applied sciences

Article

Research on Control Methods for the PressureContinuous Regulation Electrohydraulic ProportionalAxial Piston Pump of an Aircraft Hydraulic System

Peng Zhang * and Yunhua Li *

School of Automation Science and Electric Engineering, Beihang University, Beijing 100191, China* Correspondence: zpbuaa1985@163.com (P.Z.); yhli@buaa.edu.cn (Y.L.)

Received: 12 February 2019; Accepted: 18 March 2019; Published: 1 April 2019�����������������

Featured Application: A structural scheme is proposed to match the pump delivery pressure andthe aircraft load. PID, LQR and backstepping sliding control method are used.

Abstract: The objective of this paper is to design a pump that can match its delivery pressure tothe aircraft load. Axial piston pumps used in airborne hydraulic systems are required to work in aconstant pressure mode setting based on the highest pressure required by the aircraft load. However,the time using the highest pressure working mode is very short, which leads to a lot of overflow lose.This study is motivated by this fact. Pressure continuous regulation electrohydraulic proportionalaxial piston pump is realized by combining a dual-pressure piston pump with electro-hydraulicproportional technology, realizing the match between the delivery pressure of the pump and theaircraft load. The mathematical model is established and its dynamic characteristics are analyzed.The control methods such as a proportional integral derivative (PID) control method, linear quadraticregulator (LQR) based on a feedback linearization method and a backstepping sliding control methodare designed for this nonlinear system. It can be seen from the result of simulation experimentsthat the requirements of pressure control with a pump are reached and the capacity of resistingdisturbance of the system is strong.

Keywords: aviation axial piston pump; constant pressure variable displacement; electro-hydraulicproportional control; LQR; backstepping sliding control

1. Introduction

The delivery pressure of the axial piston pump used in an airborne hydraulic system shownin Figure 1 must be set according to the highest pressure required by the aircraft load. Most of thehydraulic power is consumed in the form of overflow loss, causing low pressure to account for most ofthe flight time. The main form of overflow loss is the generation of a large amount of heat, which leadsto an increase in temperature of the hydraulic system. This situation accelerates the progress of agingfor the hydraulic oil and sealing rubber. The statistic data indicate that service life of mineral oil willbe reduced by 90% when the fluid temperature is raised by 15 ◦C [1,2].

Appl. Sci. 2019, 9, 1376; doi:10.3390/app9071376 www.mdpi.com/journal/applsci

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(a) (b)

Figure 1. (a) Aircraft engine driven pump; (b) Aircraft engine motor driven pump.

The dual-pressure variable pump of 21–35 MPa (3000–5000 psi) has been used in the F/A-18E/F by the United States [4]. Related research on dual pressure pumps can be found in many references. The dynamic mathematical model of the pressure regulation mechanism has been established, and numerical simulation research has been carried out. Methods such as setting the orifice and the effective volume of the pump outlet are used to reduce pressure overshoot [5,6].

But the matching problem between the delivery pressure of the pump and the aircraft load is only partially solved. If the requirement of aircraft load is slightly above the low-pressure mode and far below the high-pressure mode, the pump needs to operate under a high-pressure mode and the power loss is still considerable.

An intelligent variable pressure pump is another method to solve the problem. In the 1980s, engineers started to conduct research on intelligent variable pressure pumps. Because the intelligent hydraulic power supply system can provide fluid according to flight profile requirements, the power loss can be decreased. Abex corporation did some experiments on the F-15 “Iron Bird” [7]. The result indicated that the intelligent pump decreases the consumed power by 39% and the discharge fluid temperature decreases between 18 °C and 24 °C [8,9]. Researchers have done a lot of research this. The intelligent pump is optimized by taking the system work efficiency as the objective function and the performance reliability index as the constraint condition. Compared to conventional constant pressure pump systems, the average efficiency of the intelligent pump system increased by 15% [10]. Some control methods such as sliding mode variable structure control, adaptive control strategy, proportional integral derivative (PID) control strategy, etc. are used for precise control of swash plate inclination and outlet pressure of intelligent pumps [11–13]. But the structure of the intelligent variable pressure pump system is complex, and sensors are needed to detect the signals of pressure, including temperature, swashplate inclination, etc.

By introducing electro-hydraulic proportional technology to the existing dual-pressure piston pump, the delivery pressure of pump can be adjusted by the proportional solenoid input voltage. The structure is relatively simple, and the original dual pressure regulation function is retained.

The same as for the ordinary constant pressure variable pump, a lot of research works have been done on the dynamic characteristics of the pressure continuous regulation electrohydraulic proportional axial piston pump. Characteristics of swashplate control analysis with pressure compensation valve [14], design of a swash plate control mechanism [15], high frequency vibration of the swash plate near the equilibrium state [16], and the overall dynamic characteristics of the variable pump [17] are included in this research.

Many scholars have studied pressure control of the hydraulic pump, Kemmetmuller et al. [18] studied the control problem of variable load and unknown load pressure, and a two-order nonlinear controller including feed forward and feedback control was designed. The stability of the pump control system was proved based on the Lyapunov theory, and the feasibility of the controller was verified by experiments. A mathematical model of an open-circuit pump was established by Shu Wang [19,20], and the controller was established based on pressure compensating, load-sensitive control and torque control respectively. Experiments were carried out to verify the effectiveness of the controllers. Koivumaki [21] proposed for the first time not using any linearization or order reduction, and an adaptive and model-based discharge pressure control design for the variable displacement axial piston pumps (VDAPPs), whose dynamical behaviors are highly nonlinear and

Figure 1. (a) Aircraft engine driven pump; (b) Aircraft engine motor driven pump.

The variable pressure airborne hydraulic system is an effective method to solve this problem.At present, there are two aircraft hydraulic system pump source forms being proposed: dual pressurevariable pump and intelligent variable pressure pump.

A dual pressure variable pump (Dual Pressure Pump) airborne system has been put forwardby the United States, Britain and other countries. During flight, the dual pressure pump operates ina high-pressure mode when required, and a low-pressure mode is used during the rest of the time.Compared to the constant pressure variable pump, energy efficiency has been greatly improved [3].

The dual-pressure variable pump of 21–35 MPa (3000–5000 psi) has been used in the F/A-18E/Fby the United States [4]. Related research on dual pressure pumps can be found in many references.The dynamic mathematical model of the pressure regulation mechanism has been established, andnumerical simulation research has been carried out. Methods such as setting the orifice and theeffective volume of the pump outlet are used to reduce pressure overshoot [5,6].

But the matching problem between the delivery pressure of the pump and the aircraft load is onlypartially solved. If the requirement of aircraft load is slightly above the low-pressure mode and farbelow the high-pressure mode, the pump needs to operate under a high-pressure mode and the powerloss is still considerable.

