Estimation Algebra

In this chapter, we will introduce the finite-dimensional filter estimation algebra technique. Since the proposal of the linear Kalman-Bucy filter, addressing nonlinear filter problems has emerged as a significant and widely discussed area of research. A key question that arises is how to assess the efficiency of different nonlinear filter solutions. Estimation algebra, as both a geometric and an algebraic technique, serves as a powerful tool to address this challenge. By utilizing estimation algebra, we can develop finite-dimensional nonlinear filters that are governed by a finite set of statistical quantities, thus enabling systematic control over their behavior. Importantly, this approach facilitates the classification of various nonlinear systems based on these statistics, paving the way for practical applications in analyzing nonlinear control systems such as observability and controllability. This represents a groundbreaking integration of geometric and algebraic methods into the realm of nonlinear filter theory.