# A Novel Sliding Mode Control Framework for Electrohydrostatic Position Actuation System

Authors: Rongrong Yang; Yongling Fu; Ling Zhang; Haitao Qi; Xu Han; Jian Fu
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

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 reduced-order model of EHSAS, a disturbance sliding mode observer (DSMO), and a new adaptive reaching law (NARL). First, a reduced-order 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 reduced-order 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 three-loop 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, steady-state accuracy, and robustness of position control are all improved.


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## Body

## 1. Introduction

Electrohydrostatic actuation system (EHSAS) is a typical self-contained electrohydraulic servo system. EHSAS does not require an extra external hydraulic oil source compared with the traditional valve-controlled 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 power-to-weight 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 [1, 2], active suspension of a vehicle [3, 4], angle control of cannon and radar, flow rate and pressure control of injection molding machine [5–7], and position and force control of robot [8–11]. Moreover, the superiority of EHSAS has become prominent with the recent emergence of several new components, such as an integrated electrohydraulic pump (IEHP) and single-rod symmetrical cylinder. In the future, improved control accuracy, higher frequency bandwidth, and stronger robustness will be the development trends of EHSAS. However, achieving these performances, which can be attributed to the following reasons, is a challenging task. First, EHSAS includes numerous uncertain parameters, such as oil-effective bulk modulus, leakage coefficient, and load mass given the influence of oil temperature, sealing, and different operating conditions. Second, the dynamic characteristics of friction and external load disturbance are difficult to exactly acquire, which have an adverse impact on control performance.In addition, EHSAS typically adopts permanent magnet synchronous motor (PMSM) as the drive motor of the pump to increase the power-to-weight 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 [12, 13], fuzzy method and structure invariance principle were used to tune the PI parameters of the three loops and compensate external disturbance. However, fuzzy rules typically depend on specialists to a large extent, thus resulting in poor adaptability in practice. In [14], a multiloop control approach was addressed by using high gain to dominate uncertainties, which result in oscillation and even instability due to light damping feature of EHSAS. In addition, other control schemes, such as quantitative feedback theory [15, 16], model reference adaptive [17], and H∞ [18], have been developed to strengthen robustness. Moreover, many nonlinear strategies were investigated to further enhance the performance of EHSAS. Literature [19] utilizes feedback linearization to cancel nonlinear functions in a system. In [20, 21], adaptive backstepping with a neural network was presented to estimate uncertainties and guarantee asymptotic stability of a system, but this method requires that all system state variables were measurable, and the neural network was excessively complex to apply in practice. In [22], a passive-based model was introduced in which state and disturbance observers were adopted to observe unmeasured state and unknown disturbance.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,(1) obtaining full system state information, (2) coping with mismatched disturbances, and (3) inherent chattering problem, should be addressed when using SMC because EHSAS is a high-order nonlinear system subjected to multiple mismatched disturbances. The first problem is commonly solved by state observer [23, 24], but the observed states cannot converge to its actual value if nonvanishing mismatched disturbance exists in the system, thereby resulting in a complicated design process. To the best of our knowledge, nearly all existing literature handles the second problem by transforming mismatched to matched disturbance [25–30]. This solution will result in undesirable consequences in which several originally measurable states cannot be utilized; instead, new unmeasurable states will be generated such that an additional state observer is required. For the last problem, abundant approaches, such as boundary layer method [31], high-order SMC [32], and adaptive SMC [33], have been explored to mitigate or eliminate chattering. In addition, the aforementioned literature that uses SMC for EHSAS ignores the control of motor pump for its fast dynamic behavior. Actually, the speed and current control dynamic performances of the motor pump have a significant effect on the position control performance of EHSAS.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 Section2, the configuration of the EHSAS and basic principle of the IEHP are introduced, and the full-order and simplified model of EHSAS are presented. In Section 3, the detained design of procedure of DSMO and its stability analysis are discussed. In Section 4, the position and d-axis current sliding mode controller of the EHSAS are designed. In Section 5, a kind of NARL is proposed, and comparative analyses with other reaching laws are conducted. In Section 6, simulation results and analysis are given. In Section 7, conclusions are drawn.


## 2. System Description and Model Simplification

### 2.1. EHSAS Description

The hydraulic schematic of EHSAS is depicted in Figure1. EHSAS consists of a bidirectional IEHP, symmetrical hydraulic cylinder, an accumulator, check valve, relief valve, mode selector valve, sensors, controller, power driver of an IEHP, and other accessories. Among these components, the IEHP is a new hydraulic power unit that is highly integrated with PMSM and axial piston pump, and its structure diagram is illustrated in Figure 2 [34, 35]. The basic operational principle is introduced as follows: DC270V voltage is connected by a power connector and is transformed into a sinusoidal AC voltage by an inverter that is utilized to generate a rotating magnetic field. The rotating magnetic field interacts with a permanent magnet of rotor surface to drive rotor rotation. Consequently, the oil suction and discharge processes are achieved through piston motion coordination with swash and valve plates. A resolver is used to detect rotor magnet pole position and measure the speed of IEHP. The structure of which is highly different, although the working principle of the IEHP is the same as that of the traditional motor pump unit. The IEHP has high efficiency, small size, and minimal noise because motor and pump share the same rotor and housing instead of the link form by couplings used in the traditional motor pump. Oil will be compressed to cause load motion by regulating the speed of the IEHP to generate flow rate. An accumulator is responsible for supplying oil to low-pressure pipe by check valve to prevent cavitation. Two relief valves are used to restrict the maximum operation pressure of the EHSAS for safety. Mode switch valve is installed to keep the piston in a free state when a fault occurs.Figure 1
Hydraulic diagram of the EHSAS.Figure 2
Structural diagram of the IEHP.


### 2.2. Mathematical Model and Simplification

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 thed- and q-axes of the IEHP can be expressed as(1)ud=Ldi˙d+Rid-pωmLqiq,(2)uq=Lqi˙q+Riq+pωmLdid+pωmψf.The electromagnetic torqueTe of the IEHP is described by(3)Te=1.5piqψf+idLd-Lq,where ud and uq are the voltages of the d- and q-axes, respectively, id and iq represent the currents of the d- and q-axes, correspondingly, Ld and Lq are the equivalent inductances of the d- and q-axes, respectively, R is the resistance of stator winding, p is the number of pole pairs, ψf represents the magnet flux of the rotor permanent magnet, and ωm is the mechanical angular speed.The oil gap friction between the stator and the rotor can be described as [36](4)TL1=2πμLr3ωmκ=b2ωm.The motion equation of the IEHP is expressed as(5)Jtω˙m=Te-DPL-Bmωm-Tf,where μ is the oil kinetic viscosity, L is the length of the rotor, r is the radius of the rotor, κ represents the thickness of the oil gap between the rotor and the stator, Tf is the other unmodeled nonlinear friction and disturbance, Jt is the moment of inertia of the IEHP, b1 is the mechanical viscous friction coefficient, b2 is the oil gap viscous friction coefficient, Bm=b1+b2 is the total viscous friction coefficient, D represents the volumetric displacement of the IEHP, and PL is the pressure difference of the two chambers of the cylinder.The flow rate equation of the two champers of the IEHP can be defined as(6)Qa=Dωm-εpa-pb-L1pa-pr-Vaβep˙a,Qb=Dωm-εpa-pb+L1pa-pr+Vbβep˙b,where ε and L1 are the internal and external leakage coefficients of the IEHP, respectively, Qa and Qb represent the inlet and outlet flow rates of the IEHP, correspondingly, pa and pb are the pressures of the two chambers of the IEHP, respectively, Va and Vb are the volumes of the two chambers of the IEHP, and βe is the effective oil bulk modulus.For the cylinder, the flow rate equation can be expressed by(7)Q1=Ax˙+V0+Axβep˙1+L2p1-p2,Q2=Ax˙-V0-Axβep˙2+L2p1-p2,where L2 is the internal leakage coefficient of the cylinder, and the external leakage is ignored, Q1 and Q2 represent the supplied and return flow rates of the cylinder, correspondingly, p1 and p2 are the pressures of the two champers of the cylinder, respectively, V0 is the initial volume of the single chamber of the cylinder, A is the effective ram area of the cylinder, and x represents the displacement of the load.The following equation can be obtained according to flow rate continuity principle:(8)Q1=Qa,Q2=Qb.The following relationships are also considered:(9)p˙1=p˙a,p˙2=p˙b,p˙1=-p˙2.The simplified flow rate continuity equation can be acquired by combining (7)–(9), as follows:(10)Dωm=Ax˙+V0+Va2βep˙L+ε+L2+L12pL=Ax˙+Vt4βep˙L+CtpL+Qf,where Vt is the total control volume and Qf is the flow rate consumption of all kinds of valves in the EHSAS.The load is assumed to be rigidly linked with rod, then the motion equation of load can be written as(11)ApL=Mx¨+Ksx+Bcx˙+Ff+FL,where M=mL+mc, M denotes the combined mass, including piston, rod, and load, mc is the mass of piston and rod, mL is the mass of load, Ks is the stiffness coefficient of the elastic load, FL is the external load force, Bc is the viscous friction coefficient, and Ff is the other unmodeled nonlinear friction and disturbance.The system state vector is defined as(12)X=x1x2x3x4x5x6T=xx˙pLωmiqidT.Then, the state equations of the EHSAS can be denoted as(13)x˙1=x2,x˙2=AMx3-KsMx1-BmMx2-FfM-FLM,x˙3=-4βeAVtx2-4βeCtVtx3+4βeDVtx4,x˙4=1.5px5Jtψf+x6Ld-Lq-BmJtx2-DJtx3-TfJt,x˙5=-pLdLqx4x6-pψfLqx4-RLqx5+uqLq,x˙6=pLqLdx4x5-RLdx6+udLd.In (13), the EHSAS is composed of a fifth- and a first-order subsystem, and uq and ud are the corresponding control inputs of the two subsystems. Our control objective is to synthesize control voltage uq for displacement x to track the desired position xd as accurately as possible in the presence of parametric variations and external disturbance.According to the principle of EHSAS, the corresponding velocityx˙ and acceleration x¨ are necessary to accomplish the control task of displacement x. Velocity x˙ is generated by flow rate that results from the speed adjustment of the IEHP, and acceleration x¨ is derived from pressure pL for the compression of oil, thereby also resulting from the flow rate of the IEHP. Therefore, the control of position is equivalent to regulating the speed ωm of the IEHP in essence. In (10), the total output flow rate of the IEHP includes three parts as follows: the first term Ax˙ represents the demanded flow rate of load movement; the second term Vt/4βep˙L is the flow rate consumption due to oil compression; and the last two terms CtpL and Qf are the leakage flow rate and consumption of various valves, respectively. In general, the first part represents a large proportion of the total flow rate; then, the fifth-order subsystem will be reduced to third order if the other two parts are considered as a lumped disturbance.The system state vector is redefined as(14)Z=z1z2z3z4T=xωmiqidT.The system state equations are rewritten by(15)z˙1=a1z2+d1,z˙2=a2z3+a3z3z4+d2,z˙3=a5z2z4+a6z2+a7z3+b1uq,z˙4=a8z2z3+a9z4+b2ud,where(16)a1=DA,d1=-1AVt4βedpLdt+CtpL+Qf,a2=1.5pψfJt,a3=1.5pJtLd-Lq,d2=-BmJtz2-DJtpL-1JtFf,a5=-pLdLq,a6=-pψfLq,a7=-RLq,b1=1Lq,a8=pLqLd,a9=-RLd,b2=1Ld.The IEHP adopts vector control with the form ofid=0, and system state variables are reselected for ease of description as(17)Ψ=ψ1ψ2ψ3z4T=z1a1z2a1a2z3z4T.Then, the simplified state equations can be formulated as(18)ψ˙1=ψ2+d1,ψ˙2=ψ3+a1d2,ψ˙3=a2a6ψ2+a7ψ3+a1a2b1uq,z˙4=a8z2z3+a9z4+b2ud.Remark 1.
(
1
) If pressure sensors are assembled in the two champers of the cylinder, then d1 and d2 are calculated partly and can be used for feedforward compensation. However, the pressure signal is easily perturbed by noise owing to its high-frequency bandwidth. Moreover, d1 includes the derivative of pL, which results in a severe noise contamination problem. In addition, pressure sensors may not be used because of cost limitations, size, and installation room. (2) Fortunately, the position x of load and the speed ωm of the IEHP are typically measured by several digital sensors, such as grating scale, encoder, and resolver. Therefore, if these clean signals are employed to design an observer to estimate d1 and d2, then the satisfied disturbance compensation effect can be achieved. Based on the above analysis, pL is unmeasurable, while x,ωm,iq, and id are all available.


