<figure><img src="image_1.png" /></figure>

**Queensland University of Technology**
Brisbane Australia

This may be the author's version of a work that was submitted/accepted for publication in the following source:

Boukhari, Mohamed, Chaibet, Ahmed, Boukhnifer, Moussa, & Glaser, Sebastien (2019)

Two longitudinal fault tolerant control architectures for an autonomous vehicle.

*Mathematics and Computers in Simulation, 156, pp. 236-253.*

This file was downloaded from: https://eprints.qut.edu.au/123097/

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https://doi.org/10.1016/j.matcom.2018.08.002
# Two Longitudinal Fault Tolerant Control Architectures for Autonomous Vehicle

M.R. Boukhari¹, A. Chaibet², M. Boukhnifer² and S. Glaser¹

*

## Abstract

Two Fault Tolerant schemes for more reliable spacing control of an autonomous vehicle are presented in this paper. The nonlinear longitudinal model of the vehicle is tackled by mean of the Lipschitz representation. Based upon this representation, a state feedback integral controller is designed, as well as, a fault estimation observers. The Lyapunov theory is used ensuring $H_\infty$ criterion and $L_2$-gain norm in the development of the Linear Matrix Inequality constraints. Simulation results are addressed to validate the proposed FTC schemes in different autonomous vehicle scenarios.

## Keywords:

Autonomous Driving, Fault-Tolerant Control, Fault Estimation, Linear Matrix Inequality.

## 1. Introduction

Driverless vehicles have received an important consideration by the scientific community in the two last decades. In fact, the self-driving technologies offer a more effective transportation systems. Thus, the passengers and road users safety is enhanced, the traffic jam is handled more efficiently and the fuel consumption is decreased. However, such an achievements need to improve more safety and reliability in case of fault occurrence. From this perspective, a significant research efforts have been conducted and have advanced the vehicle fault tolerant control [1]. Further, the topic of longitudinal fault tolerant control appears to enhance reliability for application as the collision avoidance [2], the spacing control [3] and the variants of the adaptive cruise control [4, 5].

Indeed, the longitudinal fault tolerant control to track desired velocity of an *Automated Highway System (AHS)* in presence of actuator and parametric faults is presented in [6, 7]. Thus, a nonlinear controller is designed based upon the *Dynamic Surface Control (DSC)* to ensure closed-loop stability by passive manner. Furthermore, a switching logic, using *Fault Detection and Diagnosis (FDD)* and *Fault Management System (FMS)*, swaps from the passive to the active approach when the stability can not be achieved. In [8], authors presented a passive fault tolerant control, based upon *Sliding Mode Control (SMC)*, to overcome actuator faults affecting the engine and brake systems in *AHS* application. The aim was to use a *Constant Time Headway Policy* [9] to ensure collision avoidance in faulty scenarios, and reducing oscillations caused by slinky effects which may propagate along a string of following vehicles. In [10], a multi-sensor switching strategy for longitudinal spacing control to guarentee fault tolerance was introduced. The sensors considered were the camera, the Lidar sensor and the Radar sensor. Thus, every sensor had its corresponding estimator and feedback controller, in such a way that the loop satisfying the minimum cost function is switched on. In [11], the *Cooperative Adaptive Cruise Control (CACC)* system subjected to communication latency between vehicles
was studied. The CACC is based upon wireless communication in the platooning application, therefore, the authors aim to apprehend the latencies effect in wireless link taking advantage of the robustness of the $H_{\infty}$ passive controller.

The autonomous vehicle must be able to adjust its speed and to maintain a safe distance with the front vehicle, in such a way that the autonomous vehicle can stop safely in case of emergency. Furthermore, inter-distance policies for longitudinal control are based upon information about the vehicle velocity [12, 13, 14]. Thus, it is clear that inaccuracies or faults in the vehicle speed sensor can corrupt the inter-distance reference leading to injurious control actions. Thereby, a fault tolerant control strategy must be carried out to deal with faults in the speed sensor, and enhance vehicle safety.

The main contribution of this paper is to handle the vehicle speed sensor's additive faults, by adopting a fault tolerant control strategy. The vehicle longitudinal dynamic nonlinearity satisfied the Lipschitz condition which permitted us to use the Descriptor Observer [15, 16, 17, 18], and the Proportional and Integral Observer techniques [19, 20]. These techniques lead us to estimate accurately the sensor additive faults. Furthermore, the advantage of the proposed strategies is the avoidance of control structure reconfiguration.

The present paper is organized as follows: in section 2 vehicle longitudinal and inter-distance models are addressed. In the section 3, the description and the design of the Fault Tolerant Control (FTC) strategy is given. While the validation of these techniques by some simulation examples were devoted on the section 4. Concluding remarks and perspective of this work are presented in the section 5.

Throughout this paper, some specific notations are used for more clearness. Thus, we summarize:

* $\mathbb{I}, \mathbb{O}$ are the identity matrix and the zero matrix of appropriate dimensions respectively;

* The symbol "*" in the matrices represents the transposed element in the symmetric position;

* Subscript "T" indicates matrix transposition;

* The notation $\mathcal{H}(\cdot)$ denotes the Hermitian of the matrix, for example $\mathcal{H}(A) = A^T + A$.

## 2. Vehicle and Inter-Distance Modeling

### 2.1. Vehicle dynamics

The following assumptions are made to describe the vehicle longitudinal model [21, 22] (see Fig. 1):

* The road is supposed to be plane (no slope and no inclination);

* The vehicle longitudinal behaviour is decoupled from the lateral one, thus the steering is null;

* The yaw, pitch and roll dynamics are not considered;

* The tire radius is supposed to be constant;

* There is no longitudinal slip between the tire and the road.

