Adaptive Control of Nonlinear Time-Varying Processes Using

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Adaptive Control of Nonlinear Time-Varying Processes Using Selective Recursive Kernel Learning Method Yi Liu,† Wenlu Chen,‡ Haiqing Wang,§ Zengliang Gao,*,† and Ping Li§ †

Key Laboratory of Pharmaceutical Engineering of Ministry of Education, Institute of Process Equipment and Control Engineering, Zhejiang University of Technology, Hangzhou, 310032, China ‡ Jiangnan Institute of Computing Technique, Wuxi, 214083, China § State Key Laboratory of Industrial Control Technology, Institute of Industrial Process Control, Zhejiang University, Hangzhou, 310027, China ABSTRACT: A selective recursive kernel learning-based (SRKL) adaptive predictive controller is proposed for nonlinear timevarying processes. First, a SRKL identification model is presented with an efficient sparsification strategy which makes a trade-off between the tracking precision and the controller’s complexity. The SRKL model can be updated efficiently by introducing and/or deleting a sample via recursive learning algorithms. Consequently, the model can adjust its structure adaptively to capture the process dynamics and time-varying characteristics. On the basis of the SRKL model, a predictive controller with an adaptive modification item is designed. The novel controller can achieve better performance since the SRKL model can trace the process characteristics online. The obtained results on a laboratory-scale liquid-level process and a continuous bioreactor with time-varying parameters show that the proposed controller is superior to the traditional proportional-integral-derivative (PID) controller and related controller with an offline KL model without online updating.

1. INTRODUCTION Many chemical and biochemical processes usually exhibit nonlinear and time-varying behavior. Conventional linear control techniques and proportional-integral-derivative (PID) controllers are sometimes insufficient for control of such processes.1,2 Adaptive control is one of the effective methods to deal with model/process mismatches, which can adapt the controller to track the process’s characteristics. Therefore, many adaptive control strategies have been proposed in the literature, such as various autotuning PID controllers,3,4 adaptive generic model control,5 neural networks (NN), fuzzy systems, and other empirical models based adaptive control.6-16 Among these controllers, NN based adaptive and predictive control approaches6-13 have received considerable attention during the past 2 decades. However, the determination of network complexity and its generalization capability are still unsolved. Besides, an amount of training data is required to obtain reliable modeling performance which is generally difficult to be satisfied in practice.13 Therefore, designing a suitable controller for complex processes to improve safety and increase profitability is still a challenge in practice. Because of its good modeling performance, the support vector machine (SVM) and a number of promising kernel learning (KL) methods17,18 have found increasing applications in chemical process modeling,19-22 recently. Some SVM model based nonlinear predictive control algorithms23-36 have been also proposed. However, most SVM-based controllers23-33 adopt the offline identification model, and no online model updating occurs for new samples. Actually, only using the offline model is not suitable for process automation applications because the new information is difficult to be absorbed directly into the established model. r 2010 American Chemical Society

One characteristic which makes (bio)chemical processes difficult to control is their time-varying nature due to many factors.1,2 For nonlinear processes with time-varying behavior, it is necessary to develop controllers with efficient updating mechanism to capture process characteristics and thus obtain better control performance. Although some online SVM-based controllers34-36 have been proposed recently, the computation load is still large since the training process of online SVM converges slowly. Thus, these methods are not easily implemented for a long-term online control task. Additionally, as to the process control applications, it is desirable to keep the control strategy as simple as possible for the real-time implementation.1,2,10,32 To the best of our knowledge, few recursive KL based adaptive controllers, which can be online updated efficiently, are available for nonlinear time-varying processes up to now. In our recent work, a simple and reliable online identification method, namely, selective recursive KL (SRKL), for multiinput;multi-output (MIMO) systems with the nonlinear autoregressive with exogenous input (NARX) has been developed and shown better performance than traditional methods, e.g., NN and fuzzy systems.37 The SRKL identified model has a sparse solution and can be online updated efficiently to trace the process characteristics by introducing/deleting corresponding samples selectively. Therefore, this paper aims to develop an efficient kernel-based adaptive controller, mainly using the SRKL model, Special Issue: IMCCRE 2010 Received: March 15, 2010 Accepted: August 27, 2010 Revised: August 25, 2010 Published: September 16, 2010 2773

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which can adaptively adjust its structure to capture the process characteristics and then achieve better performance. The scheme of this paper is as follows. The SRKL identification method is given in section 2. Then, the main structure of the control strategy, including an adaptive modification item to achieve good control performance, is formulated in section 3. Simulation results on a laboratory-scale liquid-level process and a continuous bioreactor with time-varying parameters are given to show the effectiveness of the proposed control approach in section 4. Finally, the conclusions are drawn in section 5.

