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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.

In recent years, there has been a substantial increase in the number of electric vehicles (EV), due to the increase of pollution by

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 [

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 [

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

Figure

Scheme of an electric vehicle.

Longitudinal forces acting on an electric vehicle.

The ^{2}) is the motor constant,

It is important to note that, for this case, the motor current

The ^{3}) is the air density (^{3} under standard conditions), ^{2}) is the frontal area, and

The

Finally, the

It is important to note that most of the reported work [

Substituting (

For this mathematical model, the internal frictions, rotational inertias of the power train, and the inertia of the electric motor are neglected because they are small compared to the mass of the vehicle. The vehicle displacement

Finally, the nonlinear model affine to the input is given by

A TS model describes a nonlinear system using a collection of local linear time invariant (LTI) models, which are interpolated by nonlinear functions known as weighting functions. The general form of a TS model is given by

The weighting functions

There are three methods for obtaining a TS model. When it is not possible to obtain analytical models by physical principles, the most appropriate method is system identification [

Nonlinear sector transformation approach.

In order to derive a TS model, the nonlinear model of the EV in (

Two premise variables are considered, the nonlinear term

The premise variables are selected as

The premise variables are replaced in (

These premise variables comply with (

The nonlinear model of the EV represented by the TS approach is expressed as

The model parameters considered in this paper were presented in [

Parameters of the electric vehicle [

Parameter | Value | Unit |
---|---|---|

kg | ||

0.97 | — | |

N·m/A | ||

m | ||

m | ||

kg/m^{3} | ||

m^{2} | ||

— | ||

m/s^{2} |

Comparison between the general nonlinear model and the TS model in response to variations of the input current.

The fault diagnosis approach proposed in this paper is based on the generalized observer scheme. This method considers a reduced order TS observer for each of the measured outputs as shown in Figure

General scheme of the observer bank to detect sensor faults.

Each observer has the following form:

The estimation error between (

The error dynamics is given by

Replacing (

From the above expression, it follows that if the state estimation (

The stability conditions for the derivative of the error (

As can be observed from (

Nonetheless, expression (

Note that in order to perform fault diagnosis, two observers need to be designed according to Theorem

Two normalized residual signals are computed and evaluated as follows:

Numerical simulation results are presented in this section in order to illustrate the applicability of the proposed method. The EV parameters are the same as the ones presented in [

Two reduced observers are considered as discussed in the previous sections, whose gains (

The gains for observer 1 are

The gains for observer 2 are

In order to verify the convergence of both observers, the considered initial conditions are

Variation on the slope of the road (

In addition, to illustrate the fault detection and isolation method, different faults are considered to affect the two sensors in different time intervals. Sensor can fail for different reasons. Usually this behavior can be observed as additive bias, for example, offsets or calibration problems. These malfunctions can be described by ramp or step functions in order to represent abrupt or slow variation faults, as described in [

Normalized residuals vectors

The normalized residual signals are also shown in Figure

Incidence matrix.

Fault | ||
---|---|---|

1 | 0 | |

0 | 1 |

In this work, an observer-based fault detection system was designed for an electric vehicle. The EV is represented by a TS model whose weighting functions depend on the velocity and the slope of the road. This representation is more general compared with that of typical models, which consider a constant slope. Sufficient conditions, which guarantee the convergence of the TS observer, and the observer bank were proposed by a set of linear matrix inequalities. Finally, the detection of fault on the velocity and traveled-distance sensor was demonstrated through simulation by the variation of the residues compared to an incidence matrix. A variable slope was considered in order to increase the range of representativity of the nonlinear dynamics. The result shows that the TS fault diagnosis observer can detect and isolate sensor faults for different driving conditions and different types of faults. However, the model can be improved by measuring the mass of the vehicle in order to be considered as a premise variable. Future work will be done in order to consider simultaneous faults and inexact premise variables.

The authors declare that there is no conflicts of interest regarding the publication of this paper.

_{∞}sensor fault detection observer design for nonlinear systems with parameter uncertainty