Switched Systems With Multiple Invariant Sets

Dorothy, Michael; Chung, Soon‐Jo

doi:10.21236/ada625045

Cited by 4 publications

(16 citation statements)

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“…One notable example which utilizes dwell-time criteria for nonlinear systems is in [46], where it is shown that input-to-state induced norms should be bounded uniformly between switches. In terms of applications, dwell-time criteria for attaining exponential stability has been shown to be effective for robotic systems, in particular walking locomotion and flapping flight [47] as well as autonomous vehicle steering [48].…”

Section: Corollary 1 (Lévy Noise Stochastic Contraction Theorem)mentioning

confidence: 99%

Incremental Nonlinear Stability Analysis of Stochastic Systems Perturbed by Lévy Noise

Han,

Chung

2021

Preprint

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We present a theoretical framework for characterizing incremental stability of nonlinear stochastic systems perturbed by additive noise of two types: compound Poisson shot noise and bounded-measure Lévy noise. For each type, we show that trajectories of the system with distinct initial conditions and noise sample paths exponentially converge towards a bounded error ball of each other in the expected mean-squared sense under certain boundedness assumptions. The practical utilities of this study include the model-based design of stochastic controllers/observers that are able to handle a much broader class of noise than Gaussian white. We demonstrate our results using three case studies.

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Section: Corollary 1 (Lévy Noise Stochastic Contraction Theorem)mentioning

confidence: 99%

Incremental Nonlinear Stability Analysis of Stochastic Systems Perturbed by Lévy Noise

Han,

Chung

2021

Preprint

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“…It is assumed that, for each p ∈ P, the vector field f p : R n ×R m → R n is locally Lipschitz in its arguments, and that there exists a unique x * p ∈ R n with 0 = f p (x * p , 0). As in Section II-A, we allow for x * p = x * q when p = q. Analogous to Section II-A, we will require each system in the family (13) to be input-to-state stable, as defined below.…”

Section: B Switched Continuous Systemsmentioning

confidence: 99%

“…Various applications demand switching among systems that do not share a common equilibrium-such as planning motions of legged [11], [12] and aerial [13] robots, cooperative manipulation among multiple robotic arms [14], power control in multi-cell wireless networks [15], and models for nonspiking of neurons [16]. Such systems are referred to in the literature as switched systems with multiple equilibria.…”

Section: Introductionmentioning

confidence: 99%

“…The notion of modal dwell-time was introduced in [18], which provided switch-dependent dwell-time bounds, while [19] established boundedness of solutions via practical stability. The dwell-time bound of [17] was extended to discrete switched systems in [11] and to continuous switched systems with invariant sets in [13]. Yet, papers that deal with multiple equilibria/invariant sets do not study the effect of switching in the presence of disturbances.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, the size of this set grows monotonically with the supremum norm of the disturbance. With respect to prior literature on switched systems with multiple equilibria, e.g., [11], [13], [16]- [19], the results of this paper explicitly consider the effect of disturbances on the solutions of the switched system. In addition, they constitute a natural generalization of known results in the common equilibrium case; indeed, when the equilibria of the individual subsystems coalesce, ISS of the switched system can be recovered as a simple consequence of our main results, Theorems 1 and 2; see Corollaries 1 and 3.…”

Section: Introductionmentioning

confidence: 99%

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Ultimate Boundedness for Switched Systems with Multiple Equilibria Under Disturbances

Veer,

Poulakakis

2018

Preprint

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In this paper, we investigate the robustness to external disturbances of switched discrete and continuous systems with multiple equilibria. It is shown that if each subsystem of the switched system is Input-to-State Stable (ISS), then under switching signals that satisfy an average dwell-time bound, the solutions are ultimately bounded within a compact set. Furthermore, the size of this set varies monotonically with the supremum norm of the disturbance signal. It is observed that when the subsystems share a common equilibrium, ISS is recovered for solutions of the corresponding switched system; hence, the results in this paper are a natural generalization of classical results in switched systems that exhibit a common equilibrium. Additionally, we provide a method to analytically compute the average dwell time if each subsystem possesses a quadratic ISS-Lyapunov function. Our motivation for studying this class of switched systems arises from certain motion planning problems in robotics, where primitive motions, each corresponding to an equilibrium point of a dynamical system, must be composed to realize a task. However, the results are relevant to a much broader class of applications, in which composition of different modes of behavior is required.Index Terms-Switched systems with multiple equilibria; input-to-State Stability; ultimate boundedness.1 To clarify terminology, "switched systems with multiple equilibria" refers to switching among subsystems each of which exhibits a unique equilibrium which may not coincide with the equilibrium of another subsystem.

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Incremental nonlinear stability analysis of stochastic systems perturbed by Lévy noise

Han

Chung

2022

Intl J Robust & Nonlinear

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We present a theoretical framework for characterizing incremental stability of nonlinear stochastic systems perturbed by either compound Poisson shot noise or finite-measure Lévy noise. For each noise type, we compare trajectories of the perturbed system with distinct noise sample paths against trajectories of the nominal, unperturbed system. We show that for a finite number of jumps arising from the noise process, the mean-squared error between the trajectories exponentially converge toward a bounded error ball across a finite interval of time under practical boundedness assumptions. The convergence rate for shot noise systems is the same as the exponentially stable nominal system, but with a tradeoff between the parameters of the shot noise process and the size of the error ball. The convergence rate and the error ball for the Lévy noise system are shown to be nearly direct sums of the respective quantities for the shot and white noise systems separately, a result which is analogous to the Lévy-Khintchine theorem.We demonstrate both empirical and analytical computation of the error ball using several numerical examples, and illustrate how varying the parameters of the system affect the tightness of the bound.

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Switched Systems With Multiple Invariant Sets

Cited by 4 publications

References 4 publications

Incremental Nonlinear Stability Analysis of Stochastic Systems Perturbed by Lévy Noise

Incremental Nonlinear Stability Analysis of Stochastic Systems Perturbed by Lévy Noise

Ultimate Boundedness for Switched Systems with Multiple Equilibria Under Disturbances

Incremental nonlinear stability analysis of stochastic systems perturbed by Lévy noise

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