TY  - INPR
A1  - Geiselhart, Roman
A1  - Gielen, Rob H.
A1  - Lazar, Mircea
A1  - Wirth, Fