Hydraulic gap control of rolling mill based on self-tuning fuzzy PID

Abstract

A closed-loop control system for the hydraulic gap control (HGC) that is driven by electrohydraulic servo valves is developed for practical application. Based on the mathematical model estimated by a system identification technique, the control algorithm is designed, and the fuzzy inference rules are established. The simulation and field test results of the step response and the position tracking that is carried out on the HGC system of a cold rolling mill show that when compared with the conventional PID control system, the self-tuning fuzzy PID system has the characteristics of fast response, short rise time, no lag, small overshoot, and strong anti-interference ability. At the same time, the self-tuning fuzzy PID control algorithm can not only improve the position tracking ability of the HGC system, but can also tune the servo valve to overcome the nonlinearity of the HGC system.

Keywords

System identification hydraulic gap control position control self-tuning fuzzy PID

1 Introduction

Rolling strips are widely used for many fields [7]. In the development of the metallurgical industry in the last one hundred years, the high-performance, steel-rolling technology has made steel rolling the main way to mold steel in the steel industry. Steel production is faced with the challenge of market competition and sustainable development. However, in the era of the 21st century, steel will still be the main basis for global raw materials; steel rolling will still be the mainstay of steel forming technology and will play a fundamental support role in the economic development and social progress of the whole world, especially in developing countries. A rolling mill is the equipment used to accomplish the metal rolling process. It refers to the equipment used to complete the whole process of rolling production, including the main equipment, auxiliary equipment, transport equipment and ancillary equipment. However, in general, the mill often refers to the main equipment. Mills are often different in type, number, roll position and roll number [15]. The mechanical structure of a common, 4-high rolling mill is shown in Fig. 1 [6]. The basic flat rolling operation shown in Fig. 2 is being used to produce qualified plate and strip products.

As one of the most important functions of the basic automation of the rolling mill, the hydraulic gap control (HGC) is the inner loop of the automatic gauge control (AGC). Therefore, the good dynamic characteristics of the HGC system are very important to the AGC system [8]. The HGC system consists of two-acting hydraulic cylinders per stand and one per housing (housing window). The cylinder housings each rest on the top backup roll (BUR) chocks. The piston rods point upwards in direction of the stepped plates. The system composed of steeped plates, the hydraulic adjustment system and BUR chocks is held clearance-free against both mill housing yokes by the BUR balancing system. The control movements of the cylinder housing are thus transmitted directly to the top BUR chocks.

The hydraulic cylinder is generally equipped with electrohydraulic servo valve and the corresponding transducers for the measuring position, pressure, etc., so as to achieve closed-loop control. Accurate and precise position control and pressure control can be realized by the hydraulic cylinder in the modern rolling mill. Therefore, the appropriate control strategy should be adopted. From the control engineering point of view, it is necessary to achieve the rejection of external disturbances and an effective reference tracking for the step, as well as a ramp and sinusoidal reference signal [4].

A conventional PID controller has the advantages of simple realization, high reliability and good stability, but it is designed based on the precise system model. Therefore, it has poor anti-disturbance and adaptability and cannot effectively control a nonlinear and uncertain complex system [9]. When faced with a highly nonlinear, time-varying uncertainsystem like the HGC, the PID controller cannot achieve a satisfactory control effect. Therefore, it is required that the PID controller does not depend on the mathematical model, and that it realizes the real time control, which is exactly the advantage of a fuzzy controller. The fuzzy algorithm provides a good solution to solve the problem of the large range of control object parameters. By adopting the fuzzy compound control theory, the PID control strategy is introduced into the fuzzy controller to develop the fuzzy PID controller, which has been widely used [3 , 18]. According to relevant research results, most of the fuzzy controllers are based on the fuzzy PID controller. The fuzzy PID controller makes use of its advantages of flexible control, good speed and strong adaptability, as well as high control accuracy, in order to achieve the effective control of the electrohydraulic servo valve.

Based on the deep analysis of the rolling process, the fuzzy self-tuning PID controller is developed to overcome the nonlinearity and uncertainty of the HGC system. Because of the great risk and considerable expense of the online experiment, it must be taken into account in the verification of the algorithm. Therefore, the simulation is of great significance [11, 16].

