# Integral Nonsingular Terminal Sliding Mode Control of Hydraulic Servo Actuator Based on Extended State Observer

Authors: Zhenshuai Wan; Yu Fu
Journal: Shock and Vibration
 (2021)

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## Abstract

Hydraulic servo actuator always suffers from various disturbance and uncertainties, which makes it difficult to design a higher performance controller. In this paper, an integral nonsingular terminal sliding mode controller based on extended state observer (ESO-INTSM) is proposed to improve the robust performance of hydraulic servo actuator. The ESO is designed to estimate not only the parametric uncertainties but also the model disturbance. Based on the observed states of ESO, the proposed controllers could enable hydraulic servo actuator to track the desired motion trajectories. The stability of the synthesized controller is proved via Lyapunov analysis, which is very important for high-accuracy tracking control of hydraulic servo actuator. Simulation and experimental results demonstrate that the proposed control strategy can effectively attenuate the adverse influence caused by the uncertainties and apparently improve the tracking accuracy.


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## Body

## 1. Introduction

Hydraulic servo actuator plays a vital role in modern industry due to the advantages of high power to weight ratio [1], fast response [2], and high stiffness [3–5]. It is extensively employed in automotive active suspension [6], heavy engineering equipment [7], hardware-in-the-loop simulator [8], and so on. However, inherent parametric uncertainties and model disturbance reduce the tracking performance of hydraulic servo actuator [9–11]. The parametric uncertainties are mainly from unknown oil viscous damping [12], variable load stiffness [13], effective bulk modulus of fluid volumes [14], dead zone nonlinearity of valve [15], and oil temperature variations [16] while the model disturbance is mainly from uncertain external disturbance and unmodeled nonlinearities. Although traditional linear feedback control method based on the local linearization can obtain the modeling of hydraulic servo actuator, it does not consider the effect of nonlinear parameter behaviors and model disturbance [17].To deal with this issue, many advanced control methods have been proposed in recent decades, such as adaptive control [18], robust control [19], and sliding control [20]. These proposed control strategies not only addressed the control problem of hydraulic servo actuator with parametric uncertainties and model disturbance but also proved that they can achieve better performance in comparison to traditional linear control methods [21]. Yao and Fanping presented an adaptive robust control method to handle the motion control of single-rod hydraulic actuator with the parametric uncertainties and uncertain nonlinearities [22]. Tang et al. proposed an adaptive control strategy to deal with the force tracking control of electrohydraulic systems under external vibration disturbance, in which parametric uncertainties were handled by the proposed online adaptive updating law [23]. Zhao et al. designed a robust controller based on feed-forward inverse model to solve the problem of force control in electrohydraulic loading systems [24]. Guo et al. developed a backstepping controller with dynamic surface to compensate uncertain nonlinearities while guaranteeing the position tracking error within an acceptable level [25]. However, the above methods cannot ensure the position tracking performance when the disturbance exceeds certain limitation. Therefore, a method for estimating the parametric uncertainties and model disturbance using the adaptation algorithm is needed in order to improve the dynamic performance of hydraulic servo actuator.The extended state observer (ESO) not only has the capability of state observation but also can estimate the generalized disturbance between the plant and the model of the considered system in real-time [26–28]. Yao et al. employed ESO to simultaneously estimate the unmeasured states and unknown disturbance of the electrohydraulic system [29]. Zhou et al. constructed a generalized ESO to estimate the lumped disturbance of the direct-current motor servo system in real-time fashion [30]. Liu and Chen utilized ESO based on switched gain to estimate the influence of the hydraulic power and load disturbances [31]. Li et al. proposed a fixed-time ESO to estimate the external disturbances and parameter uncertainties of the quad rotor unmanned aerial vehicle [32]. However, due to the proof of convergence based on the strong prerequisite assumptions concerning system states, it can only be used when the appropriate parameters are selected for a specified control system. It suggests that strong prerequisite assumptions highly restrict the practical application of the ESO.As an effective tool to manage uncertainties and disturbance, sliding model controller suffers from the undesirable chattering phenomenon, which inevitably degrades the sliding accuracy [33, 34]. Ma et al. presented a fast terminal sliding mode (FTSM) tracking control for uncertain nonlinear systems with unknown parameters and system states combined with time-varying disturbances, which guarantees all tracking errors rapidly converge to the origin [35]. Omid Mofid et al. proposed a sliding mode disturbance observer method via adaptive scheme for synchronization of a class of fractional-order chaotic systems under time-varying disturbances [36]. Mobayen used global sliding mode control based on exponential reaching law to solve the tracking problems of underactuated systems with external disturbances, which improve the performance of the system by eliminating the reaching interval [37]. Wang et al. proposed adaptive integral terminal sliding mode (ITSM) controller to guarantee high-performance tracking control of cable-driven manipulators under complex lumped uncertainties, where a novel adaptive algorithm was designed to timely and appropriately update the gains for the controller manifold [38]. Shao et al. used adaptive recursive terminal sliding mode (ARTSM) controller to realize high-speed and high-precision control of linear motor suffering from payload uncertainties and external disturbances, where nonsingular terminal sliding function and recursive integral terminal sliding function are introduced in the controller design stage [39]. Su and Zheng integrated sliding surface and finite-time stability theory to obtain global finite-time tracking of uncertain robot manipulators [40]. However, the previously proposed controllers may lead to singularity and excessive amounts of tunable parameters, which restricts their practical applications. Motivated by the ESO and the INTSM, ESO-INTSM is proposed for high-precision control of hydraulic servo actuator. Contributions of this paper include the following. First, the ESO is introduced to the INTSM to eliminate the estimation error of uncertainties. Second, the adaptive switching control part is employed to estimate uncertainties, so the parametric uncertainties and model disturbance are no longer pre-required for controller design. Third, singularity problem is considered in the controller design procedure.The rest of this paper is arranged as follows. Section2 gives the mathematic model of hydraulic servo actuator. The proposed control scheme and the stability proving of the closed-loop control system are formulated in Section 3. Simulation and experimental comparative results are presented in Section 4. At last, Section 5 provides the resulting conclusion.


