International Journal of Vehicular Technology

Volume 2014, Article ID 829097, 9 pages

http://dx.doi.org/10.1155/2014/829097

## Traction Control of Electric Vehicles Using Sliding-Mode Controller with Tractive Force Observer

Department of Mechanical and Aerospace Engineering, Faculty of Engineering, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand

Received 24 June 2014; Revised 26 November 2014; Accepted 1 December 2014; Published 21 December 2014

Academic Editor: Nicolas Hautière

Copyright © 2014 Suwat Kuntanapreeda. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Traction control is an important element in modern vehicles to enhance drive efficiency, safety, and stability. Traction is produced by friction between tire and road, which is a nonlinear function of wheel slip. In this paper, a sliding-mode control approach is used to design a robust traction controller. The control objective is to operate vehicles such that a desired wheel slip ratio is achieved. A nonlinearity observer is employed to estimate tire tractive forces, which are used in the control law. Simulation and experimental results have illustrated the success of the proposed observer-based controller.

#### 1. Introduction

Electric vehicles (EVs) have become very attractive in replacing conventional internal combustion engine vehicles because of environmental and energy issues. They have received a great attention from the research community. Control methodologies have been actively developed and applied to EVs to improve the EVs performances [1–8].

Traction control plays an important role in vehicle motion control because it can directly enhance drive efficiency, safety, and stability [9, 10]. Traction is the vehicular propulsive force produced by friction between tire and road. The characteristics of the friction are nonlinear and uncertain, which make traction control difficult. The friction depends on many factors such as tire type, road surface, road condition, and wheel slip. Accordingly, an objective of the traction control is to operate vehicles such that a desired wheel slip ratio is obtained. The slip ratio yielding the maximum friction coefficient is usually desired because it yields the maximum torque from the propulsion system to drive the vehicle forward.

Traction control of electric vehicles has drawn extensive attention since electric motors can produce very quick and precise torques compared to conventional internal combustion engines. In [1], traction control based on a maximum transmission torque estimation (MTTE) approach was proposed. The estimation was carried out by an open-loop disturbance observer, which required only the input torque and the wheel speed. The estimated maximum transmission torque was used in the control law as a constraint to prevent the slip. Experimental results illustrated the effectiveness and practicality of the proposed control design. The MTTE approach was extended by replacing the open-loop observer with a closed-loop observer in [2]. By doing this, the robustness of the control system was markedly enhanced. In [3], traction control of electric vehicles using a sliding-mode observer to estimate the maximum friction was presented. The observer was based on the LuGre friction model. The controller used the estimated maximum friction to determine the suited maximum torque for the wheels.

Sliding-mode control has been extensively used for control of uncertain nonlinear systems because of its robustness property. The essence of the sliding-mode control is to use a switching control command to drive the controlled system’s state trajectory onto a specified sliding surface in the state space and then to keep the state trajectory moving along this surface [11, 12].

As far as sliding-mode control of vehicles is concerned, wheel slip control of electric vehicles based on a sliding-mode framework was proposed in [4]. A conditional integrator approach was employed to overcome the chattering, enabling a smooth transition to a PI control law when the slip is close to the set point. Experimental results demonstrated a good slip regulation and robustness to disturbances. In [13], a sliding-mode approach to the design of an active braking controller was proposed. The controlled variable was a convex combination of wheel deceleration and wheel slip. The approach offered advantages with respect to pure slip and deceleration control. In [14], a second-order sliding-mode traction force controller for vehicles was proposed. The traction control was achieved by maintaining the wheel slip at a desired value.

This paper presents a robust control scheme for traction control of electric vehicles. The control objective is to operate the vehicles at a desired wheel slip ratio. The paper proposes a simple approach to design a traction controller based on a sliding-mode control framework. The main motivation of the design is the robustness to uncertainties. The implementation of the control design requires tractive forces for feedback, but they are not usually available in practices. To overcome this problem, a PI observer developed in [15, 16] is used to estimate the tractive forces. The PI observer has an attractive zero-steady-state feature similar to the well-known PI controllers. This synthesis of the sliding-mode traction controller and the PI observer makes the implementation practical. At the end, the resulting observer-based controller is experimentally validated in a single-wheel test rig.

The rest of the paper is organized as follows. In the next section some preliminaries are provided. The longitudinal dynamic model of the vehicles used in the paper is presented in Section 3, followed by the controller and observer design in Section 4. Simulation results are given in Section 5. In Section 6 an experimental study is presented. The conclusions of this paper are drawn in Section 7.

#### 2. Preliminaries

##### 2.1. Sliding-Mode Control

Sliding-mode control based on the equivalent control method is summarized in this subsection. The reader is referred to [11, 12] for more details of the method.

The controlled system is expressed as where is the state variable vector, is the input vector, and and are nonlinear functions. Let be a desired sliding surface, which is usually chosen according to the control objective. Based on the equivalent control method, the control input vector is written as where and are called the equivalent control and the switching control, respectively. The equivalent control is determined based on the assumption that the system trajectory is staying on the sliding surface. Thus, it is simply obtained by setting The switching control is designed to guarantee that the system trajectory moves towards the sliding surface and stays on it. It is determined such that the reachability condition is satisfied.

##### 2.2. Nonlinearity Observer

The nonlinearity observer developed in [15, 16] is provided in this subsection. Consider the following nonlinear system: where is the state variable vector, is the control vector, is the output vector, is the system matrix, is the control input matrix, is the output matrix, and and are the constant matrices. Here, is an unknown nonlinear function whereas is a known function. The observer is designed to estimate .

The fundamental idea of the observer is to approximate by a fictitious system: By substituting (6) into (5), the system can be expressed as Thus, the observer is chosen as where and are the observer gain matrices that must be chosen such that the observer is asymptotically stable.

In a special case when and are chosen, the observer is reduced to the proportional-integral (PI) observer: and the estimated nonlinearity is given by The PI observer has been successfully applied to control problems [17, 18]. The reader is referred to [16] for a proof of the estimation convergence and an analysis of the estimation errors.

#### 3. Longitudinal Dynamic Model

A longitudinal dynamic model of two-axle vehicles is presented. The model is known as a bicycle model, which has three degrees of freedom. The model can be found in [14] and is given as where is the longitudinal velocity of the vehicle center of gravity, are the tire rotational speeds (the subscripts and stand for the front and rear axle, resp.), is the vehicle mass, are the moments of inertia, are the tire effective rolling radius, are the input torques, are the tractive forces, combining the aerodynamic drag and the rolling resistance, and and are the aerodynamic drag and rolling resistance coefficients, respectively.

The tractive forces are given by where is the friction coefficient function, is the wheel slip ratio defined as and are the normal loads given as where is the distance from the front axle to the center of gravity, is the distance from the rear axle to the center of gravity, and is the height of the center of gravity (see Figure 1).