# Fault Detection and Isolation via the Interacting Multiple Model Approach Applied to Drive-By-Wire Vehicles

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## Abstract

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## 1. Introduction

- The human–machine interfaces (HMI) are typically the steering wheel and the pedals (brake and throttle). They collect the driver’s input and provide him with haptic feedback (wheel alignment torque for instance) and eventual incentive information.
- The sensor set corresponds to a standard vehicle equipped with an ESC (electronic stability control) system: wheel speed sensors, an inertial navigation system (INS), including accelerometers and gyrometers, a steering wheel angle sensor (SAS), pedals stroke and braking pressure sensors. Moreover, a sensor for the steering angle of the front wheels is required as it is subjected to differ from the steering wheel angle.
- The actuators include the brakes (electronic or electro-hydraulic) and the electronic steering system.
- The control unit supervises the system, i.e., estimates the vehicle dynamics, identifies eventual faults, and controls the actuators consequently.

- Sensor faults: X- and Y-accelerations, vehicle yaw rate, wheel turn rates and the steering angle of front wheels.
- Actuator faults: Steering actuator and brakes.

## 2. The IMM Approach for the Fault Detection and Isolation

#### 2.1. Principles of the IMM Estimation

- First, the different estimates and their covariance matrices are mixed, according to the probabilities of activation ${\mu}_{i}$ of the model at the current time k. The predicted mode probability ${\mu}_{j}$, defined by:$${\mu}_{j,k+1|k}=\sum _{i}{\pi}_{ij}{\mu}_{i,k}$$$${\mu}_{i|j,k}=\frac{{\pi}_{ij}{\mu}_{i,k}}{{\mu}_{j,k+1|k}}$$The mixed estimates ${\widehat{x}}_{j,k-1}^{0}$ are then defined by:$${\widehat{x}}_{j,k}^{0}=\sum _{i=1}^{s}{\mu}_{i|j,k}{\widehat{x}}_{i,k}$$$${P}_{j,k}^{0}=\sum _{i=1}^{s}\left[{P}_{i,k}+\Delta {\widehat{x}}_{i|j}\right]{\mu}_{i|j,k}$$$$\Delta {\widehat{x}}_{i|j}=\left({\widehat{x}}_{j,k}^{0}-{\widehat{x}}_{i,k}\right){\left({\widehat{x}}_{j,k}^{0}-{\widehat{x}}_{i,k}\right)}^{t}$$
- Second, a probabilistic filter is performed for each mode in parallel to obtain the updated estimates ${\widehat{x}}_{j,k+1}$ and their covariance ${P}_{j,k+1}$. Starting from each mixed estimate ${\widehat{x}}_{j,k}^{0}$, an evolution model is used to predict the next state vector, and then this prediction is corrected according to the last measured data. In Section 3, different probabilistic filters are investigated: the extended Kalman filter (EKF), the unscented Kalman filter (UKF), and the (first-order) divided differences filter (DD1).
- The activation probability update is done with the computation of the likelihood ${L}_{j}$:$${L}_{j,k}=\frac{1}{{\left(2\pi \right)}^{d/2}\sqrt{det{\Lambda}_{j,k}}}exp\left[-\frac{1}{2}{\nu}_{j,k}^{\mathrm{T}}{\Lambda}_{j,k}^{-1}{\nu}_{j,k}\right]$$$${\mu}_{j,k+1}=\frac{{\mu}_{j,k+1|k}{L}_{j,k+1}}{{\sum}_{i}{\mu}_{i,k+1|k}{L}_{i,k+1}}$$
- Finally, the overall estimate ${\widehat{x}}_{k+1}$ and an overall covariance matrix ${P}_{k+1}$ are computed by weighting the different estimates by their respective probabilities. This step is optional and is required only if an overall estimate is needed, for instance for control purpose.$${\widehat{x}}_{k+1}=\sum _{j}{\mu}_{j,k+1}{\widehat{x}}_{j,k+1}$$$${P}_{k+1}=\sum _{j}{\mu}_{j,k+1}\left[{P}_{j,k+1}+\Delta {\widehat{x}}_{j}\Delta {\widehat{x}}_{j}^{t}\right]$$$$\Delta {\widehat{x}}_{j}={\widehat{x}}_{j,k+1}-{\widehat{x}}_{k+1}$$

#### 2.2. Dedicated Implementation for FDI Purpose

- Four modes (${m}_{2,3,4,5}$) (called S ${\Omega}_{FR}$, S ${\Omega}_{FL}$, S ${\Omega}_{RL}$ and S ${\Omega}_{RR}$) correspond to a fault on the wheel turn rate signals (respectively at the front right, front left, rear left and rear right wheels.
- Three modes (${m}_{6,7,8}$) are for the inertial sensor faults (S A${}_{X}$ and S A${}_{Y}$ for the longitudinal and lateral accelerations, and S YR for the yaw rate).
- One mode (${m}_{9}$) is for a fault on the front steering angle signal (S $\delta $).

