Exponential convergence of products of random matrices: application to adaptive algorithms

Moustakides, George V.

doi:10.1002/(sici)1099-1115(199811)12:7<579::aid-acs519>3.0.co;2-h

Cited by 14 publications

(41 citation statements)

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“…The mean square stability for MJLS with a countably infinite Markov chain was studied in [12], using an operator theoretic point of view. Some problems related to the mean square stability of MJLS with general state space were studied in [13], [14] under some ergodicity assumptions. The relation among several notions of stochastic stability for MJLS was studied in [11].…”

Section: Introductionmentioning

confidence: 99%

On Stability and LQ Control of MJLS With a Markov Chain With General State Space

Kordonis

Papavassilopoulos

2014

IEEE Trans. Automat. Contr.

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We study the mean square stability and the LQ control of discrete time Markov Jump Linear Systems where the Markov chain has a general state space. The mean square stability is characterized by the spectral radius of an operator describing the evolution of the second moment of the state vector. Two equivalent tests for the mean square stability are obtained based on the existence of a positive definite solution to a Lyapunov equation and a uniformity result respectively. An algorithm for testing the mean square stability is also developed based on the uniformity result. The finite and infinite horizon LQ problems are considered and their solutions are characterized by appropriate Riccati integral equations. An application to Networked Control Systems (NCS) is finally presented and a simple example is studied via simulation.Index Terms-Markov jump linear systems, stochastic optimal control, stochastic stability.

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Section: Introductionmentioning

confidence: 99%

On Stability and LQ Control of MJLS With a Markov Chain With General State Space

Kordonis

Papavassilopoulos

2014

IEEE Trans. Automat. Contr.

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“…Grande parte dos filtros adaptativos usados atualmente pode ser descrita pela recursão [28,29] W (n + 1) = W (n) + µf (e(n)) Z(n), em que µé uma constante conhecida como passo de adaptação, f (•)é uma função conhecida, e Z(n)é um vetor que depende das seqüências d(k) Uma variante muito utilizadaé o LMS normalizado (ou NLMS), obtido com o mesmo f (a) = a e com Z(n) = X(n)/( + X(n) 2 ) ( a é a norma euclideana do vetor a, e é uma constante não-negativa).…”

Section: Filtros Adaptativosunclassified

“…Outro algoritmo de grande interesse, o recursive least squares (RLS) também pode ser obtido da expressão acima com escolhas convenientes de f (•) e de Z(n) [29].d (n) = W (n) T X(n), e(n) = d(n) −d(n), W (n + 1) = W (n) + λ −1 P (n)X(n) 1 + λ −1 X(n) T P (n)X(n) e(n), P (n + 1) = λ −1 P (n) − λ −2 P (n)X(n)X(n) T P (n) 1 + λ −1 X(n) T P (n)X(n) . No caso geral, o passo de adaptação µé um parâmetro de projeto do filtro.…”

Section: (12)unclassified

“…72] Os quatro métodos possibilitam a obtenção de muitos resultadosúteis, mas atualmente têm algumas limitações importantes que são listadas a seguir. Análises teóricas segundo (a) ou (b) só existem para casos muito particulares [55,32,56,57], ou com a aproximação de adaptação "suficientemente lenta" (ou seja, para filtros com passo de adaptação infinitesimal [28,54,29,57]). Expressões para quão pequeno deve ser o passo para as análises serem válidas são raras e extremamente difíceis de serem calculadas [58,59].…”

Section: (12)unclassified

“…O método (b) pode ser estendido para fornecer o desempenho médio de filtros para o caso de passo infinitesimal[29].…”

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Análise de algoritmos para filtragem adaptativa baseados em momentos de quarta ordem.

