Free Access
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

© 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 hot-rolled strip products, plays an important role in modern tandem hot milling [13]. Among various AGC systems, the monitoring system for AGC (MN-AGC) is irreplaceable. Using the gauge meter installed at the outlet side of the last stand, the MN-AGC is a closed-loop 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 [47]. The MN-AGC is a time-delay system because there is a certain lag time in the thickness measurements. This lag time results in a time-delay link in the MN-AGC that can reduce the stability of the system and worsen the transition characteristics of the MN-AGC [810].

Solving the problem of the time-delay link in the MN-AGC 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 MN-AGC. Yin et al used the Smith predictor with a feedback-assisted iterative learning control algorithm and sliding mode variable structure control algorithm to increase the robustness of the MN-AGC [11,12]. Sun et al proposed a simple and practical fuzzy self-tuning method for the Smith predictor that results in a fast response time of the MN-AGC [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 predictor-based MN-AGC and may even cause system instability. The use of the Smith predictor cannot provide simultaneously excellent stability and robustness of the MN-AGC.

A dynamic matrix control algorithm, which is well-known 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 MN-AGC. 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 time-delay link of the MN-AGC, 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 time-delay 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 MN-AGC.

2 Parameters identification for hydraulic gap control system

2.1 Identification algorithm of the frequency-domain system

The conventional Smith predictor-based MN-AGC, which is the most widely used system in a 1700-mm tandem hot mill, is shown in Figure 1. The MN-AGC is actually a double closed-loop system and the hydraulic gap control (HGC) system is the inner loop control system of the MN-AGC. The stability of the MN-AGC 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 MN-AGC in the last stand. In this section, the model parameters of the HGC system of the #7 stand are identified in the frequency domain. Frequency-domain system identification is based on a complex curve-fitting 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 frequency-amplitude characteristics and |G () | the phase-frequency characteristics ∠G () 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() is defined as: (4) where ;;

The fitting error e(ωk) is defined as: (5) where is the measured frequency characteristic of the HGC system at the frequency of  ωk.

To avoid generating a nonlinear expression when solving the derivative of  ωk, the weight coefficient is introduced into the fitted index function J. The index J is expressed as: (6) where α0, α1 ⋯ αN, β0, β1 ⋯ βN 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 

thumbnail Fig. 1

The MN-AGC of a 1700-mm tandem hot mill.

thumbnail Fig. 2

Sine input signal and response signal.

2.2 The process of parameters identification for HGC system

The step response characteristic of the MN-AGC 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 S0 at this time as the position set referenceSref, S0 = Sref.

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 men-machine interface, and start up the sine wave input. After the HGC controller receives control instructions from men-machine interface, the sine wave 0.1sin(ωt)is modulated to the position set reference S0 as the new position set reference, Sref = S0+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 S0.

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 m-file in MATLAB2014 software; the call function of this m-file is as follows: (10) where, nn is the desired order of the numerator; nd 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 MN-AGC 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 steady-state 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 steady-state 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 MN-AGC.

Table 1

Frequency responses of the HGC system.

thumbnail Fig. 3

Experimental and simulation results.

Table 2

Comparison of dynamic characteristics.

3 Architecture of the MN-AGC

3.1 Smith prediction control strategy

The MN-AGC 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 proportional-integral (PI) controller is commonly used to achieve high-accuracy thickness control and minimize the steady-state 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 MN-AGC 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 MN-AGC 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; Gc(s) is the transfer function of the controller; G(s)eτs 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; eτs is the transfer function of the time delay; Gm(s) is the transfer function of the model; eτms is the transfer function of the estimated time delay; ΔS(s) is the change in the gap; Hτ(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 MN-AGC 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 eτs 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 MN-AGC. Hence, a CDMC strategy for the MN-AGC is proposed.

thumbnail Fig. 4

Schematic diagram of the MN-AGC.

thumbnail Fig. 5

Control diagram of the MN-AGC 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.

thumbnail Fig. 6

Diagram of the CDMC strategy for the MN-AGC.

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 unit-step 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 On-line rolling optimization

The receding-horizon 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; qi is the error weight coefficient; rj is the control weight coefficient.

