Fault Detection and Isolation of Vehicle Dynamics

Fault Detection and Isolation of Vehicle Dynamics Sensors and Actuators for an Overactuated X-by-Wire Vehicle Lok Man Ho and Daniel Ossmann German Aerospace Center (DLR) Institute of System Dynamics and Control Muenchener Str. 20, 82234 Wessling, Germany Abstract— Model-based fault detection and isolation (FDI) for an overactuated mechatronic vehicle is presented. The linear single-track model is extend to reflect the layout of the overactuated vehicle as well as its longitudinal dynamics, and sensor and actuator faults are added into the model. The DLR Fault Detection Toolbox, which makes use of rational nullspace bases computation to design residual generators, is used for the systematic design of structured residuals. A minimum set of residuals is selected according to their robustness and ability to isolate faults. The fault detection and isolation system is validated in a simulation in connection with a double track model using the parameters of the ROboMObil prototype vehicle.

Keywords: vehicle dynamics; fault detection; structured residuals; rational nullspace method I. INTRODUCTION The increasing number of available measurements in modern automobiles from sensors for vehicle dynamics control, as well as the safety relevance of vehicle dynamics related systems such as Electronic Stability Control, have stimulated interest in model-based fault diagnosis on the vehicle dynamics level. Many studies so far have focussed on model-based diagnosis of ESP sensor faults on conventional vehicle architectures with manual front steering ( [1], [2], [3]), while [4] addresses the problem for a prototype with steer-by-wire on the front wheels. The application of modelbased fault detection and isolation (FDI) to an overactuated, fully by-wire vehicle architecture with in-wheel mechatronic actuators poses interesting challenges. DLR’s ROboMObil prototype [5] employs such an architecture and acts as the application example in this paper. The work in this paper represents a part of a broader FDI scheme for this advanced vehicle architecture. As the first step of systematically finding a solution to detect all faults of interest, structural analysis was employed in [6] to divide the vehicle level FDI problem into smaller overdetermined groups of equations. Some of these groups, otherwise known as minimally structurally overdetermined (MSO) sets, are manageable in their complexity and allowed simple conversion into residual generators. In constrast, some MSO sets involving vehicle dynamics equations are more complex. The actual equations, rather than just their structure, must be considered in the fault isolation problem. This work presents a solution approach employing the DLR Fault Detection Toolbox ( [7], [8]), which designs minimal

order residual generators based on numerical nullspace computation. A subset of the computed residual generators is selected according to fault isolability and robustness critieria. The main contributions of this paper are the fault diagnosis oriented modelling of the vehicle for a constant longitudinal velocity, and the solution of the residual selection and generation problem. This paper is structured as follows. Following the introduction of the FDI scheme in Section II, the residual generation method is described in Section III. Section IV explains the modelling of the vehicle dynamics and faults, while Section V demonstrates the application and adaptation of the FDI methods to the vehicle model and illustrates its effectiveness through simulation results. Section VI concludes the paper and discusses future work. II. FAULT DETECTION AND ISOLATION SCHEME The FDI scheme described in this work is part of a hierarchical FDI setup for vehicle dynamics relevant systems in an overactuated mechatronic vehicle. The hierarchy contains three levels as illustrated in Figure 1. At the lowest level, signal and model-based local (component level) fault detection is applied within the actuator by monitoring its internal signals. On the next level, abnormal behaviour in each controlled actuator is detected by comparing the measured controlled variable with the reference signal. The highest FDI level, which contains the subject of this paper, utilises vehicle dynamics relationships in model-based fault detection approaches to detect faults in the chassis sensors and vehicle hardware. Further details of this hierarchical scheme can be found in [6]. The choice of a suitable model-based approach depends on the valid model of vehicle dynamics for a given operating regime. During low speed manoeuvring, the steering angles can be large, but due to the low forces the tyre side-slips are negligible. Therefore the vehicle dynamics are sufficiently described by a geometric model with accurate evaluations of trigonometric functions and kinematic relationships. During cruising operation, steering demands and longitudinal accelerations are low, but tyre slips become significant. The linear single-track model simulates this effect and is therefore an adequate description. The remaining operating conditions can be described as highly dynamic, for which a nonlinear double track model (DTM) is needed. The top

level of Figure 1 summarises the fault detectors based on these three models. This paper relates to the cruising operating regime, allowing the linear single-track model to be employed as a reference. The model equations contain parameters dependent on the longitudinal velocity, which is considered constant in the FDI algorithm designed in this work. This enables the use of residual generation (RG) methods for LTI systems. It can be extended to cover operation at varying speeds using LPV techniques with velocity as a scheduling variable. Vehicle level: linear single-track model-based RGs

