Zhen Wang, Aimin An,2,,*, Qianrong Li

1 College of Electrical and Information Engineering, Lanzhou University of Technology, Lanzhou 730050, China

2 Key Laboratory of Gansu Advanced Control for Industrial Processes, Lanzhou 730050, China

3 National Experimental Teaching Center of Electrical and Control Engineering, Lanzhou 730050, China

Keywords:Chemical process Distributed model predictive control Adaptive sampling mechanism Optimal sampling interval System dynamic behavior

A B S T R A C T In this work, an adaptive sampling control strategy for distributed predictive control is proposed.According to the proposed method, the sampling rate of each subsystem of the accused object is determined based on the periodic detection of its dynamic behavior and calculations made using a correlation function.Then,the optimal sampling interval within the period is obtained and sent to the corresponding sub-prediction controller, and the sampling interval of the controller is changed accordingly before the next sampling period begins.In the next control period, the adaptive sampling mechanism recalculates the sampling rate of each subsystem’s measurable output variable according to both the abovementioned method and the change in the dynamic behavior of the entire system,and this process is repeated. Such an adaptive sampling interval selection based on an autocorrelation function that measures dynamic behavior can dynamically optimize the selection of sampling rate according to the real-time change in the dynamic behavior of the controlled object. It can also accurately capture dynamic changes, meaning that each sub-prediction controller can more accurately calculate the optimal control quantity at the next moment, significantly improving the performance of distributed model predictive control (DMPC). A comparison demonstrates that the proposed adaptive sampling DMPC algorithm has better tracking performance than the traditional DMPC algorithm.

1. Introduction

With the application of distributed model predictive control(DMPC) becoming increasingly widespread, research on DMPC in the field of industrial processes is also deepening. The algorithmic concept of DMPC is to transform a large-scale online optimization problem into a small-scale distributed optimization of multiple subsystems, with different model predictive control (MPC) controllers being designed for each subsystem (Fig. 1). At the same time, each subsystem realizes the optimization operation of the whole process in a coordinated way to improve the overall control performance of the system.DMPC is capable of effectively controlling nonlinear, multivariable, and strong-coupling subsystems in a variety of large-scale industrial processes. However, due to the coupling between different subsystems, long-term work periods,and the complex working environments of industrial processes,the dynamic behavior of the subsystems changes, meaning that their dynamic behavior must be measured at different time scales[1].At the same time,various subsystems also bring new problems with respect to sampling strategies.

Fig. 1. DMPC structure.

It is well-known that the selection of the sampling interval of a system has a significant influence on the performance of the system. If the selected sampling interval is too large, important information will be lost, and the established model will be distorted. If the selected interval is too small, the performance requirements of the computer performing the process will be higher. In Ref. [2], the relationship between the characteristics of a system and the sampling periodTwere analyzed from the perspective of the characteristic equations of the system, and it was concluded that it is important to adopt different sampling times for different problems according to their actual requirements. In Ref. [3], genetic algorithms were used to optimize the sampling period.Based on the simulation results of a fast-slow coupling system,it was concluded that the stability and system performance of a multi-rate output system can be improved by properly selecting the sampling period. Because of the superiority of multi-rate sampling predictive control, multi-rate sampling remains a widely researched topic, on which there have been many representative studies. In Ref. [4], a collaborative DMPC strategy was used to study a DMPC problem consisting of large systems with multiple sampling rates. In Ref. [5], an adaptive neural network control method was used to study an output tracking problem with nonlinear objects sampled at the device layer with sampling periodT.The results in Ref.[5]demonstrated that the output of each subsystem at the device layer is able to track the decomposed set values. In Ref. [6], an MPC technique was proposed for systems with different sampling rate measurements, and a simple sub-optimal cascade filter,which can be used to significantly reduce filter gain,was also proposed. In Ref. [7], the MPC scheme was extended to cover multi-rate output sampling processes.In Ref.[8],it was proposed that the state of each local subsystem could be divided into a fast-sampling and a slow-sampling state. A network-based DMPC system was also designed to improve closed-loop performance.In Ref.[9],a direct robust predictive control algorithm with different input and output update rates and sampling rates was proposed for the interference of the actual system. In Ref. [10], a study was undertaken on the explicit MPC of piecewise linear systems whose output sampling period is several times greater than its input updating period. Based on dynamic programming, the MPC optimization problem was decomposed into a multi-stage optimization problem with a one-step optimal level. In Ref. [11],a new design method for multi-rate control systems was presented, in which the sampling interval of factory output is a multiple integral of the holding interval of the control input. A generalized predictive control method with two degrees of freedom was also proposed to account for the presence of modeling errors or disturbances in multi-rate systems.

