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Active Disturbance Rejection-Based High-Precision Temperature Control of a Semibatch Emulsion Polymerization Reactor Dazi Li,*,† Zheng Li,† Zhiqiang Gao,‡ and Qibing Jin† †

Institute of Automation, Beijing University of Chemical Technology, No.15, East Road of the North Third Ring-Road, Chao Yang District, Beijing 100029, PR China ‡ Department of Electrical and Computer Engineering, Cleveland State University, Cleveland, Ohio 44115, United States S Supporting Information *

ABSTRACT: A mathematical model can be used to mimic a real chemical, physical, or biological system. However, the process of idealizing the complicated real world into a relatively simple form requires making a set of assumptions and ignoring model variation, parameter changes, external disturbances, and noise. Therefore, no model can completely represent a real situation or process. On the other hand, if model uncertainties, parameter changes, and external disturbances are treated collectively as a “total disturbance” that is then estimated and canceled, extensive knowledge of the controlled object will no longer be required. In other words, good control performance is able to be achieved without a precise mathematical model. With this as the aim, active disturbance rejection control (ADRC), a modelfree control, presented in this paper is used to estimate and actively reject the inherent dynamic and external disturbances. In particular, it is shown in this paper that ADRC works well in the face of a nonlinear, time-varying process such as the Chylla−Haase semibatch polymerization reactor. This is because the problem of precise temperature regulation in the polymerization process can be reformulated as the problem of total disturbance rejection. The numerical results show the stability and robustness of the proposed method and better overall performance. (MPC)12−14 was proposed to address some rough problems by using a mathematical model of the process to predict the current values of the output variables and has already achieved extensively excellent control for polymerization reactors.15 However, it is not well-suited for multibatch or multiproduct situations where specialized knowledge that may not always be available is required. The aforementioned control methods rely significantly on a detailed mathematical model of a real process that may or may not be readily available. Even if a relatively accurate model can be obtained, the parameters of the model often change during different conditions and operations. To overcome such dependence on a detailed model, Han proposed and developed another kind of control method known as active disturbance rejection control (ADRC).16,17 The central idea of ADRC is to treat the collective effect of internal and external uncertainties as the total disturbance in the input channel. An extended state observer (ESO) is then used to estimate and cancel this total disturbance, thus reducing the plant to a cascade integral form that can be easily controlled by a proportional−derivative (PD) controller. Inspired by Han’s original idea, a novel cascade temperature control structure is proposed in this paper for temperature regulation in the Chylla− Haase reactor. It should be noted that the proposed method is by no means limited to this application. For example, composition and molecular weight are also of great significance in determining the quality of the polymer products, and the associated control issues have been addressed by some previous work. To keep

1. INTRODUCTION Batch and semibatch reactors1 are widely used in the chemical industry2 for the production of fine chemicals, pharmaceuticals, pigments, and polymers. For many polymerization reactions, reactor temperature has a great impact on the product quality, which may not be measured directly. As a result, temperature is selected as an indirect quality indicator and is taken into account for process control. The semibatch emulsion polymerization reactor depicted by Chylla and Haase,3 a control engineering benchmark problem, is selected as a case study, in which the temperature kinetic relations can fairly describe energy balances for the reactor as well as the recirculation loop. Consequently, the reactor temperature is the focus of study in the Chylla−Haase polymerization process. Because of process time variation, nonlinear behavior, model uncertainties, and external disturbances, temperature control of the Chylla−Haase reactor becomes an important and nontrivial problem.4,5 The traditional proportional-integral (PI) cascade control is widely used in the semibatch polymerization reactor because of the minimal requirement of process knowledge for its design. However, perfect control is unavailable because it fails to provide predictive control action to compensate for the effects of known disturbances. Although feed-forward control6 is used to improve controller performance, it can overcome only a certain type of disturbance. Online estimation7 may be accomplished using polymerization kinetic models in forms of extended Kalman filter (EKF),8 unscented Kalman filter (UKF),9 and artificial neural network (ANN)10,11 estimating the heat transfer coefficient and the reaction heat in the energy balance. These may be inadequate for the effective control of polymer properties especially in the case when the polymerization reactor exhibits either nonlinear dynamic behavior or strong interactions among the constraints on manipulated variables as well as controlled variables. Model predictive control © 2014 American Chemical Society

