Ind. E n g . Chem. Res. 1989, 28, 1308-1324
1308 w
= frequency, r a d / m i n
Literature Cited Fuentes, C.; Luyben, W. L. Control of High-Purity Distillation Column. Ind. Eng. Chem. Process Des. Deu. 1983, 22, 361. Luyben, W. L. Process Modeling, Simulation and Control for Chemical Engineers; McGraw-Hill: New York, 1973.
Moudgalya, K. M.; Luyben, W. L.; Georgakis, C. A Dual-Pulse Method for Modeling Processes with Large Time Constants. Ind. Eng. Chem. Res. 1987,26, 2498. Stahl, H. Transfer Function Synthesis Using- Frequency . . Response Data. Int. J . Control 1984,-39, 541. Received f o r review October 14, 1988 Revised manuscript received April 24, 1989 Accepted J u n e 8, 1989
Continuous Solution Polymerization Reactor Control. 1. Nonlinear Reference Control of Methyl Methacrylate Polymerization Derinola K. Adebekun and F. Joseph Schork* School of Chemical Engineering, Georgia I n s t i t u t e of Technology, Atlanta, Georgia 30332-0100
This paper focuses on nonlinear control of a solution polymerization reactor via a reference controller. A stability and convergence analysis shows that global stabilization of the reactor is possible under certain conditions. It is also demonstrated that, for certain equilibrium points, input multiplicity exists in the controlled subset of the system state, and this is detected by the controller in the closed loop.
Introduction Polymerization reactors are known to exhibit strong nonlinearities (Knorr and O’Driscoll, 1970; Hamer et al., 1981; Schmidt et al., 1984; Schmidt and Ray, 1981) and pose difficult control problems. In particular, the molecular weight distribution (MWD) also has to be maintained in order to produce polymers of acceptable quality. Typically, the MWD is characterized by the first few leading moments of the distribution. From these moments, typical measures of polymer quality, which include the number-average chain length ( F ) and the polydispersity (D), can be computed. Methods for doing this are discussed by Ray (1972). Furthermore, in continuous plants, the desired distribution may occur only at equilibrium points where the reactor is open-loop unstable. Indeed, as is demonstrated in Adebekun et al. (19881, it is possible that all equilibrium points corresponding to a given MWD are open-loop unstable. Thus, the control systems for these reactors should be able to give offset-free performance over a wide range of operating conditions. As can be expected, the control of polymerization reactors has attracted some interest in the literature. Most of the earlier works involved some form of optimal control, in particular, for batch and semibatch polymerizations. Typical case studies include the papers by Hicks et al. (19691, Osakada and Fan (1970), and Hoffman et al. (1964). In CSTR polymerization, much can still be achieved from a control standpoint. Most of the work in this area involves designs based on the traditional Taylor series linearization (or its variant) about some base point. Even so, good results are possible with this approach. Examples of these can be found in the work of Tanner et al. (1987) and Congalidis et al. (1986). More recently, advances in nonlinear systems theory have led to the development of nonlinear control synthesis techniques. Some of these results have been applied to polymerization reactor control. Thus, Alvarez et al. (1988) and Kravaris and Soroush (1988) have applied differential geometric methods to both continuous and batch polymerizations, respectively. Some other nonlinear applications *To whom all correspondence should be directed.
involve the work by Marini and Georgakis (1984). However, more experience is required in this area for rational control system design. Other approaches involve adaptive control techniques. This is typified by the work of Kwalik and Schork (1985). In this work, we approach the control problem via a reference controller approach. These ideas for nonlinear process control are becoming increasingly important and appear to have been proposed by Boye and Brogan (1986). Subsequent work on this has been presented by Bartusiak et al. (1988). Under certain conditions, we are able to guarantee global asymptotic stability. Furthermore, we establish contact again with bifurcation theory in the guise of the c a t a s t r o p h e set and demonstrate the importance of reactor “psychoanalysis” in design applications (Bilous and Amundson, 1955). The relationship to bifurcation theory is in fact not new. It was first pointed out in a control framework by Koppel (1982). Koppel’s observations have since been further generalized by Balakotaiah and Luss (1985). In particular, for polymerization reactors, the important implications of input multiplicity are first brought to light. It is interesting to note that even without integral action, the controller is able to detect the existence of input multiplicity. Indeed, this is consistent with the results of Balakotaiah and Luss (1985) in that input multiplicity is a property of the system and n o t of the controller. In part 2 of this work (Adebekun and Schork, 1989), in order to motivate realistic evaluations, we incorporate a Kalman filter in latter design stages for reconstruction of unavailable state variables. The robustness of the scheme to filter errors is then evaluated via simulations.
Model The polymerization of methyl methacrylate proceeds by a free-radical mechanism. The solvent is ethyl acetate, while the initiator is benzoyl peroxide. The reaction is highly exothermic and has been the object of some study (Schmidt and Ray, 1981). In the interest of brevity, we merely state that the following model equations hold (Schmidt and Ray, 1981; Tanner et al., 1987): VM = q ( M f- M ) - Vk,MP (1)
0888-5885/89/2628-1308$01.50/0 IC 1989 American Chemical Society
Ind. Eng. Chem. Res., Vol. 28, No. 9, 1989 1309
+ [(Dafx1Exf+ Da,x, + gtDaaWWExa) X (2a - a2)+ giDat,WE,,] W / ( l - a ) (15) X I = -xl + [(Dafx1Exf+ DasxI + gtDaaWEXa) X (a3- 3a2 + 4a) + gtDatcWExtc(a+ 2)]W/(1 - a)' (16) 1, = -x6
where a = Da$,x1
DapXlEx + DafE,fxl + gtDatWEXt+ Dasx4
(17)
and
where a=
kPM
kpM
+ kfM + kf5S+ k,P
(8)
P = (2fkd1/kt)l/' is the total concentration of live radicals, and k,(=k, + ka) is the overall termination rate constant. The ith moment of the dead polymer MWD is denoted by X i (i = 0, 1, 2), and M is the monomer concentration. T , I , and S are the reactor temperature, initiator concentration, and solvent concentration, respectively. The rate constants with the exception of kf, are assumed to follow an Arrhenius dependence on temperature. T h e Schmidt-Ray (1981) correlation for the gel effect has been employed. The relevant equations are 0.10575 exp[17.15Vf - 0.01715(T - 273.211 Vf > [0.1856 - 2.965 X 10-4(T- 273.211
gt =
k, = kt
(9)
2.3 X lo4 exp(75Vf) Vf 5 [0.1856 - 2.965 X 10-4(T - 273.2)]
where Vf is the free volume calculated from the volume fractions of monomer, polymer, and solvent in the reactor (Schmidt and Ray, 1981). It is important to state that the gel effect severely complicates the nonlinear open loop dynamics of the process. Thus, following Jaisinghani and Ray (1977), the following dimensionless variables are introduced:
The model comprised of eq 10-18 has been accepted as an accurate representation of the system dynamics. It has been demonstrated (Schmidt et al., 1984; Adebekun et al., 1988) that, over a range of residence times, the model can admit five equilibrium solutions. As noted earlier, this is mainly due to the gel effect phenomenon. Remarks are in order concerning the glass effect correlation (g,) sometimes employed for the propagation rate constant. We set this factor to unity in this work, as this is often known to be a good assumption. As is often done, we identify subsystem 1 of the reactor as consisting of the monomer, temperature, initiator, and solvent levels. Subsystem 2 consists of the MWD subsystem characterized by the moments ( A , i = 0, 1,2) and is uniquely determined by subsystem 1 at equilibrium. Some generalities are now in order. We emphasize that, even though we employ methyl methacrylate in this work, the model equations for subsystem 1 will have identical structure for many solution homopolymerizations carried out in a CSTR. Thus, the control schemes to follow are sufficiently general for most continuous solution polymerizations. Furthermore, as is sometimes done, it is possible for the dimensionless heat-transfer coefficient ((3) to be a function of the reactor medium viscosity (and, hence, the state variables of subsystem 1 as in Baillagou and Soong (1985)). Again, as long as such a function is well defined (nonzero, continuous, with strictly positive infimum and bounded on the real line), then these control schemes are also indeed feasible. Similar remarks hold for other possible gel-effect correlations (gJ. Both conditions are usually ensured by the physical nature of the polymerization process.
