# A Comparison Study of a Novel Self-Contained Electro-Hydraulic Cylinder versus a Conventional Valve-Controlled Actuator—Part 1: Motion Control

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Self-Contained Electro-Hydraulic Cylinder

_{1}and FC

_{2}, the check valves Ac

_{1}and Ac

_{2}, and the check valve CV

_{1}. The pilot-operated check valves LH

_{1}and LH

_{2}are used for passive load-holding purposes by isolating the cylinder when the 3/2 electro-valve (EV) is not actuated. The EV must be activated to enable the actuator motion, resulting in transferring the highest cylinder pressure, selected through the CV

_{3}and CV

_{4}, into the opening pilot line of LH

_{1}and LH

_{2}. Anti-cavitation valves (Ac

_{3}and Ac

_{4}) are installed between the actuator sides and the ACC to avoid cavitation in the cylinder chambers. Pressure-relief valves (RV

_{1}–RV

_{4}) on both pump ports and on both cylinder ports avoid over-pressurizations. Finally, a cooler (CO) and a low-pressure filter (F) complete the hydraulics.

#### 2.1. Nonlinear Model of the System

^{®}. Since the SCC used in this application only operates in two quadrants, the rod-side chamber is always connected to the low-pressure accumulator as visible in Figure 2b. (It is assumed that the 3/2 electro-valve is energized to enable motion).

#### 2.2. Linear Model of the System

## 3. The Valve-Controlled System

## 4. The Control Design

#### 4.1. Control Parameters for the Self-Contained Electro-Hydraulic Cylinder

#### 4.2. Control Parameters for the Valve-Controlled System

## 5. Results and Discussion

#### 5.1. Closed-Loop Step Response

#### 5.2. Representative Working Cycle

#### 5.3. Scenarios with Reduced Payload

## 6. Conclusions

- The SCC achieves significantly better position tracking (up to 66% less tracking error and 61% less overshoot) and faster response (i.e., 10 ms faster rise time and 75% faster settling time);
- The active pressure feedback in the SCC reduces the pressure oscillations more effectively since the electric drive has about 95% higher bandwidth than the control valve.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Abbreviations | |

AC | Alternating current |

ACC | Accumulator |

Ac | Anti-cavitation valve |

ALH | Active load-holding |

AV | Auxiliary valves |

BR | Brake resistor |

C | Hydraulic cylinder |

CO | Oil cooler |

CV | Check valve |

DC | Direct current |

ED | Electric drive |

EM | Electric motor |

EV | Electro-valve |

F | Low pressure oil filter |

FC | Flow compensation valve |

FOC | Field-oriented control |

Gm | Gain margin |

HPU | Hydraulic power unit |

HS | Hydraulic system |

LHV | Load-holding valve |

P | Axial piston machine (pump) |

PC | Pressure compensator |

PDCV | Proportional directional control valve |

PI | Proportional and integral |

PLC | Programmable logic controller |

PLH | Passive load-holding |

PV | Poppet valve |

PWM | Pulse-width modulation |

RMS | Root mean square |

RV | Pressure-relief valve |

SBC | Single-boom crane |

SCC | Self-contained electro-hydraulic cylinder |

SD | Servo-drive |

SM | Servo-motor |

SU | Supply unit |

V | Control valve |

VCC | Valve-controlled cylinder |

Symbols | |

${A}_{p}$ | Cylinder area on the piston-side |

${A}_{r}$ | Cylinder area on the rod-side |

$B$ | Viscous friction coefficient |

${C}_{p}$ | Piston-side capacitance |

${C}_{r}$ | Rod-side capacitance |

${D}_{p}$ | Pump displacement |

${e}_{{x}_{C}}$ | Actuator’s piston position error |

${K}_{D}$ | Gain of the mechanical-hydraulic system including direct pressure feedback |

