Journal ArticleDOI
Lyapunov techniques for the exponential stability of linear difference equations with random coefficients
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TLDR
In this article, an approach to study the exponential stability of linear difference equations with random coefficients through the use of Lyapunov stability techniques is presented. But the approach is restricted to the case where the coefficients of the difference equations are random.Abstract:
We consider an approach to studying the exponential stability of linear difference equations with random coefficients through the use of Lyapunov stability techniques. The equations we study are of a form familiar from adaptive estimation algorithms, which motivates the examination. It is necessary to define the almost sure exponential convergence of a random process, and then to derive sufficient conditions on the coefficients of the difference equations to ensure the almost sure exponential convergence of the state. We consider, in particular, two very reasonable types of random coefficients-ergodic and stationary and φ-mixing and nonstationary-which would appear to encompass many engineering situations. An example of the power of the theory is given, where it is applied to a common adaptive filtering algorithm to derive mild conditions for exponential convergence with dependent random inputs.read more
Citations
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Journal ArticleDOI
Numerically stable fast transversal filters for recursive least squares adaptive filtering
Dirk Slock,Thomas Kailath +1 more
TL;DR: A solution is proposed to the long-standing problem of the numerical instability of fast recursive least squares transversal filter (FTF) algorithms with exponential weighting, an important class of algorithms for adaptive filtering.
Journal ArticleDOI
Persistent excitation in adaptive systems
TL;DR: In this article, the authors provide a general framework for the discussion of persistent excitation and collect results in the area, which are scattered throughout the adaptive literature, and present some new results related to the uniform asymptotic stability and robustness of adaptive systems and the relation of PE to the stability properties of a class of non-linear systems.
Journal ArticleDOI
Exponential convergence of adaptive identification and control algorithms
TL;DR: Output and equation error adaptive identification algorithms and an adaptive control algorithm are shown to be exponentially convergent under a deterministic or stochastic persistently exciting condition on the reference trajectory together with some standard conditions.
Journal ArticleDOI
Unconstrained frequency-domain adaptive filter
D. Mansour,A. Gray +1 more
TL;DR: In this paper, an unconstrained frequency-domain least mean square (UFLMS) algorithm is presented, which converges to the Wiener solution without the time-domain constraint on the impulse response as proposed by Ferrara.
Journal ArticleDOI
Persistence of excitation conditions and the convergence of adaptive schemes
TL;DR: 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.
References
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Book ChapterDOI
Stationary and nonstationary learning characteristics of the LMS adaptive filter
TL;DR: It is shown that for stationary inputs the LMS adaptive algorithm, based on the method of steepest descent, approaches the theoretical limit of efficiency in terms of misadjustment and speed of adaptation when the eigenvalues of the input correlation matrix are equal or close in value.
Journal ArticleDOI
A learning method for system identification
J. Nagumo,A. Noda +1 more
TL;DR: In this paper, a method for system identification is proposed which is based on the error-correcting training procedure in learning machines, and is referred to as learning identification, which is applicable to cases where the input signal is random and nonstationary, and can be completed within a short time, so that it may be used to identify linear quasi-time-invariant systems in which some parameters vary slowly in comparison with the time required for identification.
Journal ArticleDOI
Liapunov redesign of model reference adaptive control systems
TL;DR: An alternative synthesis based on Liapunov's second method is suggested here, and is applied to the redesign of adaptive loops considered by some other authors who have all used the M.I.T.T, rule.
Journal ArticleDOI
Exponential stability of linear equations arising in adaptive identification
TL;DR: In this article, the stability properties of various time-varying linear differential equations arising in model reference adaptive identification schemes are examined and necessary and sufficient conditions for exponential stability are presented.
Journal ArticleDOI
A survey of model reference adaptive techniques-Theory and applications
TL;DR: The ''state of the art'' based on the literature published since 1964, 253 references is presented and basic properties and the classification of various types of model reference adaptive systems are paid.