Ind. Eng. Chem. Process Des. Dev. 1988, 25, 821-828
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Design of Adaptive/Inferential Control System and Its Application to Polymerization Reactors Takelchlro Takamatsu, Suteakl Shloya;
and Yoshikl Okada
Department of Chemical Engineering, Kyoto University, Kyoto, 606 Japan
I n this paper, the Adaptive/Inferential Control System (AI-CS) is proposed. I t can be used successfully for a system having deterministic disturbances and is valid for a non-minimum-phase system. This controller utilizes a secondary output for estimating the unknown deterministic disturbances. As examples of practical application, AI-CS is used to stabilize the operation of two types of polymerization reactors, that is, styrene solution polymerization and isobutylene solution polymerization. First, for the CSTR for isobutylene polymerization, the proposed control scheme returns the average molecular weight (M,) to Its set point after a disturbance and does so better than the classical PI-feedback controller and nonmodlfied Model Reference Adaptive Control System (MRACS) proposed by Landau. Second, for the CSTR for styrene polymerization, AI-CS is also utilized for controlling the M ., I n this problem, the equilibrium point to be kept constant is unstable because of the gel effect. The classical PI-feedback controller cannot control the M , at this unstable point within the practical range of the manipulating variable. However, it is shown that AI-CS can control the M, at this point sufficiently. Thus, AI-CS works successfully in controlllng the polymerization processes.
Control of a chemical process is often difficult because very little information about the process dynamics is available. It is frequently difficult to obtain an accurate nonlinear model of the process. Also, process parameters may vary due to changes in operating points, raw materials, or the environment. A PID controller with fixed gain is inadequate to control such a nonlinear chemical process. Thus, a control technique with an identification or an estimation scheme for the plant dynamics is required for controlling the chemical process. From this point of view, an adaptive controller is very appropriate for such processes. There are two schemes for parameter adaptive control: model reference adaptive controller and self-tuning regulator (Astrom, 1981). The starting point is an ordinary feedback controller with adjustable parameters. The key problem is to find a convenient way of changing the controller parameters in response to changes in process and disturbance dynamics. The schemes differ only in the way that the parameters of the controller are adjusted. In the Model Reference Adaptive System (MRAS), the specifications are given in terms of a reference model, which tells how the process output ideally should respond to the command signal. Since the MRAS was originally proposed by Whitake et al. (1958), there have been many algorithms proposed for adaptive control which will provide globally asymptotically stable controllers if some stringent conditions on the plant are met (e.g.: Monopoli, 1974; Landau, 1979; Goodwin et al., 1980; Lozano and Landau, 1981). However, there still exist some theoretical problems to be solved e.g., valid adaptive control schemes for the system with deterministic disturbances should be developed. The use of inferential control in process control has been the subject of some recent interesting studies (Brosilow and Joseph, 1978; Morari and Stephanopoulos, 1980) in which the disturbances can be estimated by using one or more outputs that are not controlled outputs. In the method proposed here, inferential control and model reference adaptive control are combined and the control scheme is applied to the control of polymerization reactors.
A polymerization reactor is a typical nonlinear chemical process and requires a sophisticated controller. An adaptive controller called Adaptive/Inferential Control System (AI-CS) is developed here and applied to the control problems of polymerization processes. The AI-CS can be used successfully for a system having deterministic disturbances and is valid for a non-minimum-phasesystem. For estimating the unknown deterministic disturbances, this controller utilizes a secondary output which is not the controlled process output. There is considerable literature on the computer control of polymerization reactors. A survey of process computer applications to continuous and batch polymerization reactors has been presented by Hoogendroorn and Shaw (1980). Tolfo (1981) described a Direct Digital Control (DDC) system which maximized the amount of product while satisfying the constraints of product quality and plant safety. Macgregor and Tidwell (1977) discussed a digital controller used for controlling the viscosity in an industrial polymerization reactor subject to stochastic disturbances, and the controller was constructed to minimize the quadratic cost function. However, references which describe an application problem of an adaptive controller to a polymerization reactor are few. Arnold et al. (1980) discussed a model reference adaptive control strategy for the production of monodisperse polymers and polymers of any predefined molecular weight distribution in a living anionic polymerization reactor. Kiparissides and Shah (1983) applied a self-tuning regulator and a globally stable model reference adaptive controller to a batch suspension PVC (poly(viny1chloride)) reactor which could be represented by a mathematical model identified from experiments. They also compared the performance of the two adaptive techniques with that of a classical PID controller and showed that the adaptive controllers always outperformed the PID controller. The aim of this paper is to develop AI-CS. The proposed AI-CS is applied to stabilize the operations of two types of polymerization reactors, that is, styrene solution polymerization and isobutylene polymerization. Adaptive/Inferential Control System
*Towhom all correspondence should be addressed.
