Yn WANG, Shengrong GUO, Hongkng DONG

a School of Transportation Science and Engineering, Beihang University, Beijing 100083, China

b Nanjing Engineer Institute of Aircraft System Jincheng, AVIC, Nanjing 211100, China

KEYWORDS

Abstract The variable pump displacement and variable motor speed electro-hydrostatic actuator(EHA), one of the three types of EHAs,has advantages such as short response time, flexible speed regulation, and high efficiency. However, the nonlinearity of its double-input single-output system poses a great challenge for system control. This study proposes a novel EHA with adaptive pump displacement and variable motor speed (EHA-APVM). A closed-loop position is realized using a servomotor. Moreover, the displacement varies with the system pressure; thus, the EHA-APVM is a single-input and single-output system. Firstly, the working principles of the EHA-APVM and the pump used in the system are introduced. Secondly, a nonlinear mathematical model of the proposed EHA-APVM control system is established, and a feedback back-stepping (FBBS)control algorithm is introduced to transform the complex nonlinear system into a linear system on the basis of the back-stepping control theory. Finally, simulation results prove that the EHAAPVM has a quick response and high robustness to variations of the load and the pump displacement.In this work,the size and weight of the motor are significantly reduced because the maximum power requirement is reduced,which is very beneficial for using the actuator in airborne equipment.?2018 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

One of the key subsystems in an electrical aircraft is the electro-hydrostatic actuator (EHA).1In general, there are three types of EHA – variable pump displacement and fixed motor speed (EHA-VPFM), fixed pump displacement and variable motor speed (EHA-FPVM),2–7and variable pump displacement and variable motor speed (EHA-VPVM).8,9As suggested by Fu et al.,10,11the EHA-FPVM has a simpler structure and higher efficiency than those of the EHAVPFM. The EHA-VPFM has a faster dynamic response than that of the EHA-FPVM; however, the efficiency of the EHAVPFM is lower. In an EHA-VPVM system, the displacement and rotation speed of the pump can be adjusted simultaneously; therefore, this system combines the advantages of the other two types of EHA systems. However, the nonlinearity of this type of double-input (control voltage of the motorUmand control voltage of the variable pumpUp)single-output (displacement of the actuatorxa) system poses a great challenge for the control of any EHA-VPVM system.

Therefore, to overcome the shortcomings of the traditional EHA-VPVM,this study proposed a novel EHA with adaptive pump displacement and variable motor speed (EHA-APVM).The closed-loop position was realized using a servomotor,and the displacement of the pump was adaptively varied with the system pressure (mechanical hydraulic control). The proposed EHA-APVM retains the advantages of the EHA-VPVM and simplifies the control system’s structure as it is a single-input(control voltage of the motorUm)single-output (displacement of the actuatorxa) system. More importantly, the EHAAPVM can operate at high speeds and light loads, which contributes to prolonging the motor’s lifetime. The EHA-APVM is a high-order nonlinear system.

Feedback linearization is a nonlinear approximation approach with dynamic characteristics. The approach does not ignore any high-order terms and can accurately provide a system model. Due to the aforementioned advantages, feedback linearization has been widely used in nonlinear systems.12–16Zhang et al.17realized the effective control of an EHA by using a bi-fuzzy sliding mode control involving feedback linearization. Yu et al.18presented a feedback linearization control algorithm that applied the Lyapunov function to a nonlinear electro-hydraulic servo system. Toulabi et al.19adopted a self-adaption strategy to realize the nonlinear control of a wind turbine by using feedback linearization.

The back-stepping control guarantees global consistency of a system and transforms a high-order complex nonlinear system into a lower-order system. Subsequently, an appropriate Lyapunov function was selected by a recursive method, and the final control rate was derived stepwise.Therefore,an effective control on the system was realized.20–24Li et al.25presented an adaptive second-order sliding mode control algorithm for a hydraulic arm on the basis of the backstepping control method. Zhou et al.26presented an adaptive back-stepping sliding-mode control algorithm that was used for an electro-hydraulic proportional propulsion system.