Table 1 Process parameters

In view of the shortcomings of multi-rate sampling predictive control, many scholars have begun to develop adaptive sampling predictive control methods and have achieved some preliminary research results.In Ref.[12],iterative learning control was applied to a periodic system using a method based on model prediction and optimization, and its trajectory was redesigned to eliminate periodic deviation. To reduce the computation of the optimization algorithm, a variable and adaptive sampling period was introduced. In Ref. [13], a dual-time-scale system was divided into two reduced-order systems,with the fast system and the slow system adopting different sampling periods. A comparison with the original system revealed the state estimation was more accurate using the design with different sampling times. In Ref. [14], the design of an adaptive output estimator for a multi-rate system with a fast uniform input update rate and a slow non-uniform output sampling rate was studied, and an adaptive output estimator was proposed.In Refs.[14]and[15],an adaptive predictive control design problem for uncertain multi-sampling rate systems was studied, and the proposed method was extended to a multi-rate sampling system.By extending the adaptive output estimator provided in Ref. [16], an adaptive predictive controller with adaptive output prediction for multi-rate systems was proposed.The results demonstrated that a stable adaptive predictive control can be designed both for multi-sampling and single-sampling systems.In Ref. [17], an aperiodic event-triggered sampling strategy was proposed for MPC of nonlinear continuous time systems with bounded disturbances. The scheme only makes evaluations at the instant of sampling times triggered by events, aiming to reduce the sensing cost and make it suitable for practical applications.Simulation examples verified the effectiveness of the scheme.

Many studies have demonstrated that research on sampling intervals can greatly improve the performance of control as well as its economic benefits for practical applications. In DMPC, the sampling interval is in most cases the same for different subsystems. However, the presence of different subsystems changes the dynamic behavior of the subsystems due to differences in operating conditions, long operating periods, coupling between different subsystems, and the complex working environments in industrial processes. Therefore, a single interval cannot accurately capture the dynamic behavior of different subsystems,and the control performance of a system using a single interval will not be ideal. The above literature reveals that researchers have been actively exploring methods for calculating the optimal sampling intervals for different subsystems. However, there have been few studies focused on ensuring that the control system deals adequately with the deviation caused by long-term operation of the equipment, complex working environments, and changes in the dynamic behavior of the subsystems caused by the coupling between subsystems.Such an investigation would help to realize the adaptive adjustment of sampling interval parameters. Based on this background,a distributed predictive control strategy based on an adaptive sampling mechanism is proposed in the present paper.

2. Distributed Model Predictive Control

In the actual production process of MPC,the controlled object is often a complex system, such as a traffic network, irrigation system, or industrial process. These controlled objects are usually restricted by one another and a number of conditions and are sometimes widely distributed and scattered. In dealing with such objects, three main control structures are generally used: centralized MPC, decentralized MPC, and DMPC [18]. Centralized MPC uses a single controller for an entire system, while decentralized MPC divides the whole system into several interconnected subsystems,with each subsystem using a single controller for control optimization (though there is no information exchange between the sub-controllers). The DMPC system is similar to the decentralized MPC system, while the difference between the two lies in the information exchange between each sub-controller and the nature of the communication.