Adaptive control systems have emerged in the attempt to avoid degradation of the dynamic performance of a
0196-4305/86/ 1125-0821$01.50/0 0 1986 American Chemical Society
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Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 3, 1986
control system when environmental variations occur. Recently, control schemes using a MRACS have been especially focused on, and attempts have been made to apply them to practical problems. However, there are several difficulties encountered when applying a MRACS to a practical process. Some of these are as follows: (1)For a system with deterministic disturbances, stability, that is, making the difference between the desired output and real output zero, cannot be guaranteed. (2) Control action becomes unstable for a non-minimum-phase system. The first problem is most frequently encountered when controlling chemical processes. For example, a parameter deviation in a nonlinear process appears as a deterministic disturbance in a linearized process model. The Adaptive/Inferential Control System (AI-CS)which is based on MRACS utilizes a secondary output for estimating the unknown deterministic disturbances. This control scheme is described below. The following singleinput-multiple(tw0)-output (SIMO) linear time-invariant system is considered
R(2-l)
= ro
+ r1x1 + ... + rn,.z-"'
and the order of R(z-l),nr, is given by nr = max ( n l - 1,nc - d l ) Using eq 9 and 1, C ( z - l ) y ( k+ d l ) = Al(z-l)S(z-l)y(k + d l ) = Bll(z-l)S(z-l)u(k) +
+ R(z-')y(k)
z-dz+d1Bl2(2-')s(Z-1)u(k)
+ R(z-')y(k) (11)
From eq 2, the disturbance u(k) can be rewritten as
When eq 12 is substituted into eq 11,
C(z-l)y(k + d l ) = + A2(z-l)y,(k)= ~ - ~ ~ B , ~ ( z - + ~ )~u- ~ ( k~)B ~ ~ ( z - '( 2) )u ( k ) A l ( ~ - ' ) y ( k=) ~ - ~ ' B l l ( z - ~ ) ~Z(-k~)' B ~ ~ ( Z - ' ) U(1) (K)
where
+ al,z-' + ... + = bo,, = bl,]z-l + ... + b,llz-mr~
A,&') = 1
B,(z-')
U~~Z-"~
i=land2
j=1and2
d, I 1 (i = 1-4)
d2 Idl
(3) (4) is derived. It is assumed that eq 13 can be rearranged and written as
+ d4
(5)
where 2-l is the backward shift operator, direpresents the plant time delay, u(k),y ( k ) ,y , ( k ) , and u ( k ) are the plant input, plant output, secondary output, and disturbance, respectively. The secondary output y , ( k ) is one of the outputs but not the variable y ( k ) being controlled. In many chemical plants, it is an acceptable assumption that a secondary output is available. It is also assumed that time delays diand the degrees of polynomials Ai(z-') and Bij(z-l)denoted ni and mij are a priori known or given. A goal of the control considered here is that the output of the plant y ( k ) approaches asymptotically the desired output value y*(k). Then, the goal of the control is proven to be accomplished if the output of the plant y ( k ) satisfies the following equation (Lozano and Landau, 1981) C(z-')e(k + d l ) = C(z-')[y(k + d l ) - y * ( k + dl)] = 0
v k 1 0 (6) where
C(z-')y(k
B,(z-')u(k) + B,(z-')y,(k)
(7)
is an output error, and C(2-1)
=1
+ clz-l + ... + cncz-nc
(8)
is an asymptotically stable ncth order polynomial which can be arbitrarily chosen. Of course, C(z-') affects the performance of the controller more or less. Then, C(2-l) should be chosen rationally for a given control problem. For the chosen C(z-l), the following uniquely determined relationship can be deduced C(2-l) = Al(z-l)S(z-l)
+ ~-~lR(z-')
(9)
+ R(z-')y(k) (14)
where
B,(z-l) = bo, + b l g l By(z-') = boy
+ bl,z-'
+ ... + b , , g m u + ... + b,z-ny
(15)
and eq 14 can also be written as C(z-')y(k + d l ) = p T @ ( k )
(16)
where p and @(k) are the plant parameter vector and plant input-output vector, respectively, which are defined as
PT =
[b~uiblui...,bmu,b~y,bly,...,bnytrOtrl,...,rn~l
(17)
a T ( k )= [u(k),u(k- l),...,~(k- mu),y,(k),y,(k11,.-,y 8 ( k - n y ) , y ( k ) , y ( k- l),-.,y(k - nr)l (18) If the plant parameters are known, the plant control input which satisfies eq 6 is obtained as
u(k) = e&) = y ( k ) - y*@)
+ dl) =
C(z-l)y*(k + d l ) - B,(z-')y,(k) - R(z-')y(k)
B,(z-')
(19)
by substituting y * ( k + d l ) for y ( k + d l ) in eq 14. When the plant parameters are unknown, plant parameter vector p is replaced by adjustable parameters f i ( k ) which are updated by the adaptation mechanism. Therefore, the control input u(k) is given by C(z-')y*(k + d l ) - B,(z-l)y,(k) - IZ(z-')y(k) u(k)= (20) B,(z-l) Parameters h,, A, and R, that is, fi(k), are estimated by the following adaptation mechanism fi(k) = @ ( k - 1) + F ( k ) @ ( k- dl)v(k) (21)
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 3, 1986
823
SECONDARY
OUTPUT ys(k+l)
)+
Figure 1. Control scheme by the AI-CS.
