Comparative Study of Different Cascade Control Configurations for a

Article pubs.acs.org/IECR

Comparative Study of Different Cascade Control Configurations for a Multiproduct Semibatch Polymerization Reactor Reddi Kamesh,†,‡ P. Swapna Reddy,‡ and K. Yamuna Rani*,†,‡ †

Academy of Scientific and Innovative Research, ‡Process Dynamics and Control Group, Chemical Engineering Division, CSIR-Indian Institute of Chemical Technology, Hyderabad 500 007, India S Supporting Information *

ABSTRACT: The present study focuses on the development of different cascade control configurations for temperature control of a multiproduct semibatch polymerization reactor. The master controller regulates the reactor temperature by manipulating the mean cooling jacket temperature, whereas the slave controller manipulates the valve opening to control the mean jacket temperature. The configurations explored are based on generic model control (GMC) and proportional integral (PI) control. The effects of different controller tuning parameters are evaluated. The effect of other unmeasured and measured disturbances, i.e., impurity factor, monomer feed flow rate, and presence of measurement noise on control performance, is evaluated for two products A and B. GMC−GMC has shown better controller performance in terms of temperature tracking with minimum rootmean-square output deviation (RMSOD) values as well as minimum normalized root-mean-square input deviation (NRMSID) values from the nominal input value, compared to GMC−PI, PI−GMC, and PI−PI.

1. INTRODUCTION Batch and semibatch reactors are widely used in the chemical industry to produce fine chemicals, pigments, polymers, and pharmaceuticals. Generally these reactions are exothermic in nature and exact temperature control is required to meet the product specifications. Many advanced control techniques have been explored for batch reactor temperature control, and among them model based control techniques have attracted recent attention. Controlling a batch/semibatch reactor along the optimal set point trajectory is one of the key problems in this area. Optimal operation of batch/semibatch reactors involves two main tasks: generation of optimal set point trajectories and controller design and synthesis for tracking of set point trajectories. Both these tasks have been handled predominantly with the help of first-principles models. The approaches available in the literature for trajectory tracking for control of batch/semibatch reactors can be classified as feed forward control, self-tuning and adaptive control, feed-back linearization based control, and other control approaches.1 There exists a considerable volume of literature on the control of batch and semibatch polymerization processes. Recent studies in nonlinear control of batch processes, based on the first-principles model as well as data-driven models, include nonlinear model predictive control (NMPC) based on the first-principles model2 for batch processes subject to input constraints and model uncertainties, inferential control based on adaptive state and parameter estimation,3 input−output linearizing control4,5 for batch processes, globally linearizing control with state estimation and designing a nonlinear model based controller,6 and adaptive and nonadaptive GMC with sensitivity compensation for data driven model and exact model based approach for semibatch processes.7,8 Katende and Jutan9 have proposed a nonlinear generalized predictive control (GPC) algorithm and showed that it is superior to constrained © XXXX American Chemical Society

self-tuning PID, generalized minimum variance control (GMV), and GPC algorithms for a batch reactor system. Chen and Huang10 have employed an artificial neural network (ANN) to model the batch reactor and used its linearized version at every sampling instant to update the tuning parameters of a PID controller. Lee and Lee11 provide an overview of iterative learning control approaches for batch process control with a particular focus on model-based quadratic iterative learning control approach. These approaches illustrate the need for application of nonlinear and advanced control strategies for batch reactors. Among the nonlinear model based control techniques, GMC12 has been one of the most widely reported nonlinear model based control approaches mainly due to its simplicity. The basic GMC of Lee and Sullivan12and some later versions of GMC13−19 are applied to many chemical engineering systems based on the dynamic model derived from first principles. Chylla and Haase20 have presented an industrial case study for temperature control of a multiproduct semibatch polymerization reactor as a challenge problem. Several control configurations have been proposed to solve this problem. This problem has been attempted using two approachesfirst as direct reactor temperature control using control valve opening as the manipulated variable and second using a cascade control approach with master loop as reactor temperature as the controlled variable and jacket temperature as the manipulated input, and slave loop as jacket temperature control using valve opening as the manipulated input. With the help of the first approach, Rani7,8 proposed novel sensitivity compensating control approaches combined with an extended external reset Received: April 11, 2014 Revised: August 25, 2014 Accepted: August 27, 2014

A

dx.doi.org/10.1021/ie501515y | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Industrial & Engineering Chemistry Research

Article

Table 1. Tuning Parameters for Different Cascade Controllersa GMC(m)−GMC(s) cascade

GMC(m)−PI(s)

parameters product A product B product A product B

master controller (m) slave controller (s)

cascade

k1 k2 kc1 kc2

1.1 1.0 0.0 0.0 2.4 2.4 0.0 0.02 PI(m)−GMC(s)

1.0 0.6 0.15 0.1 19.0 18.0 0.125 0.125 PI(m)−PI(s)

parameters product A product B product A product B

master controller (m) slave controller (s)

k1 k2 kc1 kc2

37 1.3 2.0 0.0

33 1.3 2.0 0.0

37 1.3 2.0 0.0

33 1.3 2.0 0.0

a

Fine-tuned parameter values for different cascade controllers: k1 = proportional gain for master control; k2 = integral gain for master control; kc1 = proportional gain for slave control; kc2 = integral gain for slave control.

