Li Jia*,Wendan Tan

Shanghai Key Laboratory of Power Station Automation Technology,Department of Automation,College of Mechatronics Engineering and Automation,Shanghai University,Shanghai200072,China

Keywords:Model predictive control Batch process Just-in-time learning(JITL)model

ABSTRACT Considering the two-dimension(2D)characteristic and the unknown optimal trajectory problem of the batch processes,an integrated model predictive control-iterative learning control(MPC-ILC)for batch processes is proposed in this paper.Firstly,the batch-axis information and time-axis information are combined into one quadratic performance index.It implies the integration of ILC and MPC algorithm idea,which leads to superior tracking performance and better robustness against disturbance and uncertainty.To address the problem of the unknown optimal trajectory,both time-varying prediction horizon and end product quality control are employed.Moreover,an integrated 2D just-in-time learning(JITL)model is used to improve the predictive accuracy.Furthermore,rigorous description and proof are presented to prove the convergence and tracking performance of the proposed MPC-ILC strategy.The simulation results show the effectiveness of the proposed method.

1.Introduction

Batch processes are suitable for the manufacturing of low volume,high value added products,such as special polymers,special chemicals,and pharmaceuticals[1].With the increasingly fierce competition in the international market economy,in order to obtain more profits,it is essential to improve the control efficiency of batch processes.However,compared with continuous process,the traditional optimal control methods is not suitable for batch process because it contains discrete and continuous characteristics.Therefore,new optimization and control techniques must be employed to satisfy the control demands of batch process.

Considering the repetition characteristic of batch process,iterative learning control(ILC)can be used as a control strategy of batch processes[2–4].After its initial development for industrial robot[5],ILC has been widely used in batch processes with its repetitive natures to get perfect control optimization[6,7].Xiong and Zhang presented a batch-to-batch model-based iterative optimal control strategy for batch processes to the difficulties in developing detailed mechanistic models[8].Camacho proposed a run to run optimization based on unfolded Partial Least Squares[9].But iterative learning control from the batch-axis is an open-loop control.Without real-time feedback from time-axis,it is easy to be influenced by the uncertainties and disturbances and it can't guarantee the control performance of the batch process from time-axis.Therefore,an integrated optimization control technique for batch process is necessary,in which the batch-axis information and time-axis information are combined into one system.Rogers firstly analyzed the stability and convergence of two-dimension(2D)theory[10].Chin proposed a novel RFC-ILC framework that can virtually separate ILC from the real time feedback control(RFC)action.A two-stage technique has been constructed by combining quadratic criterion-based ILC(QILC)and batch model predictive control(BMPC)methods after necessary modifications[11].Lee et al.proposed iterative learning control-based batch process control technique for integrated control of end product properties and transient profiles of process variables[12].To guarantee the convergence in batch-axis and robustness in time-axis,based on control objectives defined over a single cycle or multiple cycles,two-dimensional linear quadratic(2DLQ)optimal control schemes was developed by Shi[13,14].C.Duran-Villalobos presented an iterative learning modeling method and used it to control batch fermentation process[15].Chen J et al.through input–output feedback linearization obtain linear model,based on this model,a model predictive control scheme is derived for the case of unconstrained control as well as constrained control[16].Lu et al.proposed natural gradient method based model-free optimization to control the quality of batch process[17].However,most 2D controllers consider that the batch process is linear or can be locally linearized.But the batch process has strong nonlinearity and dynamic characteristics,which indicates that linearization and optimal control strategies are not suitable for practical industrial processes.How to design 2D controller to adapt to more general nonlinear processes is challenging.Moreover,in the above mentioned methods,most control algorithms are based on optimal reference trajectories to control output.Nevertheless,it's difficult to find a uniform optimal reference trajectory because the batch process has strong nonlinearity and inherent dynamic nature.How to solve this problem is also a challenge.

