# A Multi-Model-Particle Filtering-Based Prognostic Approach to Consider Uncertainties in RUL Predictions

## Abstract

**:**

## 1. Introduction

#### 1.1. Prognostic Approaches

#### 1.2. Uncertainties in Predictions

## 2. Developing a Prognostic Approach Considering Uncertainties

#### 2.1. General Stochastic Filtering Approaches

_{k}at time t

_{k}with k ∈ ℕ by stochastic estimates of the next state x

_{k+1}until a predefined threshold is reached. These estimates rely on the state model f in Equation (1) that considers the previous state x

_{k}and some related noise ν

_{k}.

_{k}is related to the measurement z

_{k}at time t

_{k}. This relation is modeled by the measurement model g in Equation (2), which considers again some related noise κ

_{k}.

_{k+1}is based on stochastic conditions and can be divided into two steps. In both steps, the Bayes’ theorem is used to calculate conditional probabilities for the estimation of x

_{k+1}. The detailed derivation of the states, the particular probability density functions (PDFs), and the integral formulation can be found in [57,58].

_{k+1}can only be solved analytically for a few applications. Numerical solutions are given by extended Kalman or Particle Filtering-based approaches [57].

#### 2.2. Particle Filter

_{k+1}is only available if k is dated in the past. Therefore, the second step, the update of the estimated state by a measurement, is only applicable until the current time. For future estimations, the prognostic approach strongly relies on the a priori PDF that is built upon prior knowledge. Finally, the time span of the Particle Filtering-based prognostic approach extends p steps-ahead until a criterion is fulfilled. For RUL estimation, this criterion is given by a failure threshold. In the following, “Particle Filter” is always used in the prognostics’ context, unless it is explicitly referred to a different background.

#### 2.3. Multi-Model-Particle Filter

_{k+1}. Thereby, in the developed Multi-Model-Particle Filter, each particle i is connected to one state model f

_{mi}in the state estimation process:

_{mi}enables a prediction of the next state, but in the resampling step only the states and the particular models are reused that show a high importance. While at the beginning an equal number of each state models is initialized, due to altering weights depending on the importance of the corresponding particle, the composition of used models alternates based on the resampling step. However, as long as measurement quantities are available, the particles are weighted according to their ability to accurately describe the current state. In this way, the influence of each model and the related particle is changed over the prediction steps. Once no further measurement quantities are available, equal weights of the remaining models are used to estimate the next state. After resampling, a suitable state estimation is derived from the distribution of particles. On the basis of this estimated state, it is checked whether the estimated adaptive threshold ft is exceeded by the estimated state. Based on that result, either the next state is predicted or the predicted RUL is derived. The estimation of the adaptive thresholds (see right part of Figure 2) is described in the next subsection.

#### 2.4. Estimating Thresholds Considering Uncertainties

_{s}, given as the mean of the maximum quantity Q

_{s,max}and the minimum quantity Q

_{s,min}during health state II, and the measurement quantity Q

_{e}at the end of life, as depicted in Figure 3. The three health states are separated and labelled: health state I (left side), health state II (middle), and health state III (right side). The relation between the measurement quantity Q

_{e}and the stable quantity Q

_{s}is calculated for all historical data. The mean relation is used to estimate the adaptive threshold of element j ft

_{adapt,j}when the system’s state is in health state II. Thus, depending on the current measurement in this health state, an individual threshold is estimated in Equation (5):

_{e}at the end of life are considered. Furthermore, deviations between the stable quantity of various similar systems are considered as well. The presented approach has been proven to be most suitable for the application in focus, therefore it is implemented in the prognostic approach [67].

#### 2.5. Use Case: Rubber-Metal-Elements

#### 2.5.1. Attributes and Applications

#### 2.5.2. Methods to Approximate Lifetime

#### 2.5.3. Lifetime Tests

#### 2.5.4. Developing Measurement Concepts

#### Displacement-Based Concept

#### Temperature-Based Concept

#### 2.5.5. Preprocessing and Feature Selection

#### 2.5.6. Uncertainty Analysis

_{1}…p

_{5}. These parameters are estimated individually for each RM-element through optimization. Hence, such a state model is derived for each measurement quantity of each historical element and embedded in the Multi-Model-Particle Filter. Therefore, the state x in Equation (6) is replaced by the displacement amplitude or the relative temperature. Thus, the selection of the model noise ν depends on the discrepancy between the selected model and the measurement quantity. The comparison of model and measurement is shown exemplarily Figure 9 for the relative temperature. The general course of the developing state can be described by the model. However, the deviations within the curves are not modeled by the state model.