Received: Revised: Accepted: Published: 3210

August 4, 2013 January 19, 2014 February 7, 2014 February 7, 2014 dx.doi.org/10.1021/ie402544n | Ind. Eng. Chem. Res. 2014, 53, 3210−3221

Industrial & Engineering Chemistry Research

Article

Figure 1. Reactor schematic diagram (TT = temperature transmitter, TC = temperature controller).

monomer mass, instantaneous copolymer composition, and reactor temperature constant, a nonlinear dynamical system framework was used in a semibatch emulsion copolymer reactor.18 In another work,19 a reference controller was used for a continuous solution polymerization process. In addition, to address the problem of multiobjective optimization of simultaneous control of composition and molecular weight of copolymers, a nonlinear model predictive control (NMPC)20−22 scheme was implemented in the semicontinuous emulsion copolymerization reactors. An ADRC solution to these complex control problems in polymerization reactors, as an extension to the work in this paper, will provide a refreshingly new alternative. The polymerization processes can be characterized, for example, by strong coupling among the process variables such as composition, initiator level, molecular weight, etc., and ADRC provides a natural decoupling solution that could significantly reduced the difficulty in control design and tuning, leading to a practical engineering solution in the near future. This paper is organized as follows. First, in section 2 the Chylla−Haase benchmark problem is described and the process model and process uncertainties are introduced. In section 3, the temperature control problem is reformulated as the total disturbance problem and then shown to be well-solved by ADRC. In section 4, the numerical results are shown and discussed. Finally, conclusions are presented in section 5.

Water coolant is inserted into a recirculating water stream system during the cooling mode, whereas steam is injected during the heating mode. Polymer A is produced in five subsequent batches and is removed between the batches. The reactor is cleaned once after every fifth batch. In detail, the product recipe of polymer A for each batch consists of a heating period of 30 min, followed by a feed period of 70 min and a hold period of 60 min. Accordingly, the monomer feed ṁ inM(t) is set to 7.56 g/s during the feed period and 0 g/s during other periods. In order to produce qualified polymer A, there are two consecutive control tasks: heating the reactor to a constant setpoint before the monomer feed starts and keeping the reactor temperature T within a tolerance interval of ±0.6 K during the monomer feed in the subsequent specified hold period. A reference trajectory is planned for the above control tasks as a desired reactor temperature profile determined by the following equation:

2. DESCRIPTION OF THE CHYLLA−HAASE BENCHMARK PROBLEM 2.1. Chylla−Haase Emulsion Polymerization Process. A polymerization reactor is typically equipped with a cooling jacket because of strongly exothermic reactions in it. The jacket is always capable of removing the reaction heat to keep the reaction temperature within a tolerance interval, ensuring acceptable end product. The reactor temperature is controlled by manipulating the temperature of the coolant that is circulated through the cooling jacket of the reactor (Figure 1).

In this first interval where t ≤ t , the reactor is heated to Tset. When t > theat, the reactor temperature T is kept constant at Tset. The heat-up time instant theat is set to 30 min, corresponding to the time instant when the monomer feed starts. To accomplish the tasks, a stable and robust control is required because of the nonlinear exothermic reactions in the reactor. 2.2. Polymerization Reactor Model. Industrial emulsion polymers can have many monomers, multiple surfactants, and complex reactor feeding policies resulting in even more difficult

5 ⎧ ⎛ ⎞i ⎪Tamb + (Tset − Tamb) ∑ ai⎜ t ⎟ , t ≤ t heat ⎝ t heat ⎠ ⎪ ⎪ i=3 T *(t ) = ⎨ ⎪Tset , t > t heat ⎪ ⎪(a = 10, a = −15, a = 6) ⎩ 3 4 5