Considering the above assumptions, the longitudinal vehicle dynamic can be expressed by the following equations:

$$ 
\left\{ 
\begin{aligned}
m \dot{V}_x(t) &= \sum_{i=1}^{4} F_{x_i}(t) - F_a(t) \\
\bar{J}_r \dot{\omega}_{r_i}(t) &= T_m(t) - r F_{x_i}(t) - r F_{r_i}(t) - T_{b_i}(t), \quad i = 1, 2 \\
J_{r_i} \dot{\omega}_{r_i}(t) &= -r F_{x_i}(t) - r F_{r_i}(t) - T_{b_i}(t), \quad i = 3, 4
\end{aligned}
\right. 
\quad (1) $$

Adopting a single track modeling by defining:

$$ T_{b_f} = T_{b_1} + T_{b_2}, T_{b_r} = T_{b_3} + T_{b_4}, T_{r_f} = r(F_{r_1} + F_{r_2}), T_{r_r} = r(F_{r_3} + F_{r_4}), F_{x_f} = F_{x_1} + F_{x_2}, F_{x_r} = F_{x_3} + F_{x_4}, \dot{\omega}_{r_f} = \dot{\omega}_{r_1} = \dot{\omega}_{r_2}, \dot{\omega}_{r_r} = \dot{\omega}_{r_3} = \dot{\omega}_{r_4}, \bar{J}_r = \bar{J}_{r_1} + \bar{J}_{r_2}, J_r = J_{r_1} + J_{r_2} $$
<table><caption>TABLE 1: Vehicle Parameters</caption><thead><tr><th>Notation</th><th>Definition</th><th>Unit</th></tr></thead><tbody><tr><td><i>m</i></td><td>Vehicle mass</td><td>Kg</td></tr><tr><td><i>V</i><sub>x</sub></td><td>Vehicle speed</td><td>ms<sup>-1</sup></td></tr><tr><td><i>F</i><sub>x<sub>i</sub></sub></td><td>Tire/road force of the <i>i</i><sup>th</sup> wheel</td><td>N</td></tr><tr><td><i>F</i><sub>a</sub></td><td>Aerodynamic force</td><td>N</td></tr><tr><td><i>J&#773;</i><sub>r</sub></td><td>Global inertia of the front axle</td><td>Kg.m<sup>2</sup></td></tr><tr><td><i>J&#773;</i><sub>r<sub>i</sub></sub></td><td>Inertia of <i>i</i><sup>th</sup> front wheel</td><td>Kg.m<sup>2</sup></td></tr><tr><td><i>&omega;&#775;</i><sub>r<sub>i</sub></sub></td><td>Acceleration of the <i>i</i><sup>th</sup> wheel</td><td>rad.s<sup>-2</sup></td></tr><tr><td><i>T</i><sub>m</sub></td><td>The engine torque</td><td>Nm</td></tr><tr><td><i>r</i></td><td>The tire radius</td><td>m</td></tr><tr><td><i>F</i><sub>r<sub>i</sub></sub></td><td>The rolling force of the <i>i</i><sup>th</sup> wheel</td><td>N</td></tr><tr><td><i>T</i><sub>b<sub>i</sub></sub></td><td>The braking torque of the <i>i</i><sup>th</sup> wheel</td><td>Nm</td></tr><tr><td><i>T</i><sub>r<sub>f</sub>/T<sub>r<sub>r</sub></sub></sub></td><td>The rolling torque of front/rear axle</td><td>Nm</td></tr><tr><td><i>J</i><sub>r</sub></td><td>Global inertia of the rear axle</td><td>Kg.m<sup>2</sup></td></tr><tr><td><i>J</i><sub>r<sub>i</sub></sub></td><td>Inertia of <i>i</i><sup>th</sup> rear wheel</td><td>Kg.m<sup>2</sup></td></tr></tbody></table>

We obtain the following equation:

$$ egin{cases} m\dot{V}_x(t) = F_{x_f}(t) + F_{x_r}(t) - F_a(t) \\ \bar{J}_r\dot{\omega}_{r_f}(t) = T_m(t) - rF_{x_f}(t) - T_{r_f}(t) - T_{b_f}(t) \\ J_r\dot{\omega}_{r_r}(t) = -rF_{x_r}(t) - T_{r_r}(t) - T_{b_r}(t) \end{cases} \quad (2) $$

<figure><img src="image_2.png" /><figcaption>FIGURE 1: Vehicle longitudinal dynamics</figcaption></figure>

Substituting $F_{x_f}$ and $F_{x_r}$ in equation (1) leads to:

$$ m\dot{V}_x(t) = \frac{1}{r}[T_m(t) - T_{b_f}(t) - T_{b_r}(t) - T_{r_f}(t) - T_{r_r}(t) - \bar{J}_r\dot{\omega}_{r_f}(t) - J_r\dot{\omega}_{r_r}(t)] - F_a(t) \quad (3) $$

The longitudinal slip ratio hypothesis can be written as:

$$ \lambda = \frac{r\omega_r - V_x}{\max(r\omega_r, V_x)} = 0 $$

Leads to $r\omega_r = V_x$, then $r\dot{\omega}_r = \dot{V}_x$. Substituting $\dot{\omega}_r$ in (3), we obtain:

$$ (m + \frac{\bar{J}_r + J_r}{r^2})\dot{V}_x(t) = \frac{1}{r}[T_m(t) - T_{b_f}(t) - T_{b_r}(t) - T_{r_f}(t) - T_{r_r}(t)] - F_a(t) \quad (4) $$

Denoting $T_b = T_{b_f} + T_{b_r}$, $T_r = T_{r_f} + T_{r_r}$, we get:

$$ J_{eq} \dot{V}_x(t) = T_{eq}(t) - aV_x(t) - bV_x^2(t) \quad (5) $$
Where $J_{eq} = (rm + \frac{\bar{J}_r + J_r}{r})$, $T_{eq} = T_m - T_f - T_r$, $F_a = aV_x + bV_x^2$ and a and b are aerodynamic coefficients. $T_{eq}$ is the torque given by the engine and the brake system, its dynamic is subjected to loss and to decay rate, to overcome this problem we assume a first order dynamic, with a time constant $\tau$ [23]:

$$ \dot{T}_{eq}(t) = \frac{1}{\tau}(-T_{eq}(t) + u(t)) \qquad (6) $$

Finally, the vehicle longitudinal dynamic is given in the quadratic form by combining (5) and (6) as follows:

$$ \begin{bmatrix} \dot{V}_x(t) \\ \dot{T}_{eq}(t) \end{bmatrix} = \begin{bmatrix} \frac{-a}{J_{eq}} & \frac{1}{J_{eq}} \\ 0 & \frac{-1}{\tau} \end{bmatrix} \begin{bmatrix} V_x(t) \\ T_{eq}(t) \end{bmatrix} + \begin{bmatrix} 0 \\ \frac{1}{\tau} \end{bmatrix} u(t) + \begin{bmatrix} \frac{-b}{J_{eq}} \\ 0 \end{bmatrix} V_x^2 \qquad (7) $$

The model (7) can be written in the following Lipschitz nonlinear form:

$$ \begin{cases} \dot{x}(t) = Ax(t) + Bu(t) + Gg(x(t)) \\ y(t) = Cx(t) \end{cases} \qquad (8) $$

With:

$$ x = \begin{bmatrix} V_x \\ T_{eq} \end{bmatrix}, A = \begin{bmatrix} \frac{-a}{J_{eq}} & \frac{1}{J_{eq}} \\ 0 & \frac{-1}{\tau} \end{bmatrix}, B = \begin{bmatrix} 0 \\ \frac{1}{\tau} \end{bmatrix}, G = \begin{bmatrix} \frac{-b}{J_{eq}} \\ 0 \end{bmatrix}, g(x) = V_x^2, C = \begin{bmatrix} 1 & 0 \end{bmatrix}. $$

Taking into account the exogenous disturbances:

$$ \begin{cases} \dot{x}(t) = Ax(t) + Bu(t) + Gg(x(t)) + Wd(t) \\ y(t) = Cx(t) \end{cases} \qquad (9) $$

Where $d(t)$ is the disturbance signal, and $W$ the disturbance distribution matrix of appropriate dimension.

## 2.2. The inter-distance model

The control objective is to ensure a safe distance between the leader vehicle and the follower (the controlled) one. This distance must be dynamic, in such a way that the follower vehicle adjusts automatically its speed and maintains a safe distance from the leader vehicle. Furthermore, in order to avoid possible vehicle instability due to sensor faults, a FTC control is designed to ensure the insensitivity against the occurring faults. While the fault tolerance is achieved by maintaining an acceptable performance and stability properties without changing the structure of the controller. This control technique requires the estimation of this fault. In [13], authors define the desired inter-distance $dist(V_x)$ policy as follows (see Fig. 2):

$$ dist(V_x) = d_{stop} + hV_x \qquad (10) $$

Where $d_{stop}$ represents the inter-distance while stopping and $h$ corresponds to the headway time.

<figure><img src="image_3.png" /><figcaption>FIGURE 2: The inter-distance model</figcaption></figure>
### 3. Fault Tolerant Control Strategy

The active Fault Tolerant Control accommodation strategy is based upon the separation principle. This method requires the fault estimation by an observer in order to compensate the faults effect in the closed loop control. This principle is depicted in Fig. 3. Thus, the feedback controller uses the output signal obtained by subtracting the measurement signal and the estimated faults [24].

<figure><img src="image_4.png" /><figcaption>FIGURE 3: The Fault Tolerant Control strategy</figcaption></figure>

To apply this design method, we have to synthesize a state feedback controller and an observer, for that the following assumptions are helpful:

**Assumption 1.** The nonlinear terms are considered to be a smooth Lipschitz function satisfying the following relation:

$$ \|g(x_1) - g(x_2)\| \le L\|x_1 - x_2\| \quad (11) $$

Where $x_1, x_2 \in \mathbb{R}^n$, and $L$ is a positive Lipschitz constant.

**Assumption 2.** The faults $f_s(t)$ / the disturbances $d(t)$ are assumed to be bounded additive signals affecting the measurements / the system:

$$ \begin{cases} \|f_s(t)\| \le f_{max} \\ \|d(t)\| \le d_{max} \end{cases} \quad (12) $$

Where $f_{max}$ and $d_{max}$ are constant scalars.

**Lemma 1 (Schur Complement).** Given the matrices $S \in \mathbb{R}^{n \times n}$, $M \in \mathbb{R}^{n \times m}$, and $\Gamma \in \mathbb{R}^{m \times m}$, thus the following implication holds [25]:

$$ S + M\Gamma^{-1}M^T \le 0, \Gamma < 0 \Leftrightarrow \begin{bmatrix} S & M \\ M^T & \Gamma \end{bmatrix} \le 0 \quad (13) $$

**Lemma 2.** Consider matrices A, and B, and scalar $\delta$ the following inequality holds [26]:

$$ A^T B + B^T A \le \delta A^T A + \delta^{-1} B^T B \quad (14) $$

**Lemma 3.** Consider the positive scalar $\alpha$ (called also decay rate) and the Lyapunov function $V$, the derivative of $V$ can be bounded by $\alpha$ as follows [27]:

$$ \dot{V}(t) \le -2\alpha V \quad (15) $$
### 3.1. Controller design

For the nonlinear Lipschitz model (9), the robust and stabilisable control with integral action has the following form:

$$u_n = -[K \ K_I] \begin{bmatrix} x \\ x_I \end{bmatrix} \quad (16)$$

Where $K, K_I$ are the proportional and integral gains respectively, $x$ is state vector of system (9) and $x_I = \int y_c(t) - y(t)$ ($y(t)$ is the output signal, $y_c(t)$ is the reference signal). From (9) and (16) an augmented system can be written as follows:

$$\begin{cases} \dot{\bar{X}}(t) = \bar{A}\bar{X}(t) + \bar{B}u(t) + \bar{G}g(x(t)) + \bar{W}d(t) + \Lambda y_c(t) \\ y(t) = \bar{C}\bar{X}(t) \end{cases} \quad (17)$$

