Stability and Completion of Zeno Equilibria in Lagrangian Hybrid Systems

Abstract

This paper studies Lagrangian hybrid systems, which are a special class of hybrid systems modeling mechanical systems with unilateral constraints that are 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 limit sets called Zeno equilibria. This paper derives simple conditions for stability of Zeno equilibria. Utilizing these results and the constructive 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 application of these results to the stability of unilaterally constrained motion of mechanical systems under perturbations that violate the constraint.Index Terms-Langrangian hybrid systems, stability, unilateral contact constraints, Zeno behavior.

Lyapunov analysis of rigid body systems with impacts and friction via sums-of-squares

“…Example applications of this framework to the control and analysis of hybrid mechanical systems can be found in [17], [26], and [21]. A number of numerical techniques have been presented for addressing verification, stability, and control design of general hybrid automata.…”

“…For instance, we might wish to determine the largest set of initial conditions such that the infinite horizon reachable set does not intersect some unsafe semialgebraic set U. We pose this problem in a manner similar (21), although here we do not require that V be positive definite:…”

“…Taking the total mechanical energy of the system as an initial candidate Lyapunov function, we search for a solution to (21) to find a nested set of invariant regions and verify Under the black curve is Gi, tight to Ω1 ∩ A, which we use to approximate the volume of the verified region. Under the magenta curve is Go, which must contain C. Go is parameterized as an ellipsoid in the redundant state variables, including s and (1−c), which is why it is not convex when plotted against θ.…”

Zeno phenomenon in hybrid dynamical systems

“…This phenomenon can lead to unnatural loss of stability of equilibriums and the emergence of unexpected and meaningless solutions in case of interconnected systems. Many works [2,4,5] are devoted to the question of prolongation of such solutions, however there exist no unified or commonly accepted prolongation method. In this paper we are focused on asymptotic stability properties for this kind of solutions, which is also not completely solved in the literature.…”

Controlador integrado baseado em conceito híbrido e controle geométrico para embarcações posicionadas dinamicamente.

“…Stability issues in hybrid dynamical systems are found in Decarlo et al (2000), Michel and Wang (1995), Or and Ames (2001), Sanfelice, Goebel and Teel (2007) and Ye, Michel and Hou (1998). In marine applications, Smogeli, Sorensen and Fossen (2004) developed a hybrid controller to control those thrusters that combine different operational loads.…”