�

Abstract² A fault tolerant control approach that combines

the optimal properties of Linear Quadratic Regulator (LQR)

design with an observer based fault detection scheme is

proposed for the steering control of a ground vehicle subject to

sensor faults. In the proposed approach the linear quadratic

regulator is used to steer the ground vehicle along a reference

line in the lane, while maintaining small steady state error with

minimum control cost. The system is augmented with an

observer based fault detection method to detect, identify and

accommodate sensor faults, when they occur. Computer

simulations show the effectiveness of the proposed fault tolerant

control strategy under various driving conditions. Simplicity of

the overall scheme, minimization of the required energy and the

stabilization of the system under both faulty and fault free

conditions are the main positive features of the proposed

approach.

Index Terms� Fault tolerant control, automatic steering,

ground vehicles, sensor faults.

I. INTRODUCTION

N recent years much research had been devoted to the

design of intelligent vehicles for both civilian and military

applications [1]-[3]. As those vehicles become more and

more complex and highly integrated, the vulnerability of

their components to faults/failures increases. In order to

maintain high level of performance, it is important that

failures be promptly detected and identified and appropriate

remedies be applied. Fault Tolerant Control (FTC) is the

means for determining the corrective action necessary when

a fault has been detected and isolated by an appropriate Fault

Detection and Identification (FDI) algorithm. A fault-tolerant

control (FTC) system is a control system specifically

designed with potential system component failures in mind

[4]. A growing body of research in this area has resulted in

the development of a number of effective methods [4]-[8].

In this paper, an LQR based controller is design to place the

V\VWHP¶V�eigenvalues in the stable region while operating the

dynamic system at minimum cost function. To compensate

for sensor faults/failures, the LQR design is augmented with

This work is Partially supported by the Louisiana Board of Regents

Support Fund contract number LEQSF (2012-15)-RD-A-26.

A. Fekih is with the Department of Electrical and Computer Engineering

at the University of Louisiana at Lafayette, Lafayette, LA 70504 (phone:

337-482-5333; fax: 337-482-6568; e-mail: afef.fekih@louisiana.edu).

S. Seelem is D� 0DVWHU¶V� VWXGHQW� LQ the Department of Electrical and

Computer Engineering at the University of Louisiana at Lafayette,

Lafayette, LA 70504 (e-mail:sxs1248@louisiana.edu).

a set of observers to estimate the missed sensor information.

A Fault detection scheme based on a bank of observers is

considered. The prosposed method was implemented on a

vehicle following a required path while being subject to

sensor faults. Observer based estimator schemes were

implemented to generate the residual signals, which were

processed using a deterministic fixed threshold approach.

Compared to the existing work already reported in the

literature [6]-[8], the contributions of this paper are in the

following aspects:

(i) it provides a simple and effective fault tolerant strategy

(ii) the proposed approach combines the optimal properties

of LQR design with an observer based fault detection

scheme for the steering control of a ground vehicle subject to

sensor faults.

(iii) the residual signals were processed using a deterministic

threshold approach, allowing not only the detection of sensor

faults but also the adoption of the appropriate control law in

order to compensate for the effects of faults.

The paper is organized as follows. In section 2, the dynamic

vehicle model for the automatic steering controller is

presented. The proposed observer based fault tolerant

control scheme is described in section 3. The effectiveness of

the proposed fault tolerant schema is discussed in section 4.

Finally, section 5 gives some concluding remarks.

II. VEHICLE MODEL AND PROBLEM FORMULATION

A bicycle model of the vehicle, with the front and rear

wheels lumped together into a pair of single wheels at the

center of gravity [9], is considered as shown in Fig.1.

Fig.1: Dynamic bicycle model [9]

A Fault Tolerant Control Design for Automatic

Steering Control of Ground Vehicles Afef Fekih, Senior Member, IEEE, Shankar R. Seelem

I

2012 IEEE International Conference on Control Applications (CCA)Part of 2012 IEEE Multi-Conference on Systems and ControlOctober 3-5, 2012. Dubrovnik, Croatia

978-1-4673-4505-7/12/$31.00 ©2012 IEEE 1491

Combining the lateral forces with the available slip angles

and assuming a constant longitudinal velocity, the vehicle¶V model can be defined by [9]:

G

»»»»

¼

º

««««

¬

ª

�»¼

º«¬

ª

»»»»

¼

º

««««

¬

ª

���

����

»¼

º«¬

ª

m

clm

c

r

v

vI

lcl

vI

clcl

vmv

clcl

mv

cc

r

v

ff

f

y

xz

rff

xz

ffrr

x

x

ffrr

x

rf

y

)(

)(

22�

� (1)

