Dynamic Modeling and Optimal Control of Batch Reactors, Based on

ARTICLE pubs.acs.org/IECR

Dynamic Modeling and Optimal Control of Batch Reactors, Based on Structure Approaching Hybrid Neural Networks J. Wang, L. Lin Cao,* H. Yan Wu, X. Guang Li, and Q. Bing Jin Institute of Automation, College of Information Science and Technology, Beijing University of Chemical Technology, Beijing, People’s Republic of China ABSTRACT: A novel Structure Approaching Hybrid Neural Network (SAHNN) approach to model batch reactors is presented. The Virtual SupervisorArtificial Immune Algorithm method is utilized for the training of SAHNN, especially for the batch processes with partial unmeasurable state variables. SAHNN involves the use of approximate mechanistic equations to characterize unmeasured state variables. Since the main interest in batch process operation is on the end-of-batch product quality, an extended integral square error control index based on the SAHNN model is applied to track the desired temperature profile of a batch process. This approach introduces model mismatches and unmeasured disturbances into the optimal control strategy and provides a feedback channel for control. The performance of robustness and antidisturbances of the control system are then enhanced. The simulation result indicates that the SAHNN model and model-based optimal control strategy of the batch process are effective.

1. INTRODUCTION The batch process is an important form of production in the chemical industry for realizing frequently changing production plans required by product diversity. The production of lowvolume, high-cost materials obtained via the batch mode has increased. Exothermic operation is common for batch reactors and yields good desired product properties. Furthermore, the reactor should be heated from the initial temperature as rapidly as possible, to maximize productivity in a fixed batch time. However, heating the reactor too rapidly can lead to undesired properties and significant temperature overshoot of the setpoint and can even lead to thermal runaway. Temperature overshoot is undesirable for some polymerization systems. Thus, tight control over reactor temperature is crucial for product quality control. The lack of online sensors for the rapid measurement of conversion of reactants and other properties makes direct control of the desired productivity difficult. Therefore, batch reactors are often controlled by following prespecified operating profiles such as temperature setpoint programs with special variable gain controllers. Such operating policies can be obtained using actual plant operation experience or through model-based optimization calculations, using the maximum principle or control vector parametrization techniques.13 In many of these optimization studies, the calculated temperature trajectories are suboptimal, because of modeling error, and such profiles were assumed to be well-tracked in actual operation. However, several known factors such as delayed measurements make it difficult to control the temperature in batch reactors. In light of these factors, control of reactor temperature during the initial heat-up period and the entire batch reaction should be conducted carefully and efficiently. Conventional industry reactor temperature controllers are often tuned to avoid temperature overshoot during the heatup period, resulting in slow and safe heating but long batch times. Cott et al. used a generic model control algorithm as a controller to track the reactor temperature setpoint.4 A three-term r 2011 American Chemical Society

difference equation and exponential filters are provided as the estimator to estimate the online heat release. Furthermore, Kershenbaum et al. considered the same reaction scheme that Cott et al. used and the generic model control algorithm for the controller.5 However, the extended Kalman filter was used as the online heat-release estimator. In their work, different types of controllers, such as dual-mode controllers (PI and PID) and a generic model controller, are designed to track the optimal temperature profiles of exothermic batch reactors. In the generic model control formulation, a neural network technique is used for online estimation of the heat release. Effective control, optimization, or monitoring of a process needs an accurate model to characterize the nonlinear dynamic behavior. These targets may be achievable through a reliable process model formulated classically on the basis of mass and energy balances for the chemical processes. However, development of a detailed mechanistic model is usually not feasible, because of frequently changing product grades and specifications reflecting the dynamic market demands. Thus, empirical models based on process operational dates are promising alternatives to mechanistic models. Empirical models, which must have sufficient representation capability to enable the underlying system characteristics to be approximated with an acceptable accuracy, can generally be developed very quickly without requiring detailed insight into the processes. Several types of data-based empirical models have been applied to batch reactors, including the artificial neural network (ANN) approach and the multivariate statistics approach.68 Neural networks can approximate any continuous nonlinear functions. They represent a promising tool for identifying empirical models, which may lead to improved process Received: July 19, 2010 Accepted: March 31, 2011 Revised: March 25, 2011 Published: March 31, 2011 6174

dx.doi.org/10.1021/ie1015377 | Ind. Eng. Chem. Res. 2011, 50, 6174–6186

Industrial & Engineering Chemistry Research optimization and control performance.9,10 However, the main drawbacks of ANNs, such as no guaranteed strategy for defining networks structure, long computational times, large amounts of data, and poor capacity for generalization, have been identified. The hybrid neural network is an alternative neural network modeling approach, which incorporates the available knowledge about the process being modeled into a neural network that serves as an estimator of unmeasured process parameters, which are difficult to determine. This hybrid model has better properties than standard black-box neural network models, because of maximum use of all the prior knowledge available and avoidance of time-consuming development of mechanistic models with an unknown reaction mechanism and postulated reaction orders. A distinct class of hybrid models utilizes ANNs to correct the shortcomings in available mechanistic models. A previous study proposed an integrated neural network approach involving offline training of a network to predict differences between the firstprinciples model and observed plant behavior.11 A continuous polymerization process is considered and plant input data are fed in parallel to both the first-principles and the neural network models. The networks outputs are added to the first-principles model outputs in order to make suitable corrections. A hybrid modeling approach was compared to classic linear dynamic models and nonlinear models.12 A hybrid combination of an inverse neural network with a first-principles model was studied to control a nonlinear semibatch polymerization directly.13 A gray-box modeling method was used to model and control the chemical processes with a methodology maximizing the use of prior process knowledge.14 In addition, neural networks can be used to learn the parameter functions in dynamic models. Thus, the adoption of prior process knowledge (usually in the form of conservation equations), coupled with the approximation capabilities of neural networks constitute the neural-network parameter-function modeling approach. A modeling of the batch-fed penicillin fermentation process was studied.15 An ANN is applied to correct the kinetic parameters calculated by empirical equations before these parameters are used in the balance equations. A neural-network rate-function modeling approach was proposed to obtain a reliable dynamic network model characterizing the complex MMA polymerization reactions carried out in a batch reactor.16 However, the hybrid network model developed here was built only on measurable states and did not include the state variables, which are difficult to measure, such as initiator concentration during the polymerization reaction. Furthermore, the modeling methodology of this neural-network ratefunction model was extended and a hybrid neural-network ratefunction modeling approach was proposed to tackle this problem, which is usually encountered in complex reactor systems.17 This paper concentrates on a novel hybrid neural network modeling method for a class of chemical processes called the Structure Approaching Hybrid Neural Network (SAHNN), which considers nonlinear elements of batch processes as augmented input variables. Therefore, the nonlinear process can be treated as a linear system. Based on this method, a detailed analysis of the SAHNN approach has been introduced to model the batch reactor. Moreover, a novel method called the Virtual SupervisorArtificial Immune Algorithm is presented, because many state variables in chemical