Motivated by the previous works,an integrated model predictive iterative learning control for end product quality of batch process is proposed in this paper.The batch-axis information and time-axis information are combined into one quadratic performance index and control the end output through time-varying MPC strategy.Moreover,a 2D integrated JITL model with natural adaptivity in time-varying horizon is used to improve the predictive accuracy.Just-in-Time Learning(JITL)[18–20]was recently developed as an attractive alternative for modeling the nonlinear systems.Based on the current input sample points,the just-in-time learning model selects the most similar data from the historical database of the modeling object to construct the modeling neighborhood of the current data point,and then uses the linear polynomial to combine a local model.The 2D integrated JITL model in this paper is the alteration of the traditional JITL.

Furthermore,this paper gives rigorous description and proof of the convergence analysis and tracking performance of the proposed integrated model predictive iterative learning control system.

The paper is structured as follows.Section 2 gives a brief description of a 2D integrated JITL model and the integrated model predictive iterative learning control for end product quality control system.Section 3 presents the analysis of convergence and tracking performance.The simulation example is given in Section 4.Finally the conclusion is given in Section 5.

2.Integrated MPC-ILC Strategy with a 2D JITL Model

2.1.A 2D integrated JITL model

The batch process is viewed as general nonlinear structure in the conventional just-in-time learning(JITL),without considering the characteristic of repetition of batch process.With the batch data increasing,the scale of the database becomes much bigger,which leads to the increase of the amount of calculation of JITL model.Aiming at this problem,batch-axis and time-axis integrated just-in-time learning model is employed to identify batch process in this paper.The batch-axis and time-axis integrated database of batch process is shown in Fig.1,where x,k,and t represent variable,batch,and time,respectively.x(k,t)is a subdatabase established from time-axis considering the repetitive consistency by making use of the characteristics of3D data in batch process,through which the batch axis and time axis 2D database framework is formed to improve the modeling efficiency.

According the current query data,JITL with natural adaptive ability identifies the model parameters online through rolling modeling,which makes the model parameters updated quickly and has better model precision[21].Therefore,combine JITL algorithm[22]with batch-axis and time-axis integrated database to model for batch process.The detailed algorithm is described as follows:

Step 1:Define the current query data Xq(k,t)=(yqk,t?1,yqk,t?2,uqk,t?1,uqk,t?2)Tand historical database Xi(k,t)=(yik,t?1,yik,t?2,uik,t?1,uik,t?2)T,i=1,2,N.According to time-axis information,the historical database samples can be divided into T th subdatabase{X(k,t)}={X(k,1),X(k,2),····X(k,T)}.Then,choosing the corresponding X(k,t)to model by judging to which the current query data belong.

Step 2:Compute the distance and angle between Xq(k,t)and Xi(k,t):

where,d represents the distance between the query data Xqand Xiin the entire database,ΔXq=Xq?Xq?1,ΔXi=Xi?Xi?1.

(1)Ifcos(θi)＜0,the angle between ΔxqandΔxiis really huge,and it means that these two samples have low similarity and should be abandoned.

(2)Ifcos(θi)≥ 0,the angle between Δxqand Δxiis tiny,and it means that these two samples have high similarity and should compute the similarity number si:

Step 3:Arrange all siin the descending order,the JITL will choose the l data to build a local ARX model:

Fig.1.The batch axis and time axis integrated database of batch process.

where,Ψ =[α1,α2,α3,α4]is the model parameter,l=kmax～kmin.Denote Wl=diag(s1,····sl)a diagonal weight matrix with diagonal elements being the first l largest values of si,and calculate:

where,X∈RN×4is the matrix with every row corresponding to

The local model parameters are then computed by:

Step 4:According to the validation errors[23],the optimal l is determined by:

where elis as the follow:

where yjis the j th element ofare the j th row vector of X and Pl.Then the predicted output for query data is computed as:

Remark 1.The proposed 2D JITL model is established from time axis considering the repetitive consistency by making use of the characteristics of 3-D data in batch process.Considering the characteristic of repetition of batch process,the similar modeling samples are divided into the same subdatabase.Under this framework,for the current input data,the corresponding subdatabase is chosen from time axis to set up JITL model,namely,the local model of the batch axis.The proposed 2D JITL model improve the modeling efficiency,namely,end-product quality is closer to the expectation.

