Identifiability of Linear and Linear-in-Parameters Dynamical Systems from a Single Trajectory

Certain experiments are nonrepeatable because they result in the destruction or alteration of the system under study, and thus provide data consisting of at most a single trajectory in state space. Before proceeding with parameter estimation for models of such systems, it is important to know whether the model parameters can be uniquely determined, or identified, from idealized (error-free) single trajectory data. In the case of a linear model, we provide precise definitions of several forms of identifiability, and we derive some novel, interrelated conditions that are necessary and sufficient for these forms of identifiability to arise. We also show that the results have a direct extension to a class of nonlinear systems that are linear in parameters. One of our results provides information about identifiability based solely on the geometric structure of an observed trajectory, while other results relate to whether or not there exists an initial condition that yields identifiability of a fixed but unknown coefficient matrix and depend on its Jordan structure or other properties. Lastly, we extend the relation between identifiability and Jordan structure to the case of discrete data, and we show that the sensitivity of parameter estimation with discrete data depends on a condition number related to the data’s spatial confinement.