Table 2. Overall Comparison of Control Performance for Multiproduct Semibatch Polymerization Reactor Challenge Control ProblemProducts A and B

Figure 1. Representation of Chylla Haase polymerization reactor: (a) schematic representation with cascade control strategy, TC = temperature controller, TT = temperature transmitter; (b) block diagram of cascade control strategy.

controller

feedback method to handle sensitivity and input saturation with application of GMC based on exact model and data-driven models. Further, with the same approach, Rani and Patwardhan21 have applied neural network based GMC for this system. Cascade control approach has been the most commonly applied approach for this challenge problem. Clarke-Pringle and MacGregor22 have proposed a nonlinear adaptive temperature control strategy, where some of the unknown process parameters are estimated using an extended Kalman filter to find the jacket temperature set point for the slave controller in a cascade structure. Binder et al.23 have presented an adaptive control vector parametrization strategy with the help of wavelets for optimal control for this system in order to find the set point trajectory for jacket temperature. Graichen et al.24 have applied feed forward control with online parameter estimation using EKF based on the first-principles model to find the set point trajectory for jacket temperature. Vasanthi et al.25 developed a cascade controller with a self-tuning master control loop to obtain the desired control performance for the Chylla and Haase polymerization process maintaining the reactor temperature within the tolerance interval of ±0.6K from the set point. The focus of the cascade control approach for this problem has been on using different control approaches for the master loop while retaining conventional PI controller in the slave loop. In the present study, the semibatch polymerization reactor challenge control problem of Chylla and Haase20 is considered to evaluate different combinations of cascade control configurations for master as well as slave loops. The configurations considered for comparison are master controller and slave controller by GMC [GMC−GMC], master controller by GMC and slave controller by PI [GMC−PI], master controller by PI and slave controller by GMC [PI−GMC] and both master controller and slave controller by PI [PI−PI].

product A RMSOD (master control) product A RMSOD (slave control) product A NRMSID product B RMSOD (master control) product B RMSOD (slave control) product B NRMSID

GMC (master)/ GMC (slave)

GMC (master)/PI (slave)

PI (master)/ GMC (slave)

PI (master)/ PI (slave)

0.1084

0.3416

3.5889

3.5275

3.040

5.8111

60.4306

0.1369

0.2741

0.4225

0.2101

0.1248

0.5162

4.8121

4.4896

3.998

5.6026

67.9129

0.1631

0.2274

0.4084

25.124

53.529 0.3984

by Chylla and Haase20 as a challenge problem. Achieving good temperature control in these reactors is often difficult because physical properties of the contents, such as mass, heat capacity, and heat transfer coefficient vary from run to run and within a run. The system consists of a stirred tank reactor to prepare specialty emulsion polymers. The schematic representation of the polymerization reactor is illustrated in Figure 1a. During typical reactor operation, four or five batches of a particular product would be made in succession. Afterward, the reactor would be cleaned before changing to a new product. Data has been provided for two different products (called products A and B), which are representative of the large variety of products made. The operating recipe for product A is given as follows: (1) in phase 1, place initial charge of solids and water into the reactor at ambient temperature; (2) raise temperature of the initial charge to the reaction temperature set point of 180 °F; (3) in phase 2, feed pure monomer into the reactor at 1.0 lb/min for 70 min; and (4) after the feed addition period is complete, in phase 3, hold at the reaction temperature for 60 min. For product B, the operating recipe is described as follows: (1) In phase 1, place initial charge of solids and water into the reactor at ambient temperature; (2) raise temperature of the initial charge to the reaction temperature set point of 176 °F; (3) in phase 2, feed pure monomer into the reactor at 0.8 lb/min for 60 min; (4) after the feed addition period is complete, in phase 3, hold at reaction temperature for 30 min;

2. SEMIBATCH POLYMERIZATION REACTOR An industrial case study for temperature control of a multiproduct semibatch polymerization reactor has been presented B



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Figure 2. Comparison of performance of cascade controller combinationsProduct A for summer batch 1: (a) reactor temperature profiles, (b) input profiles of master control, (c) mean jacket temperature profiles, and (d) input profiles of slave control.

and Haase20). The model equations considered for semibatch polymerization reactor are as follows: Material balance in terms of moles of monomer is given as

(5) in phase 4, feed pure monomer into the reactor at 0.8 lb/min for 40 min; and (6) after the second feed addition period is complete, in phase 5, hold at reaction temperature for 45 min. The recipe for product B consists of two feed periods; however, only the first one is considered in the present study. During the second feed, the heat transfer coefficient becomes zero and no control is possible. This observation has also been made by Clarke-Pringle and MacGregor.22 Thus, the control performance is evaluated only in the first three phases for both the products in the present study. The detailed model (consisting of four states, i.e., moles of monomer; reactor temperature; outlet jacket temperature; and inlet jacket temperature, described by four differential equations and several algebraic equations) and data as well as problem description have been defined in the original problem (Chylla

dn m = FM − RP dt

(1)

where nm = number of moles of monomer in the reactor (lb·mol), FM = molar flow rate of monomer into the reactor ((lb·mol)/min), and RP = rate of polymerization ((lb·mol)/min). The energy balance around the reactor is given as

∑ miCpi dT i

dt

= mMC pM(Tamb − T ) + R p(−ΔHP) − UA(T − Tj) − (UA)loss (T − Tamb)

C

(2)



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Figure 3. Comparison of performance of cascade controller combinationsProduct A for summer batch 5: (a) reactor temperature profiles, (b) input profiles of master control, (c) mean jacket temperature profiles, and (d) input profiles of slave control.