with
P'(k
+ 1) = Xi(k)i;Ll(k)+ X,(k)@(k - dl)aT(k - d l ) (22)
where 0 < X,(k) I1,
0 IX,(k)
< 2,
F(0) > 0,
and
~ ( k=) [ p - Ij(k)IT@(k- d l ) (23) The parameter adaptation mechanism is the same as given by Landau (1979). When the Popov's hyperstability theorem is used, it is also proved that the plant output y ( k ) approaches asymptotically the desired output y * ( k ) only when the plant time delay dl = 1or the polynomial is scalar, i.e., Ez2(z-l)= boz2for this configuration of the filter C(2-l) in eq 9. This restriction can be relaxed for another configuration of C(z-l). TI, deal with the second problem stated before, that the plant is a non-minimum-phase system, that is, E&-') has unstable zeros, the following compensator is introduced into the control scheme. As shown in Figure 1, control action u(k) given by eq 20 is modified as u(k) which is written ~ ( k ) u(k)Gj(z-') (24) where Gjlz-') is aTompensator which eliminate? unstable zeros in EJz-l). E&-') is resolved into factors bJ2-l) and b+(z-') as BJz-1) = &(z-1)6+(2-1)
(25)
where & ( z - ~ )and &+(z-l)are polynomials having unstable zeros and stable zeros, respectively. Compensator Gj(2-l) can be chosen as
Gj(z-l) = 6-(z-l)/G(z-l)
(26)
where G(2-l)is an arbitrarily given stable polynomial whose gain is equal to one; that is, IG(z-' = 1)1 = 1. Now, control action u(k)can be rewritten as
+
It will be interpreted that the terms related to y*(k dl), y ( k ) , and y , ( k ) in eq 27 correspond to the control action for the set-point change, feedback regulation, and feedforward elimination of unknown disturbances, respectively. Then, this controller developed here is a type of feedforward-feedback control system with an adaptation mechanism. Finally, the AI-CS proposed here is shown in Figure 1. Simple Example. Now let's consider the single-input-two-output (SIMO) linear discrete system with deterministic disturbance described by
[ Y ( k ) Y,(k)l =
lIx(k)
where x ( k ) , y ( k ) ,y,(k), u(k),and u(k) are state variables, output variable, secondary output variable, manipulating variable, and deterministic disturbance. The problem is to keep output y ( k ) at 0.0 by manipulating u(k) when the disturbance u ( k ) changes stepwise from 0.0 to 1.0 at the 20th step. The controlled result using AI-CS is shown in Figure 2a. The parameters and conditions used here are as follows: n = 1, m = 0, dl = d3 = d4 = 1,d2 = 2, C(2-l) = 1- 0.4%-'. The initial values of parameters and adaptive gain are as follows: boll = 0.5, bo21 = 0.0, bo12/bo22 = 0.0, all = 0.0, a12 = 0.0, F(0) = 100001where I is a unit matrix. As shown in Figure 2a, output y ( k ) deviates from the desired value 0.0 for about 10 steps, but after 10 steps, it is kept sufficiently close to 0.0. On the other hand, the controlled result, using the ordinary MRACS as shown in Figure 2b, is not adequate. Output y ( k ) frequently deviates suddenly because the model structure of the MRACS does not coincide with the real one. When a model parameter settles at a certain value, the controller gain in MRACS becomes large, and this requires high-performance parameter adaptation. However, the estimated model parameters can never coinside with real values. The gradual accumulation of output error makes the estimated model parameters deviate suddenly. Thus, the output y ( k ) frequently deviates largely from the desired value. However, the proposed
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Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 3, 1986 r
r
Figure 2. (a, left) Simulation result by AI-CS for the simple example. (b, right) Simulation result by MRACS for the simple example.
