Identification of affine dynamical systems from a single trajectory

Abstract

For dynamical systems of the form ẋ = Ax + c, henceforth called affine systems, we study the problem of determining system parameters A, c from data derived from a single trajectory. We establish necessary and sufficient conditions for identifiability of parameters from either a full trajectory or a set of discrete data points sampled from a single trajectory, expressed as geometrical conditions on the trajectory. We describe conditions under which the system has an equilibrium point and examine the problem of identifiability of the equilibrium point from data. We briefly examine the analytical upper and lower bounds for the maximal permissible uncertainty that will guarantee an inverse with specified qualitative properties of the resulting system (e.g., stability, spirality, and so on). Finally, we illustrate an application of the theory to parameter estimation for a class of nonlinear systems consisting of linear-in-parameters systems and demonstrate that affine approximations can yield more accurate estimates of parameter values than those based on finite differences.