Where:

$$\bar{X} = \begin{bmatrix} V_x \\ T_{eq} \\ x_I \end{bmatrix}, \quad \bar{A} = \begin{bmatrix} \frac{-a}{J_{eq}} & \frac{1}{J_{eq}} & 0 \\ 0 & \frac{-1}{\tau} & 0 \\ -1 & 0 & 0 \end{bmatrix}, \quad \bar{B} = \begin{bmatrix} 0 \\ \frac{1}{\tau} \\ 0 \end{bmatrix}, \quad \bar{G} = \begin{bmatrix} \frac{-b}{J_{eq}} \\ 0 \\ 0 \end{bmatrix}, \quad g(x) = V_x^2, \quad \bar{C} = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix}, \quad \Lambda = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$$

and $\bar{W} = \begin{bmatrix} W \\ 0 \end{bmatrix}$. We obtain the following augmented controller:

$$u = -\bar{K}\bar{X} \quad (18)$$

Substituting (18) in (17), one can obtain the following closed-loop model:

$$\dot{\bar{X}} = (\bar{A} - \bar{K}\bar{B})\bar{X} + \bar{G}g(x(t)) + \bar{W}d(t) \quad (19)$$

The control purpose is to design a gain $\bar{K}$ ensuring the vehicle stability, such that the unknown disturbances are minimized by the following $H_\infty$ criterion:

**Criterion 1.** The effect of the external disturbances $d$ are minimized using a criterion $\gamma$ such that [28]:

$$\int_{0}^{\infty} (\bar{X}^{T}\bar{X} - \gamma^{2}d^{T}d)dt \leq 0 \quad (20)$$

**Theorem 1.** The augmented closed-loop Lipschitz system (19) is asymptotically stable by the state feedback controller (18), if there exists a symmetric definite positive matrix $P^{-1}$, matrix $Y$, and positive scalars $\gamma, \lambda, \alpha$ and $\delta$, such that the following LMI holds:

$$\begin{bmatrix} \Omega & \mathbb{O} & \bar{W}P^{-1} & \bar{G}P^{-1} & P^{-1} & P^{-1} \\ * & -\gamma^2\mathbb{I} & \mathbb{O} & \mathbb{O} & \mathbb{O} & \mathbb{O} \\ * & * & -\lambda^2\mathbb{I} & \mathbb{O} & \mathbb{O} & \mathbb{O} \\ * & * & * & -\delta^{-1}\mathbb{I} & \mathbb{O} & \mathbb{O} \\ * & * & * & * & -\mathbb{I} & \mathbb{O} \\ * & * & * & * & * & -\mathbb{I} \end{bmatrix} \leq 0 \quad (21)$$

$$\text{Where } \Omega = \mathcal{H}(P^{-1}\bar{A}) + \mathcal{H}(\bar{B}Y) + (\delta L^2 + 1 + 2)\mathbb{I} + 2\alpha P^{-1}$$

*Proof.* Consider the following Lyapunov function, where $P = P^T > 0$ is a symmetric definite matrix:

$$V(\bar{X}) = \bar{X}^T P \bar{X} \quad (22)$$
Adopting Criterion. 1, and an L₂-gain form, minimizing the transfert from the known input u to the state vector X̄, we obtain:

$$ \dot{V}(\bar{X}) + 2\bar{X}^T \bar{X} - \gamma^2 d^T(t)d(t) - \lambda^2 u^T(t)u(t) < 0 \quad (23) $$

Using Lemma. 3:

$$ \dot{V}(\bar{X}) + 2\bar{X}^T \bar{X} - \gamma^2 d^T(t)d(t) - \lambda^2 u^T(t)u(t) + 2\alpha V < 0 \quad (24) $$

Substituting V and its derivative in (24), and using Assumption. 1 and Lemma. 2, implies the following quadratic form:

$$ \begin{bmatrix} \bar{X} \\ u \\ d \end{bmatrix}^T \begin{bmatrix} \Omega_1 & \emptyset & P\bar{W} \\ \emptyset & -\lambda^2\mathbb{I} & \emptyset \\ P\bar{W}^T P & \emptyset & -\gamma^2\mathbb{I} \end{bmatrix} \begin{bmatrix} \bar{X} \\ u \\ d \end{bmatrix} < 0 \quad (25) $$

Where $\Omega_1 = \mathcal{H}((\bar{A} - \bar{B}\bar{K})P) + \delta L^2\mathbb{I} + \delta^{-1}P\bar{G}\bar{G}^T P + 2\mathbb{I} + 2\alpha P$

Using the Schur Complement, and pre-post multiplying by diag $[P^{-1} \quad \mathbb{I} \quad \mathbb{I} \quad \mathbb{I}]$:

$$ \begin{bmatrix} \Omega_2 & \emptyset & \bar{W}P^{-1} & \bar{G}P^{-1} \\ \emptyset & -\lambda^2\mathbb{I} & \emptyset & \emptyset \\ P^{-1}\bar{W}^T & \emptyset & -\gamma^2\mathbb{I} & \emptyset \\ P^{-1}\bar{G}^T & \emptyset & \emptyset & -\delta^{-1}\mathbb{I} \end{bmatrix} < 0 \quad (26) $$

Where $\Omega_2 = \mathcal{H}(P^{-1}\bar{A}) + \mathcal{H}(\bar{B}Y) + \delta L^2 P^{-1}P^{-1} + 2P^{-1}P^{-1} + 2\alpha P^{-1}$, and $Y = KP^{-1}$.

Using twice the Schur Complement, we obtain the LMI constraint of (21).

□

## 3.2. Proportional and Integral Observer Fault Estimation (PIO)

The system (9) subjected to sensor faults is written as follows:

$$ \begin{cases} \dot{x}(t) = Ax(t) + Bu(t) + Gg(x(t)) + Wd(t) \\ y(t) = Cx(t) + Ff_s(t) \end{cases} \quad (27) $$

Where $f_s(t)$ is the sensor fault signal, and $F$ the fault distribution matrix.