^{2}for temperature measurements and 0.002 mm

^{2}for displacement measurements. The difference between these two errors is again an indication of less deviations within the displacement amplitudes. Thus, the model noise ν for predictions based on the displacement amplitude is smaller than the one for predictions based on the relative temperature. Measurement noise is often modeled as a Gaussian distribution [17], even if Particle Filtering-based approaches enable different distributions. The frequency distribution of measured temperatures in Figure 10a suggests that a Gaussian distribution is able to characterize the noise within the measurements. It is assumed that the measured displacements show similar behavior. Figure 10b depicts the histogram of the difference between the optimized model and the particular measured relative temperature for one lifetime test. The best fitted distributions are the normal and generalized extreme value (GEV) distribution. For most of the elements, the results of these two distributions are similar. Again, it is assumed that that difference shows a similar characteristic for displacement measurements. Therefore, in this work model, noise is realized as a normal distributed value around a mean value depending on the particular measurement quantity.

## 3. Evaluation of the Developed Condition Monitoring System of Rubber-Metal-Elements

#### 3.1. Predictions of RUL

_{P}as the sum of all prediction time points and n as the index of the focused time point. The rate of negative errors correlates to the rate of too large predictions that lead to an undesired breakdown of the monitored element. Therefore, the sum of events when Equation (8)

_{P}. While MAPE and the rate of negative errors are time-independent indices, the prognostic horizon (PH) takes time dependency into account. It is defined as the time between the end of life t

_{e}and the first prediction time point n from which all following predictions are within a defined threshold α. That threshold spreads a tolerance band around the actual RUL. The PH is defined by

^{5}cycles is represented by the threshold α. For the heat chamber group, a standard deviation of 5 × 10

^{4}cycles is approximated due to the small database. These thresholds are generously dimensioned, but they characterize the uncertainty within the data.

#### 3.2. Comparison of Classical One-Model Particle Filtering-based Prognostic Approach and the Developed Multi-Model-Particle Filtering-based Prognostic Approach

#### 3.3. Discussion

## 4. Conclusions

## Funding

## Conflicts of Interest

## Appendix A

Load measurements of the current system z_{1:c} until current time t_{c} |

Load state model combination or m models |

Select only models that enable a longer lifetime than time t_{c}: t_{end}(m_{i})> t_{c} |

Either define fixed failure threshold ft or estimate adaptive ft (according to Equation (5)) |

Set parameters of the particle filter based on the measurements and its uncertainty |

Initialize particles equally over m models (see Equation (4)) |

Estimate current state x_{k} based on the particles |

For x_{k} < ft |

Estimate next state x_{k+1} (according to Equation (4)) |

Is t_{c} > t_{k+1} |

Estimate (importance) weights for each particle and normalize them |

Importance resampling of particles and connected models to update next state x_{k+1} |

Else |

x_{k+1} = x_{k+1} and use the same models for the next time step k+1 |

Ek = k + 1 |

Estimate pRUL based on the time steps k and the current time t_{c} |

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**Figure 2.**Procedure of RUL prediction based on the developed Multi-Model-Particle Filter [63].

**Figure 3.**Estimating threshold (based on [63]).

**Figure 4.**Lifetime tests of Rubber-Metal-Elements: (

**a**) test bed (based on [77]) and (

**b**) displacement measurement during an element’s lifetime.

**Figure 5.**Hysteresis loops at different points of time: (

**a**) at the beginning, (

**b**) in the middle, and (

**c**) at the end of an element’s lifetime [59].

**Figure 8.**Uncertainties at the beginning of the element’s lifetime for (

**a**) displacement amplitude and (

**b**) relative temperature [63].

**Figure 9.**Comparison of model and measurement [63].

**Figure 10.**Analysis of uncertainty: (

**a**) Frequency distribution of measured temperature [63] and (

**b**) approximation of the distribution of the difference between measurement and model by a normal and a generalized extreme value (GEV) distribution.