An intelligent variable pressure pump is another method to solve the problem. In the 1980s,engineers started to conduct research on intelligent variable pressure pumps. Because the intelligenthydraulic power supply system can provide fluid according to flight profile requirements, the powerloss can be decreased. Abex corporation did some experiments on the F-15 “Iron Bird” [7]. The resultindicated that the intelligent pump decreases the consumed power by 39% and the discharge fluidtemperature decreases between 18 ◦C and 24 ◦C [8,9]. Researchers have done a lot of research this.The intelligent pump is optimized by taking the system work efficiency as the objective function andthe performance reliability index as the constraint condition. Compared to conventional constantpressure pump systems, the average efficiency of the intelligent pump system increased by 15% [10].Some control methods such as sliding mode variable structure control, adaptive control strategy,proportional integral derivative (PID) control strategy, etc. are used for precise control of swash plateinclination and outlet pressure of intelligent pumps [11–13]. But the structure of the intelligent variablepressure pump system is complex, and sensors are needed to detect the signals of pressure, includingtemperature, swashplate inclination, etc.

By introducing electro-hydraulic proportional technology to the existing dual-pressure pistonpump, the delivery pressure of pump can be adjusted by the proportional solenoid input voltage.The structure is relatively simple, and the original dual pressure regulation function is retained.

The same as for the ordinary constant pressure variable pump, a lot of research works havebeen done on the dynamic characteristics of the pressure continuous regulation electrohydraulicproportional axial piston pump. Characteristics of swashplate control analysis with pressurecompensation valve [14], design of a swash plate control mechanism [15], high frequency vibration ofthe swash plate near the equilibrium state [16], and the overall dynamic characteristics of the variablepump [17] are included in this research.

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Many scholars have studied pressure control of the hydraulic pump, Kemmetmuller et al. [18]studied the control problem of variable load and unknown load pressure, and a two-order nonlinearcontroller including feed forward and feedback control was designed. The stability of the pumpcontrol system was proved based on the Lyapunov theory, and the feasibility of the controller wasverified by experiments. A mathematical model of an open-circuit pump was established by ShuWang [19,20], and the controller was established based on pressure compensating, load-sensitivecontrol and torque control respectively. Experiments were carried out to verify the effectiveness of thecontrollers. Koivumaki [21] proposed for the first time not using any linearization or order reduction,and an adaptive and model-based discharge pressure control design for the variable displacementaxial piston pumps (VDAPPs), whose dynamical behaviors are highly nonlinear and described by afourth-order differential equation. A nonlinear supply pressure controller for a variable displacementaxial piston pump was proposed in reference [22]. A load flow disturbance observer was proposedto deal with unknown and time-varying load flow. A nonlinear feed forward controller was derivedusing the differential flatness property of the system, and a feedback controller was implemented tostabilize the system. The adaptive control design to solve the trajectory tracking problem of a Deltarobot with uncertain dynamical model was described in reference [23]. The output-based adaptivecontrol was designed within the active disturbance rejection framework. New aspects of a recursivebackstepping design methodology from both a theoretical and application point of view were exploredin reference [24]. Three main topics such as control of multivariable nonlinear systems, steering controlof light passenger vehicles and coordinated steering and braking control of commercial heavy vehicleswere investigated from a backstepping perspective.

In this paper, the control methods of pressure continuous regulation electrohydraulic proportionalaviation axial piston pump(EHPAAPP) are considered to achieve the objectives of the followingaspects: the response time of the electro-hydraulic proportional variable pump, which is defined asnot exceeding 0.05 s during the process of pressure adjustment, the instantaneous peak pressure notexceeding 135% of the rated outlet pressure, the response time is not more than 0.05 s, and the stabilitytime does not exceed 1 s.

The paper is organized as follows: Section 2 discusses the feasibility of a pressure continuousregulation of EHPAAPP; Section 3 establishes its mathematical model and dynamic characteristics areanalyzed. The PID control method, linear quadratic regulator (LQR) based on feedback linearizationmethod and backstepping sliding control method are added to this nonlinear system in Section 4,and the contents of this paper are concluded in Section 5.

2. Program of Pressure Continuous Regulation for EHPAAPP

The schematic of pressure continuous regulation for EHPAAPP is shown in Figure 2a. 1 is atwo-position three-way reversing valve, realizing the switch between a high-pressure and low-pressuremode. 2 is the regulator valve spool of the dual pressure variable flow pump. 3 is the adjustmentspring of the pump. 5 is the swash plate adjustment mechanism. The putter of proportional solenoid 6is connected with 2. When the continuous pressure regulation function is needed, solenoid valve 1 andproportion electromagnet 6 are powered on, resulting in a maximum compression of 3 and a thrustcorresponding to the current. As the force acting on the spool by pump outlet pressure is approximatelyequal to the spring force minus the proportional thrust, the pump pressure is reduced as the thrust ofproportional solenoid increases. This program retains the dual pressure regulation function, that is,in the case of the electromagnetic coil of proportional electromagnet cannot be energized, so originaldual pressure regulation can be realized. Figure 2b is a schematic of a pressure continuous regulationelectrohydraulic proportional axial piston pump.

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described by a fourth-order differential equation. A nonlinear supply pressure controller for a variable displacement axial piston pump was proposed in reference [22]. A load flow disturbance observer was proposed to deal with unknown and time-varying load flow. A nonlinear feed forward controller was derived using the differential flatness property of the system, and a feedback controller was implemented to stabilize the system. The adaptive control design to solve the trajectory tracking problem of a Delta robot with uncertain dynamical model was described in reference [23]. The output-based adaptive control was designed within the active disturbance rejection framework. New aspects of a recursive backstepping design methodology from both a theoretical and application point of view were explored in reference [24]. Three main topics such as control of multivariable nonlinear systems, steering control of light passenger vehicles and coordinated steering and braking control of commercial heavy vehicles were investigated from a backstepping perspective.

In this paper, the control methods of pressure continuous regulation electrohydraulic proportional aviation axial piston pump(EHPAAPP) are considered to achieve the objectives of the following aspects: the response time of the electro-hydraulic proportional variable pump, which is defined as not exceeding 0.05 s during the process of pressure adjustment, the instantaneous peak pressure not exceeding 135% of the rated outlet pressure, the response time is not more than 0.05 s, and the stability time does not exceed 1 s.

The paper is organized as follows: Section 2 discusses the feasibility of a pressure continuous regulation of EHPAAPP; Section 3 establishes its mathematical model and dynamic characteristics are analyzed. The PID control method, linear quadratic regulator (LQR) based on feedback linearization method and backstepping sliding control method are added to this nonlinear system in Section 4, and the contents of this paper are concluded in Section 5.

