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

Authors: Jining Guo; Haoyu Zhang; Desheng Liu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

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. The 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.


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## Body

## 1. Introduction

The 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. The 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 [3–6]. 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 finite-time 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 [9–11]. 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 [17–19]. 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. This 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 [23–27]. Research on its theory is extensive [28–30]. ADRC consists of three parts, namely, the tracker differentiator (TD), the extended state observer (ESO), and the nonlinear state error feedback (NLSEF). The 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 high-order systems and control signals, a composite control strategy combining a nonsingular terminal sliding mode with ADRC is proposed herein. The 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.This paper is organized as follows. The second section introduces the mathematical model of the hydraulic servo system of the rolling mill. The 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.


## 2. 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 Figure1, which is composed of a servo amplifier, servo valve, hydraulic cylinder, and position transducer. The 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.Figure 1
Schematic diagram of the hydraulic position servo system.The 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]. The servo valve flow equation is a typical nonlinear equation, which can be given as follows [34]:(1)Ql=cdωxrps−pl⋅sgnxrρ∂,where Ql is the load flow, cd is the valve port flow coefficient, ω is the servo valve opening gradient, xr is the servo core displacement, Ps is the hydraulic pump outlet pressure, P1 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]:(2)Ql=Apx˙+ctpl+V4βep˙l,where Ap is the hydraulic cylinder piston effective area, x is the cylinder piston displacement, ct is the hydraulic cylinder leakage coefficient, V is the total volume of the hydraulic cylinder body, and βe is the bulk modulus of elasticity. Ignoring the servo valve dynamic, the servo amplifier and servo valve equivalent proportional link can be written as follows:(3)kp=iu,ksr=xri,where ksr is the servo valve gain and i is the output current of the servo amplifier.The piston force balance equation can be given as follows [34]:(4)Appl=mx¨+Bpx˙+kx+Fl,where m is the total mass of the piston and the load, Bp is the viscous damping coefficient of the piston, k is the elastic stiffness coefficient of the piston, and Fl is the accidental load on the piston.Letx1=x, x2=x˙, and x3=x¨, and the state-space expression is as follows:(5)x˙1=x2,x˙2=x3,x˙3=a1x1+a2x2+a3+a4gxru+d,y=x1,a1=−4βeCtkmV,a2=−km−4βemVA2+CtBp,a3=−BPm−4βeCtV,a4⋅gxr=4AβeCdωmVρKPKsvps−plsgnxr,d=−Flm−4βeCtmVFl,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.


## 3. Controller Design

For further study, model (5) may be rewritten as a third-order uncertain system as follows:(6)x⃛t=fx,x˙,x¨,ωt+btut+dt,y=xt,where fx,x˙,x¨,ωt comprises the internal perturbation parameters of the system, dt is the external disturbance of the system, and fx and dt are zero and correspond to two smooth nonlinear functions, assuming that dt is bounded. The third-order active disturbance rejection control structure is shown in Figure 2.Figure 2
Control system block diagram of active disturbance rejection control (ADRC).


### 3.1. Tracking Differentiator

The role of the TD is to arrange the transition process. The differential signal of the position of the rolling mill instruction signal is given by the TD. The tracking signalx1, the first-order differential signal x2, and the second-order differential signal x3 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. The designed discrete tracking differential algorithm of the third-order TD is thus(7)u=fhanx1k,vk,x2k,r,h,x1k+1=x1k+Tx2k,x2k+1=x2k+Tx3k,x3k+1=x3k+Tu,where T is the sampling period, vk is the input signal at moment k, x1k is the tracking filter signal of vk, x2k is the first derivative of vk, x3k is the second derivative of vk, r is the speed factor, h is the filter factor of the instruction signal, and fhanx1,x2,v,r,h is the designed synthesis function of fast optimal control, and its specific algorithm is as follows:(8)d=rh,d0=hd,y=x1−v+hx2,a0=d2+8ry,a=x2+a0−d2signy,y>d0,x2+yh,y≤d0,fhan=rsinga,a>d,−rad,a≤d.The 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. The parameters to be set arer, h, and h0, which are mainly determined by the empirical method. Generally, one will choose r=0.0001/h2. As a filter factor, the larger the value of h0, the better the filtering effect. However, an excessive value will cause lag in tracking, and the two effects will be contradictory. We thus choose h0=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.Figure 3
Tracking output of the tracker differentiator (TD).


