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Sliding Mode Control Based on High Gain Observer for Electro-Hydraulic Servo System

Sliding Mode Control Based on High Gain Observer for Electro-Hydraulic Servo System Hindawi Journal of Electrical and Computer Engineering Volume 2023, Article ID 7932117, 12 pages https://doi.org/10.1155/2023/7932117 Research Article Sliding Mode Control Based on High Gain Observer for Electro-Hydraulic Servo System Zhenshuai Wan , Yu Fu , Chong Liu , and Longwang Yue School of Mechanical and Electrical Engineering, Henan University of Technology, Zhengzhou 450001, China Correspondence should be addressed to Zhenshuai Wan; wanzhenshuai@haut.edu.cn Received 18 October 2022; Revised 23 December 2022; Accepted 26 December 2022; Published 3 January 2023 Academic Editor: Chao Zhai Copyright © 2023 Zhenshuai Wan et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Te electro-hydraulic servo system is widely used in industrial automation felds for its merits of the high force to weight ratio, compact size, and fast response. However, the parameter uncertainties and external disturbances of the electro-hydraulic servo system signifcantly deteriorate the control performance of conventional linear controller in practice. To deal with this problem, sliding mode controller (SMC) that incorporates high gain observer (HGO) is proposed in this paper. HGO is used to obtain the accurate time derivative of position signal for sliding mode controller design. Te stability of the control system is guaranteed by Lyapunov stability theory. Comparation simulation is conducted to validate the efectiveness of the presented control scheme. mitigate parametric variation by using its self-learning 1. Introduction properties. Yao et al. used adaptive control to handle the Te electro-hydraulic servo system is widely applied in parametric uncertainties and nonlinear friction of hy- draulic actuators, where a continuously diferentiable industrial automation felds, such as aircraft actuator [1], shaking table [2], and construction machine [3], owing to nonlinear friction model is frst established [23]. How- the superiorities such as fast dynamic response, small size, ever, the tedious adjustment design of control law and large force/torque output [4–7]. However, inherent complicates the controller application. Due to the nonlinear friction, parameter variation, and external dis- nonlinear dynamics of hydraulic cylinder and uncertain turbance restrict the high performance trajectory tracking in fuid parameters, backstepping control has been widely practical application [8]. Hence, how to cope with the in- used in the hydraulic system. Guo et al. proposed a fuence of these drawbacks has attracted the attentions of backstepping controller with extended-state-observer to academia and industry. Although the classical proportional- compensate the unknown load disturbance and uncer- integral-derivative (PID) controller is widely used in process tain nonlinearity of the electro-hydraulic system [24]. control, it cannot achieve satisfactory tracking performance However, the “explosion of complexity” problem re- stricts its applications. Hence, dynamic surface control when facing the time-varying working condition. To im- prove the tunning processes of PID, some self-tuning and technique is integrated into backstepping control to adaptive strategies have been proposed, such as automatic eliminate abovementioned drawbacks and achieve good recalibration features and the hybrid swarm intelligent dynamic tracking performance. To measure the full state optimization-PID algorithm [9]. variables of the electro-hydraulic system, Kim et al. Recently, many advanced control methods have been designed an output feedback nonlinear controller based developed for the hydraulic servo system, such as on backstepping to deal with the unknown external load adaptive control [10–12], backstepping control [13, 14], [25]. However, the aforementioned methods assume the fuzzy logic (FL) control [15–17], neural network (NN) external load disturbance as a known value for ensuring the derivative of Lyapunov function to be negative. To control [18, 19], and SMC [20–22]. Adaptive control can 2 Journal of Electrical and Computer Engineering address unknown nonlinear dynamics of the hydraulic y Hydraulic cylinder servo system, numerous FL- or NN-based control p1 p2 schemes have been constructed. Yang et al. proposed NN-based adaptive dynamic surface controller to im- V V prove the transient tracking performance of hydraulic 1 2 manipulator, where the NN is adopted to approximate p =p -p L 1 2 the unknown joint coupling dynamics [26–28]. Despite the conceptual simplicity and easy implementation of NN approach, the convergence rates of the NN weights W W can be very slow. Shen et al. developed a novel fuzzy Servo valve robust nonlinear controller for electro-hydraulic fight motion simulator, where FL compensator is introduced to estimate nonlinear uncertain functions caused by leakage and bulk modulus [29]. However, the complex Relief Motor Pump valve design of membership function and fuzzy rules limits its practical application. As a robust control scheme, SMC can guarantee the ro- bustness of the system with external disturbances, uncer- Oil Tank tainties, and highly nonlinear characteristics [30]. Nevertheless, the inevitable chattering may lead to the high-frequency ac- Figure 1: Schematic diagram of the electro-hydraulic servo system. tivities of the servo valve and then degrade the control per- formance. To restrain chattering and improve dynamic properties, Cheng et al. presented an observer-based sliding Te dynamics of the