Model-Based Iterative Learning Control for Batch Processes Using

Article pubs.acs.org/IECR

Model-Based Iterative Learning Control for Batch Processes Using Generalized Hinging Hyperplanes Xiaodong Yu,†,‡ Zhihua Xiong,†,‡ Dexian Huang,*,†,‡ and Yongheng Jiang†,‡ †

Institute of Process Control and Engineering, Department of Automation, Tsinghua University, Beijing 100084, China Tsinghua National Laboratory for Information Science and Technology, Tsinghua University, Beijing 100084, China

‡

ABSTRACT: A model-based iterative learning control (ILC) strategy using the generalized hinging hyperplanes (GHH) is proposed to track the product quality trajectory in the batch process. As an empirical model with piecewise affine basis functions, GHH is very suitable for constructing the dynamic model of batch processes, in which its gradient information can be easily obtained due to the structure of GHH model. Based on the GHH, a quadratic-criterion-based ILC (Q-ILC) algorithm is constructed, where the input trajectory for the next batch is updated by ILC law and the output tracking error can be gradually reduced from batch to batch. The proposed strategy is demonstrated on a simulated typical batch reactor and compared with the method based on neural networks. The simulation results show the convergence of the output tracking error and the robustness of the proposed method under model−plant mismatches and unknown disturbances.

1. INTRODUCTION Generally, batch processes run intermittently and are very suitable for low-volume and high-value products, while continuous processes are proper to make high-volume products continuously.1 A batch process usually performs a given task repetitively over a fixed period of time (called a batch or trial), and it has been paid more attention for the past decade because it plays more important roles in the chemical industry.2−4 Because of the repetitiveness and periodicity in nature, batch processes are usually run based on the results and experiences of the previous batches, which is in accordance with the basic idea of iterative learning control (ILC) strategy. ILC updates control signals for the next batch based on the information from previous batches, and then the output trajectory can converge asymptotically to the reference trajectory. The initial explicit formulation of ILC was presented by Uchiyama in 1978.5 Although many significant improvements of ILC have been achieved in both industry and academia, these developments are mainly based on model-free approaches.6 At the beginning of the 21st century, model-based ILC algorithm with a quadratic criterion (Q-ILC) for time-varying linear systems was proposed.7−9 Thereafter, the ILC strategy was fused with model predictive control (MPC) to build an integrated control technique for end-product and transient profile.6 However, these above-mentioned methods are mainly based on linear models. To extend the ILC to nonlinear batch processes, neural networks (NN) and supported vector machine (SVM) methods were used to build the process model and then ILC law was formed based on the linearization of the nonlinear model.10−12 Zhang11 proposed an ILC method using a feedforward neural network model, where the network was linearized around the current batch and the control policy for the next batch was updated by ILC based on the linearized model. In our previous work,13 we combined a special controlaffine feed-forward neural network with Q-ILC scheme for batch processes, and with the help of the gradient information © 2012 American Chemical Society

obtained from the predictive model, the ILC law can be obtained theoretically. However, it is still difficult to analyze the convergence performance explicitly. The hinging hyperplanes (HH) model was first introduced by Breiman in 1993.14 The hinge functions in the HH are used as basis functions, rather than the sigmoid functions in NN. HH becomes ridge construction with an additional linear term and yields a type of piecewise-linear model.15 Compared with other nonlinear model approximation methods, there are several advantages of the HH model.14 For example, the upper bound of the approximation error can be easily determined, and all parameters employed in the HH model can be estimated using a fast and efficient least-squares (LS) method. With the help of this type of piecewise-linear model, most of the existing linear analysis techniques become available.16 By modifying the basis functions of HH, Wang et al.17 proposed a general model structure, which is called generalized hinging hyperplanes (GHH), and they also proved that GHH possessed more flexibility of nonlinear black-box modeling. Because GHH is also a type of piecewise-linear model and its performance is more accurate than NN, GHH is used here to build the model of nonlinear batch processes. In this paper, by making some compromises between the approximation capability of the model and the simplicity for controller design, the GHH model-based ILC strategy (called as GHH−ILC) for batch processes is proposed. With the predictive model of batch processes constructed by GHH, the input trajectory is updated by the quadratic-criterion-based ILC with the help of the gradient information obtained easily from the GHH model. Meanwhile, model predictions of GHH are modified by the predictive errors of the immediately previous batch, in order to improve the model accuracy. Received: Revised: Accepted: Published: 1627