Results and Discussion In the interest of brevity, only representative examples are presented. The results are organized as follows: in section I, we develop the first control scheme and show some simulation results. The solution to the control problem in this section is based on a least-squares solution to a certain problem. Some modest success is achieved. In section 11, we modify the scheme in section I somewhat to arrive at the exact linearization controller. The input multiplicity issues are also addressed in this section. Section 111 considers cases of disturbance rejection. Indirectly, some robustness issues are also addressed here, and some general results for various loops are derived. I. Least-Squares Control. In nonlinear reference model control, one is interested in deriving a control law to enable a nonlinear process to follow some desired reference model. Further details are available in the paper by Boye and Brogan (1986). Thus, we consider the problem of driving the polymerization reactor from some initial condition (x,) to some
1310 Ind. Eng. Chem. Res., Vol. 28, No. 9, 1989
open-loop equilibrium point (xd). In order t o avoid triuialities, it is assumed t h a t t h e reactor is never asked to cross t h e surface (1 + xz/yp) = 0. As usual, we choose the manipulated variables as the monomer inlet concentration (xJ, the jacket temperature (x2J, and the initiator inlet concentration (xgf). Thus, the vector of manipulated variables u E R 3. In this case, we observe that the model equations are of the form x = f(x) + Gu (19) where
The vector f(x) E R 7 in eq 19 represents the remaining terms in eq 10-17 after the above rearrangement. Material balance considerations ensure that al and a3 are nonzero. Note that because of the material balance relationship that mass fractions must sum up to unity, the solvent feed concentration xgf is not independent of xtf and x~~ and we have a three input, seven state-variable plant. Also, observe that the matrices Go and hence G always have a full column rank of three for a nonzero heat-transfer coefficient, p. Hence, the matrix GTG is always nonsingular. We demand first-order rise in all loops and hence define our reference system as X r = A(X, - xd) (20) where the matrix A is chosen to be diagonal, asymptotically stable, and of the form -kI with k > 0. Defining the error system e = x - x,, we have from eq 19 and 20 (after some arithmetic in which we add and subtract the term Ax) that B = Ae 4- [f(X) - A(x - Xd) + Gu] (21) Now we need to determine the control law such that the term in the square brackets in the right-hand side of eq 21 is zero so that, at equilibrium, we will have e 0 and x xd. Thus, we reduce to the algebraic problem of finding u such that O = f(X) - A(X - xd) + G U (22) The dimensions of the various vectors defined earlier imply that we have more equations than unknowns. Hence, we reduce to a problem in least-squares estimation in which we can compute a least-squares solution to eq 22 so that
-
-
u = -(GTG)-'GT[f(x)- A(x - xd)]
(23)
Equation 23 is t h e explicit solution for the input (control), which is best in the sense of forcing the equality in eq 22 to be "almost exact". Observe that, because of the structure of G, the state variables of subsystem 2 (the MWD subsystem) do not play a part in determining the control law. In essence, we could have obtained the same control law by merely considering subsystem 1 alone and defining a corresponding four-state reference system. Henceforth, without loss of generality, we do this. Thus, in the rest of this section, we simply take G = Go by assuming G E R 4x3 and viewing the above equality as a FORTRAN replacement. In the same spirit, we shall view xd as a vector of four components, switching it to a seven-component vector when we focus explicitly on the MWD. We now examine the convergence and stability issue in greater detail by asking the following questions: (1) In spite of the fact that exact linearization is not achieved,
is there any hope of convergence to Xd, the desired equilibrium? (2) If the linear reference plant is unstable, is the nonlinear plant necessarily unstable too? (3) Does the stability of the linear reference plant guarantee stability of the nonlinear plant? We try to resolve these issues by examining the closed-loop plant. 1.1. Convergence to the Desired Equilibrium Point, xd. If the closed-loop plant will converge to xd, then it follows that xd m u s t be an equilibrium solution of the closed-loop nonlinear plant. We claim that this is in fact true for a n y matrix A. Proposition 1. For any matrix A, xd is a solution in closed loop to the plant. Proof. From eq 19 and 23, the closed-loop plant is given by 8 = [I - G(GTG)-'GT]f(x) + G(GTG)-'GT(A(x- Xd)) (24) Substitute for xd in eq 24. Recall that, since xd is an equilibrium point in the open loop, then there exists a unique u = uoIthat satisfies eq 19 a t the equilibrium. Then, the pair (xd, uO1) satisfies eq 24 at equilibrium. It would then appear from the above result that the requirement that A in eq 20 by asymptotically stable is in fact not necessary; that is, if A is unstable, then perhaps the closed-loop nonlinear plant could still converge to xd. This is somewhat counterintuitive. In this particular instance, we disprove this conjecture if A is restricted to be of the form -kI. The following lemma is useful in this regard (Barnett, 1971): Lemma 1. If P is an idempotent matrix, then rank P = trace P. Next, if we take P to be the matrix G(GTG)-'GT, then we can compute 1+a,2
0
I
P=
[;:la3
+ a12+ as2
0
-a1a3
0 1 + aI2 a3
1
+ a12 + a32
al
0 a3 a,2
I
+ a12
(25)
and application of lemma 1 to this matrix shows that it always has a rank of three. Repeated application of the lemma to the matrix P, = I - P indicates that rank P, = 1. 1.2. Stability and Convergence When k I0. Proposition 2. Let A = -kI with k = 0. Then the solution xd is not asymptotically stable for the nonlinear plant. Proof. First note that, for k = 0, every open-loop equilibrium solution is a solution to the closed-loop plant. The proof of this follows along the lines of the proof for proposition 1. Now, we examine the linearized Jacobian (J,1) of the plant about xd given by
J,] = [ I - G(GTG)-'GT]JoI + G(GTG)-'GTA (26) where Jolis the open-loop Jacobian about x+ From lemma 1, rank P, = 1. Sylvester's theorem indicates that rank (P,Jol)5 1. Consequently, for k = 0 (i.e., A = 0), the linearized system has at least three zero eigenvalues (this is cause for serious concern, even though xd could be stable). However, an examination of the closed-loop plant (eq 24) reveals that xd is not asymptotically stable. To see this, note that the second component x 2 of x is invariant for all time; that is, once fixed, it always remains a t its initial value. Proposition 3. Let A = 4 1 with k < 0. Then all equilibrium solutions of the closed-loop plant (eq 24) are unstable.