electro-hydraulic servo system can mode control method to tackle the uncertain nonlinearities, be represented as follows [15]: external disturbances, and immeasurable states of the electro- hydraulic servo system [31]. However, the velocity and the dy dy ⎧ ⎪ equivalentpressureofthehydraulicservosystemaredifcultto ⎪ Ap � m + B + Ky ⎪ L dt dt obtainonline.Tehighgainobservercanprovideaccuratetime ⎪ derivativeinformationforagivensignal,whichisimportantfor ⎪ ⎪ dy V p practical engineering. Won et al. proposed SMC based on ⎨ t L Q � A + C p + L tc L (1) HGO for position tracking of the electro-hydraulic servo dt 4β dt system, where the velocity and load pressure were estimated by ⎪ 􏽳������������� using the position feedback [32]. Motivated by above- p − sign x 􏼁 p ⎪ s v L mentioned discussions, a novel sliding mode controller based ⎩ Q � C wx , L d v on high gain observer (SMC-HGO) is proposed to deal with parameter uncertainties and external disturbances for the where A is the efective area of the cylinder, y is the stroke electro-hydraulic servo system. of the piston position, p is the return pressure, m is the Te rest of the paper is organized as follows: Section 2 mass of the piston, K is the load spring constant, B is the describes the electro-hydraulic servo system dynamic model. viscous damping coefcient, Q is the load fow, C is the L tc Subsequently, the controller design is presented in Section 3. total leakage coefcient, V is the actuator volume, β is t e Te simulation comparisons are presented in Section 4. the efective bulk modulus, C is the fow discharge Finally, Section 5 concludes the paper. coefcient, w is the area gradient of the servo valve, ρ is the fuid oil density, p is the supply pressure, and x is the 2. Problem Formulation spool position of the servo valve. Considering equation (1) and selecting displacement Te electro-hydraulic servo system is comprised of servo y, velocity y _, and acceleration y €as the state variables, i.e, valve, hydraulic cylinder, pump, motor, and relief valve. Te T T x � [x , x , x ] � [y, y, _ y €] , the dynamics of electro-hy- 1 2 3 pump driven by motor delivers the hydraulic oil from the oil draulic servo systems can be given as the following state tank to servo valve. Te relief valve reduces the inlet pressure space description: to a required value and automatically maintains the outlet pressure by returning a required additional amount of fow x _ � x ⎧ ⎪ 1 2 to the oil tank. Te hydraulic cylinder controlled by servo x _ � x (2) valve converts hydraulic energy into mechanical energy for 2 3 driving the load movement. Te position and torque meter x � α(x) + g x 􏼁 u + d, 3 v data are collected through the diferent installed sensors. Te where α(x) � f x + f x + f x , f � − 4β C K/(mV ), control signal generated by the controller actuates the servo 1 1 2 2 3 3 1 e tc t valve spool to the proper position. Te schematic diagram of f � − K/m − 4β (A + C B)/(mV ), f � − B/m − 4β C / 2 e tc t 3 e tc 􏽰������������ � √� V , g(x ) � 4Aβ C wx K K p − p sgn(x )/(mV ρ ), the electro-hydraulic servo system is shown in Figure 1. t v e d v sv a s L v t Journal of Electrical and Computer Engineering 3 x x 1d u Sliding mode Electro-hydraulic 1 controller (SMC) servo system x ,x ,x x 1 2 3 1 High gain observe (HGO) Figure 2: Block diagram of the presented SMC-HGO control scheme. -4 -8 -12 0 26 4 8 10 12 14 16 18 20 Time (s) Reference PID -4 -8 -12 04 2 68 10 12 14 16 18 20 Time (s) Reference SMC -4 -8 -12 0 28 4 6 10 12 14 16 18 20 Time (s) Reference SMC-HGO Figure 3: Comparative tracking performance of sinusoidal signal. d is the lumped uncertainties including nonlinear charac- ˙ ˙ ˙ t t t s _ � c e _ + c € e + e � c e _ + c € e + x − x 1 2 1 2 1 1d teristic and external disturbance. (4) � c e _ + c € e + α(x) + g x 􏼁 u − x . 1d 1 2 v 3. Controller Design Te three-orderHGO based onequation (2)is as follows: Te main objective of controller design is to design a robust ⎧ ⎪ 1 controller to track a desired trajectory as closely as possible. 􏽢 x _ � x − x 􏽢 − x (t) 1 2 1 1 ⎪ ε Te HGO with fnite time convergence is used to reconstruct velocity and acceleration signal, which will be used in the control design. k _ 􏽢 x � x − x − x (t)􏼁 (5) 2 3 1 1 ⎪ 2 Te sliding mode function is defned as s � c e + c e_ + e €, (3) 1 2 ⎪ k ⎪ 3 x _ � − x 􏽢 − x (t)􏼁 . 3 1 1 where c and c are positive constants, e � x − x . 1 2 1 1d Te derivative of sliding mode function is x (mm) x (mm) x (mm) 1 1 1 4 Journal of Electrical and Computer Engineering 0.8 0.4 0.0 -0.4 -0.8 0 2 4 6 810 12 14 16 18 20 Time (s) PID 0.6 0.3 0.0 -0.3 -0.6 0 246 81 10 2 14 16 18 20 Time (s) SMC 0.6 0.4 0.2 0.0 -0.2 -0.4 0 246 810 12 14 16 18 20 Time (s) SMC-HGO Figure 4: Comparative tracking errors of sinusoidal signal. Where k , k , and k are positive constants, ε <<1. It should traditional sign function sgn (x) in g(x ) is replaced by 1 2 3 be noted that the gains of HGO largely determines the hyperbolic tangent function tanh (kx), where the positive performance of the controller. Tus, it is important to constant k is much larger than zero. properly design the gains of HGO to avoid the control input Ten, the derivative of s is written as saturation. ˙ ˙ 2 3 t t We defne h � k /ε, h � k /ε , h � k /ε , and the dif- 1 1 2 2 3 3 s _ � c e _ + c € e + e � c e _ + c € e + x − x 1 2 1 2 3 1d ferentiator is presented as _ € � c e + c e + α(x) + bu − x 1 2 1d ⎧ ⎪ x 􏽥 � x 􏽥 − h x 􏽥 ⎪ 1 2 1 1 ˙ ˙ t t _ 􏽢 􏽢 x 􏽥 � x 􏽥 − h x 􏽥 (6) � c e _ + c € e + α(x) − c e _ − c € e − α(x 􏽢) − η􏽢 s + x − x ⎪ 2 3 2 1 1 2 1 2 1d 1d x 􏽥 � − h x 􏽥 . � η􏽢 s + v(x 􏽥) + α(x) − α(x 􏽢). 