**Abstract**

This article presents a fault-tolerant control (FTC) procedure for the automotive vehicle lateral dynamics (AVLD) described by the Takagi-Sugeno (T-S) fuzzy models. This approach focuses on actuator faults, which requires knowledge of the system parameters and the faults that are occurring. For this reason, T-S fuzzy observers are suitable for simultaneously estimating system states and actuator faults. The proposed control makes it possible to maintain vehicle stability even in the presence of faults. The design of fuzzy observers and fuzzy controllers is mainly based on the one-step method of Lyapunov, which is provided in the form of linear matrix inequalities (LMIs). The simulation results clearly illustrate the effectiveness of the applied controller strategy to maintain vehicle stability.

**Keywords:** fault-tolerant control (FTC), automotive vehicle lateral dynamics (AVLD), linear matrix inequality (LMI), fault estimation (FE), continuous Takagi-Sugeno (T-S) fuzzy models

### **1. Introduction**

The vast majority of road accidents are due to faults or incorrect driving reflexes. This is why the automotive industry and researchers in the field of road safety have dedicated themselves to developing and producing vehicles that are more reliable, more relaxed, and safer, which are discussed in Refs. [1–3]. In recent decades, many new solutions have been suggested by the introduction and development of new passive safety systems such as airbags and driver-assisted active safety systems such as adaptive cruise control (ACC), antilock braking system (ABS), dynamic stability control (DSC), and electronic stability program (ESP) [4–6]. Furthermore, the dependence of the control of these systems on actuator components is becoming increasingly complicated. Generally, these systems can be exposed to certain catastrophic faults such as unknown actuator faults. It is well known that conventional control strategies are unable to adapt when system failures happen.

To address this challenge, some previous research has based on estimation techniques for vehicle dynamics, road bank angle, and faults in order to develop control laws capable of ensuring that the system maintains strict stability even when various faults happen. This task is commonly referred to as fault-tolerant control

(FTC), which is widely studied in modern control systems [7–9]. In fact, the safety of persons and the preservation of system performance are crucial requirements that have to be considered in the control design. The problem of fault tolerance has long been addressed from many angles. The FTC synthesis approaches are categorized as passive or active FTC. For the passive FTC approach, we consider possible fault situations and take them into account in the control design step; this approach is similar to the robust control design. It is pointed out in many books that this strategy is generally restrictive. While active FTC improves post-fault control performance and addresses serious faults that break the control loop, it is generally advantageous to switch to a new controller that is either online or designed offline to control the faulty plant. The FTC process is based on two theoretical steps: fault estimation (FE) and setting the controller so that the control law is reconfigured to meet the performance requirements due to the faults.

The FTC procedure was first adjusted for linear systems. Most engineering systems include vehicle dynamics that have nonlinear behaviors. The T-S fuzzy representation is widely known to be a successful solution to approximate a large class of nonlinear dynamic systems. T-S fuzzy models are nonlinear systems represented by a set of local linear models. By mixing the representations of linear systems, the global fuzzy model of the system is obtained, which makes it much easier for the observer and controller to synthesize. A major advantage is that it provides an efficient design strategy for representing a nonlinear system. As a result, many researchers have become interested in the FTC approach for T-S fuzzy systems (see [10–12]).

In this regard, great efforts have been made to improve the stability of the vehicle lateral dynamics to enhance the safety and comfort of the passengers in critical driving conditions. Since the vehicle is a very complex system, the challenge is to achieve more precise control designs and to increase the effectiveness of a vehicle dynamic control system, which necessitates an accurate knowledge of vehicle parameters, in particular, sideslip, roll, and yaw angles [12, 13].

Our main objective in this chapter is to apply a fault-tolerant control method in the vehicle lateral dynamics system. The model used contains four states: lateral slip angle, yaw rate, roll rate, and roll angle. The problem of the design of fault-tolerant active control of nonlinear systems is studied by using the T-S representation which combines the simplicity and accuracy of nonlinear behaviors. The idea is to consider a set of linear subsystems. An interpolation of all these sub-models using nonlinear functions satisfying the convex sum property gives the overall behavior of the described system over a wide operating range. The stability of the T-S models of the vehicle lateral dynamics is mainly studied using the Lyapunov function, and sufficient asymptotic stability conditions are given in the form of linear matrix inequalities (LMIs).