## 2. Mathematic Model of Hydraulic Servo Actuator

As shown in Figure1, the hydraulic servo actuator consists of motor, pump, servo valve, accumulator, symmetrical cylinder, and relief valve. The inertia load is driven by a servo valve controlled double-rod hydraulic cylinder. The goal of the hydraulic servo actuator is to have the inertia load to reproduce the desired position trajectory as closely as possible. Considering the factors of external disturbance and internal uncertain, the mathematical model of the hydraulic servo system is established by the servo valve flow equation, flow continuity equation of hydraulic cylinder, and force balance equation of inertia load.Figure 1
Schematic diagram of the hydraulic servo actuator.The force balance equation of the inertia load can be expressed as(1)ApL=my¨+By˙+Ky+F,where m and y represent the mass and displacement of the load, respectively; pL=p1−p2 is the load pressure of hydraulic cylinder; p1 and p2 are the pressures inside the two chambers of the cylinder; A is the effective area of the cylinder; B represents the damping and viscous friction forces on the load; and F represents the external disturbances.The load pressure dynamics can be written as(2)QL=Ay+CtcPL+Vt4βeP˙L,where Vt is the total control volume of chamber; βe is the effective bulk modulus; Ctc is the total leakage coefficient of the cylinder; and QL=Q1+Q2/2 represents the load flow rate, in which Q1 is the supplied flow rate to the forward chamber and Q2 is the return flow rate of chamber. QL is related to the displacement of the servo valve xv, which can be obtained by(3)QL=CdwxvPs−signxvPLρ,where Cd is the discharge coefficient; w is the area gradient of the servo valve; xv is the spool opening of the servo valve; Ps and Pr represent the supply and return pressure of cylinder, respectively; ρ is the density of hydraulic oil; and sign⋅ denotes the sign function.Since the cutoff frequency of the servo valve is far away greater than the control system bandwidth, the valve dynamics can be neglected in model construction. Hence, the relationship between the spool position and the input control current can be described as(4)xv=Ksvi,where Ksv is the gain of servo valve and i is the current of servo valve. The relationship of input voltage and output current of servo amplifier can be expressed as i = Kau. Ka is the gain of the servo amplifier, and u is the output of the servo amplifier.The state variables are defined asx=x1,x2,x3T=y,y˙,y¨. Then, the state-space representation of the hydraulic servo system is described as follows:(5)x˙1=x2,x˙2=x3,x˙3=fx+gxvu+d,where fx=f1x1+f2x2+f3x3, f1=−4βeCtcK/mVt, f2=−K/m−4βeA2+CtcB/mVt, f3=−B/m−4βeCtc/Vt, f4=4AβeCdwxvKsvKa/mVtρ, gxv=f4ps−pLsgnxv, and d=−F/m−4βeCtcF/VtIn general, the driving force of the hydraulic servo system stems from the pressure of hydraulic actuator, which is affected by parameter uncertaintiesd. Moreover, the system is also subjected to modeling uncertainties due to the variations of m, B, K, and Vt. Hence, the total disturbance is made up of parameter uncertainties and modeling uncertainties, which are defined as(6)Dx,u=Δfx+Δgxvu+d,where Δfx=Δf1x1+Δf2x2+Δf3x3 and Δgxv=Δf4ps−pLsgnxv.Then, the state-space equation can be rewritten as(7)x˙1=x2,x˙2=x3,x˙3=fx+gxvu+Dx,u,where Dx,u≤ζd and ζd is a positive constant.


## 3. Controller Design

Figure2 shows the control scheme configuration of ESO-INTSM. The ESO is constructed to estimate the parametric uncertainties and model disturbance, and then the estimation is used to compensate the dynamic model. The INSTM controller is designed to improve the tracking precision and response speed of hydraulic servo actuator.Figure 2
Block diagram of the proposed ESO-INTSM control scheme.