${k}_{f,D}$ | Gain of the direct pressure feedback |

${k}_{f,HP}$ | Gain of the high-pass filtered pressure feedback |

${K}_{MH}$ | Gain of the uncompensated mechanical-hydraulic system |

${k}_{L}$ | Combined leakage flow gain |

${k}_{I}$ | Integral controller gain |

${k}_{q}$ | Pump flow gain |

${k}_{P}$ | Proportional controller gain |

${M}_{eq}$ | Equivalent mass |

${n}_{SM}$ | Rotational speed of the servo-motor in revolutions per minute |

${Q}_{C,\mathrm{L}}$ | Internal leakage in the hydraulic cylinder |

${Q}_{P}$ | Actuator’s flow demand |

${Q}_{P,e}$ | Effective pump flow |

${Q}_{r}$ | Rod-side flow |

${Q}_{S}$ | Pump’s flow losses |

${p}_{0}$ | Fixed pressure-drop across the proportional directional control valve |

${p}_{1}$ | Piston-side pump pressure |

${p}_{2}$ | Rod-side pump pressure |

${p}_{AC,0}$ | Pre-charge pressure of the accumulator |

${p}_{LS}$ | Load-sensing pressure |

${p}_{p}$ | Actuator’s piston chamber pressure |

${p}_{r}$ | Actuator’s rod chamber pressure |

${p}_{R}$ | Return pressure |

${p}_{S}$ | Supply pressure |

${u}_{ED}$ | On/off command to enable power to the servo-motor |

${u}_{EV}$ | On/off command to open or close the 3/2 electro-valve |

${u}_{\mathrm{FC}}$ | Position feedback control signal |

${u}_{FF}$ | Velocity feedforward control signal |

${u}_{\mathrm{PF}}$ | Pressure feedback control signal |

${u}_{SM}$ | Commanded servo-motor speed |

${u}_{V}$ | Commanded opening of the control valve’s spool position |

${V}_{AC,0}$ | Effective accumulator gas volume |

${v}_{C}$ | Actuator’s piston velocity |

${v}_{C,ref}$ | Actuator’s piston velocity reference command |

${V}_{p,0}$ | Transmission lines’ volumes between the pump and the piston-side chamber |

${V}_{r,0}$ | Transmission lines’ volumes between the pump and the rod-side chamber |

${x}_{C}$ | Actuator’s piston position |

${x}_{C,0}$ | Actuator’s initial piston position |

${x}_{C,ref}$ | Actuator’s piston position reference command |

Greek symbols | |

$\beta $ | Constant bulk modulus of the hydraulic fluid |

${\kappa}_{air}$ | Adiabatic air constant |

${\tau}_{f,HP}$ | Time constant of the high-pass filtered pressure feedback |

${\varphi}_{m}$ | Phase angle |

${\omega}_{gc}$ | Gain cross over frequency |

${\omega}_{n,D}$ | Natural-frequency of mechanical-hydraulic system including direct pressure feedback |

${\omega}_{n,ED}$ | Natural-frequency of the electric drive |

${\omega}_{n,MH}$ | Natural-frequency of the uncompensated mechanical-hydraulic system |

${\omega}_{pc}$ | Phase cross over frequency |

${\omega}_{SM}$ | Angular velocity of the servo-motor in radians per second |

${\zeta}_{D}$ | Damping ratio of the mechanical-hydraulic system including direct pressure feedback |

${\zeta}_{ED}$ | Damping ratio of the electric drive |

${\zeta}_{MH}$ | Damping ratio of the uncompensated mechanical-hydraulic system |

${\zeta}_{3}$ | Damping ratio of the complex conjugate pole pair in the transfer function ${G}_{3,HP}\left(s\right)$ |

## Appendix A

**Figure A1.**Open-loop validation of the linear system model: (

**a**) commanded, simulated, and measured speed of the servo-motor; (

**b**) simulated and measured piston position; (

**c**) simulated and estimated piston velocity; and (

**d**) simulated and measured piston chamber pressure.

Parameter | Value | Parameter | Value |
---|---|---|---|

${\omega}_{n,ED}$ | $550\text{}(\mathrm{rad}/\mathrm{s})$ | ${V}_{p,0}$ | $0.88\cdot {10}^{-3}\text{}({\mathrm{m}}^{3})$ |

