Bayesian based Fault Identification for Nonlinear Mechatronic System with Backlash

Abstract: This article attempts to solve the problem of fault identification of nonlinear mechatronic system with backlash. The fault detection and isolation are carried out by evaluating the residuals and the fault signature matrix derived from the bond graph model of the system. In order to refine the fault candidates set after fault isolation, a Bayesian method is adopted where the potential faults in the fault candidates set are treated as the special states to facilitate the unknown parameters estimation. According to the estimation results, the true faults can be obtained which are useful for further maintenance purpose. Simulation studies are conducted to validate the proposed method.


Introduction
With the increase of complexity of modern industrial system, fault diagnosis becomes more and more important.It is critical to carry out fault diagnosis in a timely manner to avoid serious consequence happening in the monitored system, thus the reliability and operation safety will be enhanced.In general, there are three major steps in fault diagnosis: fault detection, fault isolation and fault identification.Fault detection attempts to indicate the occurrence of the fault by comparing the model outputs with the actual system outputs.Fault isolation tries to locate the fault after it is detected.Fault identification involves the estimation of the size and the nature of the fault.For model based fault diagnosis, the algorithm performance mainly depends on the model accuracy.Bond graph (BG) provides a systematic way to model complex system with multiple energy domains, such as mechanical, electrical, hydraulic and so on.The major advantage of BG is the causality which links the variables in BG model in a systematic manner and in turn provides an inefficient tool for fault detectabilty and isolability analysis.
Fault diagnosis of mechatronic system has received substantial attentions in recent decades.General speaking, the method used for fault diagnosis of mechanic system can be divided into two categories: signal based method and model based method [1].Signal based method utilizes the signal processing techniques to extract the fault feature for diagnosis purpose [2][3][4][5].The advantage of this method is that no deep understanding of the monitored system is required.As a result, it is relatively easy to apply this method to industrial system.In [6], a signal analysis approach for machine health monitoring using the Hilbert-Huang Transform (HHT) is proposed.It extracts instantaneous frequency components from the intrinsic-mode functions of the signal.The HHT method is not constrained by the uncertain limitations with respect to the time and frequency resolutions suffered by some time-frequency techniques, which has shown quite promising performance in terms of fault severity evaluation.
On the other hand, model based fault diagnosis usually models the system under monitoring based on physical law [7].This method is able to provide physical insights which link the fault to the component parameter variations [8].Thus, the diagnosis accuracy is better than signal based method.However, this method demands good knowledge about the concerned system which may limit its application scope, especially for complex systems.In [9], a robust fault-detection and isolation (FDI) method is developed for an electric vehicle traction system in the presence of structured and unstructured uncertainties.The BG is adopted to model the traction system with multiple energy domains.The adaptive thresholds considering uncertainties are derived based on the causality of the BG.Uncertainties raised from the parameters and structured are identified by a least-square algorithm.A quantitative hybrid bond graph (HBG) based fault diagnosis method is proposed in [10].In this method, the concept of controlled junction is utilized to capture the discrete event in hybrid dynamic systems.A set of global analytical redundancy relations (AGARRs) are established to represent the dynamic evolution of the monitored hybrid systems in a unified manner.Based on the AGARRs, fault detectabiltiy and isolability under different operating modes can be considered.
This paper proposes a BG model based fault diagnosis of nonlinear mechatronic system using Bayesian method.The mechatronic system under monitoring includes DC motor, reducer and load.Nonlinear phenomena such as backlash and friction are considered and modeled in the BG framework.After the fault is detected based on the numerical evaluations of analytical redundancy relations (ARRs), fault identification is carried out in which a Bayesian method is adopted.The nonlinear model of the mechatronic system is put in discrete time form for identification purpose.Based on the identification results, true fault can be obtained.Simulation studies are conducted to validate the proposed fault diagnosis method.

Modeling and fault diagnosis of nonlinear mechatronic system via BG
The BG is a pictorial representation of systems which is based on energy conservation law.The bond in BG refers to the half arrow line with effort and flow as energy variables [11].The causality of BG links model variables in a clear and systematic way which paves the way for efficient FDI analysis.Based on the BG model of the system, a set of dynamic constraints, called ARRs, are derived from the causal path of the graphic model.Numerical evaluations of ARRs lead to residuals which indicate the consistency of the ARR.A residual will exceed the threshold if it is sensitive to the occurred fault.
The mechatronic system consists of a DC motor, reducer and a load.The diagnostic bond graph (DBG) of the mechatronic system is shown in Figure 1, where all storage elements are put in derivative causality to avoid the initial condition, except the C element still remains in integral causality since the initial condition is known.To determine the ARRs, two sensor attached junctions 1 1 and 1 2 are considered.
The second ARR can be derived from junction 1 2 as The unknown variables in ( 2) and ( 3) can be eliminated by covering the causal paths from sensors to unknown variables.Thus Combine ( 2) ~ ( 4), two structurally independent ARRs can be expressed as Based on the two ARRs, the fault signature matrix which represents the cause and effect relation between faults and residuals can be established in Table 1.
Fault detection is carried out by evaluating the consistency of residuals of the ARRs in ( 5) and ( 6) in an online manner.A coherence vector (CV) is used to indicate the health condition of the mechatronic system.In other words, if the CV is a nonzero vector, the system is faulty, and the system is fault free when CV is zero.After a fault is detected, the obtained CV is compared with the row of the FSM in Table 1 to find the set of fault candidates which can account for the fault symptom.Since no fault in the FSM is isolable which indicates more than one fault can lead to the observed CV, fault identification is required to refine the set of fault candidates to determine the true fault.

