Model-based Sensor Fault Detection and Isolation for X-By-Wire Vehicles Using a Fuzzy Logic System with Fixed Membership Functions Pierre-François Quet, Member, IEEE and Mutasim Salman

Abstract— In this paper, a supervisory sensor diagnostics system for x-by-wire vehicles is described. The x-by-wire vehicle has brake-by-wire as well as front and rear steer-bywire subsystems. A detailed discussion of a model-based diagnostics approach that is used for detecting, as well as determining the location and type of sensor faults is given. The sensors under study are vehicle front and rear road wheel angles, lateral acceleration, yaw rate and four wheel speeds. A multi-model-based methodology is utilized for estimation of yaw rate, lateral acceleration and road wheel angle signals. Three estimates of the vehicle system behavior are generated and compared to the actual measurements. The differences between the measurements and the estimates are used to generate four residuals. Fuzzy membership functions for each of the residuals are defined and the degree-of-membership of the residuals is continuously evaluated. Fuzzy rules are developed and used to detect, isolate and determine the fault type. Specific patterns are used to detect and identify the location and type of sensors faults. The fuzzy rules implement the fault-symptoms relationships. The output of the fuzzy reasoning system is a crisp number, according to Sugeno fuzzy system, that can be interpreted as the probability of the occurrence of a specific fault.

I. INTRODUCTION

T

oday, when public interest in vehicle safety is rapidly

growing, active safety technology is becoming increasingly important. With vehicle dynamics continually being improved, a vehicle that anyone can drive safely will be forthcoming. The introduction of technologies such as chassis-by-wire in vehicles requires a good diagnostics system. The diagnostic system should be able to detect and isolate a fault in a timely fashion for safety reasons. Moreover, the diagnostics system should have minimum false alarms. There are several methods for vehicle diagnostics that depends on physical redundancy, functional (analytical) redundancy or both. Physical redundancy implies multiple-independent hardware. Physical redundancy with triple, or quadruple replication is very costly. Analytical redundancy implies the use of mathematical relations to obtain redundant Manuscript received September 11, 2006. Pierre-François Quet is with Nuvera Fuel Cells, 20 Acorn Park, Cambridge MA 02140, USA (phone: 586-883-3286; e-mail: [email protected]). Mutasim Salman is with General Motors Corp, R&D Center, Warren MI 48088, USA (e-mail: [email protected]).

measurements. For details, the reader is refereed to [1-5]. As part of any control-loop, sensors provide the controllers with necessary information about the actual state of the system and the vehicle dynamics. A successful vehicle control strategy strongly depends on the performance of the sensors. Therefore, an on-line sensor state-of-health monitoring and early fault detection (fault prognosis) system is of utmost importance. In this report, analytical redundancies are used for diagnosis of sensor faults in a x-by-wire vehicle with supervisory front and rear steer-by-wire. A multi-modelbased methodology is utilized for estimation of yaw rate, lateral acceleration and road wheel angle signals. Fuzzy membership functions for each of the residuals are defined and the degree-of-membership of the residuals is continuously evaluated. In section 2, the diagnosis approach is described. In the section, estimates of vehicle yaw rate, lateral acceleration, and road wheel angle combinations are developed. Four residuals are generated and the residual membership functions are defined. A fuzzy logic system is developed to detect, isolate and determine the fault type. In section 3, simulations are conducted to evaluate the performance of the fuzzy based diagnostic system. Finally, Concluding remarks, References and Nomenclature are given. II. DIAGNOSTICS APPROACH A. Developing Estimates In the approach adopted in this paper, linear models are used to generate estimates of vehicle states. Three model combinations are used to estimate yaw rate, lateral acceleration and road wheel angle. Table 1 shows the three equations that are used to find the estimates. The actual measurements of the other signals are also used in the estimation. To reduce numerical computation and threshold calibration, only signals from the main sensors are used as the actual measurement in the equations. In the followings, a more detailed description of the vehicle behavior estimates will be given. 1) Yaw rate estimate based on wheels speed measurements A yaw rate estimate rˆ can be derived using wheels speed measurements v −v (1) rˆ = RR RL 2t

where t is half of the vehicle track and rear right and rear left wheel speed measurements (ν RR ,ν RL ) can be obtained from the braking system, or from the motors rotational speed. To have a reasonable good estimate of the yaw rate, the wheel slips of the wheels should be small and close to each other and wheel factors that adjust for changes in the wheel radii are incorporated. 2) Lateral Acceleration Estimate Based on Yaw Rate The kinematics equation that relates the lateral acceleration

a y , yaw rate r , vehicle speed u and vehicle

lateral speed rate

(2)

