Issue 
Metall. Res. Technol.
Volume 117, Number 3, 2020



Article Number  308  
Number of page(s)  11  
DOI  https://doi.org/10.1051/metal/2020029  
Published online  01 June 2020 
Regular Article
Constrained dynamic matrix control strategy for a monitoring system for AGC of hot strip mills
Institute of Manufacturing Engineering, Hua Qiao University,
Xiamen
361021, PR China
^{*} email: yfc_ral@163.com
Received:
21
November
2019
Accepted:
22
April
2020
In the hot strip rolling process, the performance of a monitoring system for automatic gauge control (MNAGC) is influenced greatly by the model mismatch which is caused by the variation of model parameters values. A constrained dynamic matrix control (CDMC) strategy that includes a prediction model, rolling optimization, and feedback correction was used in the MNAGC. First, the conventional Smith predictionbased control strategy for the MNAGC was analyzed. Second, the performance index function and optimal control of the CDMC strategy were determined. Finally, simulations and industrial experiments were conducted. The results showed that both control strategies provided good control performance. When model mismatch occurred, the Smith predictorbased MNAGC resulted in significant overshoot or even oscillations but the control performance of the CDMCbased MNAGC was not influenced by changes in the model parameters.
Key words: hot strip rolling / automatic gauge control / monitoring system for AGC / CDMC strategy
© EDP Sciences, 2020
1 Introduction
The thickness is one of the most important quality indices in hot strip milling. Automatic gauge control (AGC), which is the most effective method for ensuring the consistent thickness of hotrolled strip products, plays an important role in modern tandem hot milling [1–3]. Among various AGC systems, the monitoring system for AGC (MNAGC) is irreplaceable. Using the gauge meter installed at the outlet side of the last stand, the MNAGC is a closedloop control of the exit thickness so that the product gauge is the same as the target gauge. However, since the gauge meter requires maintenance, it must be installed far from the last stand [4–7]. The MNAGC is a timedelay system because there is a certain lag time in the thickness measurements. This lag time results in a timedelay link in the MNAGC that can reduce the stability of the system and worsen the transition characteristics of the MNAGC [8–10].
Solving the problem of the timedelay link in the MNAGC is critical to ensure the consistent thickness of the product and stability of the system. A number of scholars have investigated solutions to this problem. One of the most common approaches is to use a Smith predictor in the MNAGC. Yin et al used the Smith predictor with a feedbackassisted iterative learning control algorithm and sliding mode variable structure control algorithm to increase the robustness of the MNAGC [11,12]. Sun et al proposed a simple and practical fuzzy selftuning method for the Smith predictor that results in a fast response time of the MNAGC [13]. However, the Smith predictor is subordinate to the classic control theory and requires an accurate system model. The model mismatch caused by the fluctuations or changes in the strip speed, hydraulic oil temperature, or plastic coefficient, which are inevitable during hot strip rolling, reduces the performance of the Smith predictorbased MNAGC and may even cause system instability. The use of the Smith predictor cannot provide simultaneously excellent stability and robustness of the MNAGC.
A dynamic matrix control algorithm, which is wellknown for its robustness, has been used to design control systems in many fields [14,15]. For instance, Wang et al. [16] designed a controller with improved dynamic control for hybrid electric vehicles and Lima et al. [17] proposed a filtered dynamic matrix controller for temperature control in a solar collector field. In this study, the constrained dynamic matrix control (CDMC) strategy is used in the MNAGC. In this strategy, the sum of the absolute values of the error between the predicted outputs and the reference trajectory sequences is used as the performance index. In order to address the timedelay link of the MNAGC, the CDMC strategy estimates a set of future control sequences at one time and the control input at the current time is expressed in a function that includes the timedelay information. The control link of the online rolling optimization and the feedback correction of the CDMC strategy provide a fast response time and good robustness of the MNAGC.
2 Parameters identification for hydraulic gap control system
2.1 Identification algorithm of the frequencydomain system