Vehicle level: geometric parity equations

Vehicle level: nonlinear DTM based RGs

Closed-loop actuator level: compare demand to measurement Actuator hardware: internal signal monitoring Figure 1: Levels of the FDI scheme for a mechatronic vehicle. The component considered in this paper is highlighted in bold.

Figure 2 shows the FDI scheme used. Each residual generator uses as input the measurement of the actuators’ controlled variables as well as vehicle sensor measurements. The residual is processed and compared to a threshold to return a decision variable. Fault isolation is achieved using a bank of residual generators such that the signature of each fault (residuals which are sensitive to it) is distinct from those of other faults.

A. Fault Detection and Isolation Problem In a deterministic setting, consider the linear continuoustime time-invariant system described by the input-output relations ( )

( ) ( )

( ) ( )

( ) ( )

(1)

Where ( ) ( ) ( ) ( ) are the Laplace-transformed vectors of the system output, control input, disturbance and ( ) ( ) ( ) are the fault vectors respectively. transfer function matrices (TFMs) from the control inputs, disturbance and faults to the outputs respectively, each of them being a rational matrix. To solve the fault detection problem with exact disturbance decoupling, we look for a linear residual generator with the general form ( )

( )[

( ) ] ( )

(2)

such that (i) ( ) when ( ) for all ( ) and ( ); and (ii) ( ) when ( ) , for 𝑖 1 … 𝑚𝑓. ( ) is the residual vector while ( ) is a stable and proper TFM. Condition (i) implies that ( ), where ( )

[

( )

( ) is the left annihilator of

( ) ]

(3)

while condition (ii) leads to the algebraic condition that the 𝑖-th column of ( ) is non-zero, where ( ) represents the fault-to-residual dynamics. ( )

( )[

( )

]

(4)

A desirable property in practical application is strong detectability, which means that persistent faults produce asymptotically persistent residuals. Enforcing this is equivalent to ensuring finite non-zero DC-gains for each column of ( ), that is Figure 2: FDI scheme with bank of residual generators

III. MODEL-BASED RESIDUAL GENERATION USING RATIONAL NULLSPACE BASIS The design and selection of residual generators are discussed in this section. Model-based residual generators can be designed using observer methods, parity space relations and nullspace bases methods, amongst others [9]. Those found using the first two methods are subsets of those computed by the nullspace based method, which has the advantage of finding the exact solution of minimal order if the problem is solveable. The advantages of minimising residual generator order are discussed in [10]. The Fault Detection Toolbox used in this paper employs proper rational bases for reliable numerical solutions based on state-space computations.

‖

()

( )‖

𝑖

1… 𝑚

(5)

To satisfy condition (i), one of the ways to determine ( ) is to first compute a minimal basis for the left nullspace of ( ), and then create a stable scalar output fault detector ( ) as a linear combination of the bases. Additionally, ( ) is chosen to provide the specified fault-to-residual dynamics where possible. Fault isolation using structured residuals makes use of a number of residuals, each being sensitive only to a subset of the faults. Full isolation of single faults requires that each fault has a unique fault signature. The fault isolation problem is solved using the same algorithm as for the fault detection problem, but the non-sensitive faults of a particular residual are considered as part of the disturbance ( ) vector instead of the fault vector ( ) [7]. This results in the desired effect of decoupling these faults from the residual.

