Chen Chen,Zhihua Xiong*,Yisheng Zhong

Department of Automation,Tsinghua University,Beijing 100084,China

Design and Analysis of Integrated Predictive Iterative Learning Control for Batch Process Based on Two-dimensional System Theory☆

Chen Chen,Zhihua Xiong*,Yisheng Zhong

Department of Automation,Tsinghua University,Beijing 100084,China

A R T I C L E I N F O

Article history:

Received 27 December 2013

Received in revised form 25 February 2014 Accepted 19March 2014

Available on line 18 June 2014

Based on the two-dimensional(2D)system theory,an integrated predictive iterative learning control(2D-IPILC) strategy for batch processes is presented.First,the output response and the error transition model predictions along the batch index can be calculated analytically due to the 2D Roesser model of the batch process.Then,an integrated framework of combining iterative learning control(ILC)and model predictive control(MPC)is form ed reasonably.The output of feed forward ILC is estimated on the basis of the predefined process2 D model.By minimizing a quadratic objective function,the feedback MPC is introduced to obtain better control performance for tracking problem of batch processes.Simulations on a typical batch reactor demonstrate that the satisfactory tracking performance as well as faster convergence speed can be achieved than traditional proportion type(P-type)ILC despite the model error and disturbances.

?2014 Chemical Industry and Engineering Society of China,and Chemical Industry Press.All rights reserved.

1.Introduction

For tracking the reference trajectory in batch processes,ILC[1]is a popular control framework in which the in formation of previous batches is used to ad just the control input for the next batch.It is widely used and applied to batch processes[2,3]owing to the key characteristic that the convergence condition is not related to the system matrix[4].However,control performance of batch processes under traditional ILC is usually degraded when the non-repetitive disturbance and uncertain dynamics such as output noise exist within a batch.Meanwhile,the convergence speed is not concerned in most cases.

It is reasonable to combine other control methods with ILC to deal with these situations.Amann[5]applied the predictive idea to the ILC scheme,referred as predictive optimal ILC,and it had been shown that forecastmay con tribute to the convergence speed.Other researches showed that encouraging results can be obtained by the combination of ILC with feedback control method such as MPC,in which the input was ad justed based on the output prediction of a predefined dynamic model.To overcome the model uncertainty and process disturbances, Ch in and Lee[6]proposed a batch MPC(BMPC)algorithm,in which not only the measurements from the current batch but also the information from the past batches was used,and later their work is extended to quadratic BMPC(QBMPC)[7]based on the quadratic ILC(Q-ILC)[8]. Xiong et al.[9,10]used sh rinking horizon model predictive control (SHMPC)within the batch,while the ILC was used between the batches to improve the tracking performance.In our previous work[11],an integrated scheme was studied further by combining a traditional P-type ILC with MPC.On the other hand,the integration of these two special control schemes should be very careful for the inconsistent predictions [12].

It is noted that these researches mentioned above are all in time domain,however,a batch process can be considered as a standard two dimensional(2D)system[13].A typical input affects not only the next time steps of ongoing batch but also the next batches while ILC is used.In view of the2D system theory,the2D dynamic of the system,referred as time domain and batch domain,can be taken into account together.Thus,it is feasible that the batch-wise feed forward controller and the time-wise feedback controller can be designed and integrated to realize better control performance.Kurek and Zaremba[14] explained the traditional P-ILC in 2D system theory and proved the convergence condition in the 2D framework.Based on a controlled auto-regressive integrated moving-average(CARIMA)model,Shi and Gao[15]presented an integrated robust learning control framework by using a 2D Roesser model and linear matrix inequalities in order to deal with the uncertain perturbation.In their later works[16,17],an integrated scheme,referred as 2D-GPILC,was proposed to combinegeneralized predictive control(GPC)with ILC.Recently,Mo et al.[18] presented a 2D dynamic matrix control(2D-DMC)algorithm and the sufficient conditions of convergence and robustness of the method are discussed.