The conventional Smith predictorbased MNAGC, which is the most widely used system in a 1700mm tandem hot mill, is shown in Figure 1. The MNAGC is actually a double closedloop system and the hydraulic gap control (HGC) system is the inner loop control system of the MNAGC. The stability of the MNAGC is affected by the parameters of the HGC system. However, owing to the high nonlinearity of the hydraulic system, it is difficult to calculate the parameters of the HGC system precisely using mathematical means.
In a tandem hot mill, the last stand is the most important stand for strip AGC and the thickness of the finished strip is determined by the control precision of the MNAGC in the last stand. In this section, the model parameters of the HGC system of the #7 stand are identified in the frequency domain. Frequencydomain system identification is based on a complex curvefitting algorithm; its operating principles are as follows:
When a certain frequency ω of the test input signal is added to the position reference of the HGC system, the dynamic output response of the system is determined by a displacement sensor. The relationship between the test input signal x(t) = Asin(ωt) and the dynamic output response x ’ (t) = A ’ sin(ωt + ϕ) is shown in Figure 2.
According to the definition of frequency characteristic, the frequencyamplitude characteristics and G (jω)  the phasefrequency characteristics ∠G (jω) of the system are expressed as follows: (1) (2) where, A is the amplitude of input signal, A^{’} is the amplitude of output signal, T is the period of input signal, τ is the delay time between input signal and output signal, j is the imaginary unit.
The frequency characteristic of the system is also expressed as: (3) where, P is the real frequency characteristic, P = A (ω) cosϕ (ω); Q is the imaginary frequency characteristic, Q = A (ω) sinϕ (ω).
The test input signal at different frequencies ω is added to the position reference of the HGC system; A (ω) and ϕ (ω) of the HGC system at different frequencies ω are calculated by using the input signal and response signal and P and Q are obtained.
Based on the complex curve fitting method, the model parameters of HGC system can be estimated by the frequency response data. The frequency characteristic of the HGC system G(jω) is defined as: (4) where ;; ;
The fitting error e(ω_{k}) is defined as: (5) where is the measured frequency characteristic of the HGC system at the frequency of ω_{k}.
To avoid generating a nonlinear expression when solving the derivative of ω_{k}, the weight coefficient is introduced into the fitted index function J. The index J is expressed as: (6) where α_{0}, α_{1} ⋯ α_{N}, β_{0}, β_{1} ⋯ β_{N} are the indefinite parameters of the HGC system, which ensures that the fitted index function J is at a minimum.
The covariance matrix in the identification algorithm is expressed as: (7) (8)
Next, we use the derivative of J with respect to all unknown parameters in and ensure that ; the HGC system in the form of a linear equation is then obtained as: (9) where
Fig. 1 The MNAGC of a 1700mm tandem hot mill. 
Fig. 2 Sine input signal and response signal. 
2.2 The process of parameters identification for HGC system
The step response characteristic of the MNAGC in the #7 stand was tested using the automatic data recording system that is part of the 1700 mm tandem hot mill. The proposed experiment will involve the following steps:
Step 1: For reproducing the running state of HGC system as closely as possible, press down the roll by hand until the pressures up to 200 tons, the position feedback signal S_{0} at this time as the position set referenceS_{ref}, S_{0} = S_{ref}.
Step 2: The HGC system mainly works in the low frequency band, so 24 frequency points in the range from 0.1 to 10 Hz were collected by equal space between principle for testing, and the amplitude of testing sine signals are chosen as 0.1 mm .
Step 3: The tested frequency is entered from menmachine interface, and start up the sine wave input. After the HGC controller receives control instructions from menmachine interface, the sine wave 0.1sin(ωt)is modulated to the position set reference S_{0} as the new position set reference, S_{ref} = S_{0}+0.1sin(ωt). The input and output data of HGC system are recorded, and the number of sample points are also calculated. When the number of sample points reaches the set number, the position set reference restores to S_{0}.
Step 4: Entering the next tested frequency, the Step 3 is executed repeatedly until all the 24 frequency points are collected.
The sampling period was 10 ms and the data were fitted to a sine curve. The frequency characteristics H of the HGC system at different frequencies ω of the test signal are shown in Table 1.