B. Residual Evaluation and Decision Making

𝛾

The 𝑖-th residual signal is further processed in the evaluation stage by calculating an approximation of its signal 1-norm ( ). This can be computed online efficiently using the filter [11]. 𝛼 and 𝛽 determine the relative weights on past and instantaneous residual values respectively, while 𝛾 is the time constant of the first order filter. The choice of parameters is a trade-off between short detection delay, and robustness against noise and transient disturbances. ̇( ) ( )

𝛾 ( ) 𝛽 | ( )| ( ) 𝛼| ( )| 𝛾 𝛽𝛼

(6)

The 𝑖-th element of the observed signature vector is set to 1 when the evaluation signal i exceeds the chosen threshold 𝐽 ℎ,i. The resulting signature vector is matched against that of each fault to determine the occurred fault. C. Residual Generator Selection This section addresses the selection of structured residuals. There is generally redundancy in the residual generators with respect to fault isolation, meaning that some can be left out without diminishing the ability to distinguish between faults. It is desirable to choose a set with the lowest number of robust residual generators which provides the highest degree of achievable fault isolation. In the first step, all achievable residual signatures were generated, which indicates the faults to which a residual is sensitive. This residual signatures are stored in the form of a matrix 𝑆, with 1 𝑖𝑓 𝑓 𝑖 { 1 𝑖𝑓 𝑓 𝑖 𝑖𝑓 𝑓 𝑖

𝑆

𝑖 𝑖

(7)

𝑖

where 𝑖 and 𝑓𝑗 represent the 𝑖-th residual and 𝑗-th fault respectively. From this set of achievable residual signatures, a reduced subset is selected as follows. The rows of 𝑆 are first rearranged according to the preference for the corresponding residual generator. Those less sensitive to model uncertainty are preferred. In order to evaluate this sensitivity, consider ( ). The residual the fault-free system with an uncertain generator is presented in (8) in its internal form, with transfer functions 𝑢 and . The time-invariant uncertain parameter vector Δ has nominal value Δ0 and is assumed to belong to a bounded set Ρ. ( ) ] ( ) ( ) ( )[ |

( )

( )[

[

(

)|

( ) ( ) ][ ] ( ) ( ) ( )] [ ] ( )

(8)

Due to the choice of ( ) as the left annihilator of ( ), ( Δ ) ( Δ ) and . The sensitivity of ( ) to uncertainty in Δ can be determined as the maximum value of the infinity norm of 𝑢 as Δ varies:

Ρ

‖

𝑢(

)‖

(9)

The residual generators and the corresponding rows of 𝑆 are sorted in ascending values of 𝛾, then the reduced subset is selected. The selection process begins with an empty set of residual generators. Starting from the first row of 𝑆, a residual generator is added to the set if at least one additional pair of faults can be distinguished from each other through the newly added residual signature. Computationally this corresponds to a reduction in the number of entries in the fault isolation matrix, where a 1 as the 𝑖𝑗-th element indicates that the signature produced by the occurrence of fault 𝑖 can be interpreted as fault 𝑗, with 0 indicating the opposite. When all the rows of 𝑆 have been processed, the selected set is checked for redundant residuals, which are removed without compromising the isolation capability. The outputs of the procedure are the reduced set of residual generators and the maximum fault isolation matrix. Compared to the method in [12] which employs a greedy algorithm, this approach prioritises robustness to model uncertainty above the minimisation of the number of residual generators. When determining the fault isolation matrix, it is necessary to consider the phenomenon of partial firing. This is where some residuals sensitive to an occurred fault do not exceed their corresponding thresholds due to the small magnitude of the fault. The thresholds are set above the maximum faultfree residual value caused by noise and model uncertainties to prevent false alarms. Partial firing modifies the observed fault signature such that some 1s are turned into 0s. If the 𝑓j signature is reachable from the 𝑓i signature by only changing 1s into 0s, the observed signature when 𝑓i occurs may match the ideal one of 𝑓j due to partial firing and result in 𝑓i being misinterpreted as 𝑓j. To reflect this, the 𝑖𝑗-th element of the fault isolation matrix is set to 1, resulting in possible asymmetry in the matrix [13]. IV. VEHICLE MODEL A. Nominal Model The vehicle model is a concatenation of linear models for lateral and longitudinal dynamics. This is described by the LTI model in (10) to (15), in which the vehicle velocity 𝑣 is considered a constant parameter. The lateral dynamics model is based on the well-known linear single-track model [9], extended to account for the higher level of actuation on the studied vehicle. Rear wheel steering is modelled by the input 𝛿 , while the ability to produce the differential wheel torque on the left and right hand sides of the car is modelled by 𝑀w,L and 𝑀w,R. The auxiliary longitudinal acceleration output ̅ is added to enable the use of this acceleration signal in fault isolation; this will be described in more detail later in this section. The state vector 𝑥, input vector 𝑢 and output vector are all functions of time, but the time argument is left out for compactness.