### 3.1. ESO Controller

The task of the observer design is not only observing the unmeasured system states but also estimating the uncertainties for controller compensation in real time.The ESO is given as(8)x^˙1=x^2+k1εe,x^˙2=x^3+k2ε2e,x^˙3=x^4+k3ε3e+fx^+gxvu,x^˙4=k4ε4e,where x^1,x^2,x^3,x^4T stands for the observing states, e˜=x^1−x1 stands for the estimated error, kii=1,2,3,4 represents positive constant and satisfies Hurwitz polynomial condition, and ε stands for a positive constant.According to the structure of ESO, it is only related to the input and output information, regardless of the specific form of the mode. Ast⟶∞, x^1⟶x1, x^2⟶x2, x^3⟶x3, and x^4⟶D are obtained. Then, the total disturbance including parametric uncertainties and model disturbance is estimated.The scaled estimation error is denoted by(9)γ1=x1−x^1ε3⋅γ2=x2−x^2ε2⋅γ3=x3−x^3ε⋅γ4=D−x^4.ESO belongs to high gain observer, which is not robust enough to model disturbance and parameter uncertainties. Furthermore, the existed estimation errors would always lead to the peak phenomenon when the disturbance is not exactly known. To attenuate the effect of the peak phenomenon,ε is designed as(10)1ε=R=μ1−e−λ1t1+e−λ2t,0≤t≤tmax,μ,t≥tmax,where μ, λ1, and λ2 are positive constants.Using the newly defined variableγ, the estimation error dynamics can be rewritten as(11)εγ˙1=x˙1−x^˙1ε2=1ε2x2−x^2+k1εe=−k1ε3e+1ε2x2−x^2=−k1γ1+γ2,εγ˙2=x˙2−x^˙2ε=1εx3−x^3+k2ε2e=−k2ε3e+1εx3−x^3=−k2γ1+γ3,εγ˙3=x˙3−x^˙3=fx+gxv+D−x^4+k3ε3e+fx^+gxvu=−k3γ1+γ4,εγ˙4=εD˙−x^˙4=εD˙−k4ε4e=−k4γ1+εD˙.The state equation of observation error can be expressed as(12)εγ˙=Aγ+εBD,where γ=γ1,γ2,γ3,γ4, A=−γ1100−γ2010−γ3001−γ4000, and B=0001.For any given symmetric positive definite matrixQ, there exists a symmetric positive definite matrix P satisfying the following Lyapunov equation:(13)ATP+PA+Q=0.The Lyapunov function of observation is defined as(14)Vo=εγTPγ.The time derivation ofVo can be given by(15)Vo=εγ˙TPγ+εγTPγ˙=Aγ+εBD˙TPγ+γTPAγ+εBD˙=γTATPγ+εBD˙TPγ+γTPAγ+εγTPBD˙=γTATP+PAγ+2εγTPBD˙≤−γTQγ+2εPB•γ•D˙≤−λminQγ2+2εδdPB•γ.The convergence condition of the observation can be deduced as(16)γ≤2εδdPBλminQ.From equation (16), it can be seen that the convergence rate of observe error γ is related to the parameter ε. In fact, the dynamic system is a fast time-varying system when parameter ε is small. The smaller ε, the faster convergence of state observe. It can be inferred that the designed ESO is stable, and the state error can be made arbitrarily small by reducing the parameter ε.


