J. Cent. South Univ. (2017) 24: 2613-2623

DOI: https://doi.org/10.1007/s11771-017-3675-6

2D multi-model general predictive iterative learning control for semi-batch reactor with multiple reactions

BO Cui-mei(薄翠梅)¹, YANG Lei(杨磊)¹, HUANG Qing-qing(黄庆庆)¹, LI Jun(李俊)¹, GAO Fu-rong(高福荣)²

1. College of Electrical Engineering and Control Science, Nanjing Tech University, Nanjing 210009, China;

2. Department of Chemical and Biomolecular Engineering, The Hong Kong University of Science and Technology,Kowloon, Hong Kong, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Abstract: Batch to batch temperature control of a semi-batch chemical reactor with heating/cooling system was discussed in this study. Without extensive modeling investigations, a two-dimensional (2D) general predictive iterative learning control (2D-MGPILC) strategy based on the multi-model with time-varying weights was introduced for optimizing the tracking performance of desired temperature profile. This strategy was modeled based on an iterative learning control (ILC) algorithm for a 2D system and designed in the generalized predictive control (GPC) framework. Firstly, a multi-model structure with time-varying weights was developed to describe the complex operation of a general semi-batch reactor. Secondly, the 2D-MGPILC algorithm was proposed to optimize simultaneously the dynamic performance along the time and batch axes. Finally, simulation for the controller design of a semi-batch reactor with multiple reactions was involved to demonstrate that the satisfactory performance could be achieved despite of the repetitive or non-repetitive disturbances.

Key words: two-dimensional system; iterative learning control; general predictive control; semi-batch reactor

1 Introduction

The batch and semi-batch operating has been widely used in the industry production or laboratory research, especially chemicals, pharmaceutical, polymerization and food industries with small production volumes and variable requirements [1, 2]. The characteristics of batch processes operation and control problems are different from the continuous processes. Continuous processes tend to operate in steady-state and the continuous processes control aims at regulatory control around an optimal region, while the batch processes emphasize dynamic operation in a small scale, and aim for an optimal performance by controlling [3]. Different operation conditions may result in poor product quality or reproducibility, and consistent end-product quality is an important objective in batch process operations. However, the on-line sensors of the product quality measurement are often unavailable in current industrial practices; the end-product quality is indirectly controlled by some measured variables, such as temperature and pressure, to track some pre-assigned trajectories [4, 5]. Therefore, the research on batch process control has focused on tracking a given reference trajectory for batch processes. Considering the control of batch reactors, temperature controlling is the key to ensure the reliability and stability of the product. However, batch reactions, as one kind of batch processes, are affected by multi- process, multi-variable and batch time uncertainties [6], which is more difficult to be control. Thus, some advanced control strategies are introduced. In the early days, the most industrial control problems of the batch control were solved by gain-scheduling PID control, sometimes with apposite feed-forward compensations [7]. And for the characteristics of the repetitive operations, iterative learning control (ILC) has been proved to be an effective control technique in many batch processes [8]. The results from some references [9, 10] indicate that the control scheme with batch-wise ILC can improve the batch-wise convergence and robustness, and it can guarantee the time-wise control performance to some extent. Recently, ZULKEFLEE et al [11] presented a nonlinear model predictive controller to control the temperature of batch esterification reactors, which is effectively embedded design architecture to deliver the real-time desired output. However, the feedback controller and feed-forward ILC were separately designed [12, 13]. It is difficult to optimize the control performance since batch operations may also be affected by non-repetitive disturbances. LEE et al [14] proposed batch model predictive control (BMPC), with the regular predictive control algorithm incorporated with the feature of ILC. SHI et al [15] proposed two-dimensional generalized predictive control for the batch processes with 2D dynamics. LIU and GAO [16] presented a robust 2D-ILC for batch processes with state delay. CHEN et al [17] proposed a two-dimensional theory of integrated predictive iterative learning control (IPILC).

Due to the repeatability of batch process, it can be considered from two-dimensional (2D) point of view along the time and batch axes, where the time-wise dynamics is mainly determined by process characteristics and the dynamics along the batch axis is determined by the process repeatability. From the 2D viewpoint, the dynamics of the system along both the time and batch axes will be considered in the process modeling as well as the controller design. GAO et al [18] proposed a novel 2D-GPILC scheme composed of generalized predictive control (GPC) and iterative learning control (ILC) for batch processes, especially for injection process. The results showed that the dynamic performances of batch process was improved significantly by the time-wise generalized predictive controller combined with batch-wise ILC algorithm.