Suppose that the number of midline subsystems in an industrial process isn, and the subsystem model is as follows:

Then,the whole system can be described by the following state space model:

To achieve optimal control,the optimization objective function of subsystemiis as follows:

Here, the following constraints must be obeyed:

In each of the above subsystemsi，yiis the process output vector, yisis the set value of the process output vector,DPiis the prediction domain,DLiis the control domain, and Δuiis the input increment. Qiis the input weighting matrix, and Riis the output weighting matrix, where both have the same dimension of input and output and are positive definite diagonal matrices. Further,firepresents the dynamic behavior of the process-modelconstrained multiple variables of the subsystemi,yimin, and yimaxrepresent the upper and lower limits of the output variables,respectively; and uiminuimaxrepresent the lower and upper limit of the regulating variables, respectively.

At the same time, the objective function optimization problem of predictive control can be regarded as a constraint solution problem.At sampling timek,the estimates for x（k）and xd（k） are made available through the concepts of real-time and rolling optimization,and the control behavior of the predictive control can be generated by solving the following optimization problem:

Here, the following constraints must be obeyed:

In the above formulas,nyrepresents the number of manipulated variablesMVs,nurepresents the number of control variablesCVs,Dpis the length of the prediction domain,the subscriptjrepresents thej- thelement in the vector, （k+i｜k） represents the predicted value of the output at timek+iobtained by inputting the information at timek,and r（k）is the reference output sampled at the current time. Next,ε represents the relaxation variable introduced to improve the robust control effect, which significantly constrains the control quantity, the increment of the control quantity, and the output prediction value. After the optimization process outlined above, the optimal value ofkcan be obtained. The sequence Δu（k｜k），···，Δu（m-1+k｜k）, which represents the increment of the adjustment variable and the optimization relaxation variable ε*, is used to obtain the control quantity to be used in the actual adjustment, as shown in the following equation:

3. Adaptive Sampling Model Predictive Control Method

The idea behind self-adaptation is to be able to quickly and automatically make adjustments according to different situations and to change the behavior or control strategy of an original system to meet the requirements of new situations[19].The adaptive sampling strategy designed in this paper is based on the adoption of adaptive thinking in the implementation of adaptive sampling strategies for the sampling interval of a MPC system. It is wellknown that there are many objective uncertainties in the MPC of multi-subsystems, such as that caused by the coupling between subsystems, by deviation during long-term operation of the system, and by environmental factors. Determining how to design an appropriate adaptive strategy to deal with these objective uncertainties so that the performance indicators of the system are optimal or better is the key issue.

In this paper, the sampling interval for MPC is selected as the object of adaptive adjustment, and the optimal sampling interval is calculated according to the dynamic behavior of the subsystems.The parameters of the sampling timeTof the controller are updated before each subsequent sampling time to realize an adaptive adjustment of the MPC parameters under different working conditions.

In this respect, there were two key issues that needed to be solved:(1)how to predict the optimal sampling interval of the subsequent sampling time according to the dynamic behavior of the subsystem at the current time, and (2) how to realize adaptivity.The solutions to these two key problems are given below.

3.1. Optimal sampling interval algorithm

The practical difficulty of selecting the sampling time can be recognized by two extreme cases. First, if the sampling intervalTis too short, the data becomes highly correlated, namely y（k）≈y（k+T）. This phenomenon is called redundancy, and in such a case some of the collected datapoints are unnecessary. In contrast, ifTis too long, the adjacent data points tend to become uncorrelated, meaning that the sampled data lacks regularity and cannot be used to make a scientific estimation of the dynamic behavior of the system. This phenomenon is called uncorrelation.

