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Analysis on Data-based Integrated Learning Control for

By April Wallace,2014-09-06 21:59
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Analysis on Data-based Integrated Learning Control for

    豆丁网地址,/msn369

    Analysis on Data-based Integrated Learning Control for

    Batch Processes

    112JIA Li, CAO Luming, CHIU Minsen 5

    (1. Shanghai Key Laboratory of Power Station Automation Technology, Department of

    Automation,College of Mechatronics Engineering and Automation, Shanghai University,

    Shanghai 200072;

    2. National University of Singapore, Department of Chemical and Biomolecular Engineering,

    10 Singapore 117576)

    Abstract: A novel integrated learning control system is presented in this paper. It systematically integrates discrete-time (batch-axis) information and continuous-time (time-axis) information into one uniform frame. More specifically, the iterative learning controller is designed in the domain of batch-axis, while an adaptive single neuron predictive controller (SNPC) in the domain of time-axis. As

    15 a result, the batch process can be regulated during one batch, which leads to superior tracking performance and better robustness against disturbance and uncertainty. Moreover, the convergence and tracking performance of the proposed integrated learning control system are firstly given rigorous description and proof. Lastly, to verify the effectiveness of the proposed integrated control system, it is applied to a benchmark batch process, in comparison with ILC recently developed.

    20 Key words: batch process; integrated learning control; iterative learning control (ILC); feedback control

    0 Introduction

    Since batch process satisfies the requirements of the modern market, it have been widely used

    25 in the production of low volume and high value added products, such as special polymers, special

    [1]chemicals, pharmaceuticals, and heat treatment processes for metallic or ceramic products. For

    the purpose of deriving the maximum benefit from batch process, it is important to optimize the operation policy of batch process. Therefore, optimal control of batch process is very significant. However, with strong nonlinearity and dynamic characteristics, the optimal control of batch

    30 process is more complex than that of continuous process and thus it needs new non-traditional techniques.

    Iterative learning control (ILC) has been used in the optimization control of batch process

    [1, 2]because of its repeatability . However, in ILC system, only the batch-to-batch performance of

    the batch process is taken for consideration but not the real-time feedback performance. Thus, ILC

    35 is actually an open-loop control from the view of a separate batch because the feedback-like control just plays role between different batches. Thus it is difficult to guarantee the performance of the batch process when uncertainty and disturbance exist. Therefore, the integrated optimization control technology is required in order to derive the maximum benefit from batch process, in which the performance of time-axis and batch-axis are both analyzed synchronously. Amann

    40 proposed a composite iterative learning algorithm which combines Riccati feedback control and

    [3]ILC feedforward control . Gao presented an iterative learning control method based on internal

    [4]model control (IMC) to solve the problems of model error and delay . Lee presented a new ILC

    control method to control the temperature of the production process based on quadratic criteria by

    [5-7]using a linear time-varying model (LTV) . Lee combined model predictive control technology

    [8] 45 with ILC and thus the problem of model inaccuracy and interference problem were solved .

    Foundations: National Natural Science Foundation of China (61004019), Research Fund for the Doctoral Program of Higher Education of China (20093108120013

    Brief author introduction:JIA Li(1975-),Female,Associate professor. Research interest: Identification, control and optimization of complex nonlinear control system. E-mail: jiali@staff.shu.edu.cn

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    Xiong proposed an on-line shrinking horizon re-optimization control method to reduce the

    [9]tracking error of the batch process . Xiong and Zhang used ILC with a LTV disturbance model to

    [10, 11]guarantee the tracking error convergence when considering of external interference . Rogers

    [12]firstly analyzed the convergence of batch process by employing 2D system stability criterion .

    50 Kurek and Fang built 2D model and proposed feedback-feedforward iterative learning control

    [13, 14]strategy for batch process . Gao et al did a series of 2D optimization control based research for

    [15-17]batch process . However, analysis on the stability and robustness of batch process is troublesome. Moreover, most reported results assume that the batch process is linear in order to simplify the design procedure.