August 17, 2011 November 26, 2012 November 26, 2012 November 26, 2012 dx.doi.org/10.1021/ie201842a | Ind. Eng. Chem. Res. 2013, 52, 1627−1634

Industrial & Engineering Chemistry Research

Article

Figure 1. Hinge, hinging hyperplanes, and hinge function depicted in (a) a two-dimensional (2-D) format and (b) a three-dimensional (3-D) format.

where [1 xT] is the regression vector and x ∈ Rn, θ0 ∈ Rn+1 is the basic coefficient vector, ci = ±1, hi(x) is the basis function, and M is the number of basis functions. The basis function hi(x) is defined by

The rest of this paper is organized as follows: batch process modeling based on GHH is addressed in Section 2. In Section 3, the procedure of GHH-ILC method is presented in detail. In Section 4, the advantages and feasibility of the proposed scheme are demonstrated on a simulated typical batch reactor. Finally, the paper is concluded in Section 5.

hi(x) = max{0, [1 x T ]θi}

where θi ∈ R is the coefficient vector. Moreover, the basis functions hi(x) are known as the hinge functions with a visualizable form of an “open book” shown in Figure 1.19 Definition 1.14 A function f(x)(x ∈ C) is sufficiently smooth if the following integral is finite:

2. BATCH PROCESS MODELING BASED ON GHH 2.1. Nonlinear Representation of Batch Processes. Prior to the detailed description of the GHH−ILC method, some preliminaries should be addressed as the control and output sequences through an entire batch: Uk = [uk(0), uk(1), ..., uk(N − 1)]T

(1)

Yk = [yk (1), yk (2), ..., yk (N )]T

(2)

∫ || ω ||2 |f ̂ (ω)| dω < ∞

M

f (x) − [1 x T ]θ0 −

(3)

∑ cihi(x) i=1

≤

cH M

(7)

M

y = [1 x T ]θ0 +

∑ cihĩ (x) i=1

(8)

where h̃i(x) is the generalized hinge function and defined as hĩ (x) = max{0, [1 x T ]θi1 , ..., [1 x T ]θiK i}

(9)

where Ki ≤ n is the number of linear functions in each of the generalized hinge functions. Lemma 2.17 Assume f(x) to be a suf f iciently smooth function. There then must exist a constant cG>0, such that for any positive integer M, there exist M hinge f unctions h̃i(x) and coef f icients ci = ±1 such that

M

+

2

The hinge functions in HH are very simple, but they result in many parameters in real nonlinear identification cases.20 It was also addressed by Wang et al.17 that the HH model was inefficient in 2-D and higher-dimensional nonlinear approximations. Under these circumstances, GHH comes into being as a more general nonlinear approximation model, which can be represented by the following expression:

where φ(t) is the regression vector, and p and q are the system orders of output and input, which are usually determined by the physical insight about the system or estimated by the identification methods. 2.2. Introduction and Modification of GHH. Piecewise affine models have attracted great attention in the field of nonlinear identification, because they are the simplest extension of linear models and can describe any nonlinear dynamic process with arbitrary accuracy.18 As a canonical piecewise affine model representation, the HH model was introduced by Breiman in 1993,14 which can be expressed as y = [1

∑ cihi(x) i=1

y (̂ t ) = g (̂ φ(t ))

x T ]θ0

(6)

̂ where f(ω) is the Fourier transform of f(x), and C is a compact set in Rn. Lemma 1.14 Assume f(x) to be a suf f iciently smooth function. There then must exist a constant cH > 0, such that for any positive integer M, there exist M hinge f unctions hi(x) and coef f icients ci = ±1, such that

where Uk is the manipulated variable, Yk is the output variable, the subscript k is denoted as the batch index, and N is the number of the sampling intervals in an entire batch. The batch process considered here is assumed to run over a fixed time length (tf) and from the same initial conditions in all batches. Therefore, the control variable in each of the N intervals is kept constant. In this study, GHH is used to construct the dynamic model of an entire batch. The batch process is assumed to be addressed by a Nonlinear Auto-Regressive eXogenous model (NARX) here, which can be written as15