The methods used to determine the sampling time of nonlinear systems include information theory, correlation time, and reconstruction expansion. The basic principle behind these methods is that if the data of adjacent points is relevant, which indicates that unwelcome information is lost, the sampling time should be long enough to avoid over correlation. At the same time, the sampling time should be long enough to avoid redundant information in most of the subsequent measurements. Although the principles of these methods are intuitive, Billings and Aguirre [20] have pointed out that using autocorrelation functions to determine correlation time only takes into account the linear correlation data,while designing nonlinear autocorrelation functions ensures that the data used to calculate the sampling time is sufficiently complete,thus making the acquisition of sampling time more efficient and scientific.

The specific method developed by Billings includes the design of the following linear and nonlinear correlation functions [20]:

Here,E｛·｝ is the mathematical expectation,are the average, and φyyand φy′y′are the linear and nonlinear correlation of the data, respectively. The sampling time of the system is determined by calculating the correlation function of the time series.

First,the related functions mentioned above must be calculated based on the particular time series used. For the input and output models,the time series corresponding to the system output should be used. Then, the following is defined:

Here,τyis the time at which the linear correlation function φyyis made to obtain its minimum value for the first time,and τy′is the time at which the nonlinear correlation function φy′y′is made to obtain its minimum value for the first time.

The final sampling time is then selected according to the following principle:

In some cases, such as when the linear correlation is high, the upper limit of the sampling time can be relaxed and selected as τm/5. In Fig. 2, the orange area indicates the sampling time range of the system, while the blue area indicates the sampling time range of the system after the upper sampling time is relaxed.Together, these two parts constitute the sampling time range of the system, and a sampling time can be selected within this area.According to this method, different subsystems of the controlled object can determine the respective output sampling time. For example, supposing that the basic sampling time of the system isT, the sampling intervals of the different subsystems are thenT1＝m1×T,T2＝m2×T, ···,Tn＝mn×T, wherem1,m2, ...,mnare positive integers, as shown in Fig. 3 below. The lifting model of the subsystem is then determined based on the sampling time.

Fig.3 shows sampling procedures by using three different sampling intervals.T0is the period of the original continuous time domain sinusoidal signal, whileT1,T2, andT3are the three different sampling periods.T1is able to sample the signal four times in one signal periodT0, whileT2＝2T1, andT3＝3T1. In Fig. 3(a),sampling takes place four times in each signal cycle. Based on the sample points,it can be seen that the shape of the original analog signal is well preserved. In Fig.3(b), each signal period is sampled twice. From the collected sample points, it can be seen that some of the signals cannot be collected, while the original signal can barely be maintained. In Fig. 3(c), which shows a larger sampling interval,it can be seen that the sample points(red line)deviate significantly from the original signal (green dotted line). As a larger sampling period is used,the error will in turn increase.Altogether, this example demonstrates that the proper selection of a sampling interval is important to the stability of the system.

Fig. 2. Schematic diagram of the value range of Ts.

Fig. 3. Comparison of data collected using three different basic sampling intervals.

3.2. Adaptive sampling strategy

In the DMPC based on the adaptive sampling mechanism proposed in this paper (shown in Fig. 4), an anomaly detector and an adaptive regulator are added to the traditional feedback channel. The function of the anomaly detector is to detect the state of the output of the system. If a sampling indicates that the process has obvious fluctuations (not necessarily indicating an abnormality), then the corresponding subsystem is sent data on changes in the system’s dynamic behavior after the detection by the anomaly detector. Next, the adaptive regulator calculates the abnormal sampling state corresponding to the optimal sampling interval based on the data and modifies the sampling interval parameter of the model predictive controller’s next rolling optimization.Then,as long as the process does not display an exception, the process will sample at the sampling interval initially set [21].

Fig. 4. DMPC structure based on adaptive sampling mechanism.

Fig. 5. Cascade CSTR structure diagram.

Fig. 6. Dynamic behavior of the variables in the CSTR subsystem.

Fig. 7. Dynamic behavior of the C1 and C2 variables of the CSTR subsystem when disturbance occurred.

