Fault Detection for Automotive Shock Absorber

Fault detection for automotive semi-active shock absorbers is a challenge due to the non-linear dynamics and the strong influence of the disturbances such as the road profile. First obstacle for this task, is the modeling of the fault, which has been shown to be of multiplicative nature. Many of the most widespread fault detection schemes consider additive faults. Two model-based fault algorithms for semiactive shock absorber are compared: an observer-based approach and a parameter identification approach. The performance of these schemes is validated and compared using a commercial vehicle model that was experimentally validated. Early results shows that a parameter identification approach is more accurate, whereas an observer-based approach is less sensible to parametric uncertainty.


Motivation
Safety processes and technologies typically improve the overall equipment effectiveness and plant efficiency, with benefits that drop to the bottom line. One study also showed the following benefits from embedded safety systems: 5% productivity increase, 3% reduction in production expenses, 5% reduction in maintenance costs, 1% savings in capital expenses and 20% reduction in insurance costs, [4]. Considering that an automobile is integrated by many systems, such as, transmission, steering, exhaust, cooling, electrical, gasoline engine, fuel supply, electrical, braking and suspension system; where each system is a set of interdependent parts fulfilling a specific function and capable of functioning on its own. These systems can fail due to any number of possible failures that may occur in the sensors, actuators and devices that integrate them.
Although the automobile is not a safety-critical system, several driving conditions can lead to lose the vehicle stability with fatal consequences. The World Health Organization [19] estimates that 2.2 % of the global mortality (around 1.24 million deaths) is related to vehicle collisions caused by multiple factors, e.g. human factors, road quality, vehicle design and its maintenance, etc. The economic cost of motor vehicle collisions is estimated at over US $ 100 billion per year in middle-income countries (i.e. 1 -2 % of their gross national product). Therefore, the preservation of the safety and comfort of passengers are the most fundamental goals for vehicle manufacturers and automotive control engineers. This research will focus on the controlled shock absorbers of the Semi-Active (SA) suspension system.