As previously mentioned, the proposed high-order EHAAPVM is nonlinear.Therefore,this study proposed a feedback back-stepping (FBBS) control algorithm on the basis of feedback linearization and back-stepping control. Simulation results suggest that the EHA-APVM has a quick response and high robustness. Moreover, the proposed EHA-APVM is compact because it has a relatively low maximum power requirement.

Fig.1 Schematic of the EHA-APVM.

2. Principle of EHA-APVM

The schematic of the EHA-APVM is shown in Fig.1. The EHA-APVM consists of a servomotor, a variable pump, a refeeding circuit, a bypass valve, a safety valve, and an actuator.The system utilizes an adaptive displacement pump to realize a high speed under a light load or a low speed under a heavy load, which are very common requirements for EHAs.As shown in Fig.2, the pump does not have external control signals. Moreover, the pump’s displacement can be adjusted by changing the angle of the swash plate through a control piston, which is driven by the outlet pressure oil. To balance the force on the swash plate,a preloaded spring is mounted on the other side of the swash plate to make the pump work with the maximum displacement under a setting pressure. The relation between the displacement and the pressure is shown in Fig.3.

As shown in Fig.3, the displacement can be expressed as follows:

wherepis the pressure at the working port, whilekp=（qmax-qmin）/（pmax-pmin） and mainly depends on the stiffness of the preload spring.

Fig.2 Adaptive displacement pump.

Fig.3 Displacement-pressure curve of the variable pump.

The size and weight of the EHA-APVM are notably lower than usual,because the maximum motor power requirement is considerably lower than that of a traditional EHA; the proposed device retains the ability to deliver a high speed under a light load and a low speed under a heavy load.

3. Modeling and linearization

3.1. Nonlinear modeling

(1) Motor model. The voltage-equilibrium equation of a brushless DC motor (BLDCM) can be expressed as follows:

whereUmis the armature voltage of the servomotor,Imis the armature current,Rmis the winding resistance,Lmis the winding inductance, andEmis the counter electromotive force.

(2) The counter electromotive force is obtained from the following equation:

whereCfmis the counter electromotive force coefficient and ω is the speed of the servomotor.

(3) The electromagnetism torque of a motor can be calculated from the following equation:

whereTmis the electromagnetism torque andCtmis the electromagnetic torque coefficient.

(4) The torque equilibrium equation is

whereTldis the load toque of the motor,Cdmis the total damping coefficient of the motor and the pump, andJmis the total inertia moment.

(5)The output flow of the variable pump can be calculated as follows:

whereQis the output flow of the pump,Ctpis the total leakage coefficient, andpis the pressure at the working port.

(6) Continuity equation. The actuator applied in the EHA-APVM is a symmetrical hydraulic cylinder,and the flow continuity equation can be described as follows:

whereAis the effective operation area of the piston,xais the rod displacement, β is the bulk modulus of oil, andVis the volume of the high-pressure chamber which equals to half of the sum volume of the actuator chamber and the pipe. Moreover,Ctcyl=Cicyl+Cecyl,whereCtcylis the total leakage coefficient,Cicylis the internal leakage coefficient, andCecylis the external leakage coefficient.

(7) In general, the force equilibrium equation can be expressed as follows:

whereCvdis the viscous damping coefficient,kldis the load spring stiffness and has a value of zero here because there is no spring load in the EHA-APVM system,Flis the external disturbance force(load),andmtis the total inertia mass acting on the piston.

By combining Eqs. (2)–(8), we get the following equations:

whereCt=Ctp+Ctcyl.

Eq. (1) demonstrates that when the system pressurep＜pminandp＞pmax, the displacements areq=qmaxandq=qmin, respectively. At these two stages, the variable pump in the EHA-APVM can be regarded as a fixed pump, thus causing the EHA-APVM to be a single-input single-output system.Refer to Ref. 2 for the modeling process of the EHA-FPVM.

wherek1=Rm/Lm,k2=Cfm/Lm,a1=1/Lm,k3=Cdm/Jm,k4=1/Jm,k5=Ctm/Jm,k6=Ctβ/V,k7=Aβ/V,k8=β/V,k9=Cvd/mt,k10=A/mt,F=Fl/mt,k11=k4(qmax+kppmin),k12=k4kp,k13=k8(qmax+kppmin), andk14=k4kp.

