Detection and Isolation of Simultaneous Additive and Parametric Faults in Nonlinear Stochastic Dynamical Systems

This paper presents a new fault detection and isolation scheme for dealing with simultaneous additive and parametric faults. The new design integrates a system for additive fault detection based on Castillo and Zufiria, 2009 and a new parametric fault detection and isolation scheme inspired in Münz and Zufiria, 2008 . It is shown that the so far existing schemes do not behave correctly when both additive and parametric faults occur simultaneously; to solve the problem a new integrated scheme is proposed. Computer simulation results are presented to confirm the theoretical studies.


Introduction
Motivated by the importance of safety in modern automated systems, fault detection and isolation schemes have received an increasing attention in the last two decades 1-4 .As opposed to costly hardware redundancy approaches, information redundancy schemes make use of data processing and system modelling paradigms, leading to either data-driven or model-based approaches.Among model-based fault diagnosis schemes, the FDI Fault Detection and Isolation techniques of the control community make use of explicit analytical models for redundancy checking 5 .
The FDI analytical tools employed up to now can be classified into two main categories.On the one hand, stochastic discrete-time model-based schemes inherited from the signal estimation and linear control fields have successfully combined statistical schemes with geometrical tools in the design and characterization of detection algorithms for linear systems 1-3, 6 .Nevertheless, these schemes have limited applicability since many real-world applications are grounded on the use of nonlinear models.On the

Problem Statement
We consider the following class of nonlinear stochastic dynamical systems: ẋ t E n x t e n f x t , u t , ϑ 0 , t η t s t − T 0 φ t , y t h x t , u t , t , x 0 x 0 , 2.1 with which models, among other cases, any nth order nonlinear scalar system.Here, x t ∈ R n is the system state, which has known initial value x 0 ∈ R n ; u t ∈ R p is the control input; the known function , which accordingly satisfies the Lipschitz condition, also satisfies that for all x ∈ R n it holds f x, u, ϑ, t ≤ C 1 x , for some constant L and C, so that existence and uniqueness of solutions are guaranteed; f represents the dynamics of the nominal model and has some parameters represented by ϑ 0 ∈ R m ; the random vector η : R → R, which gathers external disturbances and modelling errors, corresponds to a stochastic process of white Gaussian noise with autocorrelation function R η t 1 , t 2 σ η δ t 1 − t 2 and noise intensity given by σ η .y t ∈ R l is the measurable output, and the nonlinear mapping h : R n × R p × R → R l can represent different output availability situations.
We assume that the pair f, h allows the construction of an observer that provides x as an accurate estimate of x, that is, sample-wise x − x ≤ x ; high gain observers 34, 35 and Lipschitz observers 36 have been successfully employed for this purpose.This paper mainly focused on the construction and the analysis of the so-called residual to be explained in the following section and addresses its estimation, the statistics of the estimator as well as the detectability and isolability conditions based on these statistics; hence, to simplify such exposition, an exact reconstruction of the state x will be considered by assuming for the remainder of the paper that x 0 i.e., x x , which is a standard assumption for most nonlinear FDI schemes, as discussed in Section 1.
Finally, the fault function φ : R → R can represent an unknown additive fault and/or a change in the parameters of the nominal part of the system, namely, φ t φ a t φ p t φ a t φ p x t , u t , ϑ 0 , ϑ 1 , t φ a t f x t , u t , ϑ 1 , t − f x t , u t , ϑ 0 , t .

2.3
Note that the possible simultaneous occurrence of both types of faults, generating complex φ t profiles, can make very difficult to unravel the fault origin.The unit step function s t − T 0 is determined by T 0 , the instant of time when the fault occurs.Note also that neither the postfailure parameter vector ϑ 1 nor the time T 0 is known.

Residual Construction
Generally speaking, a residual is any variable whose behavior changes significantly when a fault occurs in the system.In this paper context, a valid residual will be a random variable or stochastic process whose statistical properties do change after a fault.
Under the assumption of full-state availability we can create a new state variable x c t obtained from the following consistency equation: where x c t ∈ R is the consistency checking state variable, and λ > 0 is a design constant.Note that this equation makes use of the value of x n , the n-th component of state variable x, in contrast to the estimated values usually employed in the design of observers.Subtracting 2.4 from system 2.1 we get a new variable t x n t − x c t which depends on the model error and whose evolution is described by the following equation: The solution to this differential equation is where the model error t changes significantly after the occurrence of the fault t > T 0 .Due to this property, the variable t has usually been utilized as the fundamental signal to construct valid residuals for detecting single faults.The algorithms for fault detection and isolation analyze the signal t by studying its statistical properties and its similarity with other reference signals.We will see that, when simultaneous faults do occur, t requires a more elaborated processing due to its potentially complex evolution.