In this work, a 2D multi-model generalized predictive iterative learning control (2D-MGPILC) to a semi-batch chemical reactor with multiple reactions was proposed. The paper is organized as follows: in the second section, a general semi-batch reactor was considered with a typically complex operation pattern, and the multi-model structure with time-varying weights was developed to describe the nonlinear dynamic characteristic in the multi-stage operation process. In the third section, the 2D-GPILC algorithm was proposed to optimize simultaneously the dynamic performance along the time and cycle axes. In the 2D-MGPILC scheme, the quadratic cost function was minimized on the 2D multi-model weighted by time-varying functions of the control system. In the fourth section, a simulation for the controller design of a semi-batch reactor with multiple reactions was involved to verify the effectiveness of the proposed control algorithm. In the end, conclusions were given with brief discussions on the limitation of 2D-MGPILC scheme in the practical industry process application.

2 2D multi-model general predictive iterative learning control schemes (2D-MGPILC)

2.1 Mathematic description of batch process

Operation rules and multi-stage characteristics of batch or semi-batch processes are often more complex [19]. The operational modes are of typical multistage and time-varying. For instance, some inputs are manipulated over the entire batch, while others are manipulated at a specific time point or a particular time interval only. At the same time, there are several stages in entire run, such as the preheating stage, the reacting stage, and the cooling stage existing in a biochemical batch reaction. The variables related to batch processes are often classified into four kinds as follows: 1) the manipulated input variables; 2) the controlled output variables; 3) the monitoring output variables; 4) the product quality variables, which are measured only after the completion of a batch.

To accommodate such complex operation patterns, a discrete-time nonlinear model structure with time-varying dimension was considered to describe the complex semi-batch process. The input, output and disturbance sequences of the whole batch were defined as

    (1)

These variables were assumed to be decided by a nonlinear algebraic mapIf the output reference trajectory was represented by y_r, the output error can be defined as

                     (2)

2.2 Identification of multi-model with time-varying weights

Since direct identification of time-varying impulse response coefficients for complex nonlinear dynamic batch process is very difficult, in the most ideal case a mass of batch data would be in need to get all the parameters of model [20]. On the other side, due to the ILC feature embedded in the model structure and 2D-MGPILC control algorithm, it is less important of a precise model to achieve precise control for the batch process. Therefore, here, a relatively simple model can be used instead, such as combining some linear models through using time-varying weight functions, defined as the following equation:

              (3)

where T is the time-period of each batch, which is supposed to be the same duration for every batch process; t and k are the time and batch axes, respectively; u_k(t), y_k(t) and w_k(t) are respectively the input, output and disturbance of the process in the kth batch and at time t; A(q^–1) and B_i(q^–1) are both operator polynomials

          (4)

,

                         (5)

and Δ_t is the time-wise backward difference operator, i.e., In the above, B_i/A typically represents the coefficients of ARX model of batch or semi-batch process, which can be estimated through input and output data along the time axis. Uncertain dynamics is included as unknown disturbance w_k(t). And μ_i is the time-varying weight coefficient, whose function forms may be set as step, trapezoids, exponential, Gaussian curves, etc. t_i denotes the switching time coefficient of the model, which can be optimized through an iterative procedure where the sum of the squared prediction errors over the run is minimized by the recursive least-squares algorithm.

               (6)

where y(t) is the output of the process;  is the following:

                (7)

        (8)

        (9)

Once the weight coefficients were obtained, t₁…t_n would be optimized through the input and output information of many batch cycles until the J (t; θ, t₁, …, t₂) reaches the minimum.

The following identification problem was considered:

   (10)

where a₁, a₂, …, a_na and b_i,1, b_i,2, …, b_i,nb were expected to be identified by input-output information.

The following column vectors were defined:

(11)

(12)

                    (13)

                 (14)

                 (15)

Then, Eq. (10) can be rewritten in a matrix form:

    (16)

The prediction error as a measure of identification performance was considered, and the following cost function was proposed:

        (17)

where and are sets of the identified parameters.

2.3 Recursive solution

Partial derivatives of J were taken with respect tothe following sets of equations were obtained.

