Local Unidentifiability of EdgeWeights for Linear Systems with Graph Laplacian

In this paper, we analyze the local identifiability of parameters in linear systems whose system matrix is given by a graph Laplacian. Graph Laplacian is a matrix given by a graph and the weights of the edges, and the weights represent the parameter in those systems. It is important to detect whether the parameter is locally unidentifiable (non-LI) a priori, since any parameter estimation methods cannot estimate non-LI parameters. To this aim, we address a problem to find the condition so that the parameter is non-LI for any initial state. Then, we obtain a sufficient condition for the parameter to be non-LI. The condition is given based on the number of vertices and edges and the number of distinct eigenvalues of the graph Laplacian. We give a system that satisfies the condition.