Single Fault Detection Schemes
In this section, some existing schemes for the detection of single faults either additive or parametric are illustrated.The exposition is aimed to highlight those analytical aspects which will become relevant when designing the new improved scheme to be presented in Section 4.

The Single Additive Fault Case
The scheme in 24 analyzes the residual when φ t φ a t and detects the additive faults under, roughly speaking, the unique condition that E φ t > > 0 or alternatively, E φ t < < 0 for all t > T 0 see 24 for details .In addition, isolation schemes can be implemented assuming some conditions on the set of possible additive faults 28 .
In general, these existing detection schemes will not be critically affected by the occurrence of a simultaneous parametric fault.Hence, we will see that the existing algorithms can be directly integrated into the new scheme proposed in Section 4.

Analysis of the Single Parametric Fault Case
Under the assumption that a single parametric fault occurs, that is, φ t φ p t , this section presents the main results from 25 , needed for the posterior analysis of the simultaneous fault case.

Characterization of the Fault Function
The scheme in 25 constructs a residual based on the signal φ p , using also the a priori knowledge about the fault function φ p t .The knowledge of such residual is limited due the unknown value of parameter ϑ 1 as well as the unknown instant of time T 0 .
In 25 , a finite set of fault classes Θ is defined, and it is assumed that any faulty parameter vector ϑ 1 belongs to one and just one of those classes.Furthermore, there exists a known function ϕ x, u, ϑ 0 , Δϑ, t such that where Δϑ ∈ R m is a known vector specific of the fault class, and k ∈ R is an unknown constant that depends on which particular faulty parameter of the class has occurred.Note also that since the profile of ϕ x, u, ϑ 0 , Δϑ, t depends on x t it is also affected by T 0 .This last dependence can be minimized by assuming that T 0 is large enough so that the system evolves within or nearby its ω-limit set.Thus, a set of possible fault classes can be defined, and the fault function φ p t will be approximately known for each ϑ 1 except for a multiplicative constant k.
The fact that T 0 is unknown implies another limitation when computing the integrals; this fact leads to an approximation by defining 3.2 so that, for small parameter variations, the second summand of 2.6 satisfies φ p t kϕ LP t − e −λ t−T 0 kϕ LP T 0 , where lim t → ∞ φ p t − kϕ LP t 0, meaning that φ p t ∼ kϕ LP t .

3.3
As it will be shown in Section 4, alternative reference signals can be constructed to reduce the error associated with this approximation 3.3 .

Residual Generation
After dealing with the unknown quantities, one can define the residual signal 25 : This residual, called moving angle, allows to formulate hypothesis test on it: where H 0 t η t and H 1 t η t φ p t .The moving angle changes significantly when there is a change in the system conditions from H 0 no fault to H 1 fault , a behavior that corresponds to a good residual.In a practical application one can only calculate an estimation of the integral in 3.4 so the moving angle estimation is cos α ϕ t , ϕ LP T,S ϕ LP T,S T,S .

3.7
Note that such estimator is defined by a quotient of the form g X/ √ Y , where X and Y are random variables; hence its expected value and variance can be computed upon 37

3.8
Applying this result to 3.7 , with ϕ LP T,S deterministic and X ϕ LP , T,S , Y 2 T,S , we obtain the expressions shown in Table 1 where Var Then, the resulting expressions for the estimator moments under the different hypotheses are

3.9
Based on these deterministic quantities we can construct γ confidence intervals of cos α ϕ t under both hypotheses H 0 and H 1 :

Mathematical Problems in Engineering
These confidence intervals ensure that the estimator will take values on each one of them with probability γ when the system is operating under the corresponding hypotheses.The detection scheme is triggered when the residual estimator enters the interval corresponding to H 1 see 25 for details .

The New Simultaneous Fault Detection and Isolation Scheme
When simultaneous faults occur, they may disguise each other's effects, increasing the difficulty of their detection; in such case, existing schemes for a separate fault detection may not work.In this Section, the simultaneous fault case is considered, and a new scheme for addressing this problem is proposed.The proposed detection scheme integrates improved versions of the algorithms proposed in 24 for additive faults and the one presented in 25 for parametric ones.