Based on the frequency response data of the HGC system, an algorithm for identifying the parameters of the HGC system was written with an mfile in MATLAB2014 software; the call function of this mfile is as follows: (10) where, n_{n} is the desired order of the numerator; n_{d} is the desired order of the denominator; num is the numerator polynomial of the HGC system; den is the denominator polynomial of HGC the system.
The discrete HGC system model is defined as: (11)
In order to design the MNAGC control architecture, the discrete model of the HGC system was translated into a continuous model, as shown in equation (13): (12)
A simulation analysis of the step response was conducted in MATLAB/Simulink using the proposed HGC model; the experimental and simulation results are shown as Figure 3. The thickness accuracy of strip is determined directly by the rising time and steadystate error of HGC system [12]. As shown in Table 2, except for a slight overshoot in the real response, the differences in the time and steadystate error between the simulation and experiment were smaller than that of overshoot, proposed HGC model provides a good indication of the dynamic output characteristics of the system, indicating that the proposed method provides a solid foundation for analyzing the architecture of the MNAGC.
Frequency responses of the HGC system.
Fig. 3 Experimental and simulation results. 
Comparison of dynamic characteristics.
3 Architecture of the MNAGC
3.1 Smith prediction control strategy
The MNAGC of the #7 stand is shown in Figure 4. The thickness is measured by a gauge meter installed at the end of the processing line; the signal is sent to the controller and a feedback control system is used to adjust the thickness at the roll gap. A proportionalintegral (PI) controller is commonly used to achieve highaccuracy thickness control and minimize the steadystate error.
As shown in Figure 4, a thickness control loop is used to measure the time delay. When the time delay is too long, the MNAGC is unstable. The Smith predictor is an effective method to achieve a stable process when a time delay is present. The control diagram of the MNAGC with a conventional Smith predictor is shown in Figure 5.
In Figure 5, H^{∗}(s) is the reference gauge; H(s) is the feedback from the actual gauge; G_{c}(s) is the transfer function of the controller; G(s)e^{−τs} is the transfer function of the HGC system with the time delay; G(s) is the transfer function of the HGC system without a time delay; e^{−τs} is the transfer function of the time delay; G_{m}(s) is the transfer function of the model; e^{−τms} is the transfer function of the estimated time delay; ΔS(s) is the change in the gap; H_{τ}(s) is the estimated gauge based on the Smith predictor; H^{’}(s) is the actual gauge based on the Smith predictor compensation.
It is assumed that the model provides an accurate match of the actual conditions (Eq. (13)); the transfer function of the MNAGC with the conventional Smith predictor is provided in equation (14). (13) (14)
According to equation (14), when the model provides a perfect match, the time delay e^{−τs} is outside of the control loop; therefore, the influence of the time delay on the stability of the system is compensated for by the Smith predictor. However, this control strategy performs very poorly in the presence of disturbances and modeling uncertainties. For example, in the hot rolling process, the influence of calculation errors in the forward slip may produce feedback errors in the strip speed; this may result in large estimation errors of the time delay and reduce the stability of the MNAGC. Hence, a CDMC strategy for the MNAGC is proposed.
Fig. 4 Schematic diagram of the MNAGC. 
Fig. 5 Control diagram of the MNAGC with a conventional Smith predictor. 
3.2 Constrained dynamic matrix control: a brief summary
The CDMC strategy is a predictive control algorithm based on the step response model. It uses online optimization based on feedback correction instead of the traditional optimum control. The strategy consists of three parts: model prediction, online optimization, and feedback correction, as shown in Figure 6.
Fig. 6 Diagram of the CDMC strategy for the MNAGC. 
3.2.1 Model prediction
It is assumed that the roll gap S (k) is the control input and the actual strip gauge h is the control output; the unitstep sampling data of HGC system is expressed as .
According to the superposition principle, the output of the strip gauge at time k+1 is predicted as follows: (15) where m is the length of the control horizon; p is the length of the optimization horizon; ; ; ; is a dynamic matrix.
3.2.2 Online rolling optimization
The recedinghorizon objective function is used in the CDMC strategy; the control increment sequence in the future control horizon is m. The function ensures that the predictive output in the future optimization horizon is as close as possible to the expected output. The optimal control law is deduced as: (16) where is the expected strip gauge; is the predicted strip gauge, which includes an error revision; is the error weight matrix; is the control weight matrix; q_{i} is the error weight coefficient; r_{j} is the control weight coefficient.