Remark 2.The computational complexity and model accuracy are considered in this paper.After a large number of experiments,the experimental results show that two-order ARX model can achieve precision requirements and has less the amount of calculation.So we selected two-order ARX model in this paper.

2.2.An integrated MPC-ILC system for batch process

The formulation of the proposed integrated model predictive iterative learning control for end product quality of batch process is depicted in Fig.2.Although there is only one objective function in this paper,the proposed control strategy systematically integrates batch-axis information and time-axis information into one quadratic performance index.It also implies the integration of ILC and MPC algorithm idea.in Eq.(10)embodies the ILC idea which guarantees the convergence of the proposed algorithm while the rest of the two terms in the objective function make the system have good robustness against real-time disturbances and uncertainties;so,it means an integrated batch-axis and time-axis control strategy.

The k,uk(t|t)and yk(t+1)are the batch number,the real time control signal and corresponding output,respectively.ydis the desired product quality.Ptis the time-varying prediction horizon of integrated MPCILC controller.Based on the information of previous batch input and real-time feedback,the MPC optimizer can calculate an input sequence by solving an optimization problem online.The control signal uk(t|t),which is the first component of Uk(t|t),is sent to the process.Then the Uk(t+1|t+1)is recalculated as the previous instant with shrinking prediction horizon.The step is implemented repeatedly until the end of the current batch.At next batch,this procedure is repeated to let the product qualities asymptotically converge toward ydat the batch end.

As discussed above,the proposed integrated MPC-ILC control for end product quality applied in time-axis can be described by the following objective function:

Subject to

Fig.2.Integrated MPC-ILC system for end product quality of batch process.

It should be noted that above MPC prediction horizon Ptis time-varying.Ptranges from current instant t to the final instant of a batch.

The solution to the constrained optimization problem in Eq.(10)can be easily solved by classical mathematic method or intelligent algorithm such as sequential quadratic programming(SQP)algorithm[24],particle swarm optimization(PSO)[25],and genetic algorithm(GA)[26].

In summary,the following steps describe integrated MPC-ILC control strategy.

Algorithm.Step 1:According to time-axis information,the historical database samples are divided into T subdatabase.Identify 2D JITL model based on historical batch operation time-axis.Let k=1 and initialize

Step 2:At the t?th instant of the k?th batch,solve the optimization problem Eq.(10)to achieve uk(t|t)as the actual control signal,and then measure the corresponding output yk(t+1).

Step 3:If t＜T,set t=t+1 and go back to Step 2,else set k=k+1 and go to Step 2.

3.Performance Analysis

3.1.Convergence analysis

Theorem 1.Based on 2D integrated JITL model,the control sequence Ukof integrated MPC-ILC policy will converge to a constant sequence along batch cycle,whose increment corresponds to zero,namely ΔUk=Uk+1?Uk→0 as k→∞.

Proof.We can rewrite Eq.(10):

According to the objective function,we get

at the t+1 instant of the k-th batch,the Pt+1dimension solution to the integrated MPC-ILC optimization problem in Eq.(10)Uk(t+1|t+1)is the optimal solution of the t+1 instant,so we have

According to Eqs.(12)and(14),we get

By such analogy,we have

Supposed the initial condition Uk(0|0)is the same at the t=0 instant of the each batch,according Eq.(10)we can get the following inequality.

From Eqs.(16)and(17),after T times iteration,we can get the following inequality

Furthermore,we get

Form Eq.(18),we have

3.2.Tracking performance analysis

By referring to our previous research[27],the definitions of bounded tracking and zero tracking are employed in the integrated MPC-ILC control system.