Ind. Eng. Chem. Res., Vol. 28, No. 9, 1989 1311
Proof, This follows immediately from the linearized Jacobian-eq 26. Let x, be any equilibrium solution of the closed-loop system. (We allow for the possibility of extraneous solutions introduced by the controller.) It is clear (from the second row of Jcl)that Jclalways has an eigenvalue of -k. Consequently, no equilibrium solution is stable. In fact, we claim that the closed-loop plant is globally unstable if the second component x 2 of x is not initialized from its desired value (xu).To see this, examine the closed-loop system (eq 24) and observe that x 2 is governed by 1 2 =-k(Xz-X~d) v k (27) This implies x 2 is unbounded; instability is global. The above results can then be summarized in the following statement: Theorem 1. Let A be of the form -kI in the reference plant. Then a necessary condition for x d to be an asymptotically stable equilibrium solution for the closed-loop plant (eq 24) is that k > 0. 1.3. Reactor Stabilization and Tuning. The results of the previous subsection have established the necessity for k > 0 for convergence of the nonlinear plant. Sufficient analytical conditions are hard to establish, and numercial work is of some use here. The linearized Jacobian (eq 26) provides a sufficient algebraic criterion for stabilization of the reactor about the equilibrium point x d . The following corollary follows directly from the closed-loop Jacobian (eq 26): Corollary 1. Given arbitrary asymptotically stable Jcl, let JOIbe the open-loop Jacobian about x d . Then, the closed-loop plant (eq 24) is locally stabilizable about x d if there exists a k that satisfies the equation Jcl = [I - G(GTG)-lGT] J,l- kG(GTG)-lGT (28) Thus, the above stabilizability condition depends on the existence of a k that solves eq 28. This is a difficult problem in that there exist infinite choices of the asymptotically stable matrix JcIthat might not admit a solution for k . Thus, rather than specify an arbitrary Jcl,we will try to find a k that works (if indeed such a k exists). In so doing, several alternatives are possible. These include (1) a trial-and-error approach and (2) a systematic but tedious procedure of increasing k gradually from zero and determining the eigenvalues of JCI (eq 28) as a function of k until the computed eigenvalues fall in the left half plane. However, by far, the simplest approach is the application of Geshgorin’s theorem. Let us elaborate. The positive semidefiniteness of the matrix P implies that all its diagonal elements are greater than or equal to zero. Actually, exact computations (eq 25) show this to be strictly true. Consequently, by choosing k sufficiently large, we can always make all the diagonal elements of Jcl in eq 28 strictly negative. The only thing left to achieve is diagonal dominance. If, in addition, diagonal dominance is obtained (this is easy to check since all matrices involved are real), then Jclis asymptotically stable. Note that Geshgorin’s theorem leads to a conservative design, and usually, detuning (reducing k ) will almost always be necessary to ensure that input constraints are satisfied. In summary, the theoretical basis behind the approach is clear: we apply the best input in the sense of matching the state of the nonlinear plant to the reference plant and try to force convergence by ensuring that the nonlinear system be locally stable about the desired equilibrium. The most important result here is that, even though exact dynamic linearization of the nonlinear plant is not achieved, in some neighborhood of the stable equilibrium
solution, asymptotic tracking is still obtained. Furthermore, if for any k > 0, the desired equilibrium x d is the only stable closed-loop equilibrium solution (including periodic solutions), then convergence would in fact be global. Hence, the local results are actually stronger than they appear to be, and as will be seen later, numerical simulations seem to bear out this point. The above remarks relate to some previous work in closed-loop bifurcation studies in the literature (Chen and Chang, 1984; Kwalik, 1988) in the following sense. A rigorous confirmation of the range (in the gain space) of global stabilization can be achieved by a closed-loop bifurcation analysis in which the controller gain k is employed as the distinguished parameter. If desired, such a procedure can be used to rigorously delineate regions of the gain space in which global stabilization is attained. 1.4. Simulation Results. In order to verify these results, numerical experiments were performed. The results are presented as follows. First, we illustrate the application of Geshgorin’s theorem in a given instance. Only in this case study are the computational details presented in their entirety. Next, we work in different regions of reactor parameter space in order to evaluate the convergence properties of the controller. Case Study 1. An Application of Geshgorin’s Theorem. It is desired to drive the reactor to the equilibrium point Xd
= [0.149 1.2198 1.7012 x
1.3940IT (29)
which corresponds to a conversion of 91% at a solvent volume fraction (&) of 0.48. Thus, the gel-effect interactions are extremely significant. The reactor residence time is 80 min. Also, Tf (=TJis 340 K; thus, x k = 0 in the open loop. We compute a sequence of Jclmatrices when k takes on the values 0, 2, 5 , and 600, respectively. The results are
F
-1.362 491
-3.297030 -1.199069
-0.669 4959 0 -1.620083 -0.589 1942
-619.7232 0 -1499.640 -545.3915
E
-3.100 374
1 1 1 1
-1.206 8719 X lo-’ 0 2.9204637 X lo-* 1.062 1384 X
-0.6694959 -2 -2.662 746 -1.620 083 -0.968 3917 -0.589 1942
-619.0889 0 -1500.105 -544.8333
0.242 7461 0 0.587 4100 -1.786 370
-0.6694959 -5 -1.711 320 -1.620083 -0.622 3756 -0.589 1942
1
-618.1375 0 -1500.803 -543.9960
0.588 7622 0 1.424 718 -4.481 857
-522.7275
-429.4380 0 -1639.177 -377.9299
69.212 58 0 167.4908 -539.086 8
-5.707 200
e
186.9881 68.004 14
-0.669 4959 -600 -1.620083 -0.589 1942
(30)
(31)
(32)
(33)
Note that explicit computations show that, for k = 0, Jcl has two eigenvalues very near the origin. Roundoff errors preclude these from being a t the origin exactly. In any case, at k = 600, Jclin eq 33, satisfies Geshgorin’s theorem (row dominance), although a closer inspection shows that we might as well have stopped a t k = 5 since column diagonal dominance is then already attained. Thus, stabilization about x d is possible. Numerical simulations showed convergence of the closed-loop plant to x d for k = 5, but violations of input constraints occurred. Hence, a reduction in k took place until a value of 0.3 was attained. For this value of k , no input constraints were violated. The results are shown in Figures 1 and 2. Numerous other simulations conducted from different initial conditions also converged to the desired equilibrium, suggesting that the convergence is global. The results from
1312 Ind. Eng. Chem. Res., Vol. 28, No. 9, 1989
D 0
\
0
0.0
4.0
0.0
8.0
1
11.0
0
5.0
0.0
10.0
0.0
0
rh
I
0 1
SJ 0.0
4.0
8.0
12.0
16.0
10.0
,
0.0
,
,
4.0
1.0
Rrsldrncr time unltr Figure 1. Least-squares control: case study 1, k = 0.3.
.
.
