# Distributed Fault Detection and Isolation Approach for Oil Pipelines
## 2 Structural Analysis for Model-Based FDI
### Structural Diagnosability  
Analytical redundancy occurs and analytical redundancy relations are available when there are equations that are not necessary to match unknown variables in a system \(\Sigma\). Any over-determined subgraph \(G^{+}\) contains more equations than variables, and such an equation can be used for residual generation. A residual, derived from ARRs, is generated from a subgraph \(G^{+}\) using the excess equations once all unknown variables within \(G^{+}\) are expressed in terms of known variables. When an expected value of an \(ARR\) is not met, a fault is detected.  
**Definition 6** (Arr).: _A relation \(r(z,z,z,\ldots)=0\) is an Analytical Redundancy Relation (ARR) for \(\Sigma(z,x,f)\), if, for each \(z\) consistent with \(\Sigma(z,x,f)\), the relation is satisfied._  
**Definition 7** (Residual Generator for \(\Sigma(z,x,f)\)).: _A residual generator for the model \(\Sigma(z,x,f)\) is a system taking a subset of the variables \(z\) as input, and generating a scalar signal \(r\) as output if, for all \(z\) consistent with \(\Sigma(z,x,f)\), it holds that \(\lim\limits_{t\to\infty}r(t)=0\)._  
The Minimal Structurally Over-determined Set approach is a way to find ARRs. This is done by calculating the matching \(\mathcal{M}_{\text{MSO}}\subseteq\Sigma^{+}\) of an over-determined subgraph that has structural redundancy equal to 1.  
**Definition 8** (MSO sets).: _A Minimal Structurally Over-determined Subsystem is a part of \(G^{+}\) from which removal of one constraint will make the subsystem to become just-constrained [23]._  
A minimal structurally overdetermined set, for a short MSO set, can be used to generate an \(ARR\) given that the equations number exceeds the unknown variables number by one (structural redundancy 1), which means that only one residual generator can be developed. According to [38], the computation of MSO sets can be exponential according to the structural redundancy and system measurements. One way to reduce this computational cost can be by calculating only the MSO sets of interest, that is, reducing the calculation only to those impacted by faults. Hence, the concept of Fault-Driven Minimal Structurally Overdetermined (FMSO) set is useful. An FMSO set can be defined as an MSO set of \(\Sigma(z,x,f)\) whose fault support is not empty.  
Let the FMSO set \(\varphi\), where \(Z_{\varphi}\subseteq Z\) is the set of known variables of \(\varphi\), \(X_{\varphi}\subseteq X\) is the set of unknown variables of \(\varphi\) and \(F_{\varphi}\subseteq F\) is the set of fault support of \(\varphi\), below the definition of FMSO [39].  
**Definition 9** (FMSO set).: _A subset of equations \(\varphi\subseteq\Sigma(z,x,f)\) is an FMSO set of \(\Sigma(z,x,f)\) if (1) \(F_{\varphi}\
eq\emptyset\) and \(\rho_{\varphi}=1\) that means \(|\varphi|=|X_{\varphi}|+1\), and (2) no proper subset of \(\varphi\) is overdetermined._  
A complementary concept is the _Clear Minimal Structurally Overdetermined_ (CMSO) set as an MSO set of \(\Sigma(z,x,f)\) whose fault support is empty [17].  
**Definition 10** (CMSO set).: _A subset of equations \(\psi\subseteq\Sigma(z,x,f)\) is a CMSO set of \(\Sigma(z,x,f)\) if (1) \(F_{\psi}=\emptyset\) and \(\rho_{\varphi}=1\) that means \(|\psi|=|X_{\psi}|+1\), and (2) no proper subset of \(\psi\) is overdetermined._  
The algorithm of calculation of FMSO sets presented in [17] allows the calculation of all the FMSO sets related to the set of faults of interest, which can be many in the case of large-scale distributed plants. It is possible to optimize the calculation of FMSO sets taking into account some design parameters such as: less number of equations involved, less number of unknown/known variables or forcing the use of a certain subset of variables or equations. With these considerations, a distributed diagnostic algorithm can then be proposed to respond to real constraints on communication, distance, or availability of sensors and actuators among pre-defined subsystems.