2. TWO-STAGE SRKL ONLINE IDENTIFICATION METHOD FOR MIMO NARX PROCESSES 2.1. Two-Stage SRKL Online Identification Framework. Consider MIMO NARX processes governed by the following relationship:37 8 > y k ¼ f ½y k - 1 , ... , y k - ny , uk - 1 , ... , uk - nu þ ek > > > 8 > T < > > y k ¼ ½yk, 1 , ... , yk, M < ð1Þ > uk ¼ ½uk, 1 , ... , uk, L T > > > > > > : : ek ¼ ½ek, 1 , ... , ek, M T

where f( 3 ) is the wanted nonlinear model; k is the time instance; yk, uk, and ek are the process output, process input, and noise vectors at instance k (ny and nu are the corresponding lags of the output and input), respectively; yk,m and ek,m denote the mth output measurement of the process and the corresponding noise, respectively, at instance k with m = 1, ..., M, and M is the number of outputs; uk,l is the lth process input variable at instance k with l = 1, ..., L, and L is the number of inputs. A general input vector is defined as xk = [yTk-1,...,yk-nyT,uTk-1,..., uk-nuT]T, usually composed of several measured variables and combined with their corresponding delayed forms at instance k. Therefore, a general form of the “kernelized” nonlinear MIMO model can be directly formulated as a quasi-linear form:37 yk, m ¼ f ðw k, m , xk Þ þ ek, m ¼ w Tk, m jðxk Þ þ ek, m

ð2Þ

)

)

)

)

where the symbol wk,m denotes the model parameter vector of the mth subsystem at instance k. On the basis of the philosophy of statistical learning theory,17,18 the following optimization problem, using Tihonov regularization, is formulated to get the solution f:37 8 > < min Jðw k, m Þ ¼ 1 ek, m 2 þ γ w k, m 2 2 2 ð3Þ > : s:t: yi, m - w Ti, m jðx i Þ - ei, m ¼ 0, i ¼ 1, 3 3 3 , k where ek,m = [e1,m,e2,m,...,ek,m]T (γ > 0) is the regularization parameter which determines the trade-off between the model’s complexity and approximation error minimization and consequently controls the smoothness of the solution. After deriving the dual problem to solve the optimization problem above, the following solution can be obtained:37 ½K k =γ þ Ik Rk, m ¼ y k, m

ð4Þ

where rk,m = [Rk,m,1,...,Rk,m,k]T are Lagrange multipliers; yk,m = [y1,m,...,yk,m]T and Ik ∈ Rkk is a unit matrix; Kk is a kernel matrix, with its elements Kk(i,j) = Æφ(xi),φ(xj)æ, "i,j = 1, ..., k.17,18

Then, the KL model prediction of the mth subsystem at time k þ 1, can be obtained:37 k 1X ^yk þ 1, m ¼ f ðw k, m , x k þ 1 Þ ¼ Rk, m, i jðx i Þ jðxk þ 1 Þ γ i¼1 ¼