To simplify the kinetic model, the following are assumed: (1) Reaction rate constants of propagation and transfer reactions are independent of the chain length. (2) The reactor is perfectly mixed and the liquid volume V is constant. The material balances and heat balance in the polymerization reactor can be expressed as
I I TR
YJ-1
FpCP(T'"- TR)+ v A f f 3 R ~- US(TR - Tw) (32)
I
I
where the terms of reaction rate Ri are
RM = -k,XMXAo
I
Rc = -(klXA
RA
+ k,X~o)Xc R B = k1XAXc - kzXB RAO = kZXB
POLYMER PRODUCT
Figure 3. Schematic diagram of CSTR for isobutylene polymeri-
RAl = kzXB + (k3xM
+ k4XA + k5Xc)XAo -
(k4XA + k5XC)Xj.l
zation.
AI-CSis sufficient for the control of reaction processes with deterministic disturbances.
RAz = kzXB + (kaxM + k4XA + k5Xc)XAo + 2k3XMXX1 - (k4XA + k5XC)XAZ (33)
REO= ~ z X B + (k4XA + k5XC)XAO
Control for an Isobutylene Polymerization Reactor
Re1
The average molecular weight of the polymer (M,) is one of the specifications of the product. Therefore, M , is considered as the output to be controlled thoroughout this paper. Mathematical Description of the Reactor. A continuous-stirred tank reactor (CSTR) for isobutylene solution polymerization as shown in Figure 3 is considered. Isobutylene monomer, a solvent, methyl chloride, and two initiators, chloride and aluminum organic compounds, are fed into the reactor. The average molecular weight of the polymer product, M , is measured by an instrument such as a light scattering photometer (LSP). It is assumed that M, is measured every 15 min and controlled by manipulating the addition rate of initiator or the total feed rate. The reaction mechanisms of isobutylene polymerization are represented as (i) initiation
Re2
= kZXB
= kZXB + (k4XA + k5xC)xAO
(ii) propagation
ka
P,+l+
(30)
(iii) transfer
P,+
+A
P,+ + C
ki
k6
PI++ M,
(31)
P,+ + M,
A t the beginning, the reaction between initiators A and C generates a cation of PI+. Monomer M reacts with a cation of P,+ with length n and attackes to the end of the molecule Pn+,forming a new cation, P,+,+. Dead polymer of length n, M,, is generated by the transfer reaction.
+ k3xMxAO
+ (k4XA + k&C)XAO + k3xMxAO + 2k3XMX)J
= ~ Z X+B(k1X.4+ k5Xc)X~o+ ~ ~ X M X+A O 3k3X~xxi+ ~ ~ ~ X M X A Z In these equations, Xi denotes the concentration of component i and X, and Xcjdenote the j t h moment of the molecular weight distribution of the growing polymer and the j t h moment of the molecular weight distribution of both the growing and dead polymer, respectively. T, is the temperature (i = R and W) of the reactor or the coolant. P is the inlet temperature. AH3is the reaction heat evolved. U is the overall heat-transfer coefficient; S is the heat-transfer area. When moments are used, the weight average molecular weight M , [g/mol] can be given as Rt3
M w
P,+ + M
+ k4x~o)X~
= -(k,Xc
= WXtZ/Xtl
(34)
where W is the molecular weight of the isobutylene monomer. Statement of the Control Problems. In an isobutylene polymerization reactor, the following two control problems are considered. In order to keep the average molecular weight of the polymer M , at the desired value, find a manipulating scheme of inlet concentration of initiator A, q ( t ) (problem 1)or total feed rate F(t)(problem 2), where the reactor temperature T R can be used as a secondary output. Frequently, reactor temperature is taken as a controlled variable in the operation of a reactor. However, this is not done here. Since this polymerization reactor is operated around -30 "C, the pressure of the coolant must be changed if the coolant temperature is manipulated. The implementation of an actuator for the reactor temperature control will be difficult. Therefore, it is assumed that the coolant temperature is not used as a manipulating variable
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 3, 1986 825 '0 a
Table I. Parameters and Given Condition for Isobutylene Polymerization Reactor k , = 2.8 X loz3exp(-12000/TR) reaction rate constants k 2 = 3.7 X 10l6 exp(-lOOOO/TR) k 3 = 1.5 X 10" exp(-3500/T~) = 1.8 X k, = 1.3 x
k4
total feed rate, L/h reactor vol, L feed temp, K inlet concn of monomer, mol/L inlet concn of initiator A (aluminum organic compd), mol/L inlet concn of initiator C (chloride), mol/L coolant temp, K
:i--:,
y-,/ 3
loi3 eXp(-6500/T~) ioi3 eXp(-6ooO/T~)
F = 200.0 V = 400.0 = 243.0 X $ = 3.9
xi" A - 0.77 X lo-'
---PI
X-
TIME (HR)
Figure 4. Simulation results of M, by AI-CS, MRACS, and PI controllers for problem 1with a +5% disturbance of Xg from the nominal value.