φ(t ) = [y(t − 1), ..., y(t − 1), ..., u(t − q)]T

(5)

n+1

(4) 1628

dx.doi.org/10.1021/ie201842a | Ind. Eng. Chem. Res. 2013, 52, 1627−1634

Industrial & Engineering Chemistry Research M

f (x) − [1 x T ]θ0 −

2

∑ cihĩ (x) i=1

c ≤ G M

Article

As stated in our previous work,13 the accuracy of model prediction can be improved by using the model prediction error in the immediately previous batch to modify the model prediction in the current batch, which can be defined as

(10)

It can be seen from eq 9 that the n-D GHH basis functions are generated by (n+1) n-D hyperplanes continuously joining together. By comparing eq 5 with eq 9, it is easy to find that a hinge function is essentially a 1-D GHH basis function. It then comes to the conclusion that a GHH model is the extension of a HH model in n dimensions. Accordingly, GHH can approximate continuous nonlinear functions arbitrarily well on a compact set.20 Three numerical benchmark problems of nonlinear dynamic systems were studied further by Wen et al.20 Comparisons of the identification facility and the model structure flexibility between a special type of GHH and three models of other nonlinear identification were given by Xu et al.21 Oblak et al.22 also employed this type of approximation model to the MPC for a pH neutralization process, and the results exhibited that the proposed method outperformed the robust nonlinear H∞ method. The similar error bound of GHH model in eq 10 is also given by Lemma 2. Note that the HH model with M hinge functions is a proper subset of the GHH model with M GHH basis functions. It has also been proved that the error bound (cH)/ (M) in eq 7 is looser than cG/M in eq 10.17,20 In order to estimate the dynamic model of an entire batch by using GHH, the expression of the GHH model is transformed as follows:

Yk̃ + 1 = Yk̂ + 1 + εk

where Ỹk+1 is the modified prediction of GHH model in the (k +1)th batch, and εk is the prediction error in the kth batch.

3. GHH−ILC METHOD The main objective of the ILC strategy is to find an optimal input profile Uk+1 for a new batch based on the information of the previous batches, so that the measured output Yk+1 can converge asymptotically toward the desired trajectory Yr. Using the modified prediction errors upon completion of the kth batch to update the input profile for the (k+1)th batch, the quadratic objective function of the GHH-ILC method proposed here can be expressed as13 min Jk + 1 = Uk + 1

ek̃ + 1 = Yr − Yk̃ + 1

∑ ci max{0, χk (t )T θi1 , ..., χk (t )T θiKi} i=1

= χk (t )T Ψk(t )

where ŷk(t + 1) is the model prediction of GHH, Ψk(t) is the parameter set, and χk(t) is the regression vector, respectively. Moreover, Ψk(t) and χk(t) are defined as Ψk(t ) = θ0 +

i=1

,..., χk (t ) θiK i},

T

θij

(θi0 = 0)

(12)

χk (t ) = [1, yk (t ), yk (t − 1),..., yk (t − p + 1), uk(t − 1) ,..., uk(t − q + 1)]T (13)

where p and q are the system orders defined in eq 3. During the procedure of training the GHH model, the initial modeling data set is divided into two sets. The first hinge is found by using the hinge finding algorithm (HFA) presented by Breiman,14 which is actually a Newton algorithm for function minimization under a sum of squared error criterion.19 Then, the HFA over the residuals is run further to find the second hinge and all the obtained hinges are refitted based on the new residuals. Thus, by adding a new hinge, this procedure is repeated until the global convergence criterion is satisfied.19 After the GHH model of an entire batch is trained properly, it can be used to predict the output recursively when the initial operation conditions and the entire control profile are given. The model prediction error εk+1 between the measured output Yk+1 and the prediction Ŷk+1 for the (k + 1)th batch is calculated by εk + 1 = Yk + 1 − Yk̂ + 1

(18)

and Q and R are weighting matrices with appropriate dimensions, respectively. Generally, Q and R can be selected as the diagonal matrices with different elements.26 In order to simplify the problem, weighting matrices Q and R in eq 16 are taken as Q = I, R = Λ = λIN here. As mentioned in our previous papers,11,13 subtracting the time-varying nominal trajectories from the process operation trajectories may remove the majority of the process nonlinearity and allow linear modeling methods to perform well on the resulting perturbation variables.23 The linearization of nonlinear batch process around the nominal trajectory is a good approximation to the real process as the input change is typically small.13 Here, a linearized model of the GHH is also obtained to limit the deterioration of control performance due to model-plant mismatches and unknown disturbances. According to the piecewise-linear model of GHH, the model prediction for the (k+1)th batch can be expanded using the first-order Taylor series model at the operating point of the kth batch as follows:

