# Quantitative Fault Diagnostics of Hydraulic Cylinder Using Particle Filter

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

- (1)
- A physics-based quantitative fault diagnostics problem is formally formulated, and a joint state-parameter estimation-based architecture is proposed.
- (2)
- The fault modes and their impact on the hydraulic cylinder are analytically identified and revealed through the establishment of a nonlinear dynamic model.
- (3)
- A particle filter-based quantitative fault diagnostics method is proposed and validated, enabling accurate quantitative diagnosis of multiple faults for hydraulic cylinders.

## 2. Quantitative Fault Diagnostics Approach

#### 2.1. Problem Formulation

**x**

_{k},

**θ**

_{k}). Under these circumstances, to give a meaningful estimation, it is important to obtain a probability distribution of the state-parameter estimation as opposed to a single point of estimation. Thus, the goal of quantitative fault diagnostics is to estimate the probability of the state and parameter given the input-output data, i.e., $p\left(\left.{\mathit{x}}_{k},{\mathit{\theta}}_{k}\right|{\mathit{u}}_{0:k-1},{\mathit{z}}_{0:k-1}\right)$.

#### 2.2. Quantitative Fault Diagnostics Architecture

**u**’ is used as the input to the physical plant, and the noisy output measurement ‘

**z**’ is generated as the output. When the same input signal is applied to the system model, an estimate of the output $\hat{\mathit{z}}$ can be obtained. With the output measurement, estimated output, and information about measurement noise taken into consideration, the probability distribution of the states and time-varying parameters can be calculated by the joint state-parameter estimator. The probability distribution of the states and time-varying parameters is then passed to the fault degree function $DF\left(\xb7\right)$, where a quantitative description of the degree of fault can be obtained.

## 3. Modeling and Analysis of the System

^{®}SimScape model. By utilizing the system model, we analyze the impact of time-varying parameters, enabling us to theoretically identify the fault modes of the hydraulic cylinder.

#### 3.1. Modeling of the System

- (1)
- The temperature, viscosity, and bulk modulus of the working fluid remain constant.
- (2)
- Leakage flows are characterized as laminar.
- (3)
- The pressure within the working chambers of the cylinder is uniformly distributed.

#### 3.2. Verification of the Model

^{®}SimScape model of the corresponding system was constructed. By employing the identical time-invariant parameters outlined in Table 2, along with the designated input signals, a comparison was made between the output results of the analytical model and the SimScape model. This comparison is presented in Figure 3. The evident agreement between the analytical model and the SimScape model serves to validate the efficacy of the analytical model. The small differences are given by the numerical integrator choice.

#### 3.3. Impact of the Time-Varying Parameters

_{1}functions as the working pressure, and P

_{2}acts as the back pressure. Conversely, in the retraction stroke, P

_{1}switches roles to become the back pressure, while P

_{2}becomes the working pressure. Throughout both strokes, the friction force consistently opposes the direction of motion. As a result, the impact of B on P

_{1}and P

_{2}demonstrates a contrasting pattern. During the extension stroke, an increase in B leads to a rise in P

_{1}and a decrease in P

_{2}. Conversely, in the retraction stroke, an increase in B causes P

_{1}to decrease and P

_{2}to increase. In magnitude, variations in B have a notably substantial impact on x, v, and P

_{1}, while the effect on P

_{2}is comparatively minor.

_{il}is directly connected to the rate of internal leakage flow via Equation (16). This influence of C

_{il}is depicted in Figure 4b. The effect of C

_{il}on both x and v closely resembles that of B. An increase in C

_{il}corresponds to a decrease in both extension and retraction speeds, consequently leading to an extended duration for extension/retraction. With an increase in C

_{il}, both P

_{1}and P

_{2}exhibit slight increments during the extension stroke, while they decrease in the retraction stroke. Notably, the influence of C

_{il}on P

_{2}is slightly more pronounced than on P

_{1}. The variation in C

_{il}has a more prominent effect on P

_{1}and P

_{2}when the piston is at rest compared to when it is in motion.

_{el}corresponds to the rate of external leakage flow, as shown in Equation (17). Referring to Figure 4c, an increase in C

_{el}has a minor effect on the extension speed, but significantly reduces the retraction speed. This disparity in impact can be attributed to the occurrence of external leakage exclusively at the piston rod seal. Consequently, the influence on the pressure in the rod chamber (P

_{2}) is notably greater than that on the pressure in the piston chamber (P

_{1}).