Fault identification using bayesian method
The BG model of the mechatronic system is nonlinear and linear parameter estimation methods might not be used.
In this article, a Bayesian estimation method called particle filter (PF) is adopted to identify the unknown fault parameters.PF is a sequential Monte Carlo method which is based on probability theory [13].This method aims to approximate the posterior probability density function (PDF) of the state using a set of samples (or particles) with associated weights.The main advantage of PF is that it can be applied to nonlinear systems with non-Gaussian noises.
Since the PF operates in discrete time domain, the mechatronic system model needs to be put in discrete form as follows + where to be the state vector, After fault isolation, the set of fault candidates, denoted as vector θ , can be obtained.In order to realize parameters estimation under the PF framework, the state of the system is augmented as As a result, the PF can be used for joint state and unknown parameters estimation [14].The main purpose of PF is to represent the posterior PDF of the augmented state z as where w is the particle weight, and δ is the Dirac delta function.
The particle weights are put in a recursive form as [15] If the importance density function is chosen as the transitional prior , the weights update process can be rewritten as The posterior distribution p z y can be represented by resampled particles from the systematic resampling method where ˆi k z is the particle after resampling.

Simulation study
To investigate the effectiveness of the proposed fault identification method, simulation experiment is conducted.The physical parameter values of the mechatronic system in simulation are set as: .This set of fault candidates is used for the state augmentation of PF for the purpose of joint state and unknown parameters estimation.In PF, the particle number is 0 =1000 N .The estimation is carried out during the time interval [2500,2700].Figure 3 shows the estimation states versus the sensor measurements.It is not hard to find that the PF can track the system states in a smooth way based on the available observations.value which means a fault condition.The mean estimate =0.5187Nms/r ad m f which matches the designed one.As a result, the true fault in m f is found and estimated by the PF algorithm.Figure 7 shows the distribution of the estimated parameters where the central part with more particle number represents the approximated values of PF estimation.From the simulation results, it is concluded that the fault estimation method can accurately find the true fault which leads to the observed CV.

Conclusion
In this work, a PF based fault identification method is developed for nonlinear mechatronic system with backlash.The fault detection is realized by online evaluation of ARRs.After a nonzero CV is detected, the set of fault candidates can be established based on the FSM.In order to further refine the set of fault candidates, a Bayesian based fault estimation method is developed where the PF is adopted for joint state and unknown parameters estimation.Simulation results validate the proposed methodology.Future works will be devoted to the implementations of the proposed fault identification method in the real test bed including nonlinear friction and backlash.
The motor is modeled by friction m R and inertia m J with input torque T .The friction m R is a nonlinear function which includes both Coulomb friction mc F and viscous friction m f .The C element represents the stiffness 0 K of the transmission shaft.The motor reducer is modeled by the TF element in BG with parameter N .The load part is represented by friction s R with parameters sc F and s f and load inertia s J .There are two incremental encoders representing sensors to measure the positions whose derivations are modeled by BG flow sensors D :f m θ  and D : f s θ  .The modulated effort sources m d and s d model the disturbance torques caused by the backlash atthe input and output reducer shafts[12].Thus, j and γ are constants.

Figure 1 .
Figure 1.Diagnostic bond graph model of the mechatronic system. 0 in motor mechanical part is introduced at time step 2500 k = (i.e., 50s t =) in which the parameter m f is changed abruptly from its nominal value to faulty one 0.5Nms/rad .The residual responses are illustrated in Figure2, where the dot lines are thresholds with value 0.2.It is observed that the CV=[1 0] after 50s due to the friction fault.After comparing the observed CV with the FSM, a set of fault candidates can be obtained =[ ]

Figure 4 ,F 2 ARRFigure 2 .
Figure 4, 5 and 6 demonstrate the PF estimates of unknown parameters in θ where the dashed lines are 95% confidence interval.It is found that the mean estimates