̇

1 (𝑀 𝑚

(10)

where 𝛽 [ ̇]

𝑚𝑣 [ [

𝐽

𝐽

𝐽 1

] ]

𝐽

(13)

𝐽

𝑚𝑣

(14) ]

𝑚

[ with 𝛽 ̇ ̅ 𝛿𝐹 (𝛿 ) 𝑀 ,𝐿 (𝑀

1 𝑚 ]

Chassis side-slip angle Chassis yaw rate

,

Chassis lateral acceleration Auxiliary longitudinal acceleration (see (17)) Steering angle of the front (rear) wheel ) Sum of the front and rear wheel moments on the left (right) side of the car

Descriptions of the parameters together with their values in the example application are shown in Table III in the Appendix. The valid operating region of the model is defined by the rule-of-thumb that linear single-track models provide adequate representations of lateral dynamics for lateral accelerations of up to 4 m/s. The longitudinal model used in this study has one degree of freedom, represented by the 𝑥-acceleration output 𝑥. The effects of tyre and wheel rotation dynamics on the longitudinal dynamics are minimal during cruising operation and are thus neglected. 𝑥 is related to the inputs and states by (16). Due to nonlinear dependencies on inputs and states (𝛿𝐹, 𝛿 , ̇ , 𝛽), it is not possible to entered 𝑥 into the linear system directly. Instead, by manipulating (16) we compute an auxiliary output ̅ with (17) using measurable variables. ̅ is dependent only linearly on 𝑀w,L and 𝑀w,R and can be included in the output vector.

𝑚

𝛿 )𝛿

𝑣

( 𝛽 ( 𝛽

̇ 𝑣

̇ 𝑣

𝛿 )𝛿

(16)

𝑣

𝛿 )𝛿

̇

𝛿 )𝛿

𝑣

𝑀

( 𝛽

(17)

𝑣

)

Each of the four wheel torques is a sum of the contributions due to the traction motor torque 𝑀𝑇𝑀,𝑖 and the disc brake ] representing the torque 𝑀𝐷𝐵,𝑖, where 𝑖 [𝐹𝐿 𝐹 𝐿 front-left, front-right, rear-left and rear-right wheels respectively. The application example ROboMObil has a disc brake system in which one pressure actuator in each axle applies equal pressure to the left and right wheels, and a wheel hub motor in all four wheels. Because these wheel moments are not directly measurable, 𝑀w,L is calculated from the measured traction motor currents and brake pressures of the front-left and rear-left wheels. This is shown in (18), with an analogous expression for 𝑀w,R. In the sequel, the suffix m indicates a measurement.

(15) 1 𝑚

) ̇

( 𝛽

1 (𝑀 𝑚

𝑀

𝑚

𝑚 𝑚

(12)

1

𝑚

[

̅

1 𝐽𝑣

𝑚𝑣

(11)

[̅ ] 𝑚𝑣

𝑚𝑣

𝑚

𝛽 ̇

𝛿 𝛿 𝑀 [𝑀 ]

𝑀

𝑀 𝑖

𝑀

𝑀 𝑖

𝑀

(18)

