Fault Detection and Isolation in Dynamic Systems using Statistical Local Approach and Hybrid Least Squares Algorithm

Belkacem ATHAMENA, Zina HOUHAMDI and Mohammed MUHAIRAT Software Engineering Department, Faculty of Science and Information Technology, Al-Zaytoonah University, Jordan Abstract: A fault detection and isolation (FDI) scheme for dynamic system proposed. This study deals with the design of discrete-time linear system using delta operator approach and the hybrid least squares (HLS) algorithm. A third residual generation based on statistical local approach and the derivative of the normalized residual on a small temporal window investigated. This new technique meets the desired FDI performance specifications by increasing the faults magnitude and decreasing the noise effects. Some simulation results were provided to evaluate the design.


INTRODUCTION
The increasing complexity of modern automated processes and the increasing demands for quality, reliability, availability, safety and cost efficiency require better safety management and supervision. Generally, the function of a supervisory control system is to detect and isolate faults in the system. The design of many modern fault detection and diagnosis system is based on the mathematical model of the plants. The model-based technique is where the actual behavior of the system is compared with the nominal fault free model that is driven by the same input. The result of this comparison leads to class of signal called residual. If the residual is not zero, then the system has faults, otherwise the system is normal.
In general, fault detection and diagnosis is a board and active area of research. There are a large volume of papers that deal with this subject, see, e.g., [1] and references therein. In many applications the problem of FDI is a crucial issue that has been theoretically and experimentally investigated with different types of approaches [2] .
A novel residual generation method is proposed in this paper, whereas the residual generation based on the derivation of the normalized residual in a small temporal window is developed. The proposed residual offers favorable FDI properties, in which decreasing the noise effects and increasing the faults magnitude, therefore the FDI are improved especially in the case of the small change in the physical parameters of the system.
The purpose of the paper is to describe and analyze the FDI algorithm, based on the HLS parameter estimation technique using delta operator and the statistical local approach. The delta operator, which proved to be a convenient tool for examining the asymptotic behavior of discrete-time models of continuous-time systems as the sampling period converges to zero offers several advantages as compared to the common forward shift q operator often leading to ill-conditioned processes [3] . It has been observed that for sufficiently small sampling periods the δ operator algorithms not only are much less sensitive to arithmetic round-off errors than their counterparts q -domain algorithms, but also ensure that the delta representation of a discrete-time system will converge to the corresponding continuous-time system as the sampling period tends to zero [4] . The main advantage of the statistical local approach is its ability in assessing the level of significance of discrepancies with respect to uncertainties. This approach has a wide scope of interest, since it can encompass all the types of FDI problems, for sensors, actuators and components faults.
time systems in a finite precision arithmetic can be found in [3] .
Consider a single-input single-output linear continuous-time system, Let T be the sample period, the delta operator defined by [1,11], can serve as an alternative, which is known as mediation between the differential and the shift operators. It has been shown that the delta operator offers advantages over the shift operator, in terms of numerical robustness [4] . Using the delta operator (3), an alternative discretetime representation based on the delta operator of the previous model (1) is given by, conditions the bias, but also the convergence of the estimation [4] .
have unity leading coefficients only data up to time t is required on the right-side of (7), Now define a parameter vector for (8) as, . Then the linear regression model can be written as, is the regression vector. We obtain a linear model with regard to the parameters by a transformation of the original data to the filtered data, where an analogue relation to the equation (4).
For know orders and delay, the problem consists of estimating the parameter vector θ in the linear regression model (13) from the available data. In this paper, the HLS algorithm with forgetting factors λ has been chosen, because it best fits to the problem studied and because it is easy to implement. Estimation of the parameter θ may be found by minimizing the following sum of square of the equation error, However, the parameter vector 0 θ that satisfy , is not unique. To let the estimation unique, a parameter constraint 1 = θ θ T may be used. For the recursive estimation of the parameter, the HLS algorithm with adaptive forgetting factors ) (t λ is used which is given as follows [4,5,6] , where ) (t ε is prediction error defined as, (15) and the adaptive forgetting factors ) (t λ is defined as, The forgetting factor is used mainly to put higher weights on the more recent measurements so as to facilitate the convergence rate of the estimation of timevarying parameters. FDI SCHEME FDI problem: The problem of fault detection consists in making the decision on the presence or absence of faults in a monitored system. When no fault is present in the system, the system is said in its safe mode; otherwise it is in a faulty mode. Generally speaking, faults in a dynamic system are often associated with malfunction of system components, which are reflected as changes in the system parameters. This situation can be modeled as abrupt or slowly developing parameter changes. In the above model representation (13), this type of faults can be modeled as: where 0 θ (identified with data from the safe system) and θ ∆ (same dimension as 0 θ , but with an arbitrary direction and a small magnitude) represent the nominal system parameters and the fault-induced parameter changes, respectively, and f t is the fault occurrence time. The normalization N 1 is necessary in order to get a non-trivial limit distribution.
To model the process dynamics and to generate the residual parameter it can be assumed that the equivalent input-output representation of the model (13) described as, is a vector of differential polynomials in F u , F y and θ .

