A novel sliding mode control (SMC) design framework is devoted to providing a favorable SMC design solution for the position tracking control of electrohydrostatic actuation system (EHSAS). This framework is composed of three submodules as follows: a reducedorder model of EHSAS, a disturbance sliding mode observer (DSMO), and a new adaptive reaching law (NARL). First, a reducedorder model is obtained by analyzing the flow rate continuation equation of EHSAS to avoid the use of a state observer. Second, DSMO is proposed to estimate and compensate mismatched disturbances existing in the reducedorder model. In addition, a NARL is developed to tackle the inherent chattering problem of SMC. Extensive simulations are conducted compared with the wide adoption of threeloop PID method on the cosimulation platform of EHSAS, which is built by combining AMESim with MATLAB/Simulink, to verify the feasibility and superiority of the proposed scheme. Results demonstrate that the chattering can be effectively attenuated, and the mismatched disturbance can be satisfyingly compensated. Moreover, the transient performance, steadystate accuracy, and robustness of position control are all improved.
Electrohydrostatic actuation system (EHSAS) is a typical selfcontained electrohydraulic servo system. EHSAS does not require an extra external hydraulic oil source compared with the traditional valvecontrolled electrohydraulic servo system (VEHAS), and control of position, velocity, and output force of EHSAS is implemented by regulating the speed of motor pump instead of the throttling principle of VEHAS. In contrast to electrical counterpart, that is, an electromechanical actuation system (EMAS), EHSAS has the higher powertoweight ratio and no mechanical jam fault. Thus, EHSAS demonstrates many advantages, such as high efficiency, high reliability, compact structure, and stable output force. EHSAS has been applied in a variety of fields, such as more electric actuation system of aircraft or ship [
In addition, EHSAS typically adopts permanent magnet synchronous motor (PMSM) as the drive motor of the pump to increase the powertoweight ratio. However, PMSM is nonlinear, multivariable and strong coupling objective such that the control of motor is more complex than that of servo valve in VEHAS. Furthermore, if the IEHP is used in EHSAS, because the motor is integrated into the internal part of the pump, which places the motor in a fully fresh circumstance, then the uncertainties will be more complicated.
Thus far, various control methods have been proposed to address these problems for improving the performance of EHSAS. Among these schemes, the control structure of three loops (i.e., position, motor pump speed, and current control loops) based on PID is dominated in practice because of its simplicity. In [
Sliding mode control (SMC) is a powerful robust control strategy for the linear or nonlinear system due to its insensibility to uncertainties and laconic design procedure. However, three main problems, that is,
In this paper, first, a simplified model is presented by considering leakage and oil compression flow rate as a lumped disturbance; then, mechanical and hydraulic subsystems and motor pump are regarded as a whole for SMC design in which the voltage control signal of the motor pump is obtained directly. State observers are unnecessary to design after simplification because the system states used are all measurable. Second, a finite time disturbance sliding mode observer (DSMO) is designed to estimate the mismatched disturbance and its derivative that are integrated into a sliding mode surface to guarantee the asymptotic convergence of position tracking error. In addition, a kind of new adaptive reaching law (NARL), which not only ensures faster reaching speed but also achieves an improved chattering attenuation effect, is presented.
The remainder of this paper is organized as follows. In Section
The hydraulic schematic of EHSAS is depicted in Figure
Hydraulic diagram of the EHSAS.
Structural diagram of the IEHP.
The shape of magnetic flux density remains nearly sinusoidal, although an oil gap instead of gas gap exists between the stator and the rotor of the IEHP. Thus, the voltage equations of the
The electromagnetic torque
The oil gap friction between the stator and the rotor can be described as [
The motion equation of the IEHP is expressed as
The flow rate equation of the two champers of the IEHP can be defined as
For the cylinder, the flow rate equation can be expressed by
The following equation can be obtained according to flow rate continuity principle:
The following relationships are also considered:
The simplified flow rate continuity equation can be acquired by combining (
The load is assumed to be rigidly linked with rod, then the motion equation of load can be written as
The system state vector is defined as
Then, the state equations of the EHSAS can be denoted as
In (
According to the principle of EHSAS, the corresponding velocity
The system state vector is redefined as
The system state equations are rewritten by
The IEHP adopts vector control with the form of
Then, the simplified state equations can be formulated as
In (
The desired position signal is assumed to satisfy
Then, the error dynamic equations can be obtained as
The sliding mode variable is selected as
Taking the derivative of
The constant rate reaching law (CRRL) is used; that is,
The Lyapunov function is defined as
Evidently,
By simplifying (
In (
However,
In this section, a new DSMO is presented to estimate
In addition, the dynamic behavior of
According to the aforementioned analysis, the disturbance observers can be designed as
Equation (
The detailed design procedures are given in the following.
Selecting a sliding mode variable
Apparently, if appropriate variable
Defining the Lyapunov function
According to the equivalent control principle of the SMC,
Based on the acquired
If
Continuing selecting sliding mode variable
If
Two sliding variables
If
Finally,
Schematic of the DSMO.
The proposed DSMO first observes the observation error of disturbance instead of disturbance itself, unlike the traditional sliding mode observer [
In the proposed DSMO, the estimation process includes two sequential stages. First, disturbance observation errors were obtained. Second, the finite time convergence of
The linear term and terminal attractor of observation error are added and can guarantee a rapid convergence speed during the whole reaching stage. Then, the switch gains
The existing nonlinear [
Based on the DSMO proposed in Section
The sliding mode variable
Control law
If the sliding mode surface in (
First, the reachability of
Thus, the Lyapunov function is defined as
Note that
Evidently,
Next, it is proved that
The simplification of (
According to the results presented in Section
The mismatched disturbances that are transformed into matched ones are not adopted in this work to handle mismatched disturbances
In the proposed NSMC, mechanical, hydraulic subsystems and IEHP are considered a synthesis in which control law
Although
The
Taking the derivative of
The CRRL is selected; that is,
The SMC includes the sliding motion and reaching stage two parts and the reaching law method is prevailing to ensure the reachability of the sliding mode variable. In general, the reaching stage not only requires faster reaching rate but also maintains smaller chattering magnitude.The CRRL used in the previous section is commonly employed, nevertheless, which is difficult to reconcile the contradiction between reaching speed and chattering. In this section, a NARL will be introduced to handle this conflict and compared with several commonly used reaching laws for exhibiting its advantages. For ease of analysis, a double integrator system is selected as the study object; that is,
The CRRL can be denoted as follows:
Integrating (
From the view of discrete system, the amplitude width of the chattering near the sliding surface can be acquired approximately. The sample period is assumed as
In Figure
Phase trajectories. (a) CRRL. (b) TVRRL and IVRRL. (c) Proposed adaptive reaching law.
Another reaching law that is used frequently is TVRRL, which can be expressed as
Similarly, the reaching time
In (
Recently, the IVRRL is developed in [
Similarly, the corresponding reaching time
The IVRRL can dynamically adjust the switch gain
NARL is proposed to solve the problems in the aforementioned reaching laws and can be designed by
The reaching time
In (
Compared with the CRRL,
Thus, the CRRL in (
The whole position control block diagram of the EHSAS is illustrated in Figure
Complete control block diagram of the EHSAS.
The cosimulation platform of the EHSAS is constructed by utilizing AMESim and MATLAB/Simulink to verify the feasibility and effectiveness of the proposed NSMC method. Hydraulic, mechanical subsystems and IEHP are set up with AMESim, and the DSMO, NARL, and NSMC are implemented with MATLAB/Simulink. The cosimulation model is depicted in Figure
Simulation parameters.
Parameter  Value 


