Interval matrix robust control of a MMA polymerization reactor

European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.

755

Interval Matrix Robust Control of a M M A Polymerization Reactor Andrea Silva B., Antonio Flores T. * and Guillermo Fern~dez A. Universidad Iberoamericana, Prolongaci6n Paseo de la Reforma 880 M6xico DF, 01210, M6xico In this work results on the robust stability of interval matrices are applied to the robust closedloop control of a MMA polymerization reactor. Reactor operation is analyzed around an optimal design point where the system exhibits output multiplicities, besides the operation point is practically a limit point. A robust linear MIMO control law is synthesized where plant and input disturbances are captured as interval matrices. The closed-loop system performance was analyzed in presence of input disturbances and model uncertainties.

1. INTRODUCTION The closed-loop control of polymerization reactors is an important task in the chemical processing industry. Such control is sometimes complicated due to the fact that strong nonlinearities are commonly exhibited. Among these nonlinearities the presence of input/output multiplicities, isolas and disjoint bifurcations can be mentioned. In addition one of the most important end use polymer properties, molecular weight distribution, normally cannot be on-line measured for closed-loop control purposes. Because most of the polymerization reactions kinetic parameters are seldom known with enought precision, and they can be time variant, the impact of modeling errors on the closed-loop performance must be taken into account. In the control literature there has been some ways of taking into account robustness issues. One of such approaches consists in representing model uncertainty as upper and lower bounds on the nominal plant description. Such upper and lower bounds define and interval system where the plant to be controlled can be any plant located between such bounds. In this work we propose a control synthesis technique to address robust stability issues by using a constant gain linear multivariable control law. The synthesis technique is partially based on previous work [ 1] based on capturing modeling errors as upper and lower bounds on the state-space representation of a model, and from this plant representation to derive theoretical conditions to guarantee closedloop robust stability for linear systems. However, no synthesis technique was proposed in [1]. Hence, the main contribution of this work consists in the design of the control law for systems described in terms of multivariable linear interval plants.

2. MATHEMATICAL MODEL The polymerization system addressed was the free-radical bulk methyl methacrylate (MMA) polymerization using AIBN as initiator and toluene as solvent. The set of polymerization re*Author to whom correspondence should be addressed. E-mail: [email protected],phone/fax: +52 5 267 42 79, http://kaos.dci.uia.mxJ'aflores

756 actions takes place in a CSTR, the mathematical model of the MMA polymerization is given by:

dCm at = dCl = dt dT dt = dDo = dt dD1 = dt dTj dt =

F(Cmin -Cm) v

-(kp + ~:~) C~Po +

(1)

-ktCt + (FICtin- FC1) V F(r,. - ~) (-AH)kpCm UA pc~ P o - pcov~ ( r - rj) +

(2)

(0.5ktc + ktd)Po2 + k f m C m P o

(4)

-

(3)

vo-----Z~ V

Mm(kp + k:=) C~eo- FO----11

(5)

V

Fcw(Two- Tj) UA (~ Vo + pwCpw-----~o

~)

(6)

where,

/2:*C,k, Po = Vk-~d~tc,

kr--Are

, r = p, fm, I, td, tc

the polymer average molecular weight is defined as the ratio D1/Do. Under nominal conditions, the system model exhibits three steady-states (see [2] for model notation and for a complete description of parameter values and steady-state solutions). The low conversion (7.8 %) steadystate is used as the nominal operating point throughout this study. It is important to mention that even though the operation point is stable, it is practically a limit point; that is, it almost corresponds to the point in which the slope sign changes. This fact may imply control problems.

3. CONTROL TECHNIQUE Through a non-linear behavior analysis of the polymerization system [2] it was expected that the control problem would not be simple. In [1 ] the authors provided a series of theoretical conditions that have to be met in order to assure robust stability of interval matrices and established that Hurwitz stability is guaranteed if condition 1 is met and at least one of the three conditions described in condition 2: 9 condition 1.

9 condition 2.


757 see [1] for notation. In this work the above results were applied to the robust closed-loop control of the MMA polymerization system. Using a proportional controller u = kx to control the plant represented by the linear description :~ = Ax + Bu, the following closed-loop expression is obtained Jc = (A + Bk)x. In order to use the stability conditions in such system, it is necessary that the (A + Bk) matrix be represented as an interval matrix. Defining the uncertainty level of matrices A and B as A: .4=A(I+A),

A_=A(1-A),

B=B(I+A),

B=B(1-A)

the interval and nominal matrices (At,Ao), respectively are given by:

a~= [a+B_k, ~+~k],

ao=a+Bk

using the above expressions it becomes possible to apply the conditions for robust stability as stated in [ 1] to design an interval robust control system. In order to use the proposed technique it is necessary to have a square system. Therefore, the selected controlled variables were monomer and initiator concentrations (Cm and Ct), and reactor and cooling system temperatures (T and Tj). The manipulated variables selected were the system inputs: monomer, initiator and cooling system flow rates (F,/~ and Fc~) and the inlet temperature (Tin). The central matrix was obtained by linearizing the plant around the optimal operating point, and the interval matrices were derived assuming a 4- 99% uncertainty in the central matrix. The controller k was synthesized, using the optimization toolbox, by solving any of the following unconstrained optimization programs:

ein{n k

( Iel

, i,j=l

I i

the closed-loop operability of the linear plant was analyzed and the results proved that the controller provides robust and performance stability characteristics. Nevertheless, we decided to prove the performance of the linear MIMO controller on the original highly non-linear plant representation. 4. RESULTS AND DISCUSSION