Residuals generation:
In the previous studies [7] , the parametric statistical approach distinguishes between two residuals [7] : primary residual, which is a function of the model parameter θ and the observations F u , F y , and normalized residual, which is the mean of the normalized sum of this primary residual. Our contribution in this study is the generation of a third residual based on the derivative of the normalized residual on a small temporal window.
Due to the modeling uncertainty and measurement errors, some stochastic assumption should be introduced in order to take them into consideration. For linear systems, noises are usually assumed to be additive in model equations. For the sake of simplicity, we assume that the stochastic model of the system is, where the noise F η is assumed be Gaussian.
For a system modeled by equation (11), assume that the parameter θ is locally identifiable at the nominal value 0 θ . Let θ be the actual parameter value for the system which generated the new data sample. The primary residual related to the identification of θ by minimizing the square of ( ) can be interpreted as an efficient score for model (18), , and is of interest for monitoring purposes. This is because changes in θ are reflected by changes in the mean value of ( ) , is given by [1,8] , Technical arguments for the N factor can be found in [2,9]. Let θ Ε be the expectation when the actual system parameter is θ , we know that [7] In this paper, we propose a third residual , which is estimated on the same temporal window N , Using the delta operator (section 2), the residual ) (θ N D can be expressed as, note that, the use of derivative of the means values on a small temporal window allows the filtering of noises, and at the same time, allows a fast resolution of all changes in residuals [10] . It is precisely this quantity that is subsequently evaluated in order to perform FDI. Under some conditions, the residual is asymptotically Gaussian distributed, and reflects a small fault by a change in its mean vector.
To decide whether 0 θ = θ holds true or not, or equivalently whether the residual ) (θ N D is significantly different from zero, requires the knowledge of the probability distribution of , which unfortunately is generally unknown. One manner to circumvent this difficulty is to assume close hypotheses, where vector θ ∆ is unknown but fixed. Note that for Matrix Σ captures the uncertainty in N D due to estimation errors: indeed the covariance matrix of the error in estimating 0 θ is this ) ( 0 θ Σ as well [7] .
in (23) is asymptotically Gaussian distributed with the same covariance matrix ) ( 0 θ Σ under both 0 H and 1 H . According to [7] , the residual ) (θ N D have the following properties, is the Jacobean matrix containing the sensitivities of the residual with respect to the model parameters, As seen in (26), a deviation θ ∆ in the system parameter θ is reflected into a change in the mean value of residual N D , which switches from zero to depend on neither the sample size N nor the fault vector θ ∆ in hypothesis 1 H . Thus they can be estimated prior to testing, using data on the safe system (exactly as the reference parameter 0 θ ).

Fault detection and detectability:
For deciding between 0 = θ ∆ and 0 ≠ θ ∆ , the decision can be taken based on the optimum test statistics τ , referred as the global test, is based on the generalized log-likelihood ratio. It can be shown [1,7] that the GLR test of 1 H against 0 H can be written as, (28) which should be compared to a threshold. If the incidence matrix M is an invertible matrix, then the global test τ can be reduces to, is asymptotically 2 χ -distributed, with a number of degrees of freedom equal to the dimension of θ . The limiting 2 χ -distribution is central under 0 H , and has the non-centrality parameter under 1 H equal to, (30) it is known [6] that the detection power of τ (the probability of successful detection for given probability of false alarm) is an increasing function of γ .
In (28), the dependence on 0 θ has been removed for simplicity. The only term which should be computed after data collection is residual N D in (24). The statistical properties of τ provide a theoretical guideline for the choice of a threshold ε λ found from a 2 χ -table, where ε is the false alarm rate specified by the users. If τ is found to be larger than the threshold value, then a change in parameter is detected. Therefore, the fault detection decision is performed by the rule, (34) which indicates that the magnitude of fault should be large enough to guarantee detectability.