1 T R kk þ 1 γ k, m

ð5Þ

where kkþ1(i) = Æφ(xi),φ(xkþ1)æ, "i = 1, ..., k is a kernel vector. In summary, the development of a KL identification model amounts to solving a set of linear equations in the feature space introduced by the kernel transform. The main appealing properties, such as the structural risk minimization principle, the kernel technique, the convex optimization problem, and few free parameters to be adjusted, are preserved in this identification framework.37 To make this KL method suitable for online identification problems, a two-stage SRKL identification framework has been proposed in our recent work. It mainly includes a forward growing stage and a backward pruning stage, both of which perform the recursive algorithms to avoid the direct computation of the matrix inverse (i.e., Hk = Kk/γ þ Ik in eq 4). The samples adopted into the SRKL model are noted as nodes.22,37 The forward learning stage is to grow the nodes by introducing new process information. The backward decremental learning (i.e., pruning), is to selectively delete the old information and keep the model trace of the process time-variant characteristics. Another intention of pruning is to keep the model simple in order to improve the learning speed and save the memory space. Both the update algorithms of the forward and backward learning stages are efficient. Consequently, the SRKL identification model has a small computation load and is suitable for online identification problems. 2.2. Sparsification Strategy for Identification and Control. The sparseness is generally regarded as good practice in the learning machine. Using a sparse model, the predictions for new samples depend only on the kernel function evaluated at a subset of the training samples. Therefore, the memory requirement and the computation time on test points can be reduced. The number of support vectors in SVM learning is also kept low to avoid the overfitting problem.17,18 However, its sparse solution is obtained by solving a quadratic programming optimization problem. Therefore, a simple sparsification strategy that can adaptively control the complexity is formulated in the SRKL identification method.37 The approximation error is defined between the actual process output and the one-step-ahead prediction of the identification model at instance k as follows:37 ek, m ¼ yk, m - ^yk, m ¼ yk, m - f ðwk - 1, m , xk Þ,

m ¼ 1, ... , M ð6Þ

If the approximation error of any output channel is significant, the model is considered not accurate enough and should be improved. That is the following condition holds:37 ðjek, 1 j > δ1 Þ or ... or ðjek, M j > δM Þ

ð7Þ

where δm, m = 1, ..., M, are the predefined small positive values and named as the prediction error bound (PEB). In this scenario, [xk,yk] will be introduced as a new node, and thus the identification model is updated. Otherwise, it is not necessary to add the node according to the well-known “Parsimony Principle”. 2774

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optimization problem. For process control, it is desirable to obtain a simple analytical control law, even though it may be suboptimal, so that the computation requirement can be greatly reduced.10 An analytical control law was originally proposed using Taylor linearization method:32 uk ¼ uk - 1 þ

Figure 1. SRKL for online identification of MIMO NARX systems.

Consequently, the identification model can be kept unaltered. A smaller PEB gets more nodes, and a larger one yields a more parsimonious but less precise model. Besides, when the process noise is large, PEB can be selected larger to reject noise.37 The sparsification strategy here extends the applications in our previous study37 because it connects the identification model to the control problem. Generally, the control task is to make the difference between the process output and the reference trajectory as small as possible. If the tracking performance of a control target is required to be more accurate, PEB can be set smaller and vice versa. Therefore, this sparsification strategy can trade off the tracking performance of the controller and the model. Furthermore, from a practical point of view, the computation load to obtain a sparse model is usually small. The flow chart of the SRKL method for online identification of MIMO NARX systems is shown in Figure 1. For further details of the algorithm implementation of SRKL, refer to the literature.37

3. ADAPTIVE PREDICTIVE CONTROLLER USING SRKL IDENTIFICATION MODEL For simplicity, we limit our discussion of controller design only to single-input;single-output nonlinear processes. The extension of the proposed control strategy to MIMO processes can be straightforward formulated. Then, eq 1 can be simplified as follows: 8 > yk þ 1 ¼ f ðyk , ... , yk - ny þ 1 , uk , ... , uk - nu þ 1 Þ > > > > : uk - 1 ¼ ðuk - 1 , ... , uk - n þ 1 Þ u yr,k is the process desired output. Then, the one-step-ahead predictive control law can be obtained by minimizing the following control performance index:10,32 Jðuk Þ ¼ ðyr, k þ 1 - yk þ 1 Þ2 þ λðuk - uk - 1 Þ2

ð9Þ

where λ (λ > 0) denotes the control effort weighting factor. It is difficult to get the optimal solution for eq 9 since it is a nonlinear

