Stability analysis of periodically switched linear systems using Floquet theory

Gokcek, Cevat

doi:10.1155/s1024123x04401069

Cited by 35 publications

(20 citation statements)

References 6 publications

(5 reference statements)

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“…Thus, switching systems are asymptotically stable if and only if

Ǎ

is Schur stable, where

Ǎ MathClass-rel= exp ((), Â_{2} (ρ MathClass-bin- δ)) exp ((), Ǎ_{1} δ) MathClass-punc,

with

Ǎ_{1} MathClass-rel= I_{N} MathClass-bin\otimes A MathClass-bin+ c diag (U) MathClass-bin\otimes BF

, and diag( U ) is the diagonal matrix with λ i , i = 2,3, ⋯ , N , being its diagonal elements. Recalling Theorem 3.1 in , one has that switching systems are asymptotically stable if and only if the N − 1 switching systems

\begin{matrix} aligned \\ right {\overset{MathClass-oṗ}{ζ}}_{i} MathClass-open(t MathClass-close) & left = MathClass-open(A + c λ_{i} BF MathClass-close) ζ_{i} MathClass-open(t MathClass-close), & right t & left \in MathClass-open[kρ, kρ + δ MathClass-close), & right \\ right {\overset{MathClass-oṗ}{ζ}}_{i} MathClass-open(t MathClass-close) & left = A ζ_{i} MathClass-open(t MathClass-close), & right t & left \in MathClass-open[kρ + δ, MathClass-open(k + 1 MathClass-close) ρ MathClass-close), k \in N, \end{matrix}

…”

Section: Consensus Seeking With Intermittent Communicationsmentioning

confidence: 99%

See 1 more Smart Citation

Distributed consensus of multi-agent systems with general linear node dynamics and intermittent communications

Wen

Duan

Ren

et al. 2013

Int. J. Robust. Nonlinear Control

231

116

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SUMMARYWithout assuming that the mobile agents can communicate with their neighbors all the time, the consensus problem of multi‐agent systems with general linear node dynamics and a fixed directed topology is investigated. To achieve consensus, a new class of distributed protocols designed based only on the intermittent relative information are presented. By using tools from matrix analysis and switching systems theory, it is theoretically shown that the consensus in multi‐agent systems with a periodic intermittent communication and directed topology containing a spanning tree can be cast into the stability of a set of low‐dimensional switching systems. It is proved that there exists a protocol guaranteeing consensus if each agent is stabilizable and the communication rate is larger than a threshold value. Furthermore, a multi‐step intermittent consensus protocol design procedure is provided. The consensus algorithm is then extended to solve the formation control problem of linear multi‐agent systems with intermittent communication constraints as well as the consensus tracking problem with switching directed topologies. Finally, some numerical simulations are provided to verify the effectiveness of the theoretical results. Copyright © 2013 John Wiley & Sons, Ltd.

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“…Thus, switching systems are asymptotically stable if and only if

Ǎ

is Schur stable, where

Ǎ MathClass-rel= exp ((), Â_{2} (ρ MathClass-bin- δ)) exp ((), Ǎ_{1} δ) MathClass-punc,

with

Ǎ_{1} MathClass-rel= I_{N} MathClass-bin\otimes A MathClass-bin+ c diag (U) MathClass-bin\otimes BF

\begin{matrix} aligned \\ right {\overset{MathClass-oṗ}{ζ}}_{i} MathClass-open(t MathClass-close) & left = MathClass-open(A + c λ_{i} BF MathClass-close) ζ_{i} MathClass-open(t MathClass-close), & right t & left \in MathClass-open[kρ, kρ + δ MathClass-close), & right \\ right {\overset{MathClass-oṗ}{ζ}}_{i} MathClass-open(t MathClass-close) & left = A ζ_{i} MathClass-open(t MathClass-close), & right t & left \in MathClass-open[kρ + δ, MathClass-open(k + 1 MathClass-close) ρ MathClass-close), k \in N, \end{matrix}

…”

Section: Consensus Seeking With Intermittent Communicationsmentioning

confidence: 99%

“…with L A 1 D I N˝A C c diag.U /˝BF , and diag.U / is the diagonal matrix with i , i D 2, 3, , N , being its diagonal elements. Recalling Theorem 3.1 in [28], one has that switching systems (11) are asymptotically stable if and only if the N 1 switching systems…”

Section: Proofmentioning

confidence: 99%

Distributed consensus of multi-agent systems with general linear node dynamics and intermittent communications

Wen

Duan

Ren

et al. 2013

Int. J. Robust. Nonlinear Control

231

116

View full text Add to dashboard Cite

show abstract

“…{C1/ k k holds for all t > . { C 1/T h. By induction, (6) holds for all i 2 N.By(6), it holds that kx .iT I ; ;´/k 6 2 i k k ; 8i 2 N. Thus, kx .iT I ; ;´/k 6 2 i k k ; 8i 2 N 0 ;…”

mentioning

confidence: 90%

Links between different stabilities of switched homogeneous systems with delays and uncertainties

Liu

2015

Intl J Robust & Nonlinear

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Summary This paper addresses the links between three stabilities (attractivity, asymptotic stability, and exponential stability) of switched homogeneous systems with delays and uncertainties. A system has a certain property over a given set double-struckS of switching signals if the property holds for all switching signals in double-struckS. It is shown that a switched homogeneous system of degree one is exponentially stable over a given set of switching signals if it is attractive or asymptotically stable over the same set. The result is then applied to switched linear systems with delays and uncertainties. Finally, an example follows to show that ‘being over a given set of switching signals’ is necessary to guarantee the equivalence between different stabilities. Copyright © 2015 John Wiley & Sons, Ltd.

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“…Because A inv (t) is a piecewise-constant matrix, we can compute the fundamental matrix X(t) using the matrix exponential (Gökçek 2004). The fundamental matrix is…”

Section: Example 2: Invasion Criteria For Interacting Structured Popumentioning

confidence: 99%

Floquet theory: a useful tool for understanding nonequilibrium dynamics

Klausmeier

2008

Theor Ecol

128

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Many ecological systems experience periodic variability. Theoretical investigation of population and community dynamics in periodic environments has been hampered by the lack of mathematical tools relative to equilibrium systems. Here, I describe one such mathematical tool that has been rarely used in the ecological literature but has widespread use: Floquet theory. Floquet theory is the study of the stability of linear periodic systems in continuous time. Floquet exponents/multipliers are analogous to the eigenvalues of Jacobian matrices of equilibrium points. In this paper, I describe the general theory, then give examples to illustrate some of its uses: it defines fitness of structured populations, it can be used for invasion criteria in models of competition, and it can test the stability of limit cycle solutions. I also provide computer code to calculate Floquet exponents and multipliers.

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Stability analysis of periodically switched linear systems using Floquet theory

Cited by 35 publications

References 6 publications

Distributed consensus of multi-agent systems with general linear node dynamics and intermittent communications

Distributed consensus of multi-agent systems with general linear node dynamics and intermittent communications

Links between different stabilities of switched homogeneous systems with delays and uncertainties

Floquet theory: a useful tool for understanding nonequilibrium dynamics

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