2. Program of Pressure Continuous Regulation for EHPAAPP

The schematic of pressure continuous regulation for EHPAAPP is shown in Figure 2a. 1 is a two-position three-way reversing valve, realizing the switch between a high-pressure and low-pressure mode. 2 is the regulator valve spool of the dual pressure variable flow pump. 3 is the adjustment spring of the pump. 5 is the swash plate adjustment mechanism. The putter of proportional solenoid 6 is connected with 2. When the continuous pressure regulation function is needed, solenoid valve 1 and proportion electromagnet 6 are powered on, resulting in a maximum compression of 3 and a thrust corresponding to the current. As the force acting on the spool by pump outlet pressure is approximately equal to the spring force minus the proportional thrust, the pump pressure is reduced as the thrust of proportional solenoid increases. This program retains the dual pressure regulation function, that is, in the case of the electromagnetic coil of proportional electromagnet cannot be energized, so original dual pressure regulation can be realized. Figure 2b is a schematic of a pressure continuous regulation electrohydraulic proportional axial piston pump.

1

sp

2

3

4

5

6vx

0Fcp

bFα

proportional solenoid

Pump

swash plate adjustment mechanism

two-position three-way reversing valve

regulator valve spool

adjustment spring

(a) (b)

Figure 2. (a) Program of a pressure continuous regulation for EHPAAPP; (b) Schematic of a pressure continuous regulation for EHPAAPP.

Figure 2. (a) Program of a pressure continuous regulation for EHPAAPP; (b) Schematic of a pressurecontinuous regulation for EHPAAPP.

3. Characteristic Analysis of EHPAAPP

In this section, the mathematical model of the electro-hydraulic proportional variable pump isestablished and its dynamic characteristics are analyzed.

3.1. Mathematical Model of the Pump

3.1.1. Model of Electro Hydraulic Proportional Regulation

The terminal voltage equation of the proportional electromagnet control coil is

us = Lddidt

+ iRs + Kvdxv

dt, (1)

where Rs is internal resistance of coil and proportional control amplifier.According to Newton’s second law, only the quality of armature components are considered, and

following equation can be obtained

F0 = KFii(t− τd)− fMsgn(

didt

)+ Fr. (2)

3.1.2. Model of the Pump Section

Summing forces in the positive direction and setting them to be equal to the time-rate-of-changeof linear momentum of the regulator valve spool, the following equation may be generally written todescribe the motion of the regulator valve spool

Ps Av + F0 = Mvd2xv

dt2 + fdxv

dt+ Ks(xv + x0) + Fst. (3)

Based on assumptions that the fluid pressure within the control chamber is homogeneousthroughout, and that the fluid inertia and viscous effects within this chamber are negligible comparedto the hydrostatic pressure effects of the fluid, the fluid pressure within the control actuator is modeledwith the following equation

Vc

β

dpc

dt= Qc + AcL

dα

dt. (4)

The volumetric flow-rate into the control actuator can be expressed as

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Qc =

CdW(u− xv)

√2ρ Pc −xm <xv < −u

CdW(u + xv)√

2ρ (Ps − Pc)− CdW(u− xv)

√2ρ (Ps − Pc) −u ≤ xv ≤ u

CdW(u + xv)√

2ρ (Ps − Pc) u < xv < xm

. (5)

The equation of motion for the swash-plate may be written as

Id2α

dt2 = −cdα

dt+ FbL− FcL−

n

∑n=1

FnL. (6)

The equation of motion for the control actuator can be written as follows

Mcd2xc

dt2 = Fc cos α− Pc Ac, (7)

where the displacement of the control chamber xc can be written as

xc = L tan α, (8)

and it can be obtained that

Fc = McLd2α

dt2 + Pc Ac. (9)

The equation of motion for the nth piston as it moves in the x-direction may be written as

Mpd2xn

dt2 = Fn cos α− Pn Ap, (10)

where xn = r tan α sin θ.So the following formula is obtained

Fn = Mpr sin θd2α

dt2 + 2Mpr cos θωdα

dt−Mpr sin θω2α + Pn Ap. (11)

Substituting the above-mentioned force expression into Equation (6), the following intermediateresult is produced,[

I + McL2 + Mpr2 N2

]d2α

dt2 + cdα

dt−(

kbL2 +N2

Mpr2ω2)

α = (Fb − Pc Ac)− AprN

∑n=1

Pn sin θ. (12)

The pressure of each piston has been studied numerically and has been shown to be fairly constantwhile the pistons are located directly over the intake and discharge ports, and that the pressure changesalmost linearly as the pistons pass over the transition slots on the valve plate. Because this is socommon, the fluid pressure within the nth piston bore is usually described using the pressure profileshown in Figure 3. In this figure the average transition-angle on the valve plate is shown by thesymbol γ, which is normally called the pressure carryover-angle. Using Figure 3, the following discreterepresentation for the fluid pressure within the nth piston bore may be written

Pn =

Pd −π

2 + γ < θ < π2

Pd −(Pd−Pi)

γ (θ − π/2) π2 < θ < π

2 + γ

Piπ2 + γ < θ < 3π

2Pi +

(Pd−Pi)γ (θ − 3π/2) 3π

2 < θ < 3π2 + γ

. (13)

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tancx L α= , (8)

and it can be obtained that 2

2c c c cdF M L P Adt

α= + . (9)

The equation of motion for the nth piston as it moves in the x-direction may be written as 2

2 cosnp n n pd xM F P Adt

α= − , (10)

where tan sinnx r α θ= . So the following formula is obtained

22

2sin 2 cos sinn p p p n pd dF M r M r M r P A

dtdtα αθ θω θω α= + − + . (11)

Substituting the above-mentioned force expression into Equation (6), the following intermediate result is produced,

( )2

2 2 2 2 22

1sin

2 2

N

c p b p b c c p nn

N d d NI M L M r c k L M r F PA A r Pdtdt

α α ω α θ=

+ + + − + = − − . (12)

The pressure of each piston has been studied numerically and has been shown to be fairly constant while the pistons are located directly over the intake and discharge ports, and that the pressure changes almost linearly as the pistons pass over the transition slots on the valve plate. Because this is so common, the fluid pressure within the nth piston bore is usually described using the pressure profile shown in Figure 3. In this figure the average transition-angle on the valve plate is shown by the symbol γ, which is normally called the pressure carryover-angle. Using Figure 3, the following discrete representation for the fluid pressure within the nth piston bore may be written

( ) ( )

( ) ( )

2 2

2 2 2

3

2 23 33 2 2 2

d

d id

n

i

d ii

P

P PP

PP

P PP

π πγ θ

π πθ π θ γγ

π πγ θ

π πθ π θ γγ

− + < <

− − − < < += + < < − + − < < +

. (13)

γ

γ

delivery pressure

intake pressure

Figure 3. Pressure profile for illustrating the essential characteristics of delivery pressure.

Because the oscillation frequency of the force is much higher than for the pump system, it is sufficient to consider the average effect for the whole system.

Summarize the analysis for the swash-plate dynamics produces the following result:

Figure 3. Pressure profile for illustrating the essential characteristics of delivery pressure.

Because the oscillation frequency of the force is much higher than for the pump system, it issufficient to consider the average effect for the whole system.