Definition 1.Bounded-tracking.If there exists a δ = δ(ε)＞ 0 for every ε＞ 0 and Uksuch that the inequalityholds when rk0+1＜δ for every k＞k0,where M*is positive.

Definition 2.Zero-tracking.If it is bounded-tracking and there exists δ＞0 and Uk+1such that the equalityholds when rk0+1＜ δ,where M*is positive.

Lemma.For the arbitrary initial control sequence Uk0at k0?th and every ε＞ 0,there exists δ = δ(ε) ＞ 0 such that the optimal solutionbatch satisfieswhen rk0+1＜δ,where

Table 1 Parameter values for the batch reactor

Theorem 2.The tracking error e(Uk,tf)of the proposed integrated MPCILC control system can asymptotically converge to compact set Θefor arbitrary initial control sequence with respect to the batch numberk,namely aswhereandis the upper bound of

(1)Perfect model

Proof.The perfect model assumption is assumed in this condition,we can rewrite Eq.(10)

According to Lemma,it is easy to know that for arbitrary initial control sequence and every ε＞ 0,there exists δ=δ(ε)＞ 0 such that the optimal solution Uk0+1at k0+1th batch satisfieswhen rk0+1＜ δ.Moreover,for k＞ k0,the inequalityholds.And according to above convergence analysis,we know thatis decreased with the increase of the batch number k,namely,

Furthermore,we have

Therefore,the tracking error e(Uk,tf)of the proposed integrated MPC-ILC control system can asymptotically converge to compact set for any Uk0.and k0.

According to the above convergence analysis,we know0,thenThus,holds.That is to say,converges to extreme value as k tends to be infinite.Because the condition Uk∈ΘU?for any k＞k0holds and there is only one global optimum solution in ΘU?,we can infer thatconverges to global optimal solution,namely

Therefore,when the optimal solution Uk∈ ΘU?,for any initial control sequence,the proposed integrated MPC-ILC control system is zerotracking.

(2)Model–plant mismatch

From Eq.(10),we get

According to the results in Eq.(1),we know thatis bounded to be smaller than ε when k＞k0by adjusting rk,namely

then

namely

Thus,we get

Remark 3.Because the model and plant cannot match completely,the system output would fall into suboptimal solution if rkis large at the beginning.Thus,rkcan be designed as monotonically decreasing function aboutAs a result,rkwould gradually decreased along with the increase of the model precision,then system output would approach the desire quality.Moreover,at the beginning,if rkis large,that would cause control signal change sharply to obtain better product quality.Therefore,in this paper,monotonically decreasing function for rkis designed as

where τ1,τ2and τ3are all penalty parameters used to adjust the weight rkfrom batch to batch.

Table 2 Final output error value based on two controller systems

Fig.3.Control trajectories at 1st,3rd,5th,7th,10th,20th batches.(Solid line:integrated MPC;dotted line:traditional ILC).

4.Example

The proposed integrated MPC-ILC control strategy for end product quality of batch processes is applied in a typical batch process reactor,in which in which an irreversible exothermic reaction A →k1B →k2C happens[28].This process can be described by the following dynamic equations:

where x1and x2respectively denote the reactant concentration of A and B,and T is the reaction temperature.The values of parameters k1,k2,E1and E2are given in Table 1.

The reaction temperature is normalized through u=(T?Tmin)/(Tmax?Tmin),where Tminand Tmaxare 298(k)and398(k),respectively.u is the control signal confined within[0,1],and x2(t)is the corresponding output.The control objective is to approximate the end-time output to an ideal value by adjusting the control signal u from batch to batch.The reaction time t in each batch satisfies 0≤t≤tf.The initial operating conditions are:x1(0)=1,x2(0)=0.

Fig.4.Output trajectories at 1st,3th,5th,7th,10th,20th batches.(Solid line:integrated MPC;dotted line:traditional ILC).