                (18)

Defined

F^–1(t)

(19)

               (20)

When t≤t₁,

                             (21)

When

                             (22)

according to matrix inversion lemma,

        (23)

Denoted the RHS of Eq. (18) as G(t),

                   (24)

Then,

 (25)

2.4 Iterative learning laws

For the repetitive characteristics of batch or semi-batch reactors, ILC law has been introduced as

                (26)

where r_k(t) is the updating law, and u₀(t) is the original value of iteration. The calculating relations of r_k(t) and u_k(t) can be expressed as

                (27)

where q_k^–1 is the batch-wise backward-shift operator. Contrast to the traditional ILC algorithm, the control input has been defined at time t of the kth batch based on the control input at the time of last batch, and the input change along the batch axis of previous time as well. The control law can be rewritten equivalently as

                 (28)

Equation (28) was substituted into model  the 2D model along both time and cycle axes was as follows:

             (29)

where r_k(t), y_k(t) and Δ_k(w_k(t)) are the input, output and disturbance changes along batch axis of the process. The model is the 2D equivalent model of the batch system. We defined the following equation

                             (30)

where

Thus, Eq. (29) can be unified and simplified as follows:

             (31)                                

2.5 2D-MGPILC algorithms

Due to the existence of GPC algorithm, the input and output information of the process may be divided into known and unknown parts according to model

at time t:

    (32)

(33)

(34)

Note that A₂ is a nonsingular matrix. With an assumption that is white noise, the estimation of the output prediction can be expressed as follows:

   (35)

                              (36)

(37)

Note that F_k(t) is based on the input and output information of current and last batch, which can be measurable. To obtain the updating law r_k(t), we defined a cost function based on the 2D process model as follows:

                  (38)

where integers n₁ and n₂ (n₁≥n₂) represent the time-wise prediction horizon and control horizon, respectively; is the output estimation at the ith step ahead in the kth batch; y_r(t), t=0, 1, …, T, is the reference trajectory. α₄(i)≥0, α₁(j)≥0, α₂(j), α₃(j)≥0 are the weight coefficients of cost terms, respectively. And then the quadratic cost function Eq. (38) can be expressed in a matrix form:

                  (39)

                      (40)

           (41)

                         (42)

                  (43)

               (44)

               (45)

               (46)

To minimize the cost function Eq. (39) based on the 2D process model, the following relationships among variable in time axis, variable in batch axis and variable were defined as follows:

   (47)

 (48)

where

               (49)

            (50)

According to the above equations, the generalized 2D prediction model was obtained as follows:

   (51)

The optimal control law was derived by minimizing the cost function based on the prediction model:

                    (52)

where K₁ and K₂ are the first rows of their corresponding matrixes; K₃ is the up-left element of corresponding matrix.

The 2D control system was constructed by 2D plant as follows:

(53)

where K₁, K₂, K₃, K₄ and K₅ are the adjustment coefficients based on the weight coefficients for output errors of last batch, time-wise control signal changes of last batch, batch-wise control signal changes of current batch, batch-wise update signal of current batch and batch-wise output changes of current batch, respectively.

2.6 Structure analysis and parameter tuning

From the two-dimension system point of view, the 2D-MGPILC law is feedback control scheme consisting of the 2D model and feedback controllers, and the signal transmitted by the real time information flow and the last

cycle information flow. The process consists of a batch process a time-wise integratorand a cycle-wise iterative learning control loop.

In the 2D-MGPILC structure, the control performance along both time and batch axes is determined by the dynamics of the process and the structure parameters as well. Guidelines of tuning the parameters are proposed as follows:

1) The larger the weight factor α₄(j) of updating law is, the better the robust stability will be got of the control system; While the smaller the value α₄(j) is, the faster the convergence rates along both time and batch axes will be obtained.

2) The weight factor α₂(j) influences the change of u_k(t) along time axis. A larger value will decrease the variable quantity of u_k(t), which results in the better robust stability along time axis, while a smaller value will be conducive for tracking performance along time axis. And it is necessary for efficient tracking that α₂(j) shall be set to 0.

3) The weight factor α₃(j) influences the change of u_k(t) along the batch. Better robust stability along the batch will result from lager values of α₃(j), while smaller values will result in faster convergence rates along the batch.

3 2D-MGPILC applied to semi-batch reactor with multiple reactions

Batch reactors are the main place of batch reactions, which are strict with temperature requirements. To increase/decrease the temperature, various heating/ cooling systems are embedded in the batch reactors. Many semi-batch reactors operate in fed-batch mode, and several materials will be fed into the reactor after being calculated over the run. In this section, a fed-batch consecutive reaction system with two reactions is considered, where the conversion and yield index of the end product become important. The two reactions with different reaction rates and activation energies are defined as follows.

Desired product C will be formed by the first reaction of reactants A and B.

                              (54)

Undesirable product D will be formed by the second reaction reactant B.

                               (55)

Because of the lager activation energy of the second reaction than the first, low temperature will result in higher yield of desired product C relative to undesired product D, which may be penalty for the conversion of A. The second reaction rate is strongly dependent on the concentration of B in the reactor, thus the higher concentration of B is in the reactor and then the faster undesired product D will be formed. As a result, selectivity may be increased by slow feeding. However, for a given conversion of A, batch time will be longer with slower feed rate.

