Cornering stability control for vehicles with active front steering system using T-S fuzzy based sliding mode control strategy

https://doi.org/10.1016/j.ymssp.2018.05.059Get rights and content

Highlights

  • The vehicle nonlinear characteristics are considered by using the T-S fuzzy approach.

  • A new sliding mode control strategy is proposed based on the T-S fuzzy approach.

  • The effective region is considered for the bounded sector zone of the AFS system.

  • A model parameter observer is designed to obtain the real-time cornering stiffness.

Abstract

Active front steering (AFS) has drawn considerable attention due to its superiority in improving the vehicle safety. This paper proposes a new Takagi-Sugeno (T-S) fuzzy based sliding mode control (SMC) strategy for the AFS system to improve the cornering stability of vehicles. Different from the conventional SMC strategy, the control law of the proposed control strategy is designed based on the T-S fuzzy approach which deals with the nonlinearity of the tire in a simplified but effective way. In the design of the control strategy, the bounded sector zone in the T-S fuzzy approach is determined based on the working region of the AFS system, and a model parameter observer is constructed to obtain the real-time cornering stiffness of the vehicle. To evaluate the cornering performance, simulations are conducted via open-loop and closed-loop tests. Simulation results show that the proposed T-S fuzzy based SMC strategy, which considers the nonlinearity of the tire and multi-objective control, is able to improve the cornering stability of the vehicle as compared with the conventional SMC strategy.

Introduction

In recent decades, investigations on the cornering stability have become attractive topics in the field of vehicle dynamics. The cornering stability mainly concerns with the yaw motion and roll motion in steering conditions of the vehicle [1]. Its performance directly affects the handling stability and active safety of the vehicle. In order to ease the burden of drivers and enhance the active safety of the vehicle, this work aims to study a new system to enhance the vehicle cornering stability.

Among various techniques used for improving cornering stability, the active front steering (AFS) is regarded as a new technology of the vehicle steering system. With the AFS system, it is possible to improve the cornering stability by actively introducing an additional steering angle which attains a superposition of the additional steering angle with the steering wheel angle that is prescribed by the driver [2], [3]. Moreover, the AFS system also has three more advantages in addition to enhancing the cornering stability, which are: (1) steering comfort by reducing driver’s steering effort; (2) quick response to the driver’s input; and (3) active safety as well as maneuver stability [3]. In view of preponderances of the AFS system, an investigation on the AFS system is a desirable work.

The AFS system can enhance the vehicle cornering stability by adjusting the yaw moment of the vehicle. To be more specific, the AFS system is regarded as an indirect yaw moment control method focusing on the steering angle compensation via the active steering actuator [4], [5]. However, owing to the nonlinear characteristics of the vehicle, the AFS system cannot generate satisfactory yaw moment in some situations. For instance, under limited working conditions, the high lateral acceleration may lead that the yaw moment becomes insensitive to the change of the steering input [6]. In other words, an effective working region of the AFS system should be defined to ensure the effectiveness of the AFS system under various working conditions. Moreover, most of the existing control methods focus on the yaw moment control by integrating the AFS system with other systems via multi-inputs and multi-outputs strategies [7], [8]. As a result, the conflict between the AFS system and other yaw moment control systems has not been fully considered based on the working region. Hence, the working region of AFS is also studied in this work.

As the core component of the AFS system, the controller plays a critical role in the improvement of cornering stability. In previous researches, various control approaches have been developed, such as fractional-order PID control [9], yaw rejection control [10], sliding mode control (SMC) [11], adaptive sliding mode control [12], mu-control [13], quantitative feedback theory control [14], composite nonlinear feedback control [15] and model predictive control (MPC) [16], [17]. However, most of the above strategies only take the yaw rate into account to ensure the cornering stability. Actually, the sideslip angle is also a key factor that influences the cornering stability and the dynamic behavior of the vehicle is determined by both the yaw rate and the sideslip angle. When the sideslip angle is small, the dynamic behavior of the vehicle is determined by the yaw angle which can be obtained by integrating the yaw rate with respect to time. However, when a severe sideslip occurs, the sideslip angle increases quickly, and it is not accurate to describe the dynamic behavior of the vehicle solely by the yaw rate. Therefore, two dependent control objectives, namely yaw rate and sideslip angle, are proposed to be considered together in improving the cornering stability in this work.

To accomplish the multi-objective control, several algorithms have been studied, such as the linear quadratic regulator (LQR), MPC and SMC [11], [12]. For the LQR strategy, it is usually used in linear or nonlinear systems with more than two objectives [18], while the MPC strategy is generally applied to look for the optimum for each uncorrelated control objective. However, it is difficult to solve a objective function with a complicated model when applying the MPC method [19]. Different from the LQR and MPC strategies, SMC behaves well in dealing with the multi-objective control, especially for controlling two dependent objectives [12]. In this case, the sideslip angle and yaw rate are two dependent objectives, and the SMC strategy is therefore considered in this study.

In the open literatures, most of the SMC strategies have been designed based on a two-degree-of-freedom (2-DOF) linear vehicle model, in which the nonlinearities of the model are ignored so as to ease the computation burden [13]. Undoubtedly, the model accuracy and performance of the controller is not satisfied in this case. To improve the performance of the conventional SMC strategy, the nonlinearities of the vehicle model should be carefully considered when designing the controller, especially for the nonlinearity of the tire. Even though the vehicle model with nonlinear tire characteristics (e.g. by creating the tire model via Dugoff or Pacejka mehtod) can be adopted in the controller design, a tedious computational process may be caused which definitely increases the response time in real-time control [19]. Therefore, a simplified but effective strategy should be proposed in the controller design. Upon the aforesaid concerns, it is essential to develop a new SMC controller to deal with the tradeoff between the model accuracy and simplified control process by considering the nonlinear characteristics of the tire.