State of the art
Failures are inevitable and unpredictable in vehicle suspensions. The considered fault as case of study is represented as a loss of oil in an automobile shock absorber. There are different aspects that cause a damper leakage [20], which are related to a bad installation, an excessive use, a damage by external devices or a bad design of control system that saturates the manipulation. The direct consequence of a faulty shock absorber is the loss of damping force that causes a performance reduction in the suspension capabilities (comfort and safety for passengers) at any driving situation.
There are few papers dealing with Fault Detection and Isolation (FDI) for SA shock absorbers, and most of them are based on methods that inherently handle additive faults. These methods have the advantage of low computational requirements and do not need permanent excitation.
FDI modules based on analytical redundancy are the most accepted strategies for fault diagnosis and estimation [28]; especially parity-space approaches, because of its applicability to nonlinear systems, speed of detection, isolability property, robustness and computational complexity. Some FDI approaches, based on parity space theory, have been proposed for automotive suspensions [8], [16] to diagnose faults in actuators or sensors. However, these approaches have been designed for Linear Time Invariant (LTI) systems without consideration of model uncertainties or noisy measurements.
[18] explored an optimized parity space approach for actuator FDI. The parity spaces spans all the parity relations that quantify the analytical redundancies available between the sensor outputs and the actuator inputs. A transformation matrix is then optimized to transform these parity relations into residuals that are especially sensitive to specific actuator faults. Actuator faults cause the variance of parity space residuals to increase. A CUmulative SUMmation (CUSUM) procedure is used to determine when residual variance has changed sufficiently to indicate an actuator fault. The added advantage of this design procedure is that it uses less computational power than some other FDI systems such as a bank of extended Kalman filters.
In [24] the parity space method is used to detect the damper fault. However, the SA damper nonlinearities are not considered nor model uncertainties. An extension of the FDI module based on parity space theory by adding robustness to exogenous unknown disturbances is proposed by [23]. A novel Fault-Tolerant Controller is proposed for an automotive suspension system based on aQuarter of Vehicle QoV model. The design is divided in a robust Linear Parameter Varying (LPV) controller used to isolate vibrations from external disturbances and in a compensation mechanism used to accommodate actuator faults. The compensation mechanism is based on a robust fault detection and estimation scheme that reconstructs a fault on the SA damper; this information is used to reduce the failure effect into the vertical dynamics to achieve good control performances. Results show the effectiveness of the faulttolerant SA damper versus an uncontrolled damper. [1,2] presented a fault detection method for suspension system using continuous wavelet transform in which Morlet wave function is used to determine the natural frequencies and amplitudes of signal components and include maximum contribution to the signal energy at the specific scale. Fault diagnosis has been investigated for dampers and upper damper bushings. After determination of faulty parts of suspension system by using signal energy distribution, by observing the changes in the natural frequencies and corresponding energy amplitudes, damper fault is distinguished from damper bushing fault. [27] presents an application of a diagnosis and a fault tolerant control method for an active suspension system. The method detects and identifies an actuator fault and reconfigures the controller to maintain the ride comfort and vehicle performance in the presence of road disturbances. The proposed scheme consists of a diagnosis module and a controller reconfiguration module. The diagnosis module is based on the Unknown Input Observer (UIO)approach to detect and identify actuator faults. A bank of observers is designed to generate residual signals such that each of them matches the system in a defined actuator fault mode. The controller reconfiguration module consists of a bank of controllers, each corresponding to a fault mode of the system. The module changes the control law after the detection and the identification of an actuator fault. In [25] a Lyapunov based restoring force observer that leads to the design of a robust fault detection system is proposed. The fault estimation method provides a direct estimate of fault (i.e. stiffness and damping variations).
[29] designed a fault detection filter in finite frequency domain. A sufficient condition for residual system with the prescribed H ∞ performance index is derived based on the generalized Kalman-Yakubovich-Popov lemma. The fault detection filter is designed in middle-frequency domain such that the residual systems are asymptotically stable. Results are validated with a numerical example.
Even more focus on control, [6] demonstrated the efficiency of the LPV framework for vehicle dynamics control, when some actuators may be in failure (e.g. suspension and braking actuators failures). A LPV /H ∞ fault tolerant gain-scheduled vehicle dynamic control involving the steering actuators, rear brakes and four suspension systems is proposed for yaw stability, lateral and vertical  [17] introduced a new Fault Tolerant Control (FTC) for a faulty semi-active suspension system (e.g. oil leakages damper). A fast adaptive fault algorithm is used to estimate the fault. Then, the LPV-FTC is designed to ensure the damper dissipativity constraint and to achieve the designed performance objectives in both comfort and road holding. The controller solution is based on the LMI solution for polytopic systems Some FDI approaches based on parametric identification have been also proposed. These approaches have the advantage of providing deeper insight into the process; however, most online estimation methods need excitation and other operating requirements which may be restrictive when the road profile is unknown. [26] presented two simple models for the vertical dynamics. A two-step scheme for fault diagnosis is proposed, which consists of a detection of the fault in the first step and a fault diagnosis in the second. Results with measured data, drawn from a test rig showed that fault detection with parameter estimation leads to good results. Several different faults can be distinguished due to their unique patterns of the estimated parameter changes. Also, parity equations enable detection of some faults and require only little computational effort. Weak dampers can be detected.
Artificial Intelligence approaches had been exploited too. For automatic fault detection, estimated parameters were mapped to the different faults, [3]. Neuronal networks (e.g. feedforward perceptron networks) were proven for such tasks, the generalization error was below 4 %. Even a FDI system was implemented with parameter estimation, the process has to be excited all the time to guarantee convergence of the estimation algorithm. Then, parity equations were combined because they do not need permanent excitation and require less computational effort. Also, [7] designed the FDI system by developing the parity equations based on simple kinematical equations and on models of the lateral vehicle dynamics. The calculated residuals are then used for fuzzy logic fault diagnosis. The detection and diagnosis of sensor and process faults, as well as sensor reconfiguration, were presented for lateral and vertical vehicle dynamics. Small faults could be detected with a good isolation performance.
Two different approaches are applied to SA shock absorbers to estimate faults: the observer-based approach and the parameter estimation approach. The advantages and disadvantages of each method are analysed in order to determine which one is more suitable to estimate actuator faults in a SA suspension systems.

Quarter of Vehicle model
A QoV model is the most basic automotive suspension, Fig. 1. Its use assumes an equivalent load distribution among the four corners and a linear dependency with respect to the translational and rotational chassis motions. The lateral and longitudinal wheel dynamics is not considered, while the wheel road contact is ensured.
This very simple one-dimensional model consists of a spring-damper-system representing the suspension strut and a single spring replacing the tire. This friction element is added to estimate offset values to detect offset faults, or increased friction. The damper contributes the drive safety and the drive quality to the same extent. Its tasks are the prevention of an amplification of the body and the prevention of a skipping wheel. A non-skipping wheel is the condition for a good road-contact. The task of the spring is to carry the body-mass and to isolate the body from road disturbances maximally. A QoV model is a well-known simplification for vertical dynamics analysis of vehicle, [3]. Faults occurring in the damping system can be as a result of one or more of the following factors (numbered in Fig. 1): (1) Damage of leak of the damper, (2) Weak springs, (3) Worn bushing and broken of support member assembly, (4) the looseness joins, (5) under-inflation or over-inflation of tyres and (6) wear of the tyre.
The dynamic behavior of a QoV model with a SA suspension is described by: where z s , z us and z r are the sprung mass, unsprung mass and road profile vertical positions, respectively; m s is the sprung mass which represents the chassis, m us is the unsprung mass which represents the wheel, tire, etc.; k s and k t are the suspension and the tire stiffness, and F D represents the damping force.
The SA damper force behavior may be represented according to, [10]: where z de f = z s − z us is the suspension deflection, and υ is the damper control input (i.e electric current for a MR damper), f c is the force due to the input, c p is the viscous damping coefficient, k p is the stiffness coefficient, a v and a d are coefficients which characterize the hysteretic behavior, Table 1. The considered model for multiplicative fault is : where α represents the effectiveness of the damper, i.e. α = 1 is a healthy damper, α = 0 represents a 100 % damper failure. The aim is to estimate α using the masses accelerations and the suspension deflection, while the road profile z r is an unknown perturbation input, because it is expensive to measure.

