Computation and verification of contraction metrics for exponentially stable equilibria

Giesl, Peter; Hafstein, Sigurður; Mehrabinezhad, Iman

doi:10.1016/j.cam.2020.113332

Cited by 9 publications

(3 citation statements)

References 28 publications

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“…In Giesl, Hafstein & Mehrabinezhad 2021 [66], a different strategy is followed to compute contraction metrics for nonlinear systems with an exponentially stable equilibrium. First, a semidefinite optimization problem is proposed, of which every feasible solution delivers a contraction metric for the system.…”

Section: Control Systemsmentioning

confidence: 99%

Review on contraction analysis and computation of contraction metrics

Giesl¹,

Hafstein²,

Kawan³

2023

JCD

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Contraction analysis considers the distance between two adjacent trajectories. If this distance is contracting, then trajectories have the same long-term behavior. The main advantage of this analysis is that it is independent of the solutions under consideration. Using an appropriate metric, with respect to which the distance is contracting, one can show convergence to a unique equilibrium or, if attraction only occurs in certain directions, to a periodic orbit.Contraction analysis was originally considered for ordinary differential equations, but has been extended to discrete-time systems, control systems, delay equations and many other types of systems. Moreover, similar techniques can be applied for the estimation of the dimension of attractors and for the estimation of different notions of entropy (including topological entropy).This review attempts to link the references in both the mathematical and the engineering literature and, furthermore, point out the recent developments and algorithms in the computation of contraction metrics.

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Section: Control Systemsmentioning

confidence: 99%

Review on contraction analysis and computation of contraction metrics

Giesl¹,

Hafstein²,

Kawan³

2023

JCD

Self Cite

View full text Add to dashboard Cite

show abstract

“…In this paper we will present such a verification and, in addition, show that the verification can be combined with the procedure from [4] to deliver a method that is able to compute a contraction metric for any system with an exponentially stable periodic orbit. The main idea is similar to [8], in which we have provided a computation and verification method for contraction metrics in case of exponentially stable equilibrium points. As in [8] we show that our novel method is successful in computing a metric if sufficiently many points are used in the collocation (as in [4]) and sufficiently small simplices in the verification.…”

Section: Introductionmentioning

confidence: 99%

“…The main idea is similar to [8], in which we have provided a computation and verification method for contraction metrics in case of exponentially stable equilibrium points. As in [8] we show that our novel method is successful in computing a metric if sufficiently many points are used in the collocation (as in [4]) and sufficiently small simplices in the verification. However, in contrast to the case of an equilibrium, the contraction condition involves the restriction to the (n − 1)-dimensional subspace perpendicular to f (x) at each point x, which requires a more sophisticated argumentation.…”

Section: Introductionmentioning

confidence: 99%