A new class of universal Lyapunov functions for the control of uncertain linear systems

Blanchini, Franco; Miani, Stefano

doi:10.1109/9.751368

Cited by 129 publications

(101 citation statements)

References 21 publications

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“…The main idea is then to seek for a (homogeneous) polynomialW whose level sets approximate, in some suitable sense, those of W and, finally to show thatW is also a CLF. A related result has also been obtained in [6], using an intermediate approximation with polyhedral functions (this step is implicit in our proof) and starting from the case of discrete approximating systems (so-called Euler approximating systems).…”

Section: Theorem 1bismentioning

confidence: 96%

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Common Polynomial Lyapunov Functions for Linear Switched Systems

Mason¹,

Boscain²,

Chitour³

2006

SIAM J. Control Optim.

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In this paper, we consider linear switched systemsẋ(t) = A u(t) x(t), x ∈ R n , u ∈ U , and the problem of asymptotic stability for arbitrary switching functions, uniform with respect to switching (UAS for short). We first prove that, given a UAS system, it is always possible to build a common polynomial Lyapunov function. Then our main result is that the degree of that common polynomial Lyapunov function is not uniformly bounded over all the UAS systems. This result answers a question raised by Dayawansa and Martin. A generalization to a class of piecewise-polynomial Lyapunov functions is given.

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Section: Theorem 1bismentioning

confidence: 96%

“…Remark 4 In [6], a concept similar to those introduced in the previous definition was provided and it was called "universal class of Lyapunov functions".…”

Section: Sets Of Functions Sufficient To Check Guesmentioning

confidence: 99%

Common Polynomial Lyapunov Functions for Linear Switched Systems

Mason¹,

Boscain²,

Chitour³

2006

SIAM J. Control Optim.

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“…Les questions de stabilité dans le cas où X et Y sont linéaires, en dimension deux ou plus, ont produit une littérature importante ( [3,6,8,16,20,22]). La réponse complète à ces questions en dimension 2 a été donnée dans [8].…”

Section: Q(t) = U(t)x(q(t)) + (1 − U(t))y (Q(t)unclassified

Stabilité des systèmes à commutations du plan

Boscain¹,

Charlot²,

Sigalotti³

2010

Séminaire De Théorie Spectrale Et Géométrie

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“…By Lemma 4, there exists a set of full column rank matrices M i such that V i = M i x ∞ , i ∈ m, are valid Lyapunov functions. By Blanchini and Miani (1999), M i x 2p converges uniformly as p approaches Ý. Therefore, V i may be chosen as V i = M i x 2p for some p ∈ N. Moreover, it is easy to verify that V α i is also a valid Lyapunov function for any α ∈ N, and therefore we can take V i = M i x 2p 2p .…”

Section: It Is Not Difficult To See Thatmentioning

confidence: 99%

Stability analysis of discrete-time switched systems: a switched homogeneous Lyapunov function method

Liu

Zhao

2015

International Journal of Control

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This paper addresses the stability issue of discrete-time switched systems with guaranteed dwelltime. The approach of switched homogeneous Lyapunov function of higher order is formally proposed. By means of this approach, a necessary and sufficient condition is established to check the exponential stability of the considered system. With the observation that switching signal is actually arbitrary if the dwell time is one sample time, a necessary and sufficient condition is also presented to verify the exponential stability of switched systems under arbitrary switching signals. Using the augmented argument, a necessary and sufficient exponential stability criterion is given for discretetime switched systems with delays. A numerical example is provided to show the advantages of the theoretical results.

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A new class of universal Lyapunov functions for the control of uncertain linear systems

Cited by 129 publications

References 21 publications

Common Polynomial Lyapunov Functions for Linear Switched Systems

Common Polynomial Lyapunov Functions for Linear Switched Systems

Stabilité des systèmes à commutations du plan

Stability analysis of discrete-time switched systems: a switched homogeneous Lyapunov function method

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