Decay rates for stabilization of linear continuous-time systems with random switching

Colonius, Fritz; Mazanti, Guilherme

doi:10.3934/mcrf.2019002

Cited by 5 publications

(3 citation statements)

References 32 publications

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“…These numbers give the exponential growth rate of a linear Random Dynamical System (see Arnold [1] for the definition). In [9], the authors show that the process Y from (1.10) together with the canonical shift on the space Ω of cadlag functions from R + to E is a linear ergodic random dynamical system satisfying the integrability conditions of Osedelet's Multiplicative Ergodic Theorem (see [1,Theorem 3.4.1] or [14,Proposition 3.12]). According to this theorem, there exist…”

Section: A Results On Linear Switched Systemsmentioning

confidence: 99%

On a Predator-Prey System with Random Switching that Never Converges to its Equilibrium

Hening¹,

Strickler²

2019

SIAM J. Math. Anal.

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We study the dynamics of a predator-prey system in a random environment. The dynamics evolves according to a deterministic Lotka-Volterra system for an exponential random time after which it switches to a different deterministic Lotka-Volterra system. This switching procedure is then repeated. The resulting process is a Piecewise Deterministic Markov Process (PDMP). In the case when the equilibrium points of the two deterministic Lotka-Volterra systems coincide we show that almost surely the trajectory does not converge to the common deterministic equilibrium. Instead, with probability one, the densities of the prey and the predator oscillate between 0 and ∞. This proves a conjecture of Takeuchi et al (J. Math. Anal. Appl 2006).The proof of the conjecture is a corollary of a result we prove about linear switched systems. Assume (Y t , I t ) is a PDMP that evolves according to dYt dt = A It Y t where A 0 , A 1 are 2 × 2 matrices and I t is a Markov chain on {0, 1} with transition rates k 0 , k 1 > 0. If the matrices A 0 and A 1 are not proportional and are of the formwith α 2 i + β i γ i < 0, then there exists λ > 0 such that lim t→∞ log Yt t = λ.

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Section: A Results On Linear Switched Systemsmentioning

confidence: 99%

On a Predator-Prey System with Random Switching that Never Converges to its Equilibrium

Hening¹,

Strickler²

2019

SIAM J. Math. Anal.

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“…In analogy with [8], one could further generalize the above construction by allowing λ to be a uniformly positive function depending on the indices L n and L n+1 and on the point Q n . Another possible type of generalization, following [15], would consist in allowing more general probability densities than f , supported in [τ, ∞). We prefer to restrict the framework in order to keep notations reasonably simple.…”

Section: Piecewise Deterministic Dwell-time Random Processesmentioning

confidence: 99%

Dwell-time control sets and applications to the stability analysis of linear switched systems

Boarotto

Sigalotti

2020

Journal of Differential Equations

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We propose an extension of the theory of control sets to the case of inputs satisfying a dwell-time constraint. Although the class of such inputs is not closed under concatenation, we propose a suitably modified definition of control sets that allows to recover some important properties known in the concatenable case. In particular we apply the control set construction to dwell-time linear switched systems, characterizing their maximal Lyapunov exponent looking only at trajectories whose angular component is periodic. We also use such a construction to characterize supports of invariant measures for random switched systems with dwell-time constraints.

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“…The bang–bang control principle based on the Pontryagin maximum principle, the sliding mode control, variable structure control, multi-controller, and multi-model control, which have been commonly used in electrical systems, all represent the application of the switching idea in the engineering field. 15–18 The reason for researching the switching systems is mainly because most of the actual systems represent the multi-state systems, such as temperature control systems, power systems, 19 motor systems, 20 intelligent interaction systems, precision device processing systems, 21–24 and others. In the image signal processing field, the use of switching rules can simultaneously meet the requirements for real-time and accurate gesture interaction.…”

Section: Introductionmentioning

confidence: 99%

Contactless interactive control technology with a switching system based on Butterworth filter and modified strong tracking Kalman filter

Fang

Chen

et al. 2021

Proceedings of the Institution of Mechanical Engineers, Part I:

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In the human–computer interaction field, a contactless interaction with large screens through gestures is very representative, and the recognition and filtering of gesture images are very important tasks. Aiming at solving the problems of interference and positioning drift of three-dimensional lidar sensors, this article proposes a contactless interactive control system based on switching filtering algorithm, which selects the Butterworth filtering and the modified strong tracking Kalman filter to be used in the filtering process. The proposed interactive system extracts and optimizes user gestures, maps the gestures to the screen, simulates mouse operations, and enables operations such as selection, sliding, zooming in and out, and others. This switching filtering algorithm effectively solves the accuracy problem of a single filtering algorithm and the rapidity of complex filtering algorithms in the signal processing step, and greatly improves the interaction accuracy without sacrificing too much processing time. The experimental results show that by applying the proposed switching filtering algorithm to a contactless human–computer interaction system, the system can achieve smooth gesture interaction. The proposed system can perform real-time interaction with multiple people, which fully verifies the effectiveness and superiority of the proposed algorithm.

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Decay rates for stabilization of linear continuous-time systems with random switching

Cited by 5 publications

References 32 publications

On a Predator-Prey System with Random Switching that Never Converges to its Equilibrium

On a Predator-Prey System with Random Switching that Never Converges to its Equilibrium

Dwell-time control sets and applications to the stability analysis of linear switched systems

Contactless interactive control technology with a switching system based on Butterworth filter and modified strong tracking Kalman filter

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