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Figure 1. Batch reactor system.

processes are difficult or impossible to be measured. By assuming that the rates of unmeasured state variables are located along a state trajectory by a collection of timedependent experimental data from different batch reactors, the SAHNN is trained to model the system with partial unmeasured state variables. To determine the optimal temperature profile that maximizes the conversion of the desired product, an optimal control strategy is proposed with a fixed batch time. The profile thus obtained is used as the setpoint to be tracked by the controller with an expanded integrated square error (EISE) index based on the SAHNN model. Finally, a model-based optimal control scheme with EISE index for batch processes is proposed in order to overcome the detrimental effects of model mismatches and unknown disturbances. Since the calculated real-time model error is introduced into the control index, a feedback channel is added to the modelbased controller, enhancing the robustness and anti-interference ability.

2. DYNAMIC MODELING BASED ON THE STRUCTURE APPROACHING HYBRID NEURAL NETWORKS (SAHNN) MODEL 2.1. Batch Reactor. The exothermic batch reactor in the Cott et al.4 report consisted of a batch reactor and a jack cooling system. A typical diagram of this system is shown in Figure 1. A proportional control system for the jacket temperature has already been designed. A split range control27 was used to switch the cooling water and steam, according to the range of manipulated variables. The reactor temperature (Tr) is controlled by manipulating the setpoint of the jacket temperature (Tjsp). This is a type of cascade control, and the purpose of this work is to design the outer-loop controller. Two exothermic reactions of four components (A, B, C, and D) are described as follows: k1

ðRxn 1Þ

k2

ðRxn 2Þ

A þ B sf C A þ C sf D

where A and B are raw materials, and C and D are the desirable product and undesirable byproduct, respectively. The rate constants k1 and k2 are temperature dependent, according to the Arrhenius relation. The general operating objective is to achieve a good conversion of product C while minimizing the undesired production of D. 6175

dx.doi.org/10.1021/ie1015377 |Ind. Eng. Chem. Res. 2011, 50, 6174–6186

Industrial & Engineering Chemistry Research

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Table 1. Process Parameters MWA = 30 kg kmol1

ΔH2 = 25105 kJ kmol1

1

k11 k21 k12 k22

MWB = 100 kg kmol

1

MWC = 130 kg kmol

1

MWD = 160 kg kmol

1

ΔH1 = 41840 kJ kmol

Fj = 0.348 m3 min1

U = 40.842 kJ (min m2 °C)1

3

= 20.9057

F = 1000 kg m

Cp A = 75.31 kJ (kmol °C)1

= 10000

r = 0.5 m

Cp B = 167.36 kJ (kmol °C)1 Cp C = 217.57 kJ (kmol °C)1

3

Vj = 0.6912 m

= 38.9057

1

Cp D = 334.73 kJ (kmol °C)1

Cp j = 1.8828 kJ (kg °C)

= 17000

The material balance equations for the reactions can be constructed as follows: dCA dt dCB dt dCC dt dCD dt

¼ k1 CA CB k2 CA CC ¼ k1 CA CB ¼ þ k1 CA CB k2 CA CC

ð1Þ

¼ þ k 2 CA CC

The energy balance is described by the following equation: ΔH1 ðk1 CA CB Þ ΔH2 ðk2 CA CC Þ þ UAðTj Tr Þ dTr ¼ dt Mr Cpr

Figure 2. Linear representation for a nonlinear system.

Hence, the system is given as n X_ ¼ L1 ðRÞþ L2 ðUÞ, R ¼ f ðXÞ, Y ¼ L3 ðXÞ

ð2Þ Reaction rates R1 and R2 are given as follows: R1 ¼ k1 CA CB

ð3Þ

R2 ¼ k2 CA CC

ð4Þ

k1 ¼ exp

k2 ¼ exp

k11

k12

k21 Tr þ 273:15 k22 Tr þ 273:15

! ð5Þ

where X is a vector of state variables, U a vector of external inputs, Y a vector of output variables, f a nonlinear function, and R a vector of nonlinear output variables. The terms L1, L2, and L3 represent three different linear functions. Furthermore, the above model in eq 8 can also be written in state space form, where the main nonlinear elements are separated and regarded as the augmented input vector: _ XðtÞ ¼ AðUðtÞ, PðtÞÞXðtÞ þ BðUðtÞ, PðtÞÞUAug ðtÞ Y ðtÞ ¼ CðUðtÞ, PðtÞÞXðtÞ

!