In spite of these works referred above,few 2D-theory based ILC control framework involves the nature characteristic of systemsuch as the state-transition matrix and system responses.In this paper,according to the2D Roesser model,an integrated predictive iterative learning control scheme is presented not only to overcome the model error and uncertain process disturbances but also to deliver faster convergence speed than traditional P-ILC performance.In this control scheme, the traditional P-ILC is used from batch to batch,while the input is re-ad justed by involving MPCwithin the batch.Based on the prediction of the responses of P-ILC,a quadratic objective function is minimized to determine the current input changes.The 2D-IPILC algorithm presented in this paper may provide a suitable framework of combining different types of ILC with MPC.Advantages of those candidate methods can be also contained in this scheme,but the control performance may be affected by the choice of parameter in the algorithm.Simulations are presented to demonstrate control performance of the scheme.

2.Design of 2D-IPILC Method for Batch Process

2.1.Tracking control problem of batch process

In this study,a batch process is considered as a class of singleinput-single-output(SISO)linear time-invariant(LTI)system.It is assumed that the batch process operates over a finite time duration and the process can be described by the following discrete-time state-spacemodel: where k is the batch index,t is the time index and t∈[1,N],N is the number of sampling intervals,χ∈Rn,u∈R,y∈R are state,input and output variables,respectively,d(t,k)denotes output disturbance and A,B,and C are real matrices with appropriate dimensions, respectively.

The task of the proposed control method is to find the input sequence Uk=[uk(0),uk(1)…,uk(N?1)]Tover the time duration such that for a given reference trajectory Yd=[yd(1),yd(2),…,yd(N)]T,the process output tracking error e(t,k)=yd(t)?y(t,k)is satisfied with:

Let us define:

Then the above process in Eq.(1)can be described by the following two-dimensional Roesser model[18]:

Therefore,the tracking control problem is to find the input change Δu(t?1,k)in the 2D system[Eq.(4)]in order to guarantee the convergence of the tracking error e(t,k).

In this study,the 2D system[Eq.(4)]satisfies the following assumptions.

Assumption 1.All batches run from the same initial conditions, i.e.χ(0,k)=χ0,(?k＞0),such that for the 2D system[Eq.(4)]the boundary condition is satisfied with η(1,k)=0,(?k＞0).

Assumption 2.The output noise d(t,k)is bounded by a constant Bd＞0, i.e.?t,k,‖d(t,k)‖＜Bd.Hence,the next inequality holds:‖Δd(t,k)‖≤2?Bd.

2.2.Response of P-ILC

No rm ally,for the above tracking control problem of batch process,the traditional P-ILC can be used here and the change of input from batch to batch can be calculated by the following ILC law[13]:

where L denotes the learning rate.

Substituting the control law[Eq.(5)]in the P-ILC to Eq.(4),the system can be reform ed as:

Define the state-transition matrices in the 2D system[Eq.(6)] as[13]:

Then,based on the 2D theory[19],the response of system[Eq.(6)] can be described as follows:

where:

For clarity of the description,we use the following notations:

It should be noted that in the response of Eq.(8),it holds η(1,j)=0,?j＞0 according to Assumption 1.Then under the P-ILC,the general response of 2D Roesser model(Eq.(6))can be rewritten as[19]:

Furthermore,in the system response of Eq.(11),e(i,0)(?i∈[1,N]) is the initial tracking error and should have been known,and the noisechanges Δd(i,j)are also bounded according to Assumption 2.It has been proved that the convergence of tracking control can be obtained if the learning rate L satisfied[14]:

which means all the eigenvalues of the matrix I?C·B·L are inside the unit circle.

However,the convergence condition above is a sufficient condition[14],and it can be also found that even though the condition is satisfied,perfect control performance can be hard ly obtained due to the effect of disturbance d(t,k).

2.3.2D-IPILCLaw

In our previous works[9],MPC is induced to combine ILC to achieve better performance.Here the idea is still used,the input is re-ad justed on the control profile determined by the ILC from batch to batch,and the control law of2D system of Eq.(4)takes the following form:

where Δu（t?1，k+1）represents the batch-to-batch control part and ?u（t?1，k+1）represents the within-batch control part,respectively. Δu（t?1，k+1）is usually designed as:

where the learning rate L can be designed as the above normal P-ILC. Other ILC algorithm,such as Q-ILC[8],can be also used if the selected control law conforms the convergence condition of Eq.(12).