## 2.1. EHSAS Description

The hydraulic schematic of EHSAS is depicted in Figure1. EHSAS consists of a bidirectional IEHP, symmetrical hydraulic cylinder, an accumulator, check valve, relief valve, mode selector valve, sensors, controller, power driver of an IEHP, and other accessories. Among these components, the IEHP is a new hydraulic power unit that is highly integrated with PMSM and axial piston pump, and its structure diagram is illustrated in Figure 2 [34, 35]. The basic operational principle is introduced as follows: DC270V voltage is connected by a power connector and is transformed into a sinusoidal AC voltage by an inverter that is utilized to generate a rotating magnetic field. The rotating magnetic field interacts with a permanent magnet of rotor surface to drive rotor rotation. Consequently, the oil suction and discharge processes are achieved through piston motion coordination with swash and valve plates. A resolver is used to detect rotor magnet pole position and measure the speed of IEHP. The structure of which is highly different, although the working principle of the IEHP is the same as that of the traditional motor pump unit. The IEHP has high efficiency, small size, and minimal noise because motor and pump share the same rotor and housing instead of the link form by couplings used in the traditional motor pump. Oil will be compressed to cause load motion by regulating the speed of the IEHP to generate flow rate. An accumulator is responsible for supplying oil to low-pressure pipe by check valve to prevent cavitation. Two relief valves are used to restrict the maximum operation pressure of the EHSAS for safety. Mode switch valve is installed to keep the piston in a free state when a fault occurs.Figure 1
Hydraulic diagram of the EHSAS.Figure 2
Structural diagram of the IEHP.


## 2.2. Mathematical Model and Simplification

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 thed- and q-axes of the IEHP can be expressed as(1)ud=Ldi˙d+Rid-pωmLqiq,(2)uq=Lqi˙q+Riq+pωmLdid+pωmψf.The electromagnetic torqueTe of the IEHP is described by(3)Te=1.5piqψf+idLd-Lq,where ud and uq are the voltages of the d- and q-axes, respectively, id and iq represent the currents of the d- and q-axes, correspondingly, Ld and Lq are the equivalent inductances of the d- and q-axes, respectively, R is the resistance of stator winding, p is the number of pole pairs, ψf represents the magnet flux of the rotor permanent magnet, and ωm is the mechanical angular speed.The oil gap friction between the stator and the rotor can be described as [36](4)TL1=2πμLr3ωmκ=b2ωm.The motion equation of the IEHP is expressed as(5)Jtω˙m=Te-DPL-Bmωm-Tf,where μ is the oil kinetic viscosity, L is the length of the rotor, r is the radius of the rotor, κ represents the thickness of the oil gap between the rotor and the stator, Tf is the other unmodeled nonlinear friction and disturbance, Jt is the moment of inertia of the IEHP, b1 is the mechanical viscous friction coefficient, b2 is the oil gap viscous friction coefficient, Bm=b1+b2 is the total viscous friction coefficient, D represents the volumetric displacement of the IEHP, and PL is the pressure difference of the two chambers of the cylinder.The flow rate equation of the two champers of the IEHP can be defined as(6)Qa=Dωm-εpa-pb-L1pa-pr-Vaβep˙a,Qb=Dωm-εpa-pb+L1pa-pr+Vbβep˙b,where ε and L1 are the internal and external leakage coefficients of the IEHP, respectively, Qa and Qb represent the inlet and outlet flow rates of the IEHP, correspondingly, pa and pb are the pressures of the two chambers of the IEHP, respectively, Va and Vb are the volumes of the two chambers of the IEHP, and βe is the effective oil bulk modulus.For the cylinder, the flow rate equation can be expressed by(7)Q1=Ax˙+V0+Axβep˙1+L2p1-p2,Q2=Ax˙-V0-Axβep˙2+L2p1-p2,where L2 is the internal leakage coefficient of the cylinder, and the external leakage is ignored, Q1 and Q2 represent the supplied and return flow rates of the cylinder, correspondingly, p1 and p2 are the pressures of the two champers of the cylinder, respectively, V0 is the initial volume of the single chamber of the cylinder, A is the effective ram area of the cylinder, and x represents the displacement of the load.The following equation can be obtained according to flow rate continuity principle:(8)Q1=Qa,Q2=Qb.The following relationships are also considered:(9)p˙1=p˙a,p˙2=p˙b,p˙1=-p˙2.The simplified flow rate continuity equation can be acquired by combining (7)–(9), as follows:(10)Dωm=Ax˙+V0+Va2βep˙L+ε+L2+L12pL=Ax˙+Vt4βep˙L+CtpL+Qf,where Vt is the total control volume and Qf is the flow rate consumption of all kinds of valves in the EHSAS.The load is assumed to be rigidly linked with rod, then the motion equation of load can be written as(11)ApL=Mx¨+Ksx+Bcx˙+Ff+FL,where M=mL+mc, M denotes the combined mass, including piston, rod, and load, mc is the mass of piston and rod, mL is the mass of load, Ks is the stiffness coefficient of the elastic load, FL is the external load force, Bc is the viscous friction coefficient, and Ff is the other unmodeled nonlinear friction and disturbance.The system state vector is defined as(12)X=x1x2x3x4x5x6T=xx˙pLωmiqidT.Then, the state equations of the EHSAS can be denoted as(13)x˙1=x2,x˙2=AMx3-KsMx1-BmMx2-FfM-FLM,x˙3=-4βeAVtx2-4βeCtVtx3+4βeDVtx4,x˙4=1.5px5Jtψf+x6Ld-Lq-BmJtx2-DJtx3-TfJt,x˙5=-pLdLqx4x6-pψfLqx4-RLqx5+uqLq,x˙6=pLqLdx4x5-RLdx6+udLd.In (13), the EHSAS is composed of a fifth- and a first-order subsystem, and uq and ud are the corresponding control inputs of the two subsystems. Our control objective is to synthesize control voltage uq for displacement x to track the desired position xd as accurately as possible in the presence of parametric variations and external disturbance.According to the principle of EHSAS, the corresponding velocityx˙ and acceleration x¨ are necessary to accomplish the control task of displacement x. Velocity x˙ is generated by flow rate that results from the speed adjustment of the IEHP, and acceleration x¨ is derived from pressure pL for the compression of oil, thereby also resulting from the flow rate of the IEHP. Therefore, the control of position is equivalent to regulating the speed ωm of the IEHP in essence. In (10), the total output flow rate of the IEHP includes three parts as follows: the first term Ax˙ represents the demanded flow rate of load movement; the second term Vt/4βep˙L is the flow rate consumption due to oil compression; and the last two terms CtpL and Qf are the leakage flow rate and consumption of various valves, respectively. In general, the first part represents a large proportion of the total flow rate; then, the fifth-order subsystem will be reduced to third order if the other two parts are considered as a lumped disturbance.The system state vector is redefined as(14)Z=z1z2z3z4T=xωmiqidT.The system state equations are rewritten by(15)z˙1=a1z2+d1,z˙2=a2z3+a3z3z4+d2,z˙3=a5z2z4+a6z2+a7z3+b1uq,z˙4=a8z2z3+a9z4+b2ud,where(16)a1=DA,d1=-1AVt4βedpLdt+CtpL+Qf,a2=1.5pψfJt,a3=1.5pJtLd-Lq,d2=-BmJtz2-DJtpL-1JtFf,a5=-pLdLq,a6=-pψfLq,a7=-RLq,b1=1Lq,a8=pLqLd,a9=-RLd,b2=1Ld.The IEHP adopts vector control with the form ofid=0, and system state variables are reselected for ease of description as(17)Ψ=ψ1ψ2ψ3z4T=z1a1z2a1a2z3z4T.Then, the simplified state equations can be formulated as(18)ψ˙1=ψ2+d1,ψ˙2=ψ3+a1d2,ψ˙3=a2a6ψ2+a7ψ3+a1a2b1uq,z˙4=a8z2z3+a9z4+b2ud.Remark 1.
(
1
) If pressure sensors are assembled in the two champers of the cylinder, then d1 and d2 are calculated partly and can be used for feedforward compensation. However, the pressure signal is easily perturbed by noise owing to its high-frequency bandwidth. Moreover, d1 includes the derivative of pL, which results in a severe noise contamination problem. In addition, pressure sensors may not be used because of cost limitations, size, and installation room. (2) Fortunately, the position x of load and the speed ωm of the IEHP are typically measured by several digital sensors, such as grating scale, encoder, and resolver. Therefore, if these clean signals are employed to design an observer to estimate d1 and d2, then the satisfied disturbance compensation effect can be achieved. Based on the above analysis, pL is unmeasurable, while x,ωm,iq, and id are all available.


