Integral Sliding Mode Backstepping Control of an Asymmetric Electro-Hydrostatic Actuator Based on Extended State Observer

To provide high output force and to reduce the installation space, the electro-hydrostatic actuator (EHA) usually adopts asymmetric cylinder. However, comprehensive effects pr oduced by its asymmetric flow, parameter uncertainties and unknown disturbance make it difficult to achieve high-accuracy position control. This paper proposed an integral sliding mode backstepping control (ISMBC) based on extended state observer for the asymmetric EHA. Firstly, the principle of the EHA was analyzed and an EHA model was built. Further, the state space equation of the EHA was established based on flow distribution analysis. Two extended state observers (ESO) were designed to achieve real-time estimation of the unmeasured system states, unmatched and matched disturbances. The backstepping method was used to compensate the matched and unmatched disturbance, and an integrated sliding mode controller was developed to eliminate the static error and to improve the response ability. Theoretical analysis indicates that the controller can guarantee the desired tracking performance for the actuator under time-varying unmatched disturbances, and can make the tracking error asymptotically converge to zero under constant matched disturbances. Finally, simulations were performed with the designed controller, backstepping controller, and proportional-integral-derivative (PID) controller respectively. Thereafter, detailed comparisons of the control performances were provided. The results show that the proposed controller can achieve better position tracking and stronger robustness in parameter changing compared with the backstepping controller and PID controller.


Introduction
Electro-hydrostatic actuators (EHAs) are widely used in aviation, shipbuilding, automobile and other industrial fields due to their small size, light weight, high efficiency and great reliability [1][2][3]. The EHA is a highly integrated direct driven hydraulic system that integrates an electric motor, a pump, an actuator, a tank, etc [1,4]. It achieves variable power transmission of actuators by changing the rotation speed or the displacement of the pump [5]. Compared with the traditional valvecontrolled system, the EHA eliminates the throttling loss caused by the multi-way valve and the overflow loss caused by the centralized oil supplies, which significantly improves the system efficiency [6,7].
Over the past 20 years, EHAs have been applied in high-precision industries such as aviations and submarines [8,9], but only adopting actuators with symmetrical structures. However, industrial applications usually require asymmetric hydraulic actuators, having the advanteges of smaller volume and larger output force [10]. The unbalanced flow in asymmetric EHAs, caused by the unequal effective cross-section areas in two chambers has seriously affects the control accuracy and dynamic response [11,12].
To solve the this problem, many novel methods were proposed such as the development of asymmetric flow distribution pumps [13], the research of pump-valve-coordinated system [14], and the use of dual-pumps control system [15]. On the other hand, advanced control algorithms such as robust adaptive control [16][17][18], backstepping control and sliding mode control, etc.
Besides, EHA has the characteristics of nonlinearities, parametric uncertainties and external disturbances [19][20][21]. These nonlinearities include fluid compressibility, nonlinear friction, internal and external leakage [22]. Parametric uncertainties are mainly caused by model inaccuracy and system parameters variation. The external disturbances mainly consist of the variation of external load force and unmodeled load force. The nonlinear friction reduces the response speed by affecting the transient characteristics of the EHA and leads to viscous and crawling phenomena in low -speed operation. Leakage in the EHA decreases steady-state accuracy. Parametric uncertainties normally require a high gain to improve the robustness of the system, which easily lead to over-design. External disturbances reduce system stability by influencing system output [23].
To solve these problems of nonlinearities, uncertainties and external disturbances, a lot of researches have been conducted. Lin [24] regarded the nonlinear friction force as norm-bounded uncertainties, developed a robust discrete-time sliding-mode control (DT-SMC) for an EHA system. Fu [25] applied neural networks to identify uncertainties online, combined RBF neural networks with fast terminal sliding mode controller , which not only solved the problem of sliding mode control depending on system parameters but also suppressed oscillation to some extent. Alemu [26] applied the Extended State Observer (ESO) to estimate the system states, uncertainties and external disturbances, used the friction model to compensate the friction force, and designed a sliding mode controller for the system, which improved the robustness while ensuring the tracking performance. Sun [27] developed a nonlinear robust motion controller based on the extended disturbance observer to compensate the estimation error of the outer position tracking loop, while the inner pressure control loop adopted a backstepping method to achieve accurate force control. Wang [28] introduced a feedback backstepping control algorithm based on the backstepping control theory for the highorder model of the EHA system to convert the complex nonlinear system into a linear system. Yang [29] introduced a filtered error function, integrated a novel expected compensation adaptive control framework into the controller to reduces environmental noise. Shen [30] decomposed the 5th-order EHA dynamic model into four subsystems, and designed adaptive control laws respectively to solve the controller design problem of the high-order system. Yang [31] designed a linear state observer and a nonlinear disturbance observer to estimate the matched and unmatched disturbances in the system, and employed a continuously differentiable friction model to compensate the friction force.
The above research shows that combining the observer with advanced control theory is an effective method to solve the problems of nonlinearities, uncertainties and disturbances in hydraulic systems. Most scholars focus on improving the control performance of symmetric EHAs and valvecontrolled systems. Therefore, this paper proposed a novel control strategy to solve the uncertainty problem of an asymmetric EHA. The electric motor speed control system was regarded as a separate module, and PI controller was adopted. The state equation of the asymmetric EHA system was established considering nonlinear friction, parameter uncertainty and external disturbance. The Stribeck static friction force model was used to identify the friction force; the unmodeled friction force was regarded as an external disturbance. The state equation was used to judge whether the disturbance and the control law were on the same channel. The disturbances were divided into matched disturbance and unmatched disturbance, two ESOs were established for estimation. The integral sliding mode algorithm was added in the first step of the backstepping design to reduce the steady-state error and to improve the robustness of the EHA. Based on the Lyapunov theory, t he stability and effectiveness of this control method were proven. The simulation results show that the controller has good steady-state characteristics and high control accuracy.
The remainder of this paper is organized as follows. Section 2 presents the principle analysis and model building of an asymmetric EHA. In Section 3, two ESOs are designed to deal with the disturbances and its convergence is verified. In Section 4, the integral sliding mode backstepping control (ISMBC) controller is proposed and its stability is proven. Section 5 gives comparative simulation results. Finally, the conclusion is given in Section 6.

