(273c) Estimating the State of a Dynamical System: From Kalman Filtering to Moving Horizon State Estimation

This talk presents an overview of techniques for estimating the state
of a dynamical system from the system output measurements.

The Kalman filter represents one of the early triumphs of modern state
estimation for linear systems. The conditional
probability of the state given the time sequence of measurements is
shown to be a normal distribution for a linear system with normally
distributed and independent measurement and process noises.
Maximizing this conditional density produces a convenient recursion
for the mean (the optimal state estimate) and the covariance of the
distribution.

When it comes to deriving system properties, however, such as the
stability of an estimator, the statistical properties provided by the
Kalman filter recursion are of little use. The fact that a
statistically optimal estimator is not necessarily even a stable
estimator shows the divide between these two approaches to state
estimation.

We then rederive the Kalman filter as an optimization of a cost
function, and show that analysis of the optimality properties enable
us to derive a close relative of a Lyapunov function for state
estimation that provides a stability and robustness analysis.

The talk concludes by presenting some open conjectures and open
research problems in state estimation.