Dynamical State and Parameter Estimation

We discuss the problem of determining unknown fixed parameters and unobserved state variables in nonlinear models of a dynamical system using observed time series data from that system. In dynamical terms this requires synchronization of the experimental data with time series output from a model. If the model and the experimental system are chaotic, the synchronization manifold, where the data time series is equal to the model time series, may be unstable. If this occurs, then small perturbations in parameters or state variables can lead to large excursions near the synchronization manifold and produce a very complex surface in any estimation metric for those quantities. Coupling the experimental information to the model dynamics can lead to a stabilization of this manifold by reducing a positive conditional Lyapunov exponent (CLE) to a negative value. An approach called dynamical parameter estimation (DPE) addresses these instabilities and regularizes them, allowing for smooth surfaces in the space of parameters and initial conditions. DPE acts as an observer in the control systems sense, and because the control is systematically removed through an optimization process, it acts as an estimator of the unknown model parameters for the desired physical model without external control. Examples are given from several systems including an electronic oscillator, a neuron model, and a very simple geophysical model. In networks and larger dynamical models one may encounter many positive CLEs, and we investigate a general approach for estimating fixed model parameters and unobserved system states in this situation.