To obtain better dynamic performance, both the nonlinearities of the vehicle model and the simplification of the control law should be taken into account. Thereby, a Takagi-Sugeno (T-S) fuzzy based SMC strategy is proposed. The T-S fuzzy approach, which is suitable to describe a complex nonlinear or uncertain system by a weighted sum of several simple linear or affine models via local approximation [20], [21], [22], is introduced to construct the nonlinear vehicle model with the idea of sector nonlinearity. In the review of previous investigations, Li et al. proposed a nonlinear active suspension system via the T-S fuzzy approach in which the varying sprung and unsprung masses and the unknown actuator nonlinearity are considered [23], [24]. Those studies indicate that the T-S fuzzy approach is feasible to transfer the nonlinear problem into a linear problem. Furthermore, though investigations in [25], [26], [27] have developed the vehicle lateral models via the T-S fuzzy approach for simulation test, the T-S fuzzy approach has not been applied to the development of the controller yet.

In addition, it has been proved that the T-S fuzzy system can match the SMC strategy well to attenuate external disturbances and solve system uncertainties [28], [29], [30], [31]. However, the nonlinearities of systems and multi-objective control have not been fully considered in the previous work. Besides, few of previous investigation has employed the cooperative control method based on the T-S fuzzy and SMC method for complex and practical engineering problems. Motivated by the above issues, a T-S fuzzy based SMC strategy is therefore proposed to enhance the cornering stability with considering the nonlinearity of the tire and multi-objective control.

It should be noted that parameters vary during the control process due to the nonlinearities of the vehicle. The main nonlinearity of vehicle dynamics in this study is the tire force which changes along with the wheel sideslip angle [32]. In other words, the cornering stiffness of the tire varies under different conditions. By observing the variation of the cornering stiffness, it is possible to improve the cornering stability with the proposed control strategy to cater for practical situations. Thus, the construction of the concerning stiffness observer is desirable. However, there exist difficulties in observing the cornering stiffness of the vehicle: (1) lack of the analytical model; and (2) high demand of fast reasoning response. To solve these difficulties, fuzzy logic algorithm is introduced because it does not need an explicit mathematical model and is fast in terms of the computational speed. Thereby, a model parameter observer is constructed by using the fuzzy logic algorithm to observe the cornering stiffness of the vehicle.

Based on the above discussion, a new control strategy is proposed in this paper by using the T-S fuzzy based SMC algorithm. The main novelties of this paper can be summarized as follows: (1) This paper considers both the nonlinearities of the vehicle and multi-objective control in controller design via the proposed T-S fuzzy based SMC strategy; (2) A high accuracy and simplicity vehicle model is constructed via the T-S fuzzy approach by solving the tire nonlinearities; (3) The working region of the AFS system is studied via the theory behind the yaw moment, which provides a basis for determination of the bounded sector zone in the T-S fuzzy approach; (4) A model parameter observer is constructed to obtain the real-time cornering stiffness of the vehicle through fuzzy logic. It is believed that this paper can provide a guidance for the construction of the nonlinear vehicle system and offer a new idea for enhancing cornering stability in real vehicles, as well as presenting the benefits of consideration of the nonlinearity of the tire to the improvement on the effectiveness and accuracy of the control strategy. The rest of this paper is organized as follows: Section 2 formulates the problem in cornering stability. Section 3 presents the proposed T-S fuzzy based SMC strategy. Simulations with analysis are illustrated in Section 4. Finally, conclusions of the research are given in Section 5.

Section snippets

Problem formulation

To describe the lateral motion and the yaw motion, a 2-DOF vehicle model as shown in Fig. 1 is commonly employed to identify the steering characteristics of the vehicle. The model can reflect the relationship among the sideslip angle, yaw rate and steering angle by assuming a constant longitudinal speed and small wheel sideslip angles [33]. Derivation of the model can be started by applying the Newton’s second law and torque equilibrium equation. With the law and the equilibrium equation, the

Controller design

In this section, the effective working region of the AFS system is first described through the relationship between the tire lateral force and the wheel sideslip angle. Then, a T-S Fuzzy approach is designed to solve the nonlinear problem of the vehicle in the working region of the AFS system. Subsequently, the model parameter observer is constructed through fuzzy logic to observe the cornering stiffness. Thereupon, the T-S fuzzy based SMC strategy is realized. The overall workflow of the

Simulations and analysis

In order to evaluate the performance of the proposed T-S fuzzy based SMC strategy, simulations are performed by connecting MATLAB and CarSim. Since CarSim has been universally acknowledged as the preferred tool in simulating the most accurate and realistic conditions that close to the real car test, it is reasonable to conduct simulations by using CarSim. To examine the superiority of the proposed T-S fuzzy based SMC strategy, a conventional SMC strategy is taken as a comparative case.

Conclusions

In this research, a new SMC strategy is proposed based on the T-S fuzzy approach which disposes the nonlinear characteristics of the vehicle by summing several linear models with according weighting coefficients. In the implementation of the T-S fuzzy approach, this research considers the effective region of AFS system and simplifies the complex vehicle model with varying cornering stiffness of the wheels. Moreover, a model parameter observer for cornering stiffness is designed on the basis of

Acknowledgements

This work was supported by the research grant of the University of Macau under grant numbers MYRG2016-00212-FST and MYRG2017-00135-FST. This project is also supported by National Natural Science Foundation of China (Grant No. 51705084).

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