Input-to-state stability for switched systems with unstable subsystems: A hybrid Lyapunov construction

Yang, Guosong; Liberzon, Daniel

doi:10.1109/cdc.2014.7040367

Cited by 25 publications

(17 citation statements)

References 16 publications

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“…While not directly related to the proof of Theorem 1, this result is presented here to show the advantage of our method in comparison with that of [13]. More details on this result can be found in [27].…”

Section: Digression On Iss Of the Switched Systemsmentioning

confidence: 77%

A Lyapunov-based small-gain theorem for interconnected switched systems

Yang

Liberzon

2015

Systems & Control Letters

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Section: Digression On Iss Of the Switched Systemsmentioning

confidence: 77%

A Lyapunov-based small-gain theorem for interconnected switched systems

Yang

Liberzon

2015

Systems & Control Letters

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“…, q max } can be partitioned as Q = Q u ∪ Q s , where the modes q ∈ Q s characterize the stable dynamics, while the modes q ∈ Q u characterize the unstable dynamics. For this type of systems good behavior of the solutions can be guaranteed as long as the amount of activation time of the unstable modes Q u is bounded by a timeratio constraint (Muñoz de la Peña & Christofides, 2008; Yang & Liberzon, 2014), which has the form T (s, t) ≤ T 0 + η 2 (t − s), where η 2 ∈ [0, 1), T 0 ∈ R ≥0 , and where T (s, t) denotes the total activation time of modes q u ∈ Q u between times s and t. This type of time-ratio constraint can be induced by making use of an auxiliary state τ 2 ∈ R ≥0 associated to a time-ratio monitor that evolves according to the hybrid dynamicṡ…”

Section: Switched Learning Inclusions With Unstable Modesmentioning

confidence: 99%

A framework for a class of hybrid extremum seeking controllers with dynamic inclusions

2017

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a b s t r a c tThis paper presents a prescriptive framework for the design and analysis of a class of deterministic extremum seeking controllers (ESCs) based on hybrid dynamic inclusions. This type of ESCs combines continuous and discrete-time dynamics during the seeking process, and its evolution in time is characterized by differential and difference inclusions rather than standard difference and differential equations. Examples of hybrid ESCs include, but are not limited to, purely continuous ESC with optimizers described by set-valued mappings, ESC with arbitrarily fast and slow switching modes, ESC with weaklyjumping parameters, as well as distributed ESCs for multi-agent systems with time-varying graphs. Since solutions of systems described by set-valued mappings are usually not unique, we do not insist on this property, but rather we characterize the behavior of all possible solutions generated by the closed-loop system. Making use of recent results in singular perturbations, averaging, and Ω-limit sets of sets for hybrid dynamical systems, we establish a practical asymptotic stability result. Some examples of common classes of seeking dynamics described by hybrid systems are presented, as well as some numerical simulations illustrating their application in standard maximization problems and Nash seeking problems in game theoretical scenarios.

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“…Following this theory, the use of averaged models can be avoided, as the system's dynamical behavior in every switching state is inherently included in the model. Given a switched system, the Lyapunov theory is a proven tool for stability analysis [14][15][16][17]. Based on this theoretical background, refs.…”

Section: Introductionmentioning

confidence: 99%

Hysteresis-Based Current Control Approach by Means of the Theory of Switched Systems

Thielmann

Hans

2021

Applied Sciences

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In this paper, a novel hysteresis-based current control approach is presented. The basis of the developed control approach is the theory of switched systems, in particular, the system class of switched systems with multiple equilibria. The proposed approach guarantees the convergence of the state trajectory into a region around a reference trajectory by selective switching between the individual subsystems. Here, the reference trajectory is allowed to be time varying, but lies within the state space spanned by the subsystem equilibria. Since already published approaches only show convergence to a common equilibrium of all subsystems, the extension to the mentioned state space is a significant novelty. Moreover, the approach is not limited to the number of state variables, nor to the number of subsystems. Thus, the applicability to a large number of systems is given. In the course of the paper, the theoretical basics of the approach are first explained by referring to a trivial example system. Then, it is shown how the theory can be applied to a practical application of a voltage source converter that is connected to a permanent-magnet synchronous motor. After deriving the limits of the presented control strategy, a simulation study confirms the applicability on the converter system. The paper closes with a detailed discussion about the given results.

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Input-to-state stability for switched systems with unstable subsystems: A hybrid Lyapunov construction

Cited by 25 publications

References 16 publications

A Lyapunov-based small-gain theorem for interconnected switched systems

A Lyapunov-based small-gain theorem for interconnected switched systems

A framework for a class of hybrid extremum seeking controllers with dynamic inclusions

Hysteresis-Based Current Control Approach by Means of the Theory of Switched Systems

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