Well-posed bimodal piecewise linear systems do not exhibit Zeno behavior

Abstract

The phenomenon of infinitely mode transitions in a finite time interval is called Zeno behavior in hybrid systems literature. It plays a critical role in the study of numerical methods and fundamental system and control theoretic properties of hybrid systems. This paper studies Zeno behavior for bimodal piecewise linear systems with possibly discontinuous dynamics. Our treatment is inspired by the work of Imura and Van der Schaft on the well-posedness of the same type of systems. The main contribution of the paper is two folded. Firstly, we show that ImuraVan der Schaft conditions for well-posedness guarantee that Filippov solutions have certain local properties. Secondly, we employ these in order to prove that bimodal piecewise linear systems do not exhibit Zeno behavior.

Well posedness conditions for Bimodal Piecewise Affine Systems

“…In each case, the solutions are both forward and backward Carathéodory solutions. Since forward (backward) Carathéodory solutions rule out left (right) Zeno behavior as noted in [18], it follows that well-posed BPAS considered in this work do not exhibit Zeno behavior.…”

“…An interesting dynamic behavior encountered in PLS is the existence of trajectories which change mode infinite number of times in a finite time interval. This paradoxical behavior is called Zeno behavior and investigated in a series of papers [15,18,19]. Moreover, well-posedness is also an essential issue in problems such as stability, stabilizability and feedback http://dx.doi.org/10.1016/j.sysconle.2015.06.002 0167-6911/© 2015 Elsevier B.V. All rights reserved.…”

“…Imura [22] investigated well-posedness of BPAS with control inputs in R n and presented a classification of well-posed feedback systems in R n . The papers by Çamlıbel et al [15], by Çamlıbel [18], and by Thuan and Çamlıbel [19] have also provided important contributions on the Zeno behavior of well-posed PLS.…”

The effect of coupling conditions on the stability of bimodal systems inR3

“…This issue is investigated extensively in [8][9][10][11]. BPLS is also investigated in the context of stability, stabilizability and control in [12][13][14][15][16][17]7].…”

Robust Non-Zenoness of Piecewise Affine Systems with Applications to Linear Complementarity Systems

“…This technique and its extension is applied to the linear complementarity systems (LCSs) with P-property [32] and singleton properties [33], strongly regular nonlinear complementarity systems [22], differential quasi-variational inequalities [15], and Lipschitz piecewise linear/afhne systems [6,39]. Furthermore, Qarnlibel shows that a well-posed bimodal piecewise linear/afhne system with a dis continuous right-hand side is non-Zeno [8,40],…”