Control of Chemical Reactors Using Multiple-Model Adaptive Control (MMAC)

Copyright © IFAC Dynamics and Control of Chemical Reactors (DYCORD+'95). Copenhagen. Denmark, 1995

CONTROL OF CHEMICAL REACTORS USING MULTIPLE-MODEL ADAPTIVE CONTROL (MMAC) KEVIN D. SCHOTTo and B. WAYNE BEQUETTEo °Ren"e/aer Polytechnic Inltitvte, Howard P . IHrmann Department of Chemical Engineering, Troy, NY 1£180-3590, email: bequebUrpi.edu

Abstract. Multiple-Model Adaptive Control (MMAC) is shown to be an effective strategy for controlling nonlinear chemical processes. In this paper, MMAC is applied to two difficult control problems that may be encountered in continuous stirred tank reactors (CSTRs) . In the first example, product concentration is controlled in an isothermal CSTR exhibiting input multiplicities and nonminimwn phase behavior (Van de Vusse reaction) . The MMAC configuration provides disturbance rejection where a fixed-paramet.er PlO controller fails. The second example involves temperature control of a CSTR which has output multiplicit.ies . While a single PlO controller can stabilize the plant over a large operating range, its performance can be quite poor since the plant dynamics vary widely. MMAC is shown to give significantly bet.ler performance when moving from the open-loop stable regions to the open-loop unstable operating region . Key Words. Nonlinear Cont.rol, model adaptive control, multiplicities.

1. INTRODUCTION

Nonlinear behaviour is a characteristic of chemical processes which presents a challenging control problem. In spite of the wealth of recent research results that have been presented in the literature (see Bequette (1991) for a review), few nonlinear model-based control techniques have been implemented in the process industry. One reason is the lack of nonlinear models to adequately characterize the process (either due to lack of knowledge or the time to develop the models). Another reason is that the nonlinear control methods are not well-understood by many of the engineers t.hat. would implement them . The goal of this paper is to present a method, Multiple-Model Adaptive Control (MMAC), which is conceptually straightforward and easily implemented wit.hout the requirement of extremely detailed process models .

Fig . 1. MMAC system structure

focused on aircraft control (Athans et al. 1977). More recently, He et al. (1986) applied MMAC to blood pressure control and Yu et al. (1992) to simultaneous blood flow/blood pressure control in heart attack patients. Figure 1 shows a schematic of the MMAC system structure . Instead of using a single, complex model to account for all possible plant operatingconditions , a bank of less complicated, less comput.ationally intensive models are used and the need to explicitly account for all parameter changes in one model is avoided . Each model has a controller whose tuning parameters are based on that model. A weighting function chooses which model or combination of models best represents the current plant input/out.put behaviour, then produces a weighted-sum of t.he controller outputs to supply the control act.ion .

MMAC is a model-based control strategy which incorporates a set of model/ controller pairs rather than relying on a single model and controller to handle all possible operating conditions . It was developed for non linear control problems with varying parameters, particularly for situations where the plant was well known (and hence well modeled) in some operating regions. yet poorly known in others. The MMAC strategy was proposed in the early 1970's, yet is only now being applied to chemical processes (Schott and Bequette, 1994; Banerjee et al. 1994) and offers a new technique for managing difficult process control problems. Much of the early ~II\IAC work 345

This blending of models is a key feature of the MMAC method: if a precise plant model is not available for all operating regions, model-based control techniques may still be used without loss of performance in the regions that are well known. Some systems may operate around one part.icular setpoint for much of the time and can be described by just one of the models in the model bank . For this case, a single controller is chosen which by itself can adequately control the plant. However. as disturbances enter the system or different set-points are requested (such as during start.-up). the plant dynamics may change and thus reqUIre a controller which is tuned differently.

pairs. Because (4) is recursive, a zero probability for a particular model would force all subsequent probabilities for that model to zero. The probabilities are therefore bounded from reaching zero for

Pi ,A-

>

Pi,1:

fJ

Pi,l: ~ 0

Pi ,I:

and fJ

(5) 1

<-N

This bounding also limits the number of past observations contained in the current estimation of the probability. As K increases, the current difference Ci.1: has a greater effect on Pi,1: than past differences . Finally, these lower-bounded probabilities need to be removed from the final weighting , so the probabilities are renormalized by

Although the parameters in all the model/controller pairs are fixed, MMAC is adaptive since different combinations of models are used to represent the plant as the plant parameters change . Because all of the models are always available. another MMAC feature is zero start-up time for plant identification. Nearly any control methodology (PID, LQR, NL-MPC, MIMO strategies, etc.) can be used within the MMAC framework . In this work we use PID plus first-order filters for all of the controllers in the controller bank .