(1) heat

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dx.doi.org/10.1021/ie402544n | Ind. Eng. Chem. Res. 2014, 53, 3210−3221


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First, held constant during one batch yet random from batch to batch is the impurity factor i that models fluctuations in monomer kinetics and ranges from 0.8 to 1.2 in the polymerization rate Rp . Second, the fouling factor hf−1 increases from 0 to 0.704 m2 K kW−1 while the overall heat transfer coefficient U decreases from batch to batch because of a polymer film buildup on the wall, causing fouling of the reactor walls between cleaning periods. Third, the delay times θ1 and θ2 of the recirculation loop may vary by ±25% to the nominal values. In addition, the external disturbances have effects on the process, including monomer feed rate, environmental heat loss, catalyst activity, steam, and water coolant. The first three factors are relatively identified. However, the last two may change accidentally and have a much stronger influence on the polymerization process because vapor−liquid mixing is prone to causing liquid hammering, pipeline vibration, and inaccurate instrument measurement.

modeling. The simplified dynamic model of the semibatch polymerization reactor described by Chylla and Haase was dependent on some temperature-related kinetic relations. Because of these relations fairly describing the conversion and modeling the energy balance about the reactor and the recirculation loop, only temperature dynamics of the system has been considered as the control purpose.3 The temperature dynamics are described by energy balances, expressed by the following equations for the emulsion polymerization reactor and the cooling jacket: out

Tj̇

=

1 [ṁcCp,c(Tjin(t − θ1) − Tjout) + UA(T − Tj)] mcCp,c (2)

in

out

Tj̇ = Tj̇ (t − θ2) +

1 out [Tj (t − θ2) − Tjin(t ) + K p(c)] τp (3)

where the mean jacket temperature Tj is obtained by eq 4.

Tj = (Tjin + Tjout)/2

3. PROBLEM REFORMULATION AND SOLUTION 3.1. Reformulation of the Problem of Reactor Temperature Regulation. The original equation (eq 6) for describing the reactor dynamics can be rewritten as

(4)

The energy balances of the cooling jacket and the recirculation in loop with the outlet and inlet temperatures Tout j and Tj of the coolant C are given by eqs 2 and 3, respectively. From eq 3, the heating/cooling function Kp(c) is determined by an equalpercentage valve with position c(t) resulting in the following relationship:

Ṫ =

⎧ 0.8· 30−c /50(T − T in(t )), c < 50% inlet j ⎪ ⎪ c = 50% K p(c) = ⎨ 0, ⎪ ⎪ 0.15· 30(c /50 − 2)(Tsteam − Tjin(t )), c > 50% ⎩

− (UA)loss (T − Tamb) + Q rea] 1 UA ·Tj (i = M, P, W) + ∑i m i Cp,i

For c < 50%, ice−water with inlet temperature Tinlet is inserted in the cooling jacket, whereas a valve position c > 50% leads to a heating of the coolant by injecting steam of temperature Tsteam into the recirculating water−stream system. The energy balance of the emulsion polymerization reactor for the reactor temperature T can be written as follows:

f1 (T , w1 , t ) =

b1 =

(6)

The material balances for the monomer mass mM and polymer mass mP are shown in eqs 7 and 8. ṁ M =

ṁ Min(t )

− Rp

1 [ṁ Min(t )Cp,M(Tamb − T ) − UA ·T ∑i m i Cp,i

− (UA)loss (T − Tamb) + Q rea]

1 [ṁ Min(t )Cp,M(Tamb − T ) − UA(T − Tj) ∑i m i Cp,i

− (UA)loss (T − Tamb) + Q rea] (i = M, P, W)

1 UA ∑i m i Cp,i

Ṫ = f1 (T , w1 , t ) + b1·Tj

(8)

Therefore, T̈ can be obtained as

Reaction heat Qrea related to the rate of polymerization Rp can be calculated as

T̈ = f1′ (Ṫ , T , w1 , t ) + b1′·Tj

Q rea = −ΔH ·R p

(11)

(12)

Therefore, eq 10 can be written as

(7)

ṁ P = R p

(10)