Where yv� is the rate of change of lateral velocity and r� is

the yaw rate of the vehicle and yv , xv are the lateral and

longitudinal velocity, respectively. All other variables and

parameters are defined in the Appendix (Table I). Here, the

state variables are expressed in terms of position orientation

error since our objective is to develop a steering control

system for automatic lane keeping. Now, considering a

vehicle traveling with constant velocity on a road of a large

radius with curvature k and assuming a constant longitudinal

velocity, the rate of change of the desired orientation of the

vehicle is defined by:

kvr xdes � (2)

Where desr� is the desired yaw rate and k is the curvature of

the road. The desired path lateral acceleration of the vehicle

can be written as: 2xy kvv � (3)

Define e as the distance of the center of gravity of the vehicle

from the center line of the path and 1e as the yaw angle error

of the vehicle with respect to the path.

We have:

)( desxy rrvve ������ �� (4)

)( desxy rrvve �� �� (5)

desrre � 1 (6)

The state space model in tracking error variables is therefore

given by:

desrBBAxx ��21 �� G (7)

Where:

»»»»»»»

¼

º

«««««««

¬

ª

��

��

����

xz

rrff

z

rrff

xz

ffrr

x

rrffrf

x

rf

vI

clcl

I

clcl

vI

clcl

mv

clcl

m

cc

mv

cc

A

)(1

1000

)(0

0010

22

»»»»»»

¼

º

««««««

¬

ª

m

cl

m

c

B

ff

f

0

0

1

des

xz

rrff

x

x

ffrr

r

vI

clcl

vmv

clcl

B �

»»»»»»»

¼

º

«««««««

¬

ª

��

��

)(

0

0

22

2

> @Teeeex 11��

The output vector of the system consists of measurements

from the two sets of sensors. Sensor failures are modeled as

additive signals to the sensor output which is given by:

> @ FfCxyyyT � 21 (8)

Where 1y and 2y are the measurements from the two

sensors which measure the lateral deviation of the vehicle

and f is the fault signal, which is a function of time and state

x and C is the output matrix defined as:

»¼

º«¬

ª

� »

¼

º«¬

ª

001

001

2

1

2

1

d

d

C

CC . Where d1 and d2 are

respectively the distances from the center of gravity of the

vehicle to the front and rear bumpers where the sensors are

located. > @TF 01 in the event of failure of sensor 1 and

> @TF 10 in the event of failure of sensor 2, respectively.

Here we consider that the lateral sensing system consists of

two sets of sensors which provide the information of the

lateral deviation.

III. FAULT TOLERANT CONTROL APPROACH

A. Observer based FDI technique:

The observability properties of the vehicle imply that we can

build two observers, each of which is being driven by a

single sensor output. In order to avoid a wrong estimate of

the state under sensor failures, we fuse the sensor output and

the estimated output from the other observer before they

enter the observer. Fusion blocks play the role of switches,

which select the healthy signal. The post-filters are designed

such that the transfer functions from fault signals to residuals have consistent behavior and facilitate fault identification.

Output fusion is a convex combination of the sensor output

and the estimated output from the observer. The fused output

fiy is defined by:

jiiiifi yyy Ö)1( OO �� (9)

Where jiyÖ is the estimate of the i

th output from the j

th

observer, and i, j =1, 2 and iO a weight function. The weights

�iO [0, 1] are adjusted using a weight adjustment algorithm.

When there is no fault, 0 iO and the fused output fiy is

identical to the sensor output iy . When faults occur, the

corresponding iO increases towards one. When iO =l the

sensor output is incorrect and therefore is not being taken

into account at all. The observers switch between two

configurations according to the relative size of the weights as

follows:

If 1O < 2O , then observers are defined by:

)ÖÖ()Ö(ÖÖ21111

1111111 xxCLyyLBxAx f ����� OG� (10)

)Ö(ÖÖ 2222122 yyLBxAx f ��� G� (11)

If 1O > 2O then:

)Ö(ÖÖ 1111111 yyLBxAx f ��� G� (12)

)ÖÖ()Ö(ÖÖ12222

2222122 xxCLyyLBxAx f ����� OG� (13)

Where L1 and L2 are the solutions of the characteristic

equation defined by det(sI-A+L1C1) and det(sI-A+L2C2),

respectively. Observers (10)-(13) are variations of the

Luenberger observer where the fused outputs replace the

sensor outputs.

