- Email: [email protected]
Fault Detection for an Automotive MR Damper
Proceedings of the 14th IFAC Symposium on Information Control Problems in Manufacturing Bucharest, Romania, May 23-25, 2012
Fault Detection for an Automotive MR Damper Jorge Lozoya-Santos ∗ Juan C. Tud´ on-Mart´ınez ∗ Ruben Morales-Menendez ∗ Ricardo Ram´ırez-Mendoza ∗ Arturo Molina Gutierrez ∗ ∗
Tecnol´ ogico de Monterrey, Campus Monterrey, Av. Garza Sada 2501, 64849, Monterrey N.L., M´exico e-mail: {jorge.lozoya, jc.tudon.phd.mty, rmm, ricardo.ramirez, armolina }@itesm.mx
Abstract: Driving with faulty dampers can throw the car off balance sending it out of control. Common faults of Magneto Rheological (MR) dampers are loss of oil volume because oil leakage. A fault detection system for monitoring the damper condition is proposed. The proposal monitors the transmissibility of the semi-active suspension of a Quarter of Vehicle (QoV ) based on the frequency of the road. A frequency estimator for the road based on the deflection of the suspension is the key component of the fault detection system. In order to improve the efficiency of the detection system an observable domain switch was introduced. This switch identifies the most sensible transmissibility domain in order to improve the efficiency of the proposal. Intensive randomly generated tests with less than 36 % average in detection error using a semi-active MR damper validates the transmissibility as a good indicator of MR damper condition. Keywords: Fault detection, MR damper, Oil leakage, Shock absorber. 1. INTRODUCTION An early detection of abnormal events in automotive suspension systems can reduce damages in driving situations. In Venkatasubramanian et al. [2003], advanced Fault Detection and Isolation (FDI ) methods are classified into two major groups, those which do not assume any form of model information and those which use accurate dynamic process models. Basically, the FDI selection depends on the availability to get the right information through of a reliable model or process sensors. Faults in engineering processes belong to one of the following categories, Gertler [1998]: faults in sensor and actuator called soft faults or, process faults where some plant parameters change because of physical or mechanical problems. In Automotive Suspension Control Systems (ASCS ), the instrumentation is vulnerable to malfunctions. Shock absorbers could fail when it losses oil volume, generating a decrease in its damping factor. This loss is generated due to the damaged seals in the damper housing. The spring, in the suspension system, has little variations during vehicle life, Ferreira et al. [2009]. There are some research works in FDI systems for ASCS. In Kim and Lee [2011] a residual analysis based on parity equations is proposed for fault detection in vertical accelerometers of the sprung mass. In Wang and Song [2011], a FDI system based on a H∞ robust filter is proposed for sensor faults into a semi-active suspension system, the residuals are used to accommodate the malfunction by a fuzzy controller. Similarly, a FDI filter based on the Lin⋆ Thanks to CONACyT, Mexico for the support under the Postgraduate Cooperation Project 2007-2011.
978-3-902661-98-2/12/$20.00 © 2012 IFAC
ear Parameter-Varying framework is proposed in G´asp´ar et al. [2010]; sensor and actuator faults are diagnosed and accommodated in the controller to guarantee road holding and to keep the pitch and roll stability in a vehicle model. A methodology for controlling the damping ratio of semiactive dampers in a Quarter of Vehicle (QoV ) based on estimation of parameters is proposed in Fischer and Isermann [2004]; however, the computing time is prohibitive. In Vidal et al. [2010], a FDI system based on a referencemodel observer is proposed; stiffness and damping variations in a suspension are implemented as additive faults. A novel methodology that monitors the condition of the automotive dampers through a transmissibility function is introduced by Ferreira et al. [2009]. Even experiments validate results, more complex design for a MagnetoRheological (MR) damper is needed. An MR damper is a hydraulic device whose oil contains metallic micro size particles that change the rheological properties of the fluid when a magnetic field is applied; an electric current supplied through the damper coil is used to manipulate the magnetic field. The variation of the oil viscosity modifies the damping ratio. Thus, the damping coefficient not only depends on the excitation frequency, but it also depends on the electric current. This paper generalizes the work presented by Ferreira et al. [2009] by adding the electric current and the deflection velocity effects into the transmissibility function that detects malfunctions of the semi-active damper. The outline of this paper is as follows: in the next section, the model of a QoV model with an MR damper is described. Section 3 reviews the original method proposed
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Table 2. QoV parameters.
in Ferreira et al. [2009], while the Section 4 introduces an extended version. Results are presented in Section 5, and conclusions in Section 6. All variables are alphabetically described in Table 1.
Component of the Quarter of Vehicle Sprung mass (ms ) Unsprung mass (mus ) Spring stiffness (ks ) Tire stiffness (kt )
Table 1. Variables description
fr fn fˆ, fˆ(·)
z¨ ϵ f µ , fψ ζ, ζM R |||z|||∞ , |||z||| ˙ ∞ ρ
i-k 1500 1000
zs cp + c MR
ks Unsprung mass Road
m
us kt
z
us
zr
Fig. 1. QoV model with a semi-active MR damper.
0 .
.
z yield
−1000 −1500 −0.5 −0.4 −0.3−0.2 −0.1
0
0.1 0.2 0.3 0.4 0.5
Piston velocity(=)m/s
Fig. 2. Damping force vs the maximum deflection velocity with different damping coefficient.
A QoV model is used for testing the proposed fault detection system, Figure 1. The suspension has an MR damper between the sprung and unsprung masses. It is assumed that the wheel-road contact is ensured. The parameters of a QoV correspond to a 2008 CadillacT M , Table 2. The dynamic system is given by equations (1): ms
500
−500
2. QoV MODEL USING AN MR DAMPER
Sprung mass
. . F = cp z . ____ z 1 +Ic MR MR . |||z||| i-k
. . F 1 = cp z + I c z . ____ MR MR . |||z|||
8
z˙saturation
The MR damper model is an extension of the maximum deflection velocity model based on the variation of the damping coefficients cp and cM R according to the characteristic diagram of a semi-active damper and the applied electric current. Figure 2 shows how the damping coefficient changes with maximum velocity, narrow lines are the damping coefficient paths that depend on the speed interval, z. ˙
{
z˙cubic
(1)
A commercially MR damper ACDelco T M was included. This device varies its damping factor CD according to a supplied electric current, I. It is designed to perform a damping ratio variation with independence of the piston velocity, z. ˙ The stroke is 60 mm. The maximum absolute force is 2,500 N with a temperature range of (−40, 150)o C.