At steady state, the v& is small, and the lateral acceleration estimate can be approximated by having yaw rate measurement and vehicle speed estimate, as (3) aˆ y = ru. 3) Estimation of Road Wheel Angles Combinations The front road wheel angle δ f , rear right road wheel

δ rr

and rear left road wheel angle

δ rl

can be

indirectly determined either from the rack position or motor angle position sensors. We can form the following estimate of road wheel angles combination based on lateral acceleration and yaw rate measurements

δf −

δ rr + δ rl 2

=

l r + ka y u

(4)

where l = a + b with a and b being the distance from the center of gravity of the vehicle to the front and rear axle respectively, and k is the vehicle understeer coefficient. B. Residual Generation In this approach, four residuals are generated. The first three residuals are based on the three estimation equations (in table 1). The first three residuals are the difference between the estimated values (obtained from the models) and the actual measurements as shown in Figure 1. The fourth residual is based on combined estimation models and is given as follows:     v + v FL v + v FL R = −  v RR − FR > Th1  − 0.5 v RL − FR > Th1  2 2       l   −  δ rr −  δ f − r − ka y  > Th2  ⋅ [δ rr − δ rl > Th3 ] u       l   − 0.5 δ rl −  δ f − r − ka y  > Th2  ⋅ [δ rr − δ rl > Th3 ] u    

[

Th3 = 0.75 deg ,

Th2 = 1.2 deg ,

]

+ 0.5 Ra y > Th4 . [ Rr ≤ Th5 ]  v + v FL v + v FL    + [ Rr > Th5 ]. NOR  v RR − FR > Th1 ,  v RL − FR > Th1   2 2    

Th4 = 0.18 g

and

Th5 = 3.5 deg/ s . The diagnostic residuals are shown in Table 2. C. Residual Classification Using Fuzzy Logic According to the fuzzy logic system, membership functions “0”, “+”, and “-” are defined for each of the residuals Ra , (Rr ), and Rδ −δ , as well as “0.5”, “1”, “0”,

(

( )

v& can be given as a y = v& + ru.

angle

with the convention that [a > b] has a value 1 if a > b and 0 otherwise, and NOR is the joint-denial operator. For the purpose of the initial simulation, the following values are chosen for the parameters in R: Th1 = 6 km / h ,

y

f

r

)

“-1”, and “-0.5” for the residual (R ) . Figure 2 shows the membership functions for each of the residuals. The degrees of membership values of each of the residuals are then determined. A set of fuzzy rules define the fuzzy implementation of the fault-symptoms relationships. Whenever a sensor fails and of a certain type, a unique pattern of residuals occurs. The specific pattern can be used to determine the source, location and type of fault. For example, “ a y

+ ∆a y ” means that the lateral acceleration

sensor is faulty and exhibits a positive offset. “d” stands for “don’t care”: it does not matter what the value is. Table 3 shows the residual pattern for each sensor fault and type. The outputs of the Sugeno fuzzy system are crisp numbers that can be interpreted as the probability that the corresponding fault has occurred. The diagnostic system thus concludes as faulty the sensor corresponding to the rule having the fuzzy system maximum output and with high probability (see table 4). As an example, the lateral acceleration sensor fault a y

+ ∆a y would be characterized

by the fulfillment of the rule: If Ra = "+" and (Rr = "0") and

(

y

)

(R

δ f −δ r

)