Approaches
Two complementary model-based fault estimation schemes for SA shock absorbers are proposed: an observer-based approach, which is intended to estimate additive faults; and a parameter identification approach, which is intended to estimate multiplicative faults.

Observer-based Fault Estimation
A FDI system based on an UIO and the QoV model (1) is proposed. The idea is to synthesize a residual, which is sensitive to the damper fault and insensitive to disturbances (i.e. road profile). The damper fault is first modelled as an additive fault: whereF D is the nominal force (i.e. the force of a healthy damper) and F δ is the loss of force due to the fault. By using (1), (3) and (4), a state space system is: where k = k s + k p , u(t) = υρ(t) and ρ(t) is the nonlinear term of the SA damper model: Note that system (5) is comprised by a linear subsystem in a closed loop with a memoryless nonlinear function υρ(t).
An important result of applying the UIO theory to this domain is, without road information, the system is non-observable and non-detectable [11]. This will affect the state estimations.
An approximated UIO observer can be obtained by decoupling the system (5) from the road surface using the procedure reported in [14]. The decoupled system can be written as: For the state estimation the decoupled system (7) with the output y 2 will be used since this output does not depend on the fault. For the residual generation, y 3 , which is affected by the fault, will be used.
If (Ā,C 1 ) are observable or detectable, an observer can be designed for the decoupled system to estimate the states. The observability matrix of system (7), has rank 2; thus, two of the observer poles cannot be located arbitrarily. For the system (7) the non-observable subset is related to two poles in s = 0; therefore, the system is non-observable and non-detectable. A possible solution consists on generating a stable system which approximates the frequency response of the UIO observer, [13].
An observer is a dynamical system whose inputs are the process input and output vectors. For the decoupled system (7), the observer poles in s = 0 are related to the input contribution to the state estimation. Since these poles are not stable regardless the observer gain, the best estimation effort is the open-loop simulation of these states. In general the state estimation based only in the simulation of the process (which is equivalent to an observer with observer gain equal to zero) is not advised. However, the input related with these integrators is the unsprung-mass acceleration, which is indeed a measured output, but it was considered as an input to achieve the decoupling of the system from the road profile. Thus, the non-observable states are estimated via the output, like it is done by using high-gain observers. The observer gain matrix L is designed to place the two observable poles at (−20.38 ± 1.09i). The resulting observer combines the closed-loop estimation dynamics and the integration of acceleration measurements. For the states, which are estimated by numerical integration, it is well known that low frequency noise in the measurements induces a divergence called drift [22]. Without additional measurements, it is not possible to eliminate the drift by modifying the observer gain, [11,15]. Since the suspension deflection is limited and the masses velocities converge to zero, it is possible to use high-pass filters to eliminate the drift. Then, the observer is complemented by adding a pair of high-pass filters g f (s) = s/(s + ω f ) for each of the integrators. Only the non-detectable states are filtered, while the observable ones are estimated by a closed-loop observer. The cut-frequency ω f of the filter is set at 0.14 Hz.
By using the estimated states with the UIO observer and the measurement of the sprung mass acceleration, the residual can be computed to detect a fault (loss of damping force): then the magnitude of the fault can be estimated asF δ = [D f 2 ] † r(t) where † stands for the Moore Penrose pseudo inverse.

Parameter Estimation Approach
By considering the multiplicative model of the fault in (3), the system can be represented in state-space form as: Although the fault affects both the sprung and unsprung mass dynamics, for this estimator only the sprung mass dynamic will be considered since it is independent of the unmeasured road profile. Two measurements are used:z s and z de f . From (9), the discrete-time representation of the sprung mass dynamic can be written in the regression form as and θ = α. The damper fault can be online estimated by using the Recursive Least Squares (RLS) with forgetting factor λ : whereθ is the estimation of θ . The measurement ofż de f is not available, therefore an estimation is required to implement the parameter estimation.ż de f may be estimated either by the numerical derivative approximation of z de f or by an state-observer. In this case, an approximation based on the bandwidthlimited derivative is used. For a sufficiently high sampling rate, the discrete estimator will approximate the continuous time model parameters. For this system the simulations show that 500 Hz was a sufficiently high sampling rate to have a good approximation.