_ XðtÞ ¼ f ðXðtÞ, UðtÞ, PðtÞ, tÞ

"

ð6Þ UAug ðtÞ ¼

where Ci (where i = A, B, C, D) is the concentration of the reactant components, Tr the reactor temperature, and Tj the jacket temperature. The model parameters are given in Table 1, and more details about this model are given in the work of Cott et al.4 The initial conditions of CA, CB, CC, and CD are 12, 12, 0, and 0 kmol, respectively. The initial values of the jacket and reactor temperatures are set to 20 °C. The manipulated variable (Tj) and the controlled variable (Tr) generally vary in the range of 20120 °C. Generally, most chemical processes describing the various process components can often be represented using the principles of conservation of mass and energy. In many cases, process modeling can be described in state space form: ð7Þ

where X ∈ Rm1 is a vector of state variables, U ∈ Rn1 is a vector of external inputs, and P ∈ Rrs is a vector of parameters. For some chemical processes, the main nonlinear elements that are often in static form can be separated and treated as the insider variables. Thus, the system can be represented in linear form, as shown in Figure 2.

ð8Þ

RðXðtÞ, PðtÞÞ UðtÞ

ð9Þ

# ð10Þ

where UAug(t) is the augmented input. Thus, treating the nonlinear elements in eqs 3 and 4 as the augmented input variable, the batch reactor model in eqs 1 and 2 can be written in linear state space form: 32 3 2 2: 3 0 0 0 0 0 CA 76 C A 7 6 6: 7 0 0 0 0 0 7 6 CB 7 6 6CB 7 76 7 6 6: 7 76 C 7 6 0 0 0 0 0 6CC 7 ¼ 6 76 C 7 6: 7 6 0 0 0 0 0 7 6C 7 6C 7 6 4 :D 5 UA 7 54 D 5 4 0 0 0 0 Tr Tr M r Cp r 3 2 1 1 0 72 3 6 0 0 7 R 6 1 7 17 6 1 1 0 76 6 7 þ6 ð11Þ 74 R2 5 6 7 6 0 1 0 Tj 7 6 ΔH ΔH UA 5 4 1 2 M r Cp r M r Cp r M r Cp r 6176



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Figure 3. Topology of the Structure Approaching Hybrid Neural Network (SAHNN).

2.2. Modeling Based on Structure Approaching Hybrid Neural Networks. Based on the above analysis, a novel hybrid

neural network called the Structure Approaching Hybrid Neural Network (SAHNN) is proposed, as illustrated in Figure 3. The main idea of the SAHNN model is to obtain the structural strategy of neural networks by making maximum use of all the prior process knowledge available. Based on eqs 3 and 4, the main nonlinear elements of the system are often in static form. Fully utilizing the structural information of the system, a static B-spline neural network18,19 is selected as NN1, by which the nonlinear part is modeled. Considering the dynamics behavior of the system (eq 9) and the diagonal form of the linear matrix A, a diagonal recursive neural network20 is selected as NN2 to describe the linear dynamic part. The characteristics and detailed analysis of SAHNN modeling approach is given in Cao et al.21 Based on Figure 3, the SAHNN model can be presented as follows: ( Xðk þ 1Þ ¼ AXðkÞ þ W H W I UðkÞ ð12Þ Y ðkÞ ¼ W O XðkÞ where X = [x1,x2, 3 3 3 ,xn]T and Y = [y1,y2, 3 3 3 ,yp]T are partial states and output variables, respectively. Input variables U = [u1, 3 3 3 ,us,usþ1, 3 3 3 ,um]T consist of nonlinear elements u1∼s and external actual input usþ1∼m. Ann = diag{a1, 3 3 3 ,an} is the diagonal weight of NN2. WI ∈ Rhm,WH ∈ Rnh, and WO ∈ Rpn are weight matrices of the input layer, hidden layer, and output layer of NN2, respectively. h is the node number of hidden layer of NN2, and Bn is the number of B-spline base functions. Every state variable xl (for l = 1, ..., n) in the SAHNN model (eq 12) is rewritten as xl ðk þ 1Þ ¼ al xl ðkÞ þ þ

h

s

h

m

∑ ∑ dj, i, l ui ðkÞ j¼1 i¼s þ 1

Bn

Δ gj, i, b, l Bb ðxðkÞÞ ¼ jTl ðkÞθl ðfor l ¼ 1, :::, nÞ ∑ ∑ ∑ j¼1 i¼1 b¼1

ð13Þ yp ðkÞ ¼

n

∑l WlO, pxl ðkÞ

ð14Þ

where θl ¼ ½al , d1, s þ 1, l , 3 3 3 , d1, m, l , d2, s þ 1, l , 3 3 3 , d2, m, l , 3 3 3 , dh, s þ 1, l , 3 3 3 , dh, m, l , g1, 1, 1, l , 3 3 3 , g1, 1, Bn , l , 3 3 3 , g1, s, 1, l , 3 3 3 , g1, s, Bn , l , 3 3 3 , gh, 1, 1, l , 3 3 3 , gh, s, Bn , l T

ð15Þ

jl ðkÞ ¼ ½xl ðkÞ, us þ 1 ðkÞ, 3 3 3 , um ðkÞ, 3 3 3 , us þ 1 ðkÞ, 3 3 3 , um ðkÞ, B1 ðXðkÞÞ, 3 3 3 , BBn ðXðkÞÞ, 3 3 3 , B1 ðXðkÞÞ, 3 3 3 , BBn ðXðkÞÞT ð16Þ As mentioned in Cao et al.,21 if the output variables (some state variables) are measurable, the estimated parameter vector θl and WO can be identified using a recursive least-squares methodology. 2.3. Modeling with Partial Unmeasured State. However, for some chemical processes, partial output variables (state variables) cannot be measured or be measured with delay under some conditions, such as the concentration of some reactants and the rate constant. The neural network has no supervisor to learn from and the conventional training way is ineffective. Thus, the rates of state variables are assumed to be located along a state trajectory x~= g(t), using a collection of time-dependent experimental data of different batch reactors, as studied in Edgar et al.22 A more functional form of g(t) could be found in that work that was adopted as a potential candidate. Usually, the initial and final state of the key species can be obtained in each batch experiment.23 Furthermore, the infrequent measurements of key species may be available. Thus, the general tendency of key states can be determined by adopting the proper potential candidate and infrequent measurements. Using this general tendency trajectory as virtual supervisors, an improved training algorithm for the SAHNN model based on an artificial immune algorithm called Virtual SupervisorsArtificial Immune Algorithm (VS-AIA) is developed. In this algorithm, the vaccine is initially set as the special individual with optimal performance. During the iterative search for optimal solutions, each individual obtains information from the vaccine and is altered. At the same time, the vaccine is replaced by the individual with the optimal performance in the new generation of population. Thus, all individuals and the vaccine are updated and became closer to optimal solutions. With this method, VS-AIA is developed to train the SAHNN model with partially unmeasured states. 6177