where T = reactor temperature (°F), Tamb = ambient temperature (°F), U = overall heat transfer coefficient (Btu/ (ft2·min·°F)), mi = mass of component i in the reactor (lb), i = M for monomer, W for water, and S for solids, mM = mass flow rate of monomer (lb/min), (UA)loss is heat loss to environment per unit temperature (Btu/(min·°F)), Tj = average jacket temperature (°F), −ΔHP is heat of polymerization (Btu/(lb· mol)), A = jacket heat transfer area (ft2), c(t) = slave controller output (0−100%), Cp = specific heat of component i (Btu/ (lb·°F)), and MWM = molecular weight of the monomer mix (lb/(lb·mol)). The exit temperature of the jacket is given by

mcC pc

dTjout(t ) dt

= mcC pc[Tjin(t − θ1) − Tjout(t )] + UA(T − Tj) (3)

where mc = mass of coolant in jacket (lb), Cpc = heat capacity of coolant in jacket (Btu/(lb·°F)), θ1 = transport delay in jacket (min), and θ2 = transport delay in recirculation loop (min). The inlet jacket temperature is the delayed exit jacket temperature given by the equation dTjin(t ) dt

=

dTjout(t − θ2) dt

+ [Tjout(t − θ2) − Tjin(t )] +

Kp τp (4)

D



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Table 3. Effect of Other Disturbances on Control Performance for Multiproduct Semibatch Polymerization Reactor Challenge Control Problem Using GMC−GMCProduct A and Product B effect of unmeasured disturbances

disturbances

product A

variations

i = 1; no load disturbance; no noise; θ1d = 3; θ2d = 2 impurity factor in rate i = 0.9 (−10%) expression i = 1.1(+10%) i = 0.8 (−20%) i = 1.2(+20%) load disturbance in −10% flow disturbance monomer feed flow rate +10% flow disturbance −20% flow disturbance +20% flow disturbance delay times in recirculation θ1d = 4; θ2d = 2 loop θ1d = 3; θ2d = 1 θ1d = 3; θ2d = 3 effect of measured disturbances

base case

disturbances

variations

RMSOD (slave control)

0.0961 0.179 0.1625 0.3618 0.2569 0.0888 0.1078 0.0838 0.1242 0.1421 0.2375 0.1014 product A

RMSOD (master control)

i = 0.9(−10%) i = 1.1(+10%) i = 0.8(−20%) i = 1.2(+20%) load disturbance in monomer −10% flow feed flow rate disturbance +10% flow disturbance −20% flow disturbance +20% flow disturbance presence of measurement noise in reactor temperature impurity factor in rate expression

NRMSID



NRMSID

2.5436

0.1376

0.1016

3.0404

0.1607

2.506 2.3432 2.3854 2.3994 2.1932 2.893 1.8534 3.2910 2.7801 5.4173 2.4632

0.137 0.1354 0.1190 0.1187 0.1241 0.1414 0.1105 0.1269 0.0890 0.3812 0.0922

0.188 0.205 0.4053 0.3246 0.0999 0.1138 0.1044 0.1382 0.1998 0.2981 0.1920

2.3158 3.9511 1.8320 5.0881 2.5004 3.8641 2.1396 5.1297 3.2374 3.5123 3.9758 product B

0.148 0.1656 0.1383 0.1715 0.147 0.1671 0.1414 0.1781 0.1833 0.1990 0.2242


NRMSID



NRMSID

0.0901 0.0981 0.0817 0.0998 0.0845

2.329 2.6183 2.0688 2.6894 2.1646

0.1279 0.1392 0.1147 0.1204 0.1251

0.0924 0.1163 0.0878 0.1373 0.0913

2.2835 3.9634 1.7729 5.1375 2.3562

0.1496 0.1635 0.1371 0.1704 0.1462

0.1052

2.8072

0.1338

0.1209

4.1402

0.1684

0.0749

1.8604

0.1109

0.0852

1.9428

0.1405

0.1233

3.3302

0.1267

0.1499

5.6913

0.1798

white noisenormal distribution with standard deviation of 0.02 white noisenormal distribution with standard deviation of 0.5

product A 0.0987 0.5868

where Tj =

2

and

FM =

mM MWM

R p = i(knm)

⎛ 6400 ⎞ 0.4 ⎟μ k = k 0 exp⎜ ⎝ T + 460 ⎠

product B

2.675 11.4121

a=

T jin + T jout

product B


0.1493 0.3694

1000 ; T + 460

f (t ) =

0.1053 0.5371

3.1221 11.8458

solids(t ) batch weight(t )

solids(t ) = solids(t = 0) − X (t )·total monomer added(t )

(5)

batch weight(t ) = solids(t ) + total monomer added(t )(1 − X (t )) + initial water h = 143.4 exp(−5.13 × 10−3μwall )

(6)

μwall = μ(Twall);

Twall =

T + Tj

2 1 = {0.000, 0.001, 0.002, 0.003, 0.004} hf for batch numbers {1, 2, 3, 4, 5}

(7)

where Kp = heating/cooling process gain (°F/%), k = first order

(10)

kinetic constant (1/min), k0 = pre-exponential factor (1/min), μ = product viscosity (cP), and i = impurity factor. The viscosity relation for product A is given as μ = 0.032e(16.4f )10[2.3(a − 1.563)]

The overall heat transfer coefficient is then given by the equation (8)

U=

and for product B, it is given as μ = 0.032e(119.1f )10[2.3(a − 1.563)]