Results
Data from a commercial vehicle was used, Table 1. Six tests were used to assess the performance of the FDI proposals. These tests consider a) an ISO Type D at a vehicle speed of 30 Km/h, and b) a series of bumps as road profile signals, which represents a normal driving conditions. Table 2 summarizes the tests characteristics. Although both strategies are used to estimate faults of different nature (i.e. additive or multiplicative), a comparison can be made by calculating the equivalence between them. From (4) and (3) the relation between the multiplicative and the additive faults can be obtained as . Figure 2 (left plot) shows the estimation of the loss of force F δ in the Test #1. An abrupt damper fault is modelled as a step change in α; however, the equivalent additive fault also depends on F D , therefore even if α is constant, F δ is not. This figure shows the Least Squares Estimator (LEE) error converges to zero for a step change in α, while the Observer-Based Estimator (OBE) error does not. Figure 2 (right plot) shows the multiplicative equivalent fault of the Test #1, this confirms the last discussion. The LEE is better posed for constant multiplicative faults than the OBE. Since real damper faults are more similar to multiplicative faults, only this type of faults will be discussed. Figure 3 (left plot) shows that both approaches are able to estimate gradual variations for multiplicative faults, Test # 2. When the multiplicative fault α has a constant slope, neither of the estimator errors converge to zero. The LEE has a lower overall error. In Fig. 3 (right plot) the estimation of α is compared in the same fault conditions as in Test #2, but, in this case a varying damper input ( [1,3] Amp) is considered. This is an important observation because this characteristic, is essential for SA suspension systems. This figure also shows that the LEE has lower sensitivity to damper input variation than the OBE.
Test #4 is used to asses the sensitivity of the fault estimators to sudden road profile changes, which can represent road abnormalities such as potholes. The road decoupling of the OBE is not perfect since a perfect UIO was not possible to design due to observability issues. This is more evident when the road presents sudden changes, inducing a greater level of error. The LEE is not visibly affected by these road characteristics, Fig. 4 (left plot).    Figure 4 (right plot) is used to assess the effect of sprung mass uncertainty on the estimated fault, a sprung mass 20% heavier was considered. The results show, that the mass uncertainty induces steady state error in both algorithms. The LEE has a greater error. Figure 5 (left plot) shows the estimated fault in the Test #1 under other road condition, the series of bumps. As observed in this figure, the LEE is less sensitive to the road profile and it presents a lower error level. The LEE is better posed for constant multiplicative faults than the OBE. In Fig. 5 (right plot), it can be observed that he LEE has lower sensitivity to damper input variation than the OBE. Test #6 is used to evaluate the effect of tire stiffness uncertainty on the estimated fault. This variation may occur due to variations in the tire inflation pressure. A 20% lower tire stiffness was considered, which corresponds to a variation of 5 psi in the inflation pressure. The results in Fig. 6 show that both methods are robust to the uncertainty in this parameter. The OBE error is slightly higher, 10 % with respect to the unperturbed case, whereas the estimation of the LEE is not affected since it is independent of this parameter. Finally, a quantitative comparison of the approaches was made using the Root Mean Square Error (RMSE) index. Table 3 summarizes the results. It can be observed that the LEE presents an overall lower level of error; for instance, in Test #2 the LEE has 493% lower RMSE in the estimation of the loss of force F δ . If the fault is represented in the multiplicative form, the LEE also yields a lower RMSE, but the differences are smaller. For example, in test #2 the LEE is also lower but 57%. Only when sprung-mass uncertainty is considered the OBE has a lower index, in this case 55% lower for F δ .

Conclusions
A comparison of two fault detection algorithms for the estimation of semi-active shock absorber faults is presented. One method is based on classical least square parametric identification to estimate a multiplicative fault. The second method is based on an Unknown Input Observer, which is used to calculate a residual and to estimate the loss of force due to an additive fault. It has been reported that shock absorber faults are normally of multiplicative nature; nonetheless, additive fault estimation methods are more widely reported. Since it is possible to derive the multiplicative equivalent from the additive estimation, both methods were compared through different scenarios, including mass uncertainty, shock absorber input sensitivity and road profile sensitivity. The results show that the Least Squares Estimator achieves better qualitative and quantitative performance in all cases, with the exception of mass uncertainty. This also suggests that, for this system, identification algorithms based on regression models are highly sensible to process parameters changes. It can be concluded that the two estimators present complementary characteristics which could be exploited depending on the operating conditions to improve the overall system performance. The results also show that for this application a simple fault estimation algorithm, such as Least Squares Estimator, is enough as long as a multiplicative fault structure is used and a low level of parameter uncertainty is present.