The issue of optimal placement of sensors and actuators have been studied for several system design objectives (Jiang & Doraiswami, 1990; Padula & Kincaid, 1999; Xu & Jiang, 2000; Li, 2011). For a FDD system, it is very important to select the suitable set of sensors so that the data collected from the plant are sensitive to changes caused by the faults and that the sensory data can uniquely distinguish different fault conditions. A sensor placement model usually involves three major steps or components: 1) system model; 2) sensor selection criteria; and 3) optimization. A review of optimal sensor placement methods for FDD systems can be found in (Li, 2011).
The system model describes cause-effect relationships between faults and sensors. It is the basis to derive figure of merits as the sensor placement criteria. Methods that have been used for system modeling include fault trees (Lambert, 1977), directed graph (DG) or digraph (Raghuraj, Bhushan, & Rengaswamy, 1999; Li & Upadhyaya, 2011), signed digraph (SDG) (Bhushan & Rengaswamy, 2000a; Bagajewicz, Fuxman, & Uribe, 2004;
Zhang, 2005; Li, 2011), and bond graph (Narasimhan, Mosterman, & Biswas, 1998). DG is the most popular choice for this purpose. It provides a useful tool to model whether a sensor will respond to a particular fault scenario. In a DG, faults and sensors are the nodes and the edges/arcs register the sensitivities between the faults and sensors (Raghuraj et al., 1999; Li, 2011). A bipartite graph can be built from the DG. The bipartite graph contains a set of nodes with all the faults and a set of nodes with all the sensors. To achieve fault observability, based on the bipartite graph, a minimal subset of the sensor nodes is chosen so that all fault nodes are covered by the selected sensor nodes. It is a minimal set covering problem that can be solved using optimization procedures such as greedy search. The problem of achieving diagnosability of all faults can be converted to a fault observability problem with a more complicated bipartite graph, which is derived from the original bipartite graph by adding additional nodes (Raghuraj et al., 1999; Li, 2011). Each additional node corresponds to a pair of faults, which contains the set of sensors that are sensitive to one fault but not to the second fault. It has shown that performing minimum set covering search on the more
complicated bipartite graph is a solution to achieve diagnosability for all faults (Raghuraj et al., 1999; Li, 2011).
As to the sensor selection criteria, three mostly chosen objectives are: 1) fault
observability or detectability; 2) fault resolution or diagnosability; and 3) sensor network reliability (Ali & Narasimhan, 1993; Ali & Narasimhan, 1995; Bhushan & Rengaswamy, 2000a; Li, 2011). Fault observability deals with the ability to distinguish fault conditions from normal conditions using data collected from a set of sensors. Fault resolution deals with the ability to distinguish different fault conditions from one another. Sensor network reliability is to make sure that fault diagnosis is still guaranteed considering sensor faults and unavailability of certain sensors. The general objectives of sensor placement models are that all fault conditions can be detected from the sensory data, that all fault conditions can be uniquely discriminated, or that certain minimum level of sensor network
reliability is achieved. Additional factors are often also considered in the FOM, such as the number of sensors, sensor cost, and sensor reliability. Various FOMs have been defined in the literature such as minimal networks (Bagajewicz & Sanchez, 1999), maximum diagnosability (Namburu, Azam, Luo, Choi, & Pattipati, 2007), maximum
resolution (Bhushan & Rengaswamy, 2000b), minimum cost (Bagajewicz, 1997; Bagajewicz & Sanchez, 2000), maximum reliability (Ali & Narasimhan, 1993; Ali & Narasimhan, 1995), and model prediction accuracy (Musulin, Benqlilou, Bagajewicz, & Puigjaner, 2005). Those FOMs are often formulated as constrained optimization
problems (Bhushan & Rengaswamy, 2000a; Bhushan & Rengaswamy, 2000b; Bhushan & Rengaswamy, 2002; Chmielewski, Palmer, & Manousiouthakis, 2002; Bagajewicz et al., 2004).
The final selection of sensors is usually determined by optimization of the FOMs, using methods such as simulated annealing, Tabu search method, genetic algorithm (Namburu et al., 2007; Casillas, Puig, Garza-Castanon, & Rosich, 2013), greedy search (Raghuraj et al., 1999), particle swarm optimization, and mixed integer programming (Bagajewicz, 1997; Bagajewicz & Sanchez, 2000; Bagajewicz & Cabrera, 2002; Bagajewicz et al., 2004).
However, in a practical situation, a fault diagnosis system is usually to be implemented based on availabilities of existing sensors already installed in a system for process control and monitoring. This situation is different from a standard sensor placement problem. In this case, several issues should be addressed. The first is the need to determine whether all the faults are diagnosable with the existing sensors. If the system is already diagnosable, there is no need to search for additional measurements. It is a less important problem to determine a smaller number of sensors that can achieve
diagnosability. When analyses show that the existing sensors are not sufficient to achieve fault diagnosability, additional sensors need to be put in place to enhance fault
diagnosability. It is desirable to select a minimum set of additional sensors, where it is advantageous to choose additional sensors with high sensitivities to particular non- diagnosable faults and it is indispensable to be able to quantify the influences of
installing a specific sensor to the fault diagnosability. However, to the best knowledge of the author, there is no sensor placement model specifically developed to address those practical issues.
The existing methods have several limitations to use for this purpose. First, binary entries are used to represent sensitivities between two nodes in a DG model; thus, different degrees of sensor sensitivities to a fault are not modeled. A consequence is that, if one sensor can respond to a fault but with relatively low sensitivity, it can separate two faults in theory, but the low sensitivity could be insufficient to reliably distinguish the faults in reality. In addition, to list all the sensors that could possibly consider for a sensor
placement problem will result in a graphical model with excessive complexity. What’s more, building a graph for a complex system with a potentially large number of faults and sensors requires a lot of engineering judgement and system specific experiences, which could be a challenge for practical applications. Furthermore, it is not intuitive to use a DG to guide selection of the additional sensors. One reason is that a fault
propagates in a system due to physical couplings among different variables or system states. The selection of additional sensors would be more straightforward if sensitivities between the faults and system states were known, because the additional sensors can be strategically selected to measure the system states with high sensitivities. However, the sensitivities between faults and the system states are not modeled explicitly in current methods. Another reason is that it is not flexible to add new sensors to or remove sensors from the graphical models to test the influence of a particular sensor for diagnosability of certain faults.