Multiple fault isolation

Samantaray, A. K., Ghoshal, S. K. (2007). Sensitivity bond graph approach to multiple fault isolation through parameter estimation. Proceedings of the Institution ofMechanical Engineers Part I Journal of Systems and Control Engineering, 221(4), 577-587. 

Multiple Fault Isolation by Least Squares ARR Residuals Minimisation... 

If component parameters in a system mode share a fault signature in the FSM then a fault can be detected but not isolated by simple inspection of the FSM. In case online fault detection provides a coherence vector that matches with more than one row in the FSM in a system mode, the result is a set of potential fault candidates. One way to identify multiple faults is to perform parameter estimation by Gauss-Newton least squares output error minimisation. Bond graph modelling can support this approach to multiple fault isolation by providing ARRs. Their residuals are used in the functional to be minimised. This has been discussed in Chap. 6. 

The models used can be either fixed or adaptive and parametric or non-parametric models. These methods have different performances depending on the kind of fault to be treated i.e., additive or multiplicative faults). Analytical model-based approaches require knowledge to be expressed in terms of input-output models or first principles quantitative models based on mass and energy balance equations. These methodologies give a consistent base to perform fault detection and isolation. The cost of these advantages relies on the modeling and computational efforts and on the restriction that one places on the class of acceptable models. 

The approach is able to detect, isolate, and identify a wide class of failures of sensors, actuators, and process. Sufficient conditions for residuals convergence, detectability, and isolability of faults have been derived. In detail, detection is guaranteed under mild assumptions on the magnitude of model uncertainties and disturbances, whereas correct isolation may not be achieved if multiple faults of the same nature (i.e., sensor faults and process/actuator faults) occur during the same batch operation. 

The parameter estimation presented in the previous section is based on a least squares minimisation of the errors between measured system outputs and outputs of a system model evaluated by using estimated parameter values. If the real system is replaced by a model in preferred integral causality, measured outputs can be obtained by solving the model equations for given initial conditions and can be used for offline parameter estimation in order to isolate multiple faults deliberately introduced into the system model. In real-time FDI, initial conditions are either not known or difficult to obtain. Therefore, in online parameter estimation, they have to be considered as additional unknowns that are to be estimated. 

Alternatively, multiple faults may be isolated by least squares minimisation of ARR residuals. The latter are indicators for the errors between measurements from a faulty system and outputs of a model computed by using estimated parameters. ARRs obtained from a DBG do not depend on initial conditions but use derivatives of measurements with respect to time which entails the drawback that differentiation carried out in discrete time amplifies noise if not properly filtered. 

Multiple parameter fault isolation by means of minimisation of least squares of ARR residuals needs residual parameter sensitivity functions if a gradient search based method is used. If ARRs can be derived in closed symbolic form from a bond graph, their analytical expressions can be used in the formulation of the least squares cost function and can be differentiated with respect to the vector of targeted parameters either numerically or residuals as functions of the targeted parameters can be differentiated symbolically. If ARRs are not available in symbolic form, they can be numerically computed by solving the equations of a DBG. 

A cost function to be minimised in an iterative parameter estimation procedure may be formulated by using either differences between outputs from a real system and computed outputs from a model or by means of ARR residuals. As output errors, as well as ARR residuals are generally nonlinear functions of the component parameters, multiple fault parameter isolation becomes a well-known nonlinear least squares problem. For real-time FDI, ARR residuals obtained from a DBG have the advantage that they make the parameter estimation independent of any initial conditions of the process that are hardly known and will have to be estimated along with component parameters. In off-line simulation, the real system may be replaced by a behavioural model. Measured data is then generated by assuming realistic consistent initial conditions and by solving the equations of the behavioural model. 

For these reasons, in the following, it is assumed that no parametric faults happen during system mode identification and that an initial system mode is known. System mode identification in the presence of faults is more difficult. In [4], Arogeti et al. present an advanced method for this more general case that categorises ARRs into different types and provides a refined set of fault candidates to the fault parameter estimation procedure. Multiple fault detection, isolation and identification for hybrid systems with no available information on the nature of faults (abrupt or incipient) and on system mode changes has been recently addressed in [5],... 