, 12.0
, 1 1.0
11.0
Rrsldrncr tlmr unltr
some other initial condition with k = 0.2 are displayed in Figure 3. Case Study 2. Output Multiplicity. Now, we work in a region of unstable equilibria. The reactor residence time is fixed a t 53.33 min, and reactor parameters are chosen such that three steady states are possible. The solvent volume fraction, q5* is fixed at 64%. The monomer feed concentration, Mf, is 4.50 mol/L (xlf = 1.28571). Under these conditions, we enumerate all possible equilibria of the open-loop system in physical parameter space. These are xel = [0.363 0.9777 8.7360 X
E
1.8653JT
1
n I
xeP = [0.522 0.8092 1.1373
1.8653IT
X
xe3= [1.208 0.0823 1.4242
X
(34)
1.8653IT
As usual, the intermediate equilibrium point (xeJ is open-loop unstable. The eigenvalues of the open-loop Jacobian a t this intermediate solution are -1, -1.153 613, 4.405091, and 2.68156. Starting from the same arbitrarily chosen initial point (Figure 4),the controller (with k = 0.5) takes us to each of the three open-loop equilibria without the need for returning and with good dynamics. The inputs corresponding to Figure 4 are shown in Figure 5, and it should be observed that no constraints on all inputs (in particular, xgf < 0 ) are violated. In effect, we are able to stabilize the plant about open-loop-unstable equilibria. Simulations conducted with numerous different initial conditions with this and several other values of k produced similar results, indicating that, for each of these k values, convergence could indeed by global. As is seen in all these figures, there is a loss in dynamic performance in the third loop but the convergence of the system is not affected.
;I, 0.0
p
,
J
,
,
4.0
8.0
12.0
18.0
20.0
4.0
8.0
12.0
11.0
20.0
.
,
,
,
I 0-
0.0
Rorldenca tlmr unltr Figure 2. Input corresponding to Figure 1, k = 0.3.
Ind. Eng. Chem. Res., Vol. 28, No. 9, 1989 1313
.O
1.0
10.0
11.0
25.0
10.0
10.0
31.0
I\
I
40.0
,O
1.0
10.0
40.0
0.0
LO
10.0
11.0
20.0
26.0
10.0
S5.0
4
18.e
20.0
25.0
10.0
10.0
4
9 .
3 3: 29 e-:
/
0 -
Q? O
L
.
gz-
c V
.
:?
= :?
z , , , , , . , , , . , , , , , 1.0
0.0
10.0
16.0
20.0
10.0
21.0
J6.0
Roaldrncr tlmr unlta Figure 3. Least-squares control: case study 1, k = 0.2.
Realdrnce timr units
I
O 1
o l
c-
2 .-... .... ................................. ........ 3 .......................
0--
I.
0
-
0-
:I
I
0
,
.
0.0
'i
I
,
,
.
1.0
4.0
,
1
,
,
I
11.0
,
~,
.
,
18.0
,
n
0.0
4.0
1.0
11.0
.
11.0
,
1 10.0
0.0
i. o
8.0
ii.0
ti.0
20.0
0.0
4.0
B.0
12.0
18.0
I
1 10.0
1.0
Realdrncr tlmr unltr Realdencr tlmr unltr Figure 4. Least-squares control: equilibrium point corresponding to curve 2 (xBz)is open-loop unstable.
Case Study 3. Inputs and Tuning Parameter. Here, we show the effect of the tuning parameter, It, on the input.
We attempt to drive the reactor to the unstable equilibrium point, xe2,of case study 2 by employing a sequence
Ind. Eng. Chem. Res., Vol. 28, No. 9, 1989
1314
of the catastrophe set is implicit in the global stabilization results. We then show some simulation results. Finally, we demonstrate the existence of the catastrophe set, trace its origin, and propose methods to deal with this problem as it arises. 11.1. Exact Linearization. Here we direct our attention toward controlling a subset of the state vector by defining a reference system for the first three state variables (monomer, temperature, and initiator levels) of subsystem 1 as
xr1 = Al(xr1 - xdl)
' I ( ,
I
,
,
,
,
,
,
, I
c)
5.0
0.0
10.0
15.0
20.0
25.0
0
where A, E R3x3and both xrl and xdl E R3. The vector xdl is the portion of the desired equilibrium point consisting of the desired levels of monomer, temperature, and initiator, respectively. Again, A, is chosen to be of the form -kI with k > 0. Thus, in this scheme, we neglect the dynamics of the solvent loop. But we hope that, at steady state, there will be no offset in all state variables. We will present conditions that guarantee this shortly. Proceeding as in section I, the dynamics of the error system el(=xl - xrl,E R 3 ) are then given by el = Alel + [fl(x)- Al(x, - xdl) + G,u]
3:
.____
.'..__'.
of k values. In so doing, we implement k values of 0.5,0.6, 0.7, 0.8, 0.9, and 1.0 in the control law. As is shown in Figure 6, the closed-loop plant converges to the desired equilibrium. Note that, even though only the inputs and x4 are shown in this figure, this information (together with the facts that the temperature loop is linearized exactly in closed loop and that, for k = 0.5, convergence of the closed-loop plant has been established (case study 1))is sufficient to conclude that the desired operating point is indeed attained at equilibrium. Furthermore, as can be predicted from the structure of the control law, the inputs are more severe as k is increased. Indeed, for values of k greater than 0.8, initiator feed constraints are violated. 11. Exact Linearization of a Subset of the State. The controller in section I yielded excellent closed-loop response, and it is based on controlling a plant with more outputs than inputs. Let us now see how this approach could be modified in order to give, perhaps, even stronger analytical results. In so doing, we will focus on a subset of the state, specifically the first three state variables. We shall show that, under certain conditions, global stabilization of the entire plant can be achieved. In particular, we will establish formal contact again with bifurcation theory from the viewpoint of singular surfaces in the control input space. This contact arises from the existence of what Koppel (1982) refers to as the catastrophe set. Thus, the section is organized as follows: first, we develop the control scheme. Next, we prove the stability of the closed-loop plant and argue a condition for global stability. The notion
(35)
(36)
where x1 fR represents the vector defining the first three state variables (monomer, temperature, and initiator levels) of the reactor. The vector fl(x) consists of the first three rows of f(x) in eq 19. We emphasize that this vector depends on all the state variables of subsystem 1-the dependency on x4 comes in through the gel-effect correlation, as is evident in eq 9. The nonsingular matrix G1( E R3x3) consists of the first three rows of matrix Go in eq 19. The important point here is that eq 36 can be solved exactly for the control u (defined for all time) required to make the error system el asymptotically stable. By equating the term in the square bracket in eq 36 to zero, we have U =
-Gl-'[fl(x) - Al(x1 - Xdl)]
(37)
Furthermore, applying the input u in eq 37 to the plant shows that the controlled subset of the state is now linear (in the closed loop) with coefficient matrix A,! It remains to establish (1) the stability of the entire subsystem 1 (including x4) and (2) a condition for global asymptotic stability. 11.1.1. Proof of the Stability of Subsystem 1. The control law (eq 37) is well defined for all time and guarantees that the first three state variables of subsystem 1 are bounded (first-order response). In addition, these state variables go to some desired values. Thus, note from the model eq 10-12 for subsystem 1 that, in order for the system to be stable, it suffices to show that xlf is not unbounded in closed loop (since x4 is BIB0 stable). We claim that xlf is not unbounded. If this were not so, then the term Win eq 10 (and which appears in the control law, eq 37) would be unbounded. This will happen if and only if the gel-effect correlation is such that, for some 7 E [0,m], limt-7 g, = 0, which is impossible, both from the physics of the problem ( Vf # 0) and the exponential correlation for g, (eq 9). Hence, xlf is bounded. (For the same reason, x~~ is bounded.) Next, we argue trivially that xgf is bounded. This is evident from eq 12 and 37, which indicate that xgf is always a sum of bounded quantities.