## 3. Design of a Novel DSMO

### 3.1. Problem Statement

In (18), d1,d2, and uq are placed in different channels; thus, the simplified third-order system remains a typical perturbed system with mismatched disturbance. The traditional SMC is robust only for matched disturbance, which can be interpreted by the following analysis.The desired position signal is assumed to satisfyxd∈C3, and tracking error variables are defined as(19)e1=ψ1-xd,e2=ψ2-x˙d,e3=ψ3-x¨d.Then, the error dynamic equations can be obtained as(20)e˙1=ψ˙1-x˙d=ψ2-x˙d+d1=e2+d1,e˙2=ψ˙2-x¨d=ψ3-x¨d+a1d2=e3+a1d2,e˙3=ψ˙3-x⃛d=a2a6ψ2+a7ψ3+a1a2b1uq-x⃛d.The sliding mode variable is selected ass=e3+c2e2+c1e1, where c1 and c2 meet Hurwitz condition.Taking the derivative ofs with respect to time and considering (20) yield(21)s˙=e˙3+c2e˙2+c1e˙1=a2a6ψ2+a7ψ3+a1a2b1uq-x⃛d+c2e3+a1d2+c1e2+d1.The constant rate reaching law (CRRL) is used; that is,s˙=-ηsgn⁡s. Then, control law uq can be obtained as(22)uq=-a1a2b1-1a2a6ψ2+a7ψ3-x⃛d+c2e3+c1e2+ηsgn⁡s,where η>0 is the switch gain.The Lyapunov function is defined asV=0.5s2, and the derivative of V is obtained. Then, substituting (22) into it yields(23)V˙=ss˙=s-ηsgn⁡s+c1d1+c2a1d2=-ηs+sc3d1+c2a1d2≤-ηs+sc3D1+c2a1D2=sc3D1+c2a1D2-η.Evidently,V˙<0 can be guaranteed if switch gain η satisfies condition η>c3D1+c2D2. After s reaches sliding mode surface s=0, and noting (20), we obtain(24)e3+c2e2+c1e1=e˙2-a1d2+c2e˙1-d1+c1e1=e¨1+c2e˙1+c1e1-c2d1-d˙1-a1d2=0.By simplifying (24), the system dynamic performance is decided by the following equation:(25)e¨1+c2e˙1+c1e1=c2d1+d˙1+a1d2.In (25), position tracking error e1 cannot converge asymptotically to 0 given the existence of existing of d1,d˙1, and d2. Fortunately, the asymptotic convergence of e1 can be achieved if a new sliding mode variable σ is designed as(26)σ=e3+c2e2+c1e1+c2d1+d˙1+a1d2.However,d1,d˙1, and d2 are unknown; thus, a disturbance observer is required to estimate d1,d˙1, and d2, which will be presented in the next section.


### 3.2. Design of the DSMO

In this section, a new DSMO is presented to estimated1,d˙1, and d2. Generally, the design procedure of observer includes two steps. First, the dynamic equations of the system are copied. Second, some correction terms are added to the dynamic equations of the system. In general, these correction terms are a function of the error of measurable output variable and its observation value, which can ensure unmeasurable state variable convergence to its real value. The simplified model in (18) is rewritten as the following form based on the design idea of the observer:(27)ψ˙1=ψ2+d1,d˙1=d˙1,d¨1=d¨1,(28)ψ˙2=ψ3+a1d2,d˙2=d˙2.Owing to (14) and (17), the following relationships for ψ1, ψ2, and ψ3 are considered: ψ1=z1=x,ψ2=a1z2=a1ω=D/Aω, and ψ3=a1a2z3=D/A1.5pψf/Jtiq, where x, ω, and iq are all measurable, and parameters a1 and a2 can also be acquired by using the mechanical, hydraulic, and electrical design parameters of the EHSAS. The difference between approximate and actual values can be seen as a parametric uncertainty lumped into d1 and d2, even though the obtained values of a1 and a2 are inaccurate. Therefore, we can claim that a1 and a2 are known, thereby implying that ψ1, ψ2, and ψ3 are also known. Then, ψ1 and ψ2 in (27) are considered measurable output and input variables for estimating d1 and d˙1. Similarly, ψ2 and ψ3 in (28) are regarded as measurable output and input variables that are used to observe d2.In addition, the dynamic behavior ofd1, d˙1, d¨1, and d2 is required when designing an observer that is impossible to be acquired in practice. Thus, a prevailing method in the existing literature is to assume that d1, d˙1, d¨1, and d2 are bounded. Actually, in (27) and (28), d1, d˙1, d¨1, and d2 depend on the velocity x˙, acceleration x¨, jerk x⃛, and acceleration ω¨ of the IEHP, respectively, and these variables are related to the speed ω of the IEHP, maximum operation pressure Psmax of the EHSAS, maximum output torque Temax of the IEHP, viscosity μ and bulk modulus βe of fluid oil, and enclosed volume Vt of the EHSAS. The above-mentioned variables are all constrained. Hence, the bounded assumption about d1, d˙1, d¨1, and d2 is reasonable. Thus, the following assumption is presented.Assumption 2.
d
1
∈
C
2
,
d
2
∈
C
1; moreover, d1≤D1,d˙1≤D1d,d¨1≤D1dd,d2≤D2, and d˙2≤D2d, where D1, D1d, D1dd, D2, and D2d are known constants.According to the aforementioned analysis, the disturbance observers can be designed as(29)ψ⌢˙1=ψ2+d⌢1+μ1,d⌢˙1=μ2+d⌢1d,d⌢˙1d=μ3,ψ⌢˙2=ψ3+a1d⌢2+μ4,d⌢˙2=μ5,where ψ⌢1, ψ⌢2, d⌢1, d⌢1d, and d⌢2 denote the observed values of ψ1, ψ2, d1, d˙1, and d2, respectively, and μ1, μ2, μ3, μ4, and μ5 are the auxiliary correction terms.Equation (18) is subtracted from (29), then, the error dynamic equations of the observer can be obtained by(30)ψ~˙1=d~1-μ1,d~˙1=d˙1-μ2+d~1d,d~˙1d=d¨1-μ3,ψ~˙2=a1d~2-μ4,d~˙2=d˙2-μ5,where ψ~1, ψ~2, d~1, d~1d, and d~2 represent the observer errors of ψ1, ψ2, d1, d˙1, and d2, correspondingly.The detailed design procedures are given in the following.Step 1.
Selecting a sliding mode variables1=ψ~1 and taking derivative of s1 yields(31)s˙1=ψ~˙1=d~1-μ1.Apparently, if appropriate variableμ1 is designed such that ψ~˙1=0, then d~1 will be equal to μ1. Here, μ1 is constructed as(32)μ1=ξ1s1+ξ2s10.5sgn⁡s1+η1sgn⁡s1.Defining the Lyapunov functionV1=0.5s12, differentiating V1, and substituting (32) into it yield(33)V˙1=s1s˙1=-ξ1s12-ξ2s11.5-η1s1+s1d~1≤-η1s1+s1d~1=-2V10.5η1-d~1,where ξ1, ξ2, and η1 are all positive constants. Evidently, s1 will reach a sliding mode surface s1=0 in finite time t1 only if η1 meets condition η1>d~1.According to the equivalent control principle of the SMC,s1=s˙1=0 will hold once sliding mode surface s1=0 is reached, and s1=ψ~1 is considered. Therefore, d~1 will approach μ1 in finite time t1.Step 2.
Based on the acquiredd~1 in Step 1, a new sliding mode variable s2=d~1 is selected, and μ2 is designed as the following form:(34)μ2=ξ3s2+ξ4s20.5sgn⁡s2+η2sgn⁡s2,where ξ3,ξ4, and η2 are all positive constants.Ifη2 is selected to meet the condition η2>D1d+d~1d, then d~1 and s2 will converge to 0 in finite time t2 after undergoing finite time t1, thereby also indicating that d⌢1 converges to d1 at in finite time t2. Thus, d~1d will also approach μ2 in finite time t2 based on equivalent control principle.Step 3.
Continuing selecting sliding mode variables3=d~1d, then μ3 is constructed as(35)μ3=ξ5s3+ξ6s30.5sgn⁡s3+η3sgn⁡s3,where ξ5,ξ6, and η3 are all positive constants.Ifη3 is selected to satisfy η3>D1dd, then d~1d and s3 can converge to 0 in finite time t3 after undergoing finite time t1+t2, thereby indicating that d⌢1d converges to d˙1 in finite time t3.Step 4.
Two sliding variabless4=ψ~2 and s5=d~2 are selected; similarly, μ4 and μ5 are synthesized as(36)μ4=ξ7s4+ξ8s40.5sgn⁡s4+η4sgn⁡s4,μ5=ξ9s5+ξ10s50.5sgn⁡s5+η5sgn⁡s5,where ξ7,ξ8,ξ9,ξ10,η4, and η5 are all positive constants.Ifη5 and η6 are selected as η4>a1d~2,η5>D2d by conducting the same analysis as the foregoing steps, then ψ~2 and d~2 will successively converge to 0 in finite time t4 beginning at the initial moment t=0; that is, d⌢2 converges to d2 in the same time t4.Finally,d⌢1, d⌢1d, and d⌢2 are all estimated after undergoing a finite time tr, which satisfies tr=max⁡t1+t2+t3,t4.The schematic of the DSMO is illustrated in Figure 3. The diagrams of d⌢1 and d⌢1d are only provided for the limitation of space.Figure 3
Schematic of the DSMO.Remark 3.
The proposed DSMO first observes the observation error of disturbance instead of disturbance itself, unlike the traditional sliding mode observer [37]. The smaller switch gains ηi,i=1,2,3,4, can be permitted because the observation error is typically less than the upper bound of disturbance, which is beneficial for suppression of chattering.Remark 4.
In the proposed DSMO, the estimation process includes two sequential stages. First, disturbance observation errors were obtained. Second, the finite time convergence ofd~1, d~1d, and d~2 is ensured. The extra-low-pass filter generally used in the traditional sliding mode observer is not required because the dynamic equations of d~1, d~1d, and d~2 possess filtering effects, which can be exploited to smooth chattering.Remark 5.
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 gainsηi,i=1,2,3,4, are only required to be greater than the observation error or the upper bound of disturbance, thereby contributing to chattering restraint to some extent.Remark 6.
The existing nonlinear [38] and extended disturbance observers [39] can achieve the asymptotic convergence of the observation error only if the derivative of disturbance is equal to 0, whereas the bounded estimation result is only obtained when the derivative of disturbance is bounded. By contrast, the presented DSMO can realize the finite time estimation of disturbance and its derivative.