This paper is organized as follows: In Section 2, the model structure of the HGC system is analyzed and proposed. In Section 3, the mathematical model of the HGC system was identified by an autoregressive, exogenous (ARX) approach on the foundation of the theoretical model. In Section 4, after analyzing the advantages and disadvantages of the PID controller and the fuzzy controller, the self-tuning fuzzy PID controller is designed for the HGC system of the rolling mill. In Section 5, the simulation and field test results prove that the method is effective in improving the dynamic property and decreasing the steady-state error of the HGC system. Conclusions are discussed in Section 6.

2 Modeling of HGC system

2.1 Structure of HGC system

As shown in Fig. 3, the HGC system consists of several parts, such as the controller, the servo valve, the hydraulic cylinder, the pressure transducer and the position transducer. The HGC reference can be set by several auto sequences, including roll change, gap zeroing, rolling setup and the mill modulus test. While rolling, AGC provides position offsets to maintain strip thickness. The primary position feedback is created from a magnescale, linear position transducer that measures the extension of the piston from the cylinder bore. The sensor outputs up or down count pulses that reflect the change in position. A hardware counter on an input card counts the pulses and reports the total count to the controller. The pulse count is scaled and offset for a reference position with the cylinder collapsed.

By comparing the reference with the feedback, the controller calculates the position error e, which is based on when the corresponding closed loop control is carried out. Also, by setting a dead band, the controller will act only when the error e exceeds the specified range.

The resulting error e is then multiplied by a proportional-position control gain. If mill modulus control (MMC) is on, the gain of the regulator is increased by a factor that restores the overall gain of the regulator based on the mill modulus, strip modulus and MMC gain. The gain of the regulator is also adjusted by an automatic flow-gain compensation that maintains a constant regulator response as the pressure in the cylinder changes. There is also a cylinder-position gain compensation that maintains a constant regulator response as the cylinder position changes.

The servo valves typically have a mechanical bias set to cause the cylinder to slowly collapse when no electrical control signal is applied. This means that the electrical control signal must be non-zero to maintain the cylinder position. This bias is commonly referred to as null-bias. Therefore, the compensation for this effect is called the null-bias compensation. This compensation is a very slow integrator that adds to the servo valve command signal so that the cylinder stops when the regulator error is zero. The integrator is only enabled when the regulator is trying to hold a position. The final error signal is scaled and clamped to match the control signal requirements of the servo valve.

2.2 Mathematical model

The control block of the HGC system can be seen in Fig. 4, where r denotes the HGC reference and x_p denotes the actuator piston position.

The operation principle of a valve-controlled hydraulic cylinder is shown in Fig. 5, where p₀ and p_s is the return pressure and supply pressure, respectively. A₁ and A₂ are the areas of the cavity and the rod cavity, respectively; p₁ and p₂ are the pressures of the cavity and the rod cavity, respectively.

The load flow Q_L of an ideal servo valve is given as [5, 13]: $Q_{L} = C_{d} {wx}_{v} \sqrt{\frac{p_{s} - p_{L}}{ρ}}$ (1)

where C_d is a discharge coefficient, w is the spool-valve area gradient, x_v is spool displacement, p_L is load pressure, and ρ is fluid density.

Because of the natural, nonlinearity of the servo valve, the linear approximation is used to facilitate the analysis [10]: ${\begin{matrix} Δ Q_{L} = Q_{sv 0} - k_{c} Δ p_{L} \\ Q_{sv 0} = k_{sv} I_{c} \end{matrix}$ (2)

where ΔQ_L is the change of load flow, Q_sv₀ is the idle load flow, k_c is the pressure flow factor, Δp_L is the change of load pressure, k_sv is the static flow amplification factor and I_c is the input current.

