## An Alternative Converse Lyapunov Theorem for Discrete-Time Systems

Please always quote using this URN: urn:nbn:de:bvb:20-opus-78512

- This paper presents an alternative approach for obtaining a converse Lyapunov theorem for discrete–time systems. The proposed approach is constructive, as it provides an explicit Lyapunov function. The developed converse theorem establishes existence of global Lyapunov functions for globally exponentially stable (GES) systems and semi–global practical Lyapunov functions for globally asymptotically stable systems. Furthermore, for specific classes of sys- tems, the developed converse theorem can be used to establish non–conservatism of aThis paper presents an alternative approach for obtaining a converse Lyapunov theorem for discrete–time systems. The proposed approach is constructive, as it provides an explicit Lyapunov function. The developed converse theorem establishes existence of global Lyapunov functions for globally exponentially stable (GES) systems and semi–global practical Lyapunov functions for globally asymptotically stable systems. Furthermore, for specific classes of sys- tems, the developed converse theorem can be used to establish non–conservatism of a particular type of Lyapunov functions. Most notably, a proof that conewise linear Lyapunov functions are non–conservative for GES conewise linear systems is given and, as a by–product, tractable construction of polyhedral Lyapunov functions for linear systems is attained.…

Author: | Roman Geiselhart, Rob H. Gielen, Mircea Lazar, Fabian R. Wirth |
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URN: | urn:nbn:de:bvb:20-opus-78512 |

Document Type: | Preprint |

Faculties: | Fakultät für Mathematik und Informatik / Institut für Mathematik |

Language: | English |

Year of Completion: | 2013 |

Dewey Decimal Classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |

GND Keyword: | Ljapunov-Funktion |

Tag: | conewise linear systems; converse Lyapunov theorems; discrete-time systems; stability analysis |

MSC-Classification: | 39-XX DIFFERENCE AND FUNCTIONAL EQUATIONS / 39Axx Difference equations (For dynamical systems, see 37-XX, for dynamic equations on time scales, see 34N05) |

93-XX SYSTEMS THEORY; CONTROL (For optimal control, see 49-XX) / 93Cxx Control systems | |

93-XX SYSTEMS THEORY; CONTROL (For optimal control, see 49-XX) / 93Dxx Stability | |

Release Date: | 2013/06/25 |

Licence (German): | Deutsches Urheberrecht |