A Reproducing Kernel Hilbert Space Approach to Spline Problems with Applications in Estimation and Control

摘要

The solutions to several optimization problems involve generalized spline functions. Existing algorithms for calculating splines cannot be easily updated with the addition of new data, and are therefore not suitable for real-time computation when data are acquired sequentially. In the thesis, recursive algorithms are developed for the problems of optimal interpolation and smoothing, and optimal approximation of linear functionals, when the underlying space of functions is a reproducing kernel Hilbert space. It is shown that these deterministic problems have equivalent stochastic least-squares estimation problems, and that the recursive solution of each deterministic problem corresponds to the recursive solution of the associated stochastic problem in which a discrete innovation sequence is computed. In addition, it is shown that the problem of computing the minimum-energy control of a linear time-varying system that yields an output satisfying certain functional constraints is a spline problem and can be solved recursively using the above methods. (Author)