Load force analysis of a micro crane
In this case study, the research object is the micro crane, as shown in Figure 1. The dimensions of the micro-crane was measured and its 3D model was created in Solidworks. After that, the dynamic model of the crane was built by exporting a CAD assembly from Solidworks and importing into Matlab/Simulink. The output force of the EHA system installed on the crane mainly depends on the torque, angular acceleration According to Newton's second law, the torque balance equation of the boom can be written as: where 2 2 ⁄ is the angular acceleration; J is the rotational inertia of boom. Decomposed equation (1) to get the output force equation of the hydraulic cylinder [15] where Cyl is the output force of cylinder; is the mass of boom; is the distance between the centre of mass and the joint; is the angle between the centre of mass and the r eference coordinate vertical axis; is gravitational acceleration; the load is connected to the crane by a hook and chain, and therefore, the load force is always perpendicular to the ground, such that can be defined as load mass including the mass of load, hook and the chain; is the distance from the load acting on the arm to the joint; is the angle between the connection about the joint and load with the reference coordinate vertical axis; 1 is the distance between the cylinder base and the joint; is the angle between the cylinder and the joint.
Next, the working principle of asymmetric EHA was analyzed, and the state-space equation was established for the EHA.

Principle analysis of EHA
The schematic of the EHA control system is shown in Figure 2. The system includes the hydraulic system, the electric motor and the controller. The variable speed electric motor is controlled by PI controller, which drives a bidirectional fixed pump. Two pilot-operated check valves are used to balance the flow of the asymmetric cylinder. Two relief valves are used for safety purpose. The EHA, installed on the micro-crane as shown in Figure 1, runs in two operating conditions including extending and retracting with positive load, as shown in Figure 3. Therefore, the load pressure is always greater than zero, it was given by (3) where 1 and 2 are the pressure of the piston chamber and the rod chamber of the, respectively; is the area ratio of cylinder, = 2 1 ⁄ ; 1 and 2 are the areas of the piston chamber and the rod chamber of the cylinder, m 2  the EHA assistive re traction.
Due to the different effective areas of the asymmetric cylinder, EHA requires unequal oil flow rate during moving. To prevent cavitation, hydraulic oil is replenished from the oil tank through the pilot-operated check valve 4.1, as shown in Figure 3 (a). During retracting, the excess oil flow from the piston chamber backs to the oil tank through the pilot-operated check valve 4.1, as shown in Figure 3 (b).