The proportional and Integral observer of the system (27) are given as the form proposed in [29]:

$$ \begin{cases} \dot{\hat{x}}(t) = A\hat{x}(t) + Gg(\hat{x}(t)) + Bu(t) + L_p(y - C\hat{x} - F\hat{f}_s) \\ \hat{f}(t) = L_I(y - C\hat{x} - F\hat{f}_s) \end{cases} \quad (28) $$

The aim is to calculate the observer's proportional and integral gains ensuring the asymptotic convergence. For that we consider the states error $e = x - \hat{x}$, the faults error $e_f = f - \hat{f}$, and the free fault case residual signal
$r = N(y - \hat{y}) = NC_0e(t)$ ($N$ is a weighting matrix, and $C_0 = [C \ \mathbb{O}]$). Implies the error dynamics (29).

$$ egin{cases} \dot{e}(t) = (A - L_p C)e(t) + G \tilde{g} + Wd(t) - L_p F e_f(t) \\ \dot{e}_f(t) = \dot{f}_s(t) - L_I C e(t) - L_I F e_f(t) \end{cases} \tag{29} $$

Where $\tilde{g} = g(x(t)) - g(\hat{x}(t))$. Based upon (28) and (29), the augmented system (30) is written.

$$ egin{cases} \dot{Z}_1(t) = (A^* - L^*C^*)Z_1(t) + G^*\tilde{g} + W_1d(t) \\ \dot{Z}_2(t) = A^*Z_2(t) + L^*C^*Z_1(t) + G^*g(\hat{x}(t)) + B^*u(t) + W_2\dot{f}_s(t) \end{cases} \tag{30} $$

Where $Z_1(t) = [e(t) \ e_f(t)]$, $Z_2(t) = [\hat{x}(t) \ \hat{f}_s(t)]$, $A^* = [A \ \mathbb{O}]$, $L^* = [L_p \ L_I]$, $C^* = [C \ F]$, $B^* = [\mathbb{B} \ \mathbb{O}]$, $G^* = [\mathbb{G} \ \mathbb{O}]$, $W_1 = [\mathbb{W}]$, $W_2 = [\mathbb{O}]$

The signal $r$ must be insensitive to external disturbance signal $d$, for that we adopt the following criterion:

**Criterion 2.** The $H_\infty$ criterion ensuring the disturbances rejection is written as follows:

$$ \int_0^\infty (r^T r - \gamma^2 d^T d) dt \le 0 \tag{31} $$

Therefore, the PIO (28) and the dynamics (29) are asymptotically stable via the Criterion. 2, if the LMI condition summarized in the Theorem. 2 holds.

**Theorem 2.** The nonlinear Lipschitz Proportional and Integral Observer (28) is asymptotically stable, if there exist positive definite matrices $P_1$ and $P_2$, matrices $N$ and $U_1$, and positive scalars $\delta$, $\lambda$ and $\gamma$, such that the following LMI is verified:

$$ \begin{bmatrix} \Delta_1 & C^*L^*P_2 & \mathbb{O} & \mathbb{O} & P_1W_1 & P_1G^* & C_0^T N^T & \mathbb{O} \\ * & \Delta_2 & P_2B^* & P_2W_2 & \mathbb{O} & \mathbb{O} & \mathbb{O} & P_2G^* \\ * & * & -\lambda^2\mathbb{I} & \mathbb{O} & \mathbb{O} & \mathbb{O} & \mathbb{O} & \mathbb{O} \\ * & * & * & \mathbb{O} & \mathbb{O} & \mathbb{O} & \mathbb{O} & \mathbb{O} \\ * & * & * & * & -\gamma^2\mathbb{I} & \mathbb{O} & \mathbb{O} & \mathbb{O} \\ * & * & * & * & * & -\delta^{-1}\mathbb{I} & \mathbb{O} & \mathbb{O} \\ * & * & * & * & * & * & -\mathbb{I} & \mathbb{O} \\ * & * & * & * & * & * & * & -\delta^{-1}\mathbb{I} \end{bmatrix} < 0 \tag{32} $$

Where:

$$ \Delta_1 = \mathcal{H}(P_1A^* - U_1C^*) + \delta L^2\mathbb{I} + \mathbb{I} $$

$$ \Delta_2 = \mathcal{H}(P_2A^*) + \delta L^2\mathbb{I} + \mathbb{I} $$

*Proof.* Consider the following multiple Lyapunov function, where $P_1 = P_1^T > 0$ and $P_2 = P_2^T > 0$ are symmetric definite positive matrices:
$$V = Z_1^T P_1 Z_1 + Z_2^T P_2 Z_2 \quad (33)$$

Deriviting (33) and using Criterion. 2, and the L₂-gain form, we get:

$$\dot{V} + r^T r + Z_2^T Z_2 - \lambda^2 u^T u - \gamma^2 d^T d \le 0 \quad (34)$$

Using Assumption. 1, and Lemma. 2, one have the quadratic form of (35).

$$\begin{bmatrix} Z_1 \\ Z_2 \\ u \\ \dot{f}_s \\ d \end{bmatrix}^T \begin{bmatrix} \Delta'_1 & C^{*T} L^{*T} P_2 & \mathbb{O} & \mathbb{O} & P_1 W_1 \\ * & \Delta'_2 & P_2 B^* & P_2 W_2 & \mathbb{O} \\ * & * & -\lambda^2 \mathbb{I} & \mathbb{O} & \mathbb{O} \\ * & * & * & \mathbb{O} & \mathbb{O} \\ * & * & * & * & -\gamma^2 \mathbb{I} \end{bmatrix} \begin{bmatrix} Z_1 \\ Z_2 \\ u \\ \dot{f}_s \\ d \end{bmatrix} \le 0 \quad (35)$$

Where:

$$\begin{align}
\Delta'_1 &= \mathcal{H}(P_1(A^* - L^*C^*)) + \delta L^2 \mathbb{I} + \delta^{-1} P_1 G^* G^{*T} P_1 + C_0^T N^T N C_0 + \mathbb{I} \\
\Delta'_2 &= \mathcal{H}(P_2 A^*) + \delta L^2 \mathbb{I} + \delta^{-1} P_2 G^* G^{*T} P_2 + \mathbb{I} \tag{36}
\end{align}$$

Using three times the Schur Complement, and denoting $U_1 = P_1 L^*$, yields the LMI constraint of (32).

