A Reproducing Kernel Hilbert Space Approach to Spline Problems with Applications in Estimation and ControlThe 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). |
Contents
INTRODUCTION | 1 |
THE REPRODUCING KERNEL HILBERT SPACE | 9 |
REFORMULATION AND SOLUTION OF THE INTERPOLATION PROBLEM | 15 |
5 other sections not shown
Common terms and phrases
analog filter approximation of linear Boor and Lynch cardinal spline coefficients compute the spline congruence relation covariance data are acquired defined definition degree m-1 Equation estimate of y(t estimation problem evaluation functionals Example 4.1 fact given by Eq Golomb and Weinberger Gram-Schmidt procedure Green's function H₁ implies inner product interpolating spline interpolation constraints Jerome and Schumaker K₁ kernel Hilbert space Kimeldorf and Wahba Lg-spline interpolating linear functionals linear least-squares estimate linearly independent matrix minimizing Eq minimum norm minimum-energy control problem null space optimal approximation orthonormal basis polynomial of degree projection theorem Proof proved random variables real-time computation recursive algorithm recursive solution reformulate representer reproducing kernel Hilbert reproducing property satisfy Eq Schumaker 14 smoothing problem solution to Eq space H spline function spline interpolation spline problem spline that interpolates subspace t₁ t₂ Theorem 3.3 time-varying system unique