The term in the square brackets and its multiplier factor in the right-hand side of eq 10 are denoted as f1(T,w1,t), while the multiplier factor of Tj is denoted as b1. Their detailed expressions are given by the following equations:

(5)

Ṫ =

1 [ṁ Min(t )Cp,M(Tamb − T ) − UA ·T ∑i m i Cp,i

(9)

(13)

(14)

where T is the output and Tj is the input; w1 represents external disturbances, and f1′ (Ṫ , T, w1, t) (notation f1′ ) is denoted as the “total disturbance” composed of model internal dynamics and external disturbances. f′1 is a dynamic function of T, Ṫ , and w1 over time. As it can be properly canceled, the parameter b′1 will then be the ratio between the output’s second derivative and the input. It is assumed that f1′ is differentiable; then eq 14 can also be written as

More specific information of the above equations is given in eqs A1−A9 of Supporting Information. The parameter values relating to the reactor model and polymer product A are all available in Tables S1 and S2 of Supporting Information. 2.3. Process Uncertainties. In this work, we took the emulsion polymerization process uncertainties into account. 3212



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Figure 2. Reactor temperature control scheme based on ADRC2−PI.

⎧ x1̇ = x 2 ⎪ ⎪ ⎪ x 2̇ = x3 + b1′·Tj ⎨ ⎪ x3̇ = f1′̇ ⎪ ⎪T = x ⎩ 1

⎧ d = rh1 ⎪ ⎪ d0 = dh1 ⎪ ⎪ y(k) = v1(k) − v(k) + h1v2(k) ⎪ ⎨ a = d 2 + 8r |y(k)| ⎪ 0 ⎪ ⎧ a −d ⎪ sign(y(k)), |y(k)| > d0 ⎪ v2(k) + 0 2 ⎪a = ⎨ ⎪ ⎪ |y(k)| ≤ d0 ⎩ v2(k) + y(k)/h1 , ⎩

(15)

As can be seen from eq 15, x1 and x2 represent the output T and its derivative Ṫ , respectively. The total disturbance is innovatively defined as an augmented state x3 = f1′. Also, the original nonlinear system described by eq 10 turns now into the form of a linear system described by eq 15 via some effective substitution and estimation. Remarks: (1) The significance of the reformulation is to transform the issue of modeling into the issue of estimation; in other words, precise temperature control task is now reformulated as a total disturbance rejection problem to address. (2) The linear transformation system can accurately follow the dynamics of the original nonlinear system if f′1 is accurately estimated. (3) Note that f′1 is defined as a state to be estimated; thus, there is no need to know its specific mathematical expression. 3.2. Brief Description of the ADRC Components. The ADRC technique was first proposed in the control framework17 by Han. It consists of tracking differentiator, extended state observer, nonlinear feedback, and total disturbance rejection, which are briefly described below. 3.2.1. Tracking Differentiator. The tracking differentiator (TD) in the discrete form is given as ⎧ ⎪ v1(k + 1) = v1(k ) + h · v2(k ) ⎨ ⎪ ⎩ v2(k + 1) = v2(k) + h·fhan(v1(k) − v(k), v2(k), r , h1)

The basic function of TD is to provide a filtered version of the input signal v and its derivative, which are used by the controller. 3.2.2. Extended State Observer. From the plant in eq 15, a third-order extended state observer (ESO) in the discrete form with a sampling period of h is given as ⎧ z1(k + 1) = z1(k) + h·(z 2(k) − β ·e(k)) 1 ⎪ ⎪ + = + · − ·g (e(k)) + b1′·Tj(k)) β z ( k 1) z ( k ) h ( z ( k ) ⎪ 2 2 3 2 1 ⎨ ⎪ z 3(k + 1) = z 3(k) − h·β3 ·g2(e(k)) ⎪ ⎪ e(k) = z (k) − x (k) ⎩ 1 1