### 3.2. INTSM Controller

The control objective is to ensure that the load can track a desired time-varying trajectoryx1d in spite of parametric uncertainties and model disturbance. By defining the tracking error vectors and their derivatives, the error dynamics can be given as(17)e=x1−x1d,e˙=x˙1−x˙1d,e¨=x¨1−x¨1d,e…=fx+gxvu+Dx,u−x…1d.The INTSM manifold is selected as follows:(18)s=e+∫0tc1sgne¨e¨α1+c2sgne˙e˙α2+c3sgneeα3dt,where cii=1,2,3 are positive constants, α1∈1,2, and αi=αi−1/2−αi−1 for i=2,3.The surface is given as follows:(19)σ=s˙+λs=e˙+c1sgne¨e¨α1+c2sgne˙e˙α2+c3sgneeα3+λe+∫0tc1sgne¨e¨α1+c2sgne˙e˙α2+c3sgneeα3dt,where λ is a positive tuning constant.Combining the state-space equation and the sliding surface, the controller can be obtained as follows:(20)σ˙=s¨+λs˙=e¨+c1α1e…e¨α1−1+c2α2e¨e˙α2−1+c3α3e˙eα3−1+λe˙+c1sgne¨e¨α1+c2sgne˙e˙α2+c3sgneeα3=e¨+λe˙+c1sgne¨e¨α1+c2sgne˙e˙α2+c3sgneeα3+c2α2e¨e˙α2−1+c3α3e˙eα3−1+c1α1e…e¨α1−1.Rearranging controller (11) as(21)σ˙=e¨+λe˙+c1sgne¨e¨α1+c2sgne˙e˙α2+c3sgneeα3+c2α2e¨e˙α2−1+c3α3e˙eα3−1+c1α1e¨α1−1fx+gxvu+D−x…1d,and applying the σ˙=0, we have(22)u=−g−1xve¨1−α1c1α1ge¨+λe˙+c1sgne¨e¨α1+c2sgne˙e˙α2+c3sgneeα3+c2α2e¨e˙α2−1+c3α3e˙eα3−1fx−x…1d+ζd+ηsgnσ.The controller consists of an equivalent control partueq and a switching control part usw which is defined as follows:(23)ueq=−g−1xve¨1−α1c1α1ge¨+λe˙+c1sgne¨e¨α1+c2sgne˙e˙α2+c3sgneeα3+c2α2e¨e˙α2−1+c3α3e˙eα3−1fx−x…1d,usw=−g−1xvζd+ηsgnσ.The equivalent control part determines the dynamic performance of the system on the sliding surface since it carries the system state vector over the reference trajectory. The switching control part determines the capacity of resisting disturbance for the system subjected to parametric uncertainties and modeling uncertainties, which is important to the performance of the controller.To analyze the control stability, a Lyapunov candidate function is defined as(24)V=12σ2.Then, derivative of Lyapunov candidate function is(25)V˙=σσ˙=σc1α1e¨α1−1fx+gxv+D−x…1d+e¨+c2α2e¨e˙α2−1+c3α3e˙eα3−1λ+e˙+c1sgne¨e¨α1+c2sgne˙e˙α2+c3sgneeα3,V˙=σc1α1e¨α1−1dx−ζd+ηsgnσ≤−c1α1e¨α1−1ησ.Sincec1α1e¨α1−1>0, then V˙=≤−ησ. Based on the Lyapunov stability criteria, the proposed INTSM converges to zero as the time tends to infinity according to the established control law. It should be noted that the sign function used in switching part can cause a chattering phenomenon. In order to attenuate and eliminate the instability phenomenon, the discontinuous sign function is replaced with a continuous hyperbolic tangent function.


## 3.1. ESO Controller

The task of the observer design is not only observing the unmeasured system states but also estimating the uncertainties for controller compensation in real time.The ESO is given as(8)x^˙1=x^2+k1εe,x^˙2=x^3+k2ε2e,x^˙3=x^4+k3ε3e+fx^+gxvu,x^˙4=k4ε4e,where x^1,x^2,x^3,x^4T stands for the observing states, e˜=x^1−x1 stands for the estimated error, kii=1,2,3,4 represents positive constant and satisfies Hurwitz polynomial condition, and ε stands for a positive constant.According to the structure of ESO, it is only related to the input and output information, regardless of the specific form of the mode. Ast⟶∞, x^1⟶x1, x^2⟶x2, x^3⟶x3, and x^4⟶D are obtained. Then, the total disturbance including parametric uncertainties and model disturbance is estimated.The scaled estimation error is denoted by(9)γ1=x1−x^1ε3⋅γ2=x2−x^2ε2⋅γ3=x3−x^3ε⋅γ4=D−x^4.ESO belongs to high gain observer, which is not robust enough to model disturbance and parameter uncertainties. Furthermore, the existed estimation errors would always lead to the peak phenomenon when the disturbance is not exactly known. To attenuate the effect of the peak phenomenon,ε is designed as(10)1ε=R=μ1−e−λ1t1+e−λ2t,0≤t≤tmax,μ,t≥tmax,where μ, λ1, and λ2 are positive constants.Using the newly defined variableγ, the estimation error dynamics can be rewritten as(11)εγ˙1=x˙1−x^˙1ε2=1ε2x2−x^2+k1εe=−k1ε3e+1ε2x2−x^2=−k1γ1+γ2,εγ˙2=x˙2−x^˙2ε=1εx3−x^3+k2ε2e=−k2ε3e+1εx3−x^3=−k2γ1+γ3,εγ˙3=x˙3−x^˙3=fx+gxv+D−x^4+k3ε3e+fx^+gxvu=−k3γ1+γ4,εγ˙4=εD˙−x^˙4=εD˙−k4ε4e=−k4γ1+εD˙.The state equation of observation error can be expressed as(12)εγ˙=Aγ+εBD,where γ=γ1,γ2,γ3,γ4, A=−γ1100−γ2010−γ3001−γ4000, and B=0001.For any given symmetric positive definite matrixQ, there exists a symmetric positive definite matrix P satisfying the following Lyapunov equation:(13)ATP+PA+Q=0.The Lyapunov function of observation is defined as(14)Vo=εγTPγ.The time derivation ofVo can be given by(15)Vo=εγ˙TPγ+εγTPγ˙=Aγ+εBD˙TPγ+γTPAγ+εBD˙=γTATPγ+εBD˙TPγ+γTPAγ+εγTPBD˙=γTATP+PAγ+2εγTPBD˙≤−γTQγ+2εPB•γ•D˙≤−λminQγ2+2εδdPB•γ.The convergence condition of the observation can be deduced as(16)γ≤2εδdPBλminQ.From equation (16), it can be seen that the convergence rate of observe error γ is related to the parameter ε. In fact, the dynamic system is a fast time-varying system when parameter ε is small. The smaller ε, the faster convergence of state observe. It can be inferred that the designed ESO is stable, and the state error can be made arbitrarily small by reducing the parameter ε.


