Estimation Strategies for Constrained and Hybrid Dynamical Systems

The estimation approaches examined in this dissertation focus on manipulating system dynamical models to allow the well-known form of the continuous-discrete extended Kalman filter (CDEKF) to accommodate constrained and hybrid systems. This estimation algorithm filters sequential discrete measurements for nonlinear continuous systems modeled with ordinary differential equations. The aim of the research is to broaden the class of systems for which this common tool can be easily applied.

Equality constraints, holonomic or nonholonomic, or both, are commonly found in the system dynamics for vehicles, spacecraft, and robotics. These systems are frequently modeled with differential algebraic equations. In this dissertation, three tools for adapting the dynamics of constrained systems for implementation in the CDEKF are presented. These strategies address (1) constrained systems with quasivelocities, (2) kinematically constrained redundant coordinate systems, and (3) systems for which an equality constraint can be broken. The direct linearization work for constrained systems modeled with quasi-velocities is demonstrated to be particularly useful for systems subject to nonholonomic constraints. Concerning redundant coordinate systems, the "constraint force" perspective is shown to be an effective approximation for facilitating implementation of the CDEKF while providing similar performance to that of the fully developed estimation scheme. For systems subject to constraint violation, constraint monitoring methods are presented that allow the CDEKF to autonomously switch between constrained and unconstrained models. The efficacy of each of these approaches is shown through illustrative examples.

Hybrid dynamical systems are those modeled with both finite- and infinite-dimensional coordinates. The associated governing equations are integro-partial differential equations. As with constrained systems, these governing equations must be transformed in order to employ the CDEKF. Here, this transformation is accomplished through two finite-dimensional representations of the infinite-dimensional coordinate. The application of these two assumed modes methods to hybrid dynamical systems is outlined, and the performance of the approaches within the CDEKF are compared. Initial simulation results indicate that a quadratic assumed modes approach is more advantageous than a linear assumed modes approach for implementation in the CDEKF.

The dissertation concludes with a direct estimation methodology that constructs the Kalman filter directly from the system kinematics, potential energy, and measurement model. This derivation provides a straightforward method for building the CDEKF for discrete systems and relates these direct estimation ideas to the other work presented throughout the dissertation.

Together, this collection of estimation strategies provides methods for expanding the class of systems for which a proven, well-known estimation algorithm, the extended Kalman filter, can be applied. The accompanying illustrative examples and simulation results demonstrate the utility of the methods proposed herein.