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In the general genetic algorithm, an individual is composed of undetermined weights θl in the SAHNN model, and the vaccine has the latest optimal value of weights. The population is composed of many individuals. The VS-AIA process is shown as follows. Step 1. Initializing Virtual Supervisors: Only general changing trends (ascending or descending) based on prior knowledge for unmeasured states are given, and sets of them are given as the initial values of the supervisors. Step 2. SAHNN Training: In the training, the measured states are used directly, while the virtual supervisors are used to place the unmeasured states. The SAHNN model is then trained with the normal method to get the weights, which are seen as the vaccine. Step 3. Individuals Evolving: Individuals in the population are altered by crossover, mutation, and vaccination. In vaccination, na individuals (where na is less than or equal to the population size) obtain information partly or fully from the vaccine. Some or all bits in individual genes become identical with that of the vaccine. Step 4. Selection Individuals: The performance of the altered individuals is tested by data validation of measured states xms(t), i.e., min J ¼

t ¼n

t ¼n

∑ jj^x ms ðtÞ xms ðtÞjj or t∑¼ 1 t¼1

^xms ðtÞ xms ðtÞ

2

ð17Þ New individuals are then chosen by immune selection, which includes tow operators: immune test and annealing selection. For the immune test, altered individuals with worse performance than that of their parent are replaced by their parent. And for the annealing selection, altered individuals become the new generation of population with a probability of Pi ¼

expðJi =Tk Þ n

∑

i¼1

expðJi =Tk Þ

ð18Þ

where Ji is the performance of individual i, n the size of population, and Tk the annealing temperature. Step 5. Updating Vaccine: The individual with the best performance is replaced as the vaccine in the new generation. Step 6. Updating Virtual Supervisors: The output of SAHNN including unmeasured states are forward calculated with training data, with the vaccine as network weights. Then, these predicted values of unmeasured states become new supervisors. Step 7. Return to Step 2 until the Stop Condition Is Fulfilled. The training algorithm shows that the virtual supervisors and weights are optimized synchronously by the swarm intelligence of populations and vaccine functions. Therefore, these merits of VS-AIA make model training with unmeasured variables feasible, and a quick convergence rate in the training is obtained.

Figure 4. Flowchart of the adaptive control.

3. OPTIMAL CONTROL FOR THE BATCH REACTOR Quality control of the final product in an industrial batch reactor is a difficult problem. Here, the optimal control strategy consists of two steps: (1) Within a fixed batch time, an optimal objective for maximizing the conversion of the desired product is proposed to find the optimal temperature profile; (2) Using the profile as the setpoint to be tracked, a controller with an EISE index based on the SAHNN model is designed, and the convergence and stability analysis of proposed controller are given. 3.1. Optimal Temperature Profile Calculation. In this type of problem, the objective is to compute the optimal temperature policy maximizing the concentration of a desired product for a given fixed batch time subject to bounds on the reactor temperature. The problem can be written mathematically as 8 > Max J ¼ Pðkf Þ > < Tr ðkÞ ~ ~ ~ s:t: P ¼ RNNðPðk 1Þ, Pðk 2Þ, Tr ðk 1Þ, Tr ðk 2Þ, Tr ðk 3ÞÞ > > : Pðk0 Þ ¼ Pð0Þ, kf ¼ kf

ð19Þ where P is the amount of desired product at a given final batch time, Tr the reactor temperature, and kf* the fixed batch time. Considering this application for plants, a recurrent neural network model replaces the mechanism equations for modeling the relationship with the main product concentration and reactor temperature. The best representations of the network model can be selected through cross validation. The optimization problem (eq 19) is a nonlinear programming problem that is solved using a hybrid particle swarm optimization (PSO)successive quadratic programming (SQP) algorithm. The basic strategy is that the SQP guides the search point to local optimums quickly and the PSO escapes from the local optimums in order to arrive at a near-global optimum.24,25 An optimal temperature profile Tr then is obtained. 3.2. Control Strategy Based on the EISE Index. Usually, unmeasured disturbances and model mismatch will cause the control error to increase in some model-based control strategies. The adaptive control strategy of retraining the neural network model is introduced to decrease the model error. The adaptive control flowchart is shown in Figure 4. Although the adaptive control strategy can revise the model error in real time and restrain the effect of disturbances or noises, 6178



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written as ~yðk þ 1Þ ¼ a~yðkÞ þ þ

its transient performance cannot satisfy the control requirements. An expanded integrated square error index (EISE) is proposed by considering setpoint tracking, noise restraining, and constraint of the control signal in practical applications, as shown in eq 20. The close-loop control strategy is shown in Figure 5. uðkÞ

1 E½ð~y ðk þ 1Þ gðk þ 1ÞÞ2 2

s:t:

Bn

∑ ∑ ∑ gj, i, b Bb ðxðkÞÞ j¼1 i¼1 b¼1

ð21Þ

δJ δe1 ðk þ 1Þ δe2 ðkÞ ¼ e1 ðk þ 1Þ þ γe2 ðkÞ þ Rðuðk þ 1Þ uðkÞÞ δuðkÞ δuðkÞ δuðkÞ

uðk þ 1Þ ¼ uðkÞ

λ δe1 ðk þ 1Þ δe2 ðkÞ e1 ðk þ 1Þ þ γe2 ðkÞ 1 þ Rλ δuðkÞ δuðkÞ

ð22Þ

þ γð~yðkÞ yðkÞÞ2 þ RΔu2 ðkÞ

~yðk þ 1Þ ¼ SAHNN½~yðkÞ, uðkÞ, xðkÞ

20 e uðkÞ e 120,

s

m

Since ~xn(k) = ~y(k) and n is a certain number here, it is omitted in the subscript of eq 21. The temperature tracking error is defined as e1(k) = ~y(k) g(k) and the model error is defined as e2(k) = ~y(k) y(k). The one-step-ahead control sequence can be obtained using the gradient descent technique for the cost function given in eq 20.