The chapter is structured as follows: the second section deals with the model of vehicle lateral dynamics considering roll motion, the third section presents this model through the T-S fuzzy model, the fourth section presents the fault-tolerant control strategy based on the T-S fuzzy observer, and the fifth section is devoted to simulations and analysis of the results. Finally, the conclusion will be included in the last section.

**Notation:** A real symmetrical negative definite matrix (resp. positive definite matrix) is represented by *X* <0 (resp. *X* >0). *X*<sup>1</sup> indicates the inverse of the matrix *X*. The marking "\*" signifies the transposed element in the symmetrical position of a matrix.

#### **2. Description of the lateral dynamics of the automotive vehicle**

The automotive dynamics model is very challenging to use in controlling and monitoring applications due to its complexity and number of degrees of freedom. *T-S Fuzzy Observers to Design Actuator Fault-Tolerant Control for Automotive Vehicle Lateral… DOI: http://dx.doi.org/10.5772/intechopen.92582*

For this purpose, a nominal model for the synthesis of observers and controllers is needed. The vehicle motion is characterized by a combination of translational and rotational movements (see **Figure 1**) [14], generally six principal movements. The model used in this document outlines the lateral dynamics of the automotive vehicle, taking into account the rolling motion (**Figure 2**) [15]. This model is achieved by considering the widely known "bicycle model" with a rolled degree of freedom. Here, the lateral velocity *vy*ð Þ*t* , the yaw angle *ψ*ð Þ*t* , and the roll angle *ϕ*ð Þ*t* of the vehicle are the differential variables. The suspension is modeled as a spring and torsion damping system operating around the roll axis, as illustrated in **Figure 2**. The pitch dynamics of the vehicle is ignored or even neglected.

**Figure 1.**

*The movements of the automotive vehicle bodywork.*

**Figure 2.** *Bicycle model with roll behavior of automotive vehicle dynamics (3-DOF).*

The nonlinear model of the automotive vehicle lateral dynamics considering the roll angle to be small is given in [12] by the following simplified differential equations:

$$\begin{cases} m\dot{\boldsymbol{\nu}}\_{\boldsymbol{\mathcal{V}}}(t) = -m\dot{\boldsymbol{\nu}}(t)\boldsymbol{\nu}\_{\boldsymbol{\mathcal{V}}}(t) + 2\left(\boldsymbol{F}\_{\mathcal{H}} + \boldsymbol{F}\_{\mathcal{V}}\right) \\\ I\_{x}\dot{\boldsymbol{\nu}}(t) = 2\left(l\_{r}\boldsymbol{F}\_{\mathcal{V}} - l\_{f}\boldsymbol{F}\_{\mathcal{H}}\right) \\\ I\_{x}\ddot{\boldsymbol{\phi}}(t) + \boldsymbol{K}\_{\boldsymbol{\phi}}\dot{\boldsymbol{\phi}}(t) + \boldsymbol{C}\_{\boldsymbol{\phi}}\dot{\boldsymbol{\phi}}(t) = mh\_{\text{roll}}\left(\boldsymbol{\nu}\_{\boldsymbol{\mathcal{V}}}(t) + \boldsymbol{\nu}\_{\boldsymbol{\mathfrak{x}}}(t)\dot{\boldsymbol{\nu}}(t)\right) \end{cases} \tag{1}$$

All system parameters are defined in **Table 1**.

Lateral forces can be given according to Pacejka's magic formula referenced in [16], depending on the sliding angles of the tires. To simplify the vehicle model, we assume that the forces *Fyf* and *Fyr* are proportional to the slip angles of the front and rear tires:

$$\begin{cases} F\_{\mathcal{Y}} = \mathbf{C}\_f a\_f(t) \\ F\_{\mathcal{Y}} = \mathbf{C}\_r a\_r(t) \end{cases} \tag{2}$$

where *Cf* and *Cr* are the front and rear wheel cornering stiffness coefficients, respectively, which depend on the road adhesion coefficient *σ* and the parameters of the vehicle. The linear model functions very successfully in the case of small slip angles; however, in the case of increasing slipping, a nonlinear model should be envisaged.

### **3. T-S fuzzy representation of the automotive vehicle lateral dynamics**

The challenge in modeling vehicle dynamics accurately is that contact forces are complex to measure and to model. By using the T-S models' method proposed in [15, 17], it is a highly useful mathematical representation of nonlinear systems, as they can represent any nonlinear system, regardless of its complexity, by a simple structure based on linear models interpolated by nonlinear positive functions. They have a simple structure with some interesting properties. This makes them easily exploitable from a mathematical viewpoint and makes it possible to extend certain results from the linear domain to nonlinear systems.