where 𝑖𝑇𝑀,𝐹𝐿, 𝑖𝑇𝑀, 𝐿 are the measured motor currents in the front and rear left wheels, and 𝐹, are the measured brake pressures in the front and rear axles. 𝑇𝑀,𝐹𝐿, 𝑇𝑀, 𝐿 are the motor torque constants for the front-left and rear-left traction motors, and 𝐷𝐵,𝐹, 𝐷𝐵, are the front and rear axle brake constants, which are functions of slave brake cylinder area, disc-pad friction coefficient and brake disc friction radius. Disturbances can be modelled as forces on the vehicle centre of mass in the 𝑥- and -directions together with a yaw moment. In this work, the vehicle is assumed to be driven on a level road with constant friction properties and not influenced by side-wind. As a result, no unknown disturbance is introduced to the system. B. Fault Models Reference [14] shows that most faults can be represented as additive input or additive output faults, which is done here for the purpose of residual generator design. The model equations for the additive faults are shown in (19) and (20). Faults in the steering, braking and traction motor actuators, as well as chassis sensors, are considered. Some subsets of physical faults affect the linear single-track model through identical columns in the input matrices, making them non-isolable due to the rank condition [14]. Such subsets are grouped and considered as single faults. The affected faults are: 

Steering – the single-track model combines the effects of the left and right wheels for each axle, so the steering faults on the left and right sides cannot be isolated

[

] 1

1

𝑓 [ ] 𝑓

] 1 1 1

The fault model equations can be readily integrated into the linear state space vehicle model with the fault vector 𝑓 as an additional input.

V. APPLICATION OF FDI METHODS TO THE VEHICLE MODEL A. Residual Generator Selection and Design The model described in Section IV is used for the computation of residual generators, with the constant vehicle velocity 𝑣 set to 10 m/s. The computed 𝑆-matrix contains 67 achievable residual signatures. The residuals were sorted and selected according to Section III.B, with the parameter [𝑚 ], bounded by uncertainty vector chosen as || |] [1 [| 𝑚| | ] The signatures of the selected residuals are shown in Figure 3 while the order and input signals of each residual generator are shown in Table I. The selected residual generators were designed using the Fault Detection Toolbox and implemented as continuous-time state space models for the simulative investigations. The residual signatures are complicated by the nonlinear component of the equation for the output ̅ . For any residual which uses ̅ as an input (indicated by sensitivity to 𝑓̅ , i.e. a 1 in the last column of 𝑆), its signature is modified by adding in sensitivities to the faults {𝑓 𝑓 𝑓 𝑓 ̇ 𝑓 }, because their associated model signals are contained the expression (17) for ̅ . These added sensitivities in the residual signature matrix in Figure 3 are indicated by the characters ‘×’.

The fault inputs are scaled such that the fault-to-residual gain of different faults are comparable, which is necessary for two purposes. Firstly, determination of detectability requires the fault-to-residual gain to exceed a certain common numerical threshold for the residual to be considered sensitive to the fault. Secondly, the synthesis algorithm attempts to minimise the ratio between maximum and minimum fault-to-residual gain across all fault inputs for each residual generator. Since fault inputs can be arbitarily scaled, the fault-to-residual gain also takes on this arbitary scaling and direct comparison is not meaningful. To achieve the desired comparability, we propose that each fault is normalised to its minimum size to be detected. In this case study, we assume that they are specified according to the severity of their influence on the vehicle dynamics. This influence can be characterised by the infinity norm of the normalised output vector, in which each of the model outputs is normalised to their maximum values in normal operation, shown in Table II in the Appendix. It is reasonable to set the minimum detectable faults sizes such

𝑓

̇

C. Model Scaling 1

1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 [1 1 1 1

1 1 1 1

1 1 1

1 1

1 1 1 1

𝑓

1

𝑓 𝑓̇ 𝑓 [𝑓 ]

where ̅ ( ) is the normalised transfer function matrix from the 𝑖-th fault to the output vector. Scaling of ̅ ( ) can be achieved by appropriate scaling of the 𝑖-th columns of the 𝐵 and 𝐷 matrices of its state space representation.