## 3.1. Problem Statement

In (18), d1,d2, and uq are placed in different channels; thus, the simplified third-order system remains a typical perturbed system with mismatched disturbance. The traditional SMC is robust only for matched disturbance, which can be interpreted by the following analysis.The desired position signal is assumed to satisfyxd∈C3, and tracking error variables are defined as(19)e1=ψ1-xd,e2=ψ2-x˙d,e3=ψ3-x¨d.Then, the error dynamic equations can be obtained as(20)e˙1=ψ˙1-x˙d=ψ2-x˙d+d1=e2+d1,e˙2=ψ˙2-x¨d=ψ3-x¨d+a1d2=e3+a1d2,e˙3=ψ˙3-x⃛d=a2a6ψ2+a7ψ3+a1a2b1uq-x⃛d.The sliding mode variable is selected ass=e3+c2e2+c1e1, where c1 and c2 meet Hurwitz condition.Taking the derivative ofs with respect to time and considering (20) yield(21)s˙=e˙3+c2e˙2+c1e˙1=a2a6ψ2+a7ψ3+a1a2b1uq-x⃛d+c2e3+a1d2+c1e2+d1.The constant rate reaching law (CRRL) is used; that is,s˙=-ηsgn⁡s. Then, control law uq can be obtained as(22)uq=-a1a2b1-1a2a6ψ2+a7ψ3-x⃛d+c2e3+c1e2+ηsgn⁡s,where η>0 is the switch gain.The Lyapunov function is defined asV=0.5s2, and the derivative of V is obtained. Then, substituting (22) into it yields(23)V˙=ss˙=s-ηsgn⁡s+c1d1+c2a1d2=-ηs+sc3d1+c2a1d2≤-ηs+sc3D1+c2a1D2=sc3D1+c2a1D2-η.Evidently,V˙<0 can be guaranteed if switch gain η satisfies condition η>c3D1+c2D2. After s reaches sliding mode surface s=0, and noting (20), we obtain(24)e3+c2e2+c1e1=e˙2-a1d2+c2e˙1-d1+c1e1=e¨1+c2e˙1+c1e1-c2d1-d˙1-a1d2=0.By simplifying (24), the system dynamic performance is decided by the following equation:(25)e¨1+c2e˙1+c1e1=c2d1+d˙1+a1d2.In (25), position tracking error e1 cannot converge asymptotically to 0 given the existence of existing of d1,d˙1, and d2. Fortunately, the asymptotic convergence of e1 can be achieved if a new sliding mode variable σ is designed as(26)σ=e3+c2e2+c1e1+c2d1+d˙1+a1d2.However,d1,d˙1, and d2 are unknown; thus, a disturbance observer is required to estimate d1,d˙1, and d2, which will be presented in the next section.