3 3 1 (8) Te SMC-HGO is designed as To analyze the stability, a Lyapunov candidate function is 1 ˙t 􏽢 􏽢 u(t) � 􏼒− c e_ − c e € − α(x 􏽢) − η􏽢 s + x 􏼓, (7) 1 2 1d defned as g x 􏼁 _ € V � s . (9) 􏽢 􏽢 􏽢 􏽢 􏽢 􏽢 where e � x − x , s � c e + c e + e. 1 1d 1 2 It is important to note that the sliding mode surface s is discontinuityfunction, because it contains sgn (·)function of Te derivative of Lyapunov candidate function is pre- g(x ). For eliminating the chattering phenomena, the sented as Error (mm) Error (mm) Error (mm) Journal of Electrical and Computer Engineering 5 0.8 0.4 0.0 -0.4 -0.8 0 2 4 6 810 12 14 16 18 20 Time (s) PID 0.8 0.4 0.0 -0.4 -0.8 0 2 4 6 8 101214161820 Time (s) SMC 0.8 0.4 0.0 -0.4 -0.8 0 2 4 6 810 12 14 16 18 20 Time (s) SMC-HGO Figure 5: Control inputs of sinusoidal signal. _ Theorem 1. If V ∈ R , the solution of inequality equation V � ss _ � − ηs􏽢 s + s(v(x 􏽥) + α(x) − α(x 􏽢)) V ≤ − σV + f, ∀t ≥ t ≥0 is � − ηs(s − 􏽥 s) + s(v(x 􏽥) + α(x) − α(x 􏽢)) − c t− t − c(t− τ) ( ) V(t) ≤ e V t 􏼁 + 􏽚 e f(τ)dτ, (11) � − ηs + s(η􏽥 s + v(x 􏽥) + α(x) − α(x 􏽢)) where c is a constant. (10) 2 2 2 � − ηs + sf(x 􏽥) ≤ − ηs + 􏼐sf(x 􏽥) 􏼑 Proof of theorem 1. Let Δ(t) � V(t) + cV(t) − f, that is 2 2 � − (η − 0.5)s + 0.5f(x 􏽥) V(t) � − cV(t) + f + Δ(t). (12) According to the frst order diferential equation, the � − η V + 0.5f(x) , solution of equation (12) can be obtained as follows: where η � 2η − 1, x 􏽥 � x − x 􏽢, f(x) � η􏽥 s + v(x 􏽥) + α(x)− α(x). t t − c t− t − c(t− τ) − c(t− τ) ( ) V(t) � e V t 􏼁 + 􏽚 e f(τ)dτ + 􏽚 e Δ(τ)dτ. (13) t t 0 0 − c t− t − c(t− τ) ( 0) V(t) � e V t + 􏽚 e f(τ)dτ. (14) Due to ∀t≥ t ≥0, Δ(t) ≤0, Control input (V) Control input (V) Control input (V) 6 Journal of Electrical and Computer Engineering -5 -10 -15 08 2 4 6 10 12 14 16 18 20 Time (s) Reference Estimation -4 -8 -12 0 2 4 68 10 12 14 16 18 20 Time (s) Reference Estimation -3 -6 -9 0 2468 10 12 14 16 18 20 Time (s) Reference Estimation Figure 6: State variables estimation results of sinusoidal signal. 2.4 1.8 1.2 0.6 0.0 -0.6 -1.2 -1.8 -2.4 0 2 4 6 8 10 12 14 16 18 20 Time (s) Figure 7: Comparison errors for diferent parameters of the proposed controller. x x x 3 2 1 Error (mm) Journal of Electrical and Computer Engineering 7 Table 1: Performance indices for sinusoidal signal. Indices (mm) μ σ E PID 0.4238 0.3098 0.3523 SMC 0.2042 0.1644 0.1844 SMC-HGO 0.1507 0.1069 0.1361 -4 -8 0 2 4 68 10 12 14 16 18 20 Time (s) Reference PID -4 -8 0 2 4 68 10 12 14 16 18 20 Time (s) Reference SMC -4 -8 0 2 4 68 10 12 14 16 18 20 Time (s) Reference SMC-HGO Figure 8: Comparative tracking performance of multifrequency sinusoidal signal. Because x is exponential convergence, we obtain − σ (t− t ) 0 0 ‖x(t)‖ ≤ φ ‖x 􏽥(t )‖e , then the equation (10) can be 0 0 rearranged as 2 − σ τ− t ( ) 0 0 (15) V(t) � − η V(t) + 0.5f(x 􏽥) ≤ − η V(t) + χ(•)e , 1 1 where χ(•) is K-class function. Combining equations (14) and (15), we have x (mm) x (mm) x (mm) 1 1 1 8 Journal of Electrical and Computer Engineering 1.2 0.6 0.0 -0.6 -1.2 0 2 4 6 810 12 14 16 18 20 Time (s) PID 1.0 0.5 0.0 -0.5 -1.0 01 246 810 2 14 16 18 20 Time (s) SMC 0.8 0.4 0.0 -0.4 -0.8 0 246 81 10 2 14 16 18 20 Time (s) SMC-HGO Figure 9: Comparative tracking errors of multifrequency sinusoidal signal. − η t− t − η (t− τ) − σ τ− t ( ) ( ) 1 0 1 0 0 V(t) ≤ e V t 􏼁 + χ(•) 􏽚 e e dτ − η t− t − η t+σ t η − σ τ ( ) ( ) 1 0 1 0 0 1 0 � e V t 􏼁 + χ(•)e 􏽚 e dτ (16) χ(•) − η t− t − η t+σ t η − σ t η − σ t 1( 0) 1 0 0 ( 1 0) ( 1 0) 0 � e V t + e e − e 􏼁 􏼒 􏼓 η − σ 􏼁 1 0 χ(•) − η (t− t ) − σ (t− t ) − η (t− t ) 1 0 0 0 1 0 � e V t + e − e . 􏼁 􏼒 􏼓 η − σ 􏼁 1 0 Tus, t⟶∞, V (t) �0, and V (t) exponentially converge 4. Simulation Results to zero. Te accuracy convergence accuracy is determined by To demonstrate the efectiveness of the proposed SMC-HGO η . control scheme, a classical PID controller and a SMC Figure 2 shows the block diagram of the presented SMC- controller are conducted to compare with it. Te controller HGO control scheme. Te proposed control scheme consists parameters are defned as follows: (1) PID: Te proportional of a HGO and a SMC. Te HGO is designed to estimate the gain k �1500, the integral gain k �300, and the diferential velocity and the acceleration by using the position feedback p i gain k �0.2. Especially, the gains of PID controller are signal. Te SMC is proposed to improve the position d selected by using the intelligent optimization algorithm for tracking performance of the electro-hydraulic servo obtaining the excellent dynamic performance. (2) SMC: system. □ Error (mm) Error (mm) Error (mm) Journal of Electrical and Computer Engineering 9 -4 -8 0 246 81 10 2 14 16 18 20 Time (s) PID -2 -4 0 2 4 6 810 12 14 16 18 20 Time (s) SMC -1 -2 0 2 4 6 810 12 14 16 18 20 Time (s) SMC-HGO Figure 10: Control inputs of multifrequency sinusoidal signal. c �5, c �5. (3) SMC-HGO:c �5, c �5, k �20, k �50, gain observer. Te aforementioned simulations reveal that 1 2 1 2 1 2 k �100, ε �0.01, η �1.5. Te parameters of the electro- the presented controller has better tracking performance hydraulic servo system and controller are m �200kg, than SMC and smaller chattering in control action, which 4 3 − 4 2 K �1.5 ×10 N/m, B �2 ×10 N·s/m, A �2 ×10 m , is very important for high performance control of the 3 − 11 3 − 4 3 ρ �800kg/m , C �2.5 ×10 m /(s·Pa), V �2 ×10 m , electro-hydraulic servo system. Since the HGO is very tc t β �61.5 ×10 bar, C �0.6, ε �0.01, η �0.5, c �5, c �3, critical for the presented controller, three groups of HGO e d 1 2 h �0.5, h �0.8, h �0.4. Te sampling time for the simu- gains are chosen as follows: K �[k1, k2, k3] �[20, 50, 1 2 3 2 1 3 1 lation test is chosen as 1ms. 100], K �0.1 K , and K �10 K . Comparison errors be- Te comparative tracking performance and