On the Sigma Exponent of Discrete Linear Systems

Czornik, Adam; Mokry, Piotr; Nawrat, Aleksander

doi:10.1109/tac.2010.2045699

Cited by 20 publications

(9 citation statements)

References 12 publications

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“…Theorem 4: For sequence such that (9) and any sequence of perturbations satisfying (3), the Lyapunov exponent of (2) satisfies (10) where is the constant defined by (7). A drawback of the estimations (8) and (10) is that the right side involve constants or which are difficult to calculate.…”

Section: Preliminary Resultsmentioning

confidence: 99%

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On the Lyapunov and Bohl exponent of time-varying discrete linear system

Czornik

Nawrat

Niezabitowski

et al. 2012

2012 20th Mediterranean Conference on Control &Amp; Automation (MED)

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This note studies the problem of bounds for the Lyapunov exponent of a parameter perturbed system when the perturbation has finite average value. Such a bound is presented in terms of Bohl exponent of the unperturbed system. In particular, it has been shown that the Lyapunov exponent of perturbed system is not greater than the Bohl exponent of the unperturbed system if the average value of perturbations is zero. The obtained result is illustrated by a numerical example.

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Section: Preliminary Resultsmentioning

confidence: 99%

“…This is very transparent when one compares the review of the results for continuous-time system from Chapter V of [1], or from [6] with the results contained in monograph of Barreira and Pesin [5] devoted to Lyapunov exponents of discrete-time systems. Some recent results for discrete-time systems are presented in [7]- [10].…”

Section: Introductionmentioning

confidence: 99%

On the Lyapunov and Bohl exponent of time-varying discrete linear system

Czornik

Nawrat

Niezabitowski

et al. 2012

2012 20th Mediterranean Conference on Control &Amp; Automation (MED)

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“…There are several approaches to defining the exponential growth rate, all of which turn out to be equivalent. A trajectory based definition considers Lyapunov exponents [36][37][38][39][40][41][42][43][44][45][46][47][48] of individual trajectories, which are defined in the discrete-time case as…”

Section: Lyapunov Exponents Methodsmentioning

confidence: 99%

“…The quantity ln ρ(d) is known in the control theory literature as the greatest Lyapunov exponent. Properties of Lyapunov exponents were studied in [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51], where sufficient conditions so that Lyapunov exponents characterizes exponential stability for switched systems are shown. In particular, it was shown that the system (1) is absolutely stable if and only if sup…”

Section: Lyapunov Exponents Methodsmentioning

confidence: 99%

Stability and controllability of switched systems

Klamka¹,

Czornik²,

Niezabitowski³

2013

Bulletin of the Polish Academy of Sciences: Technical Sciences

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Abstract. The study of properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. This paper aims to briefly survey recent results on stability and controllability of switched linear systems. First, the stability analysis for switched systems is reviewed. We focus on the stability analysis for switched linear systems under arbitrary switching, and we highlight necessary and sufficient conditions for asymptotic stability. After that, we review the controllability results.

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“…It is possible that a finite shift of the characteristic exponents of the original system (1) corresponds to an arbitrarily small sup Δ(n) . In particular, it is feasible for an exponentially stable system to be perturbed by an exponentially decreasing perturbations and the resulting system is not stable [41]. The determination of the movability boundaries of different numerical characteristic under various perturbations is one of the main problems of the stability theory.…”

Section: Perturbed Systemmentioning

confidence: 99%