Analysis of the Simultaneous Fault Situation
As mentioned in Section

Additive Fault Detection Scheme
As mentioned above, the scheme in 24 analyzes the residual t and detects the additive faults under, roughly speaking, the unique condition that E φ t > > 0 or alternatively, E φ t < < 0 for all t > T 0 .In general, when φ a t satisfies the detectability condition, such that |E φ a t | |φ a | > , it is very unlikely that a parametric fault would generate a significant value of E φ p t that would precisely compensate and mask the additive term.In practice, the errors caused by initially small parameter variations imply that E φ p t ≈ 0, so that E φ t ≈ E φ a t , and the additive fault detection scheme will not be affected by such simultaneous parametric faults.
The main challenge then becomes to detect and isolate the parametric faults in such working environment E φ p t ≈ 0 .Interestingly, the profile of φ p t may allow for the fault detection and isolation, as shown below.

Parametric Fault Residuals for Simultaneous Case
As it is shown below, the parametric fault detection and isolation scheme presented in 25 are likely to be disturbed by the occurrence of simultaneous additive faults disguising parametric faults.In the following section we modify such scheme in order to reduce its sensitivity to these additive faults.
Assuming that the extra summand asymptotically behaves

4.3
So the model error under hypothesis H sim 1 tends to Hence, the moving angle takes the following asymptotic expression:

4.5
We observe that the additive fault affects both the numerator and the denominator of the moving angle.Once again this quantity has to be estimated, and its statistics are calculated.The components of the expressions of the expected value and the variance are shown in single parametric fault ; fortunately, some approximations can be made.In fact, under the hypothesis of small parameter variation E φ p t ≈ 0 , we have that even if ϕ LP t might oscillate, the ergodicity assumption justifies that ϕ LP evolves in a smaller range so that the corresponding terms can be neglected; hence, the most significant term is k 2 φ a , due to the additive fault, so that Table 2 can be simplified to Table 3.  Hypothesis Using this approximation, the expected value and variance of the estimator under hypothesis H sim 1 are

4.6
Both these quantities are considerably different compared to their counterparts under hypothesis H p 1 .Thus, the γ confidence interval under hypotheses H sim

Improvements on the Detection Scheme
The scheme presented in 25 has been improved in two directions.On the one hand, the reference signals have been obtained in a way that reduces the error associated with the approximation in 3.3 ; on the other hand, the influence of the additive error has been minimized via an appropriate filtering of t .

Improving the Reference Signal
The reference signal proposed in 25 is computed integrated from the 0 initial time, since the real value of T 0 is unknown.Nevertheless, it is possible to define a new reference signal where T 0 can be dynamically chosen, for instance, as T 0 t−T the lower bound of the interval t − T, t where the moving angle is defined.The value of T 0 is likely to be closer to T 0 than the 0 value.Hence, and, if the faults occur in the interval t − T, t T 0 , then |T 0 − T 0 | < T, and we obtain the bound so that the term e −λ t−T 0 kϕ LP,T 0 T 0 will be small.This means that the new approximation will have, in general, a smaller error than 3.3 ; this fact justifies the good performance of the newly proposed reference signals.

Eliminating the Additive Term Influence
The analysis in Section 4.3 shows that the detection scheme presented in 25 does not work correctly under hypothesis H sim 1 because of the influence of k 2 φ a .Note that equality 4.4 demonstrates that asymptotically k φ a is a constant term added to t .Thus, one way to vanish its effect is to low pass filter t .Let t be a filtered version of t : In this case, under hypothesis H sim 1 we have Since E H 0 t 0, the ergodicity assumption allows us to consider H 0 t ≈ 0. Hence, the statistics of the estimator of the moving angle are now calculated using the elements of Table 4, where

4.14
Comparing this table to Table 2, one can see that the term k φ a does not show up in any term.Finally, under the usual conditions mentioned in Section 4.2 E φ p t ≈ 0, and ergodicity , we have that ϕ LP ≈ 0, so that the terms involving ϕ LP i.e., ϕ LP , ϕ LP , ϕ LP , ϕ LP , V ϕ 2 , and V ϕϕ are negligible.Hence, the expected value and the variance of the moving angle satisfy meaning that the new detection and isolation procedures proposed here can be successfully applied.
It is worth mentioning that the new resulting scheme is applicable to simultaneous faults composed by additive and parametric faults that satisfy similar detectability and isolability conditions to the ones stated in 24, 25 , respectively.Concerning detection and isolation times, although the filtering process may slightly delay the responses, in general the detection and isolation times are similar to the original schemes times, as shown in the following example.
Finally, note that such detection and isolation times do have a clear impact on the fault accommodation strategy to be applied 13 .