    55 Motivated by the previous works, an integrated learning control system based on input-output data is proposed in this paper. The contributions of this paper are shown as follows. 1) A novel integrated learning control system is presented, which systematically integrate discrete-time (batch-axis) information and continuous-time (time-axis) information into one uniform frame, namely the iterative learning controller in the domain of batch-axis, while an adaptive single

    0 neuron predictive controller (SNPC) in the domain of time-axis. As a result, the batch process can 6

    be regulated during one batch, which leads to superior tracking performance and better robustness against disturbance and uncertainty; 2) the self-tuning algorithm of SNPC controller is derived by a rigorous analysis based on the Lyapunov method such that the predicted tracking error convergences asymptotically; 3) the convergence and tracking performance of the proposed

    65 integrated learning control system are firstly given rigorous description and proof.

    The paper is structured as follows. Section 1 presents the proposed data-driven based integrated learning control system. Section 2 presents performance analysis. Simulation example is given in Section 3, followed by the concluding remarks given in Section 4.

1 DATA-BASED INTEGRATED LEARNING CONTROL

    70 SYSTEM DESIGN FOR BATCH PROCESSES

    Based on data-based model, an integrated learning control system is designed as Fig.1. The system is divided into two parts: the iterative learning control (ILC) working as feedforward controller and adaptive single neuron predictive controller (SNPC) playing as feedback controller. More clearly, the part of ILC reflects the repetitive learning procedure of batch process in the

    75 domain of space, which guarantees the end-product quality approximate the targeted value after several iterations by fully using of the repetitive characteristic. And another part, SNPC boxed by

    dotted line reflects the dynamic characteristics of batch processes in the domain of time, and plays a role of adjusting the current batch when uncertainty and disturbance exist, without the internal product quality information of the batch process.

     80 For the convenience of discussion, the number of batch and batch length are respectively

    t defined as and t . Here the batch lengthis divided into equal intervals. Defineyas the k T f f d

    uk , t uk , t andare ILC control variable and SNPCtargeted end-product quality, ILC SNPC ;

    y k , t ;is the corresponding product quality of two control variable of time in k -th batch, t ;

    ˆ u k , t uk , t uk , t yk , t is the predicted output of data-based control actions, , ;ILC SNPC

     85 model. Since batch process is repetitive in nature, the model prediction at the end of the k-th

     batch, ˆ yk,t yk, t can be corrected by. During k-th batch, the control policy of f f

    U obtained from ILC optimization controller and the control policy of Ucomputed ILC ,k SNPC ,k

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    and is sent into batch process to improve the U from SNPC controller are summed as k ˆ Yperformance, Yand are respectively product quality variables and predicted product quality k k

    90 variables. As discussed above, the proposed integrated learning optimization control action can be

    described as: U UU(1) k ILC ,k SNPC ,k u , u , u , ktktkt(2) ;ILC SNPC low up u? u k , t ? u low up y? y k , t ? y low up low 95 where u, uare the lower and upper bound of control input sequence respectively. ,y up yare the lower and upper bound of end-product quality.

     UUk 1 k 1 u(k , t ) u(k,t)U U YILC k optimizationk k Batch Process controller U ;ˆk e?; ˆe k 1 ky d ,;Feedforward Control / k ˆY on Batch-axes k 1 qˆe k

    a

    Y ;k Data-based Model e(k , t ) u(k , t 1) ;SNPC wk;1 ;u(k , t )SNPC ;e(k , t ) ?;,;;wk;2 ;e(k, t 1) Feedback Control u(k , t ) SNPC on Time-axes ywk;de(k,t 1) 3

    Parameter adjustment

    Fig. 1 Integrated learning optimization control system of batch process 100

     It is clear that the proposed integrated learning control system not only makes the end-product quality approximate the targeted value after several iterations but also guarantees

     better robustness and convergence by the action of real-time feedback control at time axis.