Even though P-ILC may be robust to a certain amount of model error from batch to batch,it usually cannot handle the uncertain process disturbances within a batch.In order to overcome the model uncertainty and process disturbances a swell as accelerate the convergence speed, the re-ad justed part?u t?1，k+1（

）is determined by MPC for the reason that the MPC is a suitable way to overcome the model error and uncertain disturbances within the current batch[9].

Substituting the control law of Eq.(13)to the 2D system of Eq.(4), the system can be reformulated as:

Let us define:

Based on the 2D theory[19],the response of system of Eq.(15)leads to:

Therefore,using the above system response,the error transition model predictions can be calculated analytically.Based on the above response of Eq.(17),the predictions of the error transition model can be estimated as:

where^ξt+l｜t，k

（），l∈1，m[]is the prediction,m is the prediction step.In this study,the prediction step is set to be m,m∈[1,+∞),which means the predictive step in MPC is not limited and maybe larger than the batch duration N,andρ=[m/N]is defined.?(t+l,k)is the value that can be calculated by using those in formation known before the time t and batch k,which can be described as follows:

Define the following vectors:

And define a matrix G in the prediction model of Eq.(18)as:

As a result,the predictive model can be formulated in the matrix form by:

where Gmis a part of thematrix G according to the time t:

whereα,β,andγare weighting parameters and are satis fi ed with α＞0,β≥0,γ≥0.

The above cost function(Eq.(24))can be rew ritten in thematrix form as follows:

where R and Q are the weighting parameter matrices as follows:

Herewe only consider the unconstrainedcase,thus an analytical solution of the MPC within the current batch can be obtained through straightforward calculation which leads to:

Ateach time t within the current batch k,the first element of Eq.(26) is used as the re-adjusted input in the control law of Eq.(13).

3.Analysis of the 2D-IPILC Algorithm

The 2D-IPILC algorithm proposed above may provide a suitable framework of combining different types of ILC with MPC.The 2D-IPILC algorithm can be transform ed in to other special method by selecting different weighting parameter matrices Q and R in the objective function.

If the learning rate L is set to be 0 and the parameters are chosen as β=0,α＞0,γ＞0,the optimization problem of Eq.(25)is transform ed as follows[6]:

which can be considered as the method of BMPC.Furthermore,if the learning rate L is chosen like Q-ILC,then the algorithm may be turned into QBMPC[7].

If the predictive step mis set to m=N at each beginning of a new batch and the predict step is also sh rinking by time t within a batch, then the algorithm can represent the combination of SHMPC and ILC proposed by Xiong etal.[9].

If the predictive step m isselected as m=κ?N,whereκisan integer, and the other parametersare selected as L=0,β=0,γ=0,andα＞0, and the 2D-IPILC isonly adop ted when t=0,then the quadratic objective function can be reform ed as[5]:

which is similar to multi-batch predictive ILC proposed by Amann et al.[5].

It is reasonable that the algorithm may contain all properties of the candidate control methods due to the particular framework.In this framework,the input of the system is not only determined by the feed forward batch-to-batch control but also the feedback withinbatch control,and by the combination of these two different methods. Thus,the system may converge to desire trajectory more robustness and faster.

Despite the advantages of the algorithm,the tracking performance is still influenced by the values of parameters Q and R in the quadratic objective function of Eq.(25).Under the conditions that a relatively accurate process model is built or more attentions are paid to the convergence speed,it is recommended that the para meter may be selected as β=0,and α＞γ.Otherwise,for the opposite parameters,the robustness can be considered more reasonably.

In Ref.[18],a2D dynamic matrix control(2D-DMC)algorithm is presented based on an integrated model.More attention is paid on the analysis of convergence condition and robustness against repeatable and non-repeatable interval uncertain ties.Simulation results show that control performance is improved by the combination of the feedback control and the feed-forward control.Nevertheless,how the past uncertain ties transmit in the process with batch and time and how it does impact the current performance are not concerned.In this paper, based on 2D Rosser theory and specific system description,it can be found that after a 2D linear transform determined by the 2D difference of time and batch,the past in formation affects current performance by superposition principle.