### 3.2. Extended State Observer

For the uncertain system shown in equation (1), we select state variables x,x˙,x¨, and x⃛ for observation, and the nonlinear state observer is thus constructed as follows [37]:(9)ε1=z1−y,z˙1=z2−β01ε1,z˙2=z3−β02falε1,α1,δ,z˙3=z4−β03falε1,α2,δ+b0u,z˙4=−β04falε1,α2,δ,where z1, z2, and z3 are the observations of x1, x2, and x3 and z4 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.(10)falei,αi,δ=eiαisignei,ei>δ,eiδ1−αi,ei≤δ,δ>0.The parameters that the nonlinear function needs to determine areα1,α2,α3,andδ. αi determines the shape of the nonlinear function, and δ determines the magnitude of the linear interval. Generally, one takes δ0=0.02, α1=0.5,α2=0.25,andα3=0.125. β01,β02,β03,β04, and b0 are the parameters to be set. According to the bandwidth principle, this should be written as s4+β01s3+β02s2+β03s+β04=s+ω04, where ω0 is the bandwidth of the observer and b0 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. The 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 z1 and z2 have a high estimation accuracy such that z3 and z4 can also estimate the state output with a slightly lower accuracy.Figure 4
The output of the extended state observer (ESO).


### 3.3. 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., letxd be the desired position, and this has first and second derivatives. The tracking error of the system is defined as e1=x1−xd, e2=x2−x˙d, and e3=x3−x¨d. The sliding mode manifold is further defined as(11)s=e3+c2e2α2+c1e1α1,where α1 and α2 are determined by pandq, positive odd numbers. Moreover, 1<p/q<2, α1=q/2p−q, and α2=q/p. c1 and c2 are both constants greater than zero, the selection of which should guarantee that the polynomial p2+c2p+c1 meets the Hurwitz stability criterion. The 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:(12)u=−b−1−c2α2e2α2−1⋅e3−c1α1e1α1−1e2+x⃛d−z4+Msgns,where M is the controller gain.(13)V=12s2.The differential of the sliding mode manifold is as follows:(14)s˙=e˙3+c2α2e2α2−1e˙2+c1α1e1α1−1e˙1=x˙3−x⃛d+c2α2e2α2−1e3+c1α1e1α1−1e2,V˙=sx˙3−x⃛d+c2α2e2α2−1e3+c1α1e1α1−1e2,(15)V˙=sfx+bxu+dx−x⃛d+c2α2e2α2−1e3+c1α1e1α1−1e2.By substituting (13) into (15), one obtains(16)V˙=sfx+bx−b−1−c2x2e2α2−1⋅e3−c1e1α1−1e2−z4+Msgns+dx−x⃛d+c2α2e2α2−1⋅e3+c1α1e2α1−1⋅e2=sfx+dx−z4+Msgns.From (15), one can obtain a relationship in which M>f+d−z4 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.


## 3.1. Tracking Differentiator

The role of the TD is to arrange the transition process. The differential signal of the position of the rolling mill instruction signal is given by the TD. The tracking signalx1, the first-order differential signal x2, and the second-order differential signal x3 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. The designed discrete tracking differential algorithm of the third-order TD is thus(7)u=fhanx1k,vk,x2k,r,h,x1k+1=x1k+Tx2k,x2k+1=x2k+Tx3k,x3k+1=x3k+Tu,where T is the sampling period, vk is the input signal at moment k, x1k is the tracking filter signal of vk, x2k is the first derivative of vk, x3k is the second derivative of vk, r is the speed factor, h is the filter factor of the instruction signal, and fhanx1,x2,v,r,h is the designed synthesis function of fast optimal control, and its specific algorithm is as follows:(8)d=rh,d0=hd,y=x1−v+hx2,a0=d2+8ry,a=x2+a0−d2signy,y>d0,x2+yh,y≤d0,fhan=rsinga,a>d,−rad,a≤d.The 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. The parameters to be set arer, h, and h0, which are mainly determined by the empirical method. Generally, one will choose r=0.0001/h2. As a filter factor, the larger the value of h0, the better the filtering effect. However, an excessive value will cause lag in tracking, and the two effects will be contradictory. We thus choose h0=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.Figure 3
Tracking output of the tracker differentiator (TD).