Pi.1:

Wi,1:

for Pi ,1:

N

Li=1 Pi,l: 0

Wi,l:

Pi,1: ~ fJ

> fJ (6)

and only Pi)' > f, are included in the summation. The final, incremental control-signal change for the kth sample instant is N

~Ul:

+ a2Yl:-2 + .. . +boul: + b1Ul:-l + .. .

alYl:-l

(1 )

The actual plant inputs Ul:. Ul:-l, . .. and out.put.s YI:, Yl:-l, ... were used in calculating the model prediction, so the ith model prediction for time step k is given by Yi,l:

=

+ a2Yl:-2 + .. . +boul: + b 1 Ul:-l + .. .

3. EXAMPLE 1. INPUT MULTIPLICITIES For a Van de Vusse reaction the following parallel and consecut.ive reactions are taking place

alYl:-l

and

(2)

The modeling equations are given by

The "weighting box" of Figure 1 is implemented in several steps. First , the difference bet.ween the plant and each model is calculated (i,1:

= Yl: -

dXl

dt dX2

(3)

Yi,l:

dt

A recursive version of Bayes' theorem f{ = Li=lexp([exp( -cj,IeA Ci,l: )Pi,l'- d -(i.1:

Pi,l:

N

(i ,l: )Pi ,A--l ,

(7)

In both of the examples which follow, discrete velocity-form PID controllers were used, augmented with a first.-order filter. Controller tuning parameters were established using the IMCbased PlO controller-tuning method with each linear model.

In this study, the models in the model bank were developed by linearizing and discretizing the plant equations, yielding difference equations

=

Wi , I:~Ui , 1e

i=1

2. IDENTIFICATION AND CONTROL WEIGHTING

Yl:

=L

(8) (9)

where XI and X2 are state variables representing concentrations and k 1 • k2, and k3 are the rate constants. The concentrat.ion X2 is to be controlled using the dilution rate u as the manipulated variable. For this example, X2 is the plant output y. Figure 2 shows the steady-state input-output behavior of the system. Parameter values for this example are listed in Table 1.

(4)

is then used to estimate the probability that the model represents the plant at the kth sample. Here K is a parameter controlling the rate of convergence and N is the number of model/ controller ith

346

1.30

moved , they can be significantly reduced by reducing the interval between models near the peak. The results of a more comprehensive MMAC implementation using 12 models are shown in figure 5. The models are evenly spaced in ~u intervals of 5 from u = 45 to u = 100. In this case we are attempting to operate directly at the peak of the I/O curve . The first setpoint (Y'P = 1.25) falls directly on model 4 and the weighting goes primarily to that model. At t 2.5, the setpoint is changed to Y,p 1.275, above the I/O curve and unattainable by the system . The plant output responds by moving to the highest attainable output, then starting to oscillate. For the remainder of the run , the setpoint is set just slightly above the peak of the I/O curve, forcing the system to run at its highest achievable productivity. Between t = 6 and t = 8, a disturbance of XI! 9.75 enters the system . Since the difference between the setpoint and the output is large the system oscillates rapidly between the models closest to the peak. However, because the distance between the models is small, the oscillation amplitude remains small . Although the oscillations may be seen in the manipulated variable, they are imperceivable at the output since the process gain is so small near the peak . Unlike the previous case, here when the disturbance ended the system went to the right side of the I/O curve.

1.20 1.10 1.00 >.

