#
A High-Order Load Model and the Control Algorithm for an Aerospace Electro-Hydraulic Actuator^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. The System and Its Modeling

_{L}is the load flowrate from the servo-valve to the actuator, P

_{L}is the differential pressure across the piston, A

_{p}is the acting piston area, X

_{c}is the normalized command signal, X

_{p}is the piston position, and X

_{L}

_{1}is the normalized load position output of the main body in the equivalent linear form. Usually, the engine gimbaling angle is not measured in flight but in ground tests in the form of linear displacement which is converted to the angular value.

_{L}

_{1}. It is a classic aerospace actuator design, where the feedback signal is picked up via the sensor inside the piston rod rather than via the angular output sensor [2,28].

_{1}, K

_{L}

_{1}, B

_{L}

_{1}, X

_{L}

_{1}and M

_{2}, K

_{L}

_{2}, B

_{L}

_{2}, X

_{L}

_{2}are the equivalent mass, spring stiffness, viscous coefficient, linear displacement of the main engine body and the minor body, respectively, and the connected hydraulic actuator is given together.

_{v}is the ideal servo-valve flowrate output, K

_{qi}is the nominal servo-valve flowrate gain, I

_{v}is the electrical current applied, ω

_{v}is the first-order servo-valve frequency bandwidth, K

_{c}is the lumped leakage coefficient across the piston, including the internal leakage of the servo-valve, V

_{t}is the total control volume of the two actuator chambers, β is the equivalent bulk modulus of the contained oil, J is the lumped rotational load inertia, and R is the nominal rotation radius of the load.

_{L}

_{1}, ξ

_{L}

_{1}, ω

_{L}

_{2}, ξ

_{L}

_{2}) are structural resonant frequencies and the corresponding damping ratios arising for the main body and the minor body, respectively, and α is the structural stiffness ratio.

_{L}

_{1}, ω

_{L}

_{2}) and the damping ratios (ξ

_{L}

_{1}, ξ

_{L}

_{2}) are independent of any other electro-hydraulic actuator design, except for the installation geometry on which the rotation radius R depends. In ground testing, there are two measurement points, one at the piston rod as X

_{p}and the other at the engine gimbaling angular output as X

_{L}

_{1}.

_{L}

_{1}is expected, with the minor body angular output X

_{L}

_{2}eliminated, the transfer function blocks from X

_{p}to X

_{L}

_{1}are derived into Figure 5, i.e., the engine thrust vector output dynamics. With the feedback loop furtherly eliminated, the condensed form is given in Figure 6.

_{L}

_{1}, ω

_{L}

_{2}, ξ

_{L}

_{1}, ξ

_{L}

_{2}).

_{p}and X

_{L}

_{1}remained.

_{c}

_{1}, ω

_{c}

_{2}) are coupled from the structural resonant frequencies (${\omega}_{L1}^{\prime},{\omega}_{L2}^{\prime}$) and hydraulic resonant frequencies (ω

_{h}

_{1}, ω

_{h}

_{2}). Analogous to the one-mass-one-spring model [2,3,27], the first set of hydraulic and composite frequencies (ω

_{h}

_{1}, ω

_{c}

_{1}) can be given in Equation (14), similar forms of the second set of frequencies (ω

_{h}

_{2}, ω

_{c}

_{2}) are proposed in Equations (15) and (16), where k

_{ωh2}is a correction factor, and the damping ratios (ξ

_{c}

_{1}, ξ

_{c}

_{2}) have to be presented as (f

_{ξc}

_{1}, f

_{ξc}

_{2}) roughly, since a direct derivation is impossible. In fact, their precise algebraic representations are not cared much since they can be identified from experiment as shown later.

_{o}, the final normalized system model is given in Figure 10, where e(t) is the normalized position error between X

_{c}and X

_{p}.

_{o}is represented in Equation (19).

_{p}is the nominal error amplification gain, K

_{vi}is the lumped voltage-to-ampere conversion coefficient of the digital-to-analog (D/A) converter and the servo-valve coil driver, K

_{xf}is the lumped conversion coefficient of the analog-to-digital (A/D) converter and the feedback displacement sensor.