<figure><img src="image_5.png" /></figure>

### 3.3. Descriptor Observer Fault Estimation (DO)

From (9) an augmented system can be written as follows:

$$\begin{cases} E \dot{\bar{x}}(t) = \bar{A} \bar{x} + \bar{G} g(x(t)) + \bar{B} u(t) + \bar{W} d(t) + \bar{F} \bar{f}_s \\ y = \bar{C} \bar{x} = C_0 \bar{x} + \bar{f}_s \end{cases} \quad (37)$$

$\bar{x} = [x \quad f_s]$, $E = [\mathbb{I} \quad \mathbb{O}]$, $\bar{A} = [A \quad \mathbb{O}]$, $\bar{G} = [G \quad \mathbb{O}]$, $\bar{B} = [B \quad \mathbb{O}]$, $\bar{W} = [W \quad \mathbb{O}]$, $\bar{F} = [\mathbb{O} \quad \mathbb{I}]$, $\bar{f}_s = F f_s$, $\bar{C} = [C \quad F]$, and $C_0 = [C \quad \mathbb{O}]$.

The nonlinear Lipschitz Descriptor Observer leading to estimate the system's states and the sensor faults is written as follows:

$$\begin{cases} \bar{E} \dot{z} = Sz + \bar{G}g(\hat{x}(t)) + \bar{B}u(t) \\ \hat{x} = z + Ly \end{cases} \quad (38)$$

$S = [A \quad \mathbb{O}]$, $L = [\mathbb{O} \quad \mathbb{I}]$, $\bar{E} = [\mathbb{I} + \Theta C \quad \Theta]$, and $\Theta$, and $R$ are chosen in such a way that $\bar{E}$ is nonsingular [30].

Let us define the error $e = \bar{x} - \hat{x}$, and the free faults residual $r = M(y - \hat{y}) = MC_0e$ (where M is a whitening matrix), yields the following error dynamic:

$$\dot{e}(t) = \tilde{S} e(t) + \tilde{G} \tilde{g} + \tilde{W} d(t) \quad (39)$$
With $\tilde{g} = g(x(t)) - g(\hat{x}(t))$, $\tilde{S} = \tilde{E}^{-1}S = \begin{bmatrix} A + \Theta R^{-1}C & \Theta R^{-1} \\ -CA - (R^{-1} + C\Theta R^{-1})C & -R^{-1} - C\Theta R^{-1} \end{bmatrix}$, $\tilde{G} = \tilde{E}^{-1}\tilde{G} = \begin{bmatrix} G \\ -CG \end{bmatrix}$, $\tilde{W} = \tilde{E}^{-1}\tilde{W} = \begin{bmatrix} W \\ -CW \end{bmatrix}$

From (9) and (39), we have the following augmented system:

$$ \begin{bmatrix} \dot{e}(t) \\ \dot{x}(t) \end{bmatrix} = \begin{bmatrix} \tilde{S} & \emptyset \\ \emptyset & A \end{bmatrix} \begin{bmatrix} e(t) \\ x(t) \end{bmatrix} + \begin{bmatrix} \tilde{G} & \emptyset \\ \emptyset & G \end{bmatrix} \begin{bmatrix} \tilde{g} \\ g(x(t)) \end{bmatrix} + \begin{bmatrix} \emptyset \\ B \end{bmatrix} u(t) + \begin{bmatrix} \tilde{W} \\ W \end{bmatrix} d(t) \quad (40) $$

The stability of the system (40) is ensured, using the Criterion. 2, and the L₂-gain form, if the LMI condition summarized in the Theorem. 3 holds.

**Theorem 3.** The nonlinear augmented Lipschitz Descriptor System is asymptotically stable, if there exist positive definite matrices $P_{11}$, $P_{12}$, and $P_{2}$, matrices $N_{1}$, $N_{2}$, and $M$, positive scalars $\lambda$, $\gamma$, and $\delta$, such that the following LMI condition is satisfied:

$$ \begin{bmatrix} \Phi_{11} & \Phi_{12} & \emptyset & \emptyset & P_{11}W & P_{11}G & C^T M^T & \emptyset \\ * & \Phi_{21} & \emptyset & \emptyset & -P_{12}CW & -P_{12}CG & \emptyset & \emptyset \\ * & * & \Phi_{31} & P_{2}B & P_{2}W & \emptyset & \emptyset & P_{2}G \\ * & * & * & -\lambda^2 \mathbb{I} & \emptyset & \emptyset & \emptyset & \emptyset \\ * & * & * & * & -\gamma^2 \mathbb{I} & \emptyset & \emptyset & \emptyset \\ * & * & * & * & * & -\delta^{-1} \mathbb{I} & \emptyset & \emptyset \\ * & * & * & * & * & * & -\mathbb{I} & \emptyset \\ * & * & * & * & * & * & * & -\delta^{-1} \mathbb{I} \end{bmatrix} < 0 \quad (41) $$

With:

$$ \Phi_{11} = \mathcal{H}(P_{11}A) + \mathcal{H}(N_1C) + \delta L^2 \mathbb{I} + \mathbb{I} $$

$$ \Phi_{12} = N_1 - A^T C^T P_{12}^T - C^T N_2^T $$

$$ \Phi_{21} = -\mathcal{H}(N_2) + \mathbb{I} $$

$$ \Phi_{31} = \mathcal{H}(P_2A) + \delta L^2 \mathbb{I} + \mathbb{I} \quad (42) $$