errors for tween the reference and the response for diferent pa- sinusoidal signal y �10sin (0.25πt) are shown in Figures 3 rameters of the proposed controller are shown in Figure 7. and 4, respectively. As shown in Figure 3, three controllers It is clear that the tracking performance can be improved can make the actuator of hydraulic cylinder follow the by increasing the gains of HGO. However, too large HGO desired trajectory well. Specially, the maximum tracking gains will lead to a larger error. errors of PID controller and SMC controller are 0.7 mm In addition, three quantitative performance indices, i.e., and 0.5mm, respectively. In contrast, the tracking error of the average value of absolute error μ, the standard deviation SMC-HGO is only 0.4 mm, which validates the superiority of absolute error σ, and integral of time multiplied by error E of the proposed control scheme. Te comparison of are adopted to evaluate the abovementioned controllers. control inputs signal is shown in Figure 5. It can be noted Performance indices for sinusoidal signal are listed in Ta- that control input of SMC is changed drastically due to the ble 1. It shows that the SMC-HGO outperforms the other inherent chattering characteristic. Compared with SMC, two controllers in terms of all performance indices. the SMC-HGO has smooth control input signal, which is To further validate the performance of the presented due to its ability to compensate for lumped uncertainties. controller, multifrequency sinusoidal signal y �5sin Te state variables estimation results of the proposed (0.1πt)+3sin (0.45πt) is conducted. Comparative track- method are shown in Figure 6. It can be seen from Figure 6 ing performance and errors of three controllers are that the estimated states track the reference states quite shown in Figures 8 and 9, respectively. It is shown that well, which validates the estimation accuracy of the high the presented controller has the better tracking accuracy Control input (V) Control input (V) Control input (V) 10 Journal of Electrical and Computer Engineering -5 -10 -15 0 2468 10 12 14 16 18 20 Time (s) Reference Estimation -4 -8 -12 04 2 68 10 12 14 16 18 20 Time (s) Reference Estimation -3 -6 -9 0 2468 10 12 14 16 18 20 Time (s) Reference Estimation Figure 11: State variables estimation results of multifrequency sinusoidal signal. 2.0 1.6 1.2 0.8 0.4 0.0 -0.4 -0.8 -1.2 -1.6 -2.0 01 28 4 6 10 12 146 18 20 Time (s) 10 dB 20 dB 30 dB Figure 12: Comparative tracking errors of multifrequency sinusoidal signal under diferent SNR. compared with PID controller and SMC controller. Tis acceleration signal. Te comparative control inputs is because the high gain observer is introduced in con- signal of three controllers is displayed in Figure 10. It is troller design to estimate the unmeasurable velocity and clear that the presented control method is smoother than x x x 3 2 1 Error (mm) Journal of Electrical and Computer Engineering 11 Provincial Education Department (Grant no. 23A460014) Table 2: Performance indices for multifrequency sinusoidal signal. and the High Level Talent Foundation of Henan University Indices (mm) μ σ E of Technology (Grant no. 2020BS043). PID 0.0166 0.0142 0.0154 SMC 0.0097 0.0088 0.0095 References SMC-HGO 0.0080 0.0065 0.0078 [1] Q. Guo and Z. L. Chen, “Neural adaptive control of single-rod electrohydraulic system with lumped uncertainty,” Mechan- that of the other two controllers, which shows that the ical Systems and Signal Processing, vol.146, Article ID 106869, SMC-HGO provides superior control performance than PID and SMC schemes. State variables estimation results [2] G. Shen, Z. C. Zhu, X. Li, Q. G. Wang, G. Li, and Y. Tang, can be found in Figure 11. It can be found that the “Acceleration waveform replication on six-degree-of-freedom reference state variable is accurately estimated by the redundant electro-hydraulic shaking tables using an inverse high gain observer. To study the antiuncertainty of the model controller with a modelling error,” Transactions of the presented controller, the comparative tracking errors of Institute of Measurement and Control, vol. 40, no. 3, multifrequency sinusoidal signal under the diferent pp. 968–986, 2018. signal-noise ratio (SNR) are displayed in Figure 12. Te [3] S. Chen, Z. Chen, B. Yao et al., “Adaptive robust cascade force control of 1-DOF hydraulic exoskeleton for human perfor- tracking error is related to the SNR. Te larger the SNR is, mance augmentation,” IEEE-ASME Transactions on Mecha- the lower the tracking performance accuracy is. How- tronics, vol. 22, no. 2, pp. 589–600, 2017. ever, the compared tracking errors of the three SNR are [4] W. X. Deng, J. Y. Yao, Y. Y. Wang, X. W. 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Sliding Mode Control Based on High Gain Observer for Electro-Hydraulic Servo System

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10.1155/2023/7932117