Figure 5. Closed-loop control profile.

min J ¼

h

h

∑ ∑ dj, i ui ðkÞ j¼1 i¼s þ 1

20 e yðkÞ e 120

ð20Þ

where γ is the mismatch effort weight factor (γ > 0), R the control effort weight factor (R > 0), ~(k y þ 1) the one-ahead-step predictive temperature output of the neural model (given as ~(k y þ 1) = T ~r(k þ 1)), g(k þ 1) is the optimal reactor temperature profile (calculated according to eq 19), y(k) the actual temperature output of the plant at time k, Δu(k) the change rate of input Tj (defined as Δu(k) = u(k) u(k 1)), and x(k) the prediction of state variables at time k (in which the reactor temperature ~(k) y y is one of them and denoted as the last states variable ~xn(k) = ~(k)) (given as x(k) = [x~1(k), ..., ~xn1(k),y~(k)]T). The first item in eq 20 is a one-step-ahead tracking problem, which aims to restrain temperature overshoot effectively. However, this goal is difficult to achieve in practice. Another two items should be considered. The second item aims at decreasing the error ~(k) y y(k) between the measured values and model prediction caused by unknown disturbances or noises. This error will increase obviously if the system is under unknown disturbances or parameter perturbations, then the controller responds rapidly. The error g(k) y(k) between the measured values and setpoint will not change obviously in a traditional controller (such as a proportional integral differential (PID) controller); therefore, the response of a traditional controller is slower than that of the proposed method. The third item aims to constrain the dramatic changes in input signal caused by closely tracking the output. Compatible optimization of followup control and fixed setpoint control can be implemented by minimizing the control index under the input limitation conditions. As the controller tries to bring~(kþ1) y to a desired value g(k þ 1) in one step, an excessive control effort may occur. Therefore, minimization of the cost function in eq 20 with weight R > 0 to achieve a compromise between the perfect one-step-ahead control and the variation of control effort is considered. Suppose that the neural model is prespecified well by training; then, according to the neural model (eq 13), the reactor temperature, which is one of the state variables, can be

Here, λ > 0 is an optimal step where cost function J decreases fastest when one goes from u(k) in the direction of the negative gradient of J at u(k), i.e., δJ/(δu(k)). Note that the value of the step size λ is allowed to change at each iteration. The following relation should be met: δJ δJ ¼ min J uðkÞ tk J uðkÞ λ δuðkÞ δuðkÞ To simplify, define λ 1 þ Rλ

β¼

ð23Þ

Substituting eq 21 into eq 20, and set Δ

Gðk þ 1Þ ¼ gðk þ 1Þ þ a~yðkÞ þ

h

s

Bn

∑ ∑ ∑ gj, i, b Bb ðxðkÞÞ j¼1 i¼1 b¼1 ð24Þ

The tracking and model error is written as e1 ðk þ 1Þ ¼ Gðk þ 1Þ þ

h

m

h

Δ dj, i ui ðkÞ ¼ Gðk þ 1Þ þ ∑ dj uðkÞ ∑ ∑ j¼1 i¼s þ 1 j¼1

ð25Þ e2 ðkÞ ¼ GðkÞ þ gðk þ 1Þ yðkÞ þ

h

∑ djuðk 1Þ j¼1

ð26Þ

The sensitivity (δe1(k þ 1)/δu(k)) is derived from the model described by eq 21: δe1 ðk þ 1Þ ¼ δuðkÞ

h

m

h

∑j i ¼∑s þ 1 dj, i ¼Δ ∑j dj

ð27Þ

From eq 26, we have, approximately, δe2 ðkÞ δyðkÞ yðkÞ yðk 1Þ Δ ¼ ¼ ηðkÞ δuðkÞ δuðkÞ uðkÞ uðk 1Þ 6179

ð28Þ



ARTICLE

Substituting eqs 2528 into eq 22, the control sequence can be obtained and written as 2

uðk þ 1Þ ¼ uðkÞ β4Gðk þ 1Þ

h

h

∑ dj þ ∑j ¼d1j

!2

j¼1

2 ¼ 41 β

" !2 3 5 uðkÞ þ β Gðk þ 1Þ dj

h

∑j ¼ 1

h

∑

j¼1

3 uðkÞ γηðkÞ ~yðkÞ yðkÞ 5

dj γηðkÞ ~yðkÞ yðkÞ

Furthermore, the convergence of the proposed control strategy is analyzed. The derivation of the cost function described by eq 20 is written as δJ δe1 ðk þ 1Þ δuðkÞ δe2 ðkÞ δuðkÞ ¼ e1 ðk þ 1Þ þ γe2 ðkÞ δk δuðkÞ δk δuðkÞ δk δuðkÞ ð35Þ þ RΔuðkÞ δk

#

ð29Þ After training the parameters of the SAHNN model, all the terms in the right side of eq 29 are known and the manipulated variable u(k þ 1) is obtained. Note that the particular case treated is one in which the upper and lower bounds on the inputs/outputs are not active. The EISE problem is solved using the gradient descent technique, without considering the constraints. On the other hand, the third item in the EISE index, RΔu2(k), will constrain the dramatic changes in input signal caused by closely tracking the output. If the coefficient R is selected properly, the input signal value will be limited in an appropriate range as much as possible. The output then is also limited in a certain range in this system. 3.3. Convergence and Stability Analysis. First, define the Lyapunov function: V ð 3 Þ ¼ e21 ðk þ 1Þ þ Δe21 ðk þ 1Þ þ γe22 ðkÞ þ γΔe22 ðkÞ ð30Þ