105 1.1 BATCH-AXIS ITERATIVE LEARNING CONTROL [18] The predicted output of the data-based model is

     ˆ yk , t j Model y k , t j 1y k , t j 2,, y k , t 1, (3)

    u k , t j 1, k , t j 2,, u k , t 1;

    To reflect the physical limitation in batch process, the input trajectory is bounded by the

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    low up lower and upper bounds, u and u, and the final product qualities are bounded by the lower low up y y and . Thus the boundness of the input of data-based model coupled110and upper bounds, [19] with its stability guarantees the boundness of the output of data-based model. In addition, model-plant mismatches always exist, especially when the model is a data based empirical model

     developed from a limited amount of process operational data. Moreover there always exist uncertainties/disturbances that vary from batch to batch so there is no one-to-one correspondence

    between the product qualities and input variables. To limit the deterioration of control 115

    performance due to model-plant mismatches and uncertainties/disturbances, error correction is

    adopted to overcome the impact of model errors. It is described in the following.

     The model prediction error of end quality is written as:

     ˆ (4) ˆek,t y ,kt y, kt ;; ff f

     120 where y(k 1, t ) ˆ e(k, t ) and are defined as average prediction error of end quality and model f f ; correction term ; ;k k 1 1(5) ˆ ˆ ; yi,t ;ˆ ek,t ?;; ?y i,t ei,t f f f fi 1k i 1k ˆ (6) ˆ yk 1,t yk 1,t ?ek,t ; f f f ? where is error correction term parameter.

    The batch-axis iterative learning control optimization problem can be formulated as: 125 2 2 U U (7) min J U t yU , t ;y ; ILC ,k 1 k ILC ,k 1 f ILC ,k 1 f d R Q

    (8) U U U ILC ,k ILC ,k 1 k

    where Q is selected as constant matrix here, R=r I, whereQ q I, is dynamic matrix,R k T T

    rM Iis bounded and its upper bound is .is T -dimensional matrix. k r T

    The solution to the constrained optimization problem in (7) can be easily solved by using 130

    classical mathematic method or intelligent algorithm such as sequential quadratic programming

    [20][21][22](SQP) algorithm , particle swarm optimization (PSO) , and genetic algorithm (GA) .

1.2 TIME-AXIS FEEDBACK CONTROL

     The proposed SNPC is described as ;

    uk , t uk 1, t uk , t 135 (9) ;SNPC SNPC SNPC

    uk , t wk , t e k , t wk , t e k , t wk , t ? e k , t (10) ;SNPC 1 2 3

    e k , t e k , t e k , t 1(11) ;

    ? e k , t e k , t e k , t 1;(12) wk , t i 1, 2, 3is adjustable parameter. In order to apply it to practical batch process, where ?; i

    the following transformation is taken 140; k ,t i ; ?;e,wk, t 0 ;i ?; (13)w k, t i 1, 2, 3 ? i ;, k t i ;e wk, t 0 ;;i ;

     k , t where is a real number. ;i ;

    e k , t According to the characteristic of batch processes, is studied by the following two

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     different cases, which is distinguished by the threshold of ? :

    145 (1) ek,t yt y k,t , t y k , t ? ?;y d f d f

    ˆe k , t yk , t y k ,t (2) , yt y k , t ( ?; r d f ;; ;where ;

    (14) ˆˆyk,t 1 Modely k,t , y k,t 1,, y k,t 1,u k,t ,u k,t 1,,u k,t 1; r r ;;For the convenience of discussion, we denote ;;

    150 ek,t ?ek,t ek,t ? ek,t ?(15) ;; u ?; ? T(16)wk, t ?wk, t wk, t wk, t ? 1 ;2 3 ?? T k, t ?k, t k,t ?(17) k,t ;; 2 1 3 ?? ;By using Lyapunov method, the adjustment algorithm of parameters can be obtained as ;

    (18) k , t 1 k , t k , t ;

    ;? ˆ ?ˆ ˆ ?yk , t L 1?yk , t 2?yk , t 1 ?; ?; ; e k , t L e k , t 1 e k , t ?; ; ? ?r rr?u k , t ?u k , t ?u k , t ?; ?; (k , t ) ,;2 2 2 ;? ˆ ? ? ˆ ? ? ˆ ??yk , t L 1?yk , t 2?yk , t 1 ;; ? ? ? ?? ?155 k , t k , t k , t ?u ?u ?u ;;;? ? ? ? ? ?