Summarize the analysis for the swash-plate dynamics produces the following result:(I + McL2 + Mpr2 Z

2

)d2α

dt2 + cdα

dt+

(kbL2 +

Z2

Mpr2ω2)

α = AprγZ

2πPs − AcLPc + FbL. (14)

Based upon similar assumptions are is used to derive Equation (4), the fluid pressure within thedischarge actuator may be modeled with the following equation

Vhβ

dPs

dt= Qp −Qd − KPs. (15)

For a pump with a fixed input-speed, the volumetric flow-rate generated by the pump is oftenexpressed as

Qp = Gpα, (16)

where Gp =ZAprω

π .The system expression is:

.x1 =

ZAprωβπVh

x2 − kβVh

x1.x2 = x3.x3 =

−kb L2+ Z2 mpr2ω2

I+mc L2+ Z2 mpr2 x2 − c

I+mc L2+ Z2 mpr2 x3 −

Z2π Aprγ

I+mc L2+ Z2 mpr2 x1 − Ac L

I+mc L2+ Z2 mpr2 x4 +

Fb LI+mc L2+ Z

2 mpr2.x4 = Ac Lβ

Vcx2 + Qc

.x5 = x6.x6 = 1

mv

(ps Av + F0 − fv

dxvdt − Ks(xv + x0)− Fst

), (17)

and the state vector is X =[α

.α ps pc xv

.xv]T .

3.2. Response Analysis of the Pump

Based on consideration of each subsystem, a mathematical model of pressure continuousregulation for EHPAAPP is established, the dynamic response graph is obtained by MATLABsimulation. The main parameters selected are shown in Table 1, the maximum input current ofthe electromagnet is 3.3 A, and the maximum output force of the electromagnet is 170 N.

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Table 1. The table of main parameters.

Symbol Value Unit

K 4.55 × 10−2 m3/(Pa · s)β 1000 MPa

Vh 3.5 × 10−3 m3

N 9 NullAp 265 × 10−6 m2

Ac 150 × 10−6 m2

Av 12.5 × 10−6 m2

L 60 mmI 1.15 × 10−2 kg ·m2

Kq 2.85 × 10−1 m2/sKc 2.14 × 10−11 m3/(Pa · s)k 20 N/mm

The input is the current signal of proportional solenoid, and the output is the pressure of thepump. The input current changes from 0 A to 2 A. It can be seen from Figure 4 that the pressure ofthe system can reach the setting value of pressure when the current signal value changes, but thereare big shocks with pressure value of 4.5 MPa. The pressure value of the variable pump undergoesthe oscillation regulation process, and finally stabilizes at the third second and instantaneous peakpressure exceeds 125% of the rated outlet pressure.

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The input is the current signal of proportional solenoid, and the output is the pressure of the pump. The input current changes from 0 A to 2 A. It can be seen from Figure 4 that the pressure of the system can reach the setting value of pressure when the current signal value changes, but there are big shocks with pressure value of 4.5 MPa. The pressure value of the variable pump undergoes the oscillation regulation process, and finally stabilizes at the third second and instantaneous peak pressure exceeds 125% of the rated outlet pressure.

Therefore, a pressure controller should be designed to achieve the pressure control requirements of the hydraulic pump in the case of a wide range of circumstances of external load and system parameters.

Figure 4. The delivery pressure of EHPAAPP in the case of different current signals.

4. Control Method Design of the Pump

4.1. PID Control Method

A PID controller (proportional-integral-derivative controller) is a common feedback loop component in industrial control applications. It consists of proportional unit P, integral unit I and differential unit D which is simple and no precise system model is needed for its use.

Figure 5 shows the corresponding situation of the pressure command signal from 28 MPa to 21 MPa. The pressure value is stable after about 1 s, and the maximum overshoot is 21 MPa.

The situation of flow changes little in the process of pressure control, the pressure response is shown in the Figure 6 below.

Figure 5. The delivery pressure of EHPAAPP in the case of different input signals with proportional integral derivative (PID) control method.

As seen from Figure 6, there is flow disturbance at the fifth second and the pressure command signal is 25 MPa. The pressure value is stable after about 1 s, and the error with the command pressure is 0.2 MPa after flow disturbance.

4.2. LQR Control Method

Figure 4. The delivery pressure of EHPAAPP in the case of different current signals.

Therefore, a pressure controller should be designed to achieve the pressure control requirementsof the hydraulic pump in the case of a wide range of circumstances of external load andsystem parameters.

4. Control Method Design of the Pump

4.1. PID Control Method

A PID controller (proportional-integral-derivative controller) is a common feedback loopcomponent in industrial control applications. It consists of proportional unit P, integral unit I anddifferential unit D which is simple and no precise system model is needed for its use.

Figure 5 shows the corresponding situation of the pressure command signal from 28 MPa to21 MPa. The pressure value is stable after about 1 s, and the maximum overshoot is 21 MPa.

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The input is the current signal of proportional solenoid, and the output is the pressure of the pump. The input current changes from 0 A to 2 A. It can be seen from Figure 4 that the pressure of the system can reach the setting value of pressure when the current signal value changes, but there are big shocks with pressure value of 4.5 MPa. The pressure value of the variable pump undergoes the oscillation regulation process, and finally stabilizes at the third second and instantaneous peak pressure exceeds 125% of the rated outlet pressure.

Therefore, a pressure controller should be designed to achieve the pressure control requirements of the hydraulic pump in the case of a wide range of circumstances of external load and system parameters.

Figure 4. The delivery pressure of EHPAAPP in the case of different current signals.

4. Control Method Design of the Pump

4.1. PID Control Method

A PID controller (proportional-integral-derivative controller) is a common feedback loop component in industrial control applications. It consists of proportional unit P, integral unit I and differential unit D which is simple and no precise system model is needed for its use.

Figure 5 shows the corresponding situation of the pressure command signal from 28 MPa to 21 MPa. The pressure value is stable after about 1 s, and the maximum overshoot is 21 MPa.

The situation of flow changes little in the process of pressure control, the pressure response is shown in the Figure 6 below.

Figure 5. The delivery pressure of EHPAAPP in the case of different input signals with proportional integral derivative (PID) control method.

As seen from Figure 6, there is flow disturbance at the fifth second and the pressure command signal is 25 MPa. The pressure value is stable after about 1 s, and the error with the command pressure is 0.2 MPa after flow disturbance.

4.2. LQR Control Method

Figure 5. The delivery pressure of EHPAAPP in the case of different input signals with proportionalintegral derivative (PID) control method.

The situation of flow changes little in the process of pressure control, the pressure response isshown in the Figure 6 below.Appl. Sci. 2019, 9, x FOR PEER REVIEW 8 of 16

Figure 6. The delivery pressure of EHPAAPP with PID controller in the case of input signal is 25 MPa with flow disturbance at the fifth second.