ð∂f =∂uk Þjuk ¼ uk - 1 2 ½yr, k þ 1 - f ð~xk Þ ð10Þ λ þ ð∂f =∂uk Þjuk ¼ uk - 1

where ~x k = (yk,uk-1,uk-1), (∂f/∂uk)|uk=uk-1 is the input-output sensitivity function, and f(~x k) is the quasi-one-step-ahead prediction. To implement the control law described above, (∂f/∂uk)|uk=uk-1 and f(~xk) must be calculated online. It is necessary to develop efficient methods to estimate these two quantities. Although several NN based methods6,10 were used to estimate them, some disadvantages still remain aforementioned. Besides, NN models are generally not parsimonious, and hence any adaptive control scheme based on them has to deal with the issue of updating a large number of weights. In our previous study, the offline KL method was utilized to identify nonlinear processes and obtain the corresponding control law.32 Many (bio)chemical processes usually exhibit timevarying behavior, and thus it is necessary to develop recursive algorithms to update the offline KL model and the related controller. The SRKL model can describe nonlinear processes and exhibit good generalization ability, especially with few samples.37 Therefore, the SRKL method is utilized to estimate the corresponding quantities in this study. Note that SRKL(xk) is the estimation of ykþ1. Then, substituting the corresponding quantities identified by the SRKL model into eq 10 can yield the SRKL based control law: ð∂SRKL=∂uk Þjuk ¼ uk - 1 uk ¼ uk - 1 þ 2 ½yr, k þ 1 - SRKLð~xk Þ λ þ ð∂SRKL=∂uk Þjuk ¼ uk - 1 ð11Þ To compensate for both the Taylor approximation and the identification error, an adaptive modification item (AMI) μk is introduced to the control law. Then, the control law can be reformulated as μk ð∂SRKL=∂uk Þjuk ¼ uk - 1 Ek þ 1 uk ¼ uk - 1 þ ð12Þ 2 λ þ ð∂SRKL=∂uk Þjuk ¼ uk - 1 where Ekþ1 = yr,kþ1 - SRKL(~x k) is an error item of the SRKL predictive model at time k. The proposed AMI in eq 12 is much different from the adjustable parameter proposed by Gao et al.,10 although they seem somewhat similar. AMI is a time-varying parameter, while the parameter proposed by Gao et al.10 is a constant and has to be selected by simulation. Additionally, AMI can be adaptively adjusted according to the convergence analysis at every sampling time, which can guarantee better tracking performance. The convergence of a control law is important and thus was investigated in our previous study.32 It can be proved that there exists a suitable μk in eq 13 such that the control algorithm given in eq 12 can be convergent. That is, the tracking error 2775



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Figure 2. Flow sheet of the SRKL-APC strategy.

εkþ1 = yr,kþ1 - ykþ1 = 0 can be obtained. 2 λ þ ð∂SRKL=∂uk Þjuk ¼ uk - 1 ð13Þ μk ¼ ð∂SRKL=∂uk Þjuk ¼ uk - 1 ð∂SRKL=∂uk Þjuk ¼uk - 1

with CPU main frequency 1.6 GHz and 512 M memory. Besides, the integral absolute error (IAE) of set point tracking is adopted to quantify the performance of controllers. 4.1. Laboratory-Scale Liquid-Level Process. The process is identified from a liquid level system and can be described as11,12

where uk-1 ∈ [uk-1,uk]. With substitution of eq 13 into eq 12, then the adaptive control law at time k can be finally deduced Ek þ 1 uk ¼ uk - 1 þ ð14Þ ð∂SRKL=∂uk Þjuk ¼uk - 1

yðk þ 1Þ ¼ 0:9722yðkÞ þ 0:3578uðkÞ - 0:1295uðk - 1Þ

)

)

)

)

When the Gaussian kernel, K(x1,x2) = exp[- x1 - x2 /σ ] is utilized, where σ is the kernel parameter, the analytical control law can be formulated as Ek þ 1 u k ¼ uk - 1 þ N Pk 2 2 RNk , i expð - x i -x k =σ Þ xny þ 1, i -uk - 1 2

i¼1

ð15Þ where xk = [yk,uk-1,uk-1] and xnyþ1,i is the related item of the xi vector. The SRKL based predictive control strategy, including the updating structure of the SRKL model (in the dashed lines), is shown in Figure 2. Since the proposed controller is composed of two primary modules: the SRKL predictive model and the adaptive predictive controller (APC), it is abbreviated as SRKL-APC. The item GTDL is defined as a general time delay, through which both ~x k and xk can be obtained. As shown in Figure 2, the flow chart of SRKL-APC is straightforward. At instance k, the corresponding quantities (i.e., (∂SRKL/∂uk)|uk=uk-1, (∂SRKL/∂uk)|uk=uk-1, and SRKL(~x k)) are calculated by the SRKL identification model. Then, the error item Ekþ1 and the AMI μk are both introduced into the controller. Finally, the control law uk can be computed (i.e., eqs 14 and 15).