= "d " and

(R = "0.5") then ((a y + ∆a y ) = 1) . It is important to notice that this system assumes a single sensor fault at a time. The support of multiple sensor failures could be envisioned as an extension of the scheme presented here, where additional residuals need to be employed that would form unique patterns for each supported set of sensors faults. Also, the present detection scheme assumes that the faults can be characterized by offsets from the actual variable values. To reduce the sensitivity of the diagnostic system to intermittent noise and disturbances, and hence reduce false alarms, we added a timer of 50ms. The sensor will not be considered faulty unless the corresponding fuzzy diagnostic output indicates a fault for time duration of at least 50ms. Furthermore, a latch system keeps the fault flag on once a fault has been detected. Finally, in this paper, the diagnostic system is turned off for vehicle lateral accelerations greater than 0.5g. This will allow the tuning of the membership

functions so as to have good fault detection at low lateral acceleration maneuvers, while avoiding false alarm during maneuvers with large lateral accelerations. Putting such a constraint is not very limiting, since typically a vehicle does not have its lateral acceleration greater than 0.5g for long periods of time. However, further enhancement of the diagnosis system to operate during severe vehicle maneuvers would be desirable, and could be achieved for example by using adaptive membership functions that would depend on the vehicle operating conditions. III. SIMULATIONS RESULTS To evaluate the performance of the previously described the diagnostic system, simulations were conducted using VehSim with a typical vehicle model and parameters. The following vehicle parameters are used: the effective cornering stiffness for the front and rear 2 C f = C r = 86000 N / rad , I z = 2870 kg / m , M = 2269 kg ,

a = 1.4625 m and b = 1.5275 m . In order to get closer to a realistic system, a normally distributed random noise is added to each of the simulated sensor measurements (see Table 5). First order low pass filters with bandwidth 20Hz are used to filter the noisy measurements. This is performed in the signal conditioning part of the diagnostic system. A sampling period of 0.01s is appropriately chosen. A. Sensor fault detection Several simulation maneuvers were used to test the modelbased diagnostics system for the lateral acceleration, yaw rate, road wheel angles, and wheel speed sensors. The tests included straight line, lane change and double lane change maneuvers, were done at different vehicle speeds and on various road surfaces (dry to icy). Different levels of biases were injected at the sensor output to represent a fault. Table 6 describes the vehicle maneuver and type of sensor fault used in the simulations presented here. Figures 3 and 4 show the simulation results of the diagnostics system for the different types of faults. In each of the figures, vehicle yaw rate, lateral acceleration, and the x-y coordinates of its path are depicted. Fault is injected at a certain time (as shown in the Figures and in Table 6) and the fault flag with the corresponding residual values are plotted. It can be seen that for every simulation scenario the fault is correctly detected by the diagnostic system within 50ms and the corresponding fault flag is turned on. B. Sensitivity analysis 1) Robustness to sensor bias The membership functions of the fuzzy logic system shown in Figure 2 and the threshold values Th1 , Th2 , Th3 , Th4 , Th5 have been tuned to support some sensor bias. Figure 5 shows the robustness of the diagnostic system to a lateral acceleration sensor bias of 0.1g: while the vehicle is performing a lane change maneuver at 80 km/h,

the diagnostic system does not detect a false positive in the lateral acceleration sensor state-of-health. The amount of sensor bias supported with the current fuzzy logic membership functions tuning is shown in Table 7. Robustness to larger sensor biases can easily be obtained by widening the “0” membership functions and increasing some of the thresholds

Thi ; however this would worsen the faults

detectability (only larger faults will be detected). 2) Robustness to variations in vehicle understeer coefficient The front effective cornering stiffness used in the diagnostic system derivation is decreased to C f = 85274 N / rad , which for the vehicle considered for these simulations corresponds to an increase of 20% in the understeer coefficient. Figure 6 shows that the diagnostic system does not make a false detection when the vehicle is performing a severe lane change maneuver ( a y = 0.5 g ) at 70 km/h. IV. CONCLUDING REMARKS Based on vehicle simulation results, we can conclude that the model-based diagnostics methodology proposed in this paper, with appropriate fuzzy membership functions definitions, leads to early detection and isolation of faults with minimum false alarm. The use of the fuzzy logic reasoning system that gives an indication of fault probability adds flexibility to the supervisory diagnostics system. Fixed membership functions are used in this study, an approach that uses adaptive membership function that depend on the vehicle operating conditions is currently being investigated. V. NOMENCLATURE

a = distance from the center of gravity to the front axle b = distance from the center of gravity to the rear axle l = a+b

