A common Lyapunov function for stable LTI systems with commuting A-matrices

Narendra, Kumpati S.; Balakrishnan, J.

doi:10.1109/9.362846

Cited by 588 publications

(313 citation statements)

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“…As for the nonlinear case, we mention here the recent work [13] which directly generalizes the result and the proof technique of [10] to switched nonlinear systems. The main result of [13] states that a finite family of commuting vector fields that give rise to exponentially stable systems has a common Lyapunov function.…”

Section: Discussionmentioning

confidence: 99%

“…The problem of finding conditions that guarantee asymptotic stability of (1) for an arbitrary switching signal σ has recently attracted a considerable amount of attention-see the work reported in [2,3,8,9,10,11,14,15] and the references therein. Some of the aforementioned results suggest that certain properties of the Lie algebra {A p : p ∈ P} LA generated by the matrices A p may be of relevance to the question of stability of (1).…”

Section: Introductionmentioning

confidence: 99%

“…In particular, it is well known and easy to show that if these matrices commute pairwise, i.e., the Lie bracket [A p , A q ] := A p A q − A q A p equals zero for all p, q ∈ P, and if P is a finite set, then the system (1) is asymptotically stable for any switching signal σ. An explicit construction of a quadratic common Lyapunov function for the family of linear systemsẋ = A p x, p ∈ P (2) in this case is given in [10].…”

Section: Introductionmentioning

confidence: 99%

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Stability of switched systems: a Lie-algebraic condition

Liberzon

Hespanha

Morse

1999

Systems & Control Letters

602

324

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We present a sufficient condition for asymptotic stability of a switched linear system in terms of the Lie algebra generated by the individual matrices. Namely, if this Lie algebra is solvable, then the switched system is exponentially stable for arbitrary switching. In fact, we show that any family of linear systems satisfying this condition possesses a quadratic common Lyapunov function. We also discuss the implications of this result for switched nonlinear systems.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Stability of switched systems: a Lie-algebraic condition

Liberzon

Hespanha

Morse

1999

Systems & Control Letters

602

324

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“…The existence of a common quadratic Lyapunov function implies the exponential stability of the switching system (3) as discussed by Narendra & Balakrishnan (1994).…”

Section: Preliminary Resultsmentioning

confidence: 99%

A new methodology for the stability analysis of pairwise triangularizable and related switching systems

Shorten¹

2002

IMA Journal of Applied Mathematics

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“…Finally we add that since the set of stabilizing inputs U , results in a set of closed loop systems which share V(q) as a common Lyapunov function, it can be shown (see [9]) that a system whose right hand side switches between these inputs is also stabilizing, regardless of the nature of the switching sequence. This implies that we are free to choose the values of a online, in a possibly discontinuous or time varying fashion, without affecting the overall stability of the system or the completeness of the solution.…”

Section: Navigation Functionsmentioning

confidence: 99%

A method for modifying closed-loop motion plans to satisfy unpredictable dynamic constraints at runtime

Esposito

Kumar

Proceedings 2002 IEEE International Conference on Robotics and Automation (Cat. No.02CH37292)

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In this paper; the problem of motion planning in environments with both known static obstacles and unpredictable dynamic constraints is considered. A methodology is introduced in which the motion plan for the static environment is modified on-line to accommodate the unpredictable constraints in such a way that the completeness properties of the original motion plan are preserved. At the heart of the approach is the idea that Navigation functions are indeed Lyapunov functions; and that the traditional method of forcing the robot to track the negative gradient of field is not the only input which stabilizes the system. This extra freedom in selecting the input is used to accommodate the dynamic constraints. A computational method for selecting the appropriate inputs is given. The method is used to solve two sample problems. The constraints in these cases are used to model collisions with other robots and, in the second example, a team of robots traveling in formation. Finally, some preliminary work on extending the approach to nonholonomic systems is presented. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the University of Pennsylvania's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to pubs-permissions@ieee.org. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.This conference paper is available at ScholarlyCommons: http://repository.upenn.edu/meam_papers/13

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A common Lyapunov function for stable LTI systems with commuting A-matrices

Cited by 588 publications

References 6 publications

Stability of switched systems: a Lie-algebraic condition

Stability of switched systems: a Lie-algebraic condition

A new methodology for the stability analysis of pairwise triangularizable and related switching systems

A method for modifying closed-loop motion plans to satisfy unpredictable dynamic constraints at runtime

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