## 3.2. Design of the DSMO

In this section, a new DSMO is presented to estimated1,d˙1, and d2. Generally, the design procedure of observer includes two steps. First, the dynamic equations of the system are copied. Second, some correction terms are added to the dynamic equations of the system. In general, these correction terms are a function of the error of measurable output variable and its observation value, which can ensure unmeasurable state variable convergence to its real value. The simplified model in (18) is rewritten as the following form based on the design idea of the observer:(27)ψ˙1=ψ2+d1,d˙1=d˙1,d¨1=d¨1,(28)ψ˙2=ψ3+a1d2,d˙2=d˙2.Owing to (14) and (17), the following relationships for ψ1, ψ2, and ψ3 are considered: ψ1=z1=x,ψ2=a1z2=a1ω=D/Aω, and ψ3=a1a2z3=D/A1.5pψf/Jtiq, where x, ω, and iq are all measurable, and parameters a1 and a2 can also be acquired by using the mechanical, hydraulic, and electrical design parameters of the EHSAS. The difference between approximate and actual values can be seen as a parametric uncertainty lumped into d1 and d2, even though the obtained values of a1 and a2 are inaccurate. Therefore, we can claim that a1 and a2 are known, thereby implying that ψ1, ψ2, and ψ3 are also known. Then, ψ1 and ψ2 in (27) are considered measurable output and input variables for estimating d1 and d˙1. Similarly, ψ2 and ψ3 in (28) are regarded as measurable output and input variables that are used to observe d2.In addition, the dynamic behavior ofd1, d˙1, d¨1, and d2 is required when designing an observer that is impossible to be acquired in practice. Thus, a prevailing method in the existing literature is to assume that d1, d˙1, d¨1, and d2 are bounded. Actually, in (27) and (28), d1, d˙1, d¨1, and d2 depend on the velocity x˙, acceleration x¨, jerk x⃛, and acceleration ω¨ of the IEHP, respectively, and these variables are related to the speed ω of the IEHP, maximum operation pressure Psmax of the EHSAS, maximum output torque Temax of the IEHP, viscosity μ and bulk modulus βe of fluid oil, and enclosed volume Vt of the EHSAS. The above-mentioned variables are all constrained. Hence, the bounded assumption about d1, d˙1, d¨1, and d2 is reasonable. Thus, the following assumption is presented.Assumption 2.
d
1
∈
C
2
,
d
2
∈
C
1; moreover, d1≤D1,d˙1≤D1d,d¨1≤D1dd,d2≤D2, and d˙2≤D2d, where D1, D1d, D1dd, D2, and D2d are known constants.According to the aforementioned analysis, the disturbance observers can be designed as(29)ψ⌢˙1=ψ2+d⌢1+μ1,d⌢˙1=μ2+d⌢1d,d⌢˙1d=μ3,ψ⌢˙2=ψ3+a1d⌢2+μ4,d⌢˙2=μ5,where ψ⌢1, ψ⌢2, d⌢1, d⌢1d, and d⌢2 denote the observed values of ψ1, ψ2, d1, d˙1, and d2, respectively, and μ1, μ2, μ3, μ4, and μ5 are the auxiliary correction terms.Equation (18) is subtracted from (29), then, the error dynamic equations of the observer can be obtained by(30)ψ~˙1=d~1-μ1,d~˙1=d˙1-μ2+d~1d,d~˙1d=d¨1-μ3,ψ~˙2=a1d~2-μ4,d~˙2=d˙2-μ5,where ψ~1, ψ~2, d~1, d~1d, and d~2 represent the observer errors of ψ1, ψ2, d1, d˙1, and d2, correspondingly.The detailed design procedures are given in the following.Step 1.
Selecting a sliding mode variables1=ψ~1 and taking derivative of s1 yields(31)s˙1=ψ~˙1=d~1-μ1.Apparently, if appropriate variableμ1 is designed such that ψ~˙1=0, then d~1 will be equal to μ1. Here, μ1 is constructed as(32)μ1=ξ1s1+ξ2s10.5sgn⁡s1+η1sgn⁡s1.Defining the Lyapunov functionV1=0.5s12, differentiating V1, and substituting (32) into it yield(33)V˙1=s1s˙1=-ξ1s12-ξ2s11.5-η1s1+s1d~1≤-η1s1+s1d~1=-2V10.5η1-d~1,where ξ1, ξ2, and η1 are all positive constants. Evidently, s1 will reach a sliding mode surface s1=0 in finite time t1 only if η1 meets condition η1>d~1.According to the equivalent control principle of the SMC,s1=s˙1=0 will hold once sliding mode surface s1=0 is reached, and s1=ψ~1 is considered. Therefore, d~1 will approach μ1 in finite time t1.Step 2.
Based on the acquiredd~1 in Step 1, a new sliding mode variable s2=d~1 is selected, and μ2 is designed as the following form:(34)μ2=ξ3s2+ξ4s20.5sgn⁡s2+η2sgn⁡s2,where ξ3,ξ4, and η2 are all positive constants.Ifη2 is selected to meet the condition η2>D1d+d~1d, then d~1 and s2 will converge to 0 in finite time t2 after undergoing finite time t1, thereby also indicating that d⌢1 converges to d1 at in finite time t2. Thus, d~1d will also approach μ2 in finite time t2 based on equivalent control principle.Step 3.
Continuing selecting sliding mode variables3=d~1d, then μ3 is constructed as(35)μ3=ξ5s3+ξ6s30.5sgn⁡s3+η3sgn⁡s3,where ξ5,ξ6, and η3 are all positive constants.Ifη3 is selected to satisfy η3>D1dd, then d~1d and s3 can converge to 0 in finite time t3 after undergoing finite time t1+t2, thereby indicating that d⌢1d converges to d˙1 in finite time t3.Step 4.
Two sliding variabless4=ψ~2 and s5=d~2 are selected; similarly, μ4 and μ5 are synthesized as(36)μ4=ξ7s4+ξ8s40.5sgn⁡s4+η4sgn⁡s4,μ5=ξ9s5+ξ10s50.5sgn⁡s5+η5sgn⁡s5,where ξ7,ξ8,ξ9,ξ10,η4, and η5 are all positive constants.Ifη5 and η6 are selected as η4>a1d~2,η5>D2d by conducting the same analysis as the foregoing steps, then ψ~2 and d~2 will successively converge to 0 in finite time t4 beginning at the initial moment t=0; that is, d⌢2 converges to d2 in the same time t4.Finally,d⌢1, d⌢1d, and d⌢2 are all estimated after undergoing a finite time tr, which satisfies tr=max⁡t1+t2+t3,t4.The schematic of the DSMO is illustrated in Figure 3. The diagrams of d⌢1 and d⌢1d are only provided for the limitation of space.Figure 3
Schematic of the DSMO.Remark 3.
The proposed DSMO first observes the observation error of disturbance instead of disturbance itself, unlike the traditional sliding mode observer [37]. The smaller switch gains ηi,i=1,2,3,4, can be permitted because the observation error is typically less than the upper bound of disturbance, which is beneficial for suppression of chattering.Remark 4.
In the proposed DSMO, the estimation process includes two sequential stages. First, disturbance observation errors were obtained. Second, the finite time convergence ofd~1, d~1d, and d~2 is ensured. The extra-low-pass filter generally used in the traditional sliding mode observer is not required because the dynamic equations of d~1, d~1d, and d~2 possess filtering effects, which can be exploited to smooth chattering.Remark 5.
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 gainsηi,i=1,2,3,4, are only required to be greater than the observation error or the upper bound of disturbance, thereby contributing to chattering restraint to some extent.Remark 6.
The existing nonlinear [38] and extended disturbance observers [39] can achieve the asymptotic convergence of the observation error only if the derivative of disturbance is equal to 0, whereas the bounded estimation result is only obtained when the derivative of disturbance is bounded. By contrast, the presented DSMO can realize the finite time estimation of disturbance and its derivative.


## 4. Design of a New Sliding Mode Controller (NSMC)

Based on the DSMO proposed in Section3, a NSMC including a position and a d-axis current controller will be developed in this section.


### 4.1. Position Sliding Mode Controller

The sliding mode variableσ is reselected as follows:(37)σ=e3+c2e2+c1e1+c2d⌢1+d⌢1d+a1d⌢2.Control lawuq is designed as(38)uq=-a1a2b1-1a2a6ψ2+a7ψ3-x⃛d+c2e3+c1e2+c2d⌢2+c1d⌢1+c2d⌢1d+ηsgn⁡σ.Here, the following result is obtained.Theorem 7.
If the sliding mode surface in (37) and control law in (38) are adopted, then sliding mode variable σ can reach the sliding mode surface σ=0 in finite time; afterward, position tracking error e1 will asymptotically converge to 0.Proof.
First, the reachability ofσ will be proven. Taking the derivative of σ along with (20) and substituting (38) into it yield(39)σ˙=e˙3+c2e˙2+c1e˙1+c2d⌢˙1+d⌢˙1d+a1d⌢˙2=c2d~2+c1d~1+c2d⌢1d-ηsgn⁡σ+c2μ2+d⌢1d+μ3+a1μ5=c2d~2+c1d~1-ηsgn⁡σ+c2μ2+ξ5d~1d+ξ6d~1d0.5sgn⁡d~1d+η3sgn⁡d~1d+a1ξ7d~2+a1ξ8d~20.5sgn⁡d~2+a1η5sgn⁡d~2=-ηsgn⁡σ+c2d~2+c1d~1+c2μ2+ξ5d~1d+ξ6d~1d0.5sgn⁡d~1d+a1ξ7d~2+a1ξ8d~20.5sgn⁡d~2+η3sgn⁡d~1d+a1η5sgn⁡d~2.
Thus, the Lyapunov function is defined asV=0.5σ2, and then, its derivative is obtained as(40)V˙=σσ˙=-ησ+σc2d~2+c1d~1+c2d~1d+a1ξ7d~2+ξ5d~1d+ση3sgn⁡d~1d+a1η5sgn⁡d~2+σξ6d~1d0.5sgn⁡d~1d+a1ξ8d~20.5sgn⁡d~2≤-ση-c2d~2-c1d~1-c2d~1d-a1ξ7d~2-ξ5d~1d-η3-a1η5-ξ6d~1d0.5-a1ξ8d~20.5.
Note thatd~1, d~2, and d~1d will successively converge to zero in finite time tr, which satisfies tr=max⁡t1+t2+t3,t4, then(41)V˙≤-ησ+ση3+a1η5=-ση-η3+a1η5=-η-η3+a1η5V0.5.
Evidently,σ will converge to 0 in finite time if η is selected such that η>η3+a1η5.
Next, it is proved thate1 will asymptotically converge to 0, and the dynamic performance of e1 depends on σ=0 after σ reached the sliding mode surface σ=0; that is,(42)e3+c2e2+c1e1+c2d⌢1+d⌢1d+a1d⌢2=e¨1+c2e˙1+c1e-c2d1-d˙1-a1d2+c2d⌢1+d⌢1d+a1d⌢2=0.
The simplification of (42) can be expressed as(43)e¨1+c2e˙1+c1e=c2d~1+d~1d+a1d~2.
According to the results presented in Section3, d~1,d~2, and d~1d will all converge to 0 in finite time and consider that c1, c2 are Hurwitz; obviously, the position tracking error e1 is asymptotic stability.Remark 8.
The mismatched disturbances that are transformed into matched ones are not adopted in this work to handle mismatched disturbancesd1 and d2, because doing that will result in undesired consequences, where the velocity x˙ and the acceleration x¨ will be required when constructing sliding mode surface. Thus, the extra state observer must be designed, whereas to the proposed NSMC that is expressed in (18), (20), and (37), all state variables (i.e., x,ωm,iq) are measurable. Therefore, the state observer will be avoided to use.Remark 9.
In the proposed NSMC, mechanical, hydraulic subsystems and IEHP are considered a synthesis in which control lawuq can be directly acquired, which is highly distinct from the existing SMC design methods of the EHSAS. In addition, the estimated disturbances are also added to uq to alleviate chattering in a feed forward way in addition to compensating the mismatched disturbances.Remark 10.
Althoughe1 displays asymptotic convergence, e2 and e3 cannot converge to 0, thereby satisfying the following relationships:(44)limt→∞⁡e2=-d1limt→∞⁡e3=a1d2+d˙1.


### 4.2.d-Axis Current Sliding Mode Controller

Thed-axis current of the IEHP is assumed to adopt id∗=0 form of vector control. Thus, the sliding variable is defined as(45)sd=ed=id-id∗.Taking the derivative ofsd and substituting (1) into it yield(46)s˙d=pLqLdx4x5-RLdx6+udLd.The CRRL is selected; that is,s˙d=-ηdsgn⁡(sd). Thus, the following equation is easily obtained: (47)ud=-LdpLqLdx4x5-RLdx6+ηdsgn⁡sd,where ηd>0 is switch gain.


