# A Case Study of Open- and Closed-Loop Control of Hydrostatic Transmission with Proportional Valve Start-Up Process

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## Abstract

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## 1. Introduction

#### Hydrostatic Transmission Control Methods

## 2. Mathematical Model for the Serial Throttle Control Method

- The temperature of the medium, and consequently its viscosity, remains constant throughout the simulation;
- The pressure has no effect on the viscosity of the medium;
- The compressibility of the medium and the deformability of the hydraulic elements was reduced to the concentrated capacitance at particular points of the system;
- There is no air in the system;
- There is no backlash in the mechanical system;
- The load of the motor is focused on its shaft (inertia);
- There are no wave phenomena;
- There are no external leaks in the system;
- The speed of the electric motor driving the pump is constant and is independent of the pump load;

- On the section from the pump to the proportional spool valve:

- On the section from the proportional spool valve to the motor:

- Dynamic surplus pressure ${p}_{s\mathrm{max}}$ as the maximum pressure recorded during the start-up phase;
- Start-up time ${t}_{s}$ as the time after which the rotational speed reaches 95% of the set value from the moment of supplying the control signal;
- Reaction time ${t}_{r}$ as the time after which the speed of the motor reaches 5% of the set value;
- Energy ${E}_{s}$ generated by the pump during the start-up process (first 10 s).

- Overshoot parameter described by the following relation:

- Steady-state error for the ramp input measured at the end of the ramp signal:

## 3. Experimental Verification of the Mathematical Model

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

${a}_{vp}$ | pump leakage coefficient ${[\mathrm{m}}^{3}/\mathrm{Pa}\xb7\mathrm{s}]$ |

${a}_{vs}$ | motor leakage coefficient ${[\mathrm{m}}^{3}/\mathrm{Pa}\xb7\mathrm{s}]$ |

${A}_{0},{A}_{1},$ ${A}_{2},{A}_{3}$ | coefficients of the polynomial function describing the opening of the spool valve $[-]$ |

${c}_{p}$ | capacitance of the liquid and of the pipes on the section from the pump to the proportional spool valve ${[\mathrm{m}}^{3}/\mathrm{Pa}]$ |

${c}_{s}$ | capacitance of the liquid and of the pipes on the section from the proportional spool valve to the motor ${[\mathrm{m}}^{3}/\mathrm{Pa}]$ |

${e}_{r}$ | steady-state error to ramp input $\left[\mathrm{rpm}\right]$ |

${E}_{s}$ | energy generated by the pump during the first 10 s $\left[\mathrm{Wh}\right]$ |

$f$ | viscous friction coefficient ${[\mathrm{N}\xb7\mathrm{m}\xb7\mathrm{s}/\mathrm{rad}}^{2}]$ |

${G}_{n}$ | transmittance of the rotational speed measuring system |

${G}_{PI}$ | PI controller transmittance |

${G}_{RD}$ | conductivity of the spool valve $\left[{\mathrm{m}}^{3}/\mathrm{s}\sqrt{\mathrm{Pa}}\right]$ |

${G}_{RD\mathrm{max}}$ | maximum conductivity of the spool valve $\left[{\mathrm{m}}^{3}/\mathrm{s}\sqrt{\mathrm{Pa}}\right]$ |

${h}_{z}$ | amplification factor of the relief valve ${[\mathrm{m}}^{3}/\mathrm{Pa}\cdot \mathrm{s}]$ |

$i$ | gear ratio of the planetary gear $[-]$ |

j | summation index $[-]$ |

${s}_{0}$ | minimum value of the control signal $[-]$ |

${s}_{\mathrm{max}}$ | maximum value of the control signal $[-]$ |

${I}_{zr}$ | reduced mass moment of inertia of rotational masses $\left[\mathrm{kg}\cdot {\mathrm{m}}^{2}\right]$ |

${K}_{P}$ | proportional gain of the PI controller $[-]$ |

${M}_{b}$ | braking torque on the shaft of the hydrostatic motor $\left[\mathrm{N}\xb7\mathrm{m}\right]$ |

$L$ | total length of the pipes between the pump and the motor $\left[\mathrm{m}\right]$ |

$n$ | rotational speed of the hydrostatic motor $\left[\mathrm{rpm}\right]$ |

${n}_{0}$ | initial rotational speed in the controlled system $\left[\mathrm{rpm}\right]$ |

${n}_{\mathrm{max}}$ | target rotational speed in the controlled system $\left[\mathrm{rpm}\right]$ |

${n}_{r}$ | current rotational speed set point in the controlled system $\left[\mathrm{rpm}\right]$ |

${n}_{s}$ | rotational speed on the shaft of the hydrostatic motor $\left[\mathrm{rpm}\right]$ |

${p}_{0}$ | opening pressure of the relief valve $\left[\mathrm{Pa}\right]$ |

${p}_{d}$ | total pressure drop resulting from flow rate losses $\left[\mathrm{Pa}\right]$ |