0.90 0.80 0.70 0.60 0

25

SO

75

100

125

ISO

175

200

225

=

u

Fig. 2. Steady-state input-output relationship for CSTR with Van de Vusse reaction

Single PID Control. If production of B is to be maximized, then the system must operate as closely to the peak of the input-output curve as possible. However, noise or dist.urbances may cause the system to shift from one side of the input-output curve peak to the other. The major limitation when using a fixed-paramet.er PID controller on a system exhibiting input multiplicities is the controller's inability to account for the sign-change of the plant gain should the system cross the peak of the I/O curve. In this particular system, a disturbance in feed concentration shifts the I/O curve lower, forcing what was once a valid setpoint to become unat.t.ainable. In such a circumstance, a controller designed on one side of the I/O curve will itself push the system away from its setpoint because the plant and controller have opposite signs. Figure 3 shows the response of a PID controller designed at tl = 65 as the feed concentration to the reactor is suddenly changed from XI! 10 to XI! 9.75 g";ol for t 0.2 to 0.37. Once the system moves to the right side of the I/O curve (u > ,.., 77.5), the single PID drives the system in the wrong direction . Even when the disturbance is removed, the controller cannot recover and the manipulated variable continues to increase until a system constraint is encountered (u = 200).

=

=

=

=

4. EXAMPLE 2. OUTPUT MULTIPLICITIES The dimensionless modeling equations for a twostate exothermic CSTR for A - B are given by

=

dXl

dT dX2

dT liu

+ qX2! t\(X2)

MMAC. Figure 4 shows the response of the MMAC strategy using three models and subjected to the same rectangular-pulse disturbance as in Figure 3. Models were chosen corresponding to dilution rates of 45,65, and 10011- 1 . The MMAC configuration correctly identifies that the system gain has changed sign and then uses the controller with the correct gain to move the system back toward its setpoint. Since the setpoint is still unattainable, oscillations occur as the system get.s pushed back and forth between the controllers at the peak. Once the disturbance has ended . the system returns to its original set.point on the correct side of the input-output curve .

(11)

=

exp

C;2~ )

(12)

where Xl, and x';? are dimensionless concentration and reactor temperature and u is the dimensionless jacket temperature. Definitions of the dimensionless parameters and the parameters values used for this example are listed in Tables 2 and 3. Figure 6 shows the steady-state input-output behavior .

Szngle PID. Although a single PID controller can be tuned t.o function in both the open-loop stable and OL unstable operating regions, its overall performance may be quite poor. A single PID tuned for operation in an OL stable region can perform well for small setpoint changes close to where the controller is designed to operate, yet

While the oscillations cannot be completely re347

the same controller exhibits large overshoot when stepping further away into the OL unstable region (Figure 7a).

He, W.G ., H. Kaufman, and R.J. Roy (1986) . Multiple-Model Adaptive Control Procedures for Blood Pressure Control. IEEE Trans . Biomed. ElIg., BME-33 (I) , 10-19

MMAC. Four linear models were chosen , corresponding to dimensionless reactor temperatures of 1.30, 2.00 , 2.25, and 5.0 . Model one is in t he lower OL stable branch , models 2 and 3 are in the middle OL unstable region, and model 4 is in the upper, OL stable branch . Figure 7 shows that for the same setpoint change, even this simple implementation of MMAC is able to completely remove the overshoot seen in the single PID rase. While such a large setpoint change may not be made routinely, this example shows how MMAC may be used to improve system performance in nonroutine situations such as start-up , withou t. sacrificing performance at the system's usual setpoin t.

Schott , K.D. and B.W . Bequette (1994) . Control of Nonlinear Chemical Processes Using MultipleModel Adaptive Control (MMAC) . 1994 AIChE Annual Meet ing, paper 224j , San Francisco, CA Yu , C.L. , R.J . Roy, H. Kaufman , and B.W . Bequette (1992) . Multiple-Model Adaptive Control Predictive Control of Mean Arterial Pressure and Cardiac Output. IEEE Trans . Biomed. Eng., 39 (8) , 765-778

Table 1. Parameter values for Van de Vusse reaction simulations 50 .0 h- 1

5. SUMMARY AND CURREt\T WORK

k1

Presented here are basic results which show that Multiple-Model Adaptive Control is a viable strategy for controlling nonlinear chemical reactors. The use of different models for different. operat.ing conditions allows the integration of many cont.rol objectives into a single method . Such flexibility is clearly an advantage in cases such a<; start-up and batch-processing where comprehensive plant models may be difficult to obtain . Currently we are applying MMAC to a batch reaction control problem where the overall heat. transfer coefficient changes over the course of the batch and fouling causes additional batch-t.o-batch variations in t.he heat transfer coefficient . We are also applying robust performance concepts to gain more insight on estimating the number and spacing of models to achieve a pre-specified performance goal.