0.1668 0.3631

1 1 h

+

1 hf

(11)

where h = film heat transfer coefficient (Btu/(ft2·min·°F)) and

(9)

f = fraction of solids. The process gain equation is expressed as

where E



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Figure 4. Comparison of performance of cascade controller combinationsProduct A for winter batch 1: (a) reactor temperature profiles, (b) input profiles of master control, (c) mean jacket temperature profiles, and (d) input profiles of slave control. ⎧ 0.8 × (30)(−c(t )/50)(T − T in) 0 ≤ c(t ) < 50 ⎫ inlet j ⎪ ⎪ ⎪ ⎪ ⎬ K p = ⎨0 c(t ) = 50 ⎪ ⎪ in ( c ( t )/50 − 2) ⎪ 0.51 × (350 − Tj )(30) 50 < c(t ) ≤ 100 ⎪ ⎭ ⎩

For this challenge problem, the information available to the control algorithm is the product being made, and measurements of the reactor temperature, temperature of the inlet water to the jacket, temperature of the exit water from the jacket, and the monomer feed flow rate. The controller must work over a series of five consecutive batches for both the sample products under both summer and winter operation. In the present study, two basic controllers considered in different cascade combinations are GMC and PI controllers. It is assumed that the process model is available for formulation of GMC. Before presenting different combinations of cascade controller configurations, a brief overview of GMC is presented.

(12)

where c(t) is the valve opening. The reaction temperature determines the chemical composition and particle size distribution of the emulsion polymer. Precise control of the reaction temperature is required throughout the batch to produce an acceptable product. Reaction temperature is controlled by manipulating the temperature of water which is recirculated through the jacket of the reactor. The deviations in reactor temperature of less than 1 °F are considered as satisfactory (Chylla and Haase20). F



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3. GENERIC MODEL CONTROL OVERVIEW In this section, GMC is presented as a typical nonlinear control algorithm. Consider a control-affine single-input single-output (SISO) system described by the following equations: ẋ = f(x(t ), t ) + g(x(t ), t ) ·u(t )

u=

t

=

x(0) = x 0

y = h(x(t ))

z 2̇ = y(2) =

dt 2

(r )

zṙ = y

dt

r−1

r

= Lf h(x) + Lg Lf r − 1h(x) ·u (14)

to represent the first r derivatives of the output variable by defining the transformed states zi, where the definitions of different orders of Lie derivatives of the scalar function h(x) with respect to vector functions f(x) and g(x) are discussed in detail by Henson and Seborg.26 The remaining states can then be represented by the following transformed equations, zṙ + 1 = qr + 1(z) + pr + 1 (z) ·u : zṅ = qn(z) + pn (z) ·u

(15)

r * = k1(Tset − T ) + k 2

where pi(·) and qi(·) are scalar functions of the transformed variable vector z, zr+1 to zn denote the internal states, and eq 15 represents internal dynamics. For relative degree one systems, Lee and Sullivan12 have proposed GMC as a mechanistic model based control approach where the control law is derived by defining a reference rate trajectory for the output derivative and forcing the process output rate to match the reference rate trajectory defined as r * = k1(y sp − y) + k 2

∫0

t

(y sp − y) dτ

dy = Lf h(x) + Lg h(x) ·u dt

∫0

t

(Tset − T ) dτ

(20)

The process output rate for the reaction temperature is defined by eq 2. The manipulated variable, mean cooling jacket temperature, Tjset, is calculated by equating reference trajectory r* to reaction temperature rate as Tjset =

⎡ 1 ⎢ (k1(Tset − T ) + k 2 UA ⎢⎣

∫0

t

(Tset − T ) dτ )

⎛ × ⎜⎜(∑ miCp, i) − (mM0C PM(Tamb + T ) + RP0(−ΔHP) ⎝ i ⎞⎤ − UA × T − (UA)loss (T − Tamb))⎟⎟⎥ ⎠⎥⎦ (21)

(16)

where r* is the desired “rate of change” of process output, ysp is the set point, and k1 and k2 are the controller tuning parameters. The reference rate is proportional to the distance from the set point and includes integral action to eliminate offset. For systems of relative degree one, eq 14 reduces to y(1) =

(19)

4. CASCADE CONTROL CONFIGURATIONS FOR THE CHYLLA−HAASE PROBLEM The block diagram of cascade control strategy is shown in Figure1b. In the present study, different cascade configurations are developed for controlling the reaction temperature at the desired set point, i.e., (a) Master controller and slave controller as GMC [GMC− GMC], (b) Master controller as GMC and slave controller as PI [GMC−PI], (c) Master controller as PI and slave controller as GMC [PI−GMC], and (d) Master controller and slave controller as PI [PI−PI]. 4.1. Master Controller and Slave Controller As GMC [GMC−GMC]. In this control scheme, the master controller, i.e., GMC, regulates the reaction temperature T by manipulating the mean cooling jacket temperature Tjset. The slave controller, i.e., GMC manipulates the valve opening c(t) to control the mean jacket temperature Tj. The controller equations are given as follows: Master Loop. In order to derive GMC law, it is necessary to define a reference rate trajectory for the output derivative and to force the process output rate to match the reference rate trajectory. The reference trajectory is given according to eq 16 as

= Lf r − 1h(x) = zr

= b(z) + a(z) ·u

(18)

where Y(s) and Y (s) represent the transfer functions of the output variable and the set point, respectively.

