Investigation of the Hydraulic Servo System of the Rolling Mill Using Nonsingular Terminal Sliding Mode-Active Disturbance Rejection Control

In order to improve the disturbance rejection ability and tracking accuracy of the hydraulic servo system of the rolling mill, this study combines nonsingular terminal sliding mode control (NTSMC) with active disturbance rejection control (ADRC). A fourth-order extended state observer was designed to estimate the disturbance of the system in real time. *e stability of the control system was tested using the Lyapunov method. System effectiveness was verified through simulation experiments. Simulation results showed that the designed state observer can estimate the total disturbance of the system in real time and that the chattering of the control input can be eliminated by the introduction of a state observer. In terms of uncertainty in the system model caused by load changes and external interference signals, the nonsingular terminal sliding mode-active disturbance rejection control method exhibited better disturbance rejection capacity and a higher tracking accuracy than NTSMC.


Introduction
e hydraulic servo system is widely used in aerospace, industrial machinery, robotics and manipulator movement, and metallurgy [1,2]. Hydraulic servo drive control has also become a research focus for various research institutes. e fluctuation of system pressure, changes in friction force and oil compressibility, viscosity, and pressure and temperature will cause internal disturbances of the system. Changes in load condition are the main cause of the external disturbance. Consequently, the system model is nonlinear and uncertain [3][4][5][6]. Classical PID control methods describe the dynamics of the hydraulic servo system as a set of linear equations that define the design of the corresponding controller. In practical industrial applications, it is difficult to obtain satisfactory dynamic and static characteristics.
High-speed convergence is an important requirement of the hydraulic servo control system, and most existing control laws are asymptotically stable. Of these laws, the finite-time control law has clear advantages. Under optimal control conditions, a variety of finite-time control laws have been developed for first-order and second-order control systems [7,8]. In low-order systems, the continuous finitetime state feedback control is frequently adopted, which usually requires a certain system model and has no obvious effect on the disturbance input of the system [9][10][11]. For finite-time control of higher-order systems, the backstepping control is mainly adopted; however, due to the existence of powers less than 1, singular phenomena can still occur [12]. Terminal sliding mode (TSM) control is an effective finite-time control scheme [13,14], different from the traditional sliding mode control in that a nonlinear sliding mode manifold is introduced. Nonetheless, there are still two drawbacks: the first is the singular value problem, and the second occurs when the system state is far from equilibrium and its convergence speed is slow. To address the first problem, a nonsingular terminal sliding mode control (NTSMC) [15] has been proposed. For the second problem, a fast terminal sliding mode control (FTSMC) has been developed [16]. In addition, when solving the synthesis problem of very complex nonlinear systems, parameter perturbation and the invariance of external disturbances will cause high-frequency chatting of the control quantity. For practical applications, this kind of high-frequency chatting cannot be realized by any actuator. Methods to reduce chatting include adjusting the approach law, introducing a filter and interference observer, and the dynamic sliding mode [17][18][19]. At present, the terminal sliding mode control has been applied in the hydraulic position servo system. Shen et al. [20] combined state feedback with the backstepping sliding mode control to enable finite-time control of the second-order hydraulic servo system; however, when the disturbance exceeds a certain range, the state observer will show significant deviations, and the accuracy of control will be affected.
On the contrary, improving the antidisturbance performance of the system is expected by all control methods. Active disturbance rejection control (ADRC) is an effective method through which the disturbance rejection capability of uncertain systems can be improved. is method was proposed by Professor Han Jingqing on the basis of nonlinear PID control [21], which has the characteristics of not relying on an accurate model of the system, strong antidisturbance ability, a simple algorithm, and ease of implementation [22]. It is thus widely used in mechanical and electrical systems, aerospace and chemical engineering, and other fields [23][24][25][26][27]. Research on its theory is extensive [28][29][30]. ADRC consists of three parts, namely, the tracker differentiator (TD), the extended state observer (ESO), and the nonlinear state error feedback (NLSEF). e basis of this idea is to use the state observer to estimate internal parameter uncertainty and the external nondisturbance of the system in real time. By using nonlinear state error feedback to compensate, the antidisturbance capability of the system is improved on the premise that only an input and output are required of the system. Aiming to address the uncertainty of hydraulic servo control systems, Yao and Deng [31] designed an ADRC system and proved that it was asymptotically stable. Wang et al. [32] applied ADRC to the reduced-order model of the hydraulic servo system and proved the effectiveness of the method through an experimental approach. However, for sudden disturbances, systems are still prone to overshoot and chatting. Gao et al. [33] proposed an active disturbance rejection control scheme with velocity compensation, which used the extended state observer to observe system disturbance and improve anti-interference abilities, but did not analyze system stability. It is thought that terminal sliding mode control and ADRC are very effective control methods for uncertain systems. Nonetheless, in previous research, the second-order system model is the main research object. To solve the problem of terminal sliding mode control being unsuitable for buffering of highorder systems and control signals, a composite control strategy combining a nonsingular terminal sliding mode with ADRC is proposed herein. e main idea of this approach was to combine the advantage of finite-time convergence in the nonsingular terminal sliding mode control method with the ability of the ADRC to suppress external interference and model uncertainty and achieve smooth tracking of the hydraulic position servo system. Firstly, TD was used to obtain the differential signal of the input trajectory, and ESO was used to obtain estimated values of displacement, angular velocity, acceleration, and total disturbance of the hydraulic servo system. Nonlinear terminal sliding mode control law was used as the error feedback control law to compensate the total disturbance of the system, and a sliding mode controller was designed based on ADRC. Secondly, the stability of the system was proved using the Lyapunov method. Finally, system effectiveness was verified using the simulation experiments. is paper is organized as follows. e second section introduces the mathematical model of the hydraulic servo system of the rolling mill. e third section proposes the design of an ADRC based on nonsingular sliding mode control, and the stability of the system is proved using the Lyapunov method. In the fourth section, the hydraulic servo system is simulated under different working conditions. Section five presents our conclusions.