0.12 Wb 











2.9 kg 

5.0 kg 





1000 N/m/s 
Maximum stroke  0.1 m 
Maximum output force  20 kN 
Maximum speed of the IEHP  1000 rad/s 
Each control loop of the threeloop PID all uses PI controller, in which the proportional and integral gains of position loop are
First, the validity of the NARL is demonstrated by comparative simulations with reaching laws mentioned in Section
The simulation results are presented in Figures
Performance of the CRRL. (a) State response. (b) Sliding mode surface convergence process. (c) Control output. (d) Phase trajectory.
Performance of the TVRRL. (a) State response. (b) Sliding mode surface convergence process. (c) Control output. (d) Phase trajectory.
Performance of the IVRRL. (a) State response. (b) Sliding mode surface convergence process. (c) Control output. (d) Phase trajectory.
Performance of the NARL. (a) State response. (b) Sliding mode surface convergence process. (c) Control output. (d) Phase trajectory.
Cosimulation model of the EHSAS. (a) Hydraulic and mechanical submodel in AMESim. (b) NSMC and SMO in MATLAB/Simulink.
A smaller
The desired position step signal is presented as
Position step response under no load and step disturbance. (a) Step response in the case of no load. (b) Step response in the case of step disturbance. (c) Estimation of
The position step reference is set to
The steptype disturbance is replaced by a sinusoidaltype external load force, namely,
Position step response under sinusoidal disturbance. (a) Step response in the case of a sinusoidal disturbance. (b) Estimation of
Therefore, the disturbance rejection capability of the NSMC significantly outperforms PID and SMC; the proposed sliding mode design architecture is feasible and effective.
Frequency bandwidth is an important performance index that is used to evaluate the rapidness and accuracy of the control system, that is, the tracking performance of sinusoidal signal with different frequency. To this end, the position reference signal is arranged as
Position tracking performance to a sinusoidal signal in the case of no load. (a) Position tracking performance. (b) Tracking error.
Position tracking performance to a sinusoidal signal in the case of a complicated disturbance. (a) Position tracking performance. (b) Tracking error.
In practice, the load mass
Position tracking performance to sinusoidal signal in the case of parameter variation.
In (
The damping ratio
In this work, a unique SMC architecture is proposed for electrohydrostatic position actuation system, which possesses many prominent superiorities, such as excellent simplification of the design process, avoidance of state observer, effective compensation of mismatched disturbance, and remarkable effect of chattering suppression compared with the traditional SMC. The validity of the presented strategy is verified through cosimulation experiments compared with the traditional threeloop PID method. The performance of the electrohydrostatic position actuation system, such as in nonovershoot regulation to step signal, stronger rejection capability to an external disturbance, higher frequency bandwidth and tracking accuracy to sinusoidal signal, and wide stability margin to parametric variation, is significantly improved.
In addition, the developed DSMO can achieve the finite time estimation instead of the bounded estimation, and its stability proof process is easier than the existing finite time observer. The designed reaching law can simultaneously acquire the desirable performance of fast reaching rate and chattering alleviation through the adaptive regulation of switch gain. The shape of the chattering band is fusiform, which is particularly distinct from several traditional reaching laws.
The authors declare no conflicts of interest regarding the publication of this article.
This work was supported by the National Natural Science Foundation of China (Grant no. 51505016) and Aeronautical Science Foundation of China (Grant no. 20152851020).