     1 0 0 (?? T1;;? ?k , t ek , t ?w ? ? u ;;0 ? ?? 0 (? ? 2T T; ? ? ? k , t e k , t ek , t ? ? u u? ? 0 0 ( 3 ; ;;;?wk , t 0 0 ?;;1 T ;;?wk , t ?wk , t ?; ?(19) k , t 0 0w,; 2 T ?; ?; ;? k , t ? k , t 0 0 wk , t ;3 ;

    where (,(and(are learning rates that regulate the self-learning speed of SNPC, and isL 1 2 3

     prediction horizon. ;e(k, t) Thus, is selected as r ; ; yt y k , t ? ? ; ˆ yt yk , t ,d f f d ;;160 ek , ?r ˆˆ yk , t yk , t , yt y k , t ( ?;? ; r t?f d

    ;

     2 PERFORMANCE ANALYSIS

     2.1 Convergence analysis

    For the convenience of discussion, it defines that Cis an optimization controller (ILC) of 1 Cis a feedback controller (SNPC) of the time-axis, and denotes the batchG the batch-axis, 2

     165process.

    Theorem 1: if the feedback controller satisfies the condition of (20), the proposed integrated

     learning optimization control policy converges with respect to the batch number k, namely U ? 0 as k ?, . k

    (20) 1 G j)Cj) 1 2

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170 Proof: It can be seen that the final output of the single ILC system yk,t can be described ILC f

     as follows y k,tCGy ILC f 1d

    The corresponding output error Eis obtained 1 y CGy 1 CG yE 1 d 1d 1d

    of the proposed integrated learning optimization control system is 175 k, t ;The final output y f ;

     gotten C CyG 1 2 d y k , t f1 GC 2 The corresponding output error E of this situation is 2

     G C Cy1 CG y 1 2 d1d E y,; ;2 d 1 GC1 GC 22

    180 By using the condition of (20), we have

     Ej) 12 ,; ? 1Ej) 1 G j) Cj) ; ;2 1 ;;

    The conclusion of means that the tracking error of the integrated )? E)Ejj ;2 1 optimization control system is less than or equals to that of the system without real-time feedback control.

     185Therefore, the time-axis feedback controller satisfies the following inequality;

    e U?e U (21) ;k ILC ,k ;Q Q Similar to most new controller design methods developed in the literature, perfect model

     assumption is assumed in this work in order to develop the first of its kind that guarantees the convergence of control policy with the proposed integrated control scheme derived from a

     190rigorous proof. As a result, (7) can be simplified as 2 2;; U U (22)min J U , k 1e U ILC ,k 1 ILC ,k 1 ILC ,k 1 kR Q

    where eU y yk 1,t . ILC,k 1 d ILC f

    Suppose that U is an optimal solution of (22), then we got ILC ,k 1 2 22* * * (23) U UeU? J U, k 1? J U, k 1U;e ; ;ILC ,k 1 ILC ,k 1 ILC ,k 1 k ILC ,k 1 ILC ,k 1 R Q Q * * 195 is an arbitrary input of -th batch. IfU U where U k 1 is chosen to be , ILC ,k 1 ILC ,k 1 k * then U U . And (22) satisfies the following relationships ILC ,k 1k 2 2e U U e ? J U , k 1? J U , k 1 ; ;;ILC ,k 1 k ILC ,k 1 k Q Q ; 2 2U e e U ?and Thus, we have ; ;k ILC ,k 1 Q Q ; U ?e U (24) e ; k ILC ,k 1 ;Q Q

    200 From (21) and (22), we can get

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     ;

    e U?Ue e U ?(25) ;; k 1 k ILC ,k 1 ;Q Q Q ; ;

    e Uand satisfy the following equation. It is clear that eU;k ILC ,k 1 ;Q Q ;;lim e U U lim e (26) ILC ,k 1 k QQk ?, k ?, From (22) and (23), it can be obtained 222;;; U U , k 1? J U , k 1e U J e U 205(27) k 1ILC ,k 1 ILC ,k 1 k k R Q Q