Seen from the established mathematical model of the pump, it has a very complicated and high degree of nonlinearity. Although the pump can acquire the more satisfactory dynamic performance and control accuracy, it is difficult to ensure the desired performance when the system parameters change with working conditions.

Precise linearization methods are considered because of the existence of nonlinear components in this system, and the control strategy for the system is designed after precise linearization.

The following definitions are used:

1p

h

ZA ra

Vωβ

π= 2

h

kaVβ=

2 2 2

12 2

2

2

b p

c p

Zk L m rb

ZI m L m r

ω− +=

+ + 2

2 2

2c p

cb ZI m L m r=

+ + 3

2 2

2

2

p

c p

Z A rb

ZI m L m r

γπ=

+ +

42 2

2

c

c p

A Lb ZI m L m r

=+ +

05

2 2

2c p

F Lb ZI m L m r

=+ +

1c

c

A Lc

Vβ

= 1c

c

A Ld

Vβ

=

( )

( )

4

2 1 4 4

1 4

2 <

2 2

2 <

d m vc

d d vc

d v mc

c U x x x uV

d c U x x c U x u x uV

c U x x u x xV

β ϖρ

β ϖ ϖρ ρ

β ϖρ

− < −

= − − − ≤ ≤

− <

( )

( )

4

3 1 4 4

1 4

2 <

2 2

2 <

d m vc

d d vc

d v mc

c x x x uV

d c x x c U x u x uV

c x x u x xV

β ϖρ

β ϖ ϖρ ρ

β ϖρ

− < −

= − − − ≤ ≤

− <

.

(18)

So, the simplified mathematical model is:

( ) ( )( )

x f x g xy h x

= + =

, (19)

where ( )1 2 2 1

3

1 2 2 3 3 1 4 4 5

1 2 2

a x a xx

f xb x b x b x b x bd x d

−= − − − + +

and ( ) [ ]30 0 0 Tg x d= .

Whether the obtained pump outlet pressure equation satisfies the precise linearization condition should be verified first. The 1r − order Li is the derivative of pump outlet pressure output in the directions of ( )f x and ( )g x respectively.

The calculation result of each order Li derivative is:

Figure 6. The delivery pressure of EHPAAPP with PID controller in the case of input signal is 25 MPawith flow disturbance at the fifth second.

As seen from Figure 6, there is flow disturbance at the fifth second and the pressure commandsignal is 25 MPa. The pressure value is stable after about 1 s, and the error with the command pressureis 0.2 MPa after flow disturbance.

4.2. LQR Control Method

Seen from the established mathematical model of the pump, it has a very complicated and highdegree of nonlinearity. Although the pump can acquire the more satisfactory dynamic performanceand control accuracy, it is difficult to ensure the desired performance when the system parameterschange with working conditions.

Precise linearization methods are considered because of the existence of nonlinear components inthis system, and the control strategy for the system is designed after precise linearization.

The following definitions are used:

a1 =ZAprωβ

πVha2 = kβ

Vhb1 =

−kb L2+ Z2 mpr2ω2

I+mc L2+ Z2 mpr2 b2 = c

I+mc L2+ Z2 mpr2 b3 =

Z2π Aprγ

I+mc L2+ Z2 mpr2

b4 = Ac LI+mc L2+ Z

2 mpr2 b5 = F0LI+mc L2+ Z

2 mpr2 c1 = Ac LβVc

d1 = Ac LβVc

d2 =

β

Vc

(cdvU

√2ρ x4

)−xm <xv < −u

βVc

(cdvU

√2ρ (x1 − x4)− cdvU

√2ρ x4

)−u ≤ xv ≤ u

βVc

(cdvU

√2ρ (x1 − x4)

)u < xv < xm

d3 =

β

Vc

(cdv

√2ρ x4

)−xm <xv < −u

βVc

(cdv

√2ρ (x1 − x4)− cdvU

√2ρ x4

)−u ≤ xv ≤ u

βVc

(cdv

√2ρ (x1 − x4)

)u < xv < xm

.

(18)

Appl. Sci. 2019, 9, 1376 9 of 16

So, the simplified mathematical model is:{ .x = f (x) + g(x)

y = h(x), (19)

where f (x) =

a1x2 − a2x1

x3

b1x2 − b2x3 − b3x1 − b4x4 + b5

d1x2 + d2

and g(x) = [0 0 0 d3]T .

Whether the obtained pump outlet pressure equation satisfies the precise linearization conditionshould be verified first. The r − 1 order Li is the derivative of pump outlet pressure output in thedirections of f (x) and g(x) respectively.

The calculation result of each order Li derivative is:Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 16

11 1 2 1fL h x a x a x 2 2

1 3 1 2 2 2 3fL h x a x a a x a x

3 3 22 1 3 1 2 1 1 1 2

1 2 1 2 3 1 4 4 1 5 fL h x a a b x a a b a x

a a a b x a b x a b

1

1 0g fL L h x 1

2 0g fL L h x

1

31 4 2 0g fL L h x a b d .

(20)

The relative degree of the system is 4, which is equal to the number of system state variables, and the described system equations can be complete linearization.

It is proposed to adopt a linear optimal state regulator (LQR) method with quadratic performance indicators. This is because the solution of the optimal state regulator has a unified analytical expression, and a simple linear state feedback control law can be obtained. It is easy to implement feedback control, and it has a good engineering realization.

The quadratic performance index of the system is:

T Tz z 1 z 10

1 d2

J v v t

Z Q Z R . (21)

The corresponding optimal control expression is:

1 Tz z b b 1 0 2 1 3 2 4 3v k z k z k z k z R B P Z KZ . (22)

zP is non-singular symmetric coefficient matrix, and satisfy the equation

T 1 Tz z z 1 z z 0 A P P A PBR B P Q , (23)

Figure 7 shows the corresponding situation of the pressure command signal from 28 MPa to 25 MPa. The pressure value is stable after about 0.5 s, and the maximum overshoot is 0.15 MPa.

Figure 7. The delivery pressure of EHPAAPP with linear quadratic regulator (LQR) method in the case of input signal change from 28 MPa to 25 MPa.

The situation of flow suddenly changes greatly in the process of pressure control, the discharge pressure response is shown in the figure below.

(20)

The relative degree of the system is 4, which is equal to the number of system state variables, andthe described system equations can be complete linearization.

It is proposed to adopt a linear optimal state regulator (LQR) method with quadratic performanceindicators. This is because the solution of the optimal state regulator has a unified analytical expression,and a simple linear state feedback control law can be obtained. It is easy to implement feedback control,and it has a good engineering realization.

The quadratic performance index of the system is:

Jz =12

∫ ∞

0

(ZTQzZ + vT

1 Rzv1

)dt. (21)

The corresponding optimal control expression is:

v = −R−1z BTPzZb = −KZb = −k1z0 − k2z1 − k3z2 − k4z3. (22)

Pz is non-singular symmetric coefficient matrix, and satisfy the equation

ATPz + PzA− PBRz−1BT

1 Pz + Qz = 0, (23)

Figure 7 shows the corresponding situation of the pressure command signal from 28 MPa to 25MPa. The pressure value is stable after about 0.5 s, and the maximum overshoot is 0.15 MPa.