1492

Define the error vector:

> @TyI yyyyyyyye222

122

211

111

ÖÖÖÖ ����

(14)

Residuals are generated by filtering ey through post-filters

)(sM i i.e.,

yiii eMr i=1, 2, 3, 4. (15)

)(sM i define the transfer functions from the faulty signals to

the residuals such that the residuals from the two observers

are comparable in magnitude. Note that r1 and r2 are related

to sensor 1 and r3 and r4 are related to sensor 2, respectively.

Fault identification is more elaborate. Notice that observers

(10) ± (13) are coupled, i.e. faults in either of the two sensors

affect all residuals. The problem of identifying the exact fault

sensor can be solved by properly designed post-filters.

Define the state space transfer function from fault f to ey by:

FBAsICsVi �� �1)()( (16)

Where F is a matrix which represents the sensor failure as

follows:

> @> @°̄

°®­

T

T

F

F

10

01

Consider that sensor 1 has failed and 1O < 2O , then from

equation (16) we have:

)(

)(1))1(()1()( 1

11

111111sd

snLCLAsICsV ������ �OO (17)

)(

)()1())1(()1()( 31

11

111213sd

snLCLAsICsV

OOO

� ����� �

(18)

Where ))(),((( 1 sdsn and ))(),(( 3 sdsn are the co-prime pairs

of the polynomials.

¸¸¹

·¨¨©

§»¼

º«¬

ª

�»¼

º«¬

ª �

1)1(

0

10det)(

11

11

C

ILAsIsn

O

¸¸¹

·¨¨©

§»¼

º«¬

ª

�»¼

º«¬

ª �

11(

0

10det)(

21

13

C

ILAsIsn

O

Where )(1 sn and )(3 sn are also independent of 1O . Now

factorize )()()( 111 snsnsn�� and )()()( 333 snsnsn

�� where

)(sni

� and )(sni

�, i= 1,3, are the factors of )(sni which

have their roots in the left half plane.

Choose )()(

)()(

3

11

sksn

snsM

�

�

and )()(

)()(

1

33

sksn

snsM

�

�

Where k(s) is a Hurwitz polynomial such that )(1 sM and

)(3 sM are proper and stable. The determinant of )(sni and

)(sk should always be different from zero in order to avoid

matrix singularity. Then we have:

)()()()()1( 33111 sVsMsVsM �O (19)

Therefore if we choose post-filters Mi¶V� VXFK� WKDW�

33111 VMVMa and 22442 VMVMa for some real numbers

a1>0 and a2<1, we can define the following identification

rules:

- If 1O < 2O , and a fault has been detected, 31 rr ! implies

that sensor 1 has failed while 31 rr � implies that sensor 2

has failed.

- Similarly, If 1O > 2O , and a fault has been detected,

42 rr ! implies that sensor 1 has failed while 42 rr �

implies that sensor 2 has failed. In order to accommodate

faults, we feed the controller with the fused outputs 1fy and

2fy rather than the sensor outputs 1y and 2y .

- If the fault occurs, the faulty sensor output is replaced by

the observer output.

The next step after residual generation is the analysis of the

residual signal for fault detection. The residual generator

takes the sensor measurements as inputs and generates

residuals which are small, ideally zero, when there is no

fault. However, when a fault occurs the residuals are

significantly large. Due to the effect of disturbances, model

uncertainties, and measurement noise, the residuals are

different from zero even when there are no faults. A robust

residual generator should alleviate these effects while

remaining sensitive to faults. A simple threshold logic for

analysis of the residual signal is defined by:

hi Ttr �)( for no fault conditions

hi Ttr !)( for fault conditions

Where Th is the predefined threshold level.

The fault identification is based on the results of comparing

the sizes of residuals with the thresholds which are either

fixed or adaptive. Setting low thresholds results in high false

positive rates (alarms are issued under no fault conditions).

On the other hand, setting high thresholds increases the false

negative rates (alarms are missed when faults occur). Clearly

the selection of the thresholds is closely related to robustness

and sensitivity of the residual generator. Different analysis

procedures are used depending on the techniques employed

to generate the residual signal. The most widely used

approaches to analyze the residual signal generated by

observers are threshold logic, and limit monitoring [10]. The

threshold level selection methods are generally problem

specific and are not useful for a general case. To avoid

improper fault detection, threshold level selection is often

done on the basis of the designer's experience and in

response to problem requirements.