δf

= front road wheel angle

δ r = rear road wheel angle M = mass of the vehicle v = lateral velocity r = yaw rate a y = lateral acceleration u = longitudinal velocity C f = effective front tire cornering stiffness

C r = effective rear tire cornering stiffness I z = yaw moment of inertia k = is the vehicle understeer coefficient t = half of the vehicle track

VI. REFERENCES [1]

M. Börner, R. Isermann, “Supervision, fault detection, and sensor fault tolerance of passenger cars,” Proc. of Safeprocess 2003, pp. 327334, 2003. S. Amberkar and B. Murray, “Diagnostic strategies for advanced automotive systems,” SAE 2002-21-0024. H. Hu, M. Salman, M. Rizzo, J. Dickinson, D. Carlson, E. Leaphart, L. traccht, “Active Brake Control Diagnostics,” US 5,707,117, Jan. 13, 1998. E. Ding, and R. Herbst, “Sensor System with Monitoring Device,” US 6,625,527, Sept. 2003. G. Rizzoni, A. Soliman, P. Pisu, S. Amberkar and B. Murray “Modelbased fault detection and isolation system and method,” US 5,766,230B1, July 2004.

[2] [3]

[4] [5]

Equation

Yaw rate

v −v rˆ = RR RL 2t

Lateral acceleration

aˆ y = ru

Road wheel angles difference

δ

f

−

0.8

0.8 0 +

0.6

0.4

0.2

0.2

-0.2

0 Ray in g

rr

+ δ 2

rl

Inputs

=

l r + ka u

y

1.2

1.2

1

1

0.8

0.8

0.4

0.2

0.2

-2

-1

0 1 R -δ in deg δf r

Estimate

+

Equation ∧

r−r

Rδ f −δ r

R

∧

δ + δ rl   (δ + δ rl )   )  −  δ f − ( rr  δ f − ( rr  2 2         v + v FL v + v FL −  v RR − FR > Th1 − 0.5 v RL − FR > Th1 2 2       l   −  δ rr −  δ f − r − ka y  > Th 2 ⋅ [δ rr − δ rl > Th3] u       l   − 0.5 δ rl −  δ f − r − ka y  > Th2 ⋅ [δ rr − δ rl > Th3] u    

[

5

10

-1 -0.5 0 0.5 1

0 -1.5

-1

-0.5

0 R

0.5

1

1.5

Residual

-

TABLE II DIAGNOSTICS RESIDUALS

Rr

3

0 Rr deg/s

Fig 2. Fuzzy logic system membership functions

Faults

ay − ay

2

-5

0.6

0.4

Fig. 1. Construction of residuals

Ray

0 -10

0.4

0 +

0 +

Sensor signal

Sensor

Analytical model

Residual

0.2

0.6

δ

0.6

0.4

0 -3

Plant

1

0 -0.4

TABLE I MODELS FOR FAULT DETECTION AND ISOLATION Estimate

1

]

+ 0.5 Ra y > Th 4 . [ Rr ≤ Th5]  v + v FL   v + v FL  + [ Rr > Th5]. NOR  v RR − FR > Th1,  v RL − FR > Th1  2 2    

TABLE III EFFECT OF FAULTS ON RESIDUALS Residuals

Ra y

Rr

Rδ f −δ r

R

a y + ∆a y

+

0

d

0.5

a y − ∆a y r + ∆r r − ∆r δ f + ∆δ f

-

0

d

0.5

d d

+ -

d d

1 1

0

0

+

0

δ f − ∆δ f δ rr + ∆δ rr δ rr − ∆δ rr δ rl + ∆δ rl δ rl − ∆δ rl v RR + ∆v RR v RR − ∆v RR v RL + ∆v RL v RL − ∆v RL

