# Adaptive Feedforward Control of a Pressure Compensated Differential Cylinder

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background and Method

- Y = system output;
- $\theta $ = system parameters;
- X = system input;
- E = estimation error;
- $\widehat{\theta}$ = estimated parameters;
- $\gamma $ = adaptation gain, constant.

- u = control output;
- $\widehat{u}$ = adaptive control output;
- ${u}_{c}$ = command signal;
- ${y}_{m}$ = model output;
- y = plant output.

- u = controller output;
- ${k}_{p}$ = proportional gain;
- e = position error;
- ${k}_{ff}$ = feedforward gain;
- ${v}_{ref}$ = velocity reference.

- $\gamma $ = adaptation gain;
- ${z}_{ff}$ = feedforward gain.

## 3. Considered System

## 4. Modelling

- ${u}_{pc}$ = opening of compensator, $0\le {u}_{pc}\le 1$
- ${p}_{p}$ = compensated pressure at port p;
- $\Delta p$ = pressure difference between fully closed and fully open;
- ${p}_{a}$ = pressure at port a;
- ${p}_{b}$ = pressure at port b;
- ${p}_{t}$ = tank pressure;
- ${p}_{set}$ = spring pressure setting;
- ${p}_{load}$ = load pressure;
- ${u}_{spool}$ = position of the main spool, $-1\le {u}_{spool}\le 1$;
- ${Q}_{pc}$ = flow in pressure compensator;
- ${k}_{pc}$ = flow gain of compensator;
- ${p}_{i}$ = compensator inlet pressure.

- Q = flow in the valve;
- ${C}_{d}$ = discharge coefficient;
- ${A}_{d}$ = maximum discharge area;
- $\rho $ = mass density;
- ${Q}_{max}$ = maximum valve flow;

- ${u}_{a}$ = opening of valve a, $0\le {u}_{a}\le 1$;
- ${u}_{b}$ = opening of valve b, $0\le {u}_{b}\le 1$;
- ${p}_{a1}$ = pressure at valve a input side;
- ${p}_{a2}$ = pressure at valve a actuator side;
- ${p}_{b1}$ = pressure at valve b input side;
- ${p}_{b2}$ = pressure at valve b actuator side;
- ${p}_{crack,a}$ = crack pressure of valve a;
- ${p}_{crack,b}$ = crack pressure of valve b;
- $\psi $ = pilot area ratio;
- $\Delta p$ = pressure difference between fully closed and fully open.

## 5. Adaptive Control Design

- ${z}_{ff}^{+}$ = out-stroke feedforward gain;
- ${z}_{ff}^{-}$ = in-stroke feedforward gain;
- ${u}_{ff}$ = feedforward controller output.

## 6. Simulation Results

## 7. Experimental Results

- $\widehat{u}$ = compensated control signal;
- u = control signal;
- ${u}^{+}$ = Out-stroke deadband;
- ${u}^{-}$ = In-stroke deadband;
- $\tilde{u}$ = desired deadband, 0.001.

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 15.**Point-to-point path references for simulation. (

**a**) Position reference; (

**b**) Velocity reference.

**Figure 16.**Cylinder position error during MIT-rule feedforward simulation, $\gamma =200\phantom{\rule{4.pt}{0ex}}\mathrm{s}\xb7{\mathrm{m}}^{-3}$.

**Figure 17.**Feedforward states during MIT-rule feedforward simulation, $\gamma =200\phantom{\rule{4.pt}{0ex}}\mathrm{s}\xb7{\mathrm{m}}^{-3}$.

**Figure 18.**Control signals from feedforward and feedback during simulation, $\gamma =200\phantom{\rule{4.pt}{0ex}}\mathrm{s}\xb7{\mathrm{m}}^{-3}$.

**Figure 19.**Cylinder position error during sign-sign feedforward simulation, $\gamma =0.1\phantom{\rule{4.pt}{0ex}}{\mathrm{m}}^{-1}$.

**Figure 20.**Feedforward states during sign-sign feedforward simulation, $\gamma =0.1\phantom{\rule{4.pt}{0ex}}{\mathrm{m}}^{-1}$.

**Figure 23.**Position error during MIT-rule feedforward experiment, $\gamma =200\phantom{\rule{4.pt}{0ex}}\mathrm{s}\xb7{\mathrm{m}}^{-3}$.

**Figure 24.**Feedforward states during MIT-rule feedforward experiment, $\gamma =200\phantom{\rule{4.pt}{0ex}}\mathrm{s}\xb7{\mathrm{m}}^{-3}$.

**Figure 25.**Position error during sign-sign feedforward experiment, $\gamma =0.1\phantom{\rule{4.pt}{0ex}}{\mathrm{m}}^{-1}$.

**Figure 26.**Feedforward states during sign-sign feedforward experiment, $\gamma =0.1\phantom{\rule{4.pt}{0ex}}{\mathrm{m}}^{-1}$.

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

Piston diameter | ${D}_{p}$ | 0.15 m |

Piston area | A | 0.0177 m${}^{2}$ |

Rod diameter | ${D}_{r}$ | 0.1 m |

Annulus area | ${A}_{a}$ | 0.0098 m${}^{2}$ |

Piston area ratio | $\varphi =\frac{{A}_{a}}{A}$ | 0.5556 |

Valve maximum flow | ${Q}_{max}$ | 40 L/min |

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

Mass | ${m}_{k}$ | 851.972 kg |

Inertia matrix | ${I}_{k}$ | $\left[\begin{array}{ccc}579.552& 8.74629& 11.5456\\ 8.74629& 573.285& 0.174433\\ 11.5456& 0.174433& 32.2491\end{array}\right]$ kg·m${}^{2}$ |

MIT-Rule | Sign-Sign | Fixed Gain | |
---|---|---|---|

RMS error | 1.6 mm | 2.1 mm | 2.1 mm |

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

Out-stroke deadband | ${u}^{+}$ | 0.21 |

In-stroke deadband | ${u}^{-}$ | −0.31 |

MIT-Rule | Sign-Sign | Fixed Gain | |
---|---|---|---|

RMS error | 5.2 mm | 5.3 mm | 24.9 mm |

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**MDPI and ACS Style**

Jensen, K.J.; Ebbesen, M.K.; Hansen, M.R.
Adaptive Feedforward Control of a Pressure Compensated Differential Cylinder. *Appl. Sci.* **2020**, *10*, 7847.
https://doi.org/10.3390/app10217847

**AMA Style**

Jensen KJ, Ebbesen MK, Hansen MR.
Adaptive Feedforward Control of a Pressure Compensated Differential Cylinder. *Applied Sciences*. 2020; 10(21):7847.
https://doi.org/10.3390/app10217847

**Chicago/Turabian Style**

Jensen, Konrad Johan, Morten Kjeld Ebbesen, and Michael Rygaard Hansen.
2020. "Adaptive Feedforward Control of a Pressure Compensated Differential Cylinder" *Applied Sciences* 10, no. 21: 7847.
https://doi.org/10.3390/app10217847