The estimated fault is written as follows:

$$ \hat{f}_s = -(F^T F)^{-1} F^T \hat{\mathcal{F}} \quad (43) $$

Where:

$$ \hat{f}_s = [\emptyset \quad \mathbb{I}] \hat{x} \quad (44) $$

*Proof.* Consider the following Lyapunov function, where $P_1 = P_1^T > 0$ and $P_2 = P_2^T > 0$ are symmetric definite positive matrices, of appropriate dimensions:

$$ V(e(t), x(t)) = \begin{bmatrix} e \\ x \end{bmatrix}^T \begin{bmatrix} P_1 & \emptyset \\ \emptyset & P_2 \end{bmatrix} \begin{bmatrix} e \\ x \end{bmatrix} \quad (45) $$

Derivating (45) and using Criterion. 2, and the L₂-gain form, we get equation (46).

$$ \dot{V}(e(t), x(t)) + r^T r + x^T x - \gamma^2 d^T d - \lambda^2 u^T u \quad (46) $$
Using Assumption. 1, and Lemma. 2, yields the equation (47).

$$ \begin{bmatrix} e \\ x \\ u \\ d \end{bmatrix}^T \begin{bmatrix} \Gamma_1 & \mathbb{O} & \mathbb{O} & P_1 \tilde{W} \\ * & \Gamma_2 & B^T P_2 & P_2 W \\ * & * & -\lambda^2 \mathbb{I} & \mathbb{O} \\ * & * & * & -\gamma^2 \mathbb{I} \end{bmatrix} \begin{bmatrix} e \\ x \\ u \\ d \end{bmatrix} < 0 \quad (47) $$

With:

$$ \Gamma_1 = \mathcal{H}(P_1 \tilde{S}) + \delta L^2 \mathbb{I} + \delta^{-1} P_1 \tilde{G} \tilde{G}^T P_1 + C_0^T M^T M C_0 + \mathbb{I} $$

$$ \Gamma_2 = \mathcal{H}(P_2 A) + \delta L^2 \mathbb{I} + \delta^{-1} P_2 G G^T P_2 + \mathbb{I} $$

Using the Schur Complement twice, yields equation (48).

$$ \begin{bmatrix} \Gamma'_1 & \mathbb{O} & \mathbb{O} & P_1 \tilde{W} & P_1 \tilde{G} & C_0^T M^T \\ * & \Gamma_2 & B^T P_2 & P_2 W & \mathbb{O} & \mathbb{O} \\ * & * & -\lambda^2 \mathbb{I} & \mathbb{O} & \mathbb{O} & \mathbb{O} \\ * & * & * & -\gamma^2 \mathbb{I} & \mathbb{O} & \mathbb{O} \\ * & * & * & * & -\delta^{-1} \mathbb{I} & \mathbb{O} \\ * & * & * & * & * & -\mathbb{I} \end{bmatrix} < 0 \quad (48) $$

And $\Gamma'_1 = \mathcal{H}(P_1 \tilde{S}) + \delta L^2 \mathbb{I} + \mathbb{I}$.

Substituting $\tilde{S}$, $\tilde{G}$, and $\tilde{W}$ by their values, and taking $P_1 = \begin{bmatrix} P_{11} & \mathbb{O} \\ \mathbb{O} & P_{12} \end{bmatrix}$, $N_1 = P_{11} \Theta R^{-1}$, and $N_2 = P_{12}(R^{-1} + C \Theta R^{-1})$, Implies equation (49).

$$ \begin{bmatrix} \Phi_{11} & \Phi_{12} & \mathbb{O} & \mathbb{O} & P_{11} W & P_{11} G & C^T M^T \\ * & \Phi_{21} & \mathbb{O} & \mathbb{O} & \mathbb{O} & \mathbb{O} & \mathbb{O} \\ * & * & \Gamma_2 & B^T P_2 & P_2 W & \mathbb{O} & \mathbb{O} \\ * & * & * & -\lambda^2 \mathbb{I} & \mathbb{O} & \mathbb{O} & \mathbb{O} \\ * & * & * & * & -\gamma^2 \mathbb{I} & \mathbb{O} & \mathbb{O} \\ * & * & * & * & * & -\delta^{-1} \mathbb{I} & \mathbb{O} \\ * & * & * & * & * & * & -\mathbb{I} \end{bmatrix} < 0 \quad (49) $$

Using the Schur Complement yields the LMI constraint of (41).

<figure><img src="image_6.png" /></figure>

## 4. Simulation results

In this section, simulations have been performed in order to show the effectiveness of the proposed FTC structures of vehicle driving scenarios. The LMI conditions devlopped above for such the controller and the observers were resolved using the penlab solver which run under the Yalmip environment [31]. Thus, the LMIs developed are solved, and give the following solutions:

* For the Controller :

$$ K = 10^4 \begin{bmatrix} -1.7399 \\ 1.5747 \end{bmatrix}, K_I = -0.3727 * 10^4; \text{ with } P^{-1} = 10^3 * \begin{bmatrix} 1.0574 & 0.9293 & 0.1645 \\ 0.9293 & 1.2497 & 0.3350 \\ 0.1654 & 0.3350 & 0.9532 \end{bmatrix}, \lambda = 0.0055, \gamma = 0.0033, \\ \alpha = 0.0002 $$
• For the *Proportional and integral observer*:

$$
\begin{align*}
L_p &= \begin{bmatrix} -2.8605 \\ 5.6037 \end{bmatrix}, \quad L_I &= \begin{bmatrix} 22.7274 \\ -0.5799 \end{bmatrix}, \quad \text{and} \quad P_1 = \begin{bmatrix} 38.7735 & 1.7723 & 10.6408 & 0.4298 \\ 1.7723 & 15.4934 & 1.5384 & 1.9737 \\ 10.6408 & 1.5384 & 13.0298 & 1.1650 \\ 0.4298 & 1.9737 & 1.1650 & 23.1656 \end{bmatrix}, \quad \lambda = 0.3662, \gamma = 0.023, \\
P_2 &= \begin{bmatrix} 6.5789 & -4.6569 & -0.0800 & 0.0266 \\ -4.6569 & 18.8529 & -0.1052 & -0.1294 \\ -0.0800 & -0.1052 & 0.0318 & 0.4454 \\ 0.0266 & -0.1294 & 0.4454 & 17.8108 \end{bmatrix}
\end{align*}
$$

• For the *Descriptor observer*:

$$
\begin{align*}
\bar{E} &= \begin{bmatrix} 0.2968 & 0 & -0.7032 & -0.2672 \\ -1.1999 & 1.0000 & -1.1999 & 0.0240 \\ 2.3570 & 0 & 2.3570 & -0.8756 \\ -0.3328 & 0 & -0.3328 & 0.8898 \end{bmatrix}, \quad \text{and} \quad R = \begin{bmatrix} 0.4158 & 2.8839 \\ 0.7453 & -1.2691 \end{bmatrix}, \quad \Theta = \begin{bmatrix} -0.7174 & 0.2841 \\ -0.8577 & -1.9111 \end{bmatrix}, \\
P_{11} &= \begin{bmatrix} 3.4789 & 1.2250 \\ 1.2250 & 3.5056 \end{bmatrix}, \quad P_{12} = \begin{bmatrix} 4.2255 & 2.0013 \\ 2.0013 & 3.4122 \end{bmatrix}, \quad P_2 = \begin{bmatrix} 4.5470 & 2.2896 \\ 2.2896 & 3.7373 \end{bmatrix}, \quad \gamma = 0.0619, \text{ and } \lambda = 0.3833.
\end{align*}
$$

<table><caption>TABLE 2: Vehicle parameters</caption>
  <thead>
    <tr>
      <th>parameter</th>
      <th>value</th>
      <th>Unit</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>J<sub>eq</sub></td>
      <td>480</td>
      <td>Kg.m</td>
    </tr>
    <tr>
      <td>a</td>
      <td>17.45</td>
      <td>Kgm<sup>-1</sup></td>
    </tr>
    <tr>
      <td>b</td>
      <td>0.019</td>
      <td>Kgs<sup>-1</sup></td>
    </tr>
    <tr>
      <td>&tau;</td>
      <td>0.05</td>
      <td>s</td>
    </tr>
    <tr>
      <td>h</td>
      <td>1</td>
      <td>s</td>
    </tr>
  </tbody>
</table>

The additive sensor fault used in this simulation has the following form:

$$
f_s(t) = (0.01t - 2) + \sin(2\pi t) \tag{50}
$$

The *Stop and Go* maneuver is considered in this work. This scenario is summarized as follows:

The vehicle leader moves with an initial speed of 20ms⁻¹, the following vehicle evolves with initial velocity of 25ms⁻¹. Since the desired longitudinal distance is around 40m. In order to control this distance, the follower must decrease. While decelerating to reach the speed of 20ms⁻¹. At t = 15s, the additive sensor fault occurred (in the form of (50)), we can notice that the following vehicle maneuver without FTC could lead to loss of vehicle control and stability, while its behavior with FTC is kept stable, and the desired spacing is preserved. At t = 24s, the vehicle follower carries out a deceleration until to stop, and thereafter at t = 40s the vehicle accelerates to reach a velocity of 20ms⁻¹. At t = 65s, the leader decelerates until to stop.

The comparison of speed profiles are given on Fig. 6. Furthermore, the vehicle follower maneuver is carried out with success with respect to the passengers comfort where the reasonable boundaries in acceleration range are from -0.3ms⁻² to 0.3ms⁻² as depicted on Fig. 7. Moreover, the DO and PIO observers are designed to estimate the longitudinal vehicle speed the Figs. 4, 5 and the additive sensor fault Fig. 8 show the good estimation. Fig. 9 shows the additive sensor fault influence in the inter-distance, while the applied torque is depicted in Fig. 10. The Fig. 11 shows output control signal, displaying the performance of the proposed FTC control in comparison with vehicle without FTC.
<figure><img src="image_7.png" /><figcaption>FIGURE 4: Vehicle estimated speed with DO observer</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>FIGURE 6: Speed profiles</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>FIGURE 8: The additive sensor fault</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>FIGURE 5: Vehicle estimated speed with PI observer</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>FIGURE 7: Accelerations comparison</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>FIGURE 9: The vehicle inter-distance</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>FIGURE 10: The applied torque comparison</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>FIGURE 11: The output control signal comparison</figcaption></figure>

The following remarks can be summarized:

* The estimated vehicle speeds converge quickly toward the real one;

* The estimated fault converges to the real one;

* The performances obtained are good as well in dynamics as in statics;

* The observation errors are steered to zero in finite time.

The fruitful simulation results presented above, prove the effectiveness of the *FTC* strategy in managing the occurrence of additive sensor faults. In addition, the additive fault is well estimated, and the fault accommodation maintains the vehicle stability in faulty cases.

**5. Conclusion**

In this paper, the reliability of the spacing control for an autonomous vehicle was performed. The emphasis is to give on sensor fault tolerant controller scheme. Furthermore, the design of the proposed fault-tolerant scheme is based upon a state feedback integral controller ensuring robustness against disturbances, and a fault estimation observer. Indeed, two observers techniques were designed, the *Descriptor Observer* and the *Proportional and Integral Observer*. The controller and observers gains were done by solving the LMI constraints that ensure the $H_\infty$ criterion and the $L_2$-gain form. To illustrate the ability of the proposed FTC architectures, computation simulations have been carried out to give well performance of the fault detection and control law design in some scenarios of automated driving where the performances and the vehicle stability are kept with respect of the passenger comfort. In future work, experimental tests should therefore be implemented on prototype vehicle.

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