Ind. Eng. Chem. Res., Vol. 28, No. 9, 1989 1315
.
.
0
0.0
4.0
1.0
12.0
18.0
0 1
I
20.0
0.0
4.0
0.0
4.0
I
8.0
12.0
18.0
20.0
8.0
11.0
18.0
20.0
Ll 0
-E
. I
Rrsidrncr tlmr units
Rrsidrncr tlmr units
Figure 6. Least-squares control: curves A-F correspond to values of k ranging from 0.50 to 1 in steps of 0.10, respectively.
Now because all mass fractions sum to unity, x4f is some linear combination of bounded xlf, xgf,and the constant term 1. Specifically, xlf is computed from ~ 4 = f
[(I - Vi, - Vmf)~/Mw,I/Mfo
(38)
where V, and V , are linear in x~~and x l f , respectively, and involve terms that include MWi and MW,. Hence, x~~ is bounded; the system is stable. 11.1.2. A Sufficient Condition for Global Asymptotic Stability. In this discussion, we establish conditions for global asymptotic stability for subsystem 1and hence for subsystem 2 also. From the foregoing material, we have established stability. Now we argue a sufficient condition that reveals when x4 will go to its desired value. Note that, from asymptotic stability of the monomer, temperature, and initiator levels, at equilibrium, x3f attains identical values both in the closed loop and the open loop. Suppose now that, corresponding to the desired equilibrium point, exactly one value of xlf satisfies eq 10; that is, only the open-loop solution (xlfo)satisfies this equation. Then, in the closed loop, xlf xlfo. Consequently, the value of xlf in the closed loop must approach its open-loop value (xqfo),and then also, x4 xlfO,and the system would come to equilibrium a t the desired point, irrespective of the initial conditions in the system. Finally, it is trivial to argue that, even if for that input, steady-state (output) multiplicity exists (that is, in the open loop, several equilibrium points, each of which belongs to the set El defined below, correspond to that input), the controller goes to the appropriate one. The above result is important enough to be summarized as follows (noting that at equilibrium, from the dynamics, xlf uniquely determines xzc): Let El denote that set of equilibrium points for which input multiplicity in xlf does not exist in the linearized subset of state variables, that is, in xdl. Then, the control
-
law in eq 37 generates a global asymptotically stable closed-loop system for the reactor composed of subsystems 1 and 2; that is, irrespective of initial conditions, the reactor approaches the desired set point as t w. We can at present draw no conclusions about what happens if the desired equilibrium point belongs to ElC (E/El;E being the global set of equilibrium points for the plant). However, the above result begins to address directly the importance of classifying the set of equilibrium points existing for the system in global space. We return to this issue later. It is significant to point out that in polymerization, a t relatively low conversions and/or high solvent volume fractions, for the most part, g, 1. Consequently, the effect of x4 on eq 10-12 is nonexistent. Thus, at low conversions and/or high solvent levels, input multiplicities will neuer occur and asymptotic stability is global. To see this more clearly, we set g, to unity in the model eq 10-13 to observe that, once we specify xl, x 2 , and x3, then the input vector u consisting of x l f , xZc,and xSfis uniquely determined at equilibrium. 11.2. Simulation Results. We now present some simulations to evaluate the controller. Additionally, we make some comparison with the least-squares controller of section I. In order to facilitate ease of comparison, we work with the equilibrium points of section I. Case Study 4. Output Multiplicity. The simulation conditions are identical with those of case study 2 except that, here, k = 0.6. The results are shown in Figures 7 and 8. These plots are to be compared with Figures 4 and 5 , respectively. Note the linear dynamics of the initiator loop in Figure 7 . Furthermore, the differences in the transient response of the solvent loop (x4) are quite significant (that is, when compared with Figure 4). In spite of these differences, it is readily observed that, for the given input, the desired equilibrium is attained.
-
1316
Ind. Eng. Chem. Res., Vol. 28,No. 9,1989 .I
1 -,
1
--
c-
"
-1
0
L '
,.
0;-
E E :.:
%a: c
O -0 n0-. 0
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.+.-1-.
..........................................
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-
0
t
Ej
,
,
,
,
,
,
,
,
n
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,
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... ...
: I , , 0.0
,
4.0
Residence tlme u n l t r
, 8.0
,
, 12.0
,
, 18.0
,
I 20.0
Resfdance tlme unltr
Figure 7. Exact linearization: equilibrium point xe2(curve 2) is open-loop unstable, k = 0.6.
Case Study 5. Inputs and Tuning Parameter. The objective and conditions are as in case study 3. The associated inputs are shown in Figure 9. It is interesting to compare this figure and Figure 6. The corresponding transient responses of the inputs in both of these figures are vastly different, even though they are identical a t equilibrium. 11.3. Exact Linearization, Input Multiplicity, and Catastrophe Theory. In the previous subsection, we presented some results on controller convergence. We examine this question in more detail here and consider various approaches to force convergence in the event of input multiplicity in the exact linearization controller. As was discussed in the Introduction section, Koppel (1982) appears to have first examined the input multiplicity question and its relationship to catastrophe theory. Roughly speaking, input multiplicity is said to occur if given an equilibrium point, several sets of input satisfy the corresponding steady-state equations. The set of equilibrium points for which input multiplicity can occur is referred to as the catastrophe set. For a square system in which integral action is incorporated on all outputs, the existence of input multiplicity often goes undetected because, typically, only the system outputs are being monitored. Technically, our contact with this problem in this application differs somewhat from the precise definition in that, here, the input multiplicity problem is present in the subset of the subsystem 1 state, namely, xl, x 2 , and x3. Thus, if we are given equilibrium values of xl, x,, x3, and x4, we can uniquely determine the corresponding input vector u, as is evident from the dynamic equations. However, if instead only equilibrium values of the first three state variables (xl,x2, and x3) are given, then, in general, the input which satisfies eq 10-12 at steady state
(after substituting for x4, which then appears nonlinearly in the above equations) is not unique. We emphasize that the gel-effect induces this behavior. To be precise, the free volume, V,, appearing nonlinearly (exponentially) in the gel-effect correlation, g,, is a linear function of x l , xz, x3, and x4. However, a t equilibrium, x4 = xaf Furthermore, xlf is a linear combination of xlf and x3p Hence, we have a situation in which several sets of u satisfy these equations a t equilibrium. However, for this reactor, unlike in typical cases, we can detect the presence of this phenomenon by checking xq. Let us elaborate. The controller always guarantees asymptotic stability of x l , x2, and x3. Hence, no information is gained by checking these variables. However, if x4 does not go to the desired equilibrium value, then it is clear that this can arise only if the steady-state value of the inputs in the closed loop differs from the open-loop values. Thus, for the desired equilibrium point, input multiplicity exists in the subset of subsystem 1 consisting of xl, x 2 , and x3. In general, it is not a trivial matter to compute the catastrophe set, although Balakotaiah and Luss (1985) have used singularity theory techniques in a given instance. In this application, the catastrophe set will consist of all equilibrium points (xd)for which, given the values of the first three components (xl,x2, and x,) of xd, two or more sets of inputs will satisfy the dynamical eq 10-12 a t equilibrium. Thus, in essence, this set is a volume. We note that, a priori, the implicit function theorem provides us with a tool for recognizing potential difficulties via executing the following program: (a) Substitute for x4 a t equilibrium from eq 13 and 38 into the eq 10-12. (b) Assuming differentiability, compute J = det (dfi/du), with f, ( E R 3 )being the result from (a). (c) There exists a u such that J = 0 could indicate po-
Ind. Eng. Chem. Res., Vol. 28, No. 9, 1989 1317 E?