#### **Table 1.**

*Parameters of the automotive vehicle lateral dynamics system.*

*T-S Fuzzy Observers to Design Actuator Fault-Tolerant Control for Automotive Vehicle Lateral… DOI: http://dx.doi.org/10.5772/intechopen.92582*

Tire characteristics are usually assumed that the front and rear lateral forces (2) are modeled by the following rules:

$$\text{if } |a\_{\mathbf{f}}| \text{ is } M\_1 \text{ then} \\
\begin{pmatrix} F\_{\mathcal{Y}} = \mathcal{C}\_{f1} a\_f(t) \\ F\_{\mathcal{Y}} = \mathcal{C}\_{r1} a\_r(t) \end{pmatrix} \\ \tag{3}$$

$$\text{if } |a\_{\text{f}}| \text{ is } M\_2 \text{ then} \begin{pmatrix} F\_{\text{yf}} = \mathcal{C}\_{f^2} a\_f(t) \\ F\_{\text{yr}} = \mathcal{C}\_{r2} a\_r(t) \end{pmatrix} \tag{4}$$

The front and rear tire slip angles are given in this instance as cited in [12], by

$$\begin{cases} a\_f(t) \approx \delta\_f(t) - \frac{a\_f \dot{\varphi}(t)}{v\_x(t)} - \beta(t) \\\\ a\_r(t) \approx \frac{a\_r \dot{\varphi}(t)}{v\_x(t)} - \beta(t) \end{cases} \tag{5}$$

The proposed rules are only made for *α <sup>f</sup>*ð Þ*t* ; this assumption reduces the number of adhesion functions and considers the rear steering angle to be ignored; *α <sup>f</sup>*ð Þ*t* and *αr*ð Þ*t* are considered to be in the same fuzzy ensemble. The combined front and rear forces are generated by

$$\begin{cases} F\_{\mathcal{Y}} = \sum\_{i=1}^{2} h\_i(\xi(t)) \mathbf{C}\_{\text{fi}} a\_f(t) \\\\ F\_{\mathcal{Y}^r} = \sum\_{i=1}^{2} h\_i(\xi(t)) \mathbf{C}\_{ri} a\_r(t) \end{cases} \tag{6}$$

With *hi*ð Þ *i* ¼ 1*:*2 as membership functions, relating to the front tire slip angles *α <sup>f</sup>*ð Þ*t* , which are considered to be available for measurement, they satisfy the following assumptions:

$$\begin{cases} \sum^2 h\_i(\xi(t)) = \mathbf{1} \\ \displaystyle \mathbf{0} \le h\_i(\xi(t)) \le \mathbf{1}, \quad i = \mathbf{1}, 2 \end{cases} \tag{7}$$

The membership function *hi*ð Þ *ξ*ð Þ*t* expressions are the following:

$$h\_i(\xi(t)) = \frac{\alpha\_i(\xi(t))}{\sum\_{i=1}^2 \alpha\_i(\xi(t))} \text{with } : \xi(\mathbf{t}) = |a\_{\mathbf{f}}(\mathbf{t})| \tag{8}$$

and

$$o\_i(\xi(t)) = \frac{1}{\left(1 + \left|\frac{\xi(t) - c\_i}{a\_i}\right|\right)^{2b\_i}}\tag{9}$$

To determine the membership function parameters and the stiffness coefficient parameters, an identification method based on the Levenberg–Marquardt algorithm in combination with the least squares method is used as in [18], the values listed in **Table 2**.


**Table 2.**

*Nominal stiffness and membership function coefficients.*

The nonlinear lateral dynamics of the automotive vehicle, described by Eq. (1), can be written as follows:

$$\begin{cases} \dot{\mathbf{x}}(t) = \sum\_{i=1}^{2} h\_i(\xi) (A\_i \mathbf{x}(t) + B\_i \mathbf{u}(t)) \\\\ \mathbf{y}(t) = \sum\_{i=1}^{2} h\_i(\xi) \mathbf{C}\_i \mathbf{x}(t) \end{cases} \tag{10}$$