3.1 Operation pattern of a semi-batch reactor

Unlike the continuous process, the operation modes of semi-batch processes may be time-varying. Therefore, it is inherently more difficult for design and control of batch reactors than continuous stirred tank reactors. To test the effect of operation parameters on conversion, yield, and reaction rates, the reaction may be set at different temperatures, with the different ratios of the reactants, and different run time. The optimal operation of the semi-batch reactor involves the optimal operation modes and the perfect tacking performance, with economic performance criterion optimized. And the measurement periods of input or output variables are different; especially the end-product quality can be measured only after the batch in real industrial process. Therefore, accuracy of temperature control is used to evaluate the control system performance. In this research, a typical temperature operation trajectory of the semi-batch reactor is used, as shown in Fig. 1, which demonstrates the complexity during a batch process operation.

In Fig. 1, the reactant A is fed firstly into the reactor and the heat-up is followed until the temperature reaches a certain value, which is defined as pre-heating stage. And then, the reactant B starts to be fed into the reactor at a flow rate F(t) (m³/min), which can be manipulated at a function of time or a fixed value. The reaction starts at the time point and continues until the end of the batch or the reactant A is completely consumed, which is defined as reaction stage. Maintaining the final yield of product C at an expected target value is the main objective of the control scheme. The two manipulated variables, temperature of the jacket and flow rate of reactant B, are considered during the semi-batch operation.

3.2 Model identification of semi-batch reactor

The dynamic model of the semi-batch reactor is a nonlinear model consisting of a total mass balance, component balances and an energy balance for the liquid in the reactor and cooling water in the jacket, respectively (see appendix). And the values of the kinetic and process parameters are shown as Table 1. The process model for 2D-MGPILC design was identified through the simulating data using the above mechanism model for two batches. The process model was represented through the following multi-model using time-varying weights functions:

              (56)

where

             (57)

where B₁/A represents the dynamics for the initial heat-up and cooling stage; B₂/A represents the dynamics for the reaction stage; The two time-varying weighting functions μ₁(t, t₁) and μ₂(t, t₂) were defined as the following shapes as shown in Fig. 2.

The coefficients of the ARX model were estimated respectively. The method of least squares was used as the core algorithm to identify the process model. The temperature of cooling jacket, obtained from the conventional PID controller applied to semi-batch reactor, was used to get the identified models with pseudo-random binary sequence (RPBS), as shown in Fig. 3. The temperature of reactor as the identified results is shown in Fig. 4, which demonstrates the accuracy of the identification.

3.3 2D-MGPILC applied to temperature control of semi-batch reactor

In the above fed-batch reaction with two reactions, the important objective in control system design is to achieve the consistent of conversion and end-product quality. It is assumed that the on-line sensors of the product quality measurement is unavailable, therefore, the end-product quality is indirectly controlled by controlling the temperature of reactor to track some pre-assigned trajectories. The multivariate control structure of the semi-batch reactor is shown in Fig. 5. The reactive temperature T_R was controlled by adjusting the flow rate of hot water or cold water to track the pre-assigned temperature reference trajectory using the proposed 2D-MGPILC. And the PID control method was applied to other controlled variables.

Fig. 1 A typical operation pattern of a semi-batch reactor

Table 1 Kinetic and process parameters of a semi-batch reactor

Fig. 2 Time-varying weighting function

Fig. 3 Pseudo-random binary sequence of identification

Fig. 4 Identified results about temperature of reactor

The simulation of the fed-batch consecutive reaction with two reactions has been achieved by writing programs using MATLAB based on the nonlinear dynamic model. For the identified multi-model with time-varying weights, the proposed 2D-MGPILC controller was implemented to the simulation platform. The dynamic responses of the control signal and the temperature of the reactor are shown in Fig. 6, which indicates that faster batch-wise convergence rate can be got by the scheme.

The value of control signal is changed with the increasing number of batches. At the same time, the output errors are smaller and smaller. As we can see from Fig. 6, the tracking performance of the 2D-MGPILC is satisfying while the errors of pre-heating stage are slightly large. The results show that the control scheme can guarantee the tracking performance including convergence and robust along stability both the time and batch axes.

In order to further test the robustness of control strategy, both repetitive disturbances and non repetitive disturbances were introduced into the simulation system of a semi-batch reactor.

Case 1: A repetitive disturbance with decreasing 1 °C of the reactor is introduced from the fourth batch to last batch.