It should be noted that if the noise d(t,k)in the process of Eq.(1)is a kind of repetitive disturbances for all batches,the perfect tracking performance can be obtained[9].

4.Simulation on a Typical Batch Reactor

The simulated process is a typical batch reactor with temperature as the control variable which is studied by Logsdon et al.[20].The reaction scheme is A→k1B→k2C.The objective of the reactor is to maximize the product B after a period of time.The process can be described by the following differential equations:

where χ1and χ2denote the concentrations of input A and B,respectively,and the initial conditions are χ1(0)=1,χ2(0)=0. Input u=T/Trefis the dimensionless temperature where Tref= 348 K is the reference temperature and the temperature T is constrained by 298(K)≤T≤398(K).Values for other parameters in Eq.(29)are k1=4.0×10?3,k2=6.2×105,E1=2.5×103and E2= 5.0×103.The period of batch reaction is set to be 1.0 h and the duration is divided in to N=10 equal intervals.The desired reference trajectory is set to be the same as our precious work,so does the nominal input trajectory[10].

The root-sum-square-error(RMSE)of tracking error is used to illustrate the tracking performance.For comparison,P-ILC is also used to deal with the same problem.Furthermore,model disturbance is simulated to verify the performance of the algorithm.

In this study the real nonlinear differential model is assumed to be unknown.Instead,an approximate discrete linear model is established to describe the process.For such a typical batch reactor,a two-order system can be used to approach the dynamic behavior.Correspondingly,the process can be described by a difference equation:

By means of different input profiles deviating from the nominal input,three batches were simulated and are used as the historical process datasets to identify the parameters of the discrete system of Eq.(30).Then,it is easy to get a state realization of the discrete-time model:

where the disturbance is assumed to be an uniform distribution between[?0.0025,0.0025].

Fig.1.Testing performance of the discrete linear model.

From Fig.1,it can be seen that the discrete linear model can approach the dynamic behavior although model errors exist.

Considering the model error,the parameter of the special method is chosen as:m=5,L=0.4,α=1,β=0,γ=10,which means more attentions are paid to the robustness of the process rather than the convergence speed due to the inaccurate model and disturbances. Fig.2 shows the trajectories of the product concentration χ2under the 2D-IPILC algorithm.It can be found that output converges to the desired trajectory asymptotically.Figs.3 and 4 indicate the trajectories of input and each part of the control law of Eq.(13), respectively.

As a comparison,P-ILC is also used to handle the tracking control problem with the learning rate as L=0.4.Even though the P-ILC method is a non-sensitive model method,it cannot deal with the measuring noise very well.Fig.5 shows the RMSE of the tracking error of these two different methods.It is shown that the combination of ILC and MPC does make the convergence more robust.It is also noted that the MPC may not speed up obviously the convergence of tracking error due to the model mismatch.

Fig.2.Trajectories of concentration.

Furthermore,it always occurred that the kinetic parameters of batch reactor may change due to catalyst activities or impurities etc.These disturbances may cause the mutation of the process model so that the control performance is affected correspondingly.

To show the applications of the proposed framework to the unexpected mutations,a scenario,in which a kinetic parameter is changed, is considered in this simulated process.In this case,it is assumed that the parameter E2in the model is increased by 5%,from the nominal data of 5.0×103to 5.25×103.Fig.6 shows the RMSE of control performance under this disturbance.It is shown that the tracking error deteriorates sharply at the 16th batch,and then converges to thedesired trajectory asymptotically while the mutation occurred.It is also indicated that the combination of ILC and MPC is a model non-sensitive method,while the suitable parameter of the algorithm is chosen.

Fig.3.Input temperature profile.

Fig.4.Trajectories of each part of the control law.

Fig.5.RMSE of each batch in the comparison.

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☆Supported in part by the State Key Development Program for Basic Research of China(2012CB720505),and the National Natural Science Foundation of China (61174105,60874049).

*Corresponding author.

E-mailaddress:zhxiong@tsinghua.edu.cn(Z.Xiong).

Iterative learning control Model predictive control Integrated control

Batch process

Two-dimensional systems