Observability and Identifiability Methods for Structural Dynamic Systems

Abstract

An important consideration for experimental setups throughout various fields of science and engineering is whether the quantities measured suffice in determining the desired states of the underlying dynamic system (i.e., whether the measurements render these states observable). More often than not, there is uncertainty with regard to the real parameters of the dynamic system. The purpose of system identification methods is to obtain the most likely values for the parameters and the states given a set of measurements. This uncertainty with regard to the parameters of the system results in them being treated as new states in an augmented dynamic system. Consequently, even in the simplest case of a linear underlying dynamic system, the corresponding augmented system becomes nonlinear. Thus, the question of whether a system identification method could succeed for given measurements in defining the parameters and states of a system (i.e., the augmented states) becomes a problem of nonlinear observability. If only the parameters of the system are of interest, identifiability methods may, in certain cases, be used. Thus, observability and identifiability methods enable the design of experimental setups that would at least work if the measurements were free of noise and the rejection of those that would not work even in this ideal scenario. In this work, three methods for the observability and identifiability of nonlinear dynamic systems are studied and compared against each other. For a system whose state and measurement equations are analytic, geometric observability methods based on Lie Derivatives may be used. Moreover, if the equations are rational, algebraic methods are also available. For this last category of systems, identifiability methods may be used to investigate not only the finiteness of the possible parameter values, but their uniqueness as well.