In the actual MN-AGC, 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 MN-AGC is established: (18)

By substituting equation (16) into equation (18) is obtained by solving  as follows: (19)

Equation (19) is the open-loop control form of the CDMC strategy for the MN-AGC. However, the MN-AGC 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 closed-loop control form of the CDMC strategy for the MN-AGC must be established. The calculated control increment ΔS1 (k) is used as the actual control increment in the closed-loop 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; hm(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)

where (24)

After solving equation (18), ΔS1 (k+1) is obtained again using the on-line rolling optimization at k + 1 time, as described above. Hence, on-line 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 MN-AGC

The control process of the CDMC strategy for the MN-AGC is divided into the offline calculations and the online calculations. As shown in Figure 3, there is a time delay  in the MN-AGC. 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 MN-AGC. The time delay τ should be included in the response of the MN-AGC although there is no output from the system at that time. Therefore, the model vector of the MN-AGC Ad 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 MN-AGC (s).

Due to the model prediction link in the CDMC strategy, the time delay of the MN-AGC can be determined without adding a control structure. Compared with the model vector of the no-delay system , the model vector of the time-delay MN-AGC,  has more l dimensions than A and there are no additional parameters. The modeling horizon has increased because the CDMC strategy for the MN-AGC takes into account the model vector .

In the CDMC strategy for the time-delay MN-AGC, the length of the model vector Nd is N + l to correspond to the CDMC strategy for time-delay system. Because of the time delay in the MN-AGC, the optimization horizon pd is set to pd = p + l to ensure a good performance of the on-line rolling optimization. The control horizon md remains at m because the time delay in the MN-AGC has no influence on the control increment. Finally, based on the optimization horizon pd and the control horizon md, the error weight matrix  and control weight matrix  are adjusted as follows: (26) (27)

The offline calculations of the CDMC strategy for the MN-AGC are expressed as: (28)

It can be deduced from these results that the optimization horizon should be set based on the time-delay constant of the MN-AGC; the control performance of the CDMC strategy is the same for the time-delay MN-AGC as the no-delay system.

The online calculations of the CDMC strategy for the MN-AGC are conducted using the initialization module and real-time 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 thicknessh0 (k + i|k). The initialization module is changed to the real-time control module in the second step. The flow chart of the calculations of the CDMC strategy is shown in Figure 7.

thumbnail 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 predictor-based MN-AGC and CDMC-based MN-AGC. The parameters of the rolling process of the 1700 mm tandem hot rolling mill used in the simulation are given in Table 3.

Table 3

Parameters of the rolling process used in the simulations.

4.1 Simulations

The simulation for the Smith predictor-based system and CDMC-based 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 predictor-based system and CDMC-based system are shown in Table 4.

As shown in Figure 8 and Table 4, the Smith predictor-based system has a rising time of 39.2 ms and an overshoot of 8.6%. The proposed CDMC-based system provides a lower rising time of just 35.2 ms and an overshoot of only 1.6%. For the MN-AGC 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 MN-AGC.

thumbnail Fig. 8

Response curves of the exit thickness using model matching.

Table 4

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 predictor-based system and CDMC-based system for different speeds of the strip; the results are shown in Figure 9.

The dynamic characteristic parameters of Smith predictor-based and CDMC-based 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 CDMC-based MN-AGC 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 predictor-based MN-AGC, 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 MN-AGC.

thumbnail Fig. 9

Response curves of the exit thickness for different exit speeds of the strip.

Table 5

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 predictor-based and CDMC-based 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 predictor-based system increases to 567.8 ms at the lowest value of the plastic coefficient, which is unacceptable for the MN-AGC. For the CDMC-based MN-AGC, the rising time remains below 36 ms and the overshoot is less than 9% for all cases.

thumbnail Fig. 10

Response curves of the exit thickness for different values of the plastic coefficient.

Table 6

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 predictor-based and CDMC-based 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 predictor-based system increases to 2106.6 ms when the hydraulic oil temperature is80 °C; this response is not acceptable in the MN-AGC. For the CDMC-based MN-AGC, the rising time remains below 35 ms and the overshoot does not exceed 5% for all temperatures; this indicates good robustness of the CDMC-based system.

The simulation results demonstrate that good control performances of the MN-AGC can be achieved by the Smith predictor-based and CDMC-based systems. In the cases of model mismatch for different speeds of the strip, different plastic coefficient values, or different hydraulic oil temperatures, the Smith predictor-based system has larger overshoot and longer rising times; these responses are not acceptable in the MN-AGC. The CDMC-based system is relatively insensitive to changes in the rolling parameters and good control performance is achieved in all cases.

thumbnail Fig. 11

Response curves of exit thickness for different hydraulic oil temperatures.

