Parameters of the automotive vehicle lateral dynamics system.

## 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, 2, 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, 5, 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, 8, 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, 11, 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*. 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. 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

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:

All system parameters are defined in Table 1.

Parameters | Description | Unit |
---|---|---|

Vehicle total mass | [kg] | |

Vehicle speed/vehicle lateral velocity | [m/s] | |

Roll/yaw inertia moment at the center of gravity | [kg m^{2}] | |

Distance from the center of gravity to front/rear axles | [m] | |

Center of gravity height from the roll axis | [m] | |

Combined roll damping/stiffness coefficients | [N.ms/rad][N.m/rad] | |

Front/rear lateral forces | [N] | |

Front/rear tire slip angles | [rad] |

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

where

## 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.

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

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

The proposed rules are only made for

With

The membership function

and

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.

Nominal stiffness coefficients | C _{f1} | C _{f2} | C _{r1} | C _{r2} |
---|---|---|---|---|

Values | 55,234 | 15,544 | 49,200 | 13,543 |

Membership function coefficients | ||||

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

where

with

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

true if there exist matrices

## 4. Fault-tolerant control strategy based on T-S fuzzy observers

The purpose of this section is to design a faulty T-S fuzzy system for the vehicle model. Then, observers are trained to estimate system states and actuator faults. Next, a fault-tolerant control law is developed from the information provided by the observers.

### 4.1 Faulty vehicle lateral dynamics system description

From the T-S zero fault model (10), the system in the presence of actuator faults

where

### 4.2 T-S fuzzy observers design

In this subsection, the T-S fuzzy observers are constructed to simultaneously estimate states and actuator faults. Consider the following T-S observers:

where

The output error between faulty T-S fuzzy systems (11) and T-S fuzzy observers (12) is given by

The dynamics of the estimation error

with

The tracking error

and its dynamics is given by

### 4.3 Design fault-tolerant control based on T-S fuzzy observers

A fault-tolerant control is proposed in this section to ignore the impact of faults and to maintain the stability of the faulty vehicle dynamics system. The suggested control procedure is outlined in Figure 3 [20].

Consider the following control law:

where

Substituting Eq. (20) in Eq. (19) gives:

Assume that

Consider the following augmented system:

where

**Theorem 1.1.** The state tracking error

where

with

The gains of T-S fuzzy observers

**Proof:** the gains

Let us select the following quadratic Lyapunov function:

where

The time derivative of

where

Inequality (34) is negative if the following conditions are satisfied:

with

Using the lemma of congruence, we have

with

Then, this inequality is obtained

The negativity of (39) enforces that

with

which can be analyzed using the following property:

Accordingly, (39) can then be delineated as follows:

Using lemma 1, and with some manipulations, we can obtain easily (26)–(28). This completes the proof.

**Remark 1.***The calculation of the observer and controller gains is done independently in* [11, 21]*, which is restrictive. Therefore, in this study, the resolution of the LMIs is carried out in one step.*

## 5. Vehicle simulation results

In this section, the computer testing displays a critical driving situation. Initially, the system states and actuator faults will be estimated. Next, to demonstrate the effectiveness of the proposed FTC, the states of the faulty system will be simulated considering the fault-tolerant control law.

Simulations are performed with the forward steering angle profile given in Figure 4. This shows a sequence of right and left turns. The vehicle data is shown in Table 3 [22].

Parameters | Value | Parameters | Value |
---|---|---|---|

9.806 m/s^{2} | 1.18 m | ||

1832 kg | 1.77 m | ||

23 m/s | 0.90 m | ||

614 kg m^{2} | 6000 Nms/rad | ||

2988 kg m^{2} | 140,000 Nms/rad |

Each

To see the effectiveness of the proposed scheme, we compare the states of the zero-fault T-S model with the states of the faulty system both with and without the FTC strategy.

Figure 5 illustrates the time evolution of the fault

In Figures 6–9, simulations of faulty system states in the absence of FTC clearly show that the performances of the vehicle has been lost and that its states reached unsustainable levels immediately after the actuator became faulty, but applying fault-tolerant control law, the vehicle stays stable throughout the simulation without losing performances despite the presence of faults, demonstrating the success of the proposed FTC approach.

## 6. Conclusion

In this chapter, a fault-tolerant nonlinear control law scheme is proposed to reform the states of the lateral dynamics system of the automotive vehicle when it becomes faulty at the actuator side. The nonlinear model of the lateral dynamics of the automotive vehicle is first represented by a T-S fuzzy models, and then the fault-tolerant control design based on the T-S fuzzy observers is presented. The mentioned strategy is based on the use of the T-S reference models and the information provided by the T-S fuzzy observers. This control law is developed to reduce the deviation of the faulty system from the reference; it uses the steering angle, the estimation error, and the tracking error. The stability of the whole system is investigated in one step with the Lyapunov theory and by solving the LMIs constraints. The simulations of the automotive vehicle strongly demonstrate that the engineered FTC is very successful and can be adapted to the specific driving situations. These results will be applied to the CarSim software in future work.

## Conflict of interest

The authors declare no conflict of interest.