The system output is given by the following equation:

Whenpmin≤p≤pmax,the pump displacement is a function of the system pressure.Therefore,the EHA-APVM is a singleinput (Um) single-output (x4) system and contains a multiplying nonlinear system.

3.2. Feedback linearization

Eq. (10) can be modified as

where

f（x）=[f1（x），f2（x），f3（x），f4（x），f5（x）]T

g（x）=[a1， 0， 0， 0， 0]T

f1（x）=-k1x1-k2x2

f2（x）=-k3x2-k11x3+k12x23+k5x1

f3（x）=-k6x3-k14x2x3-k7x5+k13x2

f4（x）=x5

f5（x）=-k9x5+k10x3-F

x=[x1，x2，x3，x4，x5]T

andUmis the system input.

The system output is as follows:

wherex=[x1，x2，x3，x4，x5]Tare system states,andudis the input.

The feedback linearization equation is given as follows27:

The linearization form of Eq.(13)can be written as follows:

whereF（x） andG（x） are functions ofx.

As can be seen in Eq.(15),the input-output relationship of the EHA-APVM is transformed to a linear relationship, and the outputymeets the demand by controllingUm.

The virtual control variablevcan be defined as

Then Eq. (15) can be transformed into

4. Back-stepping control

The basic concept of back-stepping is to decompose a highorder complex nonlinear system into lower-order subsystems.Subsequently,the Lyapunov function and intermediate virtual control variable for every subsystem must be obtained until the control rate for the entire system is achieved.

By assuming thatv1=y,v2=y（1）,v3=y（2）,v4=y（3）,v5=y（4）, andv6=y（5）, Eq. (17) can be written as follows:

The system output is

The steps for designing a back-stepping controller are as follows.

whereyeis the expected speed of the hydraulic cylinder.

Then, the following equation is obtained:

The controlled variable α1can be defined as

wherec1＞0.

Similar toz1,z2can be defined as

The Lyapunov functionV1can be defined as

Then, the following equation can be obtained:

By substituting α1into Eq.(25),the following equation can be easily obtained:

Step 2:The Lyapunov functionV2can be defined as

Then, the following equation can be obtained:

The virtual control variable α2can be defined as

wherec2＞0.

The tracking errorz3can be defined as

By combining Eqs. (28) and (30), the following equation can be easily obtained:

Step 3:The Lyapunov functionV3can be defined as follows:

The virtual control variable α3can be defined as below:

wherec3＞0.

The tracking errorz4can be defined as follows:

By substitutingz4into Eq.(32),the following equation can be obtained:

Step 4:The Lyapunov functionV4can be defined as

Then, the following equation can be obtained:

The controlled variable α4can be defined as

wherec4＞0.

The tracking errorz5can be defined as

By substituting Eq.(39)into Eq.(37),V4can be modified as

Step 5:The Lyapunov functionV5can be defined as

Then, the following equation can be obtained:

Asv6=v, Eq. (42) is rewritten as follows:

The control rate can be expressed as follows:

wherec5＞0.

Then, the following equation can be obtained:

If the control rate is correctly designed,the system can meet the Lyapunov stability requirements. The aforementioned tracking errorsz1,z2,z3,z4, andz5become stable exponentially. This guarantees the stability of the entire system exponentially in the global perspective.

The control law of the servomotor can be obtained using Eqs. (16) and (44) as

After the control law has been derived, a simulation analysis will be conducted, as discussed in the next section.

5. Simulation analysis

A summary of the EHA-APVM parameters is presented in Table 1.