4. ILLUSTRATIVE EXAMPLES AND DISCUSSIONS Two examples, including a laboratory-scale liquid-level process and a continuous bioreactor with time-varying parameters, are utilized herein to verify the ability of the SRKL-APC method. The first example is to show how the SRKL-APC method works. Its ability to trace time-varying characteristics is investigated in the second example. The simulation environment is Matlab V7.1

- 0:3103yðkÞ uðkÞ - 0:04228y2 ðk - 1Þ þ 0:1663yðk - 1Þ uðk - 1Þ - 0:03259y2 ðkÞ yðk - 1Þ - 0:3513y2 ðkÞ uðk - 1Þ þ 0:3084yðkÞ yðk - 1Þ uðk - 1Þ þ 0:1087yðk - 1Þ uðkÞ uðk - 1Þ

ð16Þ

This process has been investigated using NN based adaptive control11 and predictive control,12 respectively. However, the controllers designed are complex, and thus this may limit their applications. The general input vector is chosen as x(k) = [y(k), y(k-1), u(k), u(k-1)], and the Gaussian kernel is utilized in all simulations. First, the set-point tracking ability of SRKL-APC is investigated. There are four parameters, i.e., [γ, σ, δ, λ], to be predefined for designing this nonlinear controller. The kernel parameter selection is still an open question in the machine learning area.17,18 Besides, there is no rigorous parameter selection theory of the online KL framework. Fortunately, the first two parameters can both work well in a wide range and a suitable pair of parameters can be chosen using some trial-and-error experiments.37 In all the simulations of this example, the regularization parameter γ = 0.0001 and the kernel parameter σ = 10 are adopted and kept unchanged to show its robustness for different situations. The PEB parameter δ = 0.02 is in accord with the precision of this control task. As mentioned above, a larger λ implies more penalties on the control effort and vice versa. With the action of AMI, λ can be selected in a wide range and the control performance will not degrade. We choose λ = 0.01 here. As shown in Figure 3, during learning of the SRKL model, the number of nodes is likely to be increasing when the process is in the transient state or the set-point changes. Besides, AMI can adjust itself adaptively according to the difference between the actual output and the desired output. Consequently, the process output can track the set points quickly, and the stable errors are zero or approximately zero. Anyhow, the stable error is far less than PEB. Another advantage of the SRKL-APC strategy, as also shown in Figure 3, is that only 13 nodes are selected out, i.e., 26% of all of 2776



Figure 3. Set-point tracking performance of SRKL-APC for liquid-level process.

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Figure 5. Performance comparison of noise and disturbance rejection for liquid-level process.

Figure 6. Schematic diagram of the continuous bioreactor.

Figure 4. Set-point tracking performance comparison of SRKL-APC and PID for liquid-level process.

the samples. It is important to point out that only these key samples are adopted in the SRKL identification model. This is different from the existing SVM based predictive controller without sparsity,31 where all of the samples are fed into the least-squares SVM model, resulting in a verbose identification model and, furthermore, a complex control law and a large computation. To provide a suitable comparison with traditional techniques, a well-tuned PID controller with parameters (Kc,Ti,Td) = (1,0.2,0.01) is utilized, in both deterministic and noise environments.12 The performance comparison of both controllers is depicted in Figure 4. Due to the learning ability of SRKL and the effect of AMI, the SRKL-APC strategy can adjust its control law adaptively. Therefore, the transient response of SRKL-APC is more satisfactory than PID and its IAE index is smaller. It is also noticed that the performance of SRKL-APC is better, compared with the obtained results of NN based controllers.11,12 To mimic an industrial environment, the process is subjected to additive noise and unmeasured disturbance. It is shown in Figure 5 that the SRKL-APC is more robust against additive noise and unknown