To determine the circumstances,the system performs an exception report and sets an abnormal limit value for the system output μ.Then,the area between the control limits is divided intoD1andD2as follows:

Here, 0 ≤μ ＜γ.

For then-th sampling, the decision rules are as follows:

If Yn（k）∈D2,which means that Yn（k）exceeds the control limit,a process anomaly signal is sent,and the output data is fed back to the adaptive regulator. The adaptive regulator then calculates the optimal sampling interval and changes the interval of the controller accordingly before the next sampling period begins. This guarantees adequate capturing of the subsystem’s future dynamic behavior.

i. If Yn（k）∈D1, the system is normal,and the anomaly detector counts the sampling period as a part of the time accumulation. During the cycle in which the accumulated time reaches five minutes, the sampling interval is optimizedviathe performance of an optimal calculation on the sampling interval.

ii. The state space model of the subsystems can be expressed as follows:

The objective functions of the different subsystems can be expressed as follows:

The sampling interval is optimized at an integer multiple ofT,and the controller parameters are modified before the next sampling.For distributed adaptive sampling systems,the lifting model of each subsystem at different sampling times can be obtained as follows [22]:

Here,Ai,Bi, and uiare the corresponding promotion models,and their forms are as follows:

Adaptive sampling algorithm:

1. Input the exception bounds values μ and γ, where0≤μ ＜r;2. for i ＝1:n;

3. InitializeD1andD2,D1＝[Yi（k）-μ，Yi（k）+μ],D2＝[Yi（k）-γ]∪[Yi（k）+γ];

4. if Yi（k）∈D2then

5. InputYi（k） into the following autocorrelation function:

6. Update τy，τy′;

7. Definition τm＝min, based on, we get thevalue ofTs;

8. Before the next sampling, update the MPC controller parameterTs;

9. end if

10. if Yi（k）∈D1then

11. The system runs according to the value ofTsof the last sampling

12. end if

13. end for

4. Stability Analysis of the Adaptive Sampling System

The distributed predictive control method using an adaptive sampling mechanism is introduced above. According to the method, different subsystems use different sampling times. At the same time,the sampling time of the subsystems is able to satisfyTi＝miT.To ensure the stability of the control system,the following lemma is introduced [23]:

Lemma:.Suppose a（·）∈Rna, b（·）∈Rnb, and S（·）∈Rna×nbfor the matrix h，y，l ∈Rna×nbcan satisfy the following:

According to the above,the state space model of the system can be expressed as follows:

At the same time:

Theorem:If ｜W ｜＞0,then Q,h,y,and l are able to satisfy the following inequality [24]:

Proof:.Using the description of the state space model(Section 3.1)of the discrete space of the original system, Eq. (19) can be expressed as:

Eq. (24) is then substituted into Eq. (26) as follows:

Substituting Eq. (10) into the above formula, the following can then be introduced:

Through Schur’s complement lemma, it can be seen that Eqs.(20)and(32)are equivalent.The theorem is thus proved,meaning that stability can be guaranteed.

5. Simulation and Analysis

5.1. Control object description

The simulation was carried out using Henson and Seborg’s example of a continuous stirred tank reactor (CSTR) (Fig. 5) [25].It was assumed that the system was completely mixed and that the physical parameters were constant. The process consisted of a feed stream, an output stream, and a cooling water stream.qcis the cooling water flow,qis the feed flow,C1andC2are the outlet concentrations, andT1andT2are the outlet temperatures. It was also assumed that the flow was constant throughout the process.The system model is as follows:

Fig. 8. Dynamic behavior of the T1 and T2 variables of the CSTR subsystem when disturbance occurred.

Fig. 9. CSTR subsystem variables φyy and φy′y′.

Fig. 10. Comparison between C1 simulation results obtained by adaptive and non-adaptive sampling.

Fig. 11. Comparison between C2 simulation results obtained by adaptive and non-adaptive sampling.