xt = 1.26 x 10-3 Tw = 235.0
and is constant. Moreover, the reactor temperature TR is to be used as a secondary output y s . Simulation Results for Problem 1. Many simulation studies of the AI-CS in Figure 1were performed by using the parameter values listed in Table I, where the compensator is taken as G@') = 1. One of the simulation results for problem 1 is shown in Figure 4. The inlet concentration of the monomer, X h was disturbed 4 h after the start and was made 5% higher than before. The influence of the disturbance on M , was reduced by manipulating the inlet initiator concentration X T , according to the AI-CS. In the figure, the results when a classical Proportional-plus-Integral (PI)feedback controller and the MRACS proposed by Landau (1979) are used for controlling M , are also given. In,Figure 5, the time courses of the manipulating variable X E and the secondary output TRby AI-CS are shown. From Figure 4, it can be seen that the proposed AI-CS works better than the other two control schemes. Figure 5 illustrates that the disturbance caused by X b can be estimateded quickly by using the secondary output TR. Simulation Results for Problem 2. Figure 6 shows the step response of M , caused by a unit step change of flow rate F, which is an inverse response. Thus, the system which describes a transfer function from F to M , should be recognized as a non-minimum-phase (NMP) system. In this case, the compensator G (2-l) of the AI-CS shown in Figure 1 works actively and will be able to eliminate unstable zeros in the system as explained before. The same parameter values and conditions as stated before are used for the simulation study of problem 2. One of the results using the AI-CS is shown in Figure 7. Figure 8 gives the result of using the AI-CS without the compensator Gf(z-l), that is, with Gf(z-') being taken as G,(z-l) = 1. As shown in Figure 7, the AI-CS again provides good performance. On the other hand, without compensator G f ( z - l ) the , manipulating variable u ( k ) diverges infinitely and the M , which should be controlled also becomes unstable as shown in Figure 8. From these examples, it can be seen that the proposed AI-CS performs adequately or better in both problems.
TIME (HR)
Figure 5. Time courses of manipulating variable X k and secondary output TRby AI-CS for problem 1.
'.''..''S '".'
200
400
TIME (MINI
Figure 6. Response of M, by step change of manipulating variable F.
assumed that M , is measured every 15 min and controlled by manipulating the total feed rate F. The reaction mechanisms of styrene polymerization are represented as (i) initiator decomposition
I2
kdl
2Ry
(35)
(ii) initiation
R1. + M -% P1.
(iii) propagation
Control for a Styrene Polymerization Reaction
(iv) termination
Mathematical Description of the Reactor. The same type of CSTR as shown in Figure 3 is used for styrene solution polymerization. Styrene monomer, toluene as the solvent, and two initiators, benzoyl peroxide and tert-butyl perbenzoate, are fed to the reactor. The reactor temperature TRand the reactant volume V are assumed to be kept constant. In this case, the average molecular weight of polymer product M , can be measured by an instrument such as a gel permeation chromatograph (GPC). It is also
Initiators I1 and I2are decomposed into radicals R1. and R2.. Radicals R1-and R2. react with monomer M, and radicals P1. are generated. Monomer M is added onto the end of radical PI-,forming a new radical Pj+l.. Termination by recombination generates dead polymer Mj.
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Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 3, 1986
Table 11. Parameters and Given Condition for Styrene Polymerization Reactor reactor rate constants
Figure 7. Simulation result of M , by AI-CS with the compensator for problem 2 with a +5% disturbance of Xi;l from the nominal value.