## 4.1. Position Sliding Mode Controller

The sliding mode variableσ is reselected as follows:(37)σ=e3+c2e2+c1e1+c2d⌢1+d⌢1d+a1d⌢2.Control lawuq is designed as(38)uq=-a1a2b1-1a2a6ψ2+a7ψ3-x⃛d+c2e3+c1e2+c2d⌢2+c1d⌢1+c2d⌢1d+ηsgn⁡σ.Here, the following result is obtained.Theorem 7.
If the sliding mode surface in (37) and control law in (38) are adopted, then sliding mode variable σ can reach the sliding mode surface σ=0 in finite time; afterward, position tracking error e1 will asymptotically converge to 0.Proof.
First, the reachability ofσ will be proven. Taking the derivative of σ along with (20) and substituting (38) into it yield(39)σ˙=e˙3+c2e˙2+c1e˙1+c2d⌢˙1+d⌢˙1d+a1d⌢˙2=c2d~2+c1d~1+c2d⌢1d-ηsgn⁡σ+c2μ2+d⌢1d+μ3+a1μ5=c2d~2+c1d~1-ηsgn⁡σ+c2μ2+ξ5d~1d+ξ6d~1d0.5sgn⁡d~1d+η3sgn⁡d~1d+a1ξ7d~2+a1ξ8d~20.5sgn⁡d~2+a1η5sgn⁡d~2=-ηsgn⁡σ+c2d~2+c1d~1+c2μ2+ξ5d~1d+ξ6d~1d0.5sgn⁡d~1d+a1ξ7d~2+a1ξ8d~20.5sgn⁡d~2+η3sgn⁡d~1d+a1η5sgn⁡d~2.
Thus, the Lyapunov function is defined asV=0.5σ2, and then, its derivative is obtained as(40)V˙=σσ˙=-ησ+σc2d~2+c1d~1+c2d~1d+a1ξ7d~2+ξ5d~1d+ση3sgn⁡d~1d+a1η5sgn⁡d~2+σξ6d~1d0.5sgn⁡d~1d+a1ξ8d~20.5sgn⁡d~2≤-ση-c2d~2-c1d~1-c2d~1d-a1ξ7d~2-ξ5d~1d-η3-a1η5-ξ6d~1d0.5-a1ξ8d~20.5.
Note thatd~1, d~2, and d~1d will successively converge to zero in finite time tr, which satisfies tr=max⁡t1+t2+t3,t4, then(41)V˙≤-ησ+ση3+a1η5=-ση-η3+a1η5=-η-η3+a1η5V0.5.
Evidently,σ will converge to 0 in finite time if η is selected such that η>η3+a1η5.
Next, it is proved thate1 will asymptotically converge to 0, and the dynamic performance of e1 depends on σ=0 after σ reached the sliding mode surface σ=0; that is,(42)e3+c2e2+c1e1+c2d⌢1+d⌢1d+a1d⌢2=e¨1+c2e˙1+c1e-c2d1-d˙1-a1d2+c2d⌢1+d⌢1d+a1d⌢2=0.
The simplification of (42) can be expressed as(43)e¨1+c2e˙1+c1e=c2d~1+d~1d+a1d~2.
According to the results presented in Section3, d~1,d~2, and d~1d will all converge to 0 in finite time and consider that c1, c2 are Hurwitz; obviously, the position tracking error e1 is asymptotic stability.Remark 8.
The mismatched disturbances that are transformed into matched ones are not adopted in this work to handle mismatched disturbancesd1 and d2, because doing that will result in undesired consequences, where the velocity x˙ and the acceleration x¨ will be required when constructing sliding mode surface. Thus, the extra state observer must be designed, whereas to the proposed NSMC that is expressed in (18), (20), and (37), all state variables (i.e., x,ωm,iq) are measurable. Therefore, the state observer will be avoided to use.Remark 9.
In the proposed NSMC, mechanical, hydraulic subsystems and IEHP are considered a synthesis in which control lawuq can be directly acquired, which is highly distinct from the existing SMC design methods of the EHSAS. In addition, the estimated disturbances are also added to uq to alleviate chattering in a feed forward way in addition to compensating the mismatched disturbances.Remark 10.
Althoughe1 displays asymptotic convergence, e2 and e3 cannot converge to 0, thereby satisfying the following relationships:(44)limt→∞⁡e2=-d1limt→∞⁡e3=a1d2+d˙1.


## 4.2.d-Axis Current Sliding Mode Controller

Thed-axis current of the IEHP is assumed to adopt id∗=0 form of vector control. Thus, the sliding variable is defined as(45)sd=ed=id-id∗.Taking the derivative ofsd and substituting (1) into it yield(46)s˙d=pLqLdx4x5-RLdx6+udLd.The CRRL is selected; that is,s˙d=-ηdsgn⁡(sd). Thus, the following equation is easily obtained: (47)ud=-LdpLqLdx4x5-RLdx6+ηdsgn⁡sd,where ηd>0 is switch gain.


## 5. NARL

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,x˙1=x2,x˙2=u, where u is the control input, and the sliding mode variable is designed as σ=x2+cx1. The following reaching laws are adopted.


### 5.1. CRRL

The CRRL can be denoted as follows:(48)σ˙=-ηsgn⁡⁡σ.Integrating (48) with respect to time, then the time ta1 of σ reaching the sliding mode surface can be obtained by(49)ta1=σ0η,where σ(0) is the initial value of σ at t=0.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 asTs. Then, (48) can discretize the following form through Euler method:(50)σk+1=σk-Tsηsgn⁡σk,where σ(k)=0+ denotes the limiting value of σ(k) approaching zeros from the σ>0 region and σ(k)=0- represents the limiting value of σ(k) approaching zeros from the σ<0 region. Thus, the resulting values of σ in the next sample period are Tsη and -Tsη, respectively, thereby implying that σ will eventually chatter in a range from -Tsη to Tsη, and this chattering bandwidth Δ1 can be calculated as(51)Δ1=2Tsη.In Figure4(a), the shape of the chattering region of the CRRL is a rectangle, which means that the state x1 cannot converge to 0. In addition, (49) indicates that the larger switch gain η must be chosen if the smaller ta1 wants to be achieved, and then the chattering bandwidth will be further enlarged. Therefore, the CRRL cannot solve the contradiction between reaching speed and chattering.Figure 4
Phase trajectories. (a)  CRRL. (b) TVRRL and IVRRL. (c) Proposed adaptive reaching law.
(a)
(b)
(c)


### 5.2. Traditional Variable Rate Reaching Law (TVRRL)

Another reaching law that is used frequently is TVRRL, which can be expressed as(52)σ˙=-ηx1sgn⁡σ.Similarly, the reaching timeta2 and the chattering bandwidth Δ2 of the TVRRL can be expressed as(53)ta2=σ0ηx1,(54)Δ2=2Tsηx1.In (53) and (54), ta2 and Δ2 are related to state x1 compared with the CRRL. The advantages of the TVRRL lie in the following: the reaching rate will increase if x1 is larger; moreover, x1 and Δ2 will all gradually approach 0 with the sliding mode motion proceeding after σ reached the sliding surface. However, the reaching rate will also be slowed with the decrease of x1, and the severe chattering may occur if x1 maintains a larger value during the initial stage of the sliding motion. Note that from Figure 4(b) the shape of the TVRRL chattering domain appears to be triangular.


### 5.3. Improved Variable Rate Reaching Law (IVRRL)

Recently, the IVRRL is developed in [40], which can be expressed as(55)σ˙=-ηx1β+1-βexp⁡-ασsgn⁡σ,where α≥1,0<β≤1.Similarly, the corresponding reaching timeta3 and chattering bandwidth Δ3 will be obtained as(56)ta3=1ηx1βσ0+1-βα1-exp⁡-ασ0,(57)Δ3=2Tsηx1.The IVRRL can dynamically adjust the switch gainη between η|x1| and ηx1/β compared with the TVRRL. The smaller η can be selected under the same reaching speed, and then the chattering is suppressed to some extent. However, the deficiencies in the TVRRL remain; thus, the reaching speed is affected by state x1 and the dramatic chattering in the incipient phase of the sliding mode. In Figure 4(c), the shape of the IVRRL chattering band is the same as that of the TVRRL.


### 5.4. NARL

NARL is proposed to solve the problems in the aforementioned reaching laws and can be designed by(58)σ˙=-ηβ+1-β+exp⁡x1/x1exp⁡⁡-ασsgn⁡σ=-fσ,x1sgn⁡σ,where α≥1,0<β≤1.The reaching timeta4 and the chattering bandwidth Δ4 can be formulated by(59)ta4=1ηβσ0+1-βα+exp⁡x1/x1α1-exp-ασ0,(60)Δ4=2ηTsx1x1+exp⁡x1.In (58), the switch gain f(σ,x1) will tend to η/β when σ is far from the sliding surface, and f(σ,x1) approaches ηx1/|x1|+expx1 when σ is near the sliding surface σ=0. Thus, the NARL can also realize an adaptive regulation in response to the distance from the sliding surface. In (59), because 1-exp-ασ0≤1 and 1-β/α+exp(|x1|)/|x1|/α≥0; thus, if α is selected as α≫1-β+exp(|x1|)/|x1|, then the following relationship can be deduced by(61)ta4<βσ0η<ta1.Compared with the CRRL,η/β is much greater than η, thereby indicating that the faster reaching speed will be available even when using the same η. The smaller η will be permitted from another perspective, with the same reaching rate, and thus the chattering will be more effectively attenuated. Compared with TVRRL and IVRRL, the reaching speed of the NARL is unaffected by state x1. Moreover, since 0≤|x1|/|x1|+exp(|x1|)≤1, the severe chattering problem during the initial sliding motion is avoided. In addition, the bandwidth of the NARL is also less than that of the CRRL, TVRRL, and IVRRL by observing (51), (54), (57), and (60). In Figure 4(d), the shape of the NARL chattering band is fusiform. In addition, since f(σ,x1)>0 in (58), the reachability condition is automatically satisfied.Thus, the CRRL in (38) and (47) is replaced by the NARL, and the new control laws can be formulated as(62)uq=-a1a2b1-1a2a6ψ2+a7ψ3-x⃛d+c2e3+c1e2+c2d⌢2+c1d⌢1+c2d⌢1d+fσ,e1sgn⁡σ,ud=-LdpLqLdx4x5-RLdx6+fsdsgn⁡sd,where (63)fσ,e1=-ηβ+1-β+exp⁡e1/e1exp⁡-ασ,fsd=-ηdβ+1-β+exp⁡sd/sdexp⁡-αsd.The whole position control block diagram of the EHSAS is illustrated in Figure5.Figure 5
Complete control block diagram of the EHSAS.


