Journal of Sensors

Volume 2018, Article ID 3291639, 9 pages

https://doi.org/10.1155/2018/3291639

## Sensor Fault Diagnosis Observer for an Electric Vehicle Modeled as a Takagi-Sugeno System

^{1}Tecnológico Nacional de México (TecNM)/Instituto Tecnológico de Tuxtla Gutiérrez, TURIX Dynamics-Diagnosis and Control Group, Carretera Panam, km 1080, Tuxtla Gutiérrez, CHIS, Mexico^{2}Tecnológico Nacional de México (TecNM)/Instituto Tecnológico de Hermosillo, TURIX-Hermosillo, Av. Tecnológico y Periférico Poniente S/N, 83170 Hermosillo, SON, Mexico^{3}HSPdigital-CA Mecatrónica, Facultad de Ingeniería Campus San Juan del Río, Universidad Autónoma de Querétaro, Río Moctezuma 249, San Cayetano, 76807 San Juan del Río, QRO, Mexico

Correspondence should be addressed to F. R. López-Estrada; xm.ude.gtti@zepolrf

Received 25 August 2017; Revised 28 November 2017; Accepted 4 December 2017; Published 28 March 2018

Academic Editor: Jing Xu

Copyright © 2018 S. Gómez-Peñate et al. 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

A sensor fault diagnosis of an electric vehicle (EV) modeled as a Takagi-Sugeno (TS) system is proposed. The proposed TS model considers the nonlinearity of the longitudinal velocity of the vehicle and parametric variation induced by the slope of the road; these considerations allow to obtain a mathematical model that represents the vehicle for a wide range of speeds and different terrain conditions. First, a virtual sensor represented by a TS state observer is developed. Sufficient conditions are given by a set of linear matrix inequalities (LMIs) that guarantee asymptotic convergence of the TS observer. Second, the work is extended to perform fault detection and isolation based on a generalized observer scheme (GOS). Numerical simulations are presented to show the performance and applicability of the proposed method.

#### 1. Introduction

In recent years, there has been a substantial increase in the number of electric vehicles (EV), due to the increase of pollution by emissions to the environment. Recent studies show that currently there are more than 800 million cars circulating every day, which represent a distribution of 400 to 800 vehicles per 1000 inhabitants [1]. As a result, vehicles are responsible for a high percentage of global energy consumption and greenhouse gas emissions. This tendency shows an accelerated growth of vehicles per inhabitants, and, while the energy consumption in other sectors decreases, the consumption due to the continuous use of transport vehicles grows [2]. On the other hand, new sensor systems and actuators on EV are increasing in complexity, and the probability for a fault taking place is high [3].

For example, recently, a Tesla driver died in a crash while using the autopilot mode because the car’s sensor system failed to distinguish a large white 18-wheel truck and trailer crossing the highway. This accident caused a severe crisis in the EV industry. Therefore, safety, reliability, and energy-saving optimization systems are a demand of the new growing industry. In line with this demand, this work is dedicated to propose a method to detect and isolate sensor faults in an electric vehicle.

An important stage in the design of the diagnosis system is the mathematical model that represents the dynamic characteristics of the EV, which is expressed by a set of nonlinear differential equations depending on exogenous nonstationary parameters [4], for example, slopes or poor conditions of the road. However, typical models of EV consider a simplified representation given by linear models, in which it is not possible to consider these nonstationary parameters [5–7]. Nonetheless, it is possible to obtain better representations when nonstationary exogenous parameters, such as the road slope, could be measured online, such that the desired diagnosis system, which also depends on these measurable parameters, is better and less conservative [8]. In typical linear time-invariant systems, it is not possible to consider these variations. However, a viable alternative is Takagi-Sugeno models that consider nonlinearities and varying parameters as part of the mathematical modeling [9, 10], which increases the physical representativity of a real physical system.

The main advantage of a TS model is its capability of describing nonlinear dynamics through a collection of local linear models that are interpolated by nonlinear functions [11]. These functions are known as weighting functions and depend on exogenous variables that can be measurable (e.g., system inputs, outputs, exogenous nonstationary parameters, velocity, and the slope of the terrain) or unmeasurable (e.g., state variables, magnetic flux, and slip angle of the tires) [12, 13]. In this work, weighting functions are considered measurable. Additionally, an important property of this type of models is that the weighting functions are convex, which allows to extend some of the tools and methods developed for linear systems to TS systems. In particular, it has been shown that a TS dynamic model obtained through the nonlinear sector transformation approach can describe the overall behavior of a highly complex nonlinear system with a high degree of accuracy [11, 14]. As a result, its applicability in designing controllers, diagnosis systems, and observers, among others, has become of high importance; see, for instance, [15–17] and references therein. Applications on vehicles can be found in the literature; for example, in [4], a predictive control strategy using a TS model is presented to control the velocity in an EV, and the authors in [18] proposed a linear parameter varying (LPV) controller in order to control the tracking of the longitudinal velocity and the yaw velocity of the EV. In [19], a TS fuzzy model is used to represent the nonlinear behavior of an electric power steering (EPS) system, and stabilization conditions for nonlinear EPS system with both constrained and saturated control input cases are proposed in terms of linear matrix inequalities. Some works related to fault diagnosis can be consulted in the following references: in [20], an observer design strategy is presented to estimate the lateral dynamics of a vehicle and the curvature of the road. The nonlinear model of vehicle dynamics is transformed into an exact TS model with weighting functions depending on unmeasured states. In [21], a TS observer is designed to detect faults and estimate states in an induction motor, in order to implement a fault-tolerant control. Recently, in [22], an observer was designed to estimate the lateral dynamics of a motorcycle represented as a quasi-LPV system. It is important to note that, in the works reported in [4, 18, 21], the slope of the road is considered constant or close to zero; nevertheless, in real driving conditions, this parameter is not constant and has a great impact on the vehicle performance and battery consumption. Unlike previous papers, this work considers the slope of the road in order to obtain an improved sensor fault diagnosis system.

In this paper, we propose the design of an observer-based fault diagnosis for an electric vehicle. The main contributions of this paper are listed as follows: (i) a Takagi-Sugeno model is developed, whose weighting functions depend on the longitudinal velocity and the slope of the terrain in order to increase the operation range of the diagnostic system; (ii) sufficient conditions are proposed in order to guarantee the asymptotic convergence of the observer that is deduced through a quadratic Lyapunov function and a set of linear matrix inequalities (LMIs); finally, (iii) a bank of observers based on a generalized observer scheme is proposed to detect and isolate sensor faults. The combination of both techniques results in a scheme for detecting faults in the traveled-distance and speed sensors at different operating and slope conditions.

The paper is organized as follows: in Section 2, the longitudinal model of the electric vehicle is presented; in Section 3, a TS model that uses the velocity and the slope as premise variables is developed; the conditions of the designed observer for the TS model of the EV are formulated in Section 4; the numerical simulation results are presented in Section 5. Finally, conclusions and perspectives of this work are presented in Section 6.

#### 2. Nonlinear Model of the Electric Vehicle

Figure 1 shows the general scheme of an EV, which is constituted mainly by an energy source (battery bank), a power inverter, an electric motor, and a transmission system coupled to the wheels. Considering Newton’s second law and the translational equilibrium principle, the longitudinal dynamics can be represented by (see Figure 2) [2] where (kg) is the total mass of the vehicle, (m/s) is the speed of the vehicle, (N) is the traction force on the wheels, (N) is the aerodynamic force, (N) is the rolling resistance, (N) is the slope resistance due to the vehicle’s weight and the road slope, and (s) is the time.