_{L}

_{1}, it is only a half-closed loop. It needs to note that the half-closed loop is a classic aerospace design [2,28], where the actuator acts as an integrated control device for the simplest design and therefore the most reliable reasons in a higher system perspective, rather than another angular sensor needed in flight, though D.V. Lazić studied the approach [4]. Inside the piston position X

_{p}loop, the fourth-order transfer function with two pairs of zeros and poles is dominant, both poorly damped, representing the effect of the outside fourth-order load dynamics on the closed position loop, called “load effect”, which is more complicated than that of an ordinary one-mass-one-spring modeled system [2,3,26].

_{c}

_{3}, ξ

_{c}

_{3}) are the corresponding third hydro-mechanical resonance frequency and its damping ratio, and (${\omega}_{L2}^{\u2033},{\xi}_{L2}^{\u2033}$, ${\omega}_{L3}^{\u2033},{\xi}_{L3}^{\u2033}$) are the derived resonance frequencies and their damping ratios in the numerator, which are different from (ω

_{L}

_{2}, ξ

_{L}

_{2}) as in a two-mass-two-spring model, since they can be not directly derived as in preceding Equation (10).

_{p}is needed to guarantee the composite hydro-mechanical frequencies are only slightly smaller than the structural frequencies, e.g., 10%. However, as for the heavy kerolox engine, since its structural natural frequencies are inevitably low, their effect on the system cannot be neglected as in common applications.

## 3. The Combined Control Algorithm

_{n}, K

_{p}(e(t)) − K

_{i}(e(t)) − K

_{d}(e(t)) and F(X

_{c}) represent the notch filter network, PID and feedforward controller, respectively.

_{n}is shown as Equation (20).

_{c}

_{1}, ξ

_{c}

_{1}) and (ω

_{c}

_{2}, ξ

_{c}

_{2}) are cancelled out by a pair of nearby poorly damped zeros (ω

_{n}

_{1}, ξ

_{n}

_{1}) and (ω

_{n}

_{2}, ξ

_{n}

_{2}) and replaced with a pair of better damped poles (ω

_{d}

_{1}, ξ

_{d}

_{1}) and (ω

_{d}

_{2}, ξ

_{d}

_{2}). In aerospace applications, the structural quality is guaranteed so that the resonance frequencies are controlled well within tolerances and notch filters can be applied in faith. On the other hand, the parameters can be optimized to change the width and depth of the notch window to accommodate permitted model variations, even to control a three resonance peak model as later shown. For non-stationary natural frequencies, Yao J. recommended adaptive notch filters [16].

_{dpf}is the gain and τ is the time constant. Therefore, a multiple-notch-filter network is an indispensable choice rather than a replaceable one.

_{K}is the bigger portion factor of the piecewise proportional gain and e

_{n}is prescribed error threshold under which the bigger gain is used.

_{in}is the nominal integral gain, K

_{i}

_{1}is the on-off switch triggered by the hydraulic power supply, P

_{s}(t) and P

_{sn}are the instant and nominal supply pressure, respectively, K

_{i}

_{2}is the on-off switch triggered by the maximum piston stroke, X

_{pmax}is the maximum piston stroke, K

_{i}

_{3}is the piecewise gain, and a and b are the constants to regulate the integral gain near and far from zero.

_{f}is the feedforward gain.

## 4. Model Identification

_{L}

_{1}and X

_{p}shown in Figure 14.

_{L}

_{1}in Figure 14. It is clear that the system has to be compensated for a bigger gain and hence better dynamics.

_{p}at the piston point from that of X

_{L}

_{1}at the load output. The resulted amplitude curve is plotted in Figure 15, where both an approximate fourth-order and sixth-order model are plotted together.

_{Lc}is the composite amplitude difference, ∆${A}_{L1}^{\prime}$, ∆${A}_{L2}^{\u2033}$ and ∆${A}_{L2}^{\prime}$ are the amplitude differences at peak and bottom frequencies, namely 70 rad/s, 80 rad/s and 100 rad/s, and 0.5, 0.2 and 0.3 are the weighting factors.

_{L}

_{2}, ξ

_{L}

_{2}) in the original fourth-order as in the preceding Figure 7, because it is a reduced form from the sixth-order.

_{po}is computed by breaking the closed loop response X

_{p}, as shown in Equations (29)–(32).