(19)

where z1 = x̂1 (x̂1, the estimate of x1), z2 = x̂2, z3 = f′1̂ , and βi (i = 1, 2, 3) are adjustable parameters. zi(k + 1) (i = 1, 2, 3) are representatives of the next states value while zi(k) (i = 1, 2, 3) represent the current states value. g(e) is defined as α ⎧ ⎪|e| sign(e), |e| > δ g (e) = fal(e , α , δ) = ⎨ ⎪ 1−α ⎩ e/δ , |e| ≤ δ

(20)

where δ is the threshold value and α is a tuning parameter. In particular, g1(e) and g2(e) are special cases of g(e) for different values of α. Note that the inputs to the ESO are the system output T and the control signal Tj; the outputs of the ESO are estimates of T, Ṫ , and f′1, which is the “extended” state representing the unknown system dynamics. Refer to ref 17 for more details. 3.2.3. Total Disturbance Rejection and Nonlinear Feedback. The total disturbance rejection is achieved by the control law

(16)

Where v (in eq 18) is the set-point, v1 is the filtered version of v, and v2 = v̇1. h is the sampling period. h1 is called “filter factor”, r is adjustable parameter, and the function f han(•) is defined in the following equations: ⎧−rsign(a), |a| > d ⎪ fhan = ⎨ a | a| ≤ d ⎪− r , ⎩ d

(18)

Tj* = u0 − z 3/b1′

(17) 3213

(21)



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Figure 3. Comparisons of system response and the ESO estimation of ADRC1, batch 1.

When the above equation is substituted into eq 14 and it is assumed that the ESO functions well, z3 will track f1′(Ṫ , T, w1, t) closely and the system (eq 14) will be reduced to the ideal integral form

which can be controlled via the nonlinear feedback in the form of

T̈ = f1′ (Ṫ , T , w1 , t ) + b1′·Tj = f1′ (Ṫ , T , w1 , t ) + b1′·Tj* = f1′ (Ṫ , T , w1 , t ) − z 3 + b′1 ·u0 ≈ b1′·u0

u0(k + 1) = β01·fal(e1(k + 1), α1 , δ) + β02 ·fal(e 2(k + 1), α2 , δ)

(22) 3214

(23)



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Figure 4. Comparisons of system response and the ESO estimation of ADRC2, batch 1.

(the convergence has been proven23) and proportionalintegral (ADRC−PI) was applied to the reactor temperature regulation. From eq 13, a first-order incremental active disturbance rejection controller (ADRC1) can be obtained as

where fal(•) is a nonlinear function with the form of eq 20 and β01 and β02 are adjustable parameters; e1 = v1 − z1 and e2 = v2 − z2. 3.3. Solution to the Problem of Reactor Temperature Regulation. In this section, a cascade control based on ADRC 3215


Industrial & Engineering Chemistry Research ⎧ e(k) = z1(k) − T (k) ⎪ ⎪ z (k + 1) = z (k) + h· (z (k) − β · g (e(k)) 1 2 1 1 ⎪ 1 *(k)) b T + · ⎪ 1 j ⎪ ⎨ ⎪ z 2(k + 1) = z 2(k) − h· β2 ·g2(e(k)) ⎪ ⎪ u0 = β01·fal(v − z1, α1, δ0) ⎪ ⎪Tj*(k + 1) = Tj*(k) + u0 − z 2/b1 ⎩

Article

(24)

From eq 14, a second-order incremental active disturbance rejection controller (ADRC2) can be obtained as ⎧ e(k) = z1(k) − T (k) ⎪ ⎪ z1(k + 1) = z1(k) + h· (z 2(k) − β1· e(k)) ⎪ ⎪ z (k + 1) = z (k) + h·(z (k) − β · g (e(k)) 2 3 2 1 ⎪ 2 ⎪ + b1′·Tj*(k)) ⎪ ⎪ ⎪ z 3(k + 1) = z 3(k) − h· β3 · g2(e(k)) ⎨ ⎪ e1(k + 1) = v1(k + 1) − z1(k + 1) ⎪ ⎪ e 2(k + 1) = v2(k + 1) − z 2(k + 1) ⎪ ⎪ u0(k + 1) = β01· fal(e1(k + 1), α1, δ) ⎪ ⎪ + β02 ·fal(e 2(k + 1), α2 , δ) ⎪ ⎪Tj*(k + 1) = Tj*(k) + u0(k + 1) − z 3(k + 1)/b1′ ⎩

Figure 5. Viscosity μ (heat period, feed period, and hold period).