In the actual MNAGC, the saturation in the roll gap adjustment must be considered; the linear increment constraint regarding is expressed as: (17)
where is the maximum amplitude of ; is the minimum amplitude of .
The mathematical description of the CDMC strategy for the MNAGC is established: (18)
By substituting equation (16) into equation (18), is obtained by solving as follows: (19)
Equation (19) is the openloop control form of the CDMC strategy for the MNAGC. However, the MNAGC is often influenced by disturbances in the production field and the control law defined in equation (19) does not have good tracking properties for the set value. Hence, the closedloop control form of the CDMC strategy for the MNAGC must be established. The calculated control increment ΔS_{1} (k) is used as the actual control increment in the closedloop CDMC, as defined in equations (20) and (21). (20) (21)
3.2.3 Feedback correction
The actual strip gauge is considered in the CDMC strategy to correct the predicted strip gauge. The predicted strip gauge with the error revision is expressed as: (22) where is the vector of the error revision; h_{m}(k) is the predicted strip gauge; h(k) is the actual strip gauge.
The initially predicted strip gauge at k + 1 time is expressed as: (23)
After solving equation (18), ΔS_{1} (k+1) is obtained again using the online rolling optimization at k + 1 time, as described above. Hence, online optimization planning can be performed during the rolling process and the optimization and feedback mechanisms are combined.
3.3 Application of CDMC strategy to the MNAGC
The control process of the CDMC strategy for the MNAGC is divided into the offline calculations and the online calculations. As shown in Figure 3, there is a time delay in the MNAGC. Based on the principle of the model vector used in the CDMC strategy, the modeling horizon N should contain the dynamic part of the response of the MNAGC. The time delay τ should be included in the response of the MNAGC although there is no output from the system at that time. Therefore, the model vector of the MNAGC A_{d} is expressed as: (25) where, , the distance between the last stand and the gauge meter (m); v is the speed of the last stand (m/s); T is the sampling period of the MNAGC (s).
Due to the model prediction link in the CDMC strategy, the time delay of the MNAGC can be determined without adding a control structure. Compared with the model vector of the nodelay system , the model vector of the timedelay MNAGC, has more l dimensions than A and there are no additional parameters. The modeling horizon has increased because the CDMC strategy for the MNAGC takes into account the model vector .
In the CDMC strategy for the timedelay MNAGC, the length of the model vector N_{d} is N + l to correspond to the CDMC strategy for timedelay system. Because of the time delay in the MNAGC, the optimization horizon p_{d} is set to p_{d} = p + l to ensure a good performance of the online rolling optimization. The control horizon m_{d} remains at m because the time delay in the MNAGC has no influence on the control increment. Finally, based on the optimization horizon p_{d} and the control horizon m_{d}, the error weight matrix and control weight matrix are adjusted as follows: (26) (27)
The offline calculations of the CDMC strategy for the MNAGC are expressed as: (28)
It can be deduced from these results that the optimization horizon should be set based on the timedelay constant of the MNAGC; the control performance of the CDMC strategy is the same for the timedelay MNAGC as the nodelay system.
The online calculations of the CDMC strategy for the MNAGC are conducted using the initialization module and realtime control module. The initialization module is designed to measure the actual thickness h(k) in the first step, where h(k) is the initial predicted value of the thicknessh_{0} (k + ik). The initialization module is changed to the realtime control module in the second step. The flow chart of the calculations of the CDMC strategy is shown in Figure 7.
Fig. 7 Flow chart of online calculations of part of the CDMC strategy. 
4 Simulation
The simulation was performed in MATLAB/Simulink software to compare the control performance of Smith predictorbased MNAGC and CDMCbased MNAGC. The parameters of the rolling process of the 1700 mm tandem hot rolling mill used in the simulation are given in Table 3.
Parameters of the rolling process used in the simulations.
4.1 Simulations