8
z˙yield
Units kg kg kN/m kN/m
ms z¨s = −ks (zs − zus ) − FM R mus z¨us = ks (zs − zus ) − kt (zus − zr ) + FM R
{
I kq , jq k 1 , k2 , k 3 , k4 ks , kt ms , mus Q3 ,Q4 r R TR , TˆR yM R zr zs , zus z¨s , z¨us z, zdef z, ˙ z˙def z˙f riction
Description MR damping coefficient, N · A/m Damping factor, CD due to I Passive damping coefficient N s/m Damping force, N MR damping force, N Force due to the change of oil viscosity caused by the electric current in N Frequency of the road surface, Hz Resonance frequency of the ms , Hz Estimated frequency, Hz Electric current, A Receding horizon indexes Receding horizon based on z˙ Stiffness coefficient: spring, wheel tire, N/m Sprung mass and unsprung mass, kg Quartile 75 % and 100 %. Frequency ratio, adim Sinusoidal amplitude, m Transmissibility and estimated TR , adim Damping coefficient depending on I, Ns/m Road profile, m Vertical position of the mass ms , mus , m Vertical acceleration of ms , mus , m/s2 Piston deflection, m Piston velocity, m/s Threshold for piston velocity, m/s with coulomb friction Threshold for piston deflection velocity, m/s before blow-off valves yield Threshold for piston deflection velocity, m/s with a cubic softening nonlinearity Threshold for piston deflection velocity, m/s with a saturation nonlinearity Piston acceleration, m/s2 Constant that bounds the ρ-value Span of algorithm detection Damping ratio, MR damping ratio Absolute maximum deflection, velocity Nonlinear term in MR damper model
Force (=)N
Variable cM R CD , CD (I) cp Fdamping FM R fI
Value 522 50 83 230
The semi-active MR damper model is: { yM R z˙ + cp zρ ˙ z˙ < z˙yield FM R = yM R zρ ˙ + cp z˙ z˙ > z˙yield
(2a)
yM R = I · cM R (2b) 1 ρ= (2c) i |||z||| ˙ ∞ i−kq + ϵ k |z| ˙ < z˙f riction 1 k2 z˙f riction < |z| ˙ < z˙yield kq = (2d) k z ˙ < | z| ˙ < z˙cubic 3 yield k4 z˙cubic < |z| ˙ < z˙saturation where ρ describes the damping force behavior due to the friction, the activation of the symmetrical blow-off valves, the softening of the damping force, and the saturation, Worden and Tomlinson [2001]. The parameters of model equation (2a) were learned with experimental data from Figure 3. Table 3 shows the
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Electric Velocity Force (N) current (A) (m/s)
identified parameters. The domain is divided in four zones where the limits are: z˙f riction , z˙yield , z˙cubic , and z˙saturation . The parameters cp and cM R changes based on the piston velocity at zyield . 0.4 0.2 0 −0.2 −0.4 2.5 2 1.5 1 0.5 1000 0
−1000 1
2
3
4. FAULT DETECTION SYSTEM The transmissibility can be monitored by online computing the damping ratio, ζ in the shock absorber and the excitation frequency, fr . The MR damper model and an estimation of the frequency, fˆdef through a deflection sensor, zdef are proposed for computing the transmissibility TR . A change in the semi-active and passive damping coefficients could represent a fault in the MR damper. The model relates the transmissibility TR with an electric current, I. A threshold for each electric current and frequency will support a fault detection system. 4.1 Heuristic frequency estimation
4
Time (=) secs
By assuming a harmonic motion of the damper piston, Worden and Tomlinson [2001], two variables of the suspension can be described by:
Fig. 3. Experimental test for model learning. Table 3. Identified parameters for the piecewise nonlinear MR damper model. k1 k2 k3 k4 cp cM R
z˙ < zyield 154 2 214 3004
z˙ > zyield 3 16 598 485
units Ns/m N/A
Domain z˙f riction = 0.02 m/s z˙yield = 0.1 m/s z˙cubic = 0.5 m/s z˙saturation = 1 m/s -
z ∼ R · sin(2π fr · t)
(6)
z˙ ∼ ω · R · cos(2π fr · t)
(7)
The signals z and z˙ are unknown and there is not possible to obtain a forecast. Using an index of the amplitude such as the Root Mean Square value in a moving window; or the infinite norm of the deflection and velocity of the piston in the last k samples, the envelopes of equations (6, 7) are: i
|z| = R ∼ |||z|||∞ i−jq
3. MONITORING THE DAMPER CONDITION
i |z| ˙ ∼ |||z||| ˙ ∞ i−jq
Based on QoV model, the transmissibility TR function given by output to input magnitude ratio between the sprung and the unsprung mass accelerations (¨ zs and z¨us ) is, Gillespie [1992]: √ z¨s 1 + (2ζ r)2 = TR = (3) z¨us (1 + r2 )2 + (2ζ r)2 This function depends on frequency, where the damping ratio ζ is: 1 CD ζ= √ (4) 2 ms ks and r is the ratio between the excitation frequency fr and the natural frequency fn of the system defined by fr with fn = r= fn
√ ks /ms 2
(5)
In conventional vehicle passive suspension systems, the spring constant ks has little variation during vehicle life. Vehicle mass ms changes with vehicle load, this effect is more pronounced in the rear axle, as the front axle load is mainly determined by the powertrain mass coefficient. The damper condition is related with the damping factor CD , so changes due to aging, wear or malfunction of the damper may be detectable by computing transmissibility function. The transmissibility is directly affected by the excitation frequency. A system that measures both, z¨s and z¨us accelerations and computes the transmissibility, TR , could monitor the damper condition, and detect faults.