Approximating yields ΔJ δe1 ðk þ 1Þ ΔuðkÞ δe2 ðkÞ ΔuðkÞ ¼ e1 ðk þ 1Þ þ γe2 ðkÞ Δk δuðkÞ Δk δuðkÞ Δk ΔuðkÞ þ RΔuðkÞ Δk Then,

δe1 ðk þ 1Þ δe2 ðkÞ þ γe2 ðkÞ ΔJ ¼ e1 ðk þ 1Þ ΔuðkÞ þ R½ΔuðkÞ2 δuðkÞ δuðkÞ

ð36Þ Substituting the control strategy described by eq 22 into eq 36, 1 ΔJ ¼ þ R ½ΔuðkÞ2 β To ensure the convergence of control strategy, it must be 1 1 ΔJ e 0 w þ R e 0 w β e β R

Clearly, the equilibrium is [e1(k),Δe1(k),e2(k),Δe2(k)] = [0,0,0,0], which means that not only the system output asymptotically tracks the optimal profile, but also the model error caused by disturbance is eliminated at the same time. To stabilize the closed system, ΔV( 3 ) e 0 should be satisfied, i.e.,

From eqs 34 and 37, the optimal coefficient should satisfy 0 1 B B1 β e minB @R ,

δV δe1 ðk þ 1Þ δuðkÞ 2½e1 ðk þ 1Þ þ Δe1 ðk þ 1Þ þ 2γ½e2 ðkÞ δk δuðkÞ δk δe2 ðkÞ δuðkÞ þ Δe2 ðkÞ δuðkÞ δk

þ Δe2 ðkÞ

ð32Þ

Then, Δe1 ðk þ 1Þ ¼

δe1 ðk þ 1Þ ΔuðkÞ, δuðkÞ

Δe2 ðkÞ ¼

δe2 ðkÞ ΔuðkÞ ð33Þ δuðkÞ

Substituting eqs 22 and 33 into eq 32, (

) 1 δe1 ðk þ 1Þ 2 δe2 ðkÞ 2 ½ΔuðkÞ2 e 0 ΔV ¼ 2 þ þγ β δuðkÞ δuðkÞ

1 δe1 ðk þ 1Þ 2 δe2 ðkÞ 2 þ þγ e0 β δuðkÞ δuðkÞ

β e "

δe1 ðk þ 1Þ δuðkÞ

h

ð ∑ dj Þ

C C C 2A

ð38Þ

j¼1

4. CASE STUDY

δe1 ðk þ 1Þ ΔuðkÞ þ 2γ½e2 ðkÞ δuðkÞ

δe2 ðkÞ ΔuðkÞ δuðkÞ

1

ð31Þ

Approximating the differential by the difference in eq 31, ΔV 2½e1 ðk þ 1Þ þ Δe1 ðk þ 1Þ

ð37Þ

1 1 1 ¼ h 2 #e δe1 ðk þ 1Þ 2 δe2 ðkÞ 2 ½ ∑ dj 2 þγ j¼1 δuðkÞ δuðkÞ

ð34Þ

4.1. Modeling with SAHNNs. 4.1.1. Modeling for a Known Reactor Mechanism (SAHNN-I). According to the analysis of the

reactor process in eq 11, the jacket temperature is selected as the manipulated variable, i.e., [u1] = [Tj], the output and state variables of the SAHNN are the same, i.e., h iT x ¼ y ¼ CA CB C C CD T r so the weight matrix WO of the output layer is an identity matrix. The nonlinear part has two dimensions (two reaction rates: R1 and R2); the input variables of SAHNN are composed of the nonlinear part and manipulated variable (Tj), h i h i u ¼ u1 u2 u3 ¼ Tj R1 R2 The reactor is integrated over a batch duration of 200 min. Input and output variables are defined with the following domain restrictions: Tj ∈ [20,120],CA ∈ [0,12],CB ∈ [0,12],CC ∈ [0,8], CD ∈ [0,2], and Tr ∈ [20,120]. In the SAHNN model, the structural parameters of B-spline network are 542 (reactant rate R1 and R2) and B-spline base function with rank 3. That of the diagonal recursive network is 31045. 6180



Figure 6. Concentration predictions by SAHNN and HNNs.13

The SAHNN model is compared with other two hybrid neural network (HNN) models. The HNN in the Cao et al. report26 is the series of B-spline networks and a diagonal recursive network, which is a typical black-box model, and a recursive prediction error algorithm, which converges faster than the BP algorithm, is applied to train the network. HNN in Ng et al.13 is a hybrid combination of a neural network with a first-principles model in which the network is applied to predict kinetic parameters (reactant rates R1 and R2). This is a type of gray-box model in which the mechanism model is supposed to be known, while two kinetic parameters are not known. The optimal architecture of SAHNN and HNNs13,26 were chosen based on the mean-square error (MSE) of training data. The training and testing data are generated by solving the highly nonlinear mathematical models of the polymerization system (i.e., the mathematical model described by eqs 1 and 2), using the Matlab 7.0 software. In this case, 25 datasets under different operating conditions were prepared: 20 for training and 5 for validation. Training data are obtained by setting the jacket inlet temperature Tj between 20 and 120 °C in a random function and a multistep function with Gaussian noise ((1 °C). With a sampling time of 0.1 min, which is about onetenth of the response time of the system, 2000 data points were collected for a period of 200 min in the batch process. The testing data are obtained with a sampling time of 1 min by setting Tj to be continuous practical operation input signals. The output signals (i.e., the concentrations Ci and the reactor temperature Tr) are subjected to 1% and 5% Gaussian noise, respectively, for them to behave in a manner similar to that of a real practical system. The input and output data are scaled by the following equation: ðxðkÞ minðXÞÞ Xnorm ðkÞ¼ ðmaxðXÞ minðXÞÞ The same training precision is adopted as network convergence criteria. From the simulation results, the SAHNN converges ∼400 samples, while HNNs converge ∼1300 and ∼1800 samples.13,26 Furthermore, the convergence speed of the SAHNN is 34 times of that of HNNs. Comparisons of the trained SAHNN and HNNs13,26 with the targets for the testing data are shown in Figures 6 and 7. The MSE values of the SAHNN model and the HNN models13,26 are given in Table 2. The SAHNN model has captured the dynamic trends of the product quality CD and reactor temperature Tr with higher precision than the HNNs do.13,26 Both results fully demonstrated that the SAHNN had better performance than some types of