where *x t*ðÞ¼ *vy ψ ϕ* \_ *<sup>ϕ</sup>*\_ � �*<sup>T</sup>* is the system state vector, *y t*ð Þ is the measured output vector, and *u t*ð Þ is the input vector. In this study, we consider that the input signal is the front wheel steering angle given by the driver *u t*ðÞ¼ *δfd*. *Ai*, *Bi* f g ,*Ci* are sub-model matrices, which are given as follows:

$$A\_{i} = \begin{bmatrix} a\_{11} & a\_{12} & \mathbf{0} & \mathbf{0} \\ a\_{21} & a\_{22} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{1} \\ a\_{41} & a\_{42} & a\_{43} & a\_{44} \end{bmatrix}; B\_{i} = \begin{bmatrix} b\_{11} \\ b\_{21} \\ \mathbf{0} \\ \mathbf{0} \\ b\_{41} \end{bmatrix}; \mathbf{C}\_{i} = \begin{bmatrix} c\_{11} & c\_{12} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{1} & \mathbf{0} & \mathbf{0} \end{bmatrix}$$

with

*<sup>a</sup>*<sup>11</sup> ¼ � <sup>2</sup> *Cfi* <sup>þ</sup> *Cri* � � *mvx* , *<sup>a</sup>*<sup>12</sup> <sup>¼</sup> <sup>2</sup> *<sup>a</sup> fCfi* � *arCri* � � *mvx* � *vx*, *<sup>a</sup>*<sup>21</sup> ¼ � <sup>2</sup> *<sup>a</sup> fCfi* <sup>þ</sup> *arCri* � � *Izvx a*<sup>22</sup> ¼ 2 *a*<sup>2</sup> *<sup>f</sup>Cfi* � *<sup>a</sup>*<sup>2</sup> *rCri* � � *Izvx* , *<sup>a</sup>*<sup>41</sup> ¼ � <sup>2</sup>*hroll Cfi* <sup>þ</sup> *Cri* � � *mvxIx* , *<sup>a</sup>*<sup>42</sup> <sup>¼</sup> <sup>2</sup>*mshroll Cfi* <sup>þ</sup> *Cri* � � *mvxIx <sup>a</sup>*<sup>43</sup> <sup>¼</sup> *hrollmsg* � *<sup>K</sup><sup>ϕ</sup> Ix* , *<sup>a</sup>*<sup>44</sup> <sup>¼</sup> *<sup>C</sup><sup>ϕ</sup> Ix* , *c*<sup>11</sup> ¼ �2 *Cfi* þ *Cri mvx* , *c*<sup>12</sup> ¼ �2 *Cfia <sup>f</sup>* � *Criar mvx <sup>b</sup>*<sup>11</sup> <sup>¼</sup> <sup>2</sup>*Cfi <sup>m</sup>* , *<sup>b</sup>*<sup>21</sup> <sup>¼</sup> <sup>2</sup>*<sup>a</sup> fCfi Iz* , *<sup>b</sup>*<sup>41</sup> <sup>¼</sup> <sup>2</sup>*mshrollCfi mIx*

**Lemma 1**. [19] *Let the matrices Nij and the condition be*

$$\begin{aligned} \sum\_{i=1}^r \sum\_{j=1}^r h\_i(\xi(t)) h\_j(\xi(t)) N\_{\vec{\imath}\vec{\imath}} &= \sum\_{i=1}^r h\_i^2(\xi(t)) N\_{\vec{\imath}\vec{\imath}} \\ &+ \sum\_{i=1}^r \sum\_{\substack{i$$

*T-S Fuzzy Observers to Design Actuator Fault-Tolerant Control for Automotive Vehicle Lateral… DOI: http://dx.doi.org/10.5772/intechopen.92582*

true if there exist matrices Ξ*ii* and Ξ*ij* such that the following conditions are fulfilled:

$$\begin{aligned} &N\_{ii} < \Xi\_{ii}, \quad i = \mathbf{1}, \ldots, r, \\ &N\_{ij} + N\_{ji} \le \Xi\_{ij} + \Xi\_{ij}^T, \quad i, j = \mathbf{1}, 2, \ldots, r, \quad i < j \\ &\begin{bmatrix} \Xi\_{11} & \Xi\_{12} & \ldots & \Xi\_{1r} \\ \ast & \Xi\_{22} & \ldots & \Xi\_{2r} \\ \vdots & \vdots & \ddots & \vdots \\ \ast & \ast & \ldots & \Xi\_{rr} \end{bmatrix} < \mathbf{0} \end{aligned}$$