## 3.2. INTSM Controller

The control objective is to ensure that the load can track a desired time-varying trajectoryx1d in spite of parametric uncertainties and model disturbance. By defining the tracking error vectors and their derivatives, the error dynamics can be given as(17)e=x1−x1d,e˙=x˙1−x˙1d,e¨=x¨1−x¨1d,e…=fx+gxvu+Dx,u−x…1d.The INTSM manifold is selected as follows:(18)s=e+∫0tc1sgne¨e¨α1+c2sgne˙e˙α2+c3sgneeα3dt,where cii=1,2,3 are positive constants, α1∈1,2, and αi=αi−1/2−αi−1 for i=2,3.The surface is given as follows:(19)σ=s˙+λs=e˙+c1sgne¨e¨α1+c2sgne˙e˙α2+c3sgneeα3+λe+∫0tc1sgne¨e¨α1+c2sgne˙e˙α2+c3sgneeα3dt,where λ is a positive tuning constant.Combining the state-space equation and the sliding surface, the controller can be obtained as follows:(20)σ˙=s¨+λs˙=e¨+c1α1e…e¨α1−1+c2α2e¨e˙α2−1+c3α3e˙eα3−1+λe˙+c1sgne¨e¨α1+c2sgne˙e˙α2+c3sgneeα3=e¨+λe˙+c1sgne¨e¨α1+c2sgne˙e˙α2+c3sgneeα3+c2α2e¨e˙α2−1+c3α3e˙eα3−1+c1α1e…e¨α1−1.Rearranging controller (11) as(21)σ˙=e¨+λe˙+c1sgne¨e¨α1+c2sgne˙e˙α2+c3sgneeα3+c2α2e¨e˙α2−1+c3α3e˙eα3−1+c1α1e¨α1−1fx+gxvu+D−x…1d,and applying the σ˙=0, we have(22)u=−g−1xve¨1−α1c1α1ge¨+λe˙+c1sgne¨e¨α1+c2sgne˙e˙α2+c3sgneeα3+c2α2e¨e˙α2−1+c3α3e˙eα3−1fx−x…1d+ζd+ηsgnσ.The controller consists of an equivalent control partueq and a switching control part usw which is defined as follows:(23)ueq=−g−1xve¨1−α1c1α1ge¨+λe˙+c1sgne¨e¨α1+c2sgne˙e˙α2+c3sgneeα3+c2α2e¨e˙α2−1+c3α3e˙eα3−1fx−x…1d,usw=−g−1xvζd+ηsgnσ.The equivalent control part determines the dynamic performance of the system on the sliding surface since it carries the system state vector over the reference trajectory. The switching control part determines the capacity of resisting disturbance for the system subjected to parametric uncertainties and modeling uncertainties, which is important to the performance of the controller.To analyze the control stability, a Lyapunov candidate function is defined as(24)V=12σ2.Then, derivative of Lyapunov candidate function is(25)V˙=σσ˙=σc1α1e¨α1−1fx+gxv+D−x…1d+e¨+c2α2e¨e˙α2−1+c3α3e˙eα3−1λ+e˙+c1sgne¨e¨α1+c2sgne˙e˙α2+c3sgneeα3,V˙=σc1α1e¨α1−1dx−ζd+ηsgnσ≤−c1α1e¨α1−1ησ.Sincec1α1e¨α1−1>0, then V˙=≤−ησ. Based on the Lyapunov stability criteria, the proposed INTSM converges to zero as the time tends to infinity according to the established control law. It should be noted that the sign function used in switching part can cause a chattering phenomenon. In order to attenuate and eliminate the instability phenomenon, the discontinuous sign function is replaced with a continuous hyperbolic tangent function.