(8)
∼ 2π fr ·
i |||z|||∞ i−jq
(9)
Solving for w using (8, 9), the instantaneous estimation of the frequency is: i
˙ ∞ i−jq 1 |||z||| fˆ = 2π |||z|||∞ ii−j
(10)
q
4.2 Estimation of Transmissibility In a shock absorber the damping factor, CD , and the damping ratio, ζ, can be obtained as a function of the damping force and excitation velocity: Fdamping (11) z˙ Combining equations (2a, 4, 11) a formula for the MR damping ratio is: CD =
1 √ I · cM R + cp ρ 2 ks ms ζM R (I, z˙def ) =
1 √ I · cM R ρ + cp 2 ks ms
z˙ < z˙yield (12) z˙ > z˙yield
The MR damping ratio ζM R depends on the electric current I, and the piston velocity z˙def . Using the estimated frequency of excitation fˆr in eqn. (5), the formula for r is:
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1
i
|||z||| ˙ ∞ i−jq
r=√ ks /ms |||z|||∞ ii−jq
(13)
INCOM 2012, May 23-25, 2012 Bucharest, Romania
i
ρω (zdef , z˙def ) =
|||z||| ˙ ∞ i−jq
(14)
i
|||z|||∞ i−jq
and combining equations (3, 12, 13), the estimated TˆR is: v ( )2 u u 2ρω [ζM R ] u 1+ √ u ks /ms TˆR (I, zdef , z˙def ) = u )2 (15) ( u[ ] 2 t ρ2ω ms 2ρω [ζM R ] 1 − ks + √ ks /ms
Following the recommendation of Boggs et al. [2006], a road sequence based in a bounce sine sweep pattern with decremental displacement amplitude according to the evaluated frequency was implemented for different electric current. Figure 4 summarizes these results for three electric current, and for several frequencies of the road sequences zr . The transmissibility TR at the natural frequency (i.e ∼ 2 Hz) exhibits the greatest values.
According to the standard ISO 8606:1995, the classified roads in types A to H corresponds to roads in good conditions (i. e. city runways). The road type A corresponds to the best pavement condition. A loss of 25 % of the oil MR damper volume in a vehicle running in a road type B was considered the fault to be detected. Figure 6 shows a histogram of the average time at different estimated frequency fˆdef when a vehicle runs in a type B road. This average was computed from 30 randomly generated simulations. Continuous line corresponds to a suspension with a MR damper free of fault (0 % lost oil volume) and dashed line is for a faulty MR damper (25 % lost oil volume). This frequency histogram confirms the application domain.
Transmissibility
z¨s / z¨ us
3.5 0A
3 2.5
Vehicle with a faulty MR damper (25 % lost oil volume)
8.12
Percentage of time
Defining:
1.25 A
6.51 Vehicle with a MR damper (0 % lost oil volume)
4.88
3.25
2 2.5 A
1.5
1.63
1 0.5 0 0.5
1 1
1.5
2
2.5
3
3.5
4
4.5
1.5
2
2.5
3
3.5
4
4.5
^ Estimated frequency f def
5
Estimated frequency from z r
Fig. 4. Transmissibility based on registered accelerations (¨ zs , z¨us ) for different electric currents and frequencies.
Fig. 6. Histogram of the average time at different frequency fˆdef that a vehicle runs in a type B road. 4.4 Threshold definition
Estimated frequency from zr [Hz]
The frequency is estimated from the suspension deflection, zdef , because there is not a frequency road sensor. Figure 5 compares the real and the estimated frequencies of the road zr . The suspension is a nonlinear system that filters some frequencies; therefore, the estimated frequency of the road zr could exhibit errors or important differences. 20
20
10
10
0
0
10 20 Frequency z r [Hz]
0
0
Based on the bandwidth of the frequency given by Figure 6, a transmissibility TR plot for a suspension with a damper free of faults was computed. This function will be used for threshold definition. Several statistics were considered for each discrete interval (0.05 Hz ) of frequency. The evaluated thresholds were TRth = µ+σ, TRth = µ+2σ, TRth = µ + 3σ, TRth = Q3 , and TRth = Q4 . Figure 7 (continuous line) shows the selected threshold, TRth = Q3 . The transmissibility TR with faulty damper, a loss of 25 % of the oil volume, was computed with the same statistics, Figure 7 (dashed line). Based on these lines, if a suspension system has a transmissibility TR > TRth , it is almost sure that the MR damper is faulty.
10 20 Estimated frequency from z def [Hz]
Fig. 5. Road frequency fˆr (left) and estimated road frequency fˆdef (right). 4.3 Fault implementation and driving conditions The considered faulty condition is a loss of oil in the MR damper because an oil leakage. This damper condition is function of the damping factor CD and may be detected by monitoring the transmissibility TR . With less MR fluid, the transmissibility ratio changes (increase), Ferreira et al. [2009].
Analyzing Figure 7, the maximum transmissibility occurs in the resonance frequency of the sprung mass (∼ 2 Hz). In the frequency interval fr = (1.8, 2.2) Hz, the suspension has higher sensibility. Around this frequency the detection system will be working better. If the frequency is higher or lower, the probability of right detection decreases because the transmissibility functions are closer. Figure 7 corresponds to a constant 1.25 A for electric current; however, Figure 4 indicates how electric current affects the transmissibility. An observable domain switch (shade area) in Figure 7 is proposed to improve the efficiency of the detection system.
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It is very important the definition of the central, fµ , and span values, fϕ for this switch. This defines if the detection system is active or not; these are key tuning parameters of the fault detection system. Observable domain
Transmissibility
4
3
Considering the detection error as the time steps having wrongly detected damper condition between the total number of time steps (in percentage), the example in Figure 9 has 12% in error detection during 20 s. If the detection error considers only the time where the detection system was active, the global detection error was 27%, Figure 11 (top plot). If a diagnosis about the volume of the lost oil is needed, more thresholds can be integrated for the isolation capability.
I = 1. 25 A
TR 25 % lost oil volume
TR Th
2
fφ
1
Figure 9 (bottom plot) shows an histogram of the frequency of the road zr that validates the random behavior of type B road around the frequencies that activates the observable domain switch of the detection system.
fµ 0 1
1.5
2
2.5
Estimated frequency
B road sequence (middle plot). Transmissibility TˆR was estimated using equation (15), and threshold TRT h was recalled based on the estimation of the frequency of road zr through the frequency of deflection of the suspension zdef .
3
fˆ def [Hz]
Damper free of faults
Fig. 7. Transmissibility vs frequency. Dashed line is for a faulty damper, lower line is for damper without faults.
Transmissibility
6
4.5 Proposed algorithm Figure 8 shows the proposed algorithm. The algorithm consists of three key elements: frequency estimator, transmissibility estimator, and observable domain switch. Based on reading the electric current I, measuring the deflection of the suspension zdef , and computing the velocity of the suspension deflection z˙def , the frequency of the road zˆr is estimated: fˆdef . Then the transmissibility TˆR is computed. If the estimated frequency fˆdef is into the observable domain (fµ − fϕ , fµ + fϕ ), the detection system is switched on and a comparison TˆR > TRth is done for fault detection; otherwise, the detection system is switched off.