k'2

100 .0 h- 1

k3

10 .0 gm~1

=

10 .0

xlf

h

9";0'

Table 2. Dimensionless variables for the CSTR model
Nominal Damkohler number

[J

Dimensionless Dimensionless Dimensionless Dimensionless

£, X2J

q

61

heat of reaction heat transfer coefficient reactor feed temperature reactor flowrate

Volume ratio Dimensionless activation energy

-y

Table 3. Parameter values for two-state exotherm ic CSTR simulations

REFERENCES Athans, M. , D. Castanon , K-P. Dunn, C.S. Greene, W.H . Lee, N.R . Sandell , and A.S. Willsky (1977) . The Stochastic Control of t.he F-8C Aircraft Using a Multiple-Model Adapti ve Cont.rol (MMAC) Method - Part I: Equilibrium Flight. IEEE Trans . Automat. Control, AC-22 (5) , 768780

0.072 8.0

6

0.3


X 2J

q

61 f

Banerjee A., Y. Arkun , B. Ogunnaike, and R . Pearson (1994) . Robust Nonlinear Control by Scheduling Multiple Model Based Controllers. 1994 AIChE Annual Meeting, paper 230a , San Francisco, CA Bequette, B.W. (1991) . Nonlinear control of chemical processes: A review . IlId. E7Ig. Chern . Res. , 30 , 1391-1413. 348

=

[J

0.0 1.0 0.3 20

1.30_-----------------, 1.25

1.0 0.9 0.8

4-------. '"

'~r

1.20

+=========;::;;::::;-;::;::::;-;::::::;====:--,

0.7

o.6

o.S

1 II ~: ~ i=::;:::;::::::;;::::::;:::::;::~~~~~~::;:::;:::.-J ~ ~ 0 .4

- · ..... 1

-

0 .3

1.10

...... 2

.... ..t.ll

11

1.0!I

0.0

1.00 4-.......--r--.---,r--.--.,.-........"'T"""...,....--,---r--.-r--,.-~ 0.6 0 .4 0.7 0 .8 0.3 O.S 0.0 0. 1 0.2

0.0

0. 1

0 .2

0 .3

0 .4

O.S

0 .6

0 .7

0.8

time

time

Fig. 4. c. Model weights ; MMAC response to the same rectangular pulse as figure 3

Fig . 3. a. Output

200

I.)() _ _- - - - - - - - - - - - - - - - - - - - - ,

180

1.21

160

1.26

140

1.24

120

.., 1.22

.- - - I

100

T

o

1.20

80

1. 11

6Oi=~~~~~~~T-~~.-~ 0.0

0.1

0.2

0.3

0.4

O.S

0 .6

0.7

1. 16

0.8

time

6

10

12

time

Fig. 3. b. Input ; Single PID response to a rectangular pulse disturbance

Fig . 5. a. Input

IOO~-----------------------------_,

1.30 ......- - - - - - - - - - - - - - - - - - . . , 1.25

90

4-------.

1.20

10-

... 1.1S

r

70

1.10

60"1-_ _ _"'"

1.0!I

0.0

0.1

0 .2

0.3

0 .4

O.S

0.6

0 .7

6

0 .8

10

12

10

12

time

time

Fig . 5. h. Output

Fig . 4. a . Input

1. 0

110

.,

0.1

1:

100

00

.~

90

0.6

eo c ...

80

8

0.2

70

604-----J 6

0 .0

0.1

0 .2

0 .3

0 .4

O.S

0 .6

0.7

0.8

time

time

Fig . 5. c. Model weight.s ; MMAC 12-model implementation

Fig . 4. h . Output

349

Fig. 6. Steady-state input-output relationship for an exothermic CSTR with output multiplicities . reactor temperature: y ; jacket temperature: u

3.0 -r------------------, 2.5

2.0

1.5

o

2

6

10

time

Fig. 7. a . Input 2.0 - 1 " " ' - - - - - - - - - - - - - - - - - - ,

"

1.0

o.s 2

0

6

8

10

time

Fig. 7. h. Output 1.0

f

(

0 .8

~

"il

] §

0.6

0.4

·····:::i I '----------------------_ -

0.2

IDOdd 3

. - model.

..

:\

0.0 0

2

4

6

8

10

time

Fig. 7. c. Model weights; Performance comparison of MMAC and a single PID for the same st.ep change in setpoint

350