${\zeta}_{ED}$ | $0.8$ | ${k}_{L}$ | $1.00\cdot {10}^{-15}\text{}({\mathrm{m}}^{3}$/$(\mathrm{s}\cdot \mathrm{Pa}))$ |

${k}_{q}$ | $2.17\cdot {10}^{-7}\text{}({\mathrm{m}}^{3}/(\mathrm{s}\cdot \mathrm{rpm}))$ | ${A}_{p}$ | $0.0033\text{}({\mathrm{m}}^{2})$ |

${C}_{p}$ | $3.60\cdot {10}^{-12}\text{}({\mathrm{m}}^{3}/\mathrm{Pa})$ | $B$ | $22500\text{}(\mathrm{N}\cdot \mathrm{s}/\mathrm{m})$ |

$\beta $ | $6600\cdot {10}^{5}\text{}(\mathrm{Pa})$ | ${M}_{eq}$ | $15500\text{}(\mathrm{kg})$ |

## Appendix B

**Figure A2.**A function block representing the modified plant model that includes the pressure feedback.

#### Appendix B.1. Direct Pressure Feedback

**Figure A3.**The effect of different damping ratios in comparison to the uncompensated system: (

**a**) Bode plots of ${G}_{4,D}\left(s\right)$ and ${G}_{{x}_{C}}\left(s\right)$; (

**b**) step response (${n}_{SM}=1000$ rpm) of ${G}_{3,D}\left(s\right)$ and ${G}_{{v}_{C}}\left(s\right)$.

$\zeta $ | Gm (dB) | ${\omega}_{gc}$ (rad/s) | Rise Time (s) | Settling Time (s) | Overshoot (mm/s) |

$0.052$ | $86.9$ | 14.0 | 0.077 | 5.39 | 85.02 |

$0.5$ | 108 | 14.7 | 0.11 | 0.55 | 16.29 |

$0.7$ | 111 | 15.0 | 0.14 | 0.40 | 4.59 |

#### Appendix B.2. High-Pass Filtered Pressure Feedback

**Figure A4.**The effect of varying the time constant of the high-pass filter: (

**a**) Bode plots of ${G}_{4,HP}\left(s\right)$ and ${G}_{{x}_{C}}\left(s\right)$; (

**b**) step response of ${G}_{3,HP}\left(s\right)$ and ${G}_{{v}_{C}}\left(s\right)$.

${\mathbf{\tau}}_{\mathbf{f},\mathbf{H}\mathbf{P}}$ $\left(\mathit{s}\right)$ | Gm (dB) | ${\mathit{\omega}}_{\mathit{g}\mathit{c}}$ (rad/s) | ${\mathit{\zeta}}_{3}$ | Rise Time (s) | Settling Time (s) | Overshoot (mm/s) |
---|---|---|---|---|---|---|

$\text{}1\text{}/{\omega}_{n}$ | $\text{}92.6\text{}$ | 4.06 | 0.319 | 0.34 | 2.93 | 36.14 |

$\text{}5\text{}/{\omega}_{n}$ | 113 | 12.2 | 0.864 | 0.27 | 1.27 | 8.61 |

$\text{}10/{\omega}_{n}\text{}$ | 110 | 14.1 | 0.713 | 0.16 | 0.41 | 3.72 |

$\text{}15/{\omega}_{n}$ | 107 | 14.3 | 0.486 | 0.12 | 0.67 | 13.99 |

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**Figure 2.**The considered application and the electro-hydraulic linear actuator: (