0

0

-

0

0

0

-

-1

0

0

+

-1

0

0

-

-0.5

0

0

+

-0.5

0

-

0

-1

0

+

0

-1

0

+

0

-0.5

0

-

0

-0.5

TABLE IV FUZZY LOGIC RULES

If If If If If If If If If If

ay ay ay ay ay ay ay ay ay ay ay

and and and and and and and and and and and and and

(Rr (Rr (Rr (Rr ( Rr ( Rr ( Rr ( Rr ( Rr ( Rr (Rr (Rr (Rr (Rr

= "0")

and

= "0")

and

(R (R (R (R (R (R (R (R (R (R (R (R (R (R

δ f −δ r δ f −δ r

= "+") and

δ f −δ r

= "−") and = "0")

and

= "0")

and

= "0")

and

= "0")

and

= "0")

and

= "0")

and

δ f −δ r δ f −δ r δ f −δ r δ f −δ r

δ f −δ r δ f −δ r

δ f −δ r

= "−") and

δ f −δ r

= "+") and

δ f −δ r

= "+") and

δ f −δ r

= "−") and

δ f −δ r

) = "d ") = "d ") = "d ") = "+") = "−") = "−") = "+") = "−") = "+") = "0") = "0") = "0") = "0") = "d "

(R = "0.5") (R = "0.5") (R = "1") (R = "1") (R = "0") (R = "0") (R = "−1") (R = "−1") (R = "−0.5") (R = "−0.5") (R = "−1") (R = "−1") (R = "−0.5") (R = "−0.5")

and and and and and and and and and and and and and and

then then then then then then then then then then then then then then

((a ((a

y

+ ∆a y ) = 1)

y

− ∆a y ) = 1)

((r + ∆r ) = 1) ((r − ∆r ) = 1)

((δ ((δ

f

+ ∆δ f ) = 1)

f

− ∆δ f ) = 1)

((δ rr + ∆δ rr ) = 1) ((δ rr − ∆δ rr ) = 1) ((δ rl + ∆δ rl ) = 1) ((δ rl − ∆δ rl ) = 1) ((v RR + ∆v RR ) = 1) ((v RR − ∆v RR ) = 1) ((v RL + ∆vRL ) = 1) ((v RL − ∆v RL ) = 1)

Lateral acceleration in g

If

ay

ay

and

Lateral acceleration in g 0.3

80 75 70 65

Sensor measurement (includes f ault) Actual vehicle Vrl.

0

2

4 6 Time in second Flag for Vrl-∆ Vrl fault

0.2 0.1 0 -0.1 -0.2

8

0

1.5

1

0.5

0

0

2

4 Time in second Ray in g

6

δ f , δ rr , δ rl v FR , v FL , v RR , v RL

-0.2

0.001 deg

0 -0.5

4 6 Time in second Flag for ay-∆ay fault

0

2

4 Time in second R

6

8

0

2

4 Time in second

6

8

1.5 1 0.5

0

0 -0.5 -1

0

2

4 Time in second

6

-1.5

8

Fig. 4. Lane change on icy road, 80km/h, µ=0.2 0

2

4 Time in second Vehicle path

6

8

Global Y coordinates in m

4

0.5

0

2

4 Time in second Ray in g

6

3

1

0.1

3

0

2

-0.1 -0.2

TABLE VI DESCRIPTION OF MANEUVER AND TYPE OF SENSOR FAULT Time Figures Maneuver Type of fault of fault