## 4. Simulation and Experiment Results

### 4.1. Simulation Results

To evaluate the tracking performance of the proposed controller, simulations have been carried out firstly on the mathematical model of the hydraulic servo actuator in Section2. The main parameters of the hydraulic servo system are listed in Table 1. For the proposed ESO-INTSM control strategy, the parameters of controller are set to c1=2, c2=4.5c3=1.6, α1=1.001, λ=2.5, ζd=10, k1=3, k2=5, k3=3, k4=1, andε=0.0001. The initial states of the hydraulic servo actuator (5) and the ESO (8) are set as zeroes.Table 1
The main parameters of hydraulic servo actuator.
ParameterValueParameterValueM120Ps7.3 × 106B1100Cd0.6K5000Ksv2 × 105Ctc2.5 × 10−14w5.4 × 10−3A5.2 × 10−4βe1.6 × 109Vt6.34 × 10−5ρ840In order to evaluate the quality of the different controllers, maximumMe, average μ, and standard deviation σe of the tracking errors are defined as follows:(26)Me=maxi=1,…Nei,μe=∑i=1NeiN,σe=∑i=1N|ei|−μe2N,where N is the number of the recorded digital signals.To validate the effectiveness of the proposed ESO-INTSM control strategy, TSM controller [33] and INTSM controller [41] are also conducted. It should be noted that parameters of these three controllers are well-tuned for the sake of best tracking performance. The tracking performance of random signal with the aforementioned controller is shown in Figure 3. It is obvious that displacement response with ESO-INTEM matches well with the demand signal compared with the situation INTSM without ESO. TSM provides the bad tracking performance in three controllers.Figure 3
Tracking performance of four controllers in simulation for the random signal.Figure4 shows the displacement tracking performance of the four controllers with square-wave demand signal y = square (±10) mm. It can be seen that the displacement of the hydraulic servo actuator based on ESO-INTSM controller matches the desired position trajectory well compared with the other three controllers. Although the tracking accuracy of four controllers has satisfying performance when the system is in steady state, the overshoot of PID (100%) and TSM (50%) controller in adjustment process is too large. The tracking error of the four controllers is shown in Figure 5. Specifically, for TSM and INTSM, the maximum errors are 12 mm and 4 mm, respectively. The tracking error of ESO-INTSM is relatively very small.Figure 4
Tracking performance of four controllers in simulation for the square signal.Figure 5
Tracking error of four controllers in simulation for the square signal.Figure6 shows the displacement tracking responses of four controllers for the reference trajectory y = 20sin (12 πt) mm, and it indicates that the proposed ESO-INTSM displays better tracking performance than the other three controllers. The tracking errors of these controllers are shown in Figure 7. It is clear that the tracking error under controller PID is relatively large, reaching to 4 mm in steady state. The reason for this phenomenon is that we choose the PID controller with higher gain for seeking shorter stability time and smaller steady state tracking error. The tracking error under ESO-INTSM controller is better than TSM and INTSM, and the maximum tracking error is about 2 mm. It illustrates that the proposed ESO-INTSM can improve the dynamic tracking performance of the hydraulic servo actuator. The corresponding performance indices are collected in Table 2. It can be seen from simulation results that the proposed ESO-INTSM controller achieves the best transient and steady state tracking performance since it employed adjustable observe-based compensation in which parametric uncertainties and model disturbance are handled by adjusting control signal.Figure 6
Tracking performance of four controllers in simulation for the sine signal.Figure 7
Tracking error of four controllers in simulation for the sine signal.Table 2
Performance indices of different controllers.
IndicesMe (mm)m (mm)σe (mm)PID2.50031.07970.8228TSM1.63550.76430.4520INTSM1.58610.68660.4332ESO-INTSM0.87900.24990.1859


### 4.2. Experimental Results

Figure8 shows the experimental setup of hydraulic servo actuator, the actuator stroke of ±100 mm, and the work frequency of servo valve of 0～50 Hz. The control algorithm is programmed by MATLAB/Simulink in the host computer, and then the compiled program is downloaded to the target computer for real-time execution. The hydraulic servo actuator displacement is measured by the displacement sensor, and the pressure is measured by the pressure sensor. The drive signal processed by data acquisition (DAQ) card drives the servo valve for generating the desired motion. The DAQ card has real-time processing capability that is to be used for real-time control and monitoring of the hydraulic servo system. The sampling time for the experiment setup is chosen as 0.001 s.Figure 8
Experimental setup of the hydraulic servo system.The initial values of the hydraulic parameters are consistent with the values used in the simulation. Considering the mechanical constraint of the cylinder displacement, the desired displacement signal is selected asy = square (±10) mm. Figures 9 and 10 show the displacement tracking performance of the proposed controller. It can be found that fairly smooth tracking performance is achieved because the modeling uncertainties and parameter uncertainties can be compensated via the introduced ESO. In particular, the PID controller cannot handle external disturbance well and a large tracking error over 2 mm is exhibited. Moreover, the displacement tracking error of the proposed controller is smaller than the INTSM. Though there are some changes in tracking error for ESO-INTSM controller, it does not change so much as the TSM and INTSM controller. The proposed ESO-INTSM can achieve better tracking performance because it applied an estimator to address the uncertainties in the system.Figure 9
Tracking performance of four controllers in experiment for the square signal.Figure 10
Tracking error of four controllers in experiment for the square signal.Figure11 shows the estimation performances of the displacement, velocity, acceleration, and lumped uncertainties for sine signal y = 20 sin (12 πt) mm. The estimated state variable tracks the actual state variables well in the displacement, the velocity, and the acceleration estimations. It is noted that ESO not only estimates the state variable but also provides an accurate approximate for parametric uncertainties and model disturbance.Figure 11
Estimation results of the proposed state observer: (a) estimation ofx1; (b) estimation of x2; (c) estimation of x3; (d) estimation of x4.
(a)(b)(c)(d)The displacement tracking performance of four controllers under the same condition is shown in Figure12, and the tracking errors are depicted in Figure 13. It is clear that ESO-INTSM exhibits better tracking performance than TSM and INTSM. Moreover, PID has the largest chattering in four controllers, and it is mainly due to the fact that the PID controller cannot eliminate the model disturbance and parameter uncertainties well. In contrast, the tracking error of ESO-INTSM during the stable stage is kept within 2 mm and that of TSM is within 3.8 mm. It can be seen that ESO-INTSM is able to deal with the external disturbance and has an improved performance in comparison to INTSM that is without ESO and TSM. The performance indices of different controllers during the stable state are collected in Table 3. It can be seen that the tracking errors of ESO-INTSM are signiﬁcantly decreased, compared with TSM and INTSM, especially for μ and σe.Figure 12
Tracking performance of four controllers in experiment for the sine signal.Figure 13
Tracking error of four controllers in experiment for the sine signal.Table 3
Performance indices of different controllers.
IndicesMe (mm)m (mm)σe (mm)PID1.94920.99400.5415TSM1.75230.87110.4899INTSM1.29200.64960.3283ESO-INTSM1.10980.42650.2135