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Hindawi Journal of Electrical and Computer Engineering Volume 2023, Article ID 7932117, 12 pages https://doi.org/10.1155/2023/7932117 Research Article Sliding Mode Control Based on High Gain Observer for Electro-Hydraulic Servo System Zhenshuai Wan , Yu Fu , Chong Liu , and Longwang Yue School of Mechanical and Electrical Engineering, Henan University of Technology, Zhengzhou 450001, China Correspondence should be addressed to Zhenshuai Wan; wanzhenshuai@haut.edu.cn Received 18 October 2022; Revised 23 December 2022; Accepted 26 December 2022; Published 3 January 2023 Academic Editor: Chao Zhai Copyright © 2023 Zhenshuai Wan et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Te electro-hydraulic servo system is widely used in industrial automation felds for its merits of the high force to weight ratio, compact size, and fast response. However, the parameter uncertainties and external disturbances of the electro-hydraulic servo system signifcantly deteriorate the control performance of conventional linear controller in practice. To deal with this problem, sliding mode controller (SMC) that incorporates high gain observer (HGO) is proposed in this paper. HGO is used to obtain the accurate time derivative of position signal for sliding mode controller design. Te stability of the control system is guaranteed by Lyapunov stability theory. Comparation simulation is conducted to validate the efectiveness of the presented control scheme. mitigate parametric variation by using its self-learning 1. Introduction properties. Yao et al. used adaptive control to handle the Te electro-hydraulic servo system is widely applied in parametric uncertainties and nonlinear friction of hy- draulic actuators, where a continuously diferentiable industrial automation felds, such as aircraft actuator [1], shaking table [2], and construction machine [3], owing to nonlinear friction model is frst established [23]. How- the superiorities such as fast dynamic response, small size, ever, the tedious adjustment design of control law and large force/torque output [4–7]. However, inherent complicates the controller application. Due to the nonlinear friction, parameter variation, and external dis- nonlinear dynamics of hydraulic cylinder and uncertain turbance restrict the high performance trajectory tracking in fuid parameters, backstepping control has been widely practical application [8]. Hence, how to cope with the in- used in the hydraulic system. Guo et al. proposed a fuence of these drawbacks has attracted the attentions of backstepping controller with extended-state-observer to academia and industry. Although the classical proportional- compensate the unknown load disturbance and uncer- integral-derivative (PID) controller is widely used in process tain nonlinearity of the electro-hydraulic system [24]. control, it cannot achieve satisfactory tracking performance However, the “explosion of complexity” problem re- stricts its applications. Hence, dynamic surface control when facing the time-varying working condition. To im- prove the tunning processes of PID, some self-tuning and technique is integrated into backstepping control to adaptive strategies have been proposed, such as automatic eliminate abovementioned drawbacks and achieve good recalibration features and the hybrid swarm intelligent dynamic tracking performance. To measure the full state optimization-PID algorithm [9]. variables of the electro-hydraulic system, Kim et al. Recently, many advanced control methods have been designed an output feedback nonlinear controller based developed for the hydraulic servo system, such as on backstepping to deal with the unknown external load adaptive control [10–12], backstepping control [13, 14], [25]. However, the aforementioned methods assume the fuzzy logic (FL) control [15–17], neural network (NN) external load disturbance as a known value for ensuring the derivative of Lyapunov function to be negative. To control [18, 19], and SMC [20–22]. Adaptive control can 2 Journal of Electrical and Computer Engineering address unknown nonlinear dynamics of the hydraulic y Hydraulic cylinder servo system, numerous FL- or NN-based control p1 p2 schemes have been constructed. Yang et al. proposed NN-based adaptive dynamic surface controller to im- V V prove the transient tracking performance of hydraulic 1 2 manipulator, where the NN is adopted to approximate p =p -p L 1 2 the unknown joint coupling dynamics [26–28]. Despite the conceptual simplicity and easy implementation of NN approach, the convergence rates of the NN weights W W can be very slow. Shen et al. developed a novel fuzzy Servo valve robust nonlinear controller for electro-hydraulic fight motion simulator, where FL compensator is introduced to estimate nonlinear uncertain functions caused by leakage and bulk modulus [29]. However, the complex Relief Motor Pump valve design of membership function and fuzzy rules limits its practical application. As a robust control scheme, SMC can guarantee the ro- bustness of the system with external disturbances, uncer- Oil Tank tainties, and highly nonlinear characteristics [30]. Nevertheless, the inevitable chattering may lead to the high-frequency ac- Figure 1: Schematic diagram of the electro-hydraulic servo system. tivities of the servo valve and then degrade the control per- formance. To restrain chattering and improve dynamic properties, Cheng et al. presented an observer-based sliding Te dynamics of the electro-hydraulic servo system can mode control method to tackle the uncertain nonlinearities, be represented as follows [15]: external disturbances, and immeasurable states of the electro- hydraulic servo system [31]. However, the velocity and the dy dy ⎧ ⎪ equivalentpressureofthehydraulicservosystemaredifcultto ⎪ Ap � m + B + Ky ⎪ L dt dt obtainonline.Tehighgainobservercanprovideaccuratetime ⎪ derivativeinformationforagivensignal,whichisimportantfor ⎪ ⎪ dy V p practical engineering. Won et al. proposed SMC based on ⎨ t L Q � A + C p + L tc L (1) HGO for position tracking of the electro-hydraulic servo dt 4β dt system, where the velocity and load pressure were estimated by ⎪ 􏽳������������� using the position feedback [32]. Motivated by above- p − sign x 􏼁 p ⎪ s v L mentioned discussions, a novel sliding mode controller based ⎩ Q � C wx , L d v on high gain observer (SMC-HGO) is proposed to deal with parameter uncertainties and