     Then ;;; 2 22 U ? e U e U e U ;e U e U e U ;; ?; ;; ?; ;; ?; ILC ,k 1kILC ,k 1 kILC ,k 1kk 1 ;?; R Q Q Q QQQ

     (28)

     Therefore, we obtain;; 2 ;;; ? lim e U ? e U e U e Ulim U ILC ,k 1 k ILC ,k 1 k 1 k RQ Q QQk ?,;k ?,; ; ;

    ;210 (29) ;e U ? lim e U = lim e Ue U ;ILC ,k 1 ILC ,k 1 k k QQQ Q k ?,k ?,;;

    ;? M ? lim e Ue U ; ILC ,k 1 k QQ k ?,;; ; ; ,e Uwhere M is the upper bound of e U ILC ,k 1 k Q Q ; ;; namely e U? M . eUILC ,k 1 k Q Q U Thus, from (26) and (29), we have the conclusion that convergences to 0, namely k R U convergences to 0. This completes the proof. Q.E.D. k

    215 2.2 Tracking performance analysis 2* * In (22), (the ideal value of e ?M is defined as the minimum of M is zero). Q [23]Motivated by our previous work , the definition of bounded tracking and zero tracking of the

    integrated learning optimization control system are defined as:

    ? ? ? ( 0Definition 1: Bounded-tracking. If there exists a for every and ? ( 0 2 *r ?220 U= U U such that the inequality ? for e U Mholds when ;k SNPC ,k ILC ,k k 1 k 0 Q

    every k ( k. 0

    Definition 2: Zero-tracking. If it is bounded-tracking and there exists and ? ( 0 2 * U= U U limsuch that the equality 0 e U Mholds when k 1 k 1 SNPC ,k 1 ILC ,k 1 Q k ?,;

    r ? . k 1 0

    225 Lemma: Denoting the initial control profile at k-th batch as U. For every , there exists? ( 0 k 0 0

    U ? ? ? ? ( 0U such that the optimal solution k 1 -th batch satisfies in 0 ILC ,k 1 1 ILC ,k 1 0 0 ; ; ; ;;2 ;??*[23] ? r, where ;.when e U ? U M ; ?k 1?1 ILC ,k1 ;ILC ,k10 0 0 Q ??;;

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    e U Theorem 2 The tracking error of the proposed integrated optimization problem described ;k U .U = Uby (22) is bounded-tracking for arbitrary initial control profiles ILC ,k SNPC ,k k 00 0 2 230Moreover, if the function of( U = U U ) is derivable and thee U k k SNPC ,k ILC ,k Q

     optimization solution is not in the boundary, it is zero-tracking for arbitrary initial control

     U U = U.profiles SNPC ,k ILC ,k k 00 0

    ? ? ? ( 0 Proof: (1) From Lemma, it is easy to know that for every , there existssuch? ( 0 2 *r ? that the optimal solution in k 1 -th batch satisfies . e U M ? when 0 k 1 ILC ,k 1 0 0 Q

     235From (25), the following formula is available for any batch k;;2 2 2 e U U ?e e U ?(30) ; k ILC ,k 1 ;ILC ,k ;Q Q Q ; 2 * * e U M e U;for any k ( k 1 . Therefore, the Then we get M ?;ILC ,k ;ILC ,k 1 0 0 2 Q ;?Q ;;;; 2 2 * * ;;;; formula M M ?e Ue Uis available for any k ( k.0 ILC ,k ILC ,k Q Q ; ;;; e UFrom (21), we know, then the following formula is available? e U ILC ,k k Q Q 2 2 * *;;; ; 240e U M e U (31) M ?k ILC ,k Q Q

     k, the proposed integrated optimization problem is Therefore, for any U at timek 0 0

    bounded-tracking. 2 * * U ) as . Considering a(2) Set the optimal point of( U = UU e U , M k k SNPC ,k ILC ,k Q 2 ;e U small neighborhood (if there exists not only one optimal solution in function ;* k U Q ;