## 5.1. CRRL

The CRRL can be denoted as follows:(48)σ˙=-ηsgn⁡⁡σ.Integrating (48) with respect to time, then the time ta1 of σ reaching the sliding mode surface can be obtained by(49)ta1=σ0η,where σ(0) is the initial value of σ at t=0.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 asTs. Then, (48) can discretize the following form through Euler method:(50)σk+1=σk-Tsηsgn⁡σk,where σ(k)=0+ denotes the limiting value of σ(k) approaching zeros from the σ>0 region and σ(k)=0- represents the limiting value of σ(k) approaching zeros from the σ<0 region. Thus, the resulting values of σ in the next sample period are Tsη and -Tsη, respectively, thereby implying that σ will eventually chatter in a range from -Tsη to Tsη, and this chattering bandwidth Δ1 can be calculated as(51)Δ1=2Tsη.In Figure4(a), the shape of the chattering region of the CRRL is a rectangle, which means that the state x1 cannot converge to 0. In addition, (49) indicates that the larger switch gain η must be chosen if the smaller ta1 wants to be achieved, and then the chattering bandwidth will be further enlarged. Therefore, the CRRL cannot solve the contradiction between reaching speed and chattering.Figure 4
Phase trajectories. (a)  CRRL. (b) TVRRL and IVRRL. (c) Proposed adaptive reaching law.
(a)
(b)
(c)


## 5.2. Traditional Variable Rate Reaching Law (TVRRL)

Another reaching law that is used frequently is TVRRL, which can be expressed as(52)σ˙=-ηx1sgn⁡σ.Similarly, the reaching timeta2 and the chattering bandwidth Δ2 of the TVRRL can be expressed as(53)ta2=σ0ηx1,(54)Δ2=2Tsηx1.In (53) and (54), ta2 and Δ2 are related to state x1 compared with the CRRL. The advantages of the TVRRL lie in the following: the reaching rate will increase if x1 is larger; moreover, x1 and Δ2 will all gradually approach 0 with the sliding mode motion proceeding after σ reached the sliding surface. However, the reaching rate will also be slowed with the decrease of x1, and the severe chattering may occur if x1 maintains a larger value during the initial stage of the sliding motion. Note that from Figure 4(b) the shape of the TVRRL chattering domain appears to be triangular.


## 5.3. Improved Variable Rate Reaching Law (IVRRL)

Recently, the IVRRL is developed in [40], which can be expressed as(55)σ˙=-ηx1β+1-βexp⁡-ασsgn⁡σ,where α≥1,0<β≤1.Similarly, the corresponding reaching timeta3 and chattering bandwidth Δ3 will be obtained as(56)ta3=1ηx1βσ0+1-βα1-exp⁡-ασ0,(57)Δ3=2Tsηx1.The IVRRL can dynamically adjust the switch gainη between η|x1| and ηx1/β compared with the TVRRL. The smaller η can be selected under the same reaching speed, and then the chattering is suppressed to some extent. However, the deficiencies in the TVRRL remain; thus, the reaching speed is affected by state x1 and the dramatic chattering in the incipient phase of the sliding mode. In Figure 4(c), the shape of the IVRRL chattering band is the same as that of the TVRRL.


## 5.4. NARL

NARL is proposed to solve the problems in the aforementioned reaching laws and can be designed by(58)σ˙=-ηβ+1-β+exp⁡x1/x1exp⁡⁡-ασsgn⁡σ=-fσ,x1sgn⁡σ,where α≥1,0<β≤1.The reaching timeta4 and the chattering bandwidth Δ4 can be formulated by(59)ta4=1ηβσ0+1-βα+exp⁡x1/x1α1-exp-ασ0,(60)Δ4=2ηTsx1x1+exp⁡x1.In (58), the switch gain f(σ,x1) will tend to η/β when σ is far from the sliding surface, and f(σ,x1) approaches ηx1/|x1|+expx1 when σ is near the sliding surface σ=0. Thus, the NARL can also realize an adaptive regulation in response to the distance from the sliding surface. In (59), because 1-exp-ασ0≤1 and 1-β/α+exp(|x1|)/|x1|/α≥0; thus, if α is selected as α≫1-β+exp(|x1|)/|x1|, then the following relationship can be deduced by(61)ta4<βσ0η<ta1.Compared with the CRRL,η/β is much greater than η, thereby indicating that the faster reaching speed will be available even when using the same η. The smaller η will be permitted from another perspective, with the same reaching rate, and thus the chattering will be more effectively attenuated. Compared with TVRRL and IVRRL, the reaching speed of the NARL is unaffected by state x1. Moreover, since 0≤|x1|/|x1|+exp(|x1|)≤1, the severe chattering problem during the initial sliding motion is avoided. In addition, the bandwidth of the NARL is also less than that of the CRRL, TVRRL, and IVRRL by observing (51), (54), (57), and (60). In Figure 4(d), the shape of the NARL chattering band is fusiform. In addition, since f(σ,x1)>0 in (58), the reachability condition is automatically satisfied.Thus, the CRRL in (38) and (47) is replaced by the NARL, and the new control laws can be formulated as(62)uq=-a1a2b1-1a2a6ψ2+a7ψ3-x⃛d+c2e3+c1e2+c2d⌢2+c1d⌢1+c2d⌢1d+fσ,e1sgn⁡σ,ud=-LdpLqLdx4x5-RLdx6+fsdsgn⁡sd,where (63)fσ,e1=-ηβ+1-β+exp⁡e1/e1exp⁡-ασ,fsd=-ηdβ+1-β+exp⁡sd/sdexp⁡-αsd.The whole position control block diagram of the EHSAS is illustrated in Figure5.Figure 5
Complete control block diagram of the EHSAS.