In case multiple potential component faults may be the cause for the start of an abnormal behaviour of an ARR residual, then parameter estimation as part of fault isolation identifies the parameters that are going to deviate from their nominal values. For each of them, failure prognosis wUl have to identify a degradation model as a first step. This identification of degradation models can be performed in parallel. As a result, multiple faults may have different degradation profiles. 

This way, the identified degradation model is directly used to predict the RUL of the faulty component. If multiple faults have been isolated, if for each of them a degradation model has been identified, and if for each fault a failure alarm threshold has been set, then the latter ones can be substituted into the solutions of the degradation models to obtain a RUL for each faulty component. 

On the other hand, when more than one fault can influence the system at the same time, advanced diagnostic methods are used. These methods are based on parameter estimation. Sensitivity bond graph formulation [12] allows real-time parameter estimation and thus it is possible not only to isolate multiple faults but also to quantify the fault severities. Parameter estimation in single fault [2] or multiple fault scenarios [12] are essential steps to be performed before fault accommodation. The parameter estimation scheme also gives the temporal evolution of system parameters. Thus, it is possible to identify and quantify different kinds of fault occurrences. A progressive fault shows gradual drift in estimated parameter values and intermittent fault shows spikes in the estimated parameter values. The advances made in the field of control theory have made it possible to develop state and parameter estimators for various classes of nonlinear systems. Analytical redundancy relations may also be used in optimization loop for parameter estimation because it avoids the need for state estimation. Interested readers may see Ref. [3] for further details and some solved examples. 

In the following, it is shown that multiple process/actuator faults (multiple sensor faults) can be detected but not correctly isolated and identified. 

Partitioning is a means for providing isolation between components to contain and/or isolate faults. Partitioning between components may be achieved within the system architecturally by allocating unique target hardware and hardware resources to each component. Alternatively, partitioning may be achieved within the component architecture to allow multiple software or hardware items to run within the same hardware platform. Partitioning should ensure ... 

Multiple failures to ground may occur in floating systems. While a ground fault on an isolated system does not cause an exit, if the fault continues, increases the possibility of occurrence of a second fault in another phase. That is because the groimd insulation of the... 

Chapter 6 addresses the case in which several faults may occur simultaneously. As long as some of them do not cancel each other, they may be detected. Isolation, however, is a problem. One way to isolate multiple simultaneous faults is to formulate an optimisation problem and to apply parameter estimation on a set of ARRs. This is illustrated by a small example and the use of Scilab function optim() [6]. 

The method is particulary suited for the detection of incipient faults as it can pinpoint small parameter changes. In Chap. 6, least squares parameter estimation is considered with to regard to the isolation of multiple parametric faults. 

This also means that faults do not cancel each other in their effect on an ARR residual. Given a single fault hypothesis, the faulty component is identified by comparing the coherence vector against the rows of the FSM, i.e. the component fault signatures. If this comparison results in a match, the faulty component is isolated. However, there may be no match, or more than one match may be obtained. That is, the faulty component cannot be isolated. In the case of multiple simultaneous faults, FDI can be performed e.g. by means of parameter estimation as is discussed in Chap. 6. 

In online fault detection, ARR residuals are close to zero for a healthy system. Generally, they are not identical to zero for various reasons such as modelling uncertainties, uncertain parameters, noise, or numerical inaccuracies. For correct online fault detection it is important that true faults are reliably detected and false alarms are avoided. To that end, residuals are fed into a fault decision procedure. The result is a coherence vector. If this vector is a null vector, then the system is healthy, no fault has happened. If some of its entries are non-zero, then the coherence vector is compared with the rows of the structural FSM. Given a single fault hypothesis, the fault is isolated if there is a match with one row of the FSM. If there is more than one match then the detected fault cannot be isolated. Also, if the number of fault candidates exceeds the number of sensors, not all faults can be isolated. Isolation of multiple simultaneous faults by means of parameter estimation is considered in Chap. 6. 

Isolation of Multiple Parametric Faults from a Hybrid Model... 

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