Model of the electric motor
Permanent magnet synchronous motors (PMSMs) are widely used in industrial equipment because of high power density, high efficiency and high reliability. Hence, in this paper, a PMSM is selected to drive the EHA. The electromagnetic torque equation of PMSM in d-q reference frame can be expressed as where E is the electromagnetic torque; n is the number of pole pairs; m is the rotor magnet flux linkage; sq is the q-axis stator currents.
Assuming the PMSM rotates at a constant speed, the torque balance equation between the PMSM and the pump can be written as where is the moment of inertia; is the viscous friction coefficient; L is the load torque from the pump; is the angular speed of the rotor. In the Laplace transformation of equation (5), ignoring the effects of the pump torque, the transfer function of the motor can be given as where is the electric motor gain, = 1.5 n ψ m / ; is the time constant, = / . The PMSM mainly adopts sd = 0 vector control; the outer loop speed control provides a reference signal for the inner current loop. This paper focuses on designing high-precision EHA system controller, so the proportional-integral (PI) was used as a speed loop controller for the electric motor.

Model of the hydraulic system
The output flow of the pump can expressed by where P is the output flow of pump, m 3 /s; P is the pump displacement, m 3 /r; P is the pump angular speed, P = m , rad/s; i is the internal leakage coefficient, (m/s)/Pa; ∆ is the differential pressure, ∆ = 1 − 2 .
Due to the pressure 2 close to zero, for simplification, assuming ∆ ≈ L , equation (7) can be rewritten as The flow-pressure equation of cylinder can be expressed as where 1 and 2 are flowrates of the piston chamber and the rod chamber of the cylinder, m 3 /s; e is the effective bulk modulus, Pa; 1 and 2 are the current volumes of the cylinder, m 3 , 1 = 01 + 1 , 2 = 02 − 2 ( − P ) ; 01 and 02 are the initial volumes of two-chambers of the cylinder; is the piston stroke, m.
Assuming that the piston is moving around the centre position, the following approximation can be given [32]: where t defines the total volume of the hydraulic cylinder, m 3 . Combine equation (7) to (10), the following equation can be obtained: where = 1 + 2 .
The force balance equation of the hydraulic cylinder can be written as: where is the total load mass of the crane; L is the load force; t represents unmodeled friction, load force and external disturbance; (̇P) denotes the Stribeck friction force and its model can be given by: where C and brk are the coulomb friction and breakaway friction; v is the speed coefficient; is the speed of the piston, =̇P; is the viscous friction coefficient; th is the critical speed. Combine equation (11) to (12), defining state variables = [ 1 , 2 , 3 ] = [ P ,̇P, L ] , the statespace equation of the EHA can be written as: 1st International Electronic Conference on Actuator Technology: Materials, Devices and Applications (IeCAT 2020) 6 For simplification, the parameters set can be denoted as 1 Since the external load force and external disturbance cannot be directly measured, two ESOs will be designed to estimate them later. To facilitate the design of the observer, the load force and external disturbance are combined into one item, 1 ( ) = − 2 L − 2 t . Due to the wear, the change of temperature and pressure in the hydraulic system, parameters e , , i , t , etc, become uncertain, which will cause internal disturbance, 2 ( ) = ∆ ω − ∆ 3 2 − ∆ 4 3 . Therefore, the equation (14) can be rewritten as: Usually, 1 ( ) is regarded as the unmatched disturbance while 2 ( ) is considered as the matched disturbance. Because 2 ( ) and the control law ω are in the same channel, but 1 ( ) is in another channel and it cannot be eliminated directly by the control law .
Next, the integral sliding mode backstepping controller will be designed to compensate for the matched disturbances and unmatched disturbances, to guarantee the cylinder actuator following smooth trajectory d = 1d .
The following assumptions are necessary for the controller design.

