A Note on Uniform Exponential Stability of Linear Periodic Time-Varying Systems

Vrabel, Robert

doi:10.1109/tac.2019.2927949

Cited by 12 publications

(5 citation statements)

References 29 publications

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“…in (9). Theorem 4 then implies µ 2 [A(t)] = α(A(t)), t ≥ t 0 , which is in compliance with Corollary 1.…”

Section: Examplesupporting

confidence: 74%

“…The various types of stability of the LTV systems can be expressed in the terms of a fundamental matrix [7] (p. 54) [8], and for periodic LTV systems, see [9]. Lemma 1.…”

Section: Notations Assumptions and Preliminary Resultsmentioning

confidence: 99%

“…that is, P = √ DW in (9) with n(•) and n(•) equal to n 2 (•). where α > 0 is a constant, then the LTV system ẋ = A(t)x, t ≥ t 0 is UAS.…”

Section: Examplementioning

confidence: 99%

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When Is σ (A(t)) ⊂ {z ∈ ℂ; ℜz ≤ −α < 0} the Sufficient Condition for Uniform Asymptotic Stability of LTV System ẋ = A(t)x?

Vrabel

2022

Mathematics

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“…in (9). Theorem 4 then implies µ 2 [A(t)] = α(A(t)), t ≥ t 0 , which is in compliance with Corollary 1.…”

Section: Examplesupporting

confidence: 74%

“…The various types of stability of the LTV systems can be expressed in the terms of a fundamental matrix [7] (p. 54) [8], and for periodic LTV systems, see [9]. Lemma 1.…”

Section: Notations Assumptions and Preliminary Resultsmentioning

confidence: 99%

See 1 more Smart Citation

When Is σ (A(t)) ⊂ {z ∈ ℂ; ℜz ≤ −α < 0} the Sufficient Condition for Uniform Asymptotic Stability of LTV System ẋ = A(t)x?

Vrabel

2022

Mathematics

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“…Numerous authors (Colaneri, 2005), (DaCunha, 2004), (Mullhaupt, Buccieri, & Bonvin, 2007), (Varbel, 2020), (Wu, 1974), (Yao, Liu, Sun, Balakrishnan, & Guo, 2012), analyze the stability of LPTV systems and refer to the examples in (Aggarwal & Infante, 1968), (Markus & Yamabe, 1960), or (Rosenbrook, 1963) without suggesting any solution. For example, by using a logarithmic norm 5 , (Varbel, 2020) derived a new criterion for uniform exponential stability of an LPTV system without finding the constant matrix 𝑅. However, (Varbel, 2020) refers to a stability analysis without solving the ODE of the LPTV system itself.…”

Section: Stability Analysis Of Lptv Systemmentioning

confidence: 99%

“…For example, by using a logarithmic norm 5 , (Varbel, 2020) derived a new criterion for uniform exponential stability of an LPTV system without finding the constant matrix 𝑅. However, (Varbel, 2020) refers to a stability analysis without solving the ODE of the LPTV system itself. 5 The logarithmic norm ,defined by…”

Section: Stability Analysis Of Lptv Systemmentioning

confidence: 99%

Analysis of Linear Time-Varying & Periodic Systems

Fivel¹

2021

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This thesis applies Floquet theory to analyze linear periodic time-varying (LPTV) systems, which are represented by a system of ordinary differential equations (ODEs) that depend on a time variable 𝑡 and have a matrix of coefficients with period 𝑇 > 0 (or equivalently, with frequency 𝜔 = 2𝜋 𝑇).According to Floquet theory, the transition matrix of an LPTV system represented by a square periodic-function matrix 𝐴(𝑡) = 𝐴(𝑡 + 𝑇) can be expressed as the product of a square periodic function matrix 𝑃(𝑡) = 𝑃(𝑡 + 𝑇) and an exponentiated square matrix of the form 𝑅𝑡, where 𝑅 is a constant matrix (independent of 𝑡). Despite the validity of Floquet theory, it is difficult to find an analytical closed form for the matrices 𝑃(𝑡) and 𝑅 when the transition matrix Φ 𝐴 (𝑡, 𝑡 0 ) is unknown. In essence, it is difficult to find an analytical solution for an LPTV system (i.e., a closed form for its transition matrix).The objective of this work is to characterize a sufficiently large family of periodic matrices 𝐴(𝑡) with a finite number of harmonics such that the periodic part 𝑃(𝑡) of the solution also has a finite number of harmonics. In addition, for each family, an effective procedure is required to find the matrices 𝑃(𝑡) and 𝑅.The research method involves studying periodic matrices 𝐴(𝑡) with frequency 𝜔 as a free parameter (i.e., not a fixed number) to obtain a sufficiently large family of LPTV systems so that their transition matrices are defined by the frequency 𝜔. In this work, we focus on cases in which the Fourier coefficients of 𝐴(𝑡) are finite polynomials in 𝜔, 𝑅 is a finite polynomial in 𝜔, and the Fourier coefficients of 𝑃(𝑡) do not depend on 𝜔.The research results show that for a given family of periodic matrices 𝐴(𝑡), we can compare the powers of 𝜔 that multiply the harmonics (i.e., 𝜔 is part of the coefficients multiplying the cosine [sine] factors in even [odd] representations or of the exponential factors in complex representations) to determine the matrices -ii-July 2021 𝑃(𝑡) and 𝑅. In addition, the results lead to relations between LPTV systems at frequency 𝜔 and the associated linear time-invariant system, which is defined by having zero frequency (𝜔 = 0). A buy-product of using the frequency 𝜔 as a free parameter is that allows the stability of the LPTV system to be determined based on how 𝑅 depends on the frequency 𝜔.

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Non‐uniformly exponential stability for time‐varying continuous‐time linear system

Hu,

Wang,

Tian

2024

Math Methods in App Sciences

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Exponential stability analysis is one of the most important problems in the study of the asymptotic behavior of solutions of continuous‐time linear systems. In comparison with the classical exponential stability, weaker criteria of (strongly) non‐uniformly exponential stability for the time‐varying linear systems are derived in this paper. By a nonsingular linear state transformation, a (strongly) non‐uniformly exponentially stable system can be equivalent to a simple one with the coefficient matrix whose measure is bounded by negative constants.

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A Note on Uniform Exponential Stability of Linear Periodic Time-Varying Systems

Cited by 12 publications

References 29 publications

When Is σ (A(t)) ⊂ {z ∈ ℂ; ℜz ≤ −α < 0} the Sufficient Condition for Uniform Asymptotic Stability of LTV System ẋ = A(t)x?

When Is σ (A(t)) ⊂ {z ∈ ℂ; ℜz ≤ −α < 0} the Sufficient Condition for Uniform Asymptotic Stability of LTV System ẋ = A(t)x?

Analysis of Linear Time-Varying & Periodic Systems

Non‐uniformly exponential stability for time‐varying continuous‐time linear system

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