The simulation for the Smith predictorbased system and CDMCbased system was carried out using model matching. A step test signal with an amplitude of 0.05 mm was added to the initial gauge reference. The simulation results are shown in Figure 8.
The dynamic characteristics of the Smith predictorbased system and CDMCbased system are shown in Table 4.
As shown in Figure 8 and Table 4, the Smith predictorbased system has a rising time of 39.2 ms and an overshoot of 8.6%. The proposed CDMCbased system provides a lower rising time of just 35.2 ms and an overshoot of only 1.6%. For the MNAGC in a tandem hot mill, the overshoot has to be less than 10% and the rising time cannot exceed 50 ms. The dynamic characteristics of both control systems meet the response requirements of the MNAGC.
Fig. 8 Response curves of the exit thickness using model matching. 
Comparison of the dynamic characteristics.
4.2 Simulation results for different exit speeds of the strip
In the hot rolling process, the exit speed of the strip is variable because there are calculation errors in the forward slip, thus there are differences between the calculated time delay and actual time delay. Simulations were conducted to investigate the control performance of the Smith predictorbased system and CDMCbased system for different speeds of the strip; the results are shown in Figure 9.
The dynamic characteristic parameters of Smith predictorbased and CDMCbased systems for speeds of the strip ranging from 4.6 to 5.2 m/s are presented in Table 5.
The results indicate that excellent control performance was achieved by the CDMCbased MNAGC for different exist speeds of the strip; the rising time remained below 40 ms and the overshoot was less than 4%. However, for the Smith predictorbased MNAGC, the maximum rising time is 93.7 ms and the maximum of overshoot is 40.1%; these values do not meet the response requirements for the dynamic characteristics of the MNAGC.
Fig. 9 Response curves of the exit thickness for different exit speeds of the strip. 
Dynamic characteristic parameters for different exit speeds of the strip.
4.3 Simulation results for different values of the plastic coefficient
The plastic coefficient of strip means the change of rolling force which is caused by the thickness variation of strip occurs 1.0 mm, and it can be expressed as: (29) where, Q is the plastic coefficient of strip, kN/mm; ΔPis the change of rolling force, kN; Δhis the thickness variation of strip, mm.
The plastic coefficient of strip changes due to temperature differences in the strip, there exits the temperature drop along the length direction. So, the plastic coefficient of strip is a variation in the actual production process, and it causes the model mismatching. When the temperature of plain carbon steel ranges from 1020 to 970 °C, the plastic coefficient of strip is in 16 000 to 10 000 kN/mm. For testing the robustness of designed system, the simulation for different values of the plastic coefficient was carried out, the results are shown in Figure 10 and Table 6.
The dynamic characteristic parameters of Smith predictorbased and CDMCbased systems for plastic coefficient values in the range of 1.0 e4 to 1.6 e4 kN/mm are shown in Table 6.
The simulation results show that the optimum control performance of both control systems is achieved when the plastic coefficient Q is 1.6 e4 kN/mm. However, the rising time of the Smith predictorbased system increases to 567.8 ms at the lowest value of the plastic coefficient, which is unacceptable for the MNAGC. For the CDMCbased MNAGC, the rising time remains below 36 ms and the overshoot is less than 9% for all cases.
Fig. 10 Response curves of the exit thickness for different values of the plastic coefficient. 
Dynamic characteristic parameters for different values of the plastic coefficient.
4.4 Simulation results for different hydraulic oil temperatures
Servo valve is one of the main components in HGC system, and its dynamic characteristics are effected by the oil temperature, oil pressure and load conditions. Oil temperature variety is great in the hot rolling process, and it causes HGC model mismatching. In this paper, HGC model was identified separately under the condition that oil temperature is 50 to 80 °C, and the control effect and dynamic characteristic parameters were compared. The simulation results for the Smith predictorbased and CDMCbased systems for different hydraulic oil temperatures are shown in Figure 11 and Table 7.
The simulation results indicate that the rising time of the Smith predictorbased system increases to 2106.6 ms when the hydraulic oil temperature is80 °C; this response is not acceptable in the MNAGC. For the CDMCbased MNAGC, the rising time remains below 35 ms and the overshoot does not exceed 5% for all temperatures; this indicates good robustness of the CDMCbased system.