= 7.53 X 10" exp(-l6050/T~) k d 2 = 2.35 X 10'' eXp(-16990/T~) k , = 7.92 X 1O1O exp(-3927/TR) kto = 4.68 x 10" eXp(-1208/T~) k , = kto[exp(-BX - C X 2 - DX3)I2 X = conversion of monomer B = 2.57 - 5.05 X 10-3TR C = 9.56 - 1.76 X lO-'TR D = -3.03 + 7.85 X 10-3TR F = 50.0 V = 300.0 Tp = 343.0 xt; = 6.98 x 10-3 kdl
total feed rate, L / h reactor vol, L reactor temp, K inlet concn of initiator 1 (benzoyl peroxide), mol/L inlet concn of initiator 2 (tert-butyl perbenzoate), mol/L
x i 12 n
-- 8.70 x
10-3
Figure 8. Simulation result of M , by AI-CS without the compensator for the same problem as the one in Figure 7.
To simplify the kinetic model of the styrene polymerization reaction, the following assumptions are introduced: (1) Reaction rate constants of propagation and termination are independent of the chain length. (2) Polymer chains are so long that it can be assumed that monomers disappear only by the propagation reaction. (3) The quasi-steady-state assumption is satisfied because radical species react very fast. (4) The reaction medium is perfectly mixed in the constant volume V. ( 5 ) The gel effect is represented as shown later by the decrease of the termination rate constant k,. The material balances in this polymerization reactor can be expressed by using these assumptions as
dX,O k, v= -FXy0 + -X,,2V dt 2
dX,2 V -= -FX,2 dt
+ ktXxo2(Y + 1 ) ( 3 ~+ 2)V
where X, denotes the j t h moment of the dead polymer chains, Xxodenotes the zeroth moment of the growing chains, and u is the kinetic chain length, expressed as u = Rp/RI (40) with reaction rate terms for propagation and initiation given by RP = ~ J M X A ORI = 2fikdiX11 + 2fzkd2X12 (41) From these moments, the weight average molecular weight
M w is given as Mw =
wx,2/x,1
(42)
2 4 6 8 1 RESIDENCE TIME (HR)
0
5
Figure 9. Relationship between residence time and conversion rate of monomer.
where W is the molecular weight of the styrene monomer. Statement of the Control Problem. In the polymerization reaction, the increase of the polymer concentration brings about high viscosity of the solution and reduces the collision rate of radicals. That is, the termination rate constant k, decreases. As a result, the ratio of the propagation rate and termination rate becomes larger than before. This accelerates the overall conversion rate from monomer to polymer. This phenomenon is called the gel effect. It can be shown that the polymer concentration corresponds to the conversion of monomer. Then the decrease of the termination rate constant k, will be represented by one in Table 11. Due to the gel effect, an unstable steady state frequently appears. Figure 9 shows the relationship between the average residence time in the reactor and the conversion rate of the monomer at the steady state in a CSTR for the styrene polymerization. The reactor temperature TR is kept at 70 "C. The relationship is represented by an Sshaped curve as shown in the figure. For the S-shaped curve, at the part of the curve which has a negative gradient, the corresponding equilibrium point proved to be unstable. For example, at the residence time designated in Figure 9, there are three equilibrium points. But the equilibrium point denoted as point A is unstable unless an effective control action is used. On the other hand, at the other two points, stable operation can be realized without taking any control action. The objective of the control considered here is to keep the M, at the value corresponding to the unstable point A in Figure 9 by manipulating the total feed rate F. For this purpose, the proposed AI-CS and a classical PI controller are used, and it is assumed that the outlet concentration of the monomer, XM,can be used for the secondary output. Simulation Results. Numerical values of parameters and conditions used for the simulations are listed in Table
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 3, 1980 827
MRACS. In this case, the result obtained by using AI-CS is not necessarily superior to the result obtained with a PI-feedback controller, which is shown in Figure 12b. Finally, it can be concluded that AI-CS is effective for the case when the response of the secondary output due to the disturbance is quicker and larger than that of the main output. Figure 10. Controlled result of M, by AI-CS with a -2% disturbance of k, from the nominal value.
I
TIME (HR)
Figure 11. Controlled result of M, by the PI-feedback controller for the same problem as the one in Figure 10.
Figure 12. (a, left) Controlled result of M, by AI-CS for the setpoint change problem in isobutylene polymerization. (b, right) Controlled result of M, by the PI-feedback controller for the same problem as the one in a.
