UC San Diego

Estimation, in general, involves the determination of a probability distribution. This probability distribution describes the likelihood that any particular point in phase space accurately represents the truth state. That is, without knowing the actual state of a system, estimation strategies attempt to represent the probability of any given state using only a time history of noisy observations and, when available, an approximate dynamic model of the system of interest. For low-dimensional linear systems with Gaussian uncertainty in the initial state, state disturbances, and measurement noise the de facto solution to the estimation problem has been the Kalman Filter, which provides a method to propagate the mean and covariance forward in time, making the appropriate updates to both upon the receipt of each new measurement. Although ubiquitous within academia and industry, since many systems of interest are either of very high dimension or cannot be described by linear dynamics with Gaussian uncertainty, the Kalman Filter is inappropriately applied in many applications. The present thesis first reviews extensions of estimation theory to high-dimensional systems and demonstrates the first successful reconstruction of 3D turbulent channel flow (Re[tau] = 100), using wall information only, via the Ensemble Kalman Filter. Then a new hybrid method of estimation is described which improves estimation results for such high-dimensional systems by employing recent machine learning techniques (specifically, the Normal- Hedge algorithm) to consistently combine multiple estimators. Lastly, since the measurement operator critically determines the quality of the estimate, a gradient-base sensor/actuator placement strategy for Linear Time Invariant systems is presented. Using a test system (the Ginzburg-Landau equation) this sensor placement strategy is demonstrated by determining the optimal location for sensors in such a way that minimizes scalar metrics of the covariance matrix. With this theory clearly established, optimal sensor placements are determined for dynamic sensors in a 2D environmental plume estimation problem