= Lf 2h(x) = z 3 dr − 1y

Lg h(x)

sp

: zṙ − 1 = y(r − 1) =

0

k s + k2 Y (s) = 2 1 Y sp(s) s + k1s + k 2

dy = Lf h(x) = z 2 dt d2y

(k1(y sp − y) + k 2 ∫ (y sp − y) dτ − Lf h(x))

The closed-loop transfer function resulting from application of the above control law is expressed as

(13)

where x represents the state variable vector of dimension n, u is the manipulated input, t denotes the present time, x0 represents the initial state vector, y represents the output variable, f(·) and g(·) are vector functions, and h(·) is a scalar function. For systems where relative degree is well-defined, the relative degree r of the output y with respect to the manipulated input u is defined as the number of times the output is to be differentiated with respect to time so as to explicitly depend on the manipulated input. Equation 13 can be successively differentiated and expressed in the following form: z1̇ = y(1) =

(r * − Lf h(x)) Lg h(x)

Slave Loop. In this case, the reference trajectory is given according to eq 16 as r1* = kc1( Tjset − 0.5(Tjin + Tjout))

(17)

The explicit form of the GMC law can be derived from eqs 16 and 17 as

+ kc2 G

∫0

t

( Tjset − 0.5(Tjin + Tjout)) dτ

(22)



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Figure 5. Comparison of performance of cascade controller combinationsProduct A for winter batch 5: (a) reactor temperature profiles, (b) input profiles of master control, (c) mean jacket temperature profiles, and (d) input profiles of slave control.

The process output rate for the mean jacket temperature Tj is defined by taking the mean of eqs 3 and 4. The manipulated variable, valve opening c(t), is calculated by equating reference trajectory r1* to the mean jacket temperature rate in two steps in eqs 23 and 24 as

Kp =

⎛ dT out ⎞⎞ dTjout(t − θ2) τp ⎛ ⎜r1* − 1 ⎜ j − − Tjout(t − θ2) − Tjin(t )⎟⎟⎟⎟ ⎜ ⎜ 0.5 ⎝ 2 ⎝ dt dt ⎠⎠

(23)

After determining the heating/cooling process gain, the valve opening c(t) can be calculated based on eq 12 and given as

⎧− 50 × log(K /(0.8 × (T − T in)))/log(30) ⎫ K p < (0.8 × (Tinlet − Tjin))/30 p inlet j ⎪ ⎪ ⎪ ⎪ (0.8 × (Tinlet − Tjin))/30 < K p < (0.15 × (350 − Tjin))/30 ⎬ c(t ) = ⎨ 50 ⎪ ⎪ ⎪ ⎪100 − 50 × log((0.15 × (350 − Tjin))/K p)/log(30) (0.15 × (350 − Tjin))/30 < K p ⎭ ⎩

H

(24)



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Figure 6. Comparison of performance of cascade controller combinationsProduct B for summer batch 1: (a) reactor temperature profiles, (b) input profiles of master control, (c) mean jacket temperature profiles, and (d) input profiles of slave control.

The lower and higher limits for Kp are defined based on substitution of c(t) value as 50 in the two expressions of eq 12 for c(t) < 50 and c(t) > 50, respectively. Further, the values of c(t) obtained are bounded between 0 and 100 using the saturation function. The procedure followed for derivation of control laws is similar in all the cases and is therefore not included in detail in the subsections below. 4.2. Master Controller as GMC and Slave Controller as PI [GMC−PI]. In this control scheme, the master controller, i.e., GMC, regulates the reaction temperature T by manipulating the mean cooling jacket temperature Tjset. The slave controller, i.e., PI manipulates the valve opening c(t) to

control the mean jacket temperature Tj. The master control equation is given by eq 21, and the slave control equation is given by c(t ) = kc1( Tjset − 0.5(T jin + T jout)) + kc2

∫0

t

( Tjset − 0.5(T jin + T jout)) dτ

(25)

4.3. Master Controller as PI and Slave Controller as GMC [PI−GMC]. In this control scheme, the master controller, i.e., PI, regulates the reaction temperature T by manipulating the mean cooling jacket temperature Tjset. The slave controller, i.e., GMC manipulates the valve opening c(t) to control the I



Article

Figure 7. Comparison of performance of cascade controller combinationsProduct B for summer batch 5: (a) reactor temperature profiles, (b) input profiles of master control, (c) mean jacket temperature profiles, and (d) input profiles of slave control.

5. RESULTS AND DISCUSSION In the present study, a challenging industrial benchmark multiproduct semibatch polymerization reactor is considered to control the reaction temperature at desired set point using different cascade controller configurations. Chylla and Haase20 have specified that solutions can be provided for the challenge problem by defining the control problem as design of a controller capable of maintaining the desired reaction temperature throughout the batch in the presence of typical disturbances, such as change in fouling factor from batch to batch, ambient temperature change due to seasonal variations, changes in the product, and variation within each batch such as heat transfer change due to increasing viscosity of product. Conditions often change from batch to batch due to increased

mean jacket temperature Tj. The slave control is given by eq 23, and the master controller equation is given by Tjset = k1(Tset − T ) + k 2

∫0

t

(Tset − T ) dτ

(26)

4.4. Master Controller and Slave Controller as PI [PI−PI]. In this control scheme, the master controller, i.e., PI, regulates the reaction temperature T by manipulating the mean cooling jacket temperature Tjset. The slave controller, i.e., PI, manipulates the valve opening c(t) to control the mean jacket temperature Tj. The master controller equation is given by eq 26, and the slave control equation is given by eq 25. J



