown in Figure 6. As can be seen, the most notable effect of clamping the action of the lift air blower is on the peak level stack gas carbon monoxide reaches after the disturbance.
5. MPC at the Operating Points With the above choice of manipulated and controlled variables and tuning constants, model predictive control was implemented at each of the operating points. Because the results of constrained MPC simulations were very similar, only unconstrained MPC results are
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presented below. In general the weighting matrices P and Tu ought to be different at each operating point in order to reflect the changing allowed variation of the variables. For the operating points considered here, the changes are not significant. Hence the same values of the tuning constants were employed in all that follows. With minor modifications of the earlier choices, the sampling time, the model horizon, the prediction horizon, and the control horizon were T = 2 min, n = 40, p = 20, and m = 15, respectively. With the initial guess weights as described in subsection 4.2, it was observed that while the inputs lift air flow rate and differential pressure moved too much, slurry recycle flow rate was very slow in reacting. This caused offsets from setpoint over the 120 min simulation in the regenerator temperature and the wet gas valve position, two outputs that identified models indicate recycle flow rates affect more than the other inputs. To rectify this situation and with a little bit of fine tuning we evolved from the initial guesses to P = diagL0.3, 0.2,0.4, 201 and Tu = 1 diagi0.4, 0.2, 4, 201. These correspond t o aco, = 10, aT, = 2, ar,, = 2, avll= 1,amset= 2, amet = 2, aFket= 0.4, aDpst= 20, and 6 = 1. The performance of the controller in rejecting the feed coking factor disturbance was examined through simulations. For the first set of runs, which we shall name set A, a model was identified at each operating point and used in MPC a t that operating point. It was observed that apart from a spike in the stack gas CO concentration immediately following the disturbance, the output responses were fairly satisfactory (Figure 7) at OP-11. CO concentration, riser temperature, regenerator temperature as well as the suction valve position all go back to the setpoints-their starting values-within 120 min. As the feed to the system is increased, the ability of the controller to reject the disturbance progressively deteriorates. At OP-IV the initial response of the system is quite oscillatory and the controller fails in keeping stack gas carbon monoxide concentrations below the environmental stipulation of 300 ppm. Thus moving the process closer to its constraints on carbon monoxide emissions degrades the ability of the control system to reject a feed disturbance. The difference in process gain seen in the open-loop step tests (Figure 4) causes the controller that is well tuned at OP-I1 to perform poorly at OP-lV. A few more runs that were carried out led to some interesting observations. For the second set of runs (set
3068 Ind. Eng. Chem. Res., Vol. 33, No. 12, 1994
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B) a single model, the one identified at OP-11, was used in MPC at all the operating points. As Figure 8 shows, the disturbance rejection characteristics for this case, where the low throughput model is used at all the operating points, are superior in almost all respects to the corresponding characteristics of the set A simulations. The outputs come back to their setpoints in a reasonable amount of time, even at the high throughput operating point OP-IV. Two more sets of three simulations each were also carried out, this time with a fured model either identified at OP-I11(set C) or at OP-IV (set D). It was observed that sets C and D performed less satisfactorily than set B, with set D doing worse than set C evident from the oscillations in the response and a longer settling time. This seems to indicate that for this process the low throughput model is better for MPC purposes than the higher throughput models. This is a curious and interesting conclusion since it goes against the common rule of thumb that in order to implement MPC, or any other controller, at a certain operating point, a model identified at that operating point will perform the best. At present, research is still ongoing in trying to explain these observations. The approach that is being taken involves examining the transfer function of the closed-loop FCCU system following the interpretation of MPC as an observer-regulator pair (Morari and Lee,
Time
Figure 9. Unconstrained MPC at OP-IV with model from OPIV-effects of different 6: 6 = 1 (solid), 6 = 2 (dashed), and 6 = 10 (dash-dotted).