The dynamic responses of the control signal and the temperature of the reactor against the repetitive disturbance (T_R is decreased by 1 °C) are shown in Fig. 7. Although overshoot is larger because of the change in the fourth batch, general predictive control plays an important role in controlling the temperature and iterative learning control overcomes the error quickly in the next batch. As we can see in figures, the values of control signal become lager from the fourth batch. After two batches, the reference trajectory is accurately tracked again, which indicates that the 2D-MGPILC scheme has good robustness along the time and fast convergence rate along the cycle.

Fig. 5 2D-MGPILC system of semi-batch reactor

Fig. 6 Response of 2D-MGPILC system for semi-batch reactor:

Case 2: A non-repetitive disturbance with decreasing 1 °C of the reactor is introduced only in the fourth batch.

The dynamic responses of the control signal and the temperature of the reactor against the non-repetitive disturbance (T_R is decreased by 1 °C) are shown in Fig. 8. It is seen from figures that the temperature response of the fourth batch is influenced by the disturbance, and the values of control signal in this batch are changed because of the generalized predictive control scheme. Therefore, the non-repetitive disturbance should be overcome by adjusting well the parameters of the designed GPC system in the current batch axis. However, the fifth batch is also influenced due to the iterative learning control scheme, resulting in bad tracking performance to some extent.

Fig. 7 Responses of 2D-MGPILC system against repetitive disturbance:

Fig. 8 Responses of 2D-MGPILC system against non-repetitive disturbance:

The simulation of 2D-GPILC method for the semi-batch reactive process was designed further to compare further the control performance of 2D-MGPILC and 2D-GPILC. The output responses of 2D-GPILC system and 2D-MGPILC system against the repetitive disturbance with decreasing 1 °C are shown in Figs. 9 and 10. In the simulation, the repetitive disturbance was overcome well at batch 10, while it was overcome at batch 10 using the 2D-MGPILC method, and had better transition performance.

Fig. 9 Response of 2D-GPILC system against repetitive disturbance

Fig. 10 Response of 2D-MGPILC system against repetitive disturbance

4 Conclusions

1) In this study, a 2D-MGPILC algorithm has been proposed for the semi-batch reactor with multiple reactions. A cycle-wise ILC is modeled and designed to a time-wise GPC system based on the multi-model with time-varying weights from a two-dimensional system view in order to ensure the tracking performance of the desired temperature profile.

2) Considering the strong nonlinear dynamic characteristic and repeatability of semi-batch reactor, the proposed 2D-MGPILC algorithm is applied successfully to the semi-batch reactive process. Multi-model using time-varying weights functions is presented and the coefficients of the ARX model are estimated respectively.

3) Both repetitive disturbances and non-repetitive disturbances are introduced into the simulation system of a semi-batch reactor, which indicates that the 2D-MGPILC scheme has good robustness along the time and fast convergence rate along the cycle.

Appendix

The nonlinear dynamic mechanism model of the fed-batch reactor consists of a total mass balance, component balances for three components, an energy balance for the liquid in the reactor, and an energy balance for the cooling water in the jacket, and their dynamic equations are shown as follows:

Total mass balance (assuming constant densities; m³/min):

                                   (1)

Component balance for A (kmol/min):

                (2)

Component balance for B (kmol/min):

                    (3)

Component balance for D (kmol/min):

                 (4)

Reactor energy balance (kJ/min):

      (5)

Cooling jacket energy balance (kJ/min):

            (6)

The heat transfer area A_hx varies with time because the volume of liquid in the vessel increases as feed is added. The instantaneous heat transfer area is calculated from the ratio of the instantaneous volume to the total volume:

                          (7)

In the fed-batch mode, an initial charge of component A (5 m³ with concentration 0.5 kmol/m³) is placed in a 7.5 m³ reactor. Component B is fed into the reactor at a flow rate F(t) (m³/min), which can be a function of time or a fixed value. The initial concentration of B in the reactor is zero, but the concentration builds up with time as pure B is fed (C_B0 5 kmol/m³).

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(Edited by YANG Hua)

Cite this article as: BO Cui-mei, YANG Lei, HUANG Qing-qing, LI Jun, GAO Fu-rong. 2D multi-model general predictive iterative learning control for semi-batch reactor with multiple reactions [J]. Journal of Central South University, 2017, 24(11): 2613–2623. DOI:https://doi.org/10.1007/s11771-017-3675-6.

Foundation item: Projects(61673205, 21727818, 61503180) supported by the National Natural Science Foundation of China; Project(2017YFB0307304) supported by National Key R&D Program of China; Project(BK20141461) supported by the Natural Science Foundation of Jiangsu Province, China

Received date: 2016-10-10; Accepted date: 2017-04-28

Corresponding author: BO Cui-mei, PhD, Professor; E-mail: lj_bcm@163.com