Network Analysis of Sensitivity Identifiability of Initial State and Parameter for Linear Systems with Graph Laplacians

If the true parameter of a system is unknown, a parameter estimation method is used to obtain the true parameter. One of the estimation methods is the minimization of the evaluation function of the parameter-dependent model and the observed data, which is implemented by an optimization method. Using Newton's method for this optimization, we can expect the quadratic convergence to the true parameter. However, if the true parameter is sensitivity unidentifiable (non-sensitivity identifiable (SI)), quadratic convergence is not guaranteed. Thus, detecting the true parameter being non-SI a priori is important. Considering the target system has a network structure, we can expect to use the network structure for this detection if we have a network condition that implies that the true parameter is non-SI. Therefore, in this paper, we consider linear systems with graph Laplacians as a network system class and address a problem to find the network condition that implies that the true parameter is non-SI. Then, we obtain a sufficient condition which implies that the true parameter is non-SI. The condition is explained by the graph's symmetric structure and proven by the decomposition of the graph's incidence matrix.