Design of ESOs
The traditional state observer can only be used to observe unknown state variables in the system, such as position P , velocity ̇P and load pressure L . However, the unmatched disturbance and the matched disturbance cannot be effectively estimated. In this paper, the system model (15) of the EHA was divided into a position-velocity subsystem and a pressure subsystem. The disturbances 1 ( ) and 2 ( ) are extended, respectively. Two ESOs were designed to estimate the unmatched disturbance and the matched disturbance in real-time, respectively. The position-velocity subsystem is expressed as: where 1 is the output of the position-velocity subsystem, e1 = 1 ( ) , ̇e 1 =̇1 ( ) = 1 . The pressure subsystem can be expressed as: where 1 is the output of the pressure subsystem, e2 = 2 ( ) ,̇e 2 =̇2 ( ) = 2 . Two ESOs for two subsystems are given as: 1st International Electronic Conference on Actuator Technology: Materials, Devices and Applications (IeCAT 2020) . Further, the estimation error is defined as * = * − * and represents as: As the matrix ε and ϵ is Hurwitz, two positive definite matrixs ε and ϵ hold the following matrix equality:

Lyapunov analysis of ESOs
Based on Assumption 1 and Assumption 2, The ESO1 was designed to observe the unmatched disturbance and the velocity of the position-velocity subsystem, and the ESO2 was used to observe the matched disturbance, respectively. Next, in view of the Lyapunov method, the stability of the designed ESOs were analyzed. The analysis method was divided into two parts, corresponding to two ESOs.

Design of the controller
In this paper, the backstepping design was applied to compensate for the disturbances of matched items and unmatched items in the EHA. In order to further reduce the tracking error and suppress the oscillation of the EHA, an integral sliding mode control algorithm is introduced into the position control term. According to the system (15), it can be known that the system feedback output is the state 1 , and the tracking trajectory is defined as d = 1d . Hence, the position tracking error 1 of the EHA can be represented as (30) Using the system (15), the derivative equation of the tracking error 1 can be expressed as: Here, the sliding mode surface is designed to ensure position tracking accuracy; An integral item is introduced to suppress the switching oscillation.
where is the integral item, = ∫ 1 0 ; 0 the integral gain that is a positive constant. The derivative of the sliding surface can be defined as: is the switching gain. For the first equation of the system (15), in which the input is the state 2 . Due to 2 cannot be obtained directly, a virtual control law 1 is designed for 2 .The error function 2 is defined as: Using equation (33), the virtual control law 1 can be designed as: (38) According to equation (35), the error 2 is unknown, since the state variable 2 cannot be measured directly. Therefore, its estimated value ̂2 is introduced from the ESO1 (18); the virtual error 2 can be split into two parts, including the computable part 2c and the non-computable part Usually the computable part 2c is used in the controller design. Based on equation (15) and (35), the derivative of the virtual error 2 can be written as: In this step, the state variable 3 is used as the virtual control input. Then, a virtual control law 2 is designed for it to improve tracking performance and to afford feed forward compensation for unmatched disturbances. Define virtual control input error 3 as: (41) The virtual control law 2 can be designed as: Combining equation (40) with (42), the dynamic of the virtual error 2 can be rewritten as: For the third equation of system (15), in which the input is ω ; ω is also the control input of the EHA. According to the definition of 2 and 3 , the virtual control error 3 can be divided into a computable part 3c and a non-computable part 3 : (44) Based on equation (15) and (44), the derivative of the virtual error 3 cann be written as: where ̇2 is defined as the dynamic of the virtual control law 2 , which can be calculated by: where ̇f(̂2) is the dynamic of the estimate of friction, which can be obtained by a filter + , and is the filter gain. ̂̇1 is the dynamic of the estimate of unmatched disturbance, which can be obtained in the same way.
In view of the ̇3 approaching 0, the resulting control law ω is designed as Substituting the control law (47) into equation (46), it follows that

Stability analysis of the controller
To prove the stability of the proposed ISMBC controller, the Lyapunov function of the controller is defined as Consider the Lyapunov function of the total system as = 1 + 2 + 3 , (50) From the equation (27), one obtains where Ψ 1 = .
According to inequality (56) and omitting integral term and sliding mode term | | , greater controller gain is required to achieve the control system stability. Therefore, it can be concluded that the introduction of integral sliding mode control into the backstepping design can achieve higher stability and better robustness.

