On the benefit of overparameterization in state reconstruction

TL;DR

This paper introduces a method for state and parameter reconstruction in dynamical systems by overparameterizing the model, enabling full trajectory estimation and extending easily to nonlinear models using neural network training techniques.

Contribution

It presents a novel overparameterization approach that unifies state and parameter estimation, with an analytical solution for linear models and easy extension to nonlinear cases.

Findings

Analytical state reconstruction formula for linear models.

Method estimates entire state trajectories, not just initial conditions.

Framework is simple, does not require initial guesses, and adaptable to nonlinear models.

Abstract

The identification of states and parameters from noisy measurements of a dynamical system is of great practical significance and has received a lot of attention. Classically, this problem is expressed as optimization over a class of models. This work presents such a method, where we augment the system in such a way that there is no distinction between parameter and state reconstruction. We pose the resulting problem as a batch problem: given the model, reconstruct the state from a finite sequence of output measurements. In the case the model is linear, we derive an analytical expression for the state reconstruction given the model and the output measurements. Importantly, we estimate the state trajectory in its entirety and do not aim to estimate just an initial condition: that is, we use more degrees of freedom than strictly necessary in the optimization step. This particular approach can be reinterpreted as training of a neural network that estimates the state trajectory from available measurements. The technology associated with neural network optimization/training allows an easy extension to nonlinear models. The proposed framework is relatively easy to implement, does not depend on an informed initial guess, and provides an estimate for the state trajectory (which incorporates an estimate for the unknown parameters) over a given finite time horizon.