According to different natural frequency ω_h of a hydraulic actuator, the dynamic characteristic of the servo valve can be described as follows: $\frac{Q_{sv 0}}{I_{c}} = {\begin{matrix} \frac{k_{sv}}{\frac{1}{ω_{sv}} s + 1}, ω_{h} \leq 50 Hz \\ \frac{k_{sv}}{\frac{1}{ω_{sv}^{2}} s^{2} + \frac{2 ζ_{sv}}{ω_{sv}} s + 1}, ω_{h} > 50 Hz \end{matrix}$ (3)

Then the load flow Q_L is expressed as: $Q_{L} = A_{1} X_{p} s + k_{ce} p_{L} + \frac{V_{t}}{4 β_{e}} P_{L} s$ (4)

where k_ce is the piston leakage coefficient, β _e is the effective bulk modulus of oil, V_t is the compressed volume, X_p and P_L are the Fourier transforms of x_p and p_L respectively.

The total force F generated from the actuator is given as: $F = M_{c} X_{p} s^{2} + B_{p} X_{p} s + k_{t} X_{p} + F_{L}$ (5)

where M_c is the total mass of the moving parts in rolls, B_p is the viscosity coefficient of moving parts, such as the piston and load, k_t is the stiffness coefficient of the elastic load and F_L is other loads acting on the piston.

According to the above deductions, the dynamic model of the HGC system is expressed as: $X_{p} = \frac{N (s)}{D (s)}$ (6)

with $N (s) = \frac{k_{sv}}{A_{1}} I_{c} - \frac{k_{ce}}{A_{1}^{2}} (1 + \frac{V_{t}}{4 β_{e} k_{ce}} s) (F_{L} + A_{2} p_{2})$ (7)

and $\begin{matrix} D (s) = \frac{M_{e} V_{t}}{4 β_{e} A_{1}^{2}} s^{3} + (\frac{M_{e} k_{ce}}{A_{1}^{2}} + \frac{B_{p} V_{t}}{4 β_{e} A_{1}^{2}}) s^{2} \\ + (1 + \frac{k_{t} V_{t}}{4 β_{e} A_{1}^{2}} + \frac{B_{p} k_{ce}}{A_{1}^{2}}) s + \frac{k_{ce} k_{t}}{A_{1}^{2}} \end{matrix}$ (8)

3 Model identification of HGC system

3.1 Technological parameters

Taking the HGC system of a 4-high, single-stand, cold rolling mill as the identified object, we carried out an identification test. The specifications of the rolling mill are listed in Table 1.

3.2 Data acquisition

The measured data of the open-loop test of the HGC system were collected by the ibaPDA (Process Data Acquisition) system. The ibaPDA system is a PC-based acquisition and analysis system for measured values. The gathered data is saved in files on the hard disk of the online PC. Figure. 6 shows the system topology with one server and multiple clients. The server is a basic process in ibaPDA which handles the data acquisition and storage. It can run independently from and without a client.

3.3 Identification tool

The input signal u(k) is summed up by three cosine functions of different amplitudes and frequencies. $u (k) = \sum_{i = 1}^{3} a_{i} cos (ω_{i} T_{s} k)$ (9)

where a_i is the amplitude, ω_i is the frequency, and T_s is the sampling period.

The ARX model is built with data measured of the input and output of a system, in which the behavior of the system can be represented through two vectors called the input regression size n+1 and outputregression size n. The coefficients a_j and b_j are parameters to be estimated that apply any kind of numeric analysis technique; for example, least squares. The term ɛ(k) is associated to disturbances of the systems not represented by the behavior of an equation but by a numerical value [2]. The ARX model is expressed as: $y (k) = \sum_{j = 1}^{n} a_{j} y (k - j) + \sum_{j = 0}^{n} b_{j} u (k - j) + ɛ (k)$ (10)

According to the mathematical model of Equation (5), (6) and (7), the HGC system is simplified as a third-order system. Therefore, the parameter n in Equation (10) is equal to 3, and the third-order ARX model is used to approximate the real plant model.

As a graphical user interface (GUI), the Matlab System Identification Toolbox is used to estimate and analyze linear and nonlinear models in the system identification [17]. This includes data import, data analysis, data preprocessing, model estimation and estimated model validation.

3.4 Identification results

Data for the model estimation is taken from the cold rolling mill with multi-frequency, sine-wave input. The input and output signals acquired by the ibaPDA system are shown in Fig. 7, from which the transfer function of the model ARX331 can be obtained. The output signal analysis shows that the fit percentage is 90.15%.