## 3.2. Extended State Observer

For the uncertain system shown in equation (1), we select state variables x,x˙,x¨, and x⃛ for observation, and the nonlinear state observer is thus constructed as follows [37]:(9)ε1=z1−y,z˙1=z2−β01ε1,z˙2=z3−β02falε1,α1,δ,z˙3=z4−β03falε1,α2,δ+b0u,z˙4=−β04falε1,α2,δ,where z1, z2, and z3 are the observations of x1, x2, and x3 and z4 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.(10)falei,αi,δ=eiαisignei,ei>δ,eiδ1−αi,ei≤δ,δ>0.The parameters that the nonlinear function needs to determine areα1,α2,α3,andδ. αi determines the shape of the nonlinear function, and δ determines the magnitude of the linear interval. Generally, one takes δ0=0.02, α1=0.5,α2=0.25,andα3=0.125. β01,β02,β03,β04, and b0 are the parameters to be set. According to the bandwidth principle, this should be written as s4+β01s3+β02s2+β03s+β04=s+ω04, where ω0 is the bandwidth of the observer and b0 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. The 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 z1 and z2 have a high estimation accuracy such that z3 and z4 can also estimate the state output with a slightly lower accuracy.Figure 4
The output of the extended state observer (ESO).


## 3.3. 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., letxd be the desired position, and this has first and second derivatives. The tracking error of the system is defined as e1=x1−xd, e2=x2−x˙d, and e3=x3−x¨d. The sliding mode manifold is further defined as(11)s=e3+c2e2α2+c1e1α1,where α1 and α2 are determined by pandq, positive odd numbers. Moreover, 1<p/q<2, α1=q/2p−q, and α2=q/p. c1 and c2 are both constants greater than zero, the selection of which should guarantee that the polynomial p2+c2p+c1 meets the Hurwitz stability criterion. The 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:(12)u=−b−1−c2α2e2α2−1⋅e3−c1α1e1α1−1e2+x⃛d−z4+Msgns,where M is the controller gain.(13)V=12s2.The differential of the sliding mode manifold is as follows:(14)s˙=e˙3+c2α2e2α2−1e˙2+c1α1e1α1−1e˙1=x˙3−x⃛d+c2α2e2α2−1e3+c1α1e1α1−1e2,V˙=sx˙3−x⃛d+c2α2e2α2−1e3+c1α1e1α1−1e2,(15)V˙=sfx+bxu+dx−x⃛d+c2α2e2α2−1e3+c1α1e1α1−1e2.By substituting (13) into (15), one obtains(16)V˙=sfx+bx−b−1−c2x2e2α2−1⋅e3−c1e1α1−1e2−z4+Msgns+dx−x⃛d+c2α2e2α2−1⋅e3+c1α1e2α1−1⋅e2=sfx+dx−z4+Msgns.From (15), one can obtain a relationship in which M>f+d−z4 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.


## 4. 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 5
Simulation model of the control system.Table 1
Simulation model of the control system.
ParameterNumerical valueViscous damping coefficient of piston Bp (N·s/m)2.25 × 106Servo gain coefficient ksv (m3/(V·s))0.01Bulk modulus of elasticityβe (Pa)7 × 108Return pressure pr (Pa)0Hydraulic pump outlet pressure Ps (Pa)3.8 × 107Hydraulic oil densityρ (kg/m3)850Effective area of the cylinder Ap (m2)0.1256Hydraulic cylinder leakage coefficient Ct (m5/(N·s))5 × 10–16Total mass of the piston and loadm (kg)1500Elastic stiffness coefficientk (N/m)2.5 × 109Valve flow coefficientCd0.61Servo valve area gradientw (m)0.025Initial volume of the hydraulic cylinderV (m3)3.768 × 10−3