(25)

Note that the track signal v is identical to the desired trajectory T* (from eq 1) in this paper. ADRC2−PI cascade structure is shown as an example in Figure 2 where the ADRC2 is designed for the outer loop and a proportional-integral (PI) controller is designed for the inner loop. Remarks: (1) Incremental ADRC has the following advantages: First, malfunction will have minor impact on the equipment of the production process because of the mere incremental output. Second, it can reduce computational cost while improving the operating speed. Finally, smoother transitions are available in spite of some changes in certain batch operations. (2) h, r, h1, α, δ, β1, β2, β3, β01, and β02 are key parameters for control performance. They can be tuned separately in accordance with the three components of ADRC. In TD, the sampling period h of around 0.1 is usually sufficient. The amplification coefficient r, corresponding to the limit of acceleration, is around 100. The filter factor h1 determines the aggressiveness of the control loop and is usually selected as 0.1 or less. In the ESO, the parameter values of α and threshold δ are around 0.5 and 0.1, respectively. The observer gains β1, β2, and β3 are selected so that ESO is fast enough but not too sensitive to noise, following the pattern of β3 < β2 < 1.0 < β1. β01 and β02 are the nonlinear feedback gains which determine the tracking performance; they are both chosen to be less than 1. For more detailed information on tuning of ADRC, see refs 17 and 24. The parameters h, r, h1, α, δ, β1, β2, β3, β01, and β02 in this work were set to 0.1, 87, 0.05, 0.5, 0.12, 10.1, 0.5, 0.15, 0.35, and 0.05, respectively.

Figure 6. Overall heat transfer coefficient U for different batches (batches 1, 3, and 5).

4. RESULTS AND DISCUSSION 4.1. Estimation of Total Disturbance. In the Chylla− Haase engineering benchmark, the values of i (∈[0.8 1.2]) and h−1 f (∈[0 0.704]) were random. As can be seen from Figure 3a−c, the system dynamic function f1 (“total disturbance”) was accurately estimated by the augmented state z2 with ADRC1, z1 excellently reproduced T (the output) while they can track the desired profile T* closely, and the reference control signal T*j provided by the ADRC1 to the PI controller was given. Also from Figure 3a−c, it was observed that apart from some small oscillations at the turning points, ADRC1 drove the system successfully. Moreover, the results with ADRC2 shown in Figure 4a−c further confirm the effectiveness of the ADRC method. It should be emphasized that ADRC is independent of a precise mathematical model of the controlled plant, and it can estimate the unavailable states online with respect to polymerization systems containing multiple channels. Therefore, ADRC can well address not only the reactor temperature T but also more ample control issues of polymerizations. 3216



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Figure 7. Responses of the closed-loop control system (batches 1 and 5).

4.2. Robustness under Model Uncertainty. The heat transfer coefficient decreased significantly during a batch because of an increasing batch viscosity (as shown in Figure 5). Also, the overall heat transfer coefficient U decreased from batch to batch which (Figure 6), while the fouling factor h−1 f increased from 0 to 0.704 m2 K kW−1. This accounted for the fact that a polymer film builds up on the wall, increasing fouling of the reactor walls between cleaning periods. The process required a stable and robust temperature control because of nonlinear exothermic reactions and changes of heat transfer characteristics. Figure 7a,b shows the profiles of the reactor temperature T, absolute error |ΔT| of the desired and actual reactor temperature (|ΔT| = |T* − T|), valve position c, jacket temperature Tj, and material mass mM and mP. As can be seen in Figure 7a,b, the major changes in the temperature error and control valve steam position happened at 30 and 100 min, while the jacket temperature increased. At 30 min, the monomer was added into the reactor. Because the monomer was at ambient temperature whereas the reactor was at an operating temperature of Tset = 355.382 K, the reactor was instantaneously cooled by the feed monomer before the reaction started. At this point, there was a need to increase reactor temperature by heating the reactor jacket. After that, the exothermic reactions caused the heat accumulation inside the