Simulation Setup
Here the correct behavior of the work presented in the previous sections is illustrated with the Van der Pol oscillator VdPO ÿ 2ωζ μy 2 − 1 ẏ ω 2 y 0 via simulations with Matlab Simulink.The election of this system has also been made in other works on deterministic system fault diagnosis 38 as well as in the study of stochastic systems 24, 25 as it is the case here.
The VdPO describes an LC oscillator with nonlinear resistive element such as a tunnel diode.The output y represents the voltage at the inductor, whereas ẏ is the current through this inductor.In this simulation, it is considered that all electrical elements are not ideal e.g., due to change of temperature but stochastically varying.Consequently, we obtain the following state space representation of the VdPO: where η i , i 1, 2 are normalized white Gaussian noise with zero mean and auto correlation R η i t 1 , t 2 δ t 1 − t 2 .We assume that both states are measurable as indicated in Section 2. The system function is f x, u, θ, t 2ωζ 1 − μx 2 1 x 2 − ω 2 x 1 with θ ω, ζ, μ T .This system presents a nice feature: it is linear in ζ and μ and nonlinear in ω.Hence, fault functions that are both linear and nonlinear in Δθ can be investigated; in this example we focus on the detection of faults on the nonlinear parameter, ω.Moreover, the oscillator runs on stable limit cycles for ω, ζ, μ > 0, which do change slightly for small parameter changes.Despite this fact, the detection scheme presented in 25 successfully detects these single faults.
A fault class is defined for the ω parameter whose corresponding representative is Note that since f is nonlinear in ω, ϕ ω is only a linearization.The consistency equation is:

5.3
The simulation parameters are presented in Table 5.Note that only small changes in the ω parameter are to be detected; in this example it will be a 25% of the maximum change in ω 0 .
The value considered for the additive fault φ a t is also small.λ has been chosen rather big in order to reduce R , and T has been chosen such that several periods of the oscillator output are included in the integration range.

Simulation Results
Figure 1 gives an overview of the behavior of the system, the representative, its mean, and the additive fault before and after the simultaneous fault these quantities are not affected by the presence of the filter .The state space values do not change significantly due to the fault.Yet, the error function suffers a significant change when the fault occurs.It can be observed that the mean of the representative function is one order of magnitude less than such representative function: this result matches the fact that this mean has been neglected in the theoretical analysis.Note that these representative values are much smaller than the abrupt additive fault function represented in the last plot; this fact supports the validity of the new scheme.On the other hand, Figure 2 shows the behavior of the estimator for the existing scheme top figure and for the new proposed scheme bottom figure .It is clear that when a simultaneous fault occurs and the old detector/isolator is employed, the estimators do change due to the parametric fault but not enough to get out the upper boundary of the decision region grey line in the figure .This undesirable situation is not encountered when the new detector/isolator is applied, as it can be seen in the bottom figure; there, the additive fault does not disguise the effect of the parametric one, and the estimators do change beyond the boundary of the decision region, demonstrating the improved behavior of the new proposed scheme.

Conclusions
A new scheme for the detection and isolation of simultaneous additive and parametric faults in nonlinear stochastic dynamical systems has been presented.A theoretical analysis has been developed to highlight the limitations of the existing detection/isolation schemes when such types of simultaneous faults occur.Based on the analytical studies, a new detector/isolator has been designed which integrates improved versions of the existing schemes.Comparative simulations have supported the theoretical results by showing the good performance of the new detector/isolator as opposed to the previously existing schemes.

1 / Δ cos α ϕ H p 1 .
This fact causes several detection problems, since the detection scheme checks if the estimator belongs to Δ cos α ϕ H p 1 to trigger the alarm; however, under the H sim 1 hypothesis it will belong to Δ cos α ϕ H sim 1 with probability γ.

Figure 1 :Figure 2 :
Figure 1: Behavior of the states x 1 , x 2 , the error function , the representative ϕ LP ω , its mean ϕ LP ω, and the additive fault function φ a t of a simulation with a parameter change in ω at T 0 15.
1, simultaneous faults are likely to occur in real-world systems.Nevertheless, most standard FDI schemes assume that only one single fault occurs at a time.In some specific cases, separation mechanisms have been developed 9 , which are not directly applicable in general.Here, we analyze the schemes presented in 24, 25 under simultaneous additive and parametric faults.If a parametric and an additive fault occur at the same time we label this hypothesis of simultaneous faults as H sim

Table 1 :
Statistics of magnitudes X and Y under the H 0 and H 1 hypotheses.

Table 2
LP τ 1 R τ 1 , τ 2 dτ 1 dτ 2 .When compared to Table1, several new additive terms show up in Table2.This fact limits the performance of the estimator under hypotheses H sim

Table 2 :
Statistics of magnitudes X and Y under the H sim

Table 3 :
Simplified statistics of magnitudes X and Y under the H sim

Table 4 :
Statistics of the magnitudes X and Y under the H sim 1 hypotheses after the filtering process.