external disturbances for the where A is the efective area of the cylinder, y is the stroke electro-hydraulic servo system. of the piston position, p is the return pressure, m is the Te rest of the paper is organized as follows: Section 2 mass of the piston, K is the load spring constant, B is the describes the electro-hydraulic servo system dynamic model. viscous damping coefcient, Q is the load fow, C is the L tc Subsequently, the controller design is presented in Section 3. total leakage coefcient, V is the actuator volume, β is t e Te simulation comparisons are presented in Section 4. the efective bulk modulus, C is the fow discharge Finally, Section 5 concludes the paper. coefcient, w is the area gradient of the servo valve, ρ is the fuid oil density, p is the supply pressure, and x is the 2. Problem Formulation spool position of the servo valve. Considering equation (1) and selecting displacement Te electro-hydraulic servo system is comprised of servo y, velocity y _, and acceleration y €as the state variables, i.e, valve, hydraulic cylinder, pump, motor, and relief valve. Te T T x � [x , x , x ] � [y, y, _ y €] , the dynamics of electro-hy- 1 2 3 pump driven by motor delivers the hydraulic oil from the oil draulic servo systems can be given as the following state tank to servo valve. Te relief valve reduces the inlet pressure space description: to a required value and automatically maintains the outlet pressure by returning a required additional amount of fow x _ � x ⎧ ⎪ 1 2 to the oil tank. Te hydraulic cylinder controlled by servo x _ � x (2) valve converts hydraulic energy into mechanical energy for 2 3 driving the load movement. Te position and torque meter x � α(x) + g x 􏼁 u + d, 3 v data are collected through the diferent installed sensors. Te where α(x) � f x + f x + f x , f � − 4β C K/(mV ), control signal generated by the controller actuates the servo 1 1 2 2 3 3 1 e tc t valve spool to the proper position. Te schematic diagram of f � − K/m − 4β (A + C B)/(mV ), f � − B/m − 4β C / 2 e tc t 3 e tc 􏽰������������ � √� V , g(x ) � 4Aβ C wx K K p − p sgn(x )/(mV ρ ), the electro-hydraulic servo system is shown in Figure 1. t v e d v sv a s L v t Journal of Electrical and Computer Engineering 3 x x 1d u Sliding mode Electro-hydraulic 1 controller (SMC) servo system x ,x ,x x 1 2 3 1 High gain observe (HGO) Figure 2: Block diagram of the presented SMC-HGO control scheme. -4 -8 -12 0 26 4 8 10 12 14 16 18 20 Time (s) Reference PID -4 -8 -12 04 2 68 10 12 14 16 18 20 Time (s) Reference SMC -4 -8 -12 0 28 4 6 10 12 14 16 18 20 Time (s) Reference SMC-HGO Figure 3: Comparative tracking performance of sinusoidal signal. d is the lumped uncertainties including nonlinear charac- ˙ ˙ ˙ t t t s _ � c e _ + c € e + e � c e _ + c € e + x − x 1 2 1 2 1 1d teristic and external disturbance. (4) � c e _ + c € e + α(x) + g x 􏼁 u − x . 1d 1 2 v 3. Controller Design Te three-orderHGO based onequation (2)is as follows: Te main objective of controller design is to design a robust ⎧ ⎪ 1 controller to track a desired trajectory as closely as possible. 􏽢 x _ � x − x 􏽢 − x (t) 1 2 1 1 ⎪ ε Te HGO with fnite time convergence is used to reconstruct velocity and acceleration signal, which will be used in the control design. k _ 􏽢 x � x − x − x (t)􏼁 (5) 2 3 1 1 ⎪ 2 Te sliding mode function is defned as s � c e + c e_ + e €, (3) 1 2 ⎪ k ⎪ 3 x _ � − x 􏽢 − x (t)􏼁 . 3 1 1 where c and c are positive constants, e � x − x . 1 2 1 1d Te derivative of sliding mode function is x (mm) x (mm) x (mm) 1 1 1 4 Journal of Electrical and Computer Engineering 0.8 0.4 0.0 -0.4 -0.8 0 2 4 6 810 12 14 16 18 20 Time (s) PID 0.6 0.3 0.0 -0.3 -0.6 0 246 81 10 2 14 16 18 20 Time (s) SMC 0.6 0.4 0.2 0.0 -0.2 -0.4 0 246 810 12 14 16 18 20 Time (s) SMC-HGO Figure 4: Comparative tracking errors of sinusoidal signal. Where k , k , and k are positive constants, ε <<1. It should traditional sign function sgn (x) in g(x ) is replaced by 1 2 3 be noted that the gains of HGO largely determines the hyperbolic tangent function tanh (kx), where the positive performance of the controller. Tus, it is important to constant k is much larger than zero. properly design the gains of HGO to avoid the control input Ten, the derivative of s is written as saturation. ˙ ˙ 2 3 t t We defne h � k /ε, h � k /ε , h � k /ε , and the dif- 1 1 2 2 3 3 s _ � c e _ + c € e + e � c e _ + c € e + x − x 1 2 1 2 3 1d ferentiator is presented as _ € � c e + c e + α(x) + bu − x 1 2 1d ⎧ ⎪ x 􏽥 � x 􏽥 − h x 􏽥 ⎪ 1 2 1 1 ˙ ˙ t t _ 􏽢 􏽢 x 􏽥 � x 􏽥 − h x 􏽥 (6) � c e _ + c € e + α(x) − c e _ − c € e − α(x 􏽢) − η􏽢 s + x − x ⎪ 2 3 2 1 1 2 1 2 1d 1d x 􏽥 � − h x 􏽥 . � η􏽢 s + v(x 􏽥) + α(x) − α(x 􏽢). 3 3 1 (8) Te SMC-HGO is designed as To analyze the stability, a Lyapunov candidate function is 1 ˙t 􏽢 􏽢 u(t) � 􏼒− c e_ − c e € − α(x 􏽢) − η􏽢 s + x 􏼓, (7) 1 2 1d defned as g x 􏼁 _ € V � s . (9) 􏽢 􏽢 􏽢 􏽢 􏽢 􏽢 where e � x − x , s � c e + c e + e. 1 1d 1 2 It is important to note that the sliding mode surface s is discontinuityfunction, because it contains sgn (·)function of Te derivative of Lyapunov candidate function is pre- g(x ). For eliminating the chattering phenomena, the sented as Error (mm) Error (mm) Error (mm) Journal of Electrical and Computer Engineering 5 0.8 0.4 0.0 -0.4 -0.8 0 2 4 6 810 12 14 16 18 20 Time (s) PID 0.8 0.4 0.0 -0.4 -0.8 0 2 4 6 8 101214161820 Time (s) SMC 0.8 0.4 0.0 -0.4 -0.8 0 2 4 6 810 12 14 16 18 20 Time (s) SMC-HGO Figure 5: Control inputs of sinusoidal signal. _ Theorem 1. If V ∈ R , the solution of inequality equation V � ss _ � − ηs􏽢 s + s(v(x 􏽥) + α(x) − α(x 􏽢)) V ≤ − σV + f, ∀t ≥ t ≥0 is � − ηs(s − 􏽥 s) + s(v(x 􏽥) + α(x) − α(x 􏽢)) − c t− t − c(t− τ) ( ) V(t) ≤ e V t 􏼁 + 􏽚 e f(τ)dτ, (11) � − ηs + s(η􏽥 s + v(x 􏽥) + α(x) − α(x 􏽢)) where c is a constant. (10) 2 2 2 � − ηs + sf(x 􏽥) ≤ − ηs + 􏼐sf(x 􏽥) 􏼑 Proof of theorem 1. Let Δ(t) � V(t) + cV(t) − f, that is 2 2 � − (η − 0.5)s + 0.5f(x 􏽥) V(t) � − cV(t) + f + Δ(t). (12) According to the frst order diferential equation, the � − η V + 0.5f(x) , solution of equation (12) can be obtained as follows: where η � 2η − 1, x 􏽥 � x − x 􏽢, f(x) � η􏽥 s + v(x 􏽥) + α(x)− α(x). t t − c t− t − c(t− τ) − c(t− τ) ( ) V(t) � e V t 􏼁 + 􏽚 e f(τ)dτ + 􏽚 e Δ(τ)dτ. (13) t t 0 0 − c t− t − c(t− τ) ( 0) V(t) � e V t + 􏽚 e f(τ)dτ. (14) Due to ∀t≥ t ≥0, Δ(t) ≤0, Control input (V) Control input (V) Control input (V) 6 Journal of Electrical and Computer Engineering -5 -10 -15 08 2 4 6 10 12 14 16 18 20 Time (s) Reference Estimation -4 -8 -12 0 2 4 68 10 12 14 16 18 20 Time (s) Reference Estimation -3 -6 -9 0 2468 10 12 14 16 18 20 Time (s) Reference Estimation Figure 6: State variables estimation results of sinusoidal signal. 2.4 1.8 1.2 0.6 0.0 -0.6 -1.2 -1.8 -2.4 0 2 4 6 8 10 12 14 16 18 20 Time (s) Figure 7: Comparison errors for diferent parameters of the proposed controller. x x x 3 2 1 Error (mm) Journal of Electrical and Computer Engineering 7 Table 1: Performance indices for sinusoidal signal. Indices (mm) μ σ E PID 0.4238 0.3098 0.3523 SMC 0.2042 0.1644 0.1844 SMC-HGO 0.1507 0.1069 0.1361 -4 -8 0 2 4 68 10 12 14 16 18 20 Time (s) Reference PID -4 -8 0 2 4 68 10 12 14 16 18 20 Time (s) Reference SMC -4 -8 0 2 4 68 10 12 14 16 18 20 Time (s) Reference SMC-HGO Figure 8: Comparative tracking performance of multifrequency sinusoidal signal. Because x is exponential convergence, we obtain − σ (t− t ) 0 0 ‖x(t)‖ ≤ φ ‖x 􏽥(t )‖e , then the equation (10) can be 0 0 rearranged as 2 − σ τ− t ( ) 0 0 (15) V(t) � − η V(t) + 0.5f(x 􏽥) ≤ − η V(t) + χ(•)e , 1 1 where χ(•) is K-class function. Combining equations (14) and (15), we have x (mm) x (mm) x (mm) 1 1 1 8 Journal of Electrical and Computer Engineering 1.2 0.6 0.0 -0.6 -1.2 0 2 4 6 810 12 14 16 18 20 Time (s) PID 1.0 0.5 0.0 -0.5 -1.0 01 246 810 2 14 16 18 20 Time (s) SMC 0.8 0.4 0.0 -0.4 -0.8 0 246 81 10 2 14 16 18 20 Time (s) SMC-HGO Figure 9: Comparative tracking errors of multifrequency sinusoidal signal. − η t− t − η (t− τ) − σ τ− t ( ) ( ) 1 0 1 0 0 V(t) ≤ e V t 􏼁 + χ(•) 􏽚 e e dτ − η t− t − η t+σ t η − σ τ ( ) ( ) 1 0 1 0 0 1 0 � e V t 􏼁 + χ(•)e 􏽚 e dτ (16) χ(•) − η t− t − η t+σ t η − σ t η − σ t 1( 0) 1 0 0 ( 1 0) ( 1 0) 0 � e V t + e e − e 􏼁 􏼒 􏼓 η − σ 􏼁 1 0 χ(•) − η (t− t ) − σ (t− t ) − η (t− t ) 1 0 0 0 1 0 � e V t + e − e . 􏼁 􏼒 􏼓 η − σ 􏼁 1 0 Tus, t⟶∞, V (t) �0, and V (t) exponentially converge 4. Simulation Results to zero. Te accuracy convergence accuracy is determined by To demonstrate the efectiveness of the proposed SMC-HGO η . control scheme, a classical PID controller and a SMC Figure 2 shows the block diagram of the presented SMC- controller are conducted to compare with it. Te controller HGO control scheme. Te proposed control scheme consists parameters are defned as follows: (1) PID: Te proportional of a HGO and a SMC. Te HGO is designed to estimate the gain k �1500, the integral gain k �300, and the diferential velocity and the acceleration by using the position feedback p i gain k �0.2. Especially, the gains of PID controller are signal. Te SMC is proposed to improve the position d selected by using the intelligent optimization algorithm for tracking performance of the electro-hydraulic servo obtaining the excellent dynamic performance. (2) SMC: system. □ Error (mm) Error (mm) Error (mm) Journal of Electrical and Computer Engineering 9 -4 -8 0 246 81 10 2 14 16 18 20 Time (s) PID -2 -4 0 2 4 6 810 12 14 16 18 20 Time (s) SMC -1 -2 0 2 4 6 810 12 14 16 18 20 Time (s) SMC-HGO Figure 10: Control inputs of multifrequency sinusoidal signal. c �5, c �5. (3) SMC-HGO:c �5, c �5, k �20, k �50, gain observer. Te aforementioned simulations reveal that 1 2 1 2 1 2 k �100, ε �0.01, η �1.5. Te parameters of the electro- the presented controller has better tracking performance hydraulic servo system and controller are m �200kg, than SMC and smaller chattering in control action, which 4 3 − 4 2 K �1.5 ×10 N/m, B �2 ×10 N·s/m, A �2 ×10 m , is very important for high performance control of the 3 − 11 3 − 4 3 ρ �800kg/m , C �2.5 ×10 m /(s·Pa), V �2 ×10 m , electro-hydraulic servo system. Since the HGO is very tc t β �61.5 ×10 bar, C �0.6, ε �0.01, η �0.5, c �5, c �3, critical for the presented controller, three groups of HGO e d 1 2 h �0.5, h �0.8, h �0.4. Te sampling time for the simu- gains are chosen as follows: K �[k1, k2, k3] �[20, 50, 1 2 3 2 1 3 1 lation test is chosen as 1ms. 100], K �0.1 K , and K �10 K . Comparison errors be- Te comparative tracking performance and errors for tween the reference and the response for diferent pa- sinusoidal signal y �10sin (0.25πt) are shown in Figures 3 rameters of the proposed controller are shown in Figure 7. and 4, respectively. As shown in Figure 3, three controllers It is clear that the tracking performance can be improved can make the actuator of hydraulic cylinder follow the by increasing the gains of HGO. However, too large HGO desired trajectory well. Specially, the maximum tracking gains will lead to a larger error. errors of PID controller and SMC controller are 0.7 mm In addition, three quantitative performance indices, i.e., and 0.5mm, respectively. In contrast, the tracking error of the average value of absolute error μ, the standard deviation SMC-HGO is only 0.4 mm, which validates the superiority of absolute error σ, and integral of time multiplied by error E of the proposed control scheme. Te comparison of are adopted to evaluate the