Appl. Sci. 2019, 9, 1376 10 of 16

Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 16

( )11 1 2 1fL h x a x a x= − ( )2 2

1 3 1 2 2 2 3fL h x a x a a x a x= − +

( ) ( ) ( )( )

3 3 22 1 3 1 2 1 1 1 2

1 2 1 2 3 1 4 4 1 5 fL h x a a b x a a b a x

a a a b x a b x a b

= − − − −

− + − + ( )

1

1 0g fL L h x = ( )1

2 0g fL L h x =

( )1

31 4 2 0g fL L h x a b d= − ≠ .

(20)

The relative degree of the system is 4, which is equal to the number of system state variables, and the described system equations can be complete linearization.

It is proposed to adopt a linear optimal state regulator (LQR) method with quadratic performance indicators. This is because the solution of the optimal state regulator has a unified analytical expression, and a simple linear state feedback control law can be obtained. It is easy to implement feedback control, and it has a good engineering realization.

The quadratic performance index of the system is:

( )T Tz z 1 z 10

1 d2

J v v t∞

= + Z Q Z R . (21)

The corresponding optimal control expression is:

1 Tz z b b 1 0 2 1 3 2 4 3v k z k z k z k z−= − = − = − − − −R B P Z KZ . (22)

zP is non-singular symmetric coefficient matrix, and satisfy the equation

T 1 Tz z z 1 z z 0−+ − + =A P P A PBR B P Q , (23)

Figure 7 shows the corresponding situation of the pressure command signal from 28 MPa to 25 MPa. The pressure value is stable after about 0.5 s, and the maximum overshoot is 0.15 MPa.

Figure 7. The delivery pressure of EHPAAPP with linear quadratic regulator (LQR) method in the case of input signal change from 28 MPa to 25 MPa.

The situation of flow suddenly changes greatly in the process of pressure control, the discharge pressure response is shown in the figure below.

Figure 7. The delivery pressure of EHPAAPP with linear quadratic regulator (LQR) method in the caseof input signal change from 28 MPa to 25 MPa.

The situation of flow suddenly changes greatly in the process of pressure control, the dischargepressure response is shown in the figure below.

As seen from the Figure 8, the pressure command signal is 28 MPa to 21 MPa, and there are flowdisturbance at the third second. The pressure value is stable after about 0.5 s, and the overshoot is0.5 MPa. Under the condition of flow interference at the third second, the error between final pumpoutlet pressure and the command pressure is 0.2 MPa.

Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 16

( )11 1 2 1fL h x a x a x= − ( )2 2

1 3 1 2 2 2 3fL h x a x a a x a x= − +

( ) ( ) ( )( )

3 3 22 1 3 1 2 1 1 1 2

1 2 1 2 3 1 4 4 1 5 fL h x a a b x a a b a x

a a a b x a b x a b

= − − − −

− + − + ( )

1

1 0g fL L h x = ( )1

2 0g fL L h x =

( )1

31 4 2 0g fL L h x a b d= − ≠ .

(20)

The relative degree of the system is 4, which is equal to the number of system state variables, and the described system equations can be complete linearization.

It is proposed to adopt a linear optimal state regulator (LQR) method with quadratic performance indicators. This is because the solution of the optimal state regulator has a unified analytical expression, and a simple linear state feedback control law can be obtained. It is easy to implement feedback control, and it has a good engineering realization.

The quadratic performance index of the system is:

( )T Tz z 1 z 10

1 d2

J v v t∞

= + Z Q Z R . (21)

The corresponding optimal control expression is:

1 Tz z b b 1 0 2 1 3 2 4 3v k z k z k z k z−= − = − = − − − −R B P Z KZ . (22)

zP is non-singular symmetric coefficient matrix, and satisfy the equation

T 1 Tz z z 1 z z 0−+ − + =A P P A PBR B P Q , (23)

Figure 7 shows the corresponding situation of the pressure command signal from 28 MPa to 25 MPa. The pressure value is stable after about 0.5 s, and the maximum overshoot is 0.15 MPa.

Figure 7. The delivery pressure of EHPAAPP with linear quadratic regulator (LQR) method in the case of input signal change from 28 MPa to 25 MPa.

The situation of flow suddenly changes greatly in the process of pressure control, the discharge pressure response is shown in the figure below.

Figure 8. The delivery pressure of EHPAAPP with LQR method in the case of different input pressuresignal with flow disturbance at the third second.

The Figure 9 shows the corresponding current change. The change trend is opposite to thepressure change trend. When the current increases, the pump outlet pressure value decreases, whichproves the accuracy of the control method.

Appl. Sci. 2019, 9, x FOR PEER REVIEW 10 of 16

Figure 8. The delivery pressure of EHPAAPP with LQR method in the case of different input pressure signal with flow disturbance at the third second.

As seen from the Figure 8, the pressure command signal is 28 MPa to 21 MPa, and there are flow disturbance at the third second. The pressure value is stable after about 0.5 s, and the overshoot is 0.5 MPa. Under the condition of flow interference at the third second, the error between final pump outlet pressure and the command pressure is 0.2 MPa.

Figure 9. Current Response of EHPAAPP with the LQR method in the case of a different input signal with flow disturbance at the third second.

The Figure 9 shows the corresponding current change. The change trend is opposite to the pressure change trend. When the current increases, the pump outlet pressure value decreases, which proves the accuracy of the control method.

4.3. Backstepping Sliding Control Method

The steps of designing the backstepping sliding mode control method for pressure tracking of the pump are as follows:

Step1. According to the pump outlet pressure 1x and the command signal 1dx , the tracking error of the system is defined as:

1 1 1de x x= − . (24)

After derivation the above pressure error, then,

1 1 1 1 2 2 1 1d de x x a x a x x= − = − − . (25)

Define the Lyapunov function of the first state variable of the system as, 2

1 10.5V e= . (26)

Derivation the Lyapunov function of the first state variable of the system,

( ) ( )1 1 1 1 1 1 1 1 2 2 1 1d dv e e e x x e a x a x x= = − = − − . (27)

Let ( )2 1 1 2 1 1 21

1dx c e a x x e

a= − − − + , if 1 0c > . 2e is virtual controller, then

( )2 2 1 1 2 1 11

1de x c e a x x

a= + − − . (28)

Substituting it into the above formula and then: 2

1 1 1 1 1 2V c e a e e= − + . (29)

Figure 9. Current Response of EHPAAPP with the LQR method in the case of a different input signalwith flow disturbance at the third second.