Feature | Reference |
---|---|

Damping work | [71,72,78] |

Dynamically stored energy | [79] |

Relative change in length | [80] |

Crack length | [81] |

Crack depth | [82] |

Rate of crack growth | [83] |

Strain amplitude | [84,85,86] |

Stiffness | [73,87,88,89] |

Tear energy | [76,90] |

**Table 2.**Performance indices for the displacement-based and the temperature-based predictions realized by the Multi-Model-Particle Filtering-based prognostic approach.

RM-Element | Displacement-Based Concept | Temperature-Based Concept | ||||
---|---|---|---|---|---|---|

Mean MAPE | Mean Rate of Negative Errors | Mean PH | Mean MAPE | Mean Rate of Negative Errors | Mean PH | |

3 | 45.3 | 11/17 | 0.15–0.95 | 14.9 | 5/17 | 0.15–0.95 |

4 | 19.9 | 8/17 | 0.15–0.95 | 20.8 | 7/17 | 0.15–0.95 |

5 | 22.7 | 4/17 | 0.15–0.95 | 41.4 | 0/17 | 0.45–0.95 |

6 | 56.3 | 0/17 | 0.45–0.95 | 66.8 | 0/17 | 0.80–0.95 |

7 | 42.6 | 1/17 | 0.70–0.95 | 82.5 | 1/17 | 0.85–0.95 |

11 | 53.2 | 0/17 | 0.70–0.95 | 43.9 | 2/17 | 0.75–0.95 |

12 | 57.8 | 17/17 | 0.15–0.95 | 26.9 | 16/17 | 0.70–0.95 |

Mean | 42.5 | 6/17 | 0.35–0.95 | 42.4 | 4/17 | 0.55–0.95 |

**Table 3.**Performance indices for the displacement-based and the temperature-based predictions realized by the classical Particle Filtering-based prognostic approach.

RM-Element | Displacement-Based Concept | Temperature-Based Concept | ||||
---|---|---|---|---|---|---|

Mean MAPE | Mean Rate of Negative Errors | Mean PH | Mean MAPE | Mean Rate of Negative Errors | Mean PH | |

3 | 66.9 | 10/17 | 0.40–0.95 | 157.5 | 14/17 | 0.60–0.95 |

4 | 44.7 | 11/17 | 0.30–0.95 | 127.4 | 14/17 | 0.50–0.95 |

5 | 43.9 | 5/17 | 0.45–0.95 | 79.5 | 7/17 | 0.30–0.95 |

6 | 49.3 | 3/17 | 0.60–0.95 | 66.3 | 3/17 | 0.50–0.95 |

7 | 79.0 | 0/17 | 0.65–0.95 | 83.3 | 0/17 | 0.60–0.95 |

11 | 43.2 | 2/17 | 0.80–0.95 | 44.7 | 0/17 | 0.55–0.95 |

12 | 27.2 | 16/17 | 0.70–0.95 | 50.6 | 17/17 | 0.15–0.95 |

Mean | 50.6 | 7/17 | 0.55–0.95 | 87.1 | 8/17 | 0.45–0.95 |

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**MDPI and ACS Style**

Bender, A.
A Multi-Model-Particle Filtering-Based Prognostic Approach to Consider Uncertainties in RUL Predictions. *Machines* **2021**, *9*, 210.
https://doi.org/10.3390/machines9100210

**AMA Style**

Bender A.
A Multi-Model-Particle Filtering-Based Prognostic Approach to Consider Uncertainties in RUL Predictions. *Machines*. 2021; 9(10):210.
https://doi.org/10.3390/machines9100210

**Chicago/Turabian Style**

Bender, Amelie.
2021. "A Multi-Model-Particle Filtering-Based Prognostic Approach to Consider Uncertainties in RUL Predictions" *Machines* 9, no. 10: 210.
https://doi.org/10.3390/machines9100210