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Figure 7. Temperature predictions by SAHNN and HNNs.26

black-box and gray-box models, such as the HNNs.13,26 The SAHNN can make full use of prior knowledge, which has the same structure as the actual system. On the other hand, the prior knowledge is the mechanistic model with some time-varying parameters substituted by a constant in the HNN model,13 which is acceptable only to a certain extent. 4.1.2. Modeling with an Unmeasured State Variable (SAHNN-II). In some cases, the complete structure of the firstprinciples model may not be available. This type of batch model is used here to study the effectiveness of the SAHNN methodology and proposed control strategy. In the reaction system described in section 2, the “desired” main reaction (Rxn 1) is assumed to be completely known, including its kinetics and parameters. The “undesired” reaction (Rxn 2) is assumed to be completely unknown. The poor partial process model is supplemented by infrequent exact measurements of CA in only a few preliminary “modeling” runs, but not in the subsequent “prediction” runs. Common in all runs, the initial [CA 0, CB 0] can be obtained. Online measurements of Tr are available. Another state space representation of the batch reactor (eq 11) is given as 2 2: 3 0 CA 6 : 7 6 60 6CB 7 6: 7 ¼ 6 60 6C 7 6 4 :C 5 4 0 Tr 2

0 0 0

0 0 0

0

0

1 6 6 1 6 þ6 1 6 ΔH 4 1 M r Cp r

32 3 0 7 CA 7 0 76 7 76 6 CB 7 0 7 6 C 7 UA 7 54 C 5 Tr M r Cp r

3 0 7" # 0 7 R 7 1 0 7 Tj UA 7 5 M r Cp r

ð39Þ

In all runs, a sampling period of 0.5 min is used to measure the temperature Tr. The concentration measurement of CA is only available in the training runs at some different sampling instants (8, 16, ..., 200 min). The concentration measurements of CB and CC can be measured offline in the training runs with negligible measurement error. According to eq 39, Tj and R1 are defined as the input vector andCA, CB, CC, and T are defined as the state vector. The 6181



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Table 2. Comparison of Mean-Square Error (MSE) Values Obtained by the SAHNN and HNNs13,26

SAHNN HNN

CA

CB

CC

CD

Tr

( 103)

( 103)

( 104)

( 104)

( 101)

3.245

4.178

8.569

4.712

6.155

from ref 26

15.62

22.96

75.34

34.86

31.27

from ref 13

13.4

19.12

54.88

21.23

82.51

Figure 9. Prediction for infrequently measured concentration via the SHANN model with Virtual Supervisors.

Figure 8. Diagram of the SAHNN model with Virtual Supervisors.

nonlinear part is one-dimensional, because only one reaction rate R1 is involved. The structural parameters of B-spline networks are 441 (reactant rate functionR1) and B-spline base-function with rank 3 and 21043 for diagonal recursive networks. Considering that CA is infrequent in exact measurements, g(t) = a1 þ a2t þ a3t2 þ a4t3 is selected as the Virtual Supervisor for its training. The detailed training procedure has been described in the last section, and the configuration of the SAHNN model with the Virtual Supervisor is shown in Figure 8. For comparison, a SAHNN model without a CA value is established, and the partial state space representation is given as 2 32 3 2: 3 0 0 0 CB 6 76 CB 7 6 0 0 0 7 6C: 7 7¼6 6C 7 6 4 :C 5 UA 7 4 54 C 5 0 0 Tr Tr Mr C p r 2

1 6 6 1 þ 6 ΔH 4 1 M r Cp r

3 0 " # 7 R1 0 7 UA 7 5 Tj Mr C p r

ð40Þ

Based on the SAHNN with Virtual Supervisors, the prediction results of the infrequently measurable variable CA and measurable variables CB, CC, and Tr are shown in Figure 911, respectively. The sum of MSE values of the SAHNN model with Virtual Supervisors and SAHNN without CA are listed in Table 3. Based on the same training procedure, the SAHNN model with Virtual Supervisors is predicted with higher accuracy than the model without them. Dealing with the problems of modeling for partially known first principles in batch processes is significant. 4.2. Optimal Control. 4.2.1. Calculation of the Optimal Temperature Profile. In this type of problem, the objective is to

Figure 10. Prediction for measurable concentrations via the SAHNN model with Virtual Supervisors.

Figure 11. Prediction for temperature via the SAHNN model with Virtual Supervisors.

Table 3. Validation Results for the MSE Values of the SAHNN CA ( 102)

CB ( 103)

CC ( 104)

Tr ( 101)

SAHNN

N/A

30.74

37.31

10.56

SAHNN-VS

8.247

4.178

7.758

3.541

compute the optimal temperature profile by maximizing the concentration of desired product under the fixed batch time, subject to the bounds on the reactor temperature Tr. The recurrent neural network model for the main product 6182



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concentration can be summarized in eq 41, and the number of hidden neurons for this network is 8. ~ ~ c ðk 1Þ, C ~ c ðk 2Þ, Tr ðk 1Þ, ~ c ðkÞ¼ RNN½C PðkÞ ¼ C

Tr ðk 2Þ, Tr ðk 3Þ

ð41Þ

The optimal control problems have been solved using the hybrid PSO-SQP method, in regard to time intervals of equal length, which varied from 1 to 20 intervals to discrete the profile. The switching time is fixed and the length of each internal is specified by dividing the batch operation time (kf) by the number of time intervals (M = 1, ..., 20). Thus, an optimal temperature value should be found in each subinterval. Results with different time intervals are reported in Table 4. The temperature profile for each case is shown in Figure 12. As shown in Table 4, when one time interval (M = 1) is used, the amount of product C obtained at the final time (kf = 200 min) is 7.0159 kmol and the optimal setpoint is 88.2 °C. Using a value Table 4. Summary of Optimization Results number of time intervals 1 amount of product C