𝑓

𝑓 𝑓 𝑓 𝑓 [𝑓

(21)

𝑓

where

𝑖𝑗

𝑓

(20)

‖ ̅ ( )‖

𝑓

(19)

‖ ̅ ( )‖

𝑓

Each element in the fault vector represent a range of physical faults. Those in represent faults such as bias, gain and freezing in the vehicle dynamics sensors. In the actuator fault vector , the steering faults 𝑓 and 𝑓 model faults in the steering angle sensors, while the disc brake fault 𝑓 models deviations in the pressure sensor measurements. Note that a disc brake actuator fault acts symmetrically on both the left and right side wheel torques. The traction motor faults 𝑓 and 𝑓 represent changes in the torque constants and frictional losses.

that their effects on this norm are equal. Normalisation of the faults to their respective minimum fault sizes leads to

𝑓



Traction motors – since the wheel torque model inputs have been combined into one input on the left and one on the right, the front and rear traction motor faults on each side cannot be isolated Friction brake – for the same reason as for the traction motors, the front and rear disc brake faults on cannot be isolated from each other and are combined into one 𝑓𝐷𝐵

𝑓



1 1

]

Figure 3: Signatures of the selected residual generators Table I: Structure of the selected residual generators (RG) RG Index

1 2 3 4 5 6 7 8

𝑄( ) order

2 2 2 1 2 1 1 2

Inputs used Model Outputs Model Inputs ̇ 𝛽 𝛿𝐹 𝛿 𝑀𝑇𝑀,𝐿 𝑀𝑇𝑀, 𝑥 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

Figure 4 shows the achieved fault isolation matrix. Ideally, any occurred fault can be uniquely isolated through a unique matching between the observed and ideal fault signatures, resulting in a diagonal matrix. This is achieved apart from the disc brake (𝑓𝐷𝐵) and the 𝑥-acceleration sensor faults (𝑓 𝑥). By manipulating (11)-(15), (17) and (19)-(20), we observe that both 𝑓𝐷𝐵 and the 𝑓 𝑥 are only present in one system output equation, which also contains the fault variables {𝑓 𝑓 𝑓 𝑓 ̇ }. Thus any residual sensitive to 𝑓 𝑥 and 𝑓𝐷𝐵 will also be sensitive to the other faults, resulting in the inability to isolate 𝑓 𝑥 and 𝑓𝐷𝐵 from other faults in this group.

𝑓 𝑓

1

𝑓

𝑓

𝑓

𝑓

𝑓

𝑓

1

𝑓 𝑓 occurred faults

𝑓

𝑓

𝑓

̇

interpreted faults

Simulations show that all the residuals respond to the faults in accordance with the specified signatures. Due to the lack of space, only one test case is described here. In this test case as shown in Figure 5, the yaw rate demand is varied [1 ] and sinusoidally in the time intervals [ ] , simulating lane change manoeuvres. Between 6.5s and 9.5s the yaw rate demand is ramped up to 0.15 rad/s and then back down the zero again. The chassis side-slip angle demand is kept at zero for the simulation leading the geometric steering controller to set equal and opposite front and rear steering angles. Due to the presence of tyre slip, the actual side-slip angle is non-zero and does not match the demand exactly. The longitudinal velocity demand is set to the design velocity of 𝑣 =10 m/s, and the maximum lateral acceleration reached is 2.9 m/s2.

1 1

1 1

1 1 1 1 1

1 1 1

1 𝑓 𝑓 𝑓̇ 𝑓 𝑓

1

[

1 1

Two faults are simulated in this test. The first is an offset fault in the yaw rate sensor (𝑓 ̇ ) of 0.1 rad/s (equal to 5.7 °/s and 25% of the normalisation value) between 2.5s and 3.5s, while the second is a torque offset of the front left traction motor (𝑓𝑇𝑀,𝐿) of 160 Nm (13% of normalisation value) between 6.5s and 7.5s.

1]

Figure 4: Fault isolation matrix of the selected residuals.

B. Residual Evaluation and Decision Making The residual is evaluated according to III.B. The constant threshold for each evaluated residual is determined using Monte Carlo Simulations in order to prevent false alarms due to parameter uncertainties and unmodelled dynamics. This involves multiple simulations of a representative driving manoeuvre with a fault-free system, in which selected physical model parameters are randomly varied between specified bounds. The maximum value of the residual norm across these simulations is taken as the minimum value for the threshold 𝐽 ℎ. An additional margin accounts for the effect of measurement noise. C. Simulation Results The double-track vehicle model used in the simulation is based on equations described in [15]. The vertical dynamics is represented by a vehicle body connected by spring-damper systems to the four wheels, each being always perpendicular to the road. This enables the simulation of vehicle roll, pitch, and heave. Horizontal tyre forces are calculated from a simplified Pacejka Magic Formula model with nonlinear dependence on wheel load. The model parameters were assigned values based on the ROboMObil prototype, which are equivalent to the nominal linear single-track model parameters shown in Table III in the Appendix. The longitudinal speed is controlled using a P-controller, while the four steering angles are computed using a geometric feed-forward controller, which assumes no tyre side-slip when calculating them from the vehicle motion demands of yaw rate and chassis side-slip angle.