_{xp}and C

_{xpo}are the closed-loop and the open-loop frequency responses of the piston position X

_{p}represented in the complex form, and A

_{xp}, θ

_{xp}, A

_{xpo}, θ

_{xpo}are corresponding amplitude and phase responses, respectively.

_{c3}is 145 rad/s, lower than the corresponding load structural frequency ${\omega}_{L3}^{\prime}$ and located in the plotted frequency scope. As shown, a sixth-order model fits the data better than a fourth-order.

## 5. Experiments

_{L}

_{1}frequency response is shown in Figure 18, where the sixth-order simulation curve better fits the data than the fourth-order in most high frequency amplitude points. It is shown that the specification has been well satisfied. The response data at the marked frequencies is listed in Table 6.

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DPF | Dynamic Pressure Feedback |

PID | Proportional, Integral and Differential |

TVC | Thrust Vector Control |

EHA | Electro-hydrostatic Actuator |

DOF | Degree of freedom |

D/A | Digital-to-Analog |

A/D | Analog-to-Digital |

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**Figure 11.**A normalized block diagram of the aerospace electro-hydraulic actuation system that incorporates a sixth-order load model.

**Figure 18.**The final output X

_{L}

_{1}dynamics of the electro-hydraulic actuation system with the combined control algorithm.

Parameter | Symbol | Value | Unit |
---|---|---|---|

Lumped engine rotational inertia | J | 1304 | kg.m^{2} |

Engine rotational arm | R | 845 | mm |

Actuator acting piston area | A_{p} | 4398 | mm^{2} |

System pressure | P_{s} | 24 | MPa |

Servo-valve bandwidth (−45°) | ω_{v} | ≥180 | rad/s |

Nominal open loop gain | K_{o} | ≈20 | rad/s |

Maximum angular output | - | 8 | degree |

Digital Control Cycle | - | 0.001 | second |

Frequency (rad/s) | Amplitude Value Tolerance (dB) |
---|---|

60 | 1 |

70 | 2 |

75 | 2 |

80 | 3 |

90 | 3 |

100 | 2 |

110 | 3 |

130 | 5 |

140 | 8 |

150 | 8 |

160 | 8 |

Parameter | Symbol | Value | Unit |
---|---|---|---|

Derived equivalent main structural resonance frequency | ${\omega}_{L1}^{\prime}$ | 70 | rad/s |

Derived equivalent main resonance damping ratio | ${\xi}_{L1}^{\prime}$ | 0.05 | – |

Derived equivalent minor structural resonance frequency | ${\omega}_{L2}^{\prime}$ | 100 | rad/s |

Derived equivalent minor resonance damping ratio | ${\xi}_{L2}^{\prime}$ | 0.186 | – |

Numerator resonance frequency | ${\omega}_{L2}^{\u2033}$ | 80 | rad/s |

Numerator resonance damping ratio | ${\xi}_{L2}^{\u2033}$ | 0.034 | – |

First composite hydro-mechanical resonance frequency | ${\omega}_{c1}$ | 67 | rad/s |

First composite hydro-mechanical damping ratio | ${\xi}_{c1}$ | 0.055 | – |

Second composite hydro-mechanical resonance frequency | ${\omega}_{c2}$ | 85 | rad/s |

Second composite hydro-mechanical resonance damping ratio | ${\xi}_{c2}$ | 0.15 | – |

Parameter | Symbol | Value | Unit |
---|---|---|---|

First derived equivalent structural resonance frequency | ${\omega}_{L1}^{\prime}$ | 70 | rad/s |

First derived equivalent resonance damping ratio | ${\xi}_{L1}^{\prime}$ | 0.05 | – |

Second derived equivalent structural resonance frequency | ${\omega}_{L2}^{\prime}$ | 100 | rad/s |

Second derived equivalent resonance damping ratio | ${\xi}_{L2}^{\prime}$ | 0.14 | – |

Third derived equivalent structural resonance frequency | ${\omega}_{L3}^{\prime}$ | 158 | rad/s |

Third derived equivalent resonance damping ratio | ${\xi}_{L3}^{\prime}$ | 0.19 | – |

