Sensor location for diagnosis in linear structured systems with disturbances

We consider here the fault detection and isolation (FDI) problem for linear systems. We design a set of observer-based residuals, in such a way that the transfer from the faults to the residuals is diagonal and the transfer from the disturbances to the residuals is zero. We deal with this problem when the system under consideration is structured, that is, the entries of the system matrices are either fixed zeros or free parameters. This problem was solved recently in terms of the graph that one can associate in a natural way to a structured system. We are concerned in this paper with the case where the FDI solvability conditions are not satisfied. We consider that some internal states can be measured at some cost and wonder if the problem is solvable with these new measurements. If this is possible we give solvability conditions at minimal cost for the considered sensor location problem. These problems are solved using the bipartite graph of the system. A particular attention is devoted to the complexity reduction of the optimization problem.