Appl. Sci. 2019, 9, 1376 11 of 16

4.3. Backstepping Sliding Control Method

The steps of designing the backstepping sliding mode control method for pressure tracking of thepump are as follows:

Step1. According to the pump outlet pressure x1 and the command signal x1d, the tracking errorof the system is defined as:

e1 = x1 − x1d. (24)

After derivation the above pressure error, then,

.e1 =

.x1 −

.x1d = a1x2 − a2x1 −

.x1d. (25)

Define the Lyapunov function of the first state variable of the system as,

V1 = 0.5e12. (26)

Derivation the Lyapunov function of the first state variable of the system,

.v1 = e1

.e1 = e1

( .x1 −

.x1d)= e1

(a1x2 − a2x1 −

.x1d). (27)

Let x2 = − 1a1

(c1e1 − a2x1 −

.x1d)+ e2, if c1 > 0. e2 is virtual controller, then

e2 = x2 +1a1

(c1e1 − a2x1 −

.x1d). (28)

Substituting it into the above formula and then:

.V1 = −c1e1

2 + a1e1e2. (29)

At this condition, if e2 = 0 and then.

V1 ≤ 0, the stability of Lyapunov is satisfied under theseconditions. In order to make e2 = 0, the next design is needed.

Step 2. Define the Lyapunov function of the second state variable of the system as:

V2 = V1 + 0.5e22. (30)

Substitute e2 = x2 +1a1

(c1e1 − a2x1 −

.x1d)

and the state space equation of x2 into the aboveequation, and then:

.V2 = −c1e1

2 + a1e1e2 + e2

(x3 +

a22

a1x1 − a2x2 −

1a1

..x1d +

1a1

c1.e1

). (31)

Let

x3 = − a22

a1x1 + a2x2 +

1a1

..x1d −

1a1

c1.e1 − a1e1 − c2e2 + e3, (32)

and

e3 =a2

2

a1x1 − a2x2 −

1a1

..x1d +

1a1

c1.e1 + a1e1 + c2e2 + x3, (33)

.V2 = −c1e1

2 − c1e22 + e2e3. (34)

At this condition, if e3 = 0 and then.

V2 ≤ 0, the stability of Lyapunov is satisfied under theseconditions. In order to make e3 = 0, the next design is needed.

Step 3. Define the Lyapunov function of the third state variable of the system as:

V3 = V2 + 0.5e32. (35)

Appl. Sci. 2019, 9, 1376 12 of 16

Derivation the Lyapunov function of the third state variable of the system, then

.V3 =

.V2 + e3

.e3 = −c1e1

2 − c2e22 + e2e3 + e3

.e3, (36)

Substitute e3 and the state space equation of x3 into the above equation, and then

.V3 = −c1e1

2 − c2e22 − c3e3

2 + e3

( (b1 + a2

2)x2 + (b2 − a2)x3 +(

b3 − a22

a1

)x1 + b4x4 + b5

− 1a1

...x 1d +

c1a1

..e1 + a1

.e1 + c2

.e2

)(37)

Letbe

x4 = − 1b4

( (b1 + a2

2)x2 + (b2 − a2)x3 +(

b3 − a22

a1

)x1

+b5 − 1a1

...x 1d +

c1a1

..e1 + a1

.e1 + c2

.e2 + e2 + c3e3

)+ e4, (38)

Then

.V3 = −c1e1

2 − c2e22 − c3e3

2 + b4e3e4, (39)

At this condition, if e4 = 0 and then.

V3 ≤ 0, the stability of Lyapunov is satisfied under theseconditions. In order to make e4 = 0, the next design is needed.

Step 4. Define the Lyapunov function of the last state variable of the system as:

V4 = V3 + 0.5σ2, (40)

whereσ = k1e1 + k2e2 + k3e3 + e4. (41)

Let

S = 1b4

[ (b1 + a2

2)x3 +(

b3 − a23

a1

)(a1x2 − a2x1) + (b2 − a2)(b1x2 + b2x3 + b3x1 + b4x4 + b5)

− 1a1

...x 1d +

c1a1

...e 1 + a1

..e + c2

..e2 +

.e2 + c3

.e3

]. (42)

Then the input of the system can be obtained

u =1d2

(−k1

.e1 + k2

.e2 + k3

.e3 − d1x3 − d3 − S− h(σ + βsgn(σ))

). (43)

Substitute e4 and the state space equation of x4 into the above equation, and then

.V4 = −c1e1

2 − c2e22 − c3e3

2 + b4e3e4 − hσ2 − hβ|σ|, (44)

where h2, β2 are positive constants. Let

Q =

hk2

1 + c1 hk1k2 hk1k3 hk1

hk1k2 hk22 + c2 hk2k3 hk2

hk1k3 hk2k3 hk23 + c3 hk3

hk1 hk2 hk3 h

, (45)

Appl. Sci. 2019, 9, 1376 13 of 16

and

eTQe = [e1e2e3e4 ]

hk2

1 + c1 hk1k2 hk1k3 hk1

hk1k2 hk22 + c2 hk2k3 hk2

hk1k3 hk2k3 hk23 + c3 hk3

hk1 hk2 hk3 h

e1

e2

e3

e4

= (hk1 + c1)e1

2 + (hk2 + c2)e22 + (hk3 + c3)e3

2 + he4

+2hk1k2e1e2 + 2hk1k3e1e3 + 2hkk1e1e4 + 2hk2k3e2e3 + 2hk2e2e4 + 2hk3e3e4

(46)

Therefore, .V4 = −eTQe− hβ|σ|, (47)

and|Q| = hc1c2c3 + h2b4c1c2k3, (48)

|Q| > 0 can be realized by taking the value of h2, k2, k3, so that Q is a positive definite matrix and.

V4 ≤ 0.As seen from the Figure 10, the pressure command signal is 28 MPa to 24 MPa, and there is flow

disturbance at the third second. The pressure value is stable after about 0.5 s, and the overshoot is 0.3MPa. Under the condition of flow interference at the third second, the error between final pump outletpressure and the command pressure is 0.15 MPa.

Appl. Sci. 2019, 9, x FOR PEER REVIEW 13 of 16

The calculated current value varies with changes in pump outlet pressure. The current value is stable after about 0.5 s, and the overshoot is 0.1 A. Under the condition of flow interference at the third second, the error is very small with the value about 0.05A.

Figure 10. The delivery pressure of EHPAAPP with backstepping sliding control method in the case of input signal change from 28 MPa to 24 MPa with flow disturbance.

Figure 11. Current Response of EHPAAPP with backstepping sliding control method in the case of input signal change from 28 MPa to 24 MPa with flow disturbance.

5. Conclusions

The analysis of EHPAAPP and the simulation results of this paper supports the following conclusions:

The principle of pressure continuous regulation electrohydraulic proportional axial piston pump was proposed. The structure is relatively simple, and the dual pressure regulation feature is retained. A contribution of this work lies in system modeling in terms of freebody diagrams of the components of the piston group together with the proportional electromagnet.