## 4. Joint State-Parameter Estimation

Algorithm 1: Sequential importance sampling with resampling particle filter | ||

Input: ${\left\{{w}_{k-1}^{i},{\tilde{\mathit{x}}}_{k-1}^{i}\right\}}_{i=1}^{N}$, ${\mathit{u}}_{k-1}$, ${\mathit{z}}_{k}$ | ||

Output: ${\left\{{w}_{k}^{i},{\tilde{\mathit{x}}}_{k}^{i}\right\}}_{i=1}^{N}$ | ||

for i = 1 to N do | ||

${\tilde{\mathit{x}}}_{k}^{i}\sim p\left({\tilde{\mathit{x}}}_{k}\left|{\tilde{\mathit{x}}}_{k-1},{\mathit{u}}_{k-1}\right.\right)\phantom{\rule{8pt}{0ex}}$# Propagate particles | ||

${\tilde{\mathit{x}}}_{k}^{i}={\tilde{\mathit{x}}}_{k}^{i}+{\mathit{Q}}_{k}^{i}\phantom{\rule{41pt}{0ex}}$# Particles evolution with process noise | ||

${w}_{k}^{i}=p\left({\mathit{z}}_{k}\left|{\tilde{\mathit{x}}}_{k}^{i}\right.\right)\phantom{\rule{34pt}{0ex}}$# Update importance weight | ||

end | ||

${w}_{sum}={\displaystyle \sum _{i=1}^{N}{w}_{k}^{i}}\phantom{\rule{50pt}{0ex}}$# Cumulative weight | ||

for i = 1 to N do | ||

${w}_{k}^{i}={w}_{k}^{i}/{w}_{sum}\phantom{\rule{37pt}{0ex}}$# Normalize weights | ||

end | ||

if ${N}_{eff}<{N}_{threshold}$ then # Resample if needed | ||

Resample N particles with replacement | ||

for i = 1 to N do | ||

${w}_{k}^{i}=1/N\phantom{\rule{36pt}{0ex}}$# Reset weights | ||

end | ||

end |

_{m}is the number of the measurement, equal to 4 in our case, and $\Sigma $ is the covariance matrix. The mean vector of this distribution is set to be equal to the estimated output ${\hat{\mathit{z}}}_{k}$, and the covariance matrix equals the covariance of the measurement noise. We evaluate this distribution at the actual measurement ${\mathit{z}}_{k}$. Then, we multiply the resulting values together to obtain a single likelihood value for the particle. Once the weights are updated for each particle, a normalization step is performed. This normalization involves dividing each particle’s weight by the cumulative sum of all the weights, ensuring that they collectively add up to one.

_{threshold}. Following resampling, the particle weights are set to equal values of 1/N.

## 5. Performance Evaluation and Analysis

_{el}converges more slowly, and the convergence speed of the internal leakage coefficient C

_{il}falls in between.

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

${P}_{1}$ | Pressure in the piston chamber, Pa |

${P}_{2}$ | Pressure in the rod chamber, Pa |

${P}_{s}$ | Supply pressure, Pa |

${P}_{b}$ | Back pressure, Pa |

${A}_{1}$ | Area of piston on the bore side, m^{2} |

${A}_{2}$ | Area of piston on the rod side, m^{2} |

${a}_{1}$ | Orifice area, m^{2} |

${a}_{2}$ | Orifice area, m^{2} |

$x$ | Displacement of the piston, m |

$\dot{x}$ | Velocity of the piston, m/s |

${V}_{pis,dead}$ | Dead volume of the bore chamber, m^{3} |

${V}_{rod,dead}$ | Dead volume of the rod chamber, m^{3} |

$\beta $ | Bulk modulus, Pa |

${Q}_{1,ext}$ | Flow rate to the piston chamber, m^{3}/s |

${Q}_{2,ext}$ | Flow rate from the rod chamber, m^{3}/s |

${Q}_{1,ret}$ | Flow rate from the piston chamber, m^{3}/s |

${Q}_{2,ret}$ | Flow rate to the rod chamber, m^{3}/s |

${Q}_{il}$ | Internal leakage, m^{3}/s |

${Q}_{el}$ | External leakage, m^{3}/s |

${L}_{s}$ | Stroke of the piston, m |

${C}_{d1}$ | Discharge coefficient |

${C}_{d2}$ | Discharge coefficient |

$\rho $ | Fluid density, kg/m^{3} |

$B$ | Viscous coefficient, N/(m/s) |

$K$ | Elastic load stiffness, N/m |

${F}_{L}$ | Loading force, N |

${F}_{c}$ | Contact force, N |

$m$ | Mass of the moving parts, kg |

${K}_{c}$ | Stiffness of the contact load, N/m |

${B}_{c}$ | Viscous coefficient of the contact load, N/(m/s) |

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**Figure 3.**Comparison between the Simscape model and the analytical model: (

**a**) position; (

**b**) velocity; (

**c**) piston chamber pressure; (

**d**) rod chamber pressure.

**Figure 4.**Influence of the time-varying parameters on the states: (

**a**) influence of B; (

**b**) influence of C

_{il}; (

**c**) influence of C

_{el.}

**Figure 5.**Hydraulic cylinder fault phenomenon: (

**a**) torn cylinder wall; (

**b**) worn piston seal; (

**c**) external leakage.

**Figure 7.**Particle initialization: (

**a**) x; (

**b**) v; (

**c**) P

_{1}; (

**d**) P

_{2}; (

**e**) B; (

**f**) C

_{il}; (

**g**) C

_{el}.