Fig. 12. Comparison between T1 simulation results obtained by adaptive and non-adaptive sampling.

5.2. Analysis of the simulation results

The following simulation results were obtained according to the system model and process parameters presented in [26].

The dynamic behaviors of the variablesC1,C2,T1, andT2are shown in Fig. 6.

Assuming that a disturbance of power of 0.05 occurred at the current time,the changes in the dynamic behavior of each parameter are shown Figs. 7 and 8.

Using the optimal sampling interval calculation method, the linear and nonlinear correlation functions of each variable in the optimal sampling interval corresponding to the dynamic behavior of the system under the disturbance condition were calculated.The simulation results are as follows.

According to the values of the linear and nonlinear correlation functions of the variablesC1,C2,T1, andT2of CSTR1 and CSTR2,the variables of the upper side corresponding to the minimum time were obtained, while the values of τyand τy′could be determined by comparing the data shown in Fig.9.By comparing the τyand τy′values corresponding to each output, the smallest value was selected as the τmvalue for the subsystem,and the sampling time range of the different subsystems was determined. The sampling time of CSTR1 was found to be 0.05,orm1＝5,while the sampling time of CSTR2 was determined to be 0.06, orm2＝6. Through the process of detection and adaptive adjustment, the optimal sampling interval of the two subsystems was calculated.This was done according to the internal calculation process of the adaptive regulator,which adjusted the sampling interval parameters of the controller before subsequent sampling times.

Fig.10 shows a comparison between the adjusted sample interval parameter and the unadjusted control row energy given the same disturbance (power = 0.02).

Fig. 13. Comparison between T2 simulation results obtained by adaptive and non-adaptive sampling.

Fig. 14. Comparison of adaptive and non-adaptive sampling simulation results for C1 at 0.05 power disturbance.

Fig. 15. Comparison of adaptive and non-adaptive sampling simulation results for C2 at 0.05 power disturbance.

Fig. 16. Comparison of adaptive and non-adaptive sampling simulation results for T1 at 0.05 power disturbance.

Fig. 17. Comparison of adaptive and non-adaptivesampling simulation results for T2 at 0.05 power disturbance.

Figs.10-13 show comparisons of the simulation results considering different subsystems,with disturbance.These were obtained both with the optimal sampling interval being adjusted through the adaptive mechanism and without the adjustment.The red line in the figure represents the system output adjusted by the adaptive sampling mechanism,and the blue line represents the system output of the original system. As can be seen in the figure, the use of the adaptive sampling mechanism outlined in this paper not only significantly enhances the ability of the CSTR system to suppress disturbance but also significantly reduces its regulation time.Thus,it can be concluded that the system’s dynamic and steady-state performance were both significantly improved.

Below is a similar comparison in the case of a higher-power disturbance (0.05):

In Figs. 14-17, it can be seen that the adjustment time of the distributed prediction control based on the adaptive sampling mechanism was reduced in the case of a higher-power disturbance.At the same time,the dynamic and steady-state performance of the system were obviously improved.

6. Summary and Outlook

In this paper, an adaptive sampling mechanism for distributed predictive control was proposed. To this end, a distributed predictive control structure using adaptive sampling was designed.According to the adaptive regulator, the autocorrelation function of the dynamic behavior data of the subsystems is calculated in different states.An optimal sampling interval is then chosen,and the controller parameters are adaptively adjusted before the next sampling instant.A simulation experiment was also carried out using a cascade CSTR system. The simulation results revealed that the adaptive sampling mechanism using the DMPC significantly improved both the dynamic and steady-state performance of the system. The correctness and feasibility of the adaptive sampling mechanism were thus verified.

Acknowledgements

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (61563032, 61963025), The Open Foundation of the Key Laboratory of Gansu Advanced Control for Industrial Processes(2019KX01),and The Project of Industrial support and guidance of Colleges and Universities in Gansu Province (2019C05).