B. FTC Strategy:

When fault is detected, the state variables are reconstructed

accordingly and the redesigned feedback controller is

defined by:

)(Ö)( txKt f� G (20)

Where Kf is the feedback controller gain when the lateral

control system enters into degraded mode, that is when the

information is lost in one of the lateral deviation sensors. A

for failure of sensor 1

for failure of sensor 2

1493

fault in any of the sensor results in a change in the output

measurements and the state variables are defined as follows:

desrBtBtAxtx ��21 )()()( �� G (21)

FftxCty � )()( (22)

Where A is the state matrix, B1 is the control matrix and C

is the output matrix, desr� is the desired yaw rate, f is the

additive fault signal and F represents the sensor fault. Now

an optimal estimator can easily be designed by solving the

relevant Riccati equation associated with the system given by

equations (38) and (39). Assuming the system is observable,

the state estimates xÖ are defined as follows:

)()()(Ö)()(Ö 1 tLytBtxCLAtx ��� G� (23)

Where L is the observer gain which is defined by 1

1� BCYL and Y is the positive semi-definite stabilizing

solution of the following algebraic Riccati equation:

0)()( 111 ����� TTBBYCYCAYYCBA (24)

In order to analyze the stability of the proposed observers,

define the estimation error E(t) as follows:

)(Ö)()( txtxtE � (25)

The error dynamics are stable if and only if all the

eigenvalues of the matrix

U= »¼

º«¬

ª

�

�

1

1

C

BLCA reside inside the unit circle. Note that the

matrix U can be written as

U = > @0110

CLBA»¼

º«¬

ª�»

¼

º«¬

ª (26)

According to equation (43), the eigenvalues of matrix

»¼

º«¬

ª

�

�

1

1

C

BLCA can be arbitrarily assigned provided that

> @¸¸¹

·¨¨©

§»¼

º«¬

ª0,

10C

BA is observable. Hence, stability of the

proposed observers is guaranteed by the proper choice of

observer gain L, which is selected in a way such that the

eigenvalues of the matrix »¼

º«¬

ª

�

�

1

1

C

BLCA reside inside the

unit circle making the system stable. Where Kf is defined by:

PABPBBRKTT

f1)( �� (27)

The objective cost function to be minimized by the control is

given by:

)()()(Ö)(Ö

0

tRttxQtxJTT GG�� ³

f

(28)

Where P satisfies the Riccati equation defined by :

QPABPBBRPBAPAAPTTTT ��� �1

1111 )( (29)

In which Q is a diagonal weighting matrix with an entry for

each state corresponding to the performance aspects

contributing to the cost function and R is the weighting value

corresponding to the control effort contributing to the cost

function. The proposed observer based fault tolerant

controller is illustrated in Fig. 2.

The proposed method uses two observers, each one is driven

by sensor outputs. The fault is detected first and then the

faulty sensor is identified. The state variables are then

constructed from the output of the healthy sensor and then

the controller is updated accordingly to provide proper

steering when faults occur.

Fig.2. Block diagram of the proposed fault tolerant control

approach

To validate the performance of the proposed approach, we

provide a series of computer experiments conducted for a

ground vehicle subject to a set of pattern faults in different

sensors.

IV. APPLICATION TO THE GROUND VEHICLE

To show the effectiveness of the proposed FTC approach,

simulation results were carried out using a D class sedan

vehicle model. Simulations are performed in the MATLAB

environment and run in CarSim [11] environment.

Fig. 3. CarSim [11]

A double lane change course is selected to illustrate the

tracking capability as well as the steering control of the

vehicle on a straight path. Figures 4 and 5 show the vehicle

following a double lane change maneuver with longitudinal

speed of 30 m/s (108 km/hr) and considering a road adhesion

of one. For comparison purposes, simulations were carried

out with and without FTC and considering that fault has

1494

occurred abruptly in the front sensor. Simulations were first

carried when 90% of the information from the sensor is lost

and the front sensor has failed at t=4 sec.

0 2 4 6 8 10 12-30

-20

-10

0

10

20

30

40

Time(sec)

Late

ral off

set(

cm

)

Lateral offset with FTC and without FTC at v=30 m/s and u=1

without FTC

with FTC

Fig. 4. Lateral offset with FTC and without FTC

0 2 4 6 8 10 12-4

-3

-2

-1

0

1

2

3

Time(sec)

Ste

ering a

ngle

(deg)

Steering angle with FTC and without FTC at v=30 m/s and u=1

with FTC

without FTC

Fig. 5. Steering angle with FTC and without FTC

Figures 6 and 7 show the response of the vehicle at

longitudinal speed of 40 m/s (144 km/hr) and considering a

road adhesion of 0.75 when 90% of the sensor information is

lost. In this case there is a reduction in the friction between

the road and the tire of the vehicle. Note that, due to the

increase in speed and decrease in road adhesion factor the

vehicle takes more time to follow the path.

