Some Extensions of Liapunov's Second Method

LaSalle, J. P.

doi:10.1109/tct.1960.1086720

Cited by 731 publications

(370 citation statements)

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“…(Note that performing a constant shift outside N p (κ) will scale ε by α p , but we could just perform the construction again with a larger ε to correct for this). However, relying on the construction of Lemma 2 may not allow for (10) to hold, and we must assume that we can find a set of Lyapunov functions which satisfy both the exponential decay property and the μ property, (10). It is left as an open problem to determine if it is generally possible to construct a set of Lyapunov functions that satisfy both conditions for any switched system.…”

Section: Lemma 2 Consider a Switched System With Lyapunov Functionsmentioning

confidence: 99%

“…Systems with bifurcation often contain multiple ω-limit sets which cannot be globally exponentially stable. Instead, results such as LaSalle's invariant set theorem [10] allow us to analyze asymptotic stability of this larger class of systems. LaSalle's theorem and much of the switched systems literature are both Lyapunov-based, and we will make use of Lyapunov functions to define all the relevant sets.…”

Section: Introductionmentioning

confidence: 99%

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Switched systems with multiple invariant sets

Dorothy

Chung

2016

Systems & Control Letters

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This paper explores dwell time constraints on switched systems with multiple, possibly disparate invariant limit sets. We show that, under suitable conditions, trajectories globally converge to a superset of the limit sets and then remain in a second, larger superset. We show the effectiveness of the dwell-time conditions by using examples of switching limit cycles commonly found in robotic locomotion and flapping flight.

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Section: Lemma 2 Consider a Switched System With Lyapunov Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Switched systems with multiple invariant sets

Dorothy

Chung

2016

Systems & Control Letters

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“…In both phases we utilize LaSalle's theorem [20], to enforce the attracttivity to this confinement. A positive definite Lyapunov candidate for agent i is proposed as…”

Section: Sliding Mode Controller (Smc)mentioning

confidence: 99%

Robust Region Tracking for Swarms via a Novel Utilization of Sliding Mode Control

Bacon¹,

Olgaç

Cepeda-Gomez³

2012

ICA

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Control of multi-agent autonomous swarms is studied for targeted flocking exercises. The desired decentralized control takes into account robustness against modeling uncertainties as well as bounded unknown forces. In this analysis, we consider the task of driving multiple agents to a moving "target region", as inter-agent repulsive forces help spread out the agents within the region. An unconventional form of sliding mode control is implemented to provide the robust attraction towards the region's center. For robustness a finite "boundary layer" is conceived, which corresponds to the desired target region. The flocking control forces are intentionally softened inside this target region, allowing agents to create a uniformly spaced formation guided by the inter-agent repulsion forces. Examples are given for moving circular and elliptical regions which illustrate the effectiveness of the proposed strategy.

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“…Recalling that asymptotic stability is equivalent to stability and convergence (Rouche et al, 1977), this result is obtained by noting that the semidefiniteness of the derivative of the Lyapunov function implies (set) stability, while convergence follows from LaSalle's invariance principle (LaSalle, 1960).…”

Section: State Feedback Set Stabilizationmentioning

confidence: 99%