${p}_{p}$ | pressure on the pump $\left[\mathrm{Pa}\right]$ |

${p}_{s}$ | pressure on the hydraulic motor $\left[\mathrm{Pa}\right]$ |

${p}_{s\mathrm{max}}$ | dynamic surplus pressure on the motor $\left[\mathrm{Pa}\right]$ |

${p}_{c}$ | pressure at the hydraulic motor corresponding to static resistance values (for $\omega \approx 0$) $\left[\mathrm{Pa}\right]$ |

$R$ | inner radius of the hydraulic pipes $\left[\mathrm{m}\right]$ |

$s$ | control signal $[-]$ |

${s}_{m}$ | modified control signal $[-]$ |

$t$ | simulation time (step) $\left[\mathrm{s}\right]$ |

${t}_{0}$ | control signal edge rise time $\left[\mathrm{s}\right]$ |

${t}_{1}$ | oil temperature in the planetary gear $\left[\xb0\mathrm{C}\right]$ |

${t}_{2}$ | oil temperature in the hydraulic system $\left[\xb0\mathrm{C}\right]$ |

${t}_{r}$ | transmission reaction time $\left[\mathrm{s}\right]$ |

${t}_{s}$ | transmission start-up time $\left[\mathrm{s}\right]$ |

${T}_{I}$ | time constant of the integrating element of the PI controller $\left[\mathrm{s}\right]$ |

${T}_{n}$ | time constant of the system for measuring the rotational speed $\left[\mathrm{s}\right]$ |

${T}_{RD}$ | time constant of the spool valve $\left[\mathrm{s}\right]$ |

${T}_{Z}$ | time constant of the relief valve $\left[\mathrm{s}\right]$ |

${q}_{s}$ | displacement of the hydraulic motor ${[\mathrm{m}}^{3}/\mathrm{rad}]$ |

${Q}_{pt}$ | theoretical pump output flow ${[\mathrm{m}}^{3}/\mathrm{s}]$ |

${Q}_{vp}$ | flow value resulting from losses in the pump ${[\mathrm{m}}^{3}/\mathrm{s}]$ |

${Q}_{Z}$ | flow through the relief valve ${[\mathrm{m}}^{3}/\mathrm{s}]$ |

${Q}_{RD}$ | flow through the proportional valve ${[\mathrm{m}}^{3}/\mathrm{s}]$ |

${Q}_{cp}$ | flow caused by compressibility in volume between the pump and the spool valve ${[\mathrm{m}}^{3}/\mathrm{s}]$ |

${Q}_{cs}$ | flow caused by the compressibility in volume between the proportional spool valve and the hydraulic motor ${[\mathrm{m}}^{3}/\mathrm{s}]$ |

${Q}_{s}$ | flow towards the hydraulic motor ${[\mathrm{m}}^{3}/\mathrm{s}]$ |

${Q}_{vs}$ | flow value resulting from losses in the hydraulic motor ${[\mathrm{m}}^{3}/\mathrm{s}]$ |

$\zeta $ | coefficient of pressure losses caused by turbulent flow $[-]$ |

${\kappa}_{n}$ | speed overshoot in a PI controlled system $[\%]$ |

$\mu $ | dynamic viscosity of the working medium $[\mathrm{N}\cdot {\mathrm{s}/\mathrm{m}}^{2}]$ |

$\rho $ | density of the working medium ${[\mathrm{kg}/\mathrm{m}}^{3}]$ |

${\omega}_{s}$ | angular velocity of the hydrostatic motor $[\mathrm{rad}/\mathrm{s}]$ |

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**Figure 1.**Diagram of the hydraulic system of a hydrostatic transmission with serial throttle control.

**Figure 3.**Opening characteristic curve of the spool valve from the data sheet with a fitted curve of model based on Equation (4). Curves 1–5 correspond to different valve pressure differential, the red dashed line is the fitted curve.

**Figure 4.**Resistance to motion of the gear motor M2C1613 and of the coupled planetary gear OH-500 as a function of angular velocity, ${\omega}_{s}$, of idle running.

**Figure 6.**Result of simulation of the waveform of pressure on the motor ${p}_{s}$ for various signal rise times, ${t}_{0}$.

**Figure 7.**Result of simulation of the waveform of rotational speed ${n}_{s}$ for various signal rise times, ${t}_{0}$.

**Figure 8.**Result of simulation of the waveform of pressure on the pump ${p}_{p}$ for various signal rise times, ${t}_{0}$.

**Figure 9.**Result of simulation of the waveform of rotational speed, ${n}_{s}$, for various parameters of the PI controller.

**Figure 10.**Result of simulation of the waveform of pressure on the motor ${p}_{s}$ for various parameters of the PI controller.

**Figure 11.**Result of simulation of the waveform of rotational speed, ${n}_{s}$, for various moments of inertia of the system (${K}_{P}=0.015,{T}_{I}=0.75\mathrm{s})$.

**Figure 12.**Result of simulation of the waveform of pressure on the motor ${p}_{s}$ for various moments of inertia of the system (${K}_{P}=0.015,{T}_{I}=0.75\mathrm{s})$.