. z def
I
Transmissibility estimator ρω
Observable domain switch
Frequency estimator
fφ fφ fµ ^
TR No faulty damper
Yes
No ^
TR > TR
2 1
Road type B [m]
5
10 Time [secs]
15
5
10 Time [secs]
15
20
0.02 0.01 0 −0.01 −0.02
Percentage of total of samples per test [%]
20
15 10 5 0 0.5
1
1.5
2
2.5
3
3.5
4
^ Estimated frequency f def
Fig. 9. Example of transmissibilities (computation and estimation) through the time for a road sequences (middle plot). In bottom plot, it is shown the histogram of this road profile.
z def
. z def
3
0
QoV model I
online estimated TR 5 TR threshold 4 (lower limit)
0 0
z def
Road
Faulty damper
Faulty damper
Fig. 8. Algorithm for monitoring of damper condition. 5. RESULTS Figure 9 (top plot) shows an example of the online estimated TˆR and threshold TRT h transmissibility for a type
A False Negative (FN ) is when there is a fault, but it is not detected; while a False Positive (FP ) is when there is no fault, but one is detected. True Positive (TP ) is when there is a fault, and it is detected; while, True Negative (TN) is when there is no fault and no fault is detected. The sum of TP and FN are the total faulty cases (TFC ) and the sum of TN and FP are total healthy cases (THC ). The probability of detection Pd = TTFPC P . Receiver while the probability of false alarms Pf a = TFHC Operating Characteristic (ROC ) curves show a relation between opportune detection probability Pd versus false alarms probability Pf a . ROC curves identify when the statistical threshold captures the normal operating conditions, Woods and Bowyer [1997]. The detection probability considers only the detection system property of indicating an abnormal event without considering the fault isolation.
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By considering only the time when the fault detection system is activated, the best performance is obtained when the third quartile (Q3 ) is used as threshold, because the probability of detection, Pd is the highest, 71%; however, the false alarm rate is 21%. On the other hand, the threshold of 1σ has a low probability of false alarms, 15%, with acceptable Pd of 65%. By using the thresholds of the fourth quartile and 3σ, the fault detection performance is very poor and close to the detection limit as Figure 10 shows. 1
Probability of detection (pd)
0.8
TR Th = Q3
0.6 0.4
ction Dete
0.2 0
0
0.1
0.2
1
0.3
0.4
Limit TR Th= Q4 0.5
0.6
0.7
0.8
0.9
1
Loss of MR oil fluid volume because an oil leakage is a common fault in this type of dampers. An improved version for monitoring a MR damper condition was proposed. The proposal monitors the transmissibility (magnitude ratio between the sprung and unsprung mass accelerations) of the semi-active suspension in a QoV model. Intensive random tests shown less than 36 % (average) in detection error. A frequency estimator for the road based on the deflection of the suspension is the crucial component of the proposal. An observable domain switch that actives (or does not active) the detection system, and the frequency of the road estimation are the key modules in the overall performance of algorithm. Even these modules can be improved, early results validates that transmissibility is a good indicator of faulty MR dampers.
_
0.8 TR = TR+2 σ Th 0.6
TR Th = TR+1 σ
Future work. Experimental validation.
_
ction Dete
0.4 0.2 0
6. CONCLUSION
Limit
REFERENCES
_
TR Th = TR+3 σ
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Probability of false alarms (pfa)
Fig. 10. ROC curve for different threshold.
Average percentage Detection error [%] of outcomes for the binary detection [%]
In order to analyze statistically the performance of the fault detection system, 30 random simulations of the vehicle running in a road profile type B were implemented, all of them include the MR damper under normal and faulty operating conditions. Figure 11 (top boxplot) shows the average detection error when the fault detection system is activated, by using the Q3 threshold. Under normal and faulty conditions, the error has similar standard deviation; however, when the MR damper is free of faults, the detection error is lower (∼ 17%). The detection error obtained in the ROC curve (100 - Pd =29%) is bounded by the global error box. Figure 11 (bottom boxplot) shows the average parameters of the ROC curve. In general, the performance is similar in rigth detections (TP and TN ); while, the major opportunity in the proposed detection algorithm is minimization of the FN values through increasing the sensitivity of the fault detection algorithm when the MR damper is faulty. 55 45 35 25 15 5 50
TR Th= Q 3
Faulty MR damper
MR damper without faults
Global TR Th= Q 3
40 30 20 10 0
TP
FN
TN
FP
C. Boggs, L. Borg, and J. Ostanek. Efficient Test Procedures for Characterizing MR Dampers. In ASME 2006 Int Mechanical Eng Congress and Exposition, 2006. C. Ferreira, P. Ventura, R. Morais, A.L.G. Valente, C. Neves, and M.C. Reis. Sensing Methodologies to Determine Automotive Damper Condition under Vehicle Normal Operation. Sensors and Actuators: Physical, 156:237–244, 2009. D. Fischer and R. Isermann. Mechatronic Semi-active and Active Vehicle Suspensions. Control Eng. Practice, 12: 1353–1367, 2004. P. G´asp´ar, Z. Szab´o, and J. Bokor. LPV Design of Faulttolerant Control for Road Vehicles. In Conf. on Control and Fault Tolerant Systems, France, pages 807–812, 2010. J. Gertler. Fault Detection and Diagnosis in Engineering Systems. CRS Press, 1st edition, 1998. T. Gillespie. Fundamentals of Vehicle Dynamics. SAE, 1st edition, 1992. J. Kim and H. Lee. Sensor Fault Detection and Isolation Algorithm for a Continuous Damping Control System. J. of Automobile Eng, 225:1347–1364, 2011. V. Venkatasubramanian, R. Rengaswamy, S. Kavuri, and K. Yin. A Review of Process Fault Detection and Diagnosis Part I Quantitative Model-Based Methods. Computers and Chemical Eng, 27:293–311, 2003. Y. Vidal, L. Acho, F. Pozo, and J. Rodellar. Fault Detection in Base-Isolation Systems Via a Restoring Force Observer. In Conf. on Control and Fault Tolerant Systems, France, pages 94–99, 2010. H. Wang and G. Song. Fault Detection and Fault Tolerant Control of a Smart Base Isolation System with MagnetoRheological Damper. Smart Mater. Struct., 20:1–9, 2011. K. Woods and K.W. Bowyer. Generating ROC Curves for Artificial Neural Networks. IEEE Transactions on Medical Imaging, 16:329–337, 1997. K. Worden and G.R. Tomlinson. Nonlinearity in Structural Dynamics. IoP, 2001.