Mathematical Model of the Hydraulic Servo System of the Rolling Mill
A schematic of the hydraulic servo system of the rolling mill is shown in Figure 1, which is composed of a servo amplifier, servo valve, hydraulic cylinder, and position transducer. e mass spring system is an external load, and the servo valve drives the movement of the hydraulic cylinder, which itself is controlled to keep the displacement of the hydraulic cylinder as close as possible to the desired trajectory, regardless of the operating point. e mathematical model consists of three main equations, i.e., the servo valve flow equation, the cylinder flow continuous equation, and the piston force balance equation [17]. e servo valve flow equation is a typical nonlinear equation, which can be given as follows [34]: where Q l is the load flow, c d is the valve port flow coefficient, ω is the servo valve opening gradient, x r is the servo core displacement, P s is the hydraulic pump outlet pressure, P 1 is the load pressure, and ρ is the hydraulic oil density. Considering leakage and compressibility of the oil, the flow continuity equation of the oil cylinder can be written as follows [34]: where A p is the hydraulic cylinder piston effective area, x is the cylinder piston displacement, c t is the hydraulic cylinder leakage coefficient, V is the total volume of the hydraulic cylinder body, and β e is the bulk modulus of elasticity. Ignoring the servo valve dynamic, the servo amplifier and 2 Mathematical Problems in Engineering servo valve equivalent proportional link can be written as follows: where k sr is the servo valve gain and i is the output current of the servo amplifier.
e piston force balance equation can be given as follows [34]: where m is the total mass of the piston and the load, B p is the viscous damping coefficient of the piston, k is the elastic stiffness coefficient of the piston, and F l is the accidental load on the piston. Let x, and the state-space expression is as follows: where u is the control input of the system, y is the output of the system, and d is the disturbance caused by load change.

Controller Design
For further study, model (5) may be rewritten as a thirdorder uncertain system as follows: x, ω(t)) comprises the internal perturbation parameters of the system, d(t) is the external disturbance of the system, and f(x) and d(t) are zero and correspond to two smooth nonlinear functions, assuming that d(t) is bounded.
e third-order active disturbance rejection control structure is shown in Figure 2.

