Nonrobustness of asymptotic stability of impulsive systems with inputs

Suitable continuity and boundedness assumptions on the function f defining the dynamics of a time-varying nonimpulsive system with inputs are known to make the system inherit stability properties from the zero-input system. Whether this type of robustness holds or not for impulsive systems was still...

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Bibliographic Details
Published in:Automatica (Oxford) 2020-12, Vol.122, p.109238, Article 109238
Main Authors: Haimovich, Hernan, Mancilla-Aguilar, José Luis
Format: Article
Language:English
Subjects:
Impulsive systems
Input-to-state stability
Nonlinear systems
Time-varying systems
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Summary:Suitable continuity and boundedness assumptions on the function f defining the dynamics of a time-varying nonimpulsive system with inputs are known to make the system inherit stability properties from the zero-input system. Whether this type of robustness holds or not for impulsive systems was still an open question. By means of suitable (counter)examples, we show that such stability robustness with respect to the inclusion of inputs cannot hold in general, not even for impulsive systems with time-invariant flow and jump maps. In particular, we show that zero-input global uniform asymptotic stability (0-GUAS) does not imply converging input converging state (CICS), and that 0-GUAS and uniform bounded-energy input bounded state (UBEBS) do not imply integral input-to-state stability (iISS). We also comment on available existing results that, however, show that suitable constraints on the allowed impulse–time sequences indeed make some of these robustness properties possible.
ISSN:0005-1098
1873-2836
DOI:10.1016/j.automatica.2020.109238