The simulation results demonstrate that good control performances of the MNAGC can be achieved by the Smith predictorbased and CDMCbased systems. In the cases of model mismatch for different speeds of the strip, different plastic coefficient values, or different hydraulic oil temperatures, the Smith predictorbased system has larger overshoot and longer rising times; these responses are not acceptable in the MNAGC. The CDMCbased system is relatively insensitive to changes in the rolling parameters and good control performance is achieved in all cases.
Fig. 11 Response curves of exit thickness for different hydraulic oil temperatures. 
Dynamic characteristic parameters for different hydraulic oil temperatures.
5 Industrial experiment
The proposed CDMCbased strategy for the MNAGC was successfully used in a 1700 mm tandem hot rolling finishing mill. The finishing mill consists of seven stands, each of which is equipped with two highresponse AC mill motors, a lowinertia looper drive system, and an HGC system. The gauge meter used for measuring the strip gauge was installed 2.5 m away from the #7 stand, as shown in Figure 12.
The last stand is the most important stand in hot strip mills because this is where the thickness precision of the finished strip is determined by the MNAGC. The MNAGC is only used in the last stand in actual hot rolling production. In order to compare the control performance of the CDMCbased system and Smith predictorbased MNAGC, the AGC control system in the other stands was switched off. Products with thicknesses of 2.5 and 5.0 mm were randomly selected. The thickness deviations of the systems are shown in Figure 13 and the thickness precision is listed in Table 8.
In the conventional Smithbased MNAGC system which is widely used in most hot rolling lines, the requirement only achieves 0.04 mm or less than for the 2.5mm products, and the requirement achieves 0.05 mm or less than for the 5.0mm products. As shown in Figure 13a, for the 2.5mm products, the thickness precision of the CDMCbased MNAGC is 99.05% and the thickness is within ±0.018 mm of the required thickness; for the 5.0mm product (Fig. 13b), the thickness precision of the CDMCbased MNAGC is 98.67% and the thickness is within ±0.022 mm of the required thickness. The results based on a large data volume obtained in the field show that the proposed CDMCbased MNAGC has a better control performance and higher precision of the strip thickness than the Smith predictorbased MNAGC.
Fig. 12 The MNAGC based on the CDMC strategy used in actual production. 
Fig. 13 Thickness deviations of the 2.5mm products (a) and the 5.0mm products (b). 
Comparison of thickness precision of the strips.
6 Conclusions
We developed an MNAGC for a 1700mm tandem hot mill and the parameters of the HGC model were determined based on an identification algorithm of the frequencydomain system.
In order to solve the problem of the time delay in the MNAGC, a Smith predictionbased control strategy was used in the MNAGC and the transfer function of the MNAGC was established.
The CDMC strategy was designed for the MNAGC to improve the robustness. Simulations were conducted using MATLAB/Simulink software; the result indicated that the CDMCbased MNAGC exhibited better control performance than the Smith predictorbased MNAGC when an HGC model mismatch occurred.
The CDMC strategy for the MNAGC was evaluated using actual production data. The results showed that the CDMCbased MNAGC had better robustness and higher thickness precision of the strip than the Smith predictorbased control strategy.
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All Tables
Dynamic characteristic parameters for different values of the plastic coefficient.
All Figures
Fig. 1 The MNAGC of a 1700mm tandem hot mill. 

In the text 
Fig. 2 Sine input signal and response signal. 

In the text 
Fig. 3 Experimental and simulation results. 

In the text 
Fig. 4 Schematic diagram of the MNAGC. 

In the text 
Fig. 5 Control diagram of the MNAGC with a conventional Smith predictor. 

In the text 
Fig. 6 Diagram of the CDMC strategy for the MNAGC. 

In the text 
Fig. 7 Flow chart of online calculations of part of the CDMC strategy. 

In the text 
Fig. 8 Response curves of the exit thickness using model matching. 

In the text 
Fig. 9 Response curves of the exit thickness for different exit speeds of the strip. 

In the text 
Fig. 10 Response curves of the exit thickness for different values of the plastic coefficient. 

In the text 
Fig. 11 Response curves of exit thickness for different hydraulic oil temperatures. 

In the text 
Fig. 12 The MNAGC based on the CDMC strategy used in actual production. 

In the text 
Fig. 13 Thickness deviations of the 2.5mm products (a) and the 5.0mm products (b). 

In the text 
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