Then, we can get the discrete, open-loop transfer function as follows: $G (k) = \frac{0.1998 z^{- 1} + 0.0911 z^{- 2} + 0.0241 z^{- 3}}{1 - 0.9140 z^{- 1} - 0.2319 z^{- 2} + 0.1462 z^{- 3}}$ (11)

4 Fuzzy PID design

The PID controller is most commonly used in industrial processes, consisting of three parts: proportional control, integral control and derivative control. It is a second-order, lead-lag correction means. PID control is usually combined with other algorithms to improve its performance. However, the basic operations still remain the same [19]. The ideal PID controller is given by the following formula: $\begin{matrix} u (k) = K_{p} e (k) + K_{i} T_{s} \sum_{i = 1}^{k} e (i) + K_{d} [e (k) \\ - e (k - 1)] / T_{s} \end{matrix}$ (12)

where k is the current sampling time, u(k) and e(k) are the control signal and the feedback error, respectively and K_p, K_i and K_d are controller parameters.

In order to improve the control performance of the PID control, the fuzzy PID controller uses the fuzzy algorithm to adjust the parameters of the PID control. The core task of the fuzzy PID control is to find out the fuzzy relationship between the three parameters of the PID control, the error e and the derivative of error ec. When the controller works, the parameters e and ec are continuously calculated. According to the predetermined fuzzy control rules, the three parameters are adjusted online to meet the different requirements of different e and ec for three parameters. The control schematic of the fuzzy PID controller is shown in Fig. 8.

As shown in Fig. 9, the feedback error e and the derivative of error ec are the two inputs of the fuzzy logic-inference engine. Each input uses seven membership functions. The fuzzy logic-inference engine has three outputs: K_p, K_i and K_d. The input and output membership functions are shown in Figs. 10 and 11, respectively. The ranges of input and output of the fuzzy logic-inference engine are set from the tests of the PID controller. These values are then substituted into Equation (12) to update the values of K_p, K_iand K_d.

${\begin{matrix} K_{p} = K_{p 0} + Δ K_{p} \\ K_{i} = K_{i 0} + Δ K_{i} \\ K_{d} = K_{d 0} + Δ K_{d} \end{matrix}$ (13)

where K_p₀, K_i₀ and K_d₀ are initial values of K_p, K_i and K_d, respectively; ΔK_p, ΔK_i and ΔK_d are defuzzified outputs of K_p, K_i and K_d, respectively.

Based on the input membership functions in Fig. 10 and the output membership functions in Fig. 11, the fuzzy control rules shown in Table 2, Table 3 and Table 4 were performed. The linguistic variables were NB (Negative Big), NM (Negative Medium), NS (Negative Small), ZO (Zero), PS (Positive Small), PM (Positive Medium) and PB (Positive Big).

The inference result obtained by the fuzzy control rules is a fuzzy vector which cannot be used directly as a control value and needs to be defuzzified. Defuzzification is a mapping from the fuzzy control action space to the precise control action space. The defuzzification technique employed in this paper is the centroid principle or center of gravity.The centroid defuzzification technique can be expressed as [14]: $z_{CEN} = \frac{\int_{z} μ_{A} (z) zdz}{\int_{z} μ_{A} (z) dz}$ (14)

where z is the value of the linguistic variable, z_CEN is the crisp output and μ _A (z) is the aggregated membership function.

5 Simulation and field test results

5.1 Simulation

The designed controller is used to control the plant based on Equation (10). The reference or position input signal is the step signal at 0th s, and the sampling time is 0.001 s. At the 100th sampling time, an interference of 1 V is added to the controller output. During this process, the controller parameters or gains are adjusted incrementally by repeatedly performing step responses. At the same time, the controller evaluates the performance error, alters the gain and repeats the step response test until the optimal gain is obtained [1]. The changes of state variables and controller parameters are illustrated in Figs. 12 and 13.

