### Abstract

Original language | English |
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Pages (from-to) | 3079-3086 |

Number of pages | 8 |

Journal | Automatica |

Volume | 44 |

Issue number | 12 |

DOIs | |

Publication status | Published - 2008 |

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*Automatica*, vol. 44, no. 12, pp. 3079-3086. https://doi.org/10.1016/j.automatica.2008.04.025

**Input-to-state stability and interconnections of discontinuous dynamical systems.** / Heemels, W.P.M.H.; Weiland, S.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - Input-to-state stability and interconnections of discontinuous dynamical systems

AU - Heemels, W.P.M.H.

AU - Weiland, S.

PY - 2008

Y1 - 2008

N2 - In this paper we will extend the input-to-state stability (ISS) framework to continuous-time discontinuous dynamical systems (DDS) adopting piecewise smooth ISS Lyapunov functions. The main motivation for investigating piecewise smooth ISS Lyapunov functions is the success of piecewise smooth Lyapunov functions in the stability analysis of hybrid systems. This paper proposes an extension of the well-known Filippov's solution concept, that is appropriate for `open' systems so as to allow interconnections of DDS. It is proven that the existence of a piecewise smooth ISS Lyapunov function for a DDS implies ISS. In addition, a (small gain) ISS interconnection theorem is derived for two DDS that both admit a piecewise smooth ISS Lyapunov function. This result is constructive in the sense that an explicit ISS Lyapunov function for the interconnected system is given. It is shown how these results can be applied to construct piecewise quadratic ISS Lyapunov functions for piecewise linear systems (including sliding motions) via linear matrix inequalities.

AB - In this paper we will extend the input-to-state stability (ISS) framework to continuous-time discontinuous dynamical systems (DDS) adopting piecewise smooth ISS Lyapunov functions. The main motivation for investigating piecewise smooth ISS Lyapunov functions is the success of piecewise smooth Lyapunov functions in the stability analysis of hybrid systems. This paper proposes an extension of the well-known Filippov's solution concept, that is appropriate for `open' systems so as to allow interconnections of DDS. It is proven that the existence of a piecewise smooth ISS Lyapunov function for a DDS implies ISS. In addition, a (small gain) ISS interconnection theorem is derived for two DDS that both admit a piecewise smooth ISS Lyapunov function. This result is constructive in the sense that an explicit ISS Lyapunov function for the interconnected system is given. It is shown how these results can be applied to construct piecewise quadratic ISS Lyapunov functions for piecewise linear systems (including sliding motions) via linear matrix inequalities.

U2 - 10.1016/j.automatica.2008.04.025

DO - 10.1016/j.automatica.2008.04.025

M3 - Article

VL - 44

SP - 3079

EP - 3086

JO - Automatica

JF - Automatica

SN - 0005-1098

IS - 12

ER -