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Figure 8. Comparison of performance of cascade controller combinationsProduct B for winter batch 1: (a) reactor temperature profiles, (b) input profiles of master control, (c) mean jacket temperature profiles, and (d) input profiles of slave control.

fouling of the reactor walls between cleaning periods and due to changes in the environmental conditions such as temperature and cooling water temperatures (e.g., summer to winter conditions). Finally, it is often the case that a given reactor will be used to produce more than one polymer grade or type. In the present study, different types of cascade control configurations are used to control the reaction temperature at the desired set point. For each controller, the tuning parameters can be different for each product but must remain the same from batch to batch for the same product. The tuning of the parameters is based on a reasonable control performance with offset free set point tracking with no constraint violations over five batches in the summer and five batches in the winter. The performance of the cascade controller is evaluated by rootmean-square output deviation (RMSOD) values from the set

point and the normalized root-mean-square input deviation (NRMSID) values from the nominal input value of 50% (no heating, no cooling) for this system, which are given by the equations t

∑t f (Tset − T )2 ref

RMSOD =

(tf − tref )

(27)

t

NRMSID =

∑t f (50 − c(t ))2 ref

(1002) × (tf − tref )

(28)

where tref is the time at which the reactor operation switches to phase 2 and tf is the total batch duration for both the products. K



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Figure 9. Comparison of performance of cascade controller combinationsProduct B for winter batch 5: (a) reactor temperature profiles, (b) input profiles of master control, (c) mean jacket temperature profiles, and (d) input profiles of slave control.

relation based on ZN tuning rules using the reaction curve method is given by

The tuning is carried out for both the products and for all control configurations, i.e., GMC−GMC, GMC−PI, PI−GMC, and PI−PI, with reference to the minimization of root-meansquare deviation of input and output. For the PI controller, for both loops, the base case tuning parameters are determined using Zeigler−Nichols tuning rules.27 In this study, the reaction curve method is followed to determine the proportional gain (Kc) and integral time (τI) value for the PI controller. From the reaction curve, the process gain Kp is determined as the ratio of the steady state change in output to the magnitude of step change in the input. The delay time (td) is calculated by the intersection of the tangent drawn at the point of inflection to the step response and the time axis. The PI control tuning

Kc =

0.9 td × s

τI = 3.33 × td

(29)

where s is the slope of tangent drawn at the point of inflection. The base case values of the tuning parameters are determined using Zeigler−Nichols tuning rules by eq 29 for both master and slave loops and are reported in Table A1 of Supporting Information. The effect of controller tuning parameters on temperature response for different cascade controller configurations is reported in Table A2 of Supporting Information. The temperature L



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Figure 10. Reactor temperature error profiles: (a) product A summer batch 1 for cascade control with master control as GMC and slave control as GMC and (b) product B winter batch 1 for cascade control with master control as GMC and slave control as GMC.

Figure 11. State variable profiles for summer batch 1 using GMC−GMC for (a) product A and (b) product B.

The final values of the tuned controller parameters of different cascade controller configurations are reported in Table 1 for the two products based on minimum RMSOD and NRMSID. A comparison of the results for different control configurations for both the products for summer and winter batches are listed in Tables A3 and A4 of Supporting Information. The results indicate that for both the products the overall average RMSOD values over 10 batches (five in the summer and five in the winter, each with a different fouling factor) for the GMC− GMC controller is smaller than those of GMC−PI, PI−GMC, and PI−PI. It may also be noted that for the GMC−GMC controller the range of controlled temperature falls well within the constraint of ±1 °F. For product A and product B, the order from best to worst case RMSOD values is found to be GMC−GMC, GMC−PI, PI−PI, and PI−GMC. From an overall point of view for both the products, Tables A3 and A4 of Supporting Information clearly illustrate that GMC−GMC has exhibited better performance compared to GMC−PI, PI− GMC, and PI−PI with respect to lower RMSOD values as well as minimum input oscillations. The results obtained in terms of the overall RMSOD and NRMSID values for summer and

response is characterized with the help of four measures, namely, the overshoot during the switching period between phase 1 and phase 2, undershoot during the switching period between phase 2 and phase 3, offset during phases 2 and 3, and oscillations in the temperature response throughout the batch. For each cascade configuration, the effect of increase and decrease in each of the tuning parameters with reference to the base case parameters is listed in Table A2 of Supporting Information by marking at the appropriate measure. For example, in GMC−GMC configuration, with GMC as master loop, high values of k1 are found to result in high overshoot in the initial time period, high undershoot at the switch between phase 2 and phase 3, and also in slightly oscillatory output profiles. On the other hand, low values of k1 are unable to lead the output to reach the set point resulting in offset during phases 2 and 3. In the case of integral gain (k2), a value of zero is found to result in the best output response. Slight increase in k2 from zero results in more oscillations in the output response. For the slave GMC loop, the effect of increase in kc1 is similar to that of decrease in k1 (master GMC loop), whereas the effect of kc2 is similar to that of k2. In the case of PI−GMC, the parameters are found to be very sensitive, leading to sustained oscillations in the temperature response with minor variations in the parameters. M



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Figure 12. Comparison of performance of GMC−GMC cascade controller with unmeasured monomer feed disturbance for summer batch 1: (a) reactor temperature profiles for product A, (b) input profiles of slave control for product A, (c) reactor temperature profiles for product B, and (d) input profiles of slave control for product B.