reactor to be large. To compensate, there had to be reduction in the jacket temperature. Because the monomer feed was stopped abruptly at 100 min, the reaction came to an end resulting in a sudden increase in reactor temperature error. Afterward, to return reactor temperature to Tset, there was an increase in the jacket temperature by means of the steam inflow. The whole process was well-regulated by the cascade control strategy proposed by this paper. Strong robustness was achieved in suppressing oscillations in the valve and in maintaining acceptable reaction temperature despite the chemical reactions involved in strongly exothermic reactions. 4.3. Robustness under Time Delay and External Disturbance. The robustness of the proposed control schemes was illustrated via a case where a 9600 min duration sinusoid with amplitude (Am) around 18% of the maximum valve position was added as a control disturbance to valve position c. In addition, a +15% mismatch took place in delay time θ1 and a +25% mismatch took place in delay time θ2. The good performance and robustness of the proposed control schemes PI−ADRCi (i = 1, 2) are shown in Figure 8a−c, while the reactor temperature T(t) stayed well within the specified tolerance interval of ±0.6 K. 4.4. Performance and Efficiency of ADRC Compared to Other Methods. Temperature control of the emulsion polymerization reactor depicted by Chylla and Haase is very 3217



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Figure 8. Results of delay times θ1 +15%, θ2 + 25%, and added control disturbance.

other two methods are significantly dependent on an exact mathematical plant model. Therefore, ADRC is more applicable and beneficial in process control. In further investigation, two commonly used measures, integral absolute error (IAE) and integral square error (ISE) were used for the performance comparison of different methods. The errors are listed in Table 1. It can be seen that ADRC2 shows the best performance (ISE and IAE are both very low) compared with other methods. Although ADRC1 has ISE and IAE higher

challenging because of the complexity of the physical mechanisms, polymerization kinetics, and strongly exothermic reactions. Several control methods have been used in some previous contributions to deal with the benchmark problem. A model predictive controller combined with an unscented Kalman filter (UKF)9 was used for the estimation of the reaction heat and heat transfer coefficient. These variables can also be estimated by using adaptive feed-forward (AFF)6 control and artificial neural network (ANN).10 Figure 9a,c shows the reactor temperature T responses with four types of advanced-control methods in batch no. 1 and 5, respectively. Subfigures enlarged partially represent the periods of 1800−2500 s and 5800−7000 s. Figure 9b,d shows absolute values |ΔT| of the temperature error with these methods in batch no. 1 and 5, respectively (where maximum absolute errors at two peaks are marked). In Figure 9a,c, it is shown that the response curves of ADRC2 are always closest to the set-point of 355.382 K among these four methods. Temperature deviation of ADRC1 is slightly larger than that of AFF but significantly smaller than that of ANN. The settling time of ADRC2 is the shortest, while that of ANN is the longest. In addition, the settling time of ADRC1 is somewhat similar to that of AFF. From Figure 9b,d, the maximum absolute errors of the four methods are all lower than 0.5. It is worth noting that ADRC is a modelfree control technique without requiring any explicit process knowledge. However, the

Table 1. Integral Square Error (ISE) and Integral Absolute Error (IAE) of the Reactor Temperature ISE

IAE

control scheme

batch 1

batch 5

batch 1

batch 5

ANNa UKFb ADRC1c AFFd ADRC2e

58.5273 54.9608 53.8874 21.9267 8.4277

75.0668 85.6100 83.4680 24.2357 9.3756

456.9057 449.0182 348.9256 323.5384 148.7140

554.2970 579.0522 514.3549 330.0724 181.7531

a ANN = artificial neural networks estimator.10 bUKF = unscented Kalman filter.9 cADRC1 = the first-order incremental active disturbance rejection controller (this paper). dAFF = adaptive feedforward control.6 eADRC2 = the second-order incremental active disturbance rejection controller (this paper).