Fig. 11. Parameters of ROC for a vehicle running in 30 randomly generated type B roads.
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Fault Detection for an Automotive MR Damper Jorge Lozoya-Santos ∗ Juan C. Tud´ on-Mart´ınez ∗ Ruben Morales-Menendez ∗ Ricardo Ram´ırez-Mendoza ∗ Arturo Molina Gutierrez ∗ ∗
Tecnol´ ogico de Monterrey, Campus Monterrey, Av. Garza Sada 2501, 64849, Monterrey N.L., M´exico e-mail: {jorge.lozoya, jc.tudon.phd.mty, rmm, ricardo.ramirez, armolina }@itesm.mx
Abstract: Driving with faulty dampers can throw the car off balance sending it out of control. Common faults of Magneto Rheological (MR) dampers are loss of oil volume because oil leakage. A fault detection system for monitoring the damper condition is proposed. The proposal monitors the transmissibility of the semi-active suspension of a Quarter of Vehicle (QoV ) based on the frequency of the road. A frequency estimator for the road based on the deflection of the suspension is the key component of the fault detection system. In order to improve the efficiency of the detection system an observable domain switch was introduced. This switch identifies the most sensible transmissibility domain in order to improve the efficiency of the proposal. Intensive randomly generated tests with less than 36 % average in detection error using a semi-active MR damper validates the transmissibility as a good indicator of MR damper condition. Keywords: Fault detection, MR damper, Oil leakage, Shock absorber. 1. INTRODUCTION An early detection of abnormal events in automotive suspension systems can reduce damages in driving situations. In Venkatasubramanian et al. [2003], advanced Fault Detection and Isolation (FDI ) methods are classified into two major groups, those which do not assume any form of model information and those which use accurate dynamic process models. Basically, the FDI selection depends on the availability to get the right information through of a reliable model or process sensors. Faults in engineering processes belong to one of the following categories, Gertler [1998]: faults in sensor and actuator called soft faults or, process faults where some plant parameters change because of physical or mechanical problems. In Automotive Suspension Control Systems (ASCS ), the instrumentation is vulnerable to malfunctions. Shock absorbers could fail when it losses oil volume, generating a decrease in its damping factor. This loss is generated due to the damaged seals in the damper housing. The spring, in the suspension system, has little variations during vehicle life, Ferreira et al. [2009]. There are some research works in FDI systems for ASCS. In Kim and Lee [2011] a residual analysis based on parity equations is proposed for fault detection in vertical accelerometers of the sprung mass. In Wang and Song [2011], a FDI system based on a H∞ robust filter is proposed for sensor faults into a semi-active suspension system, the residuals are used to accommodate the malfunction by a fuzzy controller. Similarly, a FDI filter based on the Lin⋆ Thanks to CONACyT, Mexico for the support under the Postgraduate Cooperation Project 2007-2011.
978-3-902661-98-2/12/$20.00 © 2012 IFAC
ear Parameter-Varying framework is proposed in G´asp´ar et al. [2010]; sensor and actuator faults are diagnosed and accommodated in the controller to guarantee road holding and to keep the pitch and roll stability in a vehicle model. A methodology for controlling the damping ratio of semiactive dampers in a Quarter of Vehicle (QoV ) based on estimation of parameters is proposed in Fischer and Isermann [2004]; however, the computing time is prohibitive. In Vidal et al. [2010], a FDI system based on a referencemodel observer is proposed; stiffness and damping variations in a suspension are implemented as additive faults. A novel methodology that monitors the condition of the automotive dampers through a transmissibility function is introduced by Ferreira et al. [2009]. Even experiments validate results, more complex design for a MagnetoRheological (MR) damper is needed. An MR damper is a hydraulic device whose oil contains metallic micro size particles that change the rheological properties of the fluid when a magnetic field is applied; an electric current supplied through the damper coil is used to manipulate the magnetic field. The variation of the oil viscosity modifies the damping ratio. Thus, the damping coefficient not only depends on the excitation frequency, but it also depends on the electric current. This paper generalizes the work presented by Ferreira et al. [2009] by adding the electric current and the deflection velocity effects into the transmissibility function that detects malfunctions of the semi-active damper. The outline of this paper is as follows: in the next section, the model of a QoV model with an MR damper is described. Section 3 reviews the original method proposed
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Table 2. QoV parameters.
in Ferreira et al. [2009], while the Section 4 introduces an extended version. Results are presented in Section 5, and conclusions in Section 6. All variables are alphabetically described in Table 1.
Component of the Quarter of Vehicle Sprung mass (ms ) Unsprung mass (mus ) Spring stiffness (ks ) Tire stiffness (kt )
Table 1. Variables description
fr fn fˆ, fˆ(·)
z¨ ϵ f µ , fψ ζ, ζM R |||z|||∞ , |||z||| ˙ ∞ ρ
i-k 1500 1000
zs cp + c MR
ks Unsprung mass Road
m
us kt
z
us
zr
Fig. 1. QoV model with a semi-active MR damper.
0 .
.
z yield
−1000 −1500 −0.5 −0.4 −0.3−0.2 −0.1
0
0.1 0.2 0.3 0.4 0.5
Piston velocity(=)m/s
Fig. 2. Damping force vs the maximum deflection velocity with different damping coefficient.
A QoV model is used for testing the proposed fault detection system, Figure 1. The suspension has an MR damper between the sprung and unsprung masses. It is assumed that the wheel-road contact is ensured. The parameters of a QoV correspond to a 2008 CadillacT M , Table 2. The dynamic system is given by equations (1): ms
500
−500
2. QoV MODEL USING AN MR DAMPER
Sprung mass
. . F = cp z . ____ z 1 +Ic MR MR . |||z||| i-k
. . F 1 = cp z + I c z . ____ MR MR . |||z|||
8
z˙saturation
The MR damper model is an extension of the maximum deflection velocity model based on the variation of the damping coefficients cp and cM R according to the characteristic diagram of a semi-active damper and the applied electric current. Figure 2 shows how the damping coefficient changes with maximum velocity, narrow lines are the damping coefficient paths that depend on the speed interval, z. ˙
{
z˙cubic
(1)
A commercially MR damper ACDelco T M was included. This device varies its damping factor CD according to a supplied electric current, I. It is designed to perform a damping ratio variation with independence of the piston velocity, z. ˙ The stroke is 60 mm. The maximum absolute force is 2,500 N with a temperature range of (−40, 150)o C.