Tracking Differentiator.
e role of the TD is to arrange the transition process. e differential signal of the position of the rolling mill instruction signal is given by the TD. e tracking signal x 1 , the first-order differential signal x 2 , and the second-order differential signal x 3 of the reference instruction signal are generated and can effectively alleviate overshoot phenomena caused by the excessive initial error and the contradiction between the rapid response and overshoot of the system. e designed discrete tracking differential algorithm of the third-order TD is thus

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where T is the sampling period, v(k) is the input signal at moment k, is the second derivative of v(k), r is the speed factor, h is the filter factor of the instruction signal, and fhan(x 1 , x 2 , v, r, h) is the designed synthesis function of fast optimal control, and its specific algorithm is as follows: e tracking differentiator enables the smooth approximation of the input signal and the extraction of the differential signal and has some filtering effects on the random noise signal. e parameters to be set are r, h, and h 0 , which are mainly determined by the empirical method. Generally, one will choose r � (0.0001/h 2 ). As a filter factor, the larger the value of h 0 , the better the filtering effect. However, an excessive value will cause lag in tracking, and the two effects will be contradictory. We thus choose h 0 � 20h.
For the third-order uncertain system described by equation (6), the TD designed by formulae (7) and (8) is ultimately convergent [35,36] and can enable the transition of the input signal and tracking output of the TD to a sinusoidal signal. As shown in Figure 3, there is vibration near the origin when second-order tracking signals are calculated.

Extended State
Observer. For the uncertain system shown in equation (1), we select state variables x, _ x, € x, and ẋ t for observation, and the nonlinear state observer is thus constructed as follows [37]: where z 1 , z 2 , and z 3 are the observations of x 1 , x 2 , and x 3 and z 4 is the observation of the total disturbance of the system. For system (6), the above approach is followed: a nonlinear function fal is selected to design a fourth-order ESO for real-time fast estimation of the system state variables and its total disturbance.
e parameters that the nonlinear function needs to determine are α 1 , α 2 , α 3 , and δ. α i determines the shape of the nonlinear function, and δ determines the magnitude of the linear interval. Generally, one takes δ 0 � 0.02, α 1 � 0.5, α 2 � 0.25, and α 3 � 0.125. β 01 , β 02 , β 03 , β 04 , and b 0 are the parameters to be set. According to the bandwidth principle, this should be written as s 4 + β 01 s 3 + β 02 s 2 + β 03 s + β 04 � (s + ω 0 ) 4 , where ω 0 is the bandwidth of the observer and b 0 is the estimate of parameter b, which has a physical meaning that is more important. It not only determines the control law u but also affects the interval variation of the total disturbance. e selection of its value cannot be accomplished immediately; instead, it needs to be adjusted repeatedly according to the experience after the selection of the other parameters; finally, an ideal result can be obtained.
For the third-order uncertain system described by formula (6), the ESO designed by formula (9) is adopted. In the case of uncertainty in the object model, the state estimation effect of the system is shown in Figure 4. It can be seen that the outputs of the observed values of z 1 and z 2 have a high estimation accuracy such that z 3 and z 4 can also estimate the state output with a slightly lower accuracy.

Nonsingular Terminal Sliding Mode Control.
In order to improve the tracking accuracy and speed, the nonsingular fast terminal sliding mode control method is introduced into the nonlinear feedback control law. On the basis of [38], a design method of nonsingular fast terminal sliding mode control is presented.
Firstly, we define the sliding mode manifold, i.e., let x d be the desired position, and this has first and second derivatives. e tracking error of the system is defined as where α 1 and α 2 are determined by p and q, positive odd numbers. Moreover, ((1 < p)/(q < 2)), α 1 � (q/(2p − q)), and α 2 � (q/p). c 1 and c 2 are both constants greater than zero, the selection of which should guarantee that the polynomial p 2 + c 2 p + c 1 meets the Hurwitz stability criterion. e convergence of system (16) in finite time is guaranteed for the above sliding mode manifold. For electrohydraulic servo system (2), if one selects nonsingular sliding mode manifold (10) and designs NTSMC as follows, tracking errors will converge to zero in finite time: where M is the controller gain.
e differential of the sliding mode manifold is as follows:

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_ By substituting (13) into (15), one obtains From (15), one can obtain a relationship in which M > f + d − z 4 and _ V < 0. It is known that the control system is stable and that its error can reach the sliding mode manifold and converge to zero in finite time.

