TY - JOUR
T1 - Incremental stability of hybrid dynamical systems
AU - Biemond, J.J. Benjamin
AU - Postoyan, Romain
AU - Heemels, W.P. Maurice H.
AU - Van De Wouw, Nathan
PY - 2018
Y1 - 2018
N2 - The analysis of incremental stability typically involves measuring the distance between any two solutions of a given dynamical system at the same time instant, which is problematic when studying hybrid dynamical systems. Indeed, hybrid systems generate solutions defined with respect to hybrid time instances (that consists of both the continuous time elapsed and the discrete time, which is the number of jumps experienced so far), and two solutions of the same hybrid system may not be defined at the same hybrid time instant. To overcome this issue, we present novel definitions of incremental stability for hybrid systems based on graphical closeness of solutions. As we will show, defining incremental asymptotic stability with respect to the hybrid time yields a restrictive notion, such that we also investigate incremental asymptotic stability notions with respect to the continuous time only or the discrete time only, respectively. In this manner, two (effectively dual) incremental stability notions are attained, called jump- and flow incremental asymptotic stability. To present Lyapunov conditions for these two notions, in both cases, we resort to an extended hybrid system and we prove that the stability of a well-defined set for this extended system implies incremental stability of the original system. We can then use available Lyapunov conditions to infer the set stability of the extended system. Various examples are provided throughout this paper, including an event-triggered control application and a bouncing ball system with Zeno behavior, that illustrate incremental stability with respect to continuous time or discrete time, respectively.
AB - The analysis of incremental stability typically involves measuring the distance between any two solutions of a given dynamical system at the same time instant, which is problematic when studying hybrid dynamical systems. Indeed, hybrid systems generate solutions defined with respect to hybrid time instances (that consists of both the continuous time elapsed and the discrete time, which is the number of jumps experienced so far), and two solutions of the same hybrid system may not be defined at the same hybrid time instant. To overcome this issue, we present novel definitions of incremental stability for hybrid systems based on graphical closeness of solutions. As we will show, defining incremental asymptotic stability with respect to the hybrid time yields a restrictive notion, such that we also investigate incremental asymptotic stability notions with respect to the continuous time only or the discrete time only, respectively. In this manner, two (effectively dual) incremental stability notions are attained, called jump- and flow incremental asymptotic stability. To present Lyapunov conditions for these two notions, in both cases, we resort to an extended hybrid system and we prove that the stability of a well-defined set for this extended system implies incremental stability of the original system. We can then use available Lyapunov conditions to infer the set stability of the extended system. Various examples are provided throughout this paper, including an event-triggered control application and a bouncing ball system with Zeno behavior, that illustrate incremental stability with respect to continuous time or discrete time, respectively.
KW - Hybrid systems
KW - incremental stability
KW - Lyapunov stability
UR - http://www.scopus.com/inward/record.url?scp=85045980966&partnerID=8YFLogxK
U2 - 10.1109/TAC.2018.2830506
DO - 10.1109/TAC.2018.2830506
M3 - Article
AN - SCOPUS:85045980966
SN - 0018-9286
VL - 63
SP - 4094
EP - 4109
JO - IEEE Transactions on Automatic Control
JF - IEEE Transactions on Automatic Control
IS - 12
M1 - 8350276
ER -