0 2 4 6 8 10 12-30

-20

-10

0

10

20

30

40

TIme(sec)

Late

ral O

ffset(

cm

)

Lateral offset with FTC and without FTC at v=40 m/s and u=0.75

without FTC

with FTC

Fig. 6. Steering angle with FTC and without FTC

0 2 4 6 8 10 12-4

-3

-2

-1

0

1

2

3

Time(sec)

Ste

ering a

ngle

(deg)

Steering angle with FTC and without FTC at v=40 m/s and u=0.75

with FTC

without FTC

Fig.7. Steering angle with FTC and without FTC

Computer simulations were carried out next when 70% of

sensor information is lost. Figures 8 and 9 illustrate the

response of the vehicle with longitudinal speed of 30 m/s

(108 km/hr) and considering road adhesion of one. Since in

this case the percentage of sensor information loss is less

than the one in the previous case, we can observe less

disturbance in the presence of fault when compared to the

case with 90 % information loss.

0 2 4 6 8 10 12-20

-10

0

10

20

30

40

Time(sec)

Late

ral off

set(

cm

)

Lateral offset with FTC and without FTC at v=40m/s and u=1

without FTC

with FTC

Fig. 8. Lateral offset at 70% sensor information loss

0 2 4 6 8 10 12-4

-3

-2

-1

0

1

2

3

Time(sec)

Ste

ering a

ngle

(deg)

Steering angle with FTC and without FTC at v=40m/s and u=1

without FTC

with FTC

Fig. 9: Steering angle at 70% sensor information loss

Next, we simulate the vehicle model on a circular track of

500 ft in order to show the performance of the observer

based FTC algorithm. Fig. 10 and 11 show the response of

vehicle at longitudinal speed of 30 m/s (108 km/hr) and

considering road adhesion of one when 90% of the sensor

information is lost.

1495

0 2 4 6 8 10 12-5

-2.5

0

2.5

5

7.5

10

12.5

15

Time(sec)

late

ral off

set(

cm

)

Lateral offset on circular track with v=30m/s and u=1

without FTC

with FTC

Fig. 10. Lateral offset on a circular track

0 2 4 6 8 10 120

1

2

3

4

5

6

Time(sec)

Ste

ering a

ngle

(deg)

steering control on circular track with v=30 m/s and u=1

with FTC

without FTC

Fig. 11. Steering angle on a circular track

The information from the sensor is lost after two seconds.

The vehicle requires more time to follow the path which is

observed in figure 12. The steering control is stabilized after

4 seconds and is maintained which is shown in figure 15.

Next, the simulation was carried at longitudinal speed of 30

m/s (108 km/hr) and considering road adhesion as one with

70% sensor information loss.

0 2 4 6 8 10 12-2.5

0

2.5

5

7.5

10

12.5

15

Time(sec)

late

ral off

set(

cm

)

Lateral offset on circular track with v=30 m/s and u=1

without FTC

with FTC

Fig. 12. Lateral offset at 70% sensor information loss

0 2 4 6 8 10 120

1

2

3

4

5

0

1

2

3

Time(sec)

ste

ering a

ngle

(deg)

Steering angle on circular track with v=30 m/s and u=1

with FTC

without FTC

Fig. 13. Steering angle at 70% sensor information loss

The simulation results show the effectiveness of the

proposed FTC algorithm in improving the vehicle response

under different paths and using various driving scenarios

under several faulty conditions. Note that compared to the

case without FTC algorithm, the system is not able to

recover from the sensor fault and the controller loses its

steering capability shortly after the sensor fault happens.

This is more prominent when the vehicle is following a

circular path.

V. CONCLUSION

In this paper, we proposed an observer based fault tolerant

control approach for the steering control of a ground vehicle.

The prosposed method was implemented on a vehicle

following a required path in the event of sensor faults.

Observer based estimator schemes were implemented to

generate the residual signals corresponding to the measured

and estimated variables. The residual signals were processed

using a deterministic fixed threshold approach. The

simulation results show that the proposed FTC strategy is

able to maintain the vehicle stability and acceptable

performance under both faulty and fault free conditions. In

the proposed approach, the threshold as well as the gain were

selected by trial and error; we will be working on a systemic

way to select the gains, switching threshold and other design

parameters by a more robust mathematical algorithm.

APPENDIX

TABLE I. VEHICLE PARAMETERS

m Mass of the vehicle , 1573 kg

rf cc , Cornering stiffness of front/rear wheels

2*60000 N/rad

fl Distance between front wheels and center of

gravity, 1.137 m

rl Distance between rear wheels and center of

gravity, 1.530 m

zI Yaw moment of inertia, 2753 kg m2

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1496