1
0 -
I.!
.u
I
....................................
-aE - 1
0.0
5.0
10.0
I
20.0
15.0
25.0
0
Xd,
= [0.149
1.2198 1.7012 x io-4 1.6956IT
uC1= [1.44883 0.319965 8.571 X 10-3]T
-1
, 0.0
5.0
,
, 10.0
,
, 15.0
,
, 20.0
1
,
25.0
n
0
- 1
I
W
0.0
5.0
10.0
11.0
20.0
25.0
Residence t i m e units
Figure 8. Exact linearization: input for Figure 7 (labels: as in Figure 7 ) .
tential problems in the form of the catastrophe locus. In this work, we do not attempt to generate the catastrophe envelope. Instead, we demonstrate its existence as follows. In so doing, we revisit the equilibrium point in case study 1 to which the least-squares controller was successfully applied (Figure 3). Case Study 6. The Catastrophe Set. Consider the reactor in case study 1 of section I in which a conversion of 91 70was desired. For ease of readability, the conditions corresponding to that equilibrium point are reproduced here. Thus, we have the equilibrium point ( x d ) and its corresponding input (u,J given by xd
version of 89.7% rather than the desired 91%. Thus, it would be a fallacy to have concluded that the conversion requirements are being met based on observing x1 only. This result can only be confirmed by a dynamic simulation. In so doing, we take the vector u,1 and simulate the model equations with this input (in the open loop). The result of this simulation is shown in Figure 11. Observe from this figure that, at steady state, we converge to an equilibrium point x d c given by
= [0.149 1.2198 1.7012 x
1.3940IT
u,, = [1.7142 0 8.571 X 10-3]T
(39)
As is evident in case study 1,we were successful in driving the reactor to this equilibrium point with the least-squares controller. Now we seek to achieve the same objective with the exact-linearization algorithm. Our reactor initial conditions remain invariant (as in Figure 3). Thus, we simulate the exact-linearization controller (k = 0.2) on the system, and the results are shown in Figure 10. Note from these plots that, as expected, xl, x 2 , and x 3 go to their desired values. However, a check on x 4 shows an offset. Thus, evidently, the controller has converged in the closed loop to an input (u,,)different from u,, in eq 39. This proves that the desired equilibrium point belongs to the catastrophe set. Note that, by definition, the equilibrium point corresponding to u,1 also belongs to this set. It turns out that in the closed loop, though x1 goes to its desired value, because of the input multiplicity, we obtain a con-
(40)
We now compare x d c in eq 40 to x d from eq 39 to see that we have two equilibrium points with only the fourth components differing. Evidently, in the closed loop, the system converges to x d c as seen in Figure 10. The above example clearly demonstrates the i n p u t multiplicity in the subset of subsystem 1 state variables. It typifies the difficult problems that can arise in practice. We try to fix this problem. Let us elaborate. Until now, we have chosen the tuning matrix (A,) to be of the form +I. We now utilize the additional flexibility we have in choosing Al in forcing convergence, if possible, to the desired equilibrium point. Thus, we choose Al = -[diag (kl,k2,k3)], ki > 0 V i The results are shown in Figures 12 and 13. In Figure 12, k,, k z , and h3 are 1,2.627, and 1,respectively. In Figure 13, with the exception of k2 (=2.65), all other parameters are the same as in Figure 12. We point out that, in Figure 13, only the plot for x4 together with the corresponding inputs is shown since all other state variables are essentially identical with those in Figure 12. Observe from these figures that, evidently, the desired equilibrium is unstable in the closed loop. However, as our results show, it may be possible to find a set of gains to “drive” the system there. The set of gains that will do this is not necessarily unique. Note that the system “stays” at the desired point for some time on its way to a stable equilibrium. Some remarks on these plots are in order. These results appear to indicate that we can sometimes have more than two equilibrium points (with identical first three components) belonging to the catastrophe set. Apparently, in this event as is seen in Figure 13, the system appears to have fallen into the domain of attraction of the third equilibrium point. It tries to seek for this other stable equilibrium point, which unfortunately does not belong to physical parameter space ( x >~ 0). ~ Consequently, numerical problems arose during computations, and the simulation was terminated. Whether there exists a set of stabilizing gains for any given equilibrium point is not easy to determine. Even if there exists some gains, are there relatively easy systematic methods of finding them? These and other questions are still not completely resolved. In principle, one could write the closed-loop characteristic equation about the desired point and compute the required set of gains, if they exist. Furthermore, issues relating to the domain of attraction of the desired point will then have to be addressed. 111. Disturbance Rejection/Robustness. In this section, we address some issues relating to disturbance rejection. Attention is focused only on the exact-linearization controller. In so doing, we first assess the disturbance rejection properties of the controller in its most basic form as derived in section 11. For obvious reasons, this controller is referred to as the proportional reference controller. Next, we will consider ways of improving the
1318 Ind. Eng. Chem. Res., Vol. 28, No. 9, 1989 I
- 1
I
0
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8.0
b. 0
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0.0
10.0
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18.0
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Rrsldrncr t l m r u n l t r
Rrsldrncr tlmr unltr
Figure 9. Exact linearization: inputs as a function of k (labels and k values: as in Figure 6 ) .
!I
50 n;
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0.0
10.0
10.0
10.0 40.0
50.0
10.0
70.0
DO.0 80.0 1
Resldence tlme unlts
Figure 10. Exact linearization: input multiplicity, k = 0.2.
capabilities of the controller by incorporating integral action.
111.1. Proportional Reference Control. We assume disturbances can occur in the heat-transfer coefficient ( p )
Ind. Eng. Chem. Res., Vol. 28, No. 9, 1989 1319
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$I,, 0.0
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Rrsldrncr tlme unlts
Figure 11. Exact linearization: verification of input multiplicity.
E
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,
8.0
0.0 10.0
20.0
30.0 40.0
50.0 00.0 70.0 10.0 00.0 1 1.0
Rrsldrncr tlme unlts
Figure 12. Exact linearization: input multiplicity incorporating different gains in A,.
and/or the initiator efficiency 0.A justification for this can be found in Tanner et al. (1987). Intuitively, we expect that the ability of the system to reject persistent disturb-
ances will be limited, more so with the MWD, since these state variables are not being fed back. This leads directly to the following observations:
1320 Ind. Eng. Chem. Res., Vol. 28, No. 9, 1989 E
9 0 c V
U 0 W
m
I 2i.O
10.0
0
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4
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Residence tlme units
Resldence tlme units
Figure 13. Exact linearization: input multiplicity incorporating different gains in AI.
r--.-.
