Finite-time stability of linear time-varying systems with jumps.

*(English)*Zbl 1162.93375Summary: This paper deals with the finite-time stability problem for continuous-time linear time-varying systems with finite jumps. This class of systems arises in many practical applications and includes, as particular cases, impulsive systems and sampled-data control systems. The paper provides a necessary and sufficient condition for finite-time stability, requiring a test on the state transition matrix of the system under consideration, and a sufficient condition involving two coupled differential-difference linear matrix inequalities. The sufficient condition turns out to be more efficient from the computational point of view. Some examples illustrate the effectiveness of the proposed approach.

##### MSC:

93C57 | Sampled-data control/observation systems |

93D09 | Robust stability |

93C05 | Linear systems in control theory |

##### Software:

LMI toolbox
PDF
BibTeX
XML
Cite

\textit{F. Amato} et al., Automatica 45, No. 5, 1354--1358 (2009; Zbl 1162.93375)

Full Text:
DOI

##### References:

[1] | Amato, F., Ariola, M., Carbone, M., & Cosentino, C. (2006). Finite-time output feedback control of linear systems via differential linear matrix conditions. In Proc. conf. on decision and control (pp. 5371-5375) |

[2] | Amato, F., Ariola, M., & Cosentino, C. (2005). Finite-time control of linear time-varying systems via output feedback. In Proc. American Control Conference (pp. 4722-4726) |

[3] | Amato, F.; Ariola, M.; Cosentino, C., Finite-time stabilization via dynamic output feedback, Automatica, 42, 337-342, (2006) · Zbl 1099.93042 |

[4] | Amato, F.; Ariola, M.; Dorato, P., Finite time control of linear systems subject to parametric uncertainties and disturbances, Automatica, 37, 1459-1463, (2001) · Zbl 0983.93060 |

[5] | Antsaklis, P.J., Hybrid systems II, (1995), Springer |

[6] | Boel, R.; Stremersch, G., Discrete event systems: analysis and control, (2000), Springer · Zbl 0996.00058 |

[7] | Boyd, S.; El Ghaoui, L.; Feron, E.; Balakrishnan, V., Linear matrix inequalities in system and control theory, (1994), SIAM Press · Zbl 0816.93004 |

[8] | Callier, F.M.; Desoer, C.A., Linear system theory, (1991), Springer Verlag · Zbl 0508.93011 |

[9] | Dorato, P. (1961). Short time stability in linear time-varying systems, In Proc. IRE International convention record part 4 (pp. 83-87) |

[10] | Gahinet, P.; Nemirovski, A.; Laub, A.J.; Chilali, M., LMI control toolbox, (1995), The Mathworks Inc |

[11] | Kamenkov, G., On stability of motion over a finite interval of time, Journal of applied mathematics and mechanics, 17, 529-540, (1953), (in Russian) |

[12] | Lebedev, A., On stability of motion during a given interval of time, Journal of applied mathematics and mechanics, 18, 139-148, (1954), (In Russian) |

[13] | Lebedev, A., The problem of stability in a finite interval of time, Journal of applied mathematics and mechanics, 18, 75-94, (1954), (in Russian) · Zbl 0055.32102 |

[14] | Sun, W.; Nagpal, K.; Khargonekar, P.P., \(\mathcal{H}_\infty\) control and filtering for sampled-data systems, Institute of electrical and electronic engineering transactions on automatic control, 38, 1162-1175, (1993) · Zbl 0784.93027 |

[15] | Weiss, L.; Infante, E.F., Finite time stability under perturbing forces and on product spaces, Institute of electrical and electronic engineering transactions on automatic control, 12, 54-59, (1967) · Zbl 0168.33903 |

[16] | Yang, T., Impulsive control theory, (2001), Springer |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.