**Henry D. I. Abarbanel**

University of California, San Diego

**April 13, 2010**

**Mesa Laboratory, Main Seminar Room**

**Lecture 3:30pm**

### State and Parameter Estimation in Models of Nonlinear Systems

The problem of using observed information in estimating unobserved state variables and fixed parameters in a model of a nonlinear dynamical system when the measurements are noisy, the model has errors, and the state of the system as measurements begin is uncertain is cast as an exact path integral. In this formulation the measurements are seen as a guiding “potential” directing a chaotic system to the correct region of phase space. The path integral is evaluated directly for small systems such as the Lorenz 1996 model. In the measurement window, there are indications that the distribution of state variables is nearly Gaussian. The properties of the path integral can yield an approximation to the number of required measurements.

A saddle point evaluation of the path integral is seen to be 4DVAR, and implementing this in the case where the dynamics is error free—i.e. deterministic—is shown to require regularization to achieve a smooth surface on which the estimation search can proceed. The number of observations required to regularize the search can be estimated from this requirement.

The regularized 4DVAR method is applied to experiments on a small nonlinear circuit as well as to the Lorenz 1996 model.