## 4.1. Simulation Results

To evaluate the tracking performance of the proposed controller, simulations have been carried out firstly on the mathematical model of the hydraulic servo actuator in Section2. The main parameters of the hydraulic servo system are listed in Table 1. For the proposed ESO-INTSM control strategy, the parameters of controller are set to c1=2, c2=4.5c3=1.6, α1=1.001, λ=2.5, ζd=10, k1=3, k2=5, k3=3, k4=1, andε=0.0001. The initial states of the hydraulic servo actuator (5) and the ESO (8) are set as zeroes.Table 1
The main parameters of hydraulic servo actuator.
ParameterValueParameterValueM120Ps7.3 × 106B1100Cd0.6K5000Ksv2 × 105Ctc2.5 × 10−14w5.4 × 10−3A5.2 × 10−4βe1.6 × 109Vt6.34 × 10−5ρ840In order to evaluate the quality of the different controllers, maximumMe, average μ, and standard deviation σe of the tracking errors are defined as follows:(26)Me=maxi=1,…Nei,μe=∑i=1NeiN,σe=∑i=1N|ei|−μe2N,where N is the number of the recorded digital signals.To validate the effectiveness of the proposed ESO-INTSM control strategy, TSM controller [33] and INTSM controller [41] are also conducted. It should be noted that parameters of these three controllers are well-tuned for the sake of best tracking performance. The tracking performance of random signal with the aforementioned controller is shown in Figure 3. It is obvious that displacement response with ESO-INTEM matches well with the demand signal compared with the situation INTSM without ESO. TSM provides the bad tracking performance in three controllers.Figure 3
Tracking performance of four controllers in simulation for the random signal.Figure4 shows the displacement tracking performance of the four controllers with square-wave demand signal y = square (±10) mm. It can be seen that the displacement of the hydraulic servo actuator based on ESO-INTSM controller matches the desired position trajectory well compared with the other three controllers. Although the tracking accuracy of four controllers has satisfying performance when the system is in steady state, the overshoot of PID (100%) and TSM (50%) controller in adjustment process is too large. The tracking error of the four controllers is shown in Figure 5. Specifically, for TSM and INTSM, the maximum errors are 12 mm and 4 mm, respectively. The tracking error of ESO-INTSM is relatively very small.Figure 4
Tracking performance of four controllers in simulation for the square signal.Figure 5
Tracking error of four controllers in simulation for the square signal.Figure6 shows the displacement tracking responses of four controllers for the reference trajectory y = 20sin (12 πt) mm, and it indicates that the proposed ESO-INTSM displays better tracking performance than the other three controllers. The tracking errors of these controllers are shown in Figure 7. It is clear that the tracking error under controller PID is relatively large, reaching to 4 mm in steady state. The reason for this phenomenon is that we choose the PID controller with higher gain for seeking shorter stability time and smaller steady state tracking error. The tracking error under ESO-INTSM controller is better than TSM and INTSM, and the maximum tracking error is about 2 mm. It illustrates that the proposed ESO-INTSM can improve the dynamic tracking performance of the hydraulic servo actuator. The corresponding performance indices are collected in Table 2. It can be seen from simulation results that the proposed ESO-INTSM controller achieves the best transient and steady state tracking performance since it employed adjustable observe-based compensation in which parametric uncertainties and model disturbance are handled by adjusting control signal.Figure 6
Tracking performance of four controllers in simulation for the sine signal.Figure 7
Tracking error of four controllers in simulation for the sine signal.Table 2
Performance indices of different controllers.
IndicesMe (mm)m (mm)σe (mm)PID2.50031.07970.8228TSM1.63550.76430.4520INTSM1.58610.68660.4332ESO-INTSM0.87900.24990.1859