## 6. Simulations and Analysis

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 Figure10. The proposed NSMC is compared with three-loop PID method, which is widely adopted in the control of the EHSAS. The related system parameters are listed in Table 1. In the simulation, using symbol SMC represents the method mentioned in Section 3, where the mismatched disturbances are uncompensated.Table 1
Simulation parameters.
Parameter
Value
ψ
f
0.12 Wb
J
5
e
-
4 kg⋅m2
D
3.97
e
-
7 m3/rad
B
m
6
e
-
4 Nm/rad/s
A
1.13
e
-
3 m2
V
t
4.0
e
-
4 m3
m
c
2.9 kg
m
L
5.0 kg
C
t
2.50
e
-
11 m3/s/pa
β
e
6.86
e
+
8 N/m2
B
c
1000 N/m/s
Maximum stroke
0.1 m
Maximum output force
20 kN
Maximum speed of the IEHP
1000 rad/sEach control loop of the three-loop PID all uses PI controller, in which the proportional and integral gains of position loop areKxp=10000,Kxi=20000; the speed parameters of the IEHP are set as Kvp=1.5,Kvi=2,Kqp=10,Kqi=200, and the parameters of the d-axis current loop are Kdp=25,Kdi=900. The related parameters of the NSMC are selected as c1=3600,c2=400,η=600,β=0.05,  α=10. The parameters of the DSMO are η1=0.01,ξ2=10,  η2=0.05,ξ3=50,ξ1=7,  ξ4=60,η3=0.5,  ξ5=30,ξ6=20,  η4=1,ξ7=50,  ξ8=30,η5=100,  ξ9=90000,ξ10=80000. For convenience, the three-loop PID method is denoted by PID in the sequel. Simulations are performed in the following cases.Case 1 (verify the effectiveness of the NARL).
First, the validity of the NARL is demonstrated by comparative simulations with reaching laws mentioned in Section5. For simplicity, a double integrator system is still selected as the analysis object, that is, x˙1=x2,x˙2=u, and state initial values are assumed as x1(0)=10 and x2(0)=10. Similarly, the sliding variable σ=x2+cx1 is selected, and the reaching laws introduced in Section 5 are used. Then, the resulting control law u can be obtained by CRRL:(64)u=-cx2-ηsgn⁡σ;TVRRL:(65)u=-cx2-ηx1sgn⁡σ;IVRRL:(66)u=-cx2--ηx1β+1-βexp⁡-ασsgn⁡σ;NARL:(67)u=-cx2--ηβ+1-β+exp⁡x1/x1exp⁡-ασsgn⁡s.The simulation results are presented in Figures6–9, in which (a) is the system state response curve, (b) displays the sliding variable σ convergence curves, (c) depicts the control output u, and (d) illustrates the phase trajectories. For the CRRL, the related parameters are set as η=50,c=1. In Figures 6(a) and 6(b), x1, x2, and σ are unable to approach 0, although they are convergent. Moreover, the severe chattering phenomenon occurred in σ and u, as demonstrated in Figures 6(b) and 6(c); particularly, the chattering amplitude of u has reached ±50. In Figure 6(d), the shape of the CRRL chattering band is rectangular. In terms of the TVRRL, the same parameters are selected as η=50,c=1. σ and u exhibit drastic chattering during the initial stage of the sliding mode, although the reaching speed is increased because of adding x1, and x1, x2, and σ can also be guaranteed to converge to 0, as depicted in Figures 7(b) and 7(c). The maximum amplitude chattering of u has increased to ±500, which is far greater than that of the CRRL. In Figure 7(d), the chattering band appears triangular.Figure 6
Performance of the CRRL. (a) State response. (b) Sliding mode surface convergence process. (c) Control output. (d) Phase trajectory.
(a)
(b)
(c)
(d)Figure 7
Performance of the TVRRL. (a) State response. (b) Sliding mode surface convergence process. (c) Control output. (d) Phase trajectory.
(a)
(b)
(c)
(d)Figure 8
Performance of the IVRRL. (a) State response. (b) Sliding mode surface convergence process. (c) Control output. (d) Phase trajectory.
(a)
(b)
(c)
(d)Figure 9
Performance of the NARL. (a) State response. (b) Sliding mode surface convergence process. (c) Control output. (d) Phase trajectory.
(a)
(b)
(c)
(d)Figure 10
Cosimulation model of the EHSAS. (a) Hydraulic and mechanical submodel in AMESim. (b) NSMC and SMO in MATLAB/Simulink.
(a)
(b)A smallerη in the IVRRL than in the TVRRL can be selected with the same reaching speed given the dynamic adjustment of switch gain. Thus, η is set as η=5, and other parameters are set as β=0.5,α=10. In Figures 8(b) and 8(c), the maximum amplitude chattering of u still reaches up to ±200 in the sliding mode initial stage, although chattering is suppressed given the decrease of η. Similarly, the shape of the IVRRL chattering band depicted in Figure 8(d) is an approximate triangle. The parameters of the NARL maintain the same values; that is, η=5,β=0.5,α=10. In Figures 9(b) and 9(c), the NARL achieves not only the same reaching speed by using the smaller η than the CRRL but also chattering problem in the TVRRL and IVRRL during the sliding mode initial phase is avoided. Control signal u and sliding variable σ are quite smooth throughout the sliding phase, and the chattering bandwidth of the NARL is significantly less than the CRRL, TVRRL, and IVRRL. Moreover, Figure 9(c) illustrates that the shape of the chattering band is fusiform, which is consistent with the analysis presented in Section 5.Case 2 (step signal tracking performance).
The desired position step signal is presented asxd=0.02 m at t=0 s; then, xd is changed to xd=0.08 m at t=4 s, and no external load is applied on the EHSAS. The position step response curves of the PID and NSMC are depicted in Figure 11(a). In this figure, regardless of whether small or large step reference signal, the PID generates approximately 15% overshoot, whereas the NSMC can achieve nonovershoot performance, and the settling time is obviously smaller than that of the PID.Figure 11
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 ofd1. (d) Estimation of d2. (e) Estimation of d˙1. (f) Control output.
(a)
(b)
(c)
(d)
(e)
(f)The position step reference is set toxd=0.08 m at t=0 to further verify the disturbance rejection capability of the NSMC, and the step-type external load force FL=1×103 N is suddenly exerted at t=4 s. In Figure 11(b), although PID can eventually compensate this step disturbance, the compensation speed is quite dilatory in which the maximum position difference between the desired and the actual values has reached 0.04 m, which accounts for approximately 50% of the desired value. By contrast, the maximum position difference is only 0.004 m, and disturbance is swiftly compensated when using the NSMC. Actually, PID compensates disturbance through integral action; hence, the lagged effect of compensation is inevitable. This effect may result in the larger overshoot in the case of no disturbance, although this effect can be ameliorated by increasing the value of position loop integral gain Kxi. The disturbance rejection of the NSMC is achieved by utilizing switch gain η and DSMO, which is similar to a feedforward compensation method; thus, the antidisturbance effect is necessarily superior to PID. In addition, Figure 11(b) indicates that the SMC has no compensation capability to mismatch disturbance. Moreover, the DSMO is similar to a patch such that the disturbance can be effectively compensated in the presence of disturbance but has no adverse effect on the performance in absence of disturbance. In Figures 11(c)-11(d), disturbances d1 and d2 and the derivative of d1 can be estimated exactly, and undesired chattering does not appear. The switch gain η of the CRRL under the condition of the same reaching speed is η=60000 and that of NARL is 600, it can be illustrated in Figure 11(f) that the control signal uq of NARL is very smooth, whereas that of the CRRL has large chattering.The step-type disturbance is replaced by a sinusoidal-type external load force, namely,FL=1×103sin⁡(4πt) N, to be closer to the actual situation. The corresponding results are illustrated in Figure 12; we can find that the NSMC still achieves a satisfying result in terms of disturbance compensation capability, estimation accuracy of disturbance, and suppression chattering. It is should be noted that the PID cannot compensate this sinusoidal disturbance; in fact, the integral action of PID can only handle slow- varying disturbance and can be useless to fast time- varying period disturbance. In addition, Figure 12(e) depicts the response result of the d-axis current, where the curve of the NSMC is stable throughout the response process, whereas PID displays several fluctuations.Figure 12
Position step response under sinusoidal disturbance. (a) Step response in the case of a sinusoidal disturbance. (b) Estimation ofd1. (c) Estimation of d2. (d) Estimation of d˙1. (e) id response curve. (f) Control output.
(a)
(b)
(c)
(d)
(e)
(f)Therefore, the disturbance rejection capability of the NSMC significantly outperforms PID and SMC; the proposed sliding mode design architecture is feasible and effective.Case 3 (sinusoidal signal tracking performance).
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 asxd=5×10-3sin(2πt) m at t=0 s and then changed to xd=5×10-3sin⁡(6πt) m at t=3 s; then, this signal is set to xd=5×10-3sin⁡(12πt) m at t=6 s, which is used to simulate low, intermediate, and high-frequency desired position signal, respectively. Moreover, no external load is exerted on the EHSAS. Figure 13 reveals that PID and NSMC achieve a satisfying tracking performance, and the tracking error remains smaller in the NMSC than in the PID. Furthermore, the position tracking curve of the PID appears as a flat-top phenomenon at the place of maximal position, which is caused by nonlinear friction. By contrast, the same problem is overcome in the NSMC to some extent, which is contributed to the estimation and compensation of the DSMO to disturbance d1. Next, more complicated situations are considered to further test the antidisturbance capability. A flexible load with stiffness coefficient Ks=6×105 N/m is added to simulate the internal perturbation; moreover, sinusoidal load force is exerted with an amplitude of 1000 N and frequency of 2 Hz to simulate external disturbance. In this case, Figure 14 demonstrates that the PID tracking performance is obviously deteriorated, but the NSMC still manifests a favorable control performance.Figure 13
Position tracking performance to a sinusoidal signal in the case of no load. (a)  Position tracking performance. (b) Tracking error.
(a)
(b)Figure 14
Position tracking performance to a sinusoidal signal in the case of a complicated disturbance. (a) Position tracking performance. (b) Tracking error.
(a)
(b)Case 4 (robustness to parameter variation).
In practice, the load massmL and the leakage coefficient Ct will vary with the change of load and working conditions, such as oil temperature, wear, and seal. Thus, mL and Ct are changed as mL=100 kg and Ct=2.5×10-12 m3/s/pa, respectively, to validate the robustness of the NSMC to parametric variation. The parameter η is increased as η=2000 to enhance the robustness of the NSMC, to be fair, the position loop proportional gain Kxp of PID is also increased to 30000, and the rest of the parameters remain unchanged. Figure 15 indicates that the PID nearly cannot track the given position signal and accompanies severe resonance oscillation phenomenon, with respect to the NSMC, the result remains acceptable despite the slight degradation of tracking accuracy. Therefore, the robustness of the NSMC to parametric variation is much better than PID. The reason for this phenomenon will be analyzed as follows.Figure 15
Position tracking performance to sinusoidal signal in the case of parameter variation.In (13), the transfer function between ωm and x can be deduced as(68)Xsωms=D/A•4βeA2/MVtss2+2βeCt/Vts+2βeA2/MVt,where ωn=4βeA2/MVt,ζ=βeCt/Vt•1/ωn=βeCt/VtMVt/2βeA2, ωn denotes the hydraulic natural frequency, and ζ represents the hydraulic damping ratio.The damping ratioζ equals 0.04 when mL and Ct  are mL=100 kg,Ct=2.5×10-12 m3/s/pa. It is well known that the resonance peak will be generated when the damping ratio ζ is less than 0.707; thus, the large resonance peak will certainly occur. In the formulas of ωn and ζ, ωn and ζ will be decreased with the increase of mL and decrease of Ct, which eventually lead to the significant reduction of magnitude margin (GM). For the PID, the GM will be further decreased with the increase of Kxp, which inevitably results in system oscillation and even instability. For the NSMC, because d1 includes the information of acceleration x¨, which is able to increase the hydraulic damping ratio, oscillation is restrained to some extent when d1 is added to the control law by estimating DSMO.


## 7. Conclusions

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 three-loop 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.



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*Source: 7159891-2018-01-17.xml*