∑ ci arg max{χk (t ) θi0 , χk (t ) θi1 ,..., χk (t ) θij

T

(17)

ΔUk + 1 = Uk + 1 − Uk

M T

(16)

ΔUk+1 is the difference of input profile between two adjacent batches,

(11)

T

1 (|| ek̃ + 1 ||Q2 + || ΔUk + 1 ||2R ) 2

where ẽk+1 is the tracking error of the modified GHH model prediction

M

yk̂ (t + 1) = χk (t )T θ0 +

(15)

∂Y Yk̂ + 1 = Yk̂ + ∂U

(Uk + 1 − Uk) + Dk + 1 Uk

≈ Yk̂ + Gk ΔUk + 1

(14)

(19)

where Gk is the gradient matrix with the form as given below: 1629


Industrial & Engineering Chemistry Research ⎡ ∂y (1) ∂y (1) k ⎢ k ⎢ ∂uk(0) ∂uk(1) ⎢ ⎢ ∂yk (2) ∂yk (2) Gk = ⎢⎢ ∂uk(0) ∂uk(1) ⎢ ⋮ ⋮ ⎢ ⎢ ∂yk (N ) ∂yk (N ) ⎢ ⎣ ∂uk(0) ∂uk(1)

⎤ ⎥ ∂uk(N − 1) ⎥ ⎥ ∂yk (2) ⎥ ··· ⎥ ∂uk(N − 1) ⎥ ⎥ ⋮ ⋮ ⎥ ∂yk (N ) ⎥ ··· ⎥ ∂uk(N − 1) ⎦ ···

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ΔUk + 1 = [GkTGk + Λ]−1 GkT(ek̃ − Δεk)

∂yk (1)

In eq24, the control law ΔUk+1 is updated by the tracking error ẽk of the modified predictive model. In practice, it is more direct to update ΔUk+1 by the actual tracking error ek between the reference trajectory Yr and the measured output Yk, which can be further calculated as ek = Yr − Yk (20)

= Yr − (Yk̂ + εk)

and Dk+1 is the error of the high-order terms caused by the linearization, which can be reasonably neglected for almost all the chemical and batch processes. It is obvious that the structure of Gk calculated above is restricted to a lower-block triangular form because of the causality. In this case, because of the special linear structure of GHH model, as shown in eq 11, the gradient information ∂yk/ ∂uk in eq 20 can be computed using the relation

= Yr − [(Yk̃ − εk − 1) + εk] = Yr − Yk̃ + εk − 1 − εk = ek̃ − Δεk

(25)

The above ILC law described by eq 24 then can be rewritten as ΔUk + 1 = [GkTGk + Λ]−1 GkTek

⎧ ∂y (i) ⎪ k (1 ≤ i − j ≤ q) = Ψkp + 1 + (i − j)(i) ⎪ ∂uk(j) ⎪ ∂yk (i) ⎪ ∂y (j + q) i − 1 ∂yk (l + 1) =⎨ k (q < i − j ≤ N ) ∏ ∂uk(j) ⎪ ∂uk(j) l = j + q ∂yk (l) ⎪ i−1 ⎪ = Ψ p + q + 1(j + q) ∏ Ψ2(l + 1) k k ⎪ l=j+q ⎩

(26)

Moreover, substituting eqs 23 and 24 into eq 25, the transition model of the tracking error ek+1 along the batch index k can be also obtained as ek + 1 = ek − Gk [GkTGk + Λ]−1 GkTek − Δεk + 1 = (I − Gk [GkTGk + Λ]−1 GkT)ek − Δεk + 1

(27)

13,26

Similar to our previous work, it can be found that the term (I − Gk[GTk Gk + Λ]−1GTk ) in eq 27 is related to the parameter Λ (i.e., the weighting matrix in the objective function eq16). If the parameter Λ is appropriately selected, the following condition will be satisfied for all batches:

(21)

Ψik(t)

where the subscript k is the batch index, is denoted as the ith element of the vector Ψk(t), and p and q are the system orders, respectively. The gradient matrix Gk in eq 20 then can be further rewritten as

|| I − Gk [GkTGk + Λ]−1 GkT || < 1

(28)

It has been shown that when the above condition described by eq 28 is satisfied, the tracking error ek+1 may normally converge as the batch index k approaches ∞.13,26 However, it should be noticed that because the model error of GHH always exists, the convergence rate probably degrades. In order to confirm the performance of the proposed GHH-ILC method, the convergence analysis of tracking error will be studied further in our future work.