Fault isolation and isolatability:
The problem of fault isolation poses a bigger challenge than fault detection. The fault isolation is performed only after the deliverance of a fault detection alarm. Once a change is detected from the model parameter, it may be necessary to isolate which or which set parameters have changed. The statistical approach to residual evaluation requires the knowledge of the statistical properties of the residuals. For this purpose, divide θ into, In some practice cases, we need to isolate the change in every individual parameter, which can be called complete isolation, in other cases; we just need to isolate the changes in some particular parameter subsets, which can be called partial isolation. In complete isolation cases, the test should be attribute on every individual parameter, or in other word, a θ ∆ is considered as a particular parameter. On other hand, in partial isolation cases, a θ ∆ is considered as a vector, which indicates a particular subset of parameters.
For the min-max test, the parameters in b θ are considered as nuisance and are statistically rejected. Let the Fisher information matrix according to, is the partial score in b θ .
In the case when θ ∆ = θ ∆ a (fault in all parameters), the sensitivity test a τ can be derived as, (44) and the sufficient condition for complete isolatability of the faults 0 ≠ θ ∆ i is expressed as,

MAGLEV SYSTEM
Process description: In this section, a design example will be presented to illustrate the design procedure of the proposed FDI methods. We introduced a MAGLEV system [11] to illustrate the validity of the proposed algorithm. The electromagnet-track configuration is illustrated on Fig.1 Finally, the open-loop transfer function of the MAGLEV system is expressed as, We can represent the MAGLEV system by the following input-output model, where, Hence, the incidence matrix is depicted in the following table,  Due to the nonlinear force-distance and forcecurrent characteristics, MAGLEV system is unstable in open-loop feedback of at least position through a leadlag compensator is needed to obtain stability, though feedback of vertical acceleration is commonly incorporated to gain adequate control over suspension characteristics [11]. Therefore, the open-loop poles in (48) suggest that at least one zero is needed if the system is to be stabilized by using the classical compensation techniques. The control law derived by the compensator may be expressed as, where p k , a k are the appropriate feedback gains, and ) (t c is the clearance measured with respect to the instantaneous guide-way height ) (t h , Fig.2.

Guide-way profile A ccelerom eter
(54) and, We can have the model in the following form, (57) and, can be generated by,    In order to let the reader appreciate the small parameter changes considered in our simulations, the low-pass filters can be applied to ) (t z to reduce noise and the effects of un-modeled dynamics. Many lowpass filtering techniques are available in the literature. Simple second-order Butterworth filters with cut-off frequencies at 0.1Hz have been used here to process the output, Fig.4.         ), then the fault in the model parameter 1 a is isolated but near perfect when the noise standard deviation increase. According to the incidence matrix, table 1, we can conclude that the fault in the physical parameter R is isolated.  For the same procedure, all faults are isolated, where: • Case 2: The fault in the model parameter 0 a is isolated, and according to the incidence matrix, the fault in the physical parameter w k is isolated. • Case 3: The fault in the model parameter 0 b is isolated, and according to the incidence matrix, the faults in the physical parameters L and z k are isolated. • Case 4: The fault in the model parameters 1 a and 0 a are isolated, and according to the incidence matrix, the faults in the physical parameters R , z k and i k are isolated. Fault isolatability test: The sufficient conditions for fault isolatability (42-43) results are shown in the following tables 12-13.  a is satisfied. Therefore, the condition test for fault isolatability for the physical parameter R is satisfied.

CONCLUSION
A new FDI approach has been proposed. The combination between delta operator and statistical local approach offers advantages to generate and develop a new residual. This FDI technique has been developed for on-line abrupt change in physical parameters system. The results clearly show that the residual generation and evaluation method proposed in this study, offers a great potential for detection and diagnosis of physical parameters fault. The perspectives of this work are situated in the combination with other FDI techniques and the application on nonlinear systems.