11. The result obtained by using the AI-CS is presented in Figure 10. For the disturbance, the propagation rate constant, k,, is considered to be lowered 2% 40 h after the start of the operation. Figure 10 shows that for the control action, the total feed rate, F, is manipulated in an oscillatory fashion and, therefore, M, has oscillatory motion after the disturbance occurs. However, the oscillations settle down after a while, indicating that the controller is ultimately successful. On the other hand, Figure 11shows the result when using a classical PI controller. The PI controller could not keep M, a t the equilibrium point designated as point A. From the numerical searching of parameters for P I controllers, it can be said that any PI controller cannot keep the system a t the unstable point A. Therefore, the AI-CS is shown to be very useful for achieving a stabilizing control system.
Discussion The proposed AI-CS utilizes a secondary output for estimating the disturbance. If the response of the secondary output due to the disturbance is slower than that of the main output, the AI-CS doesn't work well. This condition corresponds to eq 5; that is, dl + d4 I d2 as stated before. Further, if the steady-state gain of the secondary output due to the disturbance is smaller than that of the main output, the AI-CS also doesn't work successfully. For example, in the isobutylene polymerization reactor, the set point is changed from one value to another, but the secondary output, the temperature TR,doesn't change by as much. Figure 12a shows a simulation in which AI-CS is used. In the simulation, the monomer concentration in the feed increases by 15% at 4 h and the M, set point is changed from 170000 to 255000 a t 16 h. The secondary output, that is, the reactor temperature TR,doesn't change a t all due to the set-point change of M,. Therefore, the output M, is controlled as if it were controlled by an ordinary
Conclusion The control system called the Adaptive/Inferential Control System (AI-CS) has been proposed and applied to the problem of controlling the M, in polymerization reactors. Many simulations have shown that AI-CS can control M, well in polymerization reactors. This'kontrol system is constructed by using the fact that a secondary output can be used for estimating an unknown deterministic disturbance. That is, information about the disturbance will be used implicitly for the control of the output. The AI-CS is a type of adaptive feedforwardfeedback control system with an estimator for the disturbance. It should be stressed that the control system proposed here can be implemented and applied to many CSTR's used an industrial polymerization.
Nomenclature Ai(%-') = polynomial given by eq 3 (i = 1 and 2) (i = 1 and 2; j = 1and 2) B,(z-') = polynomial given by eq 15 B,,(z-') = polynomial given by eq 15 C(2-l) = arbitrary asymptotically stable polynomial C, = average reactant heat capacity, kcal/(g K) di = plant time delay (i = 1-4) F = total feed rate, L/h F ( k ) = adaptive gain matrix f i = efficiency of initiator G(2-l) = arbitrary asymptotically stable polynomial G&l) = compensator for non minimum phase AH3 = isobutylene polymerization reaction heat, kcal/mol k l , k2, k , k,? kS = reaction rate constants of isobutylene polymerization kdl, kd2 = initiator decomposition rate constants of styrene polymerization kill ki2 = initiation rate constants of styrene polymerization k , = propagation rate constant of styrene polymerization k , = termination rate constant of styrene polymerization mij = order of polynomial Bij(z-') ( i = 1and 2;j = 1 and 2) M, = molecular weight of polymer product, g/mol nc = order of polynomial C(2-l) ni = order of polynomial Ai(%-')(i = 1 and 2) p = plant parameter vector R(2-l) = polynomial satisfied with eq 10 Ri = reaction rate, mol/(L h) S = heat-transfer area, m2 S(2-l) = polynomial satisfied with eq 10 9 = inlet tmperature, K TR = isobutylene reactor temperature, K TW = coolant temperature, K U = overall heat-transfer coefficient, kcal/(m2 h K) u(k) = plant input V = liquid volume in reactor, L u(k) = disturbance W = molecular weight of isobutylene or styrene monomer, g/mol XA= concentration of aluminum organic compound, mol/L XB = concentration of activated complex, mol/L Xc = concentration of chloride, mol/L XI1 = concentration of benzoyl peroxide, mol/L Bij(z-l)= polynomial given by eq 4
Ind. Eng. Chem. Process Des. Dev. 1986,2 5 , 828-834
828
X 1 2= concentration of tert-butyl perbenzoate, mol/L XM = concentration of monomer, mol/L XM,,= concentration of dead polymer with length n, mol/L Xpy = concentration of radical styrene polymer with length
.I, moVL Xpn+ = concentration of cation isobutylene polymer with length n, mol/L X , = jth moment of molecular weight distribution of the dead
styrene polymer, mol/L X E j= jth moment of molecular weight distribution of both
growing and dead isobutylene polymers, mol/L XAj = jth moment of molecular weight distribution of the
growing isobutylene or styrene polymer, mol/L y(k) = plant output y,(k) = secondary output z = Z-transform operator Greek Letters Wk) = plant input-output vector Y = kinetic chain length v(k) = adaptation error p = average reactant density, g/L
Superscripts in = inlet = estimated value * = desired value
Registry No. Isobutylene (homopolymer), 9003-27-4; isobutylene, 115-11-7;styrene (homopolymer),9003-53-6;styrene,
100-42-5.