After the structure of the GHH model is obtained, the parameter set at time t for the kth batch (Ψk(t)) can be calculated based on all the data at/before time t. It is obvious that after such recursive calculation, the gradient matrix Gk in eq 22 can be calculated easily. Furthermore, substituting eqs 15 and 19 into eq 17, the tracking error transition model (i.e., an iterative relationship for the tracking error ẽk+1 along the batch index k) can be obtained as

4. APPLICATION TO A BATCH REACTOR 4.1. Process Description. In this section, a case study is discussed in the framework of the GHH-ILC strategy addressed above. This simulation plant is a typical batch reactor with temperature as the control variable, which is studied in detail by Logsdon et al.25 It is assumed that the following reactions take place in the reactor:

ek̃ + 1 = Yr − Yk̃ + 1

k1

k2

A→B→C

= Yr − [(Yk̂ + Gk ΔUk + 1) + εk]

(29)

The main objective of the batch reactor is to maximize the yield of the product B after a fixed period of reaction time. The differential equations for this batch process are demonstrated as follows:26

= Yr − (Yk̃ − εk − 1) − Gk ΔUk + 1 − εk = ek̃ − Gk ΔUk + 1 − Δεk

(24)

(23)

where Δεk = εk − εk−1 is the difference of model prediction error between two adjacent batches. After substituting eq 23 to the objective function in eq 16 and through straightforward calculation, the ILC learning law of the proposed GHH-ILC method can be derived by the following analytical form: 1630

⎛ E ⎞ dx1 = −k1 exp⎜ − 1 ⎟x12 dt ⎝ uTref ⎠

(30)

⎛ E ⎞ ⎛ E ⎞ dx 2 = k1 exp⎜ − 1 ⎟x12 − k 2 exp⎜ − 2 ⎟x 2 dt ⎝ uTref ⎠ ⎝ uTref ⎠

(31)


Industrial & Engineering Chemistry Research u=

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T , 298 K ≤ T ≤ 398 K, x1(0) = 1, x 2(0) = 0 Tref

should be noted that Figure 2 just shows the testing performance in 10 control intervals during one batch (i.e., the control interval is 6 min). 4.3. Simulation Results of the GHH-ILC. After the GHH model is constructed, the GHH−ILC strategy proposed in Section 3 can be implemented. The parameter Λ is set to be Λ = 0.1I with appropriate respective dimensions. The desired reference trajectory Y r is obtained from the literature,25 while the nominal input trajectory is selected as the initial input trajectory from our previous work.26 To investigate the performance of the proposed control strategy, several batches were simulated iteratively. Figures 3

(32)

where x1 and x2 are denoted as the concentrations of input A and product B, x1(0) and x2(0) are their initial concentrations, T is the reactor temperature, and Tref is the reference temperature, respectively. The values of all of the parameters are given in Table 1.26 Here, the batch length tf is set to be 1.0 h, and it is divided into 10 equal stages (i.e., N = 10). Table 1. Values of All Parameters for a Typical Batch Reactor parameter