8
z˙yield
Units kg kg kN/m kN/m
ms z¨s = −ks (zs − zus ) − FM R mus z¨us = ks (zs − zus ) − kt (zus − zr ) + FM R
{
I kq , jq k 1 , k2 , k 3 , k4 ks , kt ms , mus Q3 ,Q4 r R TR , TˆR yM R zr zs , zus z¨s , z¨us z, zdef z, ˙ z˙def z˙f riction
Description MR damping coefficient, N · A/m Damping factor, CD due to I Passive damping coefficient N s/m Damping force, N MR damping force, N Force due to the change of oil viscosity caused by the electric current in N Frequency of the road surface, Hz Resonance frequency of the ms , Hz Estimated frequency, Hz Electric current, A Receding horizon indexes Receding horizon based on z˙ Stiffness coefficient: spring, wheel tire, N/m Sprung mass and unsprung mass, kg Quartile 75 % and 100 %. Frequency ratio, adim Sinusoidal amplitude, m Transmissibility and estimated TR , adim Damping coefficient depending on I, Ns/m Road profile, m Vertical position of the mass ms , mus , m Vertical acceleration of ms , mus , m/s2 Piston deflection, m Piston velocity, m/s Threshold for piston velocity, m/s with coulomb friction Threshold for piston deflection velocity, m/s before blow-off valves yield Threshold for piston deflection velocity, m/s with a cubic softening nonlinearity Threshold for piston deflection velocity, m/s with a saturation nonlinearity Piston acceleration, m/s2 Constant that bounds the ρ-value Span of algorithm detection Damping ratio, MR damping ratio Absolute maximum deflection, velocity Nonlinear term in MR damper model
Force (=)N
Variable cM R CD , CD (I) cp Fdamping FM R fI
Value 522 50 83 230
The semi-active MR damper model is: { yM R z˙ + cp zρ ˙ z˙ < z˙yield FM R = yM R zρ ˙ + cp z˙ z˙ > z˙yield
(2a)
yM R = I · cM R (2b) 1 ρ= (2c) i |||z||| ˙ ∞ i−kq + ϵ k |z| ˙ < z˙f riction 1 k2 z˙f riction < |z| ˙ < z˙yield kq = (2d) k z ˙ < | z| ˙ < z˙cubic 3 yield k4 z˙cubic < |z| ˙ < z˙saturation where ρ describes the damping force behavior due to the friction, the activation of the symmetrical blow-off valves, the softening of the damping force, and the saturation, Worden and Tomlinson [2001]. The parameters of model equation (2a) were learned with experimental data from Figure 3. Table 3 shows the
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Electric Velocity Force (N) current (A) (m/s)
identified parameters. The domain is divided in four zones where the limits are: z˙f riction , z˙yield , z˙cubic , and z˙saturation . The parameters cp and cM R changes based on the piston velocity at zyield . 0.4 0.2 0 −0.2 −0.4 2.5 2 1.5 1 0.5 1000 0
−1000 1
2
3
4. FAULT DETECTION SYSTEM The transmissibility can be monitored by online computing the damping ratio, ζ in the shock absorber and the excitation frequency, fr . The MR damper model and an estimation of the frequency, fˆdef through a deflection sensor, zdef are proposed for computing the transmissibility TR . A change in the semi-active and passive damping coefficients could represent a fault in the MR damper. The model relates the transmissibility TR with an electric current, I. A threshold for each electric current and frequency will support a fault detection system. 4.1 Heuristic frequency estimation
4
Time (=) secs
By assuming a harmonic motion of the damper piston, Worden and Tomlinson [2001], two variables of the suspension can be described by:
Fig. 3. Experimental test for model learning. Table 3. Identified parameters for the piecewise nonlinear MR damper model. k1 k2 k3 k4 cp cM R
z˙ < zyield 154 2 214 3004
z˙ > zyield 3 16 598 485
units Ns/m N/A
Domain z˙f riction = 0.02 m/s z˙yield = 0.1 m/s z˙cubic = 0.5 m/s z˙saturation = 1 m/s -
z ∼ R · sin(2π fr · t)
(6)
z˙ ∼ ω · R · cos(2π fr · t)
(7)
The signals z and z˙ are unknown and there is not possible to obtain a forecast. Using an index of the amplitude such as the Root Mean Square value in a moving window; or the infinite norm of the deflection and velocity of the piston in the last k samples, the envelopes of equations (6, 7) are: i
|z| = R ∼ |||z|||∞ i−jq
3. MONITORING THE DAMPER CONDITION
i |z| ˙ ∼ |||z||| ˙ ∞ i−jq
Based on QoV model, the transmissibility TR function given by output to input magnitude ratio between the sprung and the unsprung mass accelerations (¨ zs and z¨us ) is, Gillespie [1992]: √ z¨s 1 + (2ζ r)2 = TR = (3) z¨us (1 + r2 )2 + (2ζ r)2 This function depends on frequency, where the damping ratio ζ is: 1 CD ζ= √ (4) 2 ms ks and r is the ratio between the excitation frequency fr and the natural frequency fn of the system defined by fr with fn = r= fn
√ ks /ms 2
(5)
In conventional vehicle passive suspension systems, the spring constant ks has little variation during vehicle life. Vehicle mass ms changes with vehicle load, this effect is more pronounced in the rear axle, as the front axle load is mainly determined by the powertrain mass coefficient. The damper condition is related with the damping factor CD , so changes due to aging, wear or malfunction of the damper may be detectable by computing transmissibility function. The transmissibility is directly affected by the excitation frequency. A system that measures both, z¨s and z¨us accelerations and computes the transmissibility, TR , could monitor the damper condition, and detect faults.
