Robert R. Bitmead

Asymptotically fast solution of toeplitz and related systems of linear equations

Abstract We present an inversion algorithm for the solution of a generic N X N Toeplitz system of linear equations with computational complexity O ( N log 2 N ) and storage requirements O ( N ). The algorithm relies upon the known structure of Toeplitz matrices and their inverses and achieves speed through a doubling method. All the results are derived and stated in terms of the recent concept of displacement rank, and this is used to extend the scope of the algorithm to include a wider class of matrices than just Toeplitz and also to include block Toeplitz matrices.

Design of an extended Kalman filter frequency tracker

The design of an extended Kalman filter for tracking a time-varying frequency is discussed. Its principal modes of failure are explained. The design tradeoff between balancing noise rejection and tracking at a maximal slew rate is discussed. The performance penalties for overdesign and underdesign of noise covariances are examined, and theoretically supported design guidelines are suggested.

Iterative weighted least-squares identification and weighted LQG control design

Many practical applications of control system design based on input-output measurements permit the repeated application of a system identification procedure operating on closed-loop data together with successive refinements of the designed controller. Here we develop a paradigm for such an iterative design. The key to the procedure is to account for evaluated modelling error in the control design and, equally, to let the closed-loop controller requirements determine the identification criterion. With an H-2 control problem, this is achieved by frequency weighting the linear-quadratic Gaussian (LQG) control criterion with filters that reflect the closed-loop plant/model mismatch, and by filtering the identifier signals used in a least-squares identification scheme in a logical and mutually supportive fashion.

Greatest common divisor via generalized Sylvester and Bezout matrices

We present new methods for computing the greatest common right divisor of polynomial matrices. These methods involve the recently studied generalized Sylvester and generalized Bezoutian resultant matrices, which require no polynomial operations. They can provide a row proper greatest common right divisor, test for coprimeness and calculate dual dynamical indices. The generalized resultant matrices are developments of the scalar Sylvester and Bezoutian resultants and many of the familiar properties of these latter matrices are demonstrated to have analogs with the properties of the generalized resultant matrices for matrix polynomials.

Performance of adaptive estimation algorithms in dependent random environments

We consider the convergence properties of certain algorithms arising in stochastic, discrete-time, adaptive estimation problems and operating in random environments of engineering significance. We demonstrate that the algorithms operating under ideal conditions are describable by homogeneous time-varying linear difference equations with dependent random coefficients, while in practical use, these equations are altered only through the addition of a driving term, accounting for time variation of system parameters, measurement noise, and system undermodeling. We present the concept of almost sure exponential convergence of the homogeneous difference equations as an a priori testable robustness property guaranteeing satisfactory performance in practice. For the three particular algorithms discussed, we present very mild conditions for the satisfaction of this property, and thus explain much of their observed behavior.

Brief paper: State estimation for linear systems with state equality constraints

This paper deals with the state estimation problem for linear systems with linear state equality constraints. Using noisy measurements which are available from the observable system, we construct the optimal estimate which also satisfies linear equality constraints. For this purpose, after reviewing modeling problems in linear stochastic systems with state equality constraints, we formulate a projected system representation. By using the constrained Kalman filter for the projected system and comparing its filter Riccati equation with those of the unconstrained and the projected Kalman filters, we clearly show, without using optimality, that the constrained estimator outperforms the other filters for estimating the constrained system state. Finally, a numerical example is presented, which demonstrates performance differences among those filters.

Adaptive frequency response identification

Given a stable, discrete time, single input single output system G(z), but with only the input signal and the noise corrupted output signal available for measurement, we seek to find an approximation G(z) - a finite impulse response (FIR) filter - with ||G - ¿¿|| = sup |G(ej¿) - ¿(ej¿)| ¿¿(-¿,¿] bounded and small. The infinity norm in (1) has application in control theory and signal processing; furthermore, it is a measure of the deviation in frequency response between G and ¿. Several previous papers, attempt to identify G(z) in the frequency domain; these papers fail to bound G-¿ in any norm. Central to our method of identification is interpolation. First, one estimates accurately G(z) at n equally spaced frequencies. Here, n is a design parameter one may freely choose. This estimation relies on filtering the input and output signals appropriately. Then estimates of G(eJ2¿k/n) come from a bank of n/2 decoupled least mean squares algorithms, each of two parameters; ¿(z) is then the unique FIR filter of degree n-1 with transfer function interpolating to these estimates. ¿(z) is computationally easy to evaluate. The resulting error bound has the form ||G - ¿||¿ ¿ MRn + K(1 + log2n) Here M and R are constants, dependent on G(z), with R<1; the accuracy of estimating G(z) at the interpolation points determines K.

Adaptive frequency sampling filters

We present two new structures for adaptive filters based on the idea of frequency sampling filters and gradient based estimation algorithms. These filters have a finite impulse response (FIR) and can be thought of as attempting to approximate a desired frequency response at given points on the unit circle. The filters operate in real time with no batch processing of signals as is the case when using the discrete Fourier transform. They result in a marked reduction in dimension of the time-domain problem of fitting an Nth order FIR transversal filter to a collection of length 2 transversal filters and further to a collection of N scalar filters. The advantages of this are then discussed.

A Kalman filtering approach to short-time Fourier analysis

The problem of estimating time-varying harmonic components of a signal measured in noise is considered. The approach used is via state estimation. Two methods are proposed, one involving pole-placement of a state observer, the other using quadratic optimization techniques. The result is the development of a new class of filters, akin to recursive frequency-sampling filters, for inclusion in a parallel bank to produce sliding harmonic estimates. Kalman filtering theory is applied to effect the good performance in noise, and the class of filters is parameterized by the design tradeoff between noise rejection and convergence rate. These filters can be seen as generalizing the DFT.

Persistence of excitation conditions and the convergence of adaptive schemes

In the study of the behavior of adaptive filtering algorithms, persistence of excitation of the input process arises as a sufficient condition for convergence and, perhaps more importantly, for convergence rate of the parameter estimates. In this paper the underlying nature of the persistence requirement is presented and discussed, relating its normal specification in terms of moment conditions with covariance decays, etc., to sample path properties. Deterministic and stochastic persistence conditions and persistence measures are treated, as well as, persistence conditions for output-error, equation-error, and adaptive control schemes.