Table 1 Simulation parameters.9,28

5.1. Response and robustness

To verify the performance of the EHA-APVM, the command input of the cylinder position is set to be 0.1 m. The pressure and displacement curves are shown in Fig.4. Moreover, the step response of the EHA-APVM can be obtained from Fig.5.

Fig.4 Pressure and displacement of the EHA-APVM.

Fig.5 Step response of the EHA-APVM.

As can be seen from Fig.4, the pressure ramps from zero to the maximum pressure. Moreover, the displacement correspondingly varies from 1.6 to 0.4 mL/r. Thus, the pump displacement changes automatically with the pressure.

Fig.5 indicates that the setting time of the EHA-APVM control system is 0.28 s, and the overshoot δ=0.5%. Significantly, after 0.28 s, the position remains stable even when the load and the displacement vary, which demonstrates that the system is highly robust.

5.2. Comparison with a fixed pump

The performance of the EHA-APVM is verified with an adaptive displacement pump and a fixed displacement pump that has a displacement ofqmax=1.5878 mL/r andqmin=0.4 mL/r. A comparison analysis is conducted on the dynamic performance and maximum power requirement.

5.2.1. Dynamic performance

Fig.6 presents the step responses of the three EHAs.Clearly,the EHA-APVM (setting timet2=0.283 s, overshoot δ2=0.5%,and steady-state error σ2=0.1%) has a dynamic performance similar to that of the EHA with a fixed displacementqmax(setting timet1=0.279 s,overshoot δ1=0,and steady-state error σ1=0.2%).However,the setting time of the EHA with a fixed displacementqmin(setting timet3=1.245 s, overshoot δ3=0,and steady-state error σ3=0.1%) is significantly higher than those of the other two EHAs.

5.2.2. Maximum power requirement

Fig.6 Step responses of three types of EHAs.

Fig.7 Maximum torque requirement comparison.

Fig.7 displays the maximum torque requirements of the three types of EHAs.Note that the torques of the fixed displacement pump show linear upward trends as the pressure changes,and the curve of the EHA-APVM is parabolic.The maximum torque of the EHA-FPVM withqmaxis 33.3 N·m when the pressure reaches 21 MPa, which is much higher than those of the EHA-APVM and the EHA-FPVM withqmin(both are 8.4 N·m) under the same condition. The maximum torque of the EHA-APVM occurs when the pressure is 12 MPa, and the value is 14 N·m.

The output power of the motor can be calculated by the following equation:

whereTis the torque and ω is the speed of the servomotor.

By assuming that the speeds are the same, the following equation can be obtained from Eq. (47):

wherePEHA-APVMrepresents the maximum output power of the EHA-APVM andPmaxrepresents that of the EHA-FPVM withqmax.

In summary, the simulation results demonstrate that the EHA-APVM has a quick response and high robustness to variations in the load and pump displacement.The size and weight of the motor can be minimized by reducing the maximum power requirement.

6. Conclusions

In this study, to overcome the shortcomings of traditional EHA-VPVM control systems, a novel EHA, named EHAAPVM, is introduced. Furthermore, an FBBS control algorithm is adopted to transform the complex nonlinear system into a linear system.After analyzing simulation results,conclusions can be summarized as follows:

(1) To change the displacement of the variable pump,additional control loops and actuators are widely used in traditional EHA-VPVM systems. For example, a servomotor is utilized for varying the swash plate angle.In contrast, the EHA-APVM adopts a simpler adaptive displacement regulating mechanism without any external control signals. Therefore, the EHA-APVM has higher reliability than that of traditional EHA-VPVM systems.

(2) By the FBBS control algorithm, the EHA-APVM control system realizes a quick response and high robustness even when the pump displacement and external load vary over large ranges.

(3) The maximum power requirement of the EHA-APVM can be significantly lower than that of the EHAFPVM.This enables the EHA motor to have a compact size and a light weight.

(4) Further study should be conducted on the displacementpressure relation of the EHA-APVM on the basis of performance requirements.

Acknowledgments

This study was financially supported by the National Natural Science Foundation of China (No’s. 51375029 and 51775013).