A Converse Lyapunov Theorem and Robustness for Asymptotic Stability in Probability

Teel, Andrew R.; Hespanha, João P.; Subbaraman, Anantharaman

doi:10.1109/tac.2014.2322431

Cited by 35 publications

(27 citation statements)

References 20 publications

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“…We adopt the notation of [10]. IR � o (IR>o) denotes the set of non-negative (positive) real numbers, and 2: � 0 (2:>0) denotes the set of non-negative (positive) integers.…”

Section: A Notationmentioning

confidence: 99%

“…Let As in [10], a random process Z from Z E JRn is a sequence of random variables Zi : dom Zi C 0 -+ JRn, i E 2:20, with Zo = Z for all W E 0 and dom Zi+l C dom Zi for all i E 2:20. A random process Z is adapted to the natural filtration of V if Z H I is Fi-measurable for each i E 2:>0.…”

Section: Uniform Global Asymptotic Stability In Probabilitymentioning

confidence: 99%

“…In [8] stability for discrete-time stochastic systems with unique solutions is discussed. A framework for analyzing set-valued, discrete-time stochastic systems is proposed in [9], [10]. For a class of stochastic difference inclusions satisfying basic regularity properties, sufficient conditions in terms of Lyapunov functions for guaranteeing global asymptotic stability in probability and recurrence are established in [11].…”

Section: Introductionmentioning

confidence: 99%

“…For a class of stochastic difference inclusions satisfying basic regularity properties, sufficient conditions in terms of Lyapunov functions for guaranteeing global asymptotic stability in probability and recurrence are established in [11]. In [10], [12], converse Lyapunov theorems for certain stability notions are estab lished for these systems.…”

Section: Introductionmentioning

confidence: 99%

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Discrete-time stochastic control systems: Examples of robustness to strictly causal perturbations

Grammatico

Subbaraman

Teel

2013

52nd IEEE Conference on Decision and Control

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Robust stability for discrete-time stochastic systems employing possibly discontinuous control laws is the focus of this paper. Through novel examples, we illustrate the fact that the existence of a continuous stochastic Lyapunov function implies robustness of stability to sufficiently small, state-dependent, strictly causal, worst-case perturbations, irrespective to the continuity of the stabilizing control law. We emphasize the role of strict causality through an example for which a continuous stochastic Lyapunov function is not sufficient for robustness to arbitrarily small worst-case perturbations, which are not strictly causal. Finally we illustrate our main result for a stochastic control system which admits a continuous Lyapunov function, but associated with a necessarily discontinuous control law

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“…We adopt the notation of [10]. IR � o (IR>o) denotes the set of non-negative (positive) real numbers, and 2: � 0 (2:>0) denotes the set of non-negative (positive) integers.…”

Section: A Notationmentioning

confidence: 99%

Section: Uniform Global Asymptotic Stability In Probabilitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Discrete-time stochastic control systems: Examples of robustness to strictly causal perturbations

Grammatico

Subbaraman

Teel

2013

52nd IEEE Conference on Decision and Control

Self Cite

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“…Stochastic hybrid inclusions have appeared recently in [2], [3], [4], [5], [6]; the formulation takes inspiration from non-stochastic hybrid inclusions [18], [19], [15], which are discussed briefly below. Closely related results for stochastic difference inclusions, where randomness interacts with non-uniqueness for purely discrete-time systems, have appeared in [20], [21], [22], [23], [24], [25], [26]. Earlier, independent progress has also been made on stochastic differential inclusions; see [27] or [28] for example.…”

Section: Stochastic Hybrid Inclusions a Introductionmentioning

confidence: 99%

Stochastic hybrid systems: A modeling and stability theory tutorial

Teel

Hespanha

2015

2015 54th IEEE Conference on Decision and Control (CDC)

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Stochastic hybrid systems are driven by random processes and have states that can both flow continuously and jump instantaneously. Many classes of stochastic hybrid systems, with different modeling strengths, have been considered in the literature. In this tutorial paper we first consider perhaps the simplest class of stochastic hybrid systems: those that admit unique solutions and that do not permit state conditions that force jumps. Several examples are given to illustrate the utility of this simple modeling class and Lyapunov-based sufficient conditions for various stability properties are given. The second half of the tutorial addresses a recent, more general stochastic hybrid systems modeling framework that permits state conditions to trigger jumps and that allows for non-unique solutions, via stochastic differential and difference inclusions and possibly overlapping flow and jump sets. Examples are provided to show the relevance of models that admit nonunique solutions and forced jumps. Lyapunov-based sufficient conditions for various stability properties for this class of stochastic hybrid systems are also provided.

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Recent Developments in Stability Theory for Stochastic Hybrid Inclusions

Teel

2017

Feedback Stabilization of Controlled Dynamical Systems

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A Converse Lyapunov Theorem and Robustness for Asymptotic Stability in Probability

Cited by 35 publications

References 20 publications

Discrete-time stochastic control systems: Examples of robustness to strictly causal perturbations

Discrete-time stochastic control systems: Examples of robustness to strictly causal perturbations

Stochastic hybrid systems: A modeling and stability theory tutorial

Recent Developments in Stability Theory for Stochastic Hybrid Inclusions

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