Simulation model
To verify the control performance of the proposed controller, a multi-domain model was established in Matlab/Simulink, as shown in Figure 4. The simulation model considers the dynamic response of the motor, the uncertain factors in the hydraulic model including matched disturbance and unmatched disturbance. The parameters of the EHA are shown in To prove the superiority of the designed ISMBC controller, the following three control methods were for comparison.
2) Backstepping control (BC): the control scheme is the same as the ISMBC controller but without integral sliding mode term. To verify the effectiveness of the integral sliding mode control method in the paper, let c0=0 and = 0. Other parameters are the same as those in the ISMBC controller .
3) Proportional-integral-derivative control (PID): this is a classic control algorithm, which is widely used in industrial fields. This controller realizes the trajectory tracking by tuning the three parameters, including proportional gain kp, integral gain ki and derivative gain kd. Properly increasing these gain parameters can improve the control accuracy , but the excessive gain would also cause oscillation and reduce system stability. Finally, through trial and error, parameter s were set as: kp = 28500; ki = 1000; kd = 0.  To prove the superiority of the designed ISMBC controller, the following three control methods were for comparison. 1) Integral sliding mode backstepping control (ISMBC): this is the proposed control scheme in this paper and the design is described in Section 4. The controller parameters were tuned by hand, k1 = 4500; k2 = 100; k3 = 3; k0 = 30; wo = 1000; wc = 5000; = 0.5.
2) Backstepping control (BC): the control scheme is the same as the ISMBC controller but without integral sliding mode term. To verify the effectiveness of the integral sliding mode control method in the paper, let c0=0 and = 0. Other parameters are the same as those in the ISMBC controller .
3) Proportional-integral-derivative control (PID): this is a classic control algorithm, which is widely used in industrial fields. This controller realizes the trajectory tracking by tuning the three parameters, including proportional gain kp, integral gain ki and derivative gain kd. Properly increasing these gain parameters can improve the control accuracy , but the excessive gain would also cause oscillation and reduce system stability. Finally, through trial and error, parameter s were set as: kp = 28500; ki = 1000; kd = 0.