    245 as the union of the optimal solutions), in which ( U= U U ), then we consider * k SNPC ,k ILC ,k U

     there are no other extremal solutions except global optimal solution only. Because the optimal 2 solution of( U = U U ) is not in the boundary, then we can find a sete U k SNPC ,k ILC ,k k Q ;2 ? ?* U;uncovering the boundary of . Namely, we have U M ? e * * ? ?U ILC ,k U U Q ?;?; ; U ?, and is not in the boundary of . ;* U U ;

    ? r ? , if250 According to Lemma, we know that for any forementioned , there exists a , ? ( 0 k 1 0 2 * k ( k then for any initial solution U while , namely we have;e U M ?; k 0ILC ,k 0 Q

    U ?, U, is not in the boundary of . Therefore, the optimal control * ILC ,k ILC ,k U U hence

    profile , is the extremal solution of U , k ( k 0ILC ,k; ; 2 2;; U U J U , k e U inside the open set, but not in the boundary of * ILC ,k ILC ,k ILC ,k k 1 UR Q

    255 . U

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    Therefore, when k ( k, optimal control profile U satisfies the following formula 0 ILC ,k described as: 2 ? e U; ILC ,k?J U, k ILC ,k Q; ;(32) U 0?; 2r ? U ,; ?U ?U ILC ,k ILC ,k k ILC ,k k 1

     According to (29), we have , and then we can get lim rU U 0 k ILC ,k k 1 k ?,; ;2 ;; ? e U 2ILC ,k Q ;260 e U converges to extremal solution when k tends to That is to say . lim 0 ;ILC ,k Q k ?,;?U ILC ,k U ?, infinity. Because the condition holds and there are no other extremal solutions except *ILC ,k U 2 ;g ;e U global optimal solution only in for function , we make conclusion that * ILC ,k ;U Q ;

    converges to global optimal solution, namely: 2 2 * *(33) limU M 0 lim e e U M ILC ,k 1 ILC ,k 1 Q Qk ?,;k ?,;

     265From (29), we get ;;lim e U lim e U ILC ,k 1 k QQk ?, k ?,;

     Therefore 2 ;* lim e U M 0;k 1 Q k ?,; This completes the proof. Q.E.D.

     270 3 EXAMPLE

    To demonstrate the effectiveness of the proposed scheme, this example considers the

    following batch process, in which a first-order irreversible exothermic reaction kk[24,25,26]: 1 2 takes place A ??? B ???C

    2 4000 exp 2500 / T xx 1 1

    2 5 x 4000 exp 2500 / T x 6.2 10exp 5000 / T x 2 1 2

    275 where xand xare respectively the reactant concentration of component A and B, is the reactor T 1 2

    temperature.

    TTis normalized usingT T T/ T T, in whichand are 298T Firstly, d min max min min max Kand 398Krespectively. Tis the control variable bounded between 01, and xt is the ~? d 2

    output variable. The nominal operating conditions are: x0 1 , x0 0 . 1 2

    280 xt The control objective is to maximize the endpoint concentration of , by B 2 f

    manipulating the reactor temperature, T. To proceed with the proposed method, 30 batches of d independent random signal with uniform distribution between [0, 1] are used to obtain

     input-output data for training purpose. Applying the identification procedure in [15] results in a neuro-fuzzy model with 6 fuzzy rules. The control performance of the proposed integrated control

    scheme is shown in Fig.2. Clearly, it can provide good control at the first batch quickly. For 285

    [23] comparison purpose, the iterative optimization control method is also designed to optimize the

    batch process. The performance of these two controller designs is compared in Fig.3. It is evident

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     that the iterative optimization control method cannot provide good control in first several batches. 290

     (a) Output trajectory based on integrated control 295

     (b) Input trajectory based on integrated control Fig.2 Input and output trajectories based on the proposed integrated control method

    300

305

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