(8)
∼ 2π fr ·
i |||z|||∞ i−jq
(9)
Solving for w using (8, 9), the instantaneous estimation of the frequency is: i
˙ ∞ i−jq 1 |||z||| fˆ = 2π |||z|||∞ ii−j
(10)
q
4.2 Estimation of Transmissibility In a shock absorber the damping factor, CD , and the damping ratio, ζ, can be obtained as a function of the damping force and excitation velocity: Fdamping (11) z˙ Combining equations (2a, 4, 11) a formula for the MR damping ratio is: CD =
1 √ I · cM R + cp ρ 2 ks ms ζM R (I, z˙def ) =
1 √ I · cM R ρ + cp 2 ks ms
z˙ < z˙yield (12) z˙ > z˙yield
The MR damping ratio ζM R depends on the electric current I, and the piston velocity z˙def . Using the estimated frequency of excitation fˆr in eqn. (5), the formula for r is:
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1
i
|||z||| ˙ ∞ i−jq
r=√ ks /ms |||z|||∞ ii−jq
(13)
INCOM 2012, May 23-25, 2012 Bucharest, Romania
i
ρω (zdef , z˙def ) =
|||z||| ˙ ∞ i−jq
(14)
i
|||z|||∞ i−jq
and combining equations (3, 12, 13), the estimated TˆR is: v ( )2 u u 2ρω [ζM R ] u 1+ √ u ks /ms TˆR (I, zdef , z˙def ) = u )2 (15) ( u[ ] 2 t ρ2ω ms 2ρω [ζM R ] 1 − ks + √ ks /ms
Following the recommendation of Boggs et al. [2006], a road sequence based in a bounce sine sweep pattern with decremental displacement amplitude according to the evaluated frequency was implemented for different electric current. Figure 4 summarizes these results for three electric current, and for several frequencies of the road sequences zr . The transmissibility TR at the natural frequency (i.e ∼ 2 Hz) exhibits the greatest values.
According to the standard ISO 8606:1995, the classified roads in types A to H corresponds to roads in good conditions (i. e. city runways). The road type A corresponds to the best pavement condition. A loss of 25 % of the oil MR damper volume in a vehicle running in a road type B was considered the fault to be detected. Figure 6 shows a histogram of the average time at different estimated frequency fˆdef when a vehicle runs in a type B road. This average was computed from 30 randomly generated simulations. Continuous line corresponds to a suspension with a MR damper free of fault (0 % lost oil volume) and dashed line is for a faulty MR damper (25 % lost oil volume). This frequency histogram confirms the application domain.
Transmissibility
z¨s / z¨ us
3.5 0A
3 2.5
Vehicle with a faulty MR damper (25 % lost oil volume)
8.12
Percentage of time
Defining:
1.25 A
6.51 Vehicle with a MR damper (0 % lost oil volume)
4.88
3.25
2 2.5 A
1.5
1.63
1 0.5 0 0.5
1 1
1.5
2
2.5
3
3.5
4
4.5
1.5
2
2.5
3
3.5
4
4.5
^ Estimated frequency f def
5
Estimated frequency from z r
Fig. 4. Transmissibility based on registered accelerations (¨ zs , z¨us ) for different electric currents and frequencies.
Fig. 6. Histogram of the average time at different frequency fˆdef that a vehicle runs in a type B road. 4.4 Threshold definition
Estimated frequency from zr [Hz]
The frequency is estimated from the suspension deflection, zdef , because there is not a frequency road sensor. Figure 5 compares the real and the estimated frequencies of the road zr . The suspension is a nonlinear system that filters some frequencies; therefore, the estimated frequency of the road zr could exhibit errors or important differences. 20
20
10
10
0
0
10 20 Frequency z r [Hz]
0
0
Based on the bandwidth of the frequency given by Figure 6, a transmissibility TR plot for a suspension with a damper free of faults was computed. This function will be used for threshold definition. Several statistics were considered for each discrete interval (0.05 Hz ) of frequency. The evaluated thresholds were TRth = µ+σ, TRth = µ+2σ, TRth = µ + 3σ, TRth = Q3 , and TRth = Q4 . Figure 7 (continuous line) shows the selected threshold, TRth = Q3 . The transmissibility TR with faulty damper, a loss of 25 % of the oil volume, was computed with the same statistics, Figure 7 (dashed line). Based on these lines, if a suspension system has a transmissibility TR > TRth , it is almost sure that the MR damper is faulty.
10 20 Estimated frequency from z def [Hz]
Fig. 5. Road frequency fˆr (left) and estimated road frequency fˆdef (right). 4.3 Fault implementation and driving conditions The considered faulty condition is a loss of oil in the MR damper because an oil leakage. This damper condition is function of the damping factor CD and may be detected by monitoring the transmissibility TR . With less MR fluid, the transmissibility ratio changes (increase), Ferreira et al. [2009].
Analyzing Figure 7, the maximum transmissibility occurs in the resonance frequency of the sprung mass (∼ 2 Hz). In the frequency interval fr = (1.8, 2.2) Hz, the suspension has higher sensibility. Around this frequency the detection system will be working better. If the frequency is higher or lower, the probability of right detection decreases because the transmissibility functions are closer. Figure 7 corresponds to a constant 1.25 A for electric current; however, Figure 4 indicates how electric current affects the transmissibility. An observable domain switch (shade area) in Figure 7 is proposed to improve the efficiency of the detection system.
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It is very important the definition of the central, fµ , and span values, fϕ for this switch. This defines if the detection system is active or not; these are key tuning parameters of the fault detection system. Observable domain
Transmissibility
4
3
Considering the detection error as the time steps having wrongly detected damper condition between the total number of time steps (in percentage), the example in Figure 9 has 12% in error detection during 20 s. If the detection error considers only the time where the detection system was active, the global detection error was 27%, Figure 11 (top plot). If a diagnosis about the volume of the lost oil is needed, more thresholds can be integrated for the isolation capability.
I = 1. 25 A
TR 25 % lost oil volume
TR Th
2
fφ
1
Figure 9 (bottom plot) shows an histogram of the frequency of the road zr that validates the random behavior of type B road around the frequencies that activates the observable domain switch of the detection system.
fµ 0 1
1.5
2
2.5
Estimated frequency
B road sequence (middle plot). Transmissibility TˆR was estimated using equation (15), and threshold TRT h was recalled based on the estimation of the frequency of road zr through the frequency of deflection of the suspension zdef .
3
fˆ def [Hz]
Damper free of faults
Fig. 7. Transmissibility vs frequency. Dashed line is for a faulty damper, lower line is for damper without faults.
Transmissibility
6
4.5 Proposed algorithm Figure 8 shows the proposed algorithm. The algorithm consists of three key elements: frequency estimator, transmissibility estimator, and observable domain switch. Based on reading the electric current I, measuring the deflection of the suspension zdef , and computing the velocity of the suspension deflection z˙def , the frequency of the road zˆr is estimated: fˆdef . Then the transmissibility TˆR is computed. If the estimated frequency fˆdef is into the observable domain (fµ − fϕ , fµ + fϕ ), the detection system is switched on and a comparison TˆR > TRth is done for fault detection; otherwise, the detection system is switched off.