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Figure 9. continued

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Figure 9. Comparison results for polymer A (batches 1 and 5).

thank the editors and anonymous reviewers for their valuable comments and suggestions to help improve the quality of the manuscript.

than those of AFF as well as ADRC2, it has fewer tuning parameters and a simpler algorithm and is widely applicable to most issues.

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5. CONCLUSION This first section of this paper shows how the Chylla−Haase benchmark problem can be reformulated as a problem of total disturbance rejection. Then a method based on active disturbance rejection is proposed for the emulsion polymerization process to achieve a high-precision temperature control performance via accurate state estimation. ADRC can turn the original nonlinear system into a new linear control system by transforming the total disturbance, and it supplies an effective, simple, and robust control for systems. The effectiveness and robustness of ADRC−PI cascade control in dealing with the problem of model mismatch and disturbances are then demonstrated by the numerical results. In addition, compared to other methods, ADRC is independent of detailed plant models. Finally, the discovery in this paper can be beneficial to other problems because they share one key issue in common: how to handle the unknown dynamics in control systems.

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ASSOCIATED CONTENT

S Supporting Information *

Empirical relations for the polymerization rate (eqs A1−A4), overall heat transfer coefficient (eqs A5−A8), and jacket heat transfer area (eq A9); parameter values of the reactor model (Table S1); and data of polymer A (Table S2). This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Tel: +86-10-64434930. Fax: +86-10-64437805. E-mail: lidz@ mail.buct.edu.cn. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was supported by the National Natural Science Foundation of P.R. China (Grant 61273132). The authors also 3220

NOMENCLATURE mM, mP, mW = mass amount monomer, polymer, and water (kg) mC = mass of water in the jacket (kg) ṁ inM = monomer feed rate (kg s−1) MWM = molar mass (kg mol−1) ρi (i = M, P, W, C) = specific mass of the different substances (kg m3) CP,i (i = M, P, W, C) = specific heat of the different substances (kJ kg−1 K−1) A = heat transfer area (m2) U = overall heat transfer coefficient (kW m−2 K−1) (UA)loss = heat loss coefficient (kW K−1) Tinlet = cooling water temperature (K) Tsteam = steam temperature (K) T = reactor temperature (K) Tout j = jacket outlet temperature (K) Tinj = jacket inlet temperature (K) Tj = jacket average temperature (K) Tamb = ambient temperature (K) Twall = wall average temperature (K) ΔH = heat of reaction (kJ kmol−1) Rp = reaction rate (kg s−1) Qrea = reaction heat (kW) i = impurity factor c = heating/cooling usage (%) Kp(c) = heating/cooling function (K) hf = fouling factor (kW m−2 K−1) h = heat transfer coefficient (kW m−2 K−1) μwall = wall viscosity (kg m−1 s−1) k = first-order kinetic constant (s−1) μ = batch viscosity (kg m−1 s−1) f = auxiliary variable R = natural gas constant (kJ kmol−1 K−1) B1 = reactor bottom area (kg m−3) P = jacket perimeter (m) dx.doi.org/10.1021/ie402544n | Ind. Eng. Chem. Res. 2014, 53, 3210−3221


Article

(23) Guo, B. Z.; Zhao, Z. L. On convergence of the nonlinear Active Disturbance Rejection Control for MIMO systems. SIAM Journal on Control and Optimization 2013, 51, 1727−1757. (24) Peng, C.; Tian, Y. T.; Bai, Y.; Gong, X.; Zhao, C. J.; Gao, Q. J.; Xu, D. F.ADRC trajectory tracking control based on PSO algorithm for a quad-rotor. In 2013 8th IEEE Conference on Industrial Electronics and Applications (ICIEA), Melbourne, Australia, June 19−21, 2013; 800− 805

B2 = jacket bottom area (m) τp = heating/cooling time constant (s) θ1, θ2 = time delays in jacket and recirculation loop (s) T* = desired reactor temperature trajectory (K)

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