**Figure 8.**Evolution of particles for Exp. 1: (

**a**) particles for B; (

**b**) particles for C

_{il}; (

**c**) particles for C

_{el}.

**Figure 9.**State estimation of Exp. 1: (

**a**) position; (

**b**) velocity; (

**c**) piston chamber pressure; (

**d**) rod chamber pressure.

Estimator | Model | Assumed Distribution |
---|---|---|

Kalman filter | Linear | Gaussian |

Extend Kalman filter | Locally linear | Gaussian |

Unscented Kalman filter | Non-linear | Gaussian |

Particle filter | Non-linear | Non-Gaussian |

No. | Time-Invariant Parameters | Values | No. | Time-Invariant Parameters | Values |
---|---|---|---|---|---|

1 | m | 500 kg | 8 | a_{1} | 7.854 × 10^{−5} m^{2} |

2 | K | 10 N/m | 9 | a_{2} | 7.854 × 10^{−5} m^{2} |

3 | A_{1} | 1.76 × 10^{−2} m^{2} | 10 | ρ | 870 kg/m^{3} |

4 | A_{2} | 8.45 × 10^{−3} m^{2} | 11 | L_{s} | 0.8 m |

5 | β | 7 × 10^{8} Pa | 12 | k_{c} | 1 × 10^{8} N/m |

6 | C_{d}_{1} | 0.7 | 13 | B_{c} | 1.5 × 10^{5} N/(m/s) |

7 | C_{d}_{2} | 0.7 |

Experiments | Operating Condition ^{1} | Parameters True Values ^{1} |
---|---|---|

Exp. 1 | P_{s} = 25, P_{b} = 0, F_{L}_{,ext} = 20, F_{L}_{,ret} = 0 | B = 10, C_{il} = 20 × 10^{−11}, C_{el} = 10 × 10^{−10} |

Exp. 2 | P_{s} = 25, P_{b} = 0, F_{L}_{,ext} = 20, F_{L}_{,ret} = 0 | B = 20, C_{il} = 40 × 10^{−11}, C_{el} = 15 × 10^{−10} |

Exp. 3 | P_{s} = 35, P_{b} = 0, F_{L}_{,ext} = 30, F_{L}_{,ret} = 0 | B = 10, C_{il} = 20 × 10^{−11}, C_{el} = 10 × 10^{−10} |

Exp. 4 | P_{s} = 15, P_{b} = 0, F_{L}_{,ext} = 10, F_{L}_{,ret} = 0 | B = 15, C_{il} = 25 × 10^{−11}, C_{el} = 20 × 10^{−10} |

^{1}Units: P

_{s}, P

_{b}: bar; F

_{L}

_{,ext}, F

_{L}

_{,ret}: kN; B: kN/(m/s); C

_{il}, C

_{el}: m

^{3}/(s·Pa).

State [Unit] | [Mean, Standard Deviation] | Parameter [Unit] | [Minimum, Maximum] |
---|---|---|---|

x [m] | [0, 0.0018] | B [kN/(m/s)] | [0, 70] |

v [m/s] | [0, 0.0006] | C_{il} [10^{−11} m^{3}/(s·Pa)] | [0, 50] |

P_{1} [bar] | [0, 0.03] | C_{el} [10^{−10} m^{3}/(s·Pa)] | [0, 25] |

P_{2} [bar] | [0, 0.03] |

B | C_{il} | C_{el} | |
---|---|---|---|

MAE | 0.057 | 0.149 | 0.020 |

MAPE | 0.382% | 0.595% | 0.100% |

Methods | Qualitative/Quantitative | Deterministic/Stochastic | Generalization Ability |
---|---|---|---|

Data-driven | Qualitative | Deterministic/Stochastic | Poor |

Analytical redundancy | Qualitative | Deterministic | Good |

Adaptive robust observer | Quantitative | Deterministic | Good |

Particle filter | Quantitative | Stochastic | Good |

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## Share and Cite

**MDPI and ACS Style**

Zhang, Y.; Vacca, A.; Gong, G.; Yang, H.
Quantitative Fault Diagnostics of Hydraulic Cylinder Using Particle Filter. *Machines* **2023**, *11*, 1019.
https://doi.org/10.3390/machines11111019

**AMA Style**

Zhang Y, Vacca A, Gong G, Yang H.
Quantitative Fault Diagnostics of Hydraulic Cylinder Using Particle Filter. *Machines*. 2023; 11(11):1019.
https://doi.org/10.3390/machines11111019

**Chicago/Turabian Style**

Zhang, Yakun, Andrea Vacca, Guofang Gong, and Huayong Yang.
2023. "Quantitative Fault Diagnostics of Hydraulic Cylinder Using Particle Filter" *Machines* 11, no. 11: 1019.
https://doi.org/10.3390/machines11111019