Literature Cited Arnold, K.; Johnson, A. F.; Ramsay, J. Proceedings of 4th InternationalF e d eration of Automatic Control Conference on the Instrumentation and Automation in the Paper, Rubber, Plastlcs and Polymerization Industries, Ghent, Belgium, 1980, pp 359-367. Astrom, K. J. “Preprints, 8th Triennial World Congress of InternationalFederation of Automatic Control”; Plenary Session: Kyoto, 1981; pp 28-39. Brosilow. C.; Joseph, B. AIChEJ. 1878, 24, 485-509. Goodwin, G. C.; Ramadge, P. J.; Caines, P. E. I€€€ Trans. 1880, Ac-25 (3), 449-456. Hoogendroorn, K.; Shaw, R. 4th InternationalFederation of Automatic Control Conference on the Instrumentation and Automation in the Paper, Rubber, Plastics and Polymerization Industries, Ghent, Belglum, 1980, survey paper. Kiparissides, C.; Shah, S. L. Aotomatlce 1883. 79, 225-235. Landau, Y. D. “Adaptive Control”; Marcel-Dekker: New York, 1979. Lozano, R.; Landau, I . D. Int. J . Control 1981, 33 (2), 247-268. Macgregor, J. F.; Tidwell, P. W. Proc. Inst. Electr. Eng. 1977, 124 (8), 732-734. Monopoli, R. V. IEEE Trans. 1974, AC-19 (5). 474-484. Morari, M.; Stephanopouios, G. A I C E J . ISSO, 26, 247-259. Tolfo, F. “Computerized Control and Operation of Chemical Plants”; Verein Osterreichischer Chemiker: Vlenna, 1981. Whitaker, H. P.; Yamron, J.; Kerzer, A. Report R-164, Instrumentation Laboratory, MIT, Cambridge, MA, 1958.
Received for review May 13, 1985 Accepted September 3, 1985
A
On Interaction between Ethane and Propane in Simultaneous Pyrolysis and I t s Influence on Ethylene Selectivity Zou, Renlun Hebei Academy of Sciences, Shuiazhuang, Hebei Institute of Technology, Tianjin, The People’s Republic of China
Zou, Jlnt Hebei Institute of Technology, Tianjin, The People0 Republic of China
The interaction between ethane and propane in simultaneous pyrolysis will find expression in four aspects of two pairs of contradictions, i.e., acceleration and retardation of ethane upon propane and of propane upon ethane. Ethylene selectMty is one of the comprehenstve expressions of synergetic effects of these four aspects. I t depends upon the rate of ethylene formation as well as that of reactants disappearance. This present paper shows that the ethylene selectivity in simultaneous pyrolysis is definitely lower than its addltlve selectivity.
There are utterly different points of view in the problem of ethylene selectivity in simultaneous pyrolysis (the terms “simultaneous pyrolysis” and “individual pyrolysis” used in this paper are synonymous with “copyrolysis” and “separate pyrolysis”, respectively, in some references) of the ethane-propane mixture. Froment et al. (1979) held the view of negative deviation; i.e., the ethylene selectivity in simultaneous pyrolysis is lower than its additive select In
accord with the authors’ preference, their family names are listed first.
tivity. Mol (1981) held the contrary view. Goossens (1979) also did not agree with Froment’s view, but Hofmann (1980) has supported them. Zou et al. (1986) indicated that there is a negative deviation with respect to real selectivity and a positive deviation with respect to overall selectivity. The present paper analyzes and calculates the interaction between ethane and propane in simultaneous pyrolysis and further discusses ethylene selectivity.
Calculation of Free-Radical Concentration The present paper calculates and compares three systems: ethane pyrolysis, propane pyrolysis, and ethane-
0196-4305/86/1125-0828$Ol.50/0 @ 1986 American Chemical Society