I
.-.-.-.-.-
*
1.
fl
,
0.0
, 4.0
,
,
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Figure 14. Disturbance rejection: -10% change in /3, k = 2, yz = 1; curves 1, 2, and 3 represent open loop, closed loop, and set point, respectively.
Ind. Eng. Chem. Res., Vol. 28, No. 9, 1989 1321 If the disturbance is measured, then, clearly, the only possible offset will be in the solvent loop (and hence the MWD) since the disturbance can be fed foward in time for necessary compensation. If the disturbance is unmeasured, then closed-loop stability is in general hard to guarantee. We, however, have the following result: Let an unmeasured disturbance enter loop i directly and other loops indirectly (through some other state variables). Suppose the closed-loop system is stable. Then all linearized loops (except possibly for loop i ) have zero offset as t m. Proof of this assertion follows trivially from the fact that, in all linearized loops where the disturbance does not enter directly, exact dynamic linearization still proceeds. Consequently, as in section 11, such loops exhibit zero offset at steady state, provided the closed-loop system is still stable. Thus, the above result holds for any k > 0, which renders the system stable after the disturbance enters the process. 111.2. Proportional Integral Reference Control. We now incorporate integral action into the controller in order to have excellent disturbance rejection capabilities in the controlled subset of the state. Then, the following result follows from the basic property of integral action: Let an unmeasured disturbance enter loop i directly and other loops indirectly (through some other state variables). Suppose the closed-loop system is stable and that loop i alone is fed back under integral action. Then all linearized loops exhibit zero offset as t m. We again emphasize that the above result holds if the resulting system is stable after being disturbed. The addition of integral action to the controller is achieved by state augmentation. I t is proven in the literature (Porter and Power, 1970) that for a controllable linear system to remain controllable under integral state feedback, it is necessary that
-
/
u 0
t,t
c
I.!
g?
a-
-a
-
C 40 r
a
=g P-
0.0 b.0
4.0
8.0
12.0
16.0
20.0
-
n, Im
(41)
where nl is the number of state variables being fed back under integral action and m is the number of manipulated variables. In all simulations, we adhere strictly to the above condition. The integral action is adjoined to the proportional reference controller input u in the following manner: U
= -Gl-'[fl(x) - Al(x - xd)] - r$(X
0
- xd)
dt
(42)
where ri can be viewed as a diagonal matrix of nonnegative integral gains. In this form, if the disturbance does not enter a linearized loop directly, the offset-free property in that loop is still retained. For a particular choice of A, and ri,it is, however, not clear how large the disturbance should be before the system goes unstable. In order to illustrate the power behind this extension, we shall be somewhat elaborate in our considerations of various disturbances. As will be later seen, even though the solvent loop x4 is not being controlled directly, we can obtain perfect compensation of subsystem 1 in the face of persistent disturbances in the heat-transfer coefficient, b. This is impossible when the disturbance arises from changes in the initiator efficiency, f . In the following case studies, it is assumed that the reactor is to be operated at the following equilibrium point: Xd
= [0.236 0.6621 2.3572 x
ioT3
2.1082IT
The objective is to maintain the reactor at this equilibrium
a: E
= 0.0
4.0
8.0
12.0
18.0
20.0
Residence time units
Figure 15. Disturbance rejection: input for Figure 14; observe return of xlf and x3f to their initial values.
point in the presence of persistent disturbances. In all plots, the open-loop uncontrolled behavior of the reactor is also shown. Case Study 7. Perfect Compensation for Changes in 8. In Figure 14, we introduce a 10% decrease in the heat-transfer coefficient (R). This disturbance enters the temperature loop directly and no other loop and is assumed to be unmeasured. Hence, the controls are computed based on the old value of b. The controller parameters are k = 2 and an integral gain (yz)of 1 on the temperature loop (xz). According to our result, we expect the monomer ( x , ) and initiator (x,) loops to eventually return to the desired operating point after some dynamics even without integral action. Furthermore, because of the integral action in the temperature loop, a t equilibrium, no offset exists in this loop. Consequently, we have achieved perfect disturbance rejection in the controlled subset. Also, note that the monomer and initiator loops are virtually unaffected by this disturbance. This follows from the exact linearization property of the proportional reference controller. Now, we arrive a t the most important part of this case study in the following sense. Note from Figure 14 that, ultimately, there is no offset in the solvent loop (x4). In effect, the controller has perfectly rejected the disturbance in the entire subsystem 1 (and hence, subsystem 2)!. This demands some explanation-the ultimate reason being due to exact linearization, input multiplicity, and decoupling. Let us elaborate. It is clear that, before the disturbance enters the temperature loop, the system is a t equilibrium, with the cor-
1322 Ind. Eng. Chem. Res., Vol. 28, No. 9, 1989
g
,
0.0
,
,
,
,
1.0
4.0
,
,
,
,
1a.o
11.0
1 10.0
;I, 0.0
, 4.0
,
,
.
,
,
,
,
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n
0.0
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, 1.0
,
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,
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1 10.0
Figure 16. Disturbance rejection: -10% change in respectively).
/, k
4.0
= 2, y1 = y2 = 1 (labels: 1, 2, and 3 denote open loop, closed loop, and set point,
responding input given in eq 43. Now, the offset free property of the temperature loop after the disturbance in /3 is introduced is guaranteed by the integral action. Consequently, after the disturbance propagates, as far as the monomer and initiator loops are concerned, nothing has changed in the temperature loop when equilibrium is again attained. In effect, at equilibrium both of the manipulated variables (xlf and xSf)return to the values they took on before the disturbance entered the process as in eq 43. Hence, due to material balance considerations, x4f is unaltered at equilibrium, in which case there will ultimately be no offset in x4. To see this more clearly, the input corresponding to Figure 14 is presented in Figure 15. Observe that, after some transient dynamics, the monomer and initiator feed concentrations (xlf and x3f) return to their initial values. Of course, the new jacket temperature (T,or x z c ) is correspondingly lower. The above argument as to why no offset exists in the solvent loop could break down if for that equilibrium point input multiplicity exists in xlf. The following general result can then be stated (assuming integral action is incorporated in the temperature loop only): Let E , be the set of equilibrium points for which input multiplicity in xlf does not exist in the linearized subset of the state, xdl. Suppose an unmeasured disturbance in p enters the plant, and assume the closed-loop system is stable. Then, perfect compensation of the entire reactor is attained; that is, no offsets exist in the reactor subsystems 1 and 2 as t a. Case Study 8. Disturbance Rejection for Changes in Initiator Efficiency (f). In Figure 16, we introduce a disturbance of 10% decrease in the initiator efficiency ( f ) . This disturbance enters only the monomer ( x , ) and temperature (xz) loops directly. Now, integral action is
-
0.0
Rr~ldrncrtlmr unltr
Rr~ldrncrtlmr unltr
Table I. Data" ki lit. values ref Kinetic Rate Constants kdb 1.69 X 1014exp(-30000/RT) k; 4.925 X lo5 exp(-4353/RT) kbC 9.80 X lo7 exp(-701/RT) k,' 4.92 exp(-4353/RT) Brandrup and Immergut (1975) kf,' 0.091 k a l k , 8.23 Reaction Medium Parameters f = 0.5 Schmidt and Ray, 1981 h = 135.6 cal/(m2.s.K) A, = 2.8 m2 p = 1038 g/L Rodriguez, 1982 Jaisinghani and Ray, 1977 cp = 0.4 cal/(gK) Jaisinghani and Ray, 1977 -AH = 13.8 kcal/mol "From Table 1 of Schmidt and Ray (198l), unless otherwise noted. s-l. 'In L/(mol.s).