First numerator resonance Frequency | ${\omega}_{L2}^{\u2033}$ | 80 | rad/s |

First numerator resonance damping ratio | ${\xi}_{L2}^{\u2033}$ | 0.032 | – |

Second numerator resonance frequency | ${\omega}_{L3}^{\u2033}$ | 130 | rad/s |

Second numerator resonance damping ratio | ${\xi}_{L3}^{\u2033}$ | 0.09 | – |

First composite hydro-mechanical resonance frequency | ${\omega}_{c1}$ | 67 | rad/s |

First composite hydro-mechanical damping ratio | ${\xi}_{c1}$ | 0.055 | – |

Second composite hydro-mechanical resonance frequency | ${\omega}_{c2}$ | 85 | rad/s |

Second composite hydro-mechanical resonance damping ratio | ${\xi}_{c2}$ | 0.15 | – |

Third composite hydro-mechanical resonance frequency | ${\omega}_{c3}$ | 145 | rad/s |

Third composite hydro-mechanical resonance damping ratio | ${\xi}_{c3}$ | 0.1 | – |

Parameter | Symbol | Value | Unit |
---|---|---|---|

Nominal open-loop gain | ${K}_{o}$ | 20 | rad/s |

Nominal proportional factor | ${K}_{P}$ | 1.032 | – |

The enlargement proportional factor near zero error | ${f}_{K}$ | 1.5 | – |

Nominal integral factor | ${K}_{i}$ | 0.1 | – |

Nominal differential factor | ${K}_{d}$ | 0.006 | – |

Feedforward factor | ${K}_{f}$ | 0.005 | – |

Notch filter Resonance Frequency | ${\omega}_{n1}$ | 67 | rad/s |

Notch filter Damping Ratio | ${\xi}_{n1}$ | 0.03 | – |

Notch filter Resonance Frequency | ${\omega}_{d1}$ | 70 | rad/s |

Notch filter Damping Ratio | ${\xi}_{d1}$ | 0.4 | – |

Notch filter Resonance Frequency | ${\omega}_{n2}$ | 135 | rad/s |

Notch filter Damping Ratio | ${\xi}_{n2}$ | 0.04 | – |

Notch filter Resonance Frequency | ${\omega}_{d2}$ | 135 | rad/s |

Notch filter Damping Ratio | ${\xi}_{d2}$ | 0.35 | – |

**Table 6.**The compensated load dynamic response at the specified frequencies with the combined control algorithm.

Frequency (Rad/s) | Amplitude (dB) | Phase (Degree) |
---|---|---|

2 | 0.01 | −4.31 |

4 | 0.04 | −8.07 |

6.28 | 0.05 | −12.39 |

10 | 0.13 | −18.89 |

15 | 0.49 | −28.67 |

20 | 0.8 | −39.35 |

30 | −0.2 | −72.81 |

40 | −0.58 | −103.29 |

50 | −1.54 | −135.15 |

60 | −3.51 | −177.24 |

70 | −14.95 | −208.76 |

80 | −22.49 | −178.13 |

90 | −18.05 | −220.64 |

100 | −20.4 | −255.91 |

110 | −24.62 | −267.25 |

120 | −30.38 | −289.19 |

130 | −38.91 | −285.18 |

140 | −43.49 | −255.64 |

150 | −41.15 | −180.31 |

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**MDPI and ACS Style**

Zhao, S.; Chen, K.; Zhang, X.; Zhao, Y.; Jing, G.; Yin, C.; Xiao, X.
A High-Order Load Model and the Control Algorithm for an Aerospace Electro-Hydraulic Actuator. *Actuators* **2021**, *10*, 53.
https://doi.org/10.3390/act10030053

**AMA Style**

Zhao S, Chen K, Zhang X, Zhao Y, Jing G, Yin C, Xiao X.
A High-Order Load Model and the Control Algorithm for an Aerospace Electro-Hydraulic Actuator. *Actuators*. 2021; 10(3):53.
https://doi.org/10.3390/act10030053

**Chicago/Turabian Style**

Zhao, Shoujun, Keqin Chen, Xiaosha Zhang, Yingxin Zhao, Guanghui Jing, Chuanwei Yin, and Xue Xiao.
2021. "A High-Order Load Model and the Control Algorithm for an Aerospace Electro-Hydraulic Actuator" *Actuators* 10, no. 3: 53.
https://doi.org/10.3390/act10030053