### 4.1. Simulation Analysis of the No-Load Condition

Figure6 shows the no-load situation under conditions of k = 0 and F1 = 0. The expectation of the system input sine signal is x_d = sin(t). The 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 6
Control performance comparison between NTSMC-ADRC and NTSMC when the input is a sinusoidal signal. (a) Position tracking. (b) Interference signal estimation. (c) Control signal.
(a)(b)(c)Figure7 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. The 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 7
Tracking error comparison between nonsingular terminal sliding mode control-active disturbance rejection control (NTSMC-ADRC) and NTSMC when the input is a sinusoidal signal.Figure8 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(a), the 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.Figure 8
Performance comparison between nonsingular terminal sliding mode control-active disturbance rejection control (NTSMC-ADRC) and NTSMC when the input is a step-over signal. (a) Position tracking output. (b) Control input.
(a)(b)


### 4.2. 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. The piston elastic stiffness isk=2.5×109N/m, and the external load is FL=2×106N. All other parameters are the same, and the expected value of the input is still the sinusoidal signal, which is xd=sint. The system disturbance is dt=50sin2πt. The state of the initial value is x10,x20,x30=0,0,0T. 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 9
Comparison of performance between nonsingular terminal sliding mode control-active disturbance rejection control (NTSMC-ADRC) and NTSMC when random noise is added. (a) Position tracking. (b) Partial uncertainty estimation. (c) Control input.
(a)(b)(c)Figure10 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.Figure 10
Tracking error comparison between nonsingular terminal sliding mode control-active disturbance rejection control (NTSMC-ADRC) and NTSMC when the input is a noise signal.


## 4.1. Simulation Analysis of the No-Load Condition

Figure6 shows the no-load situation under conditions of k = 0 and F1 = 0. The expectation of the system input sine signal is x_d = sin(t). The 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 6
Control performance comparison between NTSMC-ADRC and NTSMC when the input is a sinusoidal signal. (a) Position tracking. (b) Interference signal estimation. (c) Control signal.
(a)(b)(c)Figure7 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. The 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 7
Tracking error comparison between nonsingular terminal sliding mode control-active disturbance rejection control (NTSMC-ADRC) and NTSMC when the input is a sinusoidal signal.Figure8 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(a), the 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.Figure 8
Performance comparison between nonsingular terminal sliding mode control-active disturbance rejection control (NTSMC-ADRC) and NTSMC when the input is a step-over signal. (a) Position tracking output. (b) Control input.
(a)(b)


## 4.2. 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. The piston elastic stiffness isk=2.5×109N/m, and the external load is FL=2×106N. All other parameters are the same, and the expected value of the input is still the sinusoidal signal, which is xd=sint. The system disturbance is dt=50sin2πt. The state of the initial value is x10,x20,x30=0,0,0T. 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 9
Comparison of performance between nonsingular terminal sliding mode control-active disturbance rejection control (NTSMC-ADRC) and NTSMC when random noise is added. (a) Position tracking. (b) Partial uncertainty estimation. (c) Control input.
(a)(b)(c)Figure10 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.Figure 10
Tracking error comparison between nonsingular terminal sliding mode control-active disturbance rejection control (NTSMC-ADRC) and NTSMC when the input is a noise signal.


## 5. Conclusions

Aiming at addressing the uncertainty of hydraulic servo systems, this research has studied interference suppression problems. The NTSMC-ADRC controller has been designed to improve the convergence speed and anti-interference ability. The designed third-order TD can track the input signal accurately and filter random interference signals. The 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. The stability of the system is demonstrated using the Lyapunov method. Finally, NTSMC-ADRC and NTSMC were simulated under no-load and on-load conditions. The results of these simulations show that this method has faster adjustment time and a stronger anti-interference ability.



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*Source: 3467213-2020-12-10.xml*