**a**) single-boom crane; (

**b**) simplified electro-hydraulic cylinder’s circuit.

**Figure 3.**Block diagram representing the linear model of the uncompensated self-contained electro-hydraulic cylinder.

**Figure 6.**The closed-loop position step response comparison: (

**a**) commanded and measured piston positions; (

**b**) position tracking errors; (

**c**) piston chamber pressures; and (

**d**) rod chamber pressures.

**Figure 7.**Position tracking performance comparison: (

**a**) commanded and measured piston position; (

**b**) tracking error.

**Figure 8.**Commanded and measured control inputs: (

**a**) self-contained system; (

**b**) valve-controlled system.

**Figure 12.**Actuator pressures: (

**a**) piston chamber with half payload; (

**b**) piston chamber without payload; (

**c**) rod chamber with half payload; and (

**d**) rod chamber without payload.

System | $\mathbf{P}\mathbf{m}$ (deg) | $\mathbf{G}\mathbf{m}\left({\mathbf{\omega}}_{\mathbf{p}\mathbf{c}}\right)$ (dB) | ${\mathbf{\omega}}_{\mathbf{p}\mathbf{c}}$ $(\mathbf{r}\mathbf{a}\mathbf{d}/\mathbf{s})$ | ${\mathbf{k}}_{\mathbf{P}}$ $(\mathbf{r}\mathbf{a}\mathbf{d}/(\mathbf{s}\cdot \mathbf{m}\left)\right)$ | ${\mathbf{k}}_{\mathbf{I}}$ $(\mathbf{r}\mathbf{a}\mathbf{d}/({\mathit{s}}^{2}\cdot \mathbf{m}\left)\right)$ |
---|---|---|---|---|---|

${G}_{{x}_{C}}\left(s\right)$ | $45$ | $89.1$ | $13.3$ | $2.69\cdot {10}^{4}$ | $3.61\cdot {10}^{4}$ |

${G}_{4,HP}\left(s\right)$ | $45$ | $102$ | $8.93$ | $1.26\cdot {10}^{5}$ | $1.19\cdot {10}^{5}$ |

$\mathbf{P}\mathbf{m}\text{}\left(\mathbf{deg}\right)$ | $\mathbf{G}\mathbf{m}\left({\mathbf{\omega}}_{\mathbf{p}\mathbf{c}}\right)\text{}\left(\mathbf{dB}\right)$ | ${\mathbf{\omega}}_{\mathbf{p}\mathbf{c}}\text{}(\mathbf{rad}/\mathbf{s})$ | ${\mathbf{k}}_{\mathbf{P}}$$\left(1/\mathbf{m}\right)$ | ${\mathbf{k}}_{\mathbf{I}}\text{}(1/(\mathbf{m}\xb7\mathbf{s}\left)\right)$ | ${\mathbf{k}}_{\mathbf{q}}\text{}({\mathbf{m}}^{3}/\mathbf{s})$ |
---|---|---|---|---|---|

45 | 29.3 | 6.68 | 29.17 | 19.49 | $5.83\cdot {10}^{-4}$ |

System | Rise Time (s) | Settling Time (s) | Overshoot (mm) |
---|---|---|---|

VCC | 0.27 | 2.23 | 4.65 |

SCC | 0.26 | 0.57 | 1.83 |

Velocity SP (mm/s) | 20 | 75 | 120 | |||
---|---|---|---|---|---|---|

System | VCC | SCC | VCC | SCC | VCC | SCC |

RMS error (mm) | 0.17 | 0.25 | 1.26 | 0.37 | 1.5 | 0.52 |

Load Case | Max Payload | Half Payload | No Payload | |||
---|---|---|---|---|---|---|

System | VCC | SCC | VCC | SCC | VCC | SCC |

RMS error (mm) | 1.5 | 0.52 | 1.94 | 0.41 | 1.36 | 0.37 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Hagen, D.; Padovani, D.; Choux, M.
A Comparison Study of a Novel Self-Contained Electro-Hydraulic Cylinder versus a Conventional Valve-Controlled Actuator—Part 1: Motion Control. *Actuators* **2019**, *8*, 79.
https://doi.org/10.3390/act8040079

**AMA Style**

Hagen D, Padovani D, Choux M.
A Comparison Study of a Novel Self-Contained Electro-Hydraulic Cylinder versus a Conventional Valve-Controlled Actuator—Part 1: Motion Control. *Actuators*. 2019; 8(4):79.
https://doi.org/10.3390/act8040079

**Chicago/Turabian Style**

Hagen, Daniel, Damiano Padovani, and Martin Choux.
2019. "A Comparison Study of a Novel Self-Contained Electro-Hydraulic Cylinder versus a Conventional Valve-Controlled Actuator—Part 1: Motion Control" *Actuators* 8, no. 4: 79.
https://doi.org/10.3390/act8040079