value

k1 E1 Tref

4.0 × 10 2.5 × 103 348 K 3

parameter

value

k2 E2 tf

6.2 × 105 5.0 × 103 1.0 h

4.2. Model Construction. In this study, the above mechanistic model (eqs 30−32) is assumed to be unavailable. Because the interest of the reactor is to maximize the product B, a GHH model is built to represent the relationship between y (y ≅ x2) and u. For such a typical batch process, it is reasonably assumed that a two-order system can describe its dynamic behavior well. Thus, the system orders p and q are set to be p = 2 and q = 1. Correspondingly, for the GHH model, the input is [y(t − 1), y(t − 2), u(t − 1)]T and the output is y(t) in this case. Based on eqs 30−32, three batches of this process were simulated and used as the historical process datasets, in which different temperature profiles were generated randomly by means of deviating from the nominal temperature trajectory. Two batches of these data were used to train the GHH model, while the remainder was only employed for validation and was not involved in the training phase at all. The sampling time in this process is set to be 9 s, which makes 400 samples in one batch. After the GHH model is trained, it is also tested by the validation dataset, and the one-step ahead prediction performance is shown in Figure 2. It is obvious that the errors of GHH model prediction are very small and the GHH can be regarded as the dynamic predictive model for this typical batch reactor. It

Figure 3. Trajectories of the product concentration.

Figure 4. Input temperature profile using the GHH−ILC algorithm.

and 4 show the trajectories of the product concentration x2 and the input temperature profile u by using the GHH−ILC algorithm, while Figure 5 shows the differences of input temperature profiles ΔU. It can be seen from Figure 3 that the product concentration can converge to the desired trajectory after almost 10 batches, not violating the control constraints. In Figure 4, it is also demonstrated that the input temperature

Figure 2. The testing performance of the GHH model. 1631



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Figure 7. Concentration of the end-point product in each batch. Figure 5. Differences of input temperature profile using the GHH− ILC algorithm.

gradient matrix Gk mentioned above.11 The input was [y(t − 1), y(t − 2), u(t − 1)]T and output was y(t) in this case, which were the same as those in the GHH model. The training dataset and modeling error tolerance were also the same as those used in our algorithm above. The networks were trained by using the Levenberg−Marquardt optimization algorithm with regularization, which was also employed in Zhang’s work.11 After simulation and comparison of different values, the number of hidden-layer neurons was set to be 20. The gradient matrix Gk then was obtained according to the method proposed by Zhang.11 The total RMSE and the end-point product concentration of each batch are also shown in Figures 6 and 7, respectively. It is obvious that the GHH−ILC algorithm can converge more quickly and obtain better tracking performance than the NN−ILC. 4.4. Disturbance Cases. During several batch operations, batch-to-batch variations always occur. For example, raw material properties, impurities, and catalyst activities may change from batch to batch. Thus, there are often some process parametric uncertainties in the batch reactor.11 These disturbances may occur as variations of the initial condition and the plant kinetic parameter. To investigate the performance of the proposed control strategy under the presence of disturbances, two cases were considered. In Case 1, it is assumed that the variation is caused by impurities. The scenario is that, after the 16th batch, the initial concentration of A is reduced by 1‰, from its nominal value of 1 to 0.999. Figures 8 and 9 show the performance of the proposed control strategy under this type of unknown disturbance. In Case 2, it is assumed that the variation is caused by the model−plant mismatch. In addition, in the scenario that occurs after the 16th batch, the kinetic parameter k2 is increased by 1%, from its nominal value of 6.2 × 105 to 6.262 × 105. Figures 10 and 11 show the performance correspondingly. Because of the presence of these disturbances, the optimal control profile calculated at the 15th batch is no longer optimal, and then the control performance deteriorates sharply at the 16th batch, as shown in Figures 8−11. With the implementation of the proposed GHH−ILC scheme, the control performance is significantly improved after the 16th batch. At the end batch of these two disturbance cases, the total RMSE and the concentration of the end-point product almost converge to new values, respectively. However, also note that,

profile converges asymptotically. Moreover, the difference of the two adjacent input profiles ΔU almost converges to zero, as shown in Figure 5. Furthermore, it can be seen from Figure 6 that the total rootmean-square error (RMSE) of the tracking error ek of the

Figure 6. Total root-mean-square error (RMSE) of each batch in the GHH−ILC algorithm.

measured output almost converges after about 10 batches. In practice, the general interest of the batch process is the endpoint product quality, then in Figure 7, the end-point product concentrations Cend of all batches are shown sequentially and compared with the desired output concentration SPend, where SPend is set to be 0.615 selected from the literature.25 It should be also noticed that, because of model−plant mismatches, the desired optimal output of the end-point product is not achieved, even though the control system becomes stable after almost 10 batches. However, this performance is acceptable in practice. For a comparison, we also used NN instead of GHH to address the dynamic modeling problem. In this paper, a multilayer feed-forward network was introduced to represent eq 3, and its structure was so simple that it is easy to compute the 1632



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Figure 11. Concentration of the end-point product in each batch under the variation of the plant kinetic parameter.