5

10

20

7.0159 kmol 7.0259 kmol 7.0315 kmol 7.0361 kmol

Figure 12. Optimal temperature profile.

of M = 20, the amount of desired product (CC) achieved was 7.0361 kmol. Thus, CC increases with the number of intervals, because, as the number of intervals increases (more degrees of freedom in the optimization), the approximated optimal profile with piecewise constant policy becomes more similar to the actual optimal profile. 4.2.2. Control Strategy Based on the EISE Index. Considering the temperature profile (M = 10) to be tracked, the control results based on the EISE index (eq 20) and the SAHNN-VS model are determined. Here, the mismatch effort weight factor is γ = 0.1 and the control effort weighting factor is R = 0.2. Based on the trained SAHNN-VS model and eq 38, the upper limit of β is 3.37, then set β = 0.45. Experiment 1: The tracking responses based on EISE and a PID controller are given in Figure 13; the EISE index values are shown in Figure 14. Although the proposed EISE control strategy is based on the system model; a larger difference between a hybrid network model and a real system does not lead to the control results becoming worse (see Figure 13a). It is attributed to (y~(k) y(k))2 in the EISE index, which can restrict the effect caused by model mismatch very well. Comparison with the PID controller then shows that the system output using EISE control can track the desired trajectory without any overshoot or oscillation. The system responses obtained by EISE and a PID controller are very close, while a great difference in controller

Figure 14. EISE index, relative to time.

Figure 13. Effect of EISE and PID control in tracking the optimal temperature profile. 6183



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output is observed, as shown in Figure 13b. To track the desired profile perfectly, the PID controller has stronger control action, with large oscillation, than EISE. With regard to the setpoint temperature of the jacket temperature controller (Tjsp), greater oscillation action is undesired for the difficulty that is realized. To simulate a realistic situation, the system was subjected to the measurement noise, as well as some measured and unmeasured disturbances. When the system was subjected to (2.5%

Gaussian noise, good system response is still being achieved, as shown in Figure 15. Experiment 2: The reactor temperature is subjected to a step signal of þ5 °C at kf = 70200 min. The strong anti-interference performance can be demonstrated as shown in Figures 16a and 16b. It is shown that EISE has better performance, in regard to tracking response and control action, than the PID controller. Experiment 3: One parameter of the batch reactor is supposed to be perturbed. A 3.5% increase in parameter k11 (in eq 5) is given from 30 min to 200 min. Figure 17a shows that EISE has good robustness, with regard to parameter perturbation. At the point of 30 min, when the parameter changes, a larger control oscillation with the PID controller is observed before the system is adjusted to the setpoint, as shown in Figure 17b. Some system performance index values in EISE and the PID controller are listed in Table 5, such as output error sum of square (ESS) and mean-square error (MSE). It is shown that, when the system is subjected to a strong disturbance (Figure 16) or parameter perturbation (Figure 17), controller movement changes rapidly, caused by the saltation error between measured output and model prediction. Good control then is re-established with robustness, relative to variations in the disturbance or parameter. From the above experiment results, the proposed EISE strategy not only ensures that the close control system is stable, but it also ensures that the closed-loop system responds rapidly in the desired manner.

Figure 15. EISE control with (2.5% Gaussian noise.

Figure 16. EISE and PID control with þ5 °C step disturbance.

Figure 17. EISE and PID control with 3.5% variation in the k11 parameter. 6184



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Table 5. Summary of Output Response Performance Index Values ESS_EISE

ESS_PID

( 103)

( 103)

MSE_EISE

MSE_PID

Experiment 1

7.5596

8.9064

6.14

6.67

Experiment

7.6249

9.1065

6.17

6.74

7.7555

9.4883

6.22

6.88

2 Experiment 3

5. CONCLUSION Full utilization of physical insight into the structural characters and construction of a suitable model is a logical step. Therefore, a novel hybrid neural network called the Structure Approaching Hybrid Neural Network (SAHNN) has been proposed. By separating the nonlinear elements, nonlinear dynamic systems have been transformed to linear ones. The architectural similarity between the SAHNN and actual applications gives it superior ability to correct qualitative trends and logical relationships between related variables. Moreover, the Virtual Supervisor, which is used as the reference trajectory of unmeasured state variables, is fully utilized to solve the problem of partial unmeasured state variables. The exploration, effective direction, and quick convergence rate in training are fulfilled by swarm intelligence of population and optimal performance of the vaccine. The optimal reactor temperature profiles are obtained using the PSO-SQP algorithm to solve the maximum product concentration problem, based on a recurrent neural network. Considering the model mismatches and unmeasured disturbances, a novel extended integral square error (EISE) index was proposed that introduces model mismatches into the control index. The approach applies a feedback channel to control, and ot enhances the robustness and antidisturbance performance dramatically. The results of modeling and optimal control fully demonstrate that the proposed SAHNN is highly effective, which can be readily extended to other similar chemical processes. ’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT This work is supported by National Science Foundation of China (under Grant Nos. 60704011 and 60974031). ’ REFERENCES (1) Hua, X. M.; Sohrab, R.; Arthur, J. Cascade Closed-loop Optimization and Control of Batch Reactors. Chem. Eng. Sci. 2004, 59, 5695– 5708. (2) Arpornwichanop, P.; Kittisupakorn, I. M.; Mujtaba On-line Dynamic Optimization and Control Strategy for Improving the Performance of Batch Reactors. Chem. Eng. Process 2005, 44, 101–114. (3) Tian, Y.; Zhang, J.; Julian, M. Optimal Control of a Fed-batch Bioreactor Based an Augmented Recurrent Neural Network Model. Neurocomputing 2000, 48, 919–936.

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