winter runs over five batches for products A and B are summarized in Table 2. The performance of different controllers is also illustrated in Figures 2−9, where the output and input trajectories of master and slave controllers for a few typical cases for product A and product B in both the seasons are plotted. In all the figures, the output reactor temperature T and manipulated input mean jacket temperature Tjset profiles for the master controller are represented by (a) and (b), whereas the output and input profiles of the slave controller, namely, the mean jacket temperature Tj along with Tjset (set point for slave loop) and the valve opening c(t) are represented by (c) and (d). Figures 2−9 clearly illustrate that, throughout the batch duration, there are no constraint violations (requirement being maintenance within ±1 °F) with GMC−GMC, and minor constraint violations initially and

minor offset in phase 2 with GMC−PI, and the corresponding input profiles for GMC−GMC and GMC−PI have oscillations of small amplitude, whereas in the case of PI−GMC and PI−PI, reactor temperature is found to exhibit constraint violations and jacket temperature and the corresponding input profiles have oscillations of very large amplitude. The results clearly illustrate that PI−GMC has not resulted in any improvement with respect to minimizing constraint violations in reactor temperature or in set point tracking performance or in minimizing input oscillations compared to GMC−GMC and GMC−PI. Therefore, master control with PI (i.e., PI−GMC and PI−PI) is not recommended for this benchmark problem based on the results obtained. On the other hand, PI control in slave loop with master control as GMC (i.e., GMC−PI) has exhibited slightly more oscillations in output response compared to N



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Figure 13. Comparison of performance of GMC−GMC cascade controller with unmeasured impurity factor disturbance for summer batch 1: (a) reactor temperature profiles for product A, (b) input profiles of slave control for product A, (c) reactor temperature profiles for product B, and (d) input profiles of slave control for product B.

the positive direction and 0.5 °F in the negative direction from the set point (+0.083 K and −0.27 K). Recently, Vasanthi et al.25 have applied a self-tuning control based master controller with a PI based slave controller for this system where they have reported that the variation of the reactor temperature error lies within ±0.6 K (approximately ±1 °F), which is larger in amplitude compared to the present study. Analysis of this process with respect to internal dynamics is also attempted to illustrate their stability. The number of states in this system is 4. With respect to control, the temperature and mean jacket temperature are considered as the outputs for the master and slave loops, respectively. In addition to these states, the monomer composition in the reactor is the additional state dependent only on the reactor temperature and can

GMC−GMC and also the rise time for tracking reactor temperature set point is more than with GMC−GMC. Moreover, GMC as master loop as well as in slave loop gives tighter temperature control compared to all other control schemes. The possible reason for better performance of GMC−GMC is its ability to handle nonlinearity in the process dynamics, effect of delays in the recirculation loop, and the interaction between the master and slave loops. To further illustrate the performance of GMC−GMC cascade controller, reactor temperature error profiles for both product A and product B are represented in Figure 10 which show that the variation of the reactor temperature error with time for product A is in the range of ±0.5 °F from the set point (±0.27 K) whereas for product B it is in the range of 0.15 °F in O



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Figure 14. Comparison of performance of GMC−GMC cascade controller with measured monomer feed disturbance for summer batch 1: (a) reactor temperature profiles for product A, (b) input profiles of slave control for product A, (c) reactor temperature profiles for product B, and (d) input profiles of slave control for product B.

different magnitudes in both the directions (±). Further, the performance of the controllers is also evaluated for the presence of measurement noise in the reactor temperature and three cases of process/model mismatch in delay times in the recirculation loop. The RMSOD and NRMSID values for the runs considering all these disturbances are reported for products A and B in Table 3. For the purpose of comparison, the base case results are also reported in this table. The controller tuning parameters employed in all the cases are the same as for the base case. Table 3 clearly illustrates that the proposed GMC−GMC cascade control strategy is able to handle all the disturbances and is found to exhibit reasonably good performance in all the cases considered. The effect of disturbance rejection is also illustrated in Figures 12−16, where the output and input trajectories for

be considered as the state exhibiting internal dynamics. For a typical batch of products A and B using the best combination (GMC−GMC), the time-dependent profiles of the monomer composition are plotted in Figure 11, which clearly indicates that the internal dynamics is stable and also that there are no problems encountered due to the presence of internal dynamics. In addition to the variations in seasons and heat transfer coefficients expressed as batch to batch variations in the above discussion for both the products, the performance of the best controller, namely, GMC−GMC, is also evaluated in the presence of a few other disturbances for the first batch of summer for both the products. The unmeasured and measured disturbances considered are the impurity factor in the rate expression and load disturbance in monomer feed flow rate of P



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Figure 15. Comparison of performance of GMC−GMC cascade controller with measured impurity factor disturbance for summer batch 1: (a) reactor temperature profiles for product A, (b) input profiles of slave control for product A, (c) reactor temperature profiles for product B, and (d) input profiles of slave control for Product B.

measured, the responses obtained for GMC−GMC are almost similar to the base case with no disturbance, as illustrated in Figure 14a [product A], Figure 14c [product B], Figure 15a [product A], and Figure 15c [product B], since the disturbance measurement is also incorporated into the control action computation. The input profiles in all the figures (illustrated in (b) and (d) in each figure) indicate negligible variations compared to the respective base cases. Figure 16a,c illustrates that there is no visible difference in the response with measurement noise for both products A and B, whereas Figure 16b,d shows that the input oscillations are slightly more compared to the base cases. An analysis of the results obtained for the multiproduct semibatch polymerization reactor control problem shows that the cascade control schemes with GMC−GMC and GMC−PI