Simulation Analysis
For electrohydraulic control system (6), Simulink was used to establish the control system model. Figure 5 shows the simulation model of the system, and Table 1 [17,34] shows the parameters of the system. Figure 6 shows the no-load situation under conditions of k � 0 and F1 � 0. e expectation of the system input sine signal is x_d � sin(t).

Simulation Analysis of the No-Load Condition.
e system disturbance is d(t) � 50 sin(t). Comparing the control performance of NTSMC-ADRC with that of NTSMC, Figure 6(a) shows that both controls can track the expected signal, and there is a substantial error in NTSMC within the first few seconds. It can be seen from Figure 6(b) that the designed extended state observer can observe the system disturbance in real time. As shown in Figure 6(c), the NTSMC-ADRC control signal (T) changes smoothly with the periodic line of the interference signal, for which there is no single value. Buffering is eliminated due to the estimation and compensation of the ESO for uncertain parts of the system, while jitter exists in the NTSMC signal. Figure 7 compares the position tracking errors of the two control methods, and it can be clearly seen that the control performance of NTSMC-ADRC is better than that of NTSMC. e maximum position tracking error of NTSMC-ADRC is less than 0.03, whereas the maximum position tracking error of NTSMC is around 0.1. Figure 8 shows a comparison between the system input unit step signal and the performance of the two kinds of control considered herein. As shown in Figure 8   Mathematical Problems in Engineering NTSMC-ADRC control can track the position signal quickly and accurately. However, NTSMC has a large tracking error near zero and a slow stabilization time. Figure 8(b) shows the system control signal input and demonstrates that the NTSMC-ADRC control input exhibits a smoother curve of periodic change after 2 seconds, whereas the NTSMC input switches back and forth within a certain range.

Load Condition Simulation
Analysis. According to real situations, when the roll gap is close to zero, the upper and lower rollers contact each other, and parameter jumps can occur. e piston elastic stiffness is k � 2.5 × 10 9 (N/m), and the external load is F L � 2 × 10 6 N. All other parameters are the same, and the expected value of the input is still the sinusoidal signal, which is x d � sin(t).
e system disturbance is d(t) � 50 sin(2πt). e state of the initial value is [x 10 , x 20 , x 30 ] � [0, 0, 0] T . On this basis, simulating measurement noise or random noise signal interference requires addition to the input signal. As presented in Figure 9(a), the position output of the NTSMC-ADRC control is closer to the expected value. Figure 9(b) shows the estimation of interference signals by the state observer. Figure 9(c) shows the control input. It can be seen from Figure 9 that, in the case of external random interference, the amplitude of the NTSMC input fluctuates greatly, whereas that of NTSM-ADRC, due to the filtering effect of the TD and the estimation and compensation of the ESO in the total interference of the system, is smaller and shows only periodic change. Figure 10 shows the comparison of tracking errors between the two control methods when the load is not zero. NTSMC-ADRC therefore has better performance than NTSMC.

Conclusions
Aiming at addressing the uncertainty of hydraulic servo systems, this research has studied interference suppression problems. e NTSMC-ADRC controller has been designed to improve the convergence speed and anti-interference ability.
e designed third-order TD can track the input signal accurately and filter random interference signals. e designed ESO can estimate the uncertain part of the model and the external interference in real time and can effectively weaken chattering caused by sliding mode control gain. e stability of the system is demonstrated using the Lyapunov method. Finally, NTSMC-ADRC and NTSMC were simulated under no-load and on-load conditions. e results of these simulations show that this method has faster adjustment time and a stronger anti-interference ability.

Data Availability
e data used to support the findings of this study are included within the article. e data in this paper (Table 1) are from Shi et al. [17].

Conflicts of Interest
e authors declare that they have no conflicts of interest.