Figure 8. Total RMSE of each batch under the variation of the initial condition.

because of these uncertainties, the obtained performance is slightly worse than that at the 15th batch. It mainly results from the fact that the GHH model is constructed based on those normal operation data, which do not include the data under these uncertainties. Meanwhile, the optimal reference trajectories should be also different on these occasions. Therefore, this is of great potential for our future work to take into consideration the adaptive update of the predictive model and the reference trajectory under uncertainties.

5. CONCLUSIONS A novel generalized hinging hyperplanes (GHH) model-based iterative learning control (ILC) strategy (called GHH−ILC) for batch processes is proposed in this paper. With a strong approximation performance, GHH is used to construct the dynamic model of batch process. Based on a linear time-varying tracking error transition model that is derived from the linearization of the GHH model, the optimal input trajectory is updated by minimizing a quadratic objective function from batch to batch. To improve the batch-wise performance, the model predictions are also modified by the model errors of the immediately previous batch. Because of the characteristic of piecewise linearity of the GHH model, the gradient information can be computed easily and directly. It is shown that the tracking error may converge to a small constant asymptotically, while the control profile becomes stable gradually. The proposed control scheme is demonstrated on a simulated typical batch reactor, which is a single-input−single-output (SISO) system, and the simulation results show that the improvement of the tracking performance has been obtained. Furthermore, it is also shown that this control strategy can obtain good performance under the presence of model−plant mismatches and unknown disturbances. Note that our approach proposed here may also work with multiple-input−multiple-output (MIMO) systems. Assuming that there are no coupling relationships between different outputs, this control strategy is valid for MIMO systems with an augmented gradient matrix. In our future work, the convergence analysis of this algorithm and the adaptive updating mechanism of the predictive model and the reference trajectory under uncertainties will be further studied.

Figure 9. Concentration of the end-point product in each batch under the variation of the initial condition.

Figure 10. Total RMSE of each batch under the variation of the plant kinetic parameter.

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(18) Roll, J. Local and Piecewise Affine Approaches to System Identification, Ph.D. Thesis SE-581 83, Department of Electrical Engineering, Linköping University, Linköping, Sweden, 2003. (19) Pucar, P.; Sjoberg, J. On the hinge-finding algorithm for hingeing hyperplanes. IEEE Trans. Inform. Theory 1998, 44 (3), 1310− 1319. (20) Wen, C.; Wang, S.; Jin, X.; Ma, X. Identification of dynamic systems using Piecewise-Affine basis function models. Automatica 2007, 43 (10), 1824−1831. (21) Xu, J.; Huang, X.; Wang, S. Adaptive hinging hyperplanes and its applications in dynamic system identification. Automatica 2009, 45 (10), 2325−2332. (22) Oblak, S.; Skrjanc, I. Continuous-time Wiener-model predictive control of a pH process based on a PWL approximation. Chem. Eng. Sci. 2010, 65 (5), 1720−1728. (23) Russell, S. A.; Kesavan, P.; Lee, J. H.; Ogunnaike, B. A. Recursive data-based prediction and control of batch product quality. AIChE J. 1998, 44 (11), 2442−2458. (24) Zhang, X. Matrix Analysis and Applications; Publishing House of Tsinghua University: Beijing, 2005. (25) Logsdon, J. S.; Biegler, L. T. Accurate solution of differentialalgebraic optimization problems. Ind. Eng. Chem. Res. 1989, 28 (11), 1628−1639. (26) Xiong, Z.; Zhang, J. Product quality trajectory tracking in batch processes using iterative learning control based on time-varying perturbation models. Ind. Eng. Chem. Res. 2003, 42 (26), 6802−6814.

AUTHOR INFORMATION

Corresponding Author

*Tel.: +86-10-62784964. Fax: +86-10-62786911. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS

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REFERENCES

Financial supports from 973 Program of China (No. 2012CB720500), the National Natural Science Foundation of China (Nos. 60974008 and 61174105), and the National Science and Technology Major Project (No. 2011ZX02504008) are gratefully acknowledged.

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Model-Based Iterative Learning Control for Batch Processes Using

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