In this part,the 2D integrated JITL model is employed.A set of independent random signals with uniform distribution between[0,1]are used to simulate the input data.The parameters of JITL can be obtained by rolling modeling according to current query data and historical database.The input form of JITL is chosen as follows:

where x2,k(t)represents the product concentration at t-th time instant of the k-th batch.The control object is to drive the system output y=x2to approximate to yd(tf)=0.61.We can choose the dynamic parameter Rkas follows:

In the simulation,the values of τ1,τ2and τ3are chosen by trial and error.The following parameters is adopted to initialize the simulation system:U0=τ1=20,τ2=1,τ3=1 × 103,q=300=1.

The proposed integrated MPC-ILC control for final product quality in batch processes based on 2D JITL model is compared with traditional iterative learning control(ILC)strategy[24].The final output error values can be seen from Table 2.And the trajectories of two controller systems are shown in Figs.3–5 respectively.Therefore,it's clear that the proposed control strategy has better convergence accuracy and smaller final output error than those obtained by traditional iterative learning control strategy.

In order to test the robustness of the proposed control strategy,two additional cases are considered:outside disturbance and inside disturbance.

Case 1.In this case,the parameters of batch processes are varied to simulate the internal uncertainties.This case is simulated by changing parameters E1and E2at the 6th batch.

where E10and E20are the nominal value of parameters E1and E2as shown in Table 1.

Fig.5.Curve of output error based on two control strategies.

Fig.6.6th batch control and output trajectories based on two control strategies with internal uncertainties.

Case 2.The output of batch processes is corrupted by 10%?ydexternal disturbance at instant t=4 of the 5th batch in this case.

From Fig.6 to Fig.7,we can see that the proposed MPC-ILC strategy has better robustness,which can maintain good performance when disturbance exists.

In order to demonstrate the effectiveness of the proposed 2D JITL model,we add a simulation that the proposed integrated MPC-ILC control for final product quality in batch processes based on 2DJITL model is compared with neuro fuzzy model(NFM).From Fig.8 to Fig.9,we can see that the 2D JITL model proposed in this paper has higher accuracy and better control performance than those obtained by NFM.

Moreover,the proposed integrated MPC-ILC control for final product quality in batch processes based on 2D JITL model is compared with the traditional MPC strategy.And the trajectories of two controller systems are shown in Figs.10 and 11.Therefore,we can get that the proposed control strategy has smaller final output error than this obtained by MPC strategy.

Fig.7.5th batch control and output trajectories based on two control strategies with external disturbances.

Fig.8.Output trajectories at 1st,3th,7th,10th batches.(Solid line:2D JITL model;dotted line:NFM).

Fig.9.Curve of output error based on two models.

Remark 4.In order to compare two different strategies,a simulation that the proposed integrated MPC-ILC control for final product quality in batch processes based on 2D JITL model is compared with MPC strategy is added in this paper.The trajectories of two controller systems are shown in Fig.10 and Fig.11.It's obvious that the proposed control strategy has smaller final output error than this obtained by MPC strategy.Remark 5.In order to compare two different models,we add a simulation that the proposed integrated MPC-ILC control for final product quality in batch processes based on 2D JITL model is compared with neuro fuzzy model(NFM).From Fig.8 to Fig.9,we can see that the 2D JITL model proposed in this paper has higher accuracy and better control performance than those obtained by NFM.

Fig.10.Output trajectories at 1st,3th,6th,7th,10th batches.(Solid line:MPC-ILC;dotted line:MPC).

Fig.11.Curve of output error based on two models.

5.Conclusions

In this paper,considering the unknown reference trajectory and the 2D characteristic of the batch process,an integrated MPC-ILC control for batch processes based on 2D JITL model is proposed.The batch-axis information and time-axis information are combined into one quadratic performance index and the end product quality control is employed.Moreover,we can give rigorous description and proof of the convergence analysis and tracking performance of the proposed integrated model predictive iterative learning control system.Lastly,the proposed control strategy is applied in a typical batch process reactor,which demonstrates that the proposed control strategy has better control performance.