**Figure 13.**The hydrostatic transmission test stand: 1—planetary gear; 2—coupling housing; 3—hydraulic gear motor; 4—hydraulic power source.

**Figure 14.**Measurement and result of simulation of the waveform of pressure on the motor ${p}_{s}$ for the control signal rise time ${t}_{0}=1\text{}\mathrm{s}$.

**Figure 15.**Measurement and result of simulation of the waveform of rotational speed on the motor, ${n}_{s}$, for the control signal rise time ${t}_{0}=1\mathrm{s}$.

**Figure 16.**Measurement and result of simulation of the waveform of pressure on the pump ${p}_{p}$ for the control signal rise time ${t}_{0}=1\mathrm{s}$.

**Figure 17.**Measurement and result of simulation of the waveform of speed on the motor ${n}_{s}$ for the control system (${K}_{P}=0.02,{T}_{I}=0.5\mathrm{s})$. Waveforms labelled “sim.” and “sim.*” shows the difference between simulation results with and without the modified spool valve model (Equation (4)).

**Table 1.**Comparison of the control signal parameters and dynamic surplus pressure values at the inlet port of the motor, and the transmission start-up duration and reaction time. Transmission controlled with the serial throttle method without feedback.

${\mathit{t}}_{0}\left[\mathbf{s}\right]$ | ${\mathit{p}}_{\mathit{s}\mathbf{max}}\left[\mathbf{MPa}\right]$ | ${\mathit{t}}_{\mathit{s}}\text{}\left[\mathbf{s}\right]$ | ${\mathit{t}}_{\mathit{r}}\text{}\left[\mathbf{s}\right]$ | ${\mathit{E}}_{\mathit{s}}\text{}\left[\mathbf{Wh}\right]$ |
---|---|---|---|---|

0.5 | 9.29 | 3.00 | 0.48 | 5.58 |

1.0 | 8.63 | 3.12 | 0.58 | 5.62 |

2.5 | 6.99 | 3.61 | 0.81 | 5.76 |

5.0 | 5.23 | 4.60 | 1.13 | 6.08 |

8.0 | 4.03 | 5.91 | 1.44 | 6.54 |

10 | 3.53 | 6.87 | 1.65 | 6.87 |

**Table 2.**Comparison of the parameters of the PI controller and the obtained values of the dynamic surplus pressure on the motor, the start-up time, the reaction time, and the overshoot.

${\mathit{K}}_{\mathit{P}}$ | ${\mathit{T}}_{\mathit{I}}$ | ${\mathit{p}}_{\mathit{s}\mathbf{max}}\left[\mathbf{MPa}\right]$ | ${\mathit{t}}_{\mathit{s}}\text{}\left[\mathbf{s}\right]$ | ${\mathit{t}}_{\mathit{r}}\text{}\left[\mathbf{s}\right]$ | ${\mathit{\kappa}}_{\mathit{n}}\text{}[\%]$ | ${\mathit{e}}_{\mathit{r}}\text{}\left[\mathbf{rpm}\right]$ |
---|---|---|---|---|---|---|

0.020 | 0.50 | 2.70 | 9.72 | 1.49 | 102.7 | 4.4 |

0.020 | 0.75 | 2.34 | 9.84 | 1.63 | 101.8 | 6.4 |

0.020 | 1.00 | 2.15 | 9.94 | 1.73 | 100.9 | 8.8 |

0.015 | 0.50 | 2.51 | 9.79 | 1.70 | 102.7 | 5.7 |

0.015 | 0.75 | 2.19 | 9.94 | 1.87 | 101.6 | 8.6 |

0.015 | 1.00 | 2.03 | 10.09 | 2.00 | 100.5 | 11.5 |

0.010 | 0.50 | 2.28 | 9.93 | 2.05 | 102.6 | 8.5 |

0.010 | 0.75 | 2.04 | 10.16 | 2.28 | 101.2 | 12.9 |

0.010 | 1.00 | 1.91 | 10.39 | 2.45 | - | 17.4 |

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**MDPI and ACS Style**

Bury, P.; Stosiak, M.; Urbanowicz, K.; Kodura, A.; Kubrak, M.; Malesińska, A. A Case Study of Open- and Closed-Loop Control of Hydrostatic Transmission with Proportional Valve Start-Up Process. *Energies* **2022**, *15*, 1860.
https://doi.org/10.3390/en15051860

**AMA Style**

Bury P, Stosiak M, Urbanowicz K, Kodura A, Kubrak M, Malesińska A. A Case Study of Open- and Closed-Loop Control of Hydrostatic Transmission with Proportional Valve Start-Up Process. *Energies*. 2022; 15(5):1860.
https://doi.org/10.3390/en15051860

**Chicago/Turabian Style**

Bury, Paweł, Michał Stosiak, Kamil Urbanowicz, Apoloniusz Kodura, Michał Kubrak, and Agnieszka Malesińska. 2022. "A Case Study of Open- and Closed-Loop Control of Hydrostatic Transmission with Proportional Valve Start-Up Process" *Energies* 15, no. 5: 1860.
https://doi.org/10.3390/en15051860