1991). A comparison of the closed-loop pole-zero locations may yield an explanation for this phenomenon (Kalra and Georgakis, 1994). Some of the effects observed above are not seen when the controller is detuned by using larger values of the factor 6. This is seen in Figure 9 where three values 6 are used on the set A run a t OP IV. As 6 is increased from unity to 2 and further to 10, some of the oscillatory effects observed in Figure 7 go away. But this increased robustness is not without a cost. As Figure 9 shows, both the riser temperature as well as the regenerator temperature peak at higher values and take longer to settle. So while detuning the controller is an alternative, it usually takes the form of a quick-fix that does not really address the underlying cause of the problem. 6. Conclusions
Reported in this paper is a simulation-based assessment of the effects of process nonlinearities on the Performance of linear model predictive controllers. This is an issue that has to be addressed in order for linear strategies to be used to control processes close to constraints where nonlinearities could affect closedloope performance. Such a study can yield guidelines on controller tuning and the initial choice and subsequent updating of the model in linear model based control of nonlinear process such as the catalytic cracker. Using open-loop step tests on the FCCU, it was established that there is little dynamic nonlinearity between stack gas carbon monoxide concentration-the output that best helps ensure environmentally safe operation, and lift the flow rate-the input that has the most immediate effect on regenerator gas concentrations. It should however be mentioned here that step tests are not the best of indicators of high-frequency process dynamics and that nonlinearities at these frequencies would go undetected via this method. Nonlinearity was observed in the steady-state gain behavior, and this could severely restrict the effectiveness of optimizers operating higher up in the hierarchy and affect the disturbance rejection performance of linear model based controllers. Linear model predictive control was implemented on the catalytic cracker to observe how these nonlinearities manifest themselves. A set of tuning guidelines was
Ind. Eng. Chem. Res., Vol. 33, No. 12, 1994 3069 developed to address the need to approximately select the values of the weighting matrices required by MPC. I t was shown through simulations that this approach yields excellent initial guesses for these matrices, while retaining considerable flexibility for fine tuning. A n interesting phenomenon was observed in simulation experiments on the FCCU using different models for MPC. It was found that models identified at a low throughput operating point, when used in MPC performed better on the high throughput operating point than a model identified at the latter operating point. At the same time, MPC utilizing a model identified at a high throughput operating point did not perform well when used a t the low throughput operating points. Work is ongoing to explain this phenomenon through an analysis of the closed-loop catalytic cracker under MPC.
Ep = vector of predicted errors P = MPC output weighting matrix ru = MPC input weighting matrix = block diagonal output weighting matrix in eq 1 ri = block diagonal input weighting matrix in eq 1 K~p= c controller gain in MPC (unconstrained) T = sampling time n = model horizon p = prediction horizon m = control horizon yi = element on the main diagonal of the weighting matrices ai = parameter allowing an addition degree of freedom in choosing yi 6 = factor to adjust the weighting between the inputs and outputs
Acknowledgment
Bequette, B. W. Nonlinear control of chemical processes: A review. Znd. Eng. Chern. Res. 1991,30,1391. Garcia, C. E.; Prett, D. M.; Morari, M. Model predictive control: Theory and practice-a survey. Autornatica 1989,25,335. Kalra, L.; Georgakis, C. The effects of operational and design characteristics of catalytic cracking reactors on the closed-loop performance of linear model predictive controllers. In Proceedings of the ZFAC Workshop on Integration of Process Design and Control, IFAC, in press. McFarlane, R. C.; Reineman, R. C. Multivariable optimizing control of a Model IV Fluid Catalytic Cracking Unit. AICHE Spring National Meeting, Orlando, Florida, 1990. McFarlane, R. C.; Reineman, R. C.; Bartee, J.; Georgakis, C. Dynamic simulator for a Model IV Fluid Catalytic Cracking Unit. Comput. Chern. Eng. 1993,17,275. Morari, M.; Lee, J. H. Model predictive control: The good, the bad, and the ugly. In Proceedings of the Fourth International Conference on Chemical Process Control - CPC W , CACHE, 1991,419. Morari, M.; Prett, D.; Garcia, C.; Lee, J. Model Predictive Control. DraR prepublication copy.
Support of this research from the Industrial Consortium of the NSF Industry-University Cooperative Research Center in the area of Chemical Process Modeling and Control, and from a Fellowship from Hercules Incorporated are greatly appreciated. The help of Marie Campbell, a summer NSF undergraduate intern, in performing some of the reported simulations is thankfully acknowledged.
Nomenclature OP = operating point CO, = stack gas carbon monoxide concentration T, = reactor riser temperature Treg= regenerator bed temperature V11 = wet gas compressor suction valve position FgSet= lift air flow rate F3set= feed flow rate F4set= slurry recycle flow rate DpSt= reactor-regenerator differential pressure Aui = manipulated variable move at the ith sampling instant AU = vector of control moves Epj = predicted error at the j t h sampling instant
Literature Cited
Received for review April 1, 1994 Revised manuscript received September 7, 1994 Accepted September 20, 1994@
Abstract published in Advance ACS Abstracts, November 1, 1994. @