Observer verification
In view of the fact that the micro-crane mainly performs ascent and descent motions, the controller tracking trajectory was designed as a smooth curve with a max displacement of 0.3m, starting to rise at t=0.5s, and starting to fall at t=6s. The desired position of the EHA is shown as the curve x1d in Figure 6 (a). Under the ISMBC controller, the actual output position of the EHA almost overlaps with the reference position signal. The maximum error occurs when the crane just starts to descend, the value is 0.124mm; the EHA mean error only 3.93×10 -3 mm. It can be seen that the ISMBC controller can achieve high accuracy position control. On the other hand, the position estimation and position estimation error of the ESO1 are shown in Figure 5 (b). The maximum position estimation error is only 3.52×10 -3 mm. The estimation error is small enough that the observed position can be regarded as the actual output position of the EHA. The estimated values of other state are shown in Figure 6. The observation results show that the designed dual-ESOs can provide accurate feedback values for ISMBC controller and BC controller.   Figure 7 (a), it can be seen that ISMBC possesses the highest control accuracy, and the position tracking error is almost approaching to zero; BC control accuracy is second to ISMBC, which shows the integral sliding mode surface has the effect of reducing the tracking error. Meanwhile, if PID control is used to achieve higher control accuracy, the proportional gain must be increased, which would cause more severe oscillations to the system. In Figure 7 (b), the control output curves of the three controllers also prove this point. When the motor speed changes rapidly, the PID control output will oscillate violently, which brings unstable factors to the system. The control output of ISMBC controller and BC controller is smoother.
To intuitively express the control accuracy and stability of each controller, five evaluation indexes were defined to evaluate the performance of the control. Those indexes include maximum tracking error e , average tracking error e , standard deviation of the tracking error e , average controller output u and standard deviation of the controller output u . Without load, the evaluation indexes of the three controllers are listed in Table 2. It can be seen that, except for u , the ISMBC controller has the lowest indexes, so its control performance is the best, followed by BC controller and the PID control performance is the worst.   Analyzing the data in Figure 8 (a), (b), and (c) show the different control errors of PID, ISMBC and BC with loads of 100kg, 200kg, and 300kg respectively. From Section 2.3.2., the main component of the unmatched disturbance is external disturbance and load is the main source of external disturbance. The ESO1 can accurately estimate the unmatched disturbance with varying loads, to realize the compensation for the unmatched disturbance. The estimated values of unmatched disturbances with varying loads are shown in Figure 8 (d). Based on the simulation results, the three evaluation indexes e , e and e are obtained and listed in Error! Not a valid bookmark selfreference.., it can be found that as the load increases, the indexes gradually decrease with the ISMBC controller. But with the BC controller, only index e appears a downward trend, and other indexes appear an upward trend. Thus, the results indicate that the integral sliding mode surface can enhance the robustness. In the meantime, only index e of the PID controller shows a downward trend, while other indexes show an upward trend. It shows that within a certain range, larger load force can improve the control performance for the EHA. The swinging, the sudden increase and decrease of the load mass during motion would also cause external disturbance. In order to simulate the swinging, a sine force sin = 300sin(2πt) is applied. On this basis, a pulse signal with an amplitude of 3000N and a period of 2s was added to simulate the sudden increase and decrease of the load. The simulation results are shown in Figure 9. From Figure 9 (a) and (c), it can be seen that the ISMBC controller is the least affected, followed by the PID controller, and the BC controller is most affected by these external disturbances. This further proves that the integral sliding mode control has stronger anti-disturbance ability.
In summary, compared with the PID controller, the backstepping design can obtain higher control accuracy. In the first step of the backstepping design, the integral slidi ng mode surface is introduced into the position error term, which not only further improves the control accuracy, but also boots the robustness. This paper developed a novel control algorithm ISMBC that introduced integral sliding mode control into backstepping design, based on two extended state observers. The proposed control strategy was applied to solve the problems including nonlinearities, parameter uncertainties and external disturbances in the EHA. Lyapunov analysis show s that the proposed control system has higher stability and better robustness than the traditional backstepping design . A multi-domain model was established in the MATLAB/Simulink, including electric motor, hydraulic system, mechanism of a micro-crane and the proposed ISMBC controller. The following conclusions are obtained by simulation and analysis.
(1) Without load, the ISMBC controller shows the best control accuracy and fastest response. Compared with PID, the control accuracy can be increased by 89% and compared with backstepping control by 67%.
(2) With the loads of 100kg, 200kg and 300kg, the simulation results show that all control evaluation indexes of the ISMBC controller have a downward trend when load increases. With PID control, only the control accuracy index decreases slightly and the other indexes show an overall upward trend. However, all indexes of the BC controller rise.
(3) With load, sinusoidal force disturbance plus step force disturbance signals were applied to the system. The simulation results reveal that the ISMBC has the smallest position error and needs the least time to return to a stable state; the BC control has the largest error, but the oscillation during the recovery process is smaller than PID control.
The results of this study indicate that , compared with the PID controller, the BC controller can greatly improve the control accuracy of the system, but the system stability and robustness degrade. Hence, the ISMBC was proposed, by introducing the integral sliding mode control into the backstepping design. The simulation results show the proposed ISMBC can not only further improve the control accuracy, but also can enhance system stability and robustness.
Although the ISMBC control algorithm can improve the control performance of the EHA, it has only been verified by simulation, without being verified by experiment. In the next stage of work, the test platform will be established for further proof and application. On the other hand, the proposed controller relies on accurate system parameters. In the following research, we will use adaptive law to estimate system parameters and to realize adaptive control.