Asymptotic comparison of Bayesian state estimates and numerical methods for moving horizon estimation

(Master's thesis defense)

Katrin Baumgärtner

University of Freiburg

Tuesday, September 24, 2019, 11:00

Room 01-012, Georges-Köhler-Allee 102, Freiburg 79110, Germany

In order to predict and control the future behaviour of a dynamic system, model-based control typically requires knowledge of the current state of the system. As the state cannot be observed directly, it has to be estimated from noisy measurements of the system's output. One powerful method for nonlinear state estimation is Moving Horizon Estimation (MHE). With MHE, the state estimate is obtained by solving a nonlinear optimization problem that takes into account a fixed number of measurements on a moving horizon in the past.

In this thesis, we analyze MHE from two perspectives. From a Bayesian point of view, MHE can be regarded as an approximation of  maximum-a-posteriori (MAP) estimation of the state trajectory. This trajectory estimate is in practice often used as a replacement of the MAP estimate of the next state, as this estimate is in general intractable to compute. 
Starting from the observation that the two estimates are equivalent in the linear-Gaussian case, we  investigate to which extent the MAP estimate of the state trajectory approximates the MAP estimate of the next state in a more general setting as well.

On the other hand, we take a more application-oriented perspective and examine algorithms for solving the nonlinear optimization problem associated with the MHE formulation. 
One of the greatest impediments to a widespread application of MHE is still the considerable computational cost associated with the online solution of nonlinear optimization problems. We therefore propose zero-order MHE, an inexact, but fast, variant of exact MHE, which is based on the Gauss-Newton (GN) algorithm. The proposed method uses a fixed Jacobian approximation in order to avoid the computational cost induced by online sensitivity generation and factorization of the GN Hessian approximation. 
Although zero-order MHE does not yield a solution to the original MHE formulation, we can show that the estimation error produced by zero-order MHE would become zero in the absence of noise and grows linearly with the noise level. 
In addition, we present a recursive factorization approach for the GN Hessian approximation, which allows for efficient arrival cost updates within zero-order MHE.