GMC−GMC controller for product A and product B are plotted. In all the figures, the output reactor temperature T profile of the master controller and input profile of the slave controller for product A are represented by (a) and (b), whereas the output reactor temperature T profile of the master controller and input profile of the slave controller for product B are represented by (c) and (d). Figure 12a,c illustrates that there is no visible difference in the response with the variation of unmeasured disturbance in monomer feed flow rate for products A and B by GMC−GMC, whereas with the variation of unmeasured disturbance in impurity factor for products A and B, Figure 13a,c indicates that there is a partial offset in control performance by GMC−GMC (but within the acceptable bounds). On the other hand, when the variations in monomer feed flow rate and impurity factor are assumed to be Q



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Figure 16. Comparison of performance of GMC−GMC cascade controller with measurement noise for summer batch 1: (a) reactor temperature profiles for product A, (b) input profiles of slave control for product A, (c) reactor temperature profiles for product B, and (d) input profiles of slave control for product B.

controlling the reactor temperature at the desired set point, i.e., GMC−GMC, GMC−PI, PI−GMC, and PI−PI. The effects of different controller tuning parameters for master and slave controller are evaluated. The order for best performance to worst performance in terms of output tracking as well as minimum input oscillations is found to be GMC−GMC, GMC−PI, PI−PI, and PI−GMC, as illustrated by the RMSOD and NRMSID values over 10 batches (five in summer and five in winter, each with a different fouling factor).The effect of other disturbances, such as variations in impurity factor, feed flow rate, presence of measurement noise, etc., on control performance is evaluated for the both the products with GMC−GMC, and the results indicate a reasonably good control performance. For achieving a tighter reactor temperature control GMC−GMC cascade controller strategy is preferable, and the results illustrate that the variation in the

have exhibited better performance than PI−GMC and PI−PI control schemes. Further, the GMC−GMC control scheme has exhibited better performance compared to GMC−PI in all the cases considered. Although GMC has been chosen as the exact model based control scheme in the present study, the proposed schemes are quite general and can easily be extended to other nonlinear model based control approaches.

6. CONCLUSIONS In this study, set point tracking of reactor temperature of the industrial multiproduct semibatch polymerization reactor challenge problem is considered. The cascade control scheme is used to control the reaction temperature in the presence of disturbances, i.e., change in heat transfer characteristics during the run, change in ambient temperature from batch to batch, etc. Different cascade configurations are considered for R



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Greek letters

reactor temperature error over the entire batch duration is minimum. For product A it is in the range of ±0.5 °F from the set point (±0.27 K), whereas for product B it is in the range of 0.15 °F in the positive direction and 0.5 °F in the negative direction from the set point (+0.083 K and −0.27 K), falling well within the range of ±0.6 K, although the problem has been benchmarked as a challenge problem for control.

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REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

Tuning parameter details and comparison of control performances for products A and B shown in Tables A1−A4. This material is available free of charge via the Internet at http:// pubs.acs.org.

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μ = product viscosity (cP) θ1 = transport delay in jacket (min) θ2 = transport delay in recirculation loop (min) τp = heating/cooling time constant (min)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected], [email protected]. Tel.: ++9140-27193121. Fax: ++91-40-27193626. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge CSIRXII Plan Project, INDUS MAGIC, for the financial support, and the second author acknowledges CSIR, New Delhi, India, for financial support through a fellowship.

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NOMENCLATURE (UA)loss = heat loss to environment per unit temperature (Btu/(min·°F)) Tj = average jacket temperature (°F) −ΔHp = heat of polymerization (Btu/(lb·mol)) A = jacket heat transfer area (ft2) c(t) = slave controller output (0−100%) Cpi = specific heat of component i (Btu/(lb·°F)) Cpc = heat capacity of coolant in jacket (Btu/(lb·°F)) f = fraction of solids FM = molar flow rate of monomer into the reactor ((lb·mol)/ min) h = film heat transfer coefficient (Btu/(ft2·min·°F)) i = impurity factor k = first order kinetic constant (1/min) k0 = pre-exponential factor (1/min) k1 = proportional gain for master control k2 = integral gain for master control kc1 = proportional gain for slave control kc2 = integral gain for slave control Kp = heating/cooling process gain (°F/%) l/hf = fouling factor (h·ft2·°F)/Btu mc = mass of coolant in jacket (lb) mi = mass of component i in the reactor (lb), i = M for monomer, W for water, S for solids mM = mass flow rate of monomer (lb/min) MWM = molecular weight of the monomer mix (lb/(lb·mol)) nm = number of moles of monomer in the reactor (lb·mol) RP = rate of polymerization ((lb·mol)/min) T = reactor temperature (°F) Tamb = ambient temperature (°F) tf = batch time (min) U = overall heat transfer coefficient (Btu/(ft2·min·°F)) S



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(24) Graichen, K.; Hagenmeyer, V.; Zeitz, M. Feed forward control with on-line parameter estimation applied to the Chylla-Haase reactor benchmark. J. Process Control 2006, 16, 733−745. (25) Vasanthi, D.; Pranavamoorthy, B.; Pappa, N. Design of a selftuning regulator for temperature control of a polymerization reactor. ISA Trans. 2012, 51, 22−29. (26) Henson, M. A.; Seborg, D. E. Nonlinear Process Control; Prentice Hall: Englewood Cliffs, NJ, 1997. (27) Ziegler, J. G.; Nichols, N. B. Optimum settings for automatic controllers. ASME Trans. 1942, 64, 759−765.

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Comparative Study of Different Cascade Control Configurations for a

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