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Stability and Completion of Zeno Equilibria in Lagrangian Hybrid Systems

Stability and Completion of Zeno Equilibria in Lagrangian Hybrid Systems,10.1109/TAC.2010.2080790,IEEE Transactions on Automatic Control,Yizhar,Aaron

Stability and Completion of Zeno Equilibria in Lagrangian Hybrid Systems   (Citations: 2)
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This paper studies Lagrangian hybrid systems, which are a special class of hybrid systems modeling mechanical systems with unilateral constraints undergoing impacts. This class of systems naturally display Zeno behavior— an infinite number of discrete transitions that occur in finite time, leading to the convergence of solutions to limits set called Zeno equilibria. This paper derives simple conditions for stability of Zeno equilibria. Utilizing these results and the constructive proof techniques used to prove them, the paper introduces the notion of a completed hybrid system which is an extended hybrid system model allowing for the extension of solutions beyond Zeno points. A procedure for practical simulation of completed hybrid systems is outlined, and conditions guaranteeing upper bounds on the incurred numerical error are derived. Finally, we discuss an interpretation of these results to the stability of unilaterally constrained motion of mechanical systems under perturbations that violate the constraint.
Journal: IEEE Transactions on Automatic Control - IEEE TRANS AUTOMAT CONTR , vol. 56, no. 6, pp. 1322-1336, 2011
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    • ...Zeno behavior has recently gained increasing interest, in works studying conditions for existence of Zeno behavior [6], [13], [24] and its relation to asymptotic stability [1], [10], [17], [20]...
    • ...It was recently shown in [14] that a necessary and sufficient condition for existence of Zeno solutions in the vicinity of a Zeno equilibrium point is that , where is the second-order time derivative of the constraint function along trajectories of the system’s continuous-time dynamics, evaluated at . Moreover, the same condition also implies local stability of [17]...
    • ...Two fundamental questions that naturally arise are: Can one obtain an explicit expression for a neighborhood of initial conditions all leading to Zeno solutions? Can one derive bounds on the Zeno times and Zeno limit points of solutions starting at a given neighborhood? Answering these questions may prove useful for several applications, such as establishing bounds on numerical errors in simulation of Zeno solutions [2], [17], and deriving ...
    • ...In order to model the dynamics in this phase, which is not captured by the model, one has to consider the completed hybrid system model [2], [17], or alternatively, use more sophisticated complementarity formulations that describe non-smooth force-velocity relations (e.g., [3], [15])...

    Yizharet al. Zeno Stability of the Set-Valued Bouncing Ball

    • ...Traditionally, completed hybrid systems have been defined in the following manner [3, 4, 18, 22, 23, 24] (and are often termed complementary Lagrangian hybrid systems): if L is a simple hybrid Lagrangian and SH L the corresponding...
    • ...where λ is the Lagrange multiplier obtained from h .S ystems of this form have been well-studied in the above references, and conditions have been given on how to practically simulate completed hybrid systems by truncating the executions in a formal manner (see [23, 24])...

    Aaron D. Ames. Characterizing knee-bounce in bipedal robotic walking: a zeno behavior...

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