abovementioned controllers. control inputs signal is shown in Figure 5. It can be noted Performance indices for sinusoidal signal are listed in Ta- that control input of SMC is changed drastically due to the ble 1. It shows that the SMC-HGO outperforms the other inherent chattering characteristic. Compared with SMC, two controllers in terms of all performance indices. the SMC-HGO has smooth control input signal, which is To further validate the performance of the presented due to its ability to compensate for lumped uncertainties. controller, multifrequency sinusoidal signal y �5sin Te state variables estimation results of the proposed (0.1πt)+3sin (0.45πt) is conducted. Comparative track- method are shown in Figure 6. It can be seen from Figure 6 ing performance and errors of three controllers are that the estimated states track the reference states quite shown in Figures 8 and 9, respectively. It is shown that well, which validates the estimation accuracy of the high the presented controller has the better tracking accuracy Control input (V) Control input (V) Control input (V) 10 Journal of Electrical and Computer Engineering -5 -10 -15 0 2468 10 12 14 16 18 20 Time (s) Reference Estimation -4 -8 -12 04 2 68 10 12 14 16 18 20 Time (s) Reference Estimation -3 -6 -9 0 2468 10 12 14 16 18 20 Time (s) Reference Estimation Figure 11: State variables estimation results of multifrequency sinusoidal signal. 2.0 1.6 1.2 0.8 0.4 0.0 -0.4 -0.8 -1.2 -1.6 -2.0 01 28 4 6 10 12 146 18 20 Time (s) 10 dB 20 dB 30 dB Figure 12: Comparative tracking errors of multifrequency sinusoidal signal under diferent SNR. compared with PID controller and SMC controller. Tis acceleration signal. Te comparative control inputs is because the high gain observer is introduced in con- signal of three controllers is displayed in Figure 10. It is troller design to estimate the unmeasurable velocity and clear that the presented control method is smoother than x x x 3 2 1 Error (mm) Journal of Electrical and Computer Engineering 11 Provincial Education Department (Grant no. 23A460014) Table 2: Performance indices for multifrequency sinusoidal signal. and the High Level Talent Foundation of Henan University Indices (mm) μ σ E of Technology (Grant no. 2020BS043). PID 0.0166 0.0142 0.0154 SMC 0.0097 0.0088 0.0095 References SMC-HGO 0.0080 0.0065 0.0078 [1] Q. Guo and Z. L. Chen, “Neural adaptive control of single-rod electrohydraulic system with lumped uncertainty,” Mechan- that of the other two controllers, which shows that the ical Systems and Signal Processing, vol.146, Article ID 106869, SMC-HGO provides superior control performance than PID and SMC schemes. State variables estimation results [2] G. Shen, Z. C. Zhu, X. Li, Q. G. Wang, G. Li, and Y. Tang, can be found in Figure 11. It can be found that the “Acceleration waveform replication on six-degree-of-freedom reference state variable is accurately estimated by the redundant electro-hydraulic shaking tables using an inverse high gain observer. To study the antiuncertainty of the model controller with a modelling error,” Transactions of the presented controller, the comparative tracking errors of Institute of Measurement and Control, vol. 40, no. 3, multifrequency sinusoidal signal under the diferent pp. 968–986, 2018. signal-noise ratio (SNR) are displayed in Figure 12. Te [3] S. Chen, Z. Chen, B. Yao et al., “Adaptive robust cascade force control of 1-DOF hydraulic exoskeleton for human perfor- tracking error is related to the SNR. Te larger the SNR is, mance augmentation,” IEEE-ASME Transactions on Mecha- the lower the tracking performance accuracy is. How- tronics, vol. 22, no. 2, pp. 589–600, 2017. ever, the compared tracking errors of the three SNR are [4] W. X. Deng, J. Y. Yao, Y. Y. Wang, X. W. Yang, and acceptable, which further verifes the powerful capability J. H. Chen, “Output feedback backstepping control of hy- of the presented controller in suppressing lumped draulic actuators with valve dynamics compensation,” Me- uncertainties. chanical Systems and Signal Processing, vol. 158, Te performance indices for multifrequency sinusoidal pp. 107769–107818, 2021. signal are summarized by Table 2. Especially, the σ of the [5] Y. Ge, J. Zhou, W. Deng, J. Yao, and L. Xie, “Neural network presented controller is 0.0065mm, while the corresponding robust control of a 3-DOF hydraulic manipulator with as- values of the PID controller and SMC controller are ymptotic tracking,” Asian Journal of Control, 2022. [6] S. B. Wang and J. Na, “Parameter estimation and adaptive 0.0142mm and 0.0088mm, respectively. 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Te modeling uncertainties and partial state feedback,” Journal of HGO is introduced to obtain accurate velocity and accel- the Franklin Institute, vol. 355, no. 13, pp. 5893–5911, 2018. eration signal. Te stability analysis carried by the Lyapunov [9] M. Van, X. P. Do, and M. Mavrovouniotis, “Self-tuning fuzzy PID-nonsingular fast terminal sliding mode control for robust method displays an exponential convergence performance. fault tolerant control of robot manipulators,” ISA Transac- Ten, the comparative simulations are illustrated both in tions, vol. 96, pp. 60–68, 2020. diferent reference signals and controller parameters. Te [10] X. Yang, W. Deng, L. Liu, and J. Yao, “A novel adaptive-gain results show the superior tracking performance of the SMC- disturbance estimator-based asymptotic adaptive tracking HGO with high tracking precision and chattering-free in the control for uncertain nonlinear systems,” Transactions of the tracking control of the electro-hydraulic servo system. 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