disturbance. The simulations run 50 times and the average of the IAE indices are 8.43 and 10.64 of SRKL-APC and PID, respectively. 4.2. Continuous Biochemical Reactor with Time-Varying Parameters. Biochemical processes play an important role in food processing, environmental management, and the pharmaceutical industry. Bioreactor control has become very important in recent years due to the difficulty in controlling the highly nonlinear behavior associated with such systems.2 Moreover, it is difficult to define an operating point for linearizing these processes.14 Therefore, in this section the proposed SRKL-APC strategy is applied to a continuous bioreactor with time-varying kinetic parameters to validate its practicability for controlling the nonlinear biochemical processes with uncertain conditions. A schematic of this continuous bioreactor is depicted in Figure 6. The kinetics of the bioreactor is described by the following differential equations14-16 8 dX > ¼ - DX þ μX > > > dt > > > dS μX > > ¼ DðSf - SÞ > > < dt YX=S ð17Þ dP > ¼ - DP þ ðRμ þ βÞX > > dt > ! > > > P S2 > > > > : μ ¼ μm 1 - Pm S= Km þ S þ Ki 2777



Figure 7. Time-varying behavior of kinetic parameters for the bioreactor (scenario 1).

where μ is the specific growth rate, which may exhibit both substrate and product inhibition; YX/S is the cell-mass yield; R and β and are kinetic parameters; and the dilution rate D and the feed substrate concentration Sf are process inputs. Detailed nominal parameters and operating conditions of this process can be obtained from Henson and Seborg.14 The control objective is to regulate the fermentor to the maximal productivity. Generally, the dilution rate D is selected as the manipulate variable, and the effluent cell concentration X, substrate concentration S, and product concentration P are chosen as process-state variables. The cell concentration X is a reasonable choice for the control output.14 Although some other control strategies, including a nonlinear internal model controller,14 a nonlinear self-tuning controller based on a polynomial discrete time model,15 and a dynamic fuzzy model based predictive controller,16 have been applied for this bioreactor, they are complex to design. Thus, a conventional PI controller from Henson and Seborg,14 where its parameters are determined using internal model control tuning rules, is adopted here for comparison. Besides, the related controller using the offline KL model (e.g., SVM),32 namely, SVM-APC for brief, is also utilized for comparison. Here, the input vector is xk = [Xk,Dk]. The identification set with a sequence of only 50 samples, which is much less than the fuzzy method,16 is generated. The identification parameters of the offline SVM are chosen by cross-validation.17,18 To show the robustness of the tuning parameter, λ is set as 0.01 and kept unchanged. For the SRKL model, the regularization parameter γ = 0.005 and the kernel parameter σ = 20 are adopted and kept unchanged to show its robustness for different situations. The kinetics of bioreactors often involves some uncertainties during the course of fermentation.2 For many bioprocesses, the maximum specific growth rate μm and cell-mass yield YX/S exhibit significant time-varying behavior and thus can be treated as unmeasured disturbances.14 In this case study, two scenarios are designed to investigate the performances of SRKL-APC for several types of uncertainties. The nominal values of these two kinetic parameters, i.e., μm and YX/S, are 0.48 1/h and 0.4 g/g, respectively. In the first scenario, they are assumed to exhibit time-varying behavior as shown in Figure 7. Besides, the output of cell concentration is

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Figure 8. Performance comparison of SRKL-APC, SVM-APC without online updating, and PI (bioreactor scenario 1).

Figure 9. Relationship between the PEB parameter and the performance of SRKL-APC (bioreactor scenario 1).

corrupted by Gaussian noise. A combination of these effects can sometimes cause unpredictable behavior during the fermentation. In this case, the PEB parameter δ = 0.05 is adopted. The control performance comparison is shown in Figure 8. Despite of these difficulties, the SRKL-APC strategy exhibits a faster response toward different set points and shows the best performance with the smallest IAE index, compared to SVMAPC and PI. In contrast, the response of the PI controller usually exhibits oscillatory behavior for set-point changes. And the SVMAPC approach does not perform well for some zones because of its limitation without online learning. Here, the relationship between the PEB parameter δ and the tracking performance of SRKL-APC is also investigated. The choice of δ is in accord with the precision of the control task and the level of the noise. Generally, if the tracking performance of a control target is required to be more accurate, the value of δ can be set smaller and vice versa. However, it cannot be too small because many process measurements are corrupted by noise and unknown disturbance. 2778



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while the PI controller degrades markedly. Before about the 90th sampling instance, SRKL-APC is only a litter better than SVMAPC and PI because μm is still near its nominal value and in the range of [0.4, 0.5]. However, when the time-varying behavior becomes severe, e.g., the value of μm is beyond 0.5 and becomes larger, both SVM-APC and PI controllers perform worse than before. In contrast, the critical performance attributes of robustness and disturbance rejection of SRKL-APC are always approving. Therefore, from all the obtained results, it is indicated that online updating of the model is necessary, especially for timevarying processes.