Figure 5: Vehicle motion during the simulation

In both the fault cases, the residuals react as specified by their respective signatures. Figure 6 shows the response of the residuals during the simulation. From 2.5s to 3.5s, the yaw rate sensor fault triggers non-zero values in residuals 1, 2, 3, 6 and 7. Between 6.5s and 7.5s, the front left traction motor fault causes residuals 1, 3, 5, 6, 7 and 8 to respond, as expected. The time-responses to the step fault signals are in accordance to the desired fault-to-residual transfer function pole of 1 specified in the design process. During the fault-free periods, the residual magnitudes remain below the thresholds, reaching their peak values during higher lateral accelerations and input changes. These deviations occur due

to the model parameter uncertainty, higher order plant dynamics and nonlinearities that are not reproduced in the linear single-track model used in residual generator design.

Table III: Nominal vehicle model parameters Parameter

Value 0.75

Unit m

𝛼𝐹

1.2 1.2 1175 1692 31500*2

m m kg kg m2 N/rad

𝛼

27000*2

N/rad

10

m/s

𝐹

𝑚 𝐽𝑧

𝑣

Description Half of the vehicle track (distance between left and right wheels) Distance from front axle to CoG Distance from rear axle to CoG Total vehicle mass Total vehicle inertia about the z-axis Cornering stiffness – sum of front tyres Cornering stiffness – sum of rear tyres Longitudinal velocity of the vehicle

REFERENCES [1] S. X. Ding, S. Schneider, E. L. Ding and A. Rehm, "Advanced ModelBased Diagnosis of Sensor Faults in Vehicle Dynamics Control Systems," in IFAC World Congress, 2005. [2] H.-G. Schulz, "Modellgestüztzte Fehlerdiagnose der Sensoren für die Fahrzeug-Querdynamik," 2005. Figure 6: Behaviour of the residual norms (the constant thresholds are displayed as black dotted lines)

[3] D. Fischer, M. Börner, J. Schmitt and R. Isermann, "Fault detection for lateral and vertical vehicle dynamics," Control Engineering Practice, vol. 15, no. 3, pp. 315-324, 2007.

VI. CONCLUSION AND OUTLOOK

[4] C. D. Gadda, "Optimal fault-detection filter design for steer-by-wire vehicles," Stanford University, 2008.

In this paper, a scheme for detection and isolation of sensor and system faults in an overactuated mechatronic vehicle was implemented. It was shown that for a constant velocity, a bank of linear detectors was able to adequately isolate the faults of interest. The rational nullspace based method enabled a systematic way for designing structured residuals. As mentioned in the introduction, the constant velocity fault detector is a building block for the LPV FDI algorithm for variable longitudinal velocities, which will make use of extensions for the Fault Detection Toolbox [16]. Sensor fault detection for the longitudinal velocity 𝑣 must be achieved outside of this model-based FDI scheme, since it is a scheduling variable in the model equations and is assumed fault-free. A possibility is to employ the kinematic relationships with wheel-speed measurements. Once the 𝑣 sensor fault is uniquely isolatable, the 𝑣 signal can be used to improve the isolation of the 𝑥-acceleration sensor fault. VII. ACKNOWLEDGMENT The authors would like to thank Andras Varga for providing support for the DLR Fault Detection Toolbox which was extensively applied extensively for the design of structured residuals, and Tilman Bünte who advised the vehicle modelling component of the work. APPENDIX Table II: Normalising nominal values for the outputs Model output (symbol) 𝛽 𝑚 ̇ ,

𝑚

𝑥,

𝑚

Model output Side slip angle Yaw rate Lateral acceleration Longitudinal acceleration

Normalising value 5° 0.4 rad/s 4 m/s² 4 m/s²

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