The PID control method, LQR based on feedback linearization method and backstepping sliding control method were designed to control the system. The effectiveness of the control methods were verified by the MATLAB simulation platform, showed that the delivery pressure of the pump could accurately track the system input signal and the out-of-system interference such as sudden changes of flow and so on. On the other hand, control law has a great influence on system control. More efficient control law needs to be carried out.

Author Contributions: P. Z. carried out the design of the control law, modeling and simulation; and Y.L. conceived the concept of the control strategies of the aviation axial piston pump.

Funding: The two funders of this paper are National Natural Science Foundation of China (Grant no. 51475019) and National Key Basic Research Program of China (Grant no. 2014CB046403).

Acknowledgments: Authors would like to thank for the support of National Natural Science Foundation of China (Grant no. 51475019) and National Key Basic Research Program of China (Grant no. 2014CB046403).

Figure 10. The delivery pressure of EHPAAPP with backstepping sliding control method in the case ofinput signal change from 28 MPa to 24 MPa with flow disturbance.

The Figure 11 shows the corresponding current change. The change trend is opposite to thechange trend of delivery pressure. When the current increases, the pump outlet pressure valuedecreases, which proves the accuracy of the control method. When the delivery pressure is 28 MPa,the corresponding current value is 0A, and when the current is about 1.4 A, the outlet pressure of thepump is 24 MPa.

Appl. Sci. 2019, 9, x FOR PEER REVIEW 13 of 16

The calculated current value varies with changes in pump outlet pressure. The current value is stable after about 0.5 s, and the overshoot is 0.1 A. Under the condition of flow interference at the third second, the error is very small with the value about 0.05A.

Figure 10. The delivery pressure of EHPAAPP with backstepping sliding control method in the case of input signal change from 28 MPa to 24 MPa with flow disturbance.

Figure 11. Current Response of EHPAAPP with backstepping sliding control method in the case of input signal change from 28 MPa to 24 MPa with flow disturbance.

5. Conclusions

The analysis of EHPAAPP and the simulation results of this paper supports the following conclusions:

The principle of pressure continuous regulation electrohydraulic proportional axial piston pump was proposed. The structure is relatively simple, and the dual pressure regulation feature is retained. A contribution of this work lies in system modeling in terms of freebody diagrams of the components of the piston group together with the proportional electromagnet.

The PID control method, LQR based on feedback linearization method and backstepping sliding control method were designed to control the system. The effectiveness of the control methods were verified by the MATLAB simulation platform, showed that the delivery pressure of the pump could accurately track the system input signal and the out-of-system interference such as sudden changes of flow and so on. On the other hand, control law has a great influence on system control. More efficient control law needs to be carried out.

Author Contributions: P. Z. carried out the design of the control law, modeling and simulation; and Y.L. conceived the concept of the control strategies of the aviation axial piston pump.

Funding: The two funders of this paper are National Natural Science Foundation of China (Grant no. 51475019) and National Key Basic Research Program of China (Grant no. 2014CB046403).

Acknowledgments: Authors would like to thank for the support of National Natural Science Foundation of China (Grant no. 51475019) and National Key Basic Research Program of China (Grant no. 2014CB046403).

Figure 11. Current Response of EHPAAPP with backstepping sliding control method in the case ofinput signal change from 28 MPa to 24 MPa with flow disturbance.

Appl. Sci. 2019, 9, 1376 14 of 16

The calculated current value varies with changes in pump outlet pressure. The current value isstable after about 0.5 s, and the overshoot is 0.1 A. Under the condition of flow interference at the thirdsecond, the error is very small with the value about 0.05A.

5. Conclusions

The analysis of EHPAAPP and the simulation results of this paper supports thefollowing conclusions:

The principle of pressure continuous regulation electrohydraulic proportional axial piston pumpwas proposed. The structure is relatively simple, and the dual pressure regulation feature is retained.A contribution of this work lies in system modeling in terms of freebody diagrams of the componentsof the piston group together with the proportional electromagnet.

The PID control method, LQR based on feedback linearization method and backstepping slidingcontrol method were designed to control the system. The effectiveness of the control methods wereverified by the MATLAB simulation platform, showed that the delivery pressure of the pump couldaccurately track the system input signal and the out-of-system interference such as sudden changes offlow and so on. On the other hand, control law has a great influence on system control. More efficientcontrol law needs to be carried out.

Author Contributions: P.Z. carried out the design of the control law, modeling and simulation; and Y.L. conceivedthe concept of the control strategies of the aviation axial piston pump.

Funding: The two funders of this paper are National Natural Science Foundation of China (Grant no. 51475019)and National Key Basic Research Program of China (Grant no. 2014CB046403).

Acknowledgments: Authors would like to thank for the support of National Natural Science Foundation of China(Grant no. 51475019) and National Key Basic Research Program of China (Grant no. 2014CB046403).

Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design of thestudy; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision topublish the results.

Nomenclature

Ac cross-sectional area of the control actuatorAp cross-sectional area of a single piston within the pumpAv cross-sectional area of the spool valvec effective viscous-drag coefficient for the swash plateCd valve discharge-coefficientFb compressed force of the bias-spring when the swash-plate angle is zeroFc force exerted on the swash-plate by the bias actuatorfM force of electromagnetic hysteresisFn reaction force between the nth piston-slipper assembly and the swash plateF0 axial thrust of the armatureFr coulomb frictionFsp steady-state fluid force of the valveGp pump flow-gainI mass moment-of-inertia for the swash-platei input current of proportional electromagnetK pump-leakage coefficientkb spring rate for the bias-springKFi current-force gainKs spring rate for the valve-spring

Appl. Sci. 2019, 9, 1376 15 of 16

Kv coil induced back EMF coefficientL fixed moment-arm for the bias and control actuators

instantaneous moment-arm for the control actuatorinstantaneous moment-arm for the nth piston-slipper assembly

Ld coil inductanceMc mass of the control actuatorMp mass of a single piston within the pumpPc control pressurePs pump discharge pressurePi pump intake-pressurePn instantaneous pressure within the nth piston chamberQc volumetric flow-rate into the control actuatorQd volumetric flow-rate being drawn from the discharge line by the system loadQp volumetric flow-rate from the pumpr piston-pitch radius within the pumpRs internal resistance of coil and proportional control amplifieru underlapped dimension for the open-centered spool valveus input voltage of the coilVc nominal volume of the control actuatorVh volume of the discharge lineW area gradient of valve portxv displacement of the spool valvexc displacement of the control actuatorxm limit displacement of the spoolxn displacement of the nth piston-slipper assemblyτ delay timeZ number of pistons within the pumpα swash-plate angleβ fluid bulk modulus-of-elasticityγ valve plate pressure carryover-angleθ angular displacement of the nth pistonρ fluid mass-densityω input shaft-speed for the pumpEHPAAPP electrohydraulic proportional aviation axial piston pump

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