Figure 10. Time-varying behavior of kinetic parameters for bioreactor (scenario 2).

5. CONCLUSIONS The subject of designing adaptive controllers for nonlinear time-varying (bio)chemical processes is an important while difficult task because many of these processes are often difficult to model accurately. The SRKL identification method can determine its model structure adaptively to capture the nonlinear process dynamics. Besides, the AMI is adopted to guarantee to achieve good control performance. Therefore, the proposed SRKL-APC method has an analytical control law and can be efficiently updated to trace the time-varying characteristics of a process. The simulation results on a laboratory-scale liquid-level process and a nonlinear continuous bioreactor with time-varying kinetic parameters show that SRKL-APC is an alternative choice for nonlinear time-varying processes. ’ AUTHOR INFORMATION Corresponding Author

*Tel.: þ86-571-88320763. Fax: þ86-571-88320842. E-mail: [email protected].

Figure 11. Performance comparison of SRKL-APC, SVM-APC without online updating, and PI (bioreactor scenario 2).

The relationship between δ and the sparsity rate of the SRKL model is shown on the top of Figure 9. The number of nodes is increasing monotonically when δ becomes smaller. However, as shown on the bottom of Figure 9, the relationship between δ and IAE of SRKL-APC is not so distinct. As δ becomes smaller, the SRKL model can introduce more nodes and thus the controller can track the process’s characteristics. When δ becomes too small, some useless information might be introduced into the SRKL model and thus the performance of the controller might degrade. Therefore, for an actual process, a suitable choice of δ can be obtained according to these rules. To further explore the ability of SRKL-APC, these two kinetic parameters are assumed to exhibit more complex time-varying behavior. As shown in Figure 10, μm is assumed to be larger steadily and the variation of YX/S is more significant than the first scenario. Additionally, the measurement noise for the output of cell concentration is larger. In this case, the PEB parameter δ = 0.08 is adopted because the unmeasured noise is more significant. The control performance comparison is shown in Figure 11. The SRKL-APC method yields an overall better performance,

’ ACKNOWLEDGMENT This work was supported by the National Natural Science Foundation of China (Grant No. 61004136) and Zhejiang Provincial Natural Science Foundation of China (Grant No. Y4100457). We also appreciate the valuable comments and suggestions of the anonymous reviewers. ’ REFERENCES (1) Chu, J.; Su, H. Y.; Gao, F.; Wu, J. Process control: Art or practice. Annu. Rev. Control 1998, 22, 59–72. (2) Alford, J. S. Bioprocess control: Advances and challenges. Comput. Chem. Eng. 2006, 30, 1464–1475. (3) Zenger, K.; Niemi, A. J. Modelling and control of a class of timevarying continuous flow processes. J. Process Control 2009, 19, 1511–1518. (4) Yu, D. L.; Chang, T. K.; Yu, D. W. A stable self-learning PID control for multivariable time varying systems. Control Eng. Practice 2007, 15, 1577–1587. (5) Wang, D.; Zhou, D. H.; Jin, Y. H.; Qin, S. J. Adaptive generic model control for a class of nonlinear time-varying processes with input time delay. J. Process Control 2004, 14, 517–531. (6) Nørgaard, M.; Ravn, O.; Poulsen, N. K.; Hansen, L. K. Neural Networks for Modelling and Control of Dynamic Systems; Springer-Verlag: London, 2000. (7) Nikravesh, M.; Farell, A. E.; Stanford, T. G. Control of nonisothermal CSTR with time varying parameters via dynamic neural network control (DNNC). Chem. Eng. J. 2000, 76, 1–16. (8) Chen, J. H.; Huang, T. C. Applying neural networks to on-line updated PID controllers for nonlinear process control. J. Process Control 2004, 14, 211–230. 2779


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Adaptive Control of Nonlinear Time-Varying Processes Using

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