Frequency domain descriptions of linear systems

Abstract

This thesis begins by applying Lagrange interpolation to linear systems theory. More specifically, a stable, discrete time linear system, with transfer function G(z), is interpolated with an FIR transfer function at n equally spaced points around the unit circle. The L∞ error between the original system and the interpolation is bounded, the bound going to zero exponentially fast as n -> ∞. A similar result applies to unstable systems except that the interpolating function is a non-causal FIR transfer function . The thesis then considers Hilbert transforms from interpolation data. Given the real part of a stable transfer function evaluated at n equally spaced points around the unit circle, the Hilbert transform from interpolation data reconstructs the complete frequency response, real and imaginary parts, at all frequencies, to within a bounded L∞ error. The error bound goes to zero exponentially fast as n -> ∞. Also considered is the gain-phase problem from interpolation data. This is the same as the Hilbert transform from interpolation data, except that magnitude interpolation data instead of real part interpolation data is given. Two constructions for the gain phase problem from interpolation data are given , and L∞ error bounds derived . In both cases, the error bounds go to zero exponentially fast as n -> ∞. Application of Kalman filters to short-time Fourier analysis then follows. This contains a new method in Kalman filtering called covariance setting. The filters derived from covariance setting generalize the discrete Fourier transform. They offer a design trade-off between noise smoothing and transient response time, are recursive, and are of similar computational complexity to the discrete Fourier transform. Combining the Kalman filters for short-time Fourier analysis and Lagrange interpolation gives a new method of adaptive frequency response identification. A feature of this method is the L∞ error bound between the original system and the identified model. Using recent analysis on the inherent frequency weighting in identification algorithms shows the superiority of this new method over previous adaptive frequency response identification schemes. Finally, model reduction for unstable systems is considered. Given an unstable rational function of high McMillan degree, an approximation of lower McMillan degree, but with the same number of unstable poles, is constructed. An L∞ error bound between the original transfer function and approximation is derived. Such an approximation has application to control systems. Show more