The controller performance was analyzed through closed-loop dynamic simulations of the original non-linear system in presence of input disturbances, in particular changes in the monomer inlet concentration. The dynamic simulation of the closed-loop system when the inlet monomer concentration increases by 10% is shown in figure 1. Figure 1(a) shows the states deviation. In figure 1(b) the manipulated variables relative percentage deviations are shown (the notation used in the results figures is as follows. Outputs'. ACre .... ACIin -.-., A T .... ATj - - , A M W ~ . Inputs'. AF w , A F t , AFcw - - , ATin ...) . It can be observed that there is a large control action. The monomer and initiator flowrates had to be reduced by more than 40%, and the system is stable after 0.5 hours. The largest deviations in states are observed in monomer and initiator concentrations (ACre +4% and ACt -6%), while the molecular weight had a smaller deviation (+2%). Both temperatures remained at the optimal operation point (see figure 1(a)). In figure 2 closed-loop system simulation results when the propagation activation energy increases by 0.3% are shown. The output variables were always under control, but the behaviour

758

-5 o.

....................................... -IO

~_-lS

i:

~.

'.

~o . . . . . . . . . . . . . . . . . . . . . . .

o~

0'1

0~5

0'2

0~5

r~n8 (h)

0'3

o~s

014 0.~,5

-45 -15(

os

(a)

o~

o',

o',~ o'~ o~

rwne (h)

o'~ o~

o'4 o~

o~

(b)

Fig. 1. Closed-loop deviations in controlled (a) and manipulated variables (b) using a 10% disturbance.

0.3%Epunoert,~r~ . . . .

Cmi,

2O0

~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14~ 12C

g

10C

c ac

. . . . . . . . . . . . . .

Time(h)

r r n e (h)

(a)

(b)

Fig. 2. Closed-loop deviations in controled (a) and manipulated variables (b) using a 0.3% disturbance.

Ep

of the manipulated variables shows that potential saturation problems are likely to occur in the initiator flowrate control valve. This is so because the nominal initiator flowrate is small and therefore any slight modification of this flowrate might lead to saturation problems. No saturation problems were observed for the remaining manipulated variables. In order to test the interval controller under more demanding operating conditions, the monomer inlet concentration was increased by 10% and a 10 minute monomer concentration measurement delay was also considered. In figure 3 closed-loop simulation results for this case are shown. Output variables were controlled around the desired nominal operating region, however

759 lo'~C~n,'qltutt)ancamcllOrmemr,entr=endew

,

,

0

g

1o

i J

I

.

-300

,

Ot'~ Or2 0=3 014 Or5 0=5 017 ~=8 O=g

o*~ ol2 o'a 0'4 o'5 o's ol7 o s ~me (h)

"n rne (h)

(a)

o s/

(b)

Fig. 3. Closed-loop deviations in controlled (a) and manipulated variables (b) using a 10% Cmin disturbance and 10 min. concentration delay.

5

0 3 % Ep

uncertaintyen~1 0 rranconc~ntrenonaelay

loo c

!o

(a)

(b)

Fig. 4. Closed-loop deviations in controlled (a) and manipulated variables (b) using a 0.3% Ep uncertainty and 10 min. concentration delay.

wide variations in the monomer and initiator flowrates were observed. Comparing figures 1 and 3 we observe that the introduction of the concentration delay makes infeasible the use of the interval controller since saturation problems could emerge. In figure 4 the closed-loop system simulation when the propagation activation energy increases by 0.3% and there is a 10 minute monomer concentration delay is shown. Comparing figures 2 and 4 we notice that closed-loop results look similar, i.e. the introduction of the concentration delay did not make worse the controller performance. Again, potential initiator flowrate saturation problems are observed.

760 From the analyzed cases we observe that, due to the quite small values of the initiator flowrate, almost any disturbance might cause saturation problems on this input variable. Because the interval controller is a pure gain controller, offset problems are likely to occur. The problem could in principle be corrected by introducing integral action into the control system. Because most of the saturation problems occur in the initiator flowrate one might question about the utility of using the initiator flowrate as manipulated variable. In some cases product composition could be obtained by using the remaining 3 input variables. Besides, when they were obtained, acceptable closed-loop responses were observed in around 4 times the reactor residence time (0.1 h), which implies good closed-loop disturbance rejection characteristics. 5. CONCLUSIONS Polymerization systems, like the one in study, have highly non-linear behaviour specially when they operate around multiplicity regions. The nonlinear behaviour may imply control dificulties. In this case, the system was controlled around an operating point that is practically a limit point. Using the results on stability of interval matrices [1] it was posible to obtain a controller that provides robust stability on the controlled system. It is clear that some deviations were observed, but since the average molecular weight is the most important variable in a polymerization system, it can be concluded that the system was succesfully controlled since, under the considered upsets, not very large deviations in this variable were observed under the considered upsets. Besides, being the interval controller a pure gain controller, it would be easier to implement than other control structures.

REFERENCES 1. Wang, S., S.B. Lin and L.S. Shieh. Proceedings of the lASTED International Conference CONTROL' 97 (1997), 31-34. 2. Silva, A. and A. Flores Ind.Eng.Chem.Res. 38 (1999) 4790-4804.