2

0

8

Rr in deg/s

Flag for ay-∆ay fault Ray in g

-80

8

-10 -20

-0.3

0

50 100 Global X coordinates in m Rr in deg/s

Fig. 3

Lane change with large lat. acc. (0.5g), 70 km/h, dry road µ=1

3.75 s

∆a y = −0.24 g

Fig. 4

Lane change on icy road (µ=0.2), 80km/h

4.5 s

∆v RL = −7 km / h

150

1 0 -1 -2

0

2

4 Time in second Rdfdr in deg

6

-3

8

1

0

2

4 Time in second R

6

8

1.5

TABLE VII MAXIMUM SENSOR BIAS SUPPORTED

1 0.5 0.5 0

R

Rdfdr in deg

50 100 150 Global X coordinates in m Rr in deg/s

0

8

1

-0.4

6

-0.2

10

1.5

0

4 Time in second Rdfdr in deg

R

Rdfdr in deg Yaw rate in deg/s

Lateral acceleration in g

0.5

2

2

0.2

-0.4

0

-40

0.4

Yaw rate in deg/s

1

-20

20

Sensor measurement (includes fault) Actual vehicle lateral acc.

0

0

0.6

0.001 km2/s2

2

-60

2

Lateral acceleration in g

-1

0 -0.1

0.0005 g2

0 -0.5

-0.5 -1 -1

8

0 Rr in deg/s

0.2 deg2

r ay

1

Ray in g

Sensor

6

20

0.1

SENSOR NOISE Variance of sensor noise

4 Time in second Vehicle path

3

0

8

0.2

TABLE V

2

4 Global Y coordinates in m

If

) = "−") = "d ") = "d ") = "0") = "0") = "0") = "0") = "0") = "0") = "0") = "0") = "0") = "0") = "+"

ay

Rear left wheel speed in km/h

If

(R (R (R (R (R (R (R (R (R (R (R (R (R (R

Flag for Vrl-∆ Vrl fault

If

Rear left wheel speed in km/h 85

0

2

4 Time in second

6

8

-1.5

0

2

4 Time in second

6

Fig. 3. Lane change with large lat. acc. (0.5g), 70 km/h, µ=1

8

Sensors

Maximum sensor bias supported

Yaw rate

1.5 deg/s

Lateral acceleration

0.1 g

Road wheel angles

0.75 deg

Wheel speeds

0.2 km/h

Lateral acceleration in g

Yaw rate in deg/s

2

4 6 Time in second Flag for ay-∆ay fault

8

4 Time in second Vehicle path

6

0.5 0

0

2

4 Time in second Ray in g

6

0.2

2 1

0

50 100 150 Global X coordinates in m Rr in deg/s

2

Ray in g

Rr in deg/s

Ray in g

4 Time in second Rdfdr in deg

6

8

0 -1

-0.1

1

0

2

4 Time in second R

6

8

0

-1

-0.2

-1.5

0

2

4 Time in second

6

8

8

2

4 Time in second Ray in g

6

3 2

0

8

Actual path Target path

1

0

50 100 150 Global X coordinates in m Rr in deg/s

200

1 0.5 0 -0.5

0

2

4 Time in second Rdfdr in deg

6

-1

8

0

2

4 Time in second R

6

8

0

2

4 Time in second

6

8

1.5 1

-0.5

-0.1

2 4 6 Time in second Target path and actual paths

0.5 Rdfdr in deg

0

0

1

0.5

0.1

0

0 -0.05

1.5

-0.4

4

0.05

-3

0.2

0 -0.2

8

0.5

-2

0.3

R

Rdfdr in deg

2

6

0.1

1

0.05

0

4 Time in second No fault flag

0.15

0.15

0

2

1

0

200

3

0.1

0

0.2

1.5

3

0

8

-10 -20

8

No fault flag

1

-0.5

2

4 Global Y coordinates in m

Flag for ay-∆ay fault

1.5

0

Lateral acceleration in g

-5 -10

0

0.4

Global Y coordinates in m

0

0

10

Rr in deg/s

0

5

0.6

0.5 0

R

0.2

Lateral acceleration in g

20 Yaw rate in deg/s

0.4

-0.2

Yaw rate in deg/s

10 Sensor measurement (includes fault) Actual vehicle lateral acc.

Yaw rate in deg/s

Lateral acceleration in g

0.6

0 -0.5

-0.5 -1

0

2

4 Time in second

6

Fig. 5. Robustness to a lateral acceleration sensor bias of 0.1g

8

-1

0

2

4 Time in second

6

8

-1.5

Fig. 6. Robustness to variations in vehicle understeer coefficient