Table 7

Dynamic characteristic parameters for different hydraulic oil temperatures.

5 Industrial experiment

The proposed CDMC-based strategy for the MN-AGC 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 high-response AC mill motors, a low-inertia 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 MN-AGC. The MN-AGC is only used in the last stand in actual hot rolling production. In order to compare the control performance of the CDMC-based system and Smith predictor-based MN-AGC, 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 Smith-based MN-AGC system which is widely used in most hot rolling lines, the requirement only achieves 0.04 mm or less than for the 2.5-mm products, and the requirement achieves 0.05 mm or less than for the 5.0-mm products. As shown in Figure 13a, for the 2.5-mm products, the thickness precision of the CDMC-based MN-AGC is 99.05% and the thickness is within ±0.018 mm of the required thickness; for the 5.0-mm product (Fig. 13b), the thickness precision of the CDMC-based MN-AGC 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 CDMC-based MN-AGC has a better control performance and higher precision of the strip thickness than the Smith predictor-based MN-AGC.

thumbnail Fig. 12

The MN-AGC based on the CDMC strategy used in actual production.

thumbnail Fig. 13

Thickness deviations of the 2.5-mm products (a) and the 5.0-mm products (b).

Table 8

Comparison of thickness precision of the strips.

6 Conclusions

We developed an MN-AGC for a 1700-mm tandem hot mill and the parameters of the HGC model were determined based on an identification algorithm of the frequency-domain system.

In order to solve the problem of the time delay in the MN-AGC, a Smith prediction-based control strategy was used in the MN-AGC and the transfer function of the MN-AGC was established.

The CDMC strategy was designed for the MN-AGC to improve the robustness. Simulations were conducted using MATLAB/Simulink software; the result indicated that the CDMC-based MN-AGC exhibited better control performance than the Smith predictor-based MN-AGC when an HGC model mismatch occurred.

The CDMC strategy for the MN-AGC was evaluated using actual production data. The results showed that the CDMC-based MN-AGC had better robustness and higher thickness precision of the strip than the Smith predictor-based control strategy.

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Cite this article as: Yin Fang-chen, Yu Liu-Qi, Constrained dynamic matrix control strategy for a monitoring system for AGC of hot strip mills, Metall. Res. Technol. 117, 308 (2020)

All Tables

Table 1

Frequency responses of the HGC system.

Table 2

Comparison of dynamic characteristics.

Table 3

Parameters of the rolling process used in the simulations.

Table 4

Comparison of the dynamic characteristics.

Table 5

Dynamic characteristic parameters for different exit speeds of the strip.

Table 6

Dynamic characteristic parameters for different values of the plastic coefficient.

Table 7

Dynamic characteristic parameters for different hydraulic oil temperatures.

Table 8

Comparison of thickness precision of the strips.

All Figures

thumbnail Fig. 1

The MN-AGC of a 1700-mm tandem hot mill.

In the text
thumbnail Fig. 2

Sine input signal and response signal.

In the text
thumbnail Fig. 3

Experimental and simulation results.

In the text
thumbnail Fig. 4

Schematic diagram of the MN-AGC.

In the text
thumbnail Fig. 5

Control diagram of the MN-AGC with a conventional Smith predictor.

In the text
thumbnail Fig. 6

Diagram of the CDMC strategy for the MN-AGC.

In the text
thumbnail Fig. 7

Flow chart of online calculations of part of the CDMC strategy.

In the text
thumbnail Fig. 8

Response curves of the exit thickness using model matching.

In the text
thumbnail Fig. 9

Response curves of the exit thickness for different exit speeds of the strip.

In the text
thumbnail Fig. 10

Response curves of the exit thickness for different values of the plastic coefficient.

In the text
thumbnail Fig. 11

Response curves of exit thickness for different hydraulic oil temperatures.

In the text
thumbnail Fig. 12

The MN-AGC based on the CDMC strategy used in actual production.

In the text
thumbnail Fig. 13

Thickness deviations of the 2.5-mm products (a) and the 5.0-mm products (b).

In the text

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