Nascimento¹

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Esta tese apresenta um novo modelo para dois algoritmos para filtragem adaptativa baseados em momentos de quarta ordem: o least-mean fourth (LMF) e o least-mean mixed-norm (LMMN). A originalidade do modelo apresentado aquí e que não se procura calcular a média quadrática do erro de estimação do algoritmo, mas sim a probabilidade do algoritmo ter um comportamento razoável (neste caso, convergir). O trabalho mostra que o LMF e o LMMN não são estáveis na média quadrática se o regressor não for estritamente limitado (como ocorre, por exemplo, para a distribuição Gaussiana). Mesmo para a distribuição Gaussiana o LMF e o LMMN sempre têm uma probabilidade não nula de divergir, não importa quão pequeno seja o passo de adaptação. Esse resultadoé demonstrado para um filtro escalar (com umúnico coeficiente) com regressor com uma distribuição normal modificada, e verificado através de várias simulações. Além disso,é fornecido um limite superior para a probabilidade de divergência do LMF (e do LMMN), em função do comprimento do filtro, da potência dos sinais de entrada, do passo de adaptação, da variância do erroótimo, para o caso de regressores Gaussianos. Os resultados apresentados aqui fornecem ferramentas para projetistas entenderem melhor o funcionamento do algoritmo LMF, e decidir quandoé ou não conveniente o seu uso para uma dada aplicação.

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Sign‐Regressor Adaptive Filtering Algorithms for Markovian Parameters

Yin¹,

Hashemi

Wang³

2013

Asian Journal of Control

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This work is devoted to analyzing adaptive filtering algorithms with the use of sign-regressor for randomly time-varying parameters (a discrete-time Markov chain). In accordance with different adaption and transition rates, we analyze the corresponding asymptotic properties of the algorithms. When the adaptation rate is in line with the transition rate, we obtain a limit of a Markov switched differential equation. When the Markov chain is slowly changing the parameter process is almost a constant, and we derive a limit differential equation. When the Markov chain is fast varying, the limit system is again a differential equation that is an average with respect to the stationary distribution of the Markov chain. In addition to the limit dynamic systems, we obtain asymptotic properties of centered and scaled tracking errors. We obtain mean square errors to illustrate the dependence on the stepsize as well as on the transition rate. The limit distributions in terms of scaled errors are studied by examining certain centered and scaled error sequences. 0 0 , i = 1, . . . , r, and 1A is the indicator of A.Our new contributions in this paper are featured by the following distinct characteristics. First we develop signregressor algorithms for time-varying parameters modeled by a stochastic process. Moreover, we concentrate on multiscale structures. Note that the sign-regressor algorithms naturally come into play due to the application requirement. The salient feature of the time-varying parameter is highlighted in the Markov structure. In our setup, there are two inherent time scales. The Markov chain's state changes with its transition frequency described by a small parameter e. On the other hand, the stepsize of the approximation sequence is m. Interaction between e and m results multiscale structures. We use "«" or "»", the standard mathematical ("essentially less than" or "essentially greater than") notation. That is, both m → 0 and e → 0, but m → 0 much faster than e or m/e → 0. Likewise, the use of O(·) and o(·) follows standard notation. For example, e = O(m 1+d ) for some d > 0 means that e is the same order as that of m 1+d , which can also be written as e « m. In accordance with the relative size, we have three cases to consider: (a) m = O(e): the adaptation rate is in line with that of the transition rate of the parameter; (b) m » e: the time-varying parameter changes much slower than that of the stepsize; (c) m « e: the parameter process is fast changing. Corresponding to each of the cases, the asymptotic behavior is fundamentally different. We analyze the behavior of the algorithms and provide insight on their asymptotic properties.The rest of the paper is arranged as follows. Section II presents the precise formulation of the problem. Section III analyzes convergence properties of the algorithms. Section IV proceeds with mean square type estimates. Section V obtains asymptotic distributions of the algorithms and reveals the tracking ability by means of examining scaled tracking error sequences. Section VI g...

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Exponential convergence of products of random matrices: application to adaptive algorithms

Cited by 14 publications

References 14 publications

On Stability and LQ Control of MJLS With a Markov Chain With General State Space

On Stability and LQ Control of MJLS With a Markov Chain With General State Space

Análise de algoritmos para filtragem adaptativa baseados em momentos de quarta ordem.

Sign‐Regressor Adaptive Filtering Algorithms for Markovian Parameters

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