incorporated in both of these loops. The proportional and integral gains are set to values of 2 and 1, respectively. Note that there are two loops with integral action (rl= 1, y2 = 1). Observe that the disturbance is effectively rejected in the controlled subset of the state. The corresponding input is depicted in Figure 17. Notice the offset in the solvent loop in Figure 16. This ultimately leads to offsets in the MWD subsystem 2. Following the reasoning in case study 7, it is not too difficult to argue that this will always be the case. Observe that, in all these case studies, the effect of unmeasured persistent disturbances is equivalent to the robustness problem in which we design the controller based on wrong values of some parameters. It is difficult to ascertain the bounds on parameter uncertainty required for robust stability. However, it is clear that only the loops where the parameter uncertainty enters directly need be
Ind. Eng. Chem. Res., Vol. 28, No. 9, 1989 1323
, 0.0
n
4.0
,
,
,
,
,
,
I
8.0
12.0
16.0
20.0
8.0
12.0
16.0
20.0
?-
n o m 0 E
E n 0::
3 : 0
0,
0.0
4.0
Residence time units
Figure 17. Disturbance rejection: input for Figure 16; observe return of xaf to its initial value.
considered for robust design. For the moment, general theoretical results are not available with respect to stability limits for a particular set of gains. Nonetheless, it is clear that adding integral action improves this property somewhat.
Conclusions In this investigation, we have demonstrated rigorously the applicability of certain control schemes for solution polymerization reactors. In particular, we emphasize the importance of classifying equilibrium points via the catastrophe set formulation and have illustrated potential difficulties that might arise in practice. The detection of the catastrophe set is observed even without integral action in the controller, and perfect compensation is achieved for certain disturbances. We emphasize that, for the leastsquares controller, input multiplicity cannot occur for the simple reason that the null space of the matrix G (in section I) is of dimension zero. We note however that in both control schemes, because the MWD is not being fed back, modeling errors in subsystem 2 can be corrected only through reactor subsystem 1. Also, the control laws require the availability of the state variables of reactor subsystem 1. In practice, these are not always available and will have to be estimated. Consequently, issues in state estimation and the interconnection of the state feedback and the state estimator will need to be addressed. Nomenclature A, = heat-transfer area of reactor
cp = specific heat capacity of reactor contents D = polydispersity Daj = Damkohler number for species j Ej = activation energy for reaction j f = initiator efficiency g, = gel-effect factor h = heat-transfer coefficient -AH = heat of reaction I = initiator concentration in reactor If = initiator feed concentration kd = dissociation rate constant for initiator kf = rate constant for chain transfer to monomer kfs = rate constant for chain transfer to solvent k , = propagation rate constant k , = overall termination rate constant kt, = rate constant for combination termination k~ = rate constant for disproportionation k{ = frequency factor for reactor i (i = d, f, p, t) ktc,,' = zero conversion frequency factor for combination termination reaction k', = zero conversion frequency factor for disproportionation reaction k , = overall termination rate constant at zero conversion k,' = frequency factor for overall termination rate constant at zero conversion M = monomer concentration in reactor Mf = monomer feed concentration Mfo= monomer feed concentration for scaling purposes only MWi = molecular weight for initiator MW, = molecular weight for monomer MW, = molecular weight for solvent P = concentration of live polymer q = volumetric flow rate R = universal gas constant S = solvent concentration t = time T , T , = reactor temperature, jacket temperature Tf = feed temperature u = vector of manipulated variables V = volume of reactor Vf = free volume Vir = mass fraction of initiator in feed Vmf = mass fraction of monomer in feed W = dimensionless live polymer concentration xi = dimensionless reactor state variable xlf = manipulated variable (dimensionless inlet monomer concentration) xzc = manipulated variable (dimensionlessjacket temperature) xgf = manipulated variable (dimensionless inlet initiator concentration) Greek Letters
0 = dimensionless heat-transfer coefficient number-average chain length = dimensionless time A,, A,, Az = zeroth, first, and second MWD moments p = density of reacting medium p,, pi, pm = densities of solvent, initiator, and monomer, respectively
p = T
Subscripts d, f = dissociation, transfer to monomer p = propagation tc, td = termination: combination, disproportionation 1, 2, 3, 4 = monomer, temperature, initiator, solvent 5 , 6, 7 = moments: zeroth, first, second Registry No. Methyl methacrylate, 80-62-6.
Literature Cited Adebekun, A. K.; Kwalik, K. M.; Schork, F. J. Steady-State Multiplicity During Solution Polymerization of Methyl Methacrylate in a CSTR. Chem. Eng. Sci. 1988, in press.
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Received for review October 4, 1988 Revised manuscript received May 1, 1989 Accepted May 25, 1989
A Flowing Film Model for Continuous Nylon 6,6 Polymerization D. D. Steppan,' M. F. Doherty,*,l and M. F. Malone: Department of Polymer Science and Engineering and the Department of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 01003
A flowing film model useful for the design of continuous film polymerizers has been developed. The model is characterized by two dimensionless groups: the Damkohler number and the Thiele modulus. In contrast t o a stationary film model, the addition of a flow field can move the location of highest molecular weight from the gas-film interface to the wall-film interface in thin films. An idealized second-order kinetic model has significant errors compared to the nylon 6,6 kinetic model for the prediction of average outlet molecular weight, although it is adequate for the prediction of the average outlet water concentration. However, comparisons of plug, linear, and parabolic flow fields show that the linear and parabolic flows predict similar outlet water concentrations and molecular weights (*5%), whereas the plug flow field predicts significantly different values (=t10-20% 1. Process equipment which serves the dual purposes of condensate removal (separator) and molecular weight generation (reactor) is common in continuous polycondensation. An example of such a device is the flasher (US Patent 2,361,717, 1944; US Patent 3,900,450, 1975; US Patent 3,960,820, 1976), which is used in the continuous production of nylon 6,6. This unit is essentially a pipe in which a prepolymer flows along the wall in contact with a large quantity of steam; this complicated two-phase flow may result in instabilities that limit the range of steadystate operation (US Patent 3,900,450, 1975). By introducing a flow field into a stationary film model (Steppan et al., 1989), we will study the effect of distributed resiDepartment of Polymer Science a n d Engineering. * D e p a r t m e n t of Chemical Engineering.
0888-588518912628-1324$01.50/0
dence times on the performance of an idealized continuous nylon 6,6 film device and determine the key varibles that affect its design and performance. This idealized model will give us intuition and insight for continuous thin-film reactor-separators but is not intended to be a process model for any particular device. Previous studies have examined the problem of simultaneous reaction and diffusion for reversible polycondensations in simple geometries (Secor, 1969; Hoftyzer and van Krevelen, 1971; Gupta et al., 1982). In all of these studies, the effect of mass transfer was analyzed for systems with composition-independent reaction rate and equilibrium constants. However, it is well-known that many polyamidations and polyesterifications appear to change reaction order with conversion (e.g., Flory (1953) and Hiemenz (1984)). This apparent change has been 0 1989 American Chemical Society
