Analytical Redundancy Relations

Once a quantitative system model is available, different methods can be used for the generation of a fault indicator or a residual as a primary step in FDI. These methods are either based on observers or a bank of observers [26, 27], on parity relations [28, 29], or on analytical redundancy relations [30, 31], or on parameter estimation [32, 33]. In case a fault has occurred in the system, the time evolution of some residuals must deviate distinctly from that during normal healthy system operation. 

Borutzky, W. (2011). Analytical redundancy relations from bond graphs of hybrid system models. In A. Bruzzone, G. Dauphin-Tanguy, S. Junco M. A. Piera (Eds.), Proceedings of the 5th International Conference on Integrated Modeling and Analysis in Applied Control and Automation (IMAACA 2011) (pp. 43-49). Rome. 

The representation of a hybrid system model by means of a bond graph with system mode independent causalities has the advantage that a unique set of equations can be derived from the bond graph that holds for all system modes. Discrete switch state variables in these equations account for the system modes. In this chapter, this bond graph representation is used to derive analytical redundancy relations (ARRs) from the bond graph. The result of their numerical evaluation called residuals can serve as fault indicator. Analysis of the structure of ARRs reveals which system components, sensors, actuators or controllers contribute to a residual if faults in these devices happen. This information is usually expressed in a so-called structural fault signature matrix (FSM). As ARRs derived from the bond graph of a hybrid system model contain discrete switch state variables, the entries in a FSM are mode dependent. Moreover, the FSM is used to decide if a fault has occurred and whether it can unequivocally be attributed to a component. Finally, the chapter discusses the numerical computation of ARRs. 

Analytical redundancy relations (ARRs) Are mathematical equations that relate known system inputs, known parameters and quantities obtained by measurements from a real system. Their evaluation results in so-called ARR residuals that are identical to zero or close to zero in narrow limits as long as the system is healthy. Residuals that deviate distinguishably from zero serve as fault indicators. If nonlinear constitutive element equations do not permit to eliminate unknown variables in a candidate for an ARR in closed symbolic form then residuals are given implicitly and can be determined by numerically solving a set of equations. As inputs into ARRs may be time derivatives of measured quantities, measurement noise is to be filtered appropriately. The differentiation is carried out in discrete time. 

Moreover, LFT bond graphs can also support robust fault detection and isolation (FDI) of systems with uncertain parameters. The decomposition of bond graph elements leads to a derivation of analytical redundancy relations (ARRs) composed of a nominal part representing their residuals and an uncertain part due to parameter... 

Diagnosis of uncertain systems has been the subject of several recent research works [1-6]. This interest is reflected by the fact that physical systems are complex and non-stationary and require more security and performance. The bond graph model in LFT form allows the generation of analytical redundancy relations (ARRs) composed of two completely separated parts a nominal part, which represents the residuals, and an uncertain part which serves for both the calculation of adaptive thresholds and sensitivity analysis. 

The generation of robust analytical redundancy relations from a bond graph model proper and observable is summarized by the following steps ... 

Abstract Incremental true bond graphs are used for a matrix-based determination of first-order parameter sensitivities of transfer functions, of residuals of analytical redundancy relations, and of the transfer matrix of the inverse model of a linear multiple-input-multiple-output system given that the latter exists. Existing software can be used for this approach for the derivation of equations from a bond graph and from its associated incremental bond graph and for building the necessary matrices in symbolic form. Parameter sensitivities of transfer functions are obtained by multiplication of matrix entries. Symbolic differentiation of transfer functions is not needed. The approach is illustrated by means of hand derivation of results for small well-known examples. 

Keywords Incremental true bond graphs Parameter sensitivities of transfer functions Linear inverse models Fault detection and isolation Parameter sensitivities of the residuals of analytical redundancy relations... 

Analytical redundancy relations are balance equations of effort or flow variables, in which unknown variables have been replaced by input variables and measured output variables and in which parameters are known. Evaluation of an ARR provides a residual that theoretically should be zero. In practice, however, the residual of an ARR is within certain error bounds as long as no faults occur during system operation. The value is not exactly zero over some time interval due to noise in measurement, parameter uncertainties, and numerical inaccuracies. If, however, the numerical value of a residual exceeds certain thresholds, then this is an indicator to a fault in one of the system s components. Noise in measured output variables may result in residual values indicating a fault that does not exist. Hence, measured data should pass appropriate filters before being used in ARRs. 

A diagnosis procedure based on evaluation of physical constraint laws derived from bond graph models is described here. Symbolically written constraints, called analytical redundancy relations (ARRs), are expressed in terms of known variables (measurements and inputs). ARRs are static or dynamic constraints which link the time evolution of the known variables when a system operates according to its normal operation model. The error or deviation from the constraint model is called a residual. The objective of quantitative diagnosis is to evaluate the residuals and associate the fault symptoms with deviations of residuals. 

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. 

The least-squares procedure just described is an example of a univariate calibration procedure because only one response is used per sample. The process of relating multiple instrument responses to an analyte or a mixture of analytes is known as multivariate calibration. Multivariate calibration methods have become quite popular in recent years as new instruments become available that produce multidimensional responses (absorbance of several samples at multiple wavelengths, mass spectrum of chromatographically separated components, and so forth). Multivariate calibration methods are very powerful. They can be used to determine multiple components in mixtures simultaneously and can provide redundancy in measurements to improve precision. Recall that repeating a measurement N times provides a Vn improvement in the precision of the mean value. These methods can also be used to detect the presence of interferences that would not be identified in a univariate calibration. 

If there are r terms in the operator, then the separability of the variables is lost. As before, the two new Gaussians, < >, and 4>/, are combined into a single Gaussian. A method is used that substitutes die fourier transform of r S and so the matrix element becomes a linear combination of integrals over the transform variable. Although these integrals are not analytic, they are related to the error funaion, and standard algorithms exist for their solution. Since the variables do not separate, all the summations remain nested, but computational efficiency can be achieved by careful combination of terms to eliminate redundant calculations. ... 

From the data analytical standpoint, multiway arrays represent a particularly rich source of information, as they often contain a large degree of redundancy, because many signals are used to describe a single sample. Accordingly, specific mathematical and statistical tools have been developed over the years to take the maximum advantage from the analysis of these kinds of data in this respect, multiway analysis is nothing else than the analysis of multiway data [6,7]. However, its main characteristic is that, due to the peculiarity of the data structures involved, it makes use of tools which are somewhat different from, even if in some cases related to, the standard methods used for the analysis of two-way data, such as the ones discussed in Chapters 3-5. 

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