At the 200th sampling time, the error e is 0.0018 mm, which has met the requirements of the control accuracy. It can also be seen from Figs. 12 and 13, that at this point, the control system has been stable and the controller parameters are nearly constant. Therefore, we can assume that the adjustment process has been completed and the tuned controller parameters are K_p = 5.099×10^- 1, K_i = 9.0376×10^- 6, and K_d = 7.885×10^- 1.

5.2 Field test

Generally, the tuned PID controller of the HGC system is designed to have no overshoot and no steady-state error. However, sometimes in order to ensure the stability of the HGC system, we deliberately select appropriate, small values. Then the rise time and settling time of the response will be correspondingly longer.

The field test results for step response and interference response are illustrated in Fig. 14. At first, the mill system is running smoothly without interference. Then the reference of the HGC system is set from 0.0 to 0.1 at the time of 0th second. Then, in order to verify the anti-interference performance of the HGC system, an interference of 0.1 V is added to the controller output at the first 0.1 s. It can be seen from Fig. 14, that compared with the tuned PID controller without the tuning mechanism, the self-tuning fuzzy PID controller has a better control effect. This includes a faster rise time and settling time, less overshoot, less steady state error and stronger robustness. When the system suffers from greater interference, it will offer a robust response.

The field test results shown in Fig. 15 indicate that HGC system with a self-tuning, fuzzy PID controller can achieve a good position-tracking accuracy, and the deviation between the target value and the actual value is less than 5.0% over 96.2% of the entire testing period. Field test results also illustrate that 6 ms after the controller output a ramp command, the piston of the hydraulic cylinder still didn’t move.

In addition, all kinds of restriction conditions of the industrial field should be considered. Also, the action process of the controller under various extreme conditions should be tested. Only through these tests can the controller finally be put into practicalapplication.

Figure 16 illustrates the aluminum coil rolled by a cold rolling mill whose HGC systems are controlled by self-tuning fuzzy PID controllers. Of course, the rolling process also needs the cooperation of other control systems. Through the analysis of the data of the rolling process, we can find that the new HGC system has surpassed its original performancetargets. Therefore, we believe that the algorithm cancompletely meet the requirement of the HGC system. In the future, we will extend this algorithm to the field of tension control, thickness control, shape control, etc.

6 Conclusion

In this paper, we studied the self-tuning fuzzy PID controller and successfully applied it to the HGC system of the rolling mill. The ARX approach was used to obtain the controlled plant model, and the fuzzy controller was used to tune PID parameters. Through the simulation and field test results described above, one can make the conclusions as follows:

The self-tuning fuzzy PID controller can achieve good control effect and strong robustness to the rolling mill HGC system and is suitable for industrial applications.

The control algorithm of the self-tuning fuzzy PID controller is simple, does not need to identify the plant model, and is consequently easy to apply.

Compared with the tuned PID, the self-tuning fuzzy PID used in the HGC system has effective position-tracking performance.

Although the self-tuning fuzzy PID controller has good performance in the control of the rolling mill HGC system, because of the complexity of the control system, there are still many problems to be solved. This can be used as the direction of future research and are listed as follows:

The fuzzy HGC system of SISO (Single Input and Single Output) is studied in this paper. At the same time, we should pay more attention to the electrohydraulic servo system of MIMO (Multi Input and Multi Output), which is of great research value and a challenge.

The fuzzy control membership functions and control rules that are established on the basis of the experiences of engineering and technical personnel need to be optimized. Accordingly, for optimization, a lot of work needs to be done on the basis of a large number of theoretical and experimental studies.

The variation of roll speed and strip hardness during the rolling process may cause changes in the roll force, strip tension and an unbalance of mass flow of the strip. In further study, we must take into account the control of the tension and thickness in order to deal with these problems.

The electrohydraulic servo control system has many problems that need to be analyzed and studied. It is a promising and valuable subject in the field of industrial control. With the application of advanced control theory and technology in the industry field, it will have greater progress!

Footnotes

Acknowledgments

This work is partially supported by the Fundamental Research Funds for the Central Universities (No. FRF-TP-15-061A3). We thank our colleagues from National Engineering Research Center of Advanced Rolling Technology who provided insight and expertise that greatly assisted the research.

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