. z def
I
Transmissibility estimator ρω
Observable domain switch
Frequency estimator
fφ fφ fµ ^
TR No faulty damper
Yes
No ^
TR > TR
2 1
Road type B [m]
5
10 Time [secs]
15
5
10 Time [secs]
15
20
0.02 0.01 0 −0.01 −0.02
Percentage of total of samples per test [%]
20
15 10 5 0 0.5
1
1.5
2
2.5
3
3.5
4
^ Estimated frequency f def
Fig. 9. Example of transmissibilities (computation and estimation) through the time for a road sequences (middle plot). In bottom plot, it is shown the histogram of this road profile.
z def
. z def
3
0
QoV model I
online estimated TR 5 TR threshold 4 (lower limit)
0 0
z def
Road
Faulty damper
Faulty damper
Fig. 8. Algorithm for monitoring of damper condition. 5. RESULTS Figure 9 (top plot) shows an example of the online estimated TˆR and threshold TRT h transmissibility for a type
A False Negative (FN ) is when there is a fault, but it is not detected; while a False Positive (FP ) is when there is no fault, but one is detected. True Positive (TP ) is when there is a fault, and it is detected; while, True Negative (TN) is when there is no fault and no fault is detected. The sum of TP and FN are the total faulty cases (TFC ) and the sum of TN and FP are total healthy cases (THC ). The probability of detection Pd = TTFPC P . Receiver while the probability of false alarms Pf a = TFHC Operating Characteristic (ROC ) curves show a relation between opportune detection probability Pd versus false alarms probability Pf a . ROC curves identify when the statistical threshold captures the normal operating conditions, Woods and Bowyer [1997]. The detection probability considers only the detection system property of indicating an abnormal event without considering the fault isolation.
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By considering only the time when the fault detection system is activated, the best performance is obtained when the third quartile (Q3 ) is used as threshold, because the probability of detection, Pd is the highest, 71%; however, the false alarm rate is 21%. On the other hand, the threshold of 1σ has a low probability of false alarms, 15%, with acceptable Pd of 65%. By using the thresholds of the fourth quartile and 3σ, the fault detection performance is very poor and close to the detection limit as Figure 10 shows. 1
Probability of detection (pd)
0.8
TR Th = Q3
0.6 0.4
ction Dete
0.2 0
0
0.1
0.2
1
0.3
0.4
Limit TR Th= Q4 0.5
0.6
0.7
0.8
0.9
1
Loss of MR oil fluid volume because an oil leakage is a common fault in this type of dampers. An improved version for monitoring a MR damper condition was proposed. The proposal monitors the transmissibility (magnitude ratio between the sprung and unsprung mass accelerations) of the semi-active suspension in a QoV model. Intensive random tests shown less than 36 % (average) in detection error. A frequency estimator for the road based on the deflection of the suspension is the crucial component of the proposal. An observable domain switch that actives (or does not active) the detection system, and the frequency of the road estimation are the key modules in the overall performance of algorithm. Even these modules can be improved, early results validates that transmissibility is a good indicator of faulty MR dampers.
_
0.8 TR = TR+2 σ Th 0.6
TR Th = TR+1 σ
Future work. Experimental validation.
_
ction Dete
0.4 0.2 0
6. CONCLUSION
Limit
REFERENCES
_
TR Th = TR+3 σ
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Probability of false alarms (pfa)
Fig. 10. ROC curve for different threshold.
Average percentage Detection error [%] of outcomes for the binary detection [%]
In order to analyze statistically the performance of the fault detection system, 30 random simulations of the vehicle running in a road profile type B were implemented, all of them include the MR damper under normal and faulty operating conditions. Figure 11 (top boxplot) shows the average detection error when the fault detection system is activated, by using the Q3 threshold. Under normal and faulty conditions, the error has similar standard deviation; however, when the MR damper is free of faults, the detection error is lower (∼ 17%). The detection error obtained in the ROC curve (100 - Pd =29%) is bounded by the global error box. Figure 11 (bottom boxplot) shows the average parameters of the ROC curve. In general, the performance is similar in rigth detections (TP and TN ); while, the major opportunity in the proposed detection algorithm is minimization of the FN values through increasing the sensitivity of the fault detection algorithm when the MR damper is faulty. 55 45 35 25 15 5 50
TR Th= Q 3
Faulty MR damper
MR damper without faults
Global TR Th= Q 3
40 30 20 10 0
TP
FN
TN
FP
C. Boggs, L. Borg, and J. Ostanek. Efficient Test Procedures for Characterizing MR Dampers. In ASME 2006 Int Mechanical Eng Congress and Exposition, 2006. C. Ferreira, P. Ventura, R. Morais, A.L.G. Valente, C. Neves, and M.C. Reis. Sensing Methodologies to Determine Automotive Damper Condition under Vehicle Normal Operation. Sensors and Actuators: Physical, 156:237–244, 2009. D. Fischer and R. Isermann. Mechatronic Semi-active and Active Vehicle Suspensions. Control Eng. Practice, 12: 1353–1367, 2004. P. G´asp´ar, Z. Szab´o, and J. Bokor. LPV Design of Faulttolerant Control for Road Vehicles. In Conf. on Control and Fault Tolerant Systems, France, pages 807–812, 2010. J. Gertler. Fault Detection and Diagnosis in Engineering Systems. CRS Press, 1st edition, 1998. T. Gillespie. Fundamentals of Vehicle Dynamics. SAE, 1st edition, 1992. J. Kim and H. Lee. Sensor Fault Detection and Isolation Algorithm for a Continuous Damping Control System. J. of Automobile Eng, 225:1347–1364, 2011. V. Venkatasubramanian, R. Rengaswamy, S. Kavuri, and K. Yin. A Review of Process Fault Detection and Diagnosis Part I Quantitative Model-Based Methods. Computers and Chemical Eng, 27:293–311, 2003. Y. Vidal, L. Acho, F. Pozo, and J. Rodellar. Fault Detection in Base-Isolation Systems Via a Restoring Force Observer. In Conf. on Control and Fault Tolerant Systems, France, pages 94–99, 2010. H. Wang and G. Song. Fault Detection and Fault Tolerant Control of a Smart Base Isolation System with MagnetoRheological Damper. Smart Mater. Struct., 20:1–9, 2011. K. Woods and K.W. Bowyer. Generating ROC Curves for Artificial Neural Networks. IEEE Transactions on Medical Imaging, 16:329–337, 1997. K. Worden and G.R. Tomlinson. Nonlinearity in Structural Dynamics. IoP, 2001.
Fig. 11. Parameters of ROC for a vehicle running in 30 randomly generated type B roads.
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