## 4.2. Experimental Results

Figure8 shows the experimental setup of hydraulic servo actuator, the actuator stroke of ±100 mm, and the work frequency of servo valve of 0～50 Hz. The control algorithm is programmed by MATLAB/Simulink in the host computer, and then the compiled program is downloaded to the target computer for real-time execution. The hydraulic servo actuator displacement is measured by the displacement sensor, and the pressure is measured by the pressure sensor. The drive signal processed by data acquisition (DAQ) card drives the servo valve for generating the desired motion. The DAQ card has real-time processing capability that is to be used for real-time control and monitoring of the hydraulic servo system. The sampling time for the experiment setup is chosen as 0.001 s.Figure 8
Experimental setup of the hydraulic servo system.The initial values of the hydraulic parameters are consistent with the values used in the simulation. Considering the mechanical constraint of the cylinder displacement, the desired displacement signal is selected asy = square (±10) mm. Figures 9 and 10 show the displacement tracking performance of the proposed controller. It can be found that fairly smooth tracking performance is achieved because the modeling uncertainties and parameter uncertainties can be compensated via the introduced ESO. In particular, the PID controller cannot handle external disturbance well and a large tracking error over 2 mm is exhibited. Moreover, the displacement tracking error of the proposed controller is smaller than the INTSM. Though there are some changes in tracking error for ESO-INTSM controller, it does not change so much as the TSM and INTSM controller. The proposed ESO-INTSM can achieve better tracking performance because it applied an estimator to address the uncertainties in the system.Figure 9
Tracking performance of four controllers in experiment for the square signal.Figure 10
Tracking error of four controllers in experiment for the square signal.Figure11 shows the estimation performances of the displacement, velocity, acceleration, and lumped uncertainties for sine signal y = 20 sin (12 πt) mm. The estimated state variable tracks the actual state variables well in the displacement, the velocity, and the acceleration estimations. It is noted that ESO not only estimates the state variable but also provides an accurate approximate for parametric uncertainties and model disturbance.Figure 11
Estimation results of the proposed state observer: (a) estimation ofx1; (b) estimation of x2; (c) estimation of x3; (d) estimation of x4.
(a)(b)(c)(d)The displacement tracking performance of four controllers under the same condition is shown in Figure12, and the tracking errors are depicted in Figure 13. It is clear that ESO-INTSM exhibits better tracking performance than TSM and INTSM. Moreover, PID has the largest chattering in four controllers, and it is mainly due to the fact that the PID controller cannot eliminate the model disturbance and parameter uncertainties well. In contrast, the tracking error of ESO-INTSM during the stable stage is kept within 2 mm and that of TSM is within 3.8 mm. It can be seen that ESO-INTSM is able to deal with the external disturbance and has an improved performance in comparison to INTSM that is without ESO and TSM. The performance indices of different controllers during the stable state are collected in Table 3. It can be seen that the tracking errors of ESO-INTSM are signiﬁcantly decreased, compared with TSM and INTSM, especially for μ and σe.Figure 12
Tracking performance of four controllers in experiment for the sine signal.Figure 13
Tracking error of four controllers in experiment for the sine signal.Table 3
Performance indices of different controllers.
IndicesMe (mm)m (mm)σe (mm)PID1.94920.99400.5415TSM1.75230.87110.4899INTSM1.29200.64960.3283ESO-INTSM1.10980.42650.2135


## 5. Conclusions

In this paper, the INTSM controller synthesized ESO has been proposed for the high-precision control of hydraulic servo actuator considering model disturbance and parametric uncertainties. The dynamic model of the hydraulic servo actuator is constructed, and the corresponding state space is established. Based on the dynamic model, the ESO is constructed to estimate the full state and unknow parameters, which can effectively overcome the influence caused by the parameter uncertainties and model disturbance. The closed-loop stability of the proposed controller and observer is analyzed by the Lyapunov method. In addition, the sliding surface with additional integral part is designed to modify the INTSM controller for solving the chattering problem and improving the robustness of the system. In addition, the proposed controller is not just undisturbed by the singularity phenomena as terminal sliding mode methods but performs well in the presence of external disturbances and uncertainties. Simulation and experimental results together show that the proposed control scheme guarantees excellent tracking accuracy in the presence of time-varying uncertainties. Generally speaking, a faster convergence rate physically requests larger control input. Thus, actuator saturation should be derived into account. Moreover, the high-precision motion control of shaking table driving by hydraulic servo actuator will be investigated in our future work.



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*Source: 5532076-2021-09-03.xml*