- One mode (${m}_{10}$) is for a fault on the steering angle actuator (A $\delta $).
- Four modes (${m}_{11,12,13,14}$) (called A Br${}_{FR}$, A Br${}_{FL}$, A Br${}_{RL}$ and A Br${}_{RR}$) correspond to a fault on the braking system (respectively, at the front right, front left, rear left and rear right wheels).

#### 2.3. Transition Probabilities

- If ${\pi}_{ij}$ is set equal to the identity matrix, we show easily from Equations (4) and (5) that the mixed probabilities and the mixed estimates remains equal to the previously estimated ones. This results in skipping the first step of the IMM algorithm. In this case, the IMM behaves exactly like a standard multiple model approach, with the risk of late detection explained the introduction.
- If each term is set equal to 1/n (where n is the number of modes), then, in Step 1, the mixed estimates for all modes becomes equal to the previous overall estimate, and the mixed probabilities of activation becomes equal to 1/n. In this case, the previous iterations are totally forgotten, and the fault detection only relies on the current iteration. Thus, increasing the risk of false detection.

#### 2.4. Immunization to Faults

- For a total default, the easiest way is to cancel the corresponding line in the measurement matrix H.
- For a partial default, the corresponding parameter can be increased in the measurement noise covariance matrix R.

## 3. Probabilistic Vehicle State Observer

#### 3.1. The Extended Kalman Filter

- During the prediction step, the predicted state estimate $\widehat{x}(k+1|k)$ and its predicted covariance $P(k+1|k)$ are estimated following the non-linear evolution function f and its Jacobian matrix F, according to Equations (14) and (15).$${\widehat{x}}_{k+1|k}=f\left({\widehat{x}}_{k},{u}_{k}\right)$$$${P}_{k+1|k}=F{P}_{k}{F}^{t}+Q$$
- In the update stage, the predicted estimate is corrected according to the updated output vector ${y}_{k+1}$. The residual $\nu $ and its covariance $\Lambda $ are evaluated according to the measurement matrix H.$${\nu}_{k+1}={y}_{k+1}-H{\widehat{x}}_{k+1|k}$$$${\Lambda}_{k+1}=H{P}_{k+1|k}{H}^{\mathrm{T}}+R$$The filter gain K can then be calculated according to Equation (18).$${K}_{k+1}={P}_{k+1|k}{H}^{\mathrm{T}}{\Lambda}_{k+1}^{-1}$$

#### 3.2. The First-Order Divided Differences Filter

- In the prediction step, the predicted state vector ${\widehat{x}}_{k+1|k}$ and its covariance ${P}_{k+1|k}$ are computed according to:$${\widehat{x}}_{k+1|k}=f\left({\widehat{x}}_{k},{u}_{k}\right)$$$${P}_{k+1|k}=\left({S}_{x\widehat{x}}\right){\left({S}_{x\widehat{x}}\right)}^{\mathrm{T}}+\left({S}_{xv}\right){\left({S}_{xv}\right)}^{\mathrm{T}}$$The state covariance prediction can be factored using the square root decomposition ${P}_{k+1|k}={S}_{{x}_{k+1|k}}{S}_{{x}_{k+1|k}}^{\mathrm{T}}$ to yield:$${S}_{{x}_{k+1|k}}=\mathcal{H}\left(\left[\begin{array}{cc}{S}_{x\widehat{x}}& {S}_{xv}\end{array}\right]\right)$$
- In the update step, the predicted state will be corrected according to the new measured data y. The residual ${\nu}_{k+1}$ and its predicted covariance ${\Lambda}_{k+1}$ are defined by:$${\nu}_{k+1}={y}_{k+1}-H{\widehat{x}}_{k+1|k}$$$${\Lambda}_{k+1}=\left({S}_{y\widehat{x}}\right){\left({S}_{y\widehat{x}}\right)}^{\mathrm{T}}+\left({S}_{yv}\right){\left({S}_{yv}\right)}^{\mathrm{T}}$$$\Lambda $ can be decomposed into ${\Lambda}_{k+1}={S}_{y}{S}_{y}^{\mathrm{T}}$ to yield:$${S}_{y}=\mathcal{H}\left(\left[\begin{array}{cc}{S}_{y\widehat{x}}& {S}_{yv}\end{array}\right]\right)$$The DD1 gain matrix K which minimizes the trace of ${P}_{k+1}$ is defined by:$${K}_{k+1}={S}_{{x}_{k+1|k}}{S}_{y\widehat{x}}{\Lambda}_{k+1}^{-1}$$Finally, the updated state vector ${\widehat{x}}_{k+1}$ and its error covariance matrix ${P}_{k+1}$ are calculated with:$${\widehat{x}}_{k+1}={\widehat{x}}_{k+1|k}+{K}_{k+1}{\nu}_{k+1}$$$${P}_{k+1}={P}_{k+1|k}-{K}_{k+1}{\Lambda}_{k+1}{K}_{k+1}^{\mathrm{T}}$$Here, again, the error covariance matrix can be decomposed into ${P}_{k+1}={\widehat{S}}_{x}{\widehat{S}}_{x}^{\mathrm{T}}$ with:$${S}_{{x}_{k+1}}=\mathcal{H}\left(\left[\begin{array}{cc}{S}_{{x}_{k+1|k}}-{K}_{k+1}{S}_{y\widehat{x}}\phantom{\rule{2.em}{0ex}}& {K}_{k+1}{S}_{yv}\end{array}\right]\right)$$

#### 3.3. The Unscented Kalman Filter

- In the prediction step, the predicted state vector ${\widehat{x}}_{k+1|k}$ and its covariance ${P}_{k+1|k}$ are computed according to:$${\mathcal{X}}_{k+1|k}^{*}=f({\mathcal{X}}_{k},{u}_{k})$$The estimated state is the weighted centre of the sigma points.$${\widehat{x}}_{k+1|k}=\sum _{i=0}^{2n}{W}_{i}^{\left(m\right)}{\mathcal{X}}_{i,k+1|k}^{*}$$The estimated covariance is estimated from the distribution of the sigma points:$${P}_{k+1|k}=\sum _{i=0}^{2n}{W}_{i}^{\left(c\right)}\left[{\mathcal{X}}_{i}^{*}-{\widehat{x}}_{k+1|k}\right]{\left[{\mathcal{X}}_{i}^{*}-{\widehat{x}}_{k+1|k}\right]}^{\mathrm{T}}+Q$$
- In the update step, new sigma points ${\mathcal{X}}_{k+1|k}$ are drawn with the estimated covariance to estimate the predicted measurement vector ${\widehat{y}}_{k}$.$${\mathcal{X}}_{k+1|k}=\left[{\widehat{x}}_{k+1|k}\phantom{\rule{1.em}{0ex}}{\widehat{x}}_{k+1|k}\pm \gamma \sqrt{{P}_{k+1|k}}\right]$$$${\mathcal{Y}}_{k+1}=H{\mathcal{X}}_{k+1|k}$$$${\widehat{y}}_{k+1}=\sum _{i=0}^{2n}{W}_{i}^{\left(m\right)}{\mathcal{Y}}_{i,k+1}$$The residual covariance $\Lambda $ is estimated by:$${\Lambda}_{k+1}=\sum _{i=0}^{2n}{W}_{i}^{\left(c\right)}\left[{\mathcal{Y}}_{i,k+1}-{\widehat{y}}_{k+1}\right]{\left[{\mathcal{Y}}_{i,k+1}-{\widehat{y}}_{k+1}\right]}^{\mathrm{T}}+R$$The filter gain K is then computed with:$${K}_{k+1}={P}_{\widehat{x}{\widehat{y}}_{k+1}}{\Lambda}_{k+1}^{-1}$$$${P}_{\widehat{x}{\widehat{y}}_{k+1}}=\sum _{i=0}^{2n}{W}_{i}^{\left(c\right)}\left[{\mathcal{X}}_{i,k+1|k}-{\widehat{x}}_{k+1|k}\right]{\left[{\mathcal{Y}}_{i,k+1}-{\widehat{y}}_{k+1}\right]}^{\mathrm{T}}$$Finally, the state estimate and its covariance are updated according to the gain and the current measurement vector y:$${\widehat{x}}_{k+1}={\widehat{x}}_{k+1|k}+{K}_{k+1}\left({y}_{k+1}-{\widehat{y}}_{k+1}\right)$$$${P}_{k+1}={P}_{k+1|k}-{K}_{k+1}{\Lambda}_{k+1}^{-1}{K}_{k+1}^{\mathrm{T}}$$

## 4. Vehicle Dynamics Model

#### 4.1. Two-Track Vehicle Model

#### 4.2. Tyre–Road Force Estimation

#### 4.3. Actuators Models

- “normal” mode: The pressure from the master cylinder is normally transmitted to the calliper.
- “maintained” mode: The calliper pressure is maintained constant.
- “released” mode: The pump is activated to release the pressure.
- “braked” mode: The pressure is increased, without any driver action.

#### 4.4. Implementation of the Probabilistic Observers

## 5. Experimental Validation

#### 5.1. Experimental Vehicle

- an inertial measurement unit (IMU) with three-axis accelerometer and gyrometer (type Xsense MTI-G);
- four wheel turn rate sensors;
- a front wheel steering angle sensor; and
- a steering wheel angle sensor.

#### 5.2. Comparison of the Probabilistic Observers

#### 5.2.1. Observation Accuracy

#### 5.2.2. Observer Consistency

#### 5.2.3. Computational Time

#### 5.2.4. Conclusion on the Observers Comparison

#### 5.3. Sensor Fault Detection Performances

#### 5.4. Actuator Fault Detection

#### 5.5. Robustness to False Detection

#### 5.6. Fault Tolerant Velocity Estimation

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

LIVIC | Laboratory on Interactions Vehicle-Infrastructure-Driver |

IBISC | Informatique, Biologie Intégrative et Systèmes Complexes |

DBW | Drive-By-Wire |

BBW | Brake-By-Wire |

SBW | Steer-By-Wire |

FDI | Fault Detection and Isolation |

IMM | Interacting Multiple Models |

HMI | Human Machine Interface |

ESC | Electronic Stability Control |

INS | Inertial Navigation Sensor |

SAS | Steering wheel Angle Sensor |

MM | Multiple Models |

EKF | Extended Kalman Filter |

DD1 | First-order Divided Differences |

UKF | Unscented Kalman Filter |

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Sensor | ${\mathit{\omega}}_{\mathit{i}}$ | ${\mathit{a}}_{\mathit{x}}$ | ${\mathit{a}}_{\mathit{y}}$ | $\dot{\mathit{\psi}}$ | $\mathit{\delta}$ |
---|---|---|---|---|---|

EKF | 0.1261 | 0.1199 | 0.1093 | 0.0046 | 0.00045 |

UKF | 0.1215 | 0.1191 | 0.1234 | 0.0045 | 0.00046 |

DD1 | 0.1217 | 0.1198 | 0.1234 | 0.0045 | 0.00046 |

unit | rad/s | m/s${}^{2}$ | m/s${}^{2}$ | rad/s | rad |

Observer | Call of the Evolution Function per Algo Cycle | Total Computational Time |
---|---|---|

EKF | 1 | 1.37 s |

UKF | 21 | 11.5 s |

DD1 | 37 | 26.6 s |

Sensor | ${\mathit{\omega}}_{\mathit{i}}$ | ${\mathit{a}}_{\mathit{x}}$ | ${\mathit{a}}_{\mathit{y}}$ | $\dot{\mathit{\psi}}$ | $\mathit{\delta}$ |
---|---|---|---|---|---|

ampl. | 5.0 rad/s | 1.6 m/s${}^{2}$ | 2.7 m/s${}^{2}$ | 0.35 rad/s | 0.055 rad |

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## Share and Cite

**MDPI and ACS Style**

Judalet, V.; Glaser, S.; Gruyer, D.; Mammar, S. Fault Detection and Isolation via the Interacting Multiple Model Approach Applied to Drive-By-Wire Vehicles. *Sensors* **2018**, *18*, 2332.
https://doi.org/10.3390/s18072332

**AMA Style**

Judalet V, Glaser S, Gruyer D, Mammar S. Fault Detection and Isolation via the Interacting Multiple Model Approach Applied to Drive-By-Wire Vehicles. *Sensors*. 2018; 18(7):2332.
https://doi.org/10.3390/s18072332

**Chicago/Turabian Style**

Judalet, Vincent, Sébastien Glaser, Dominique Gruyer, and Saïd Mammar. 2018. "Fault Detection and Isolation via the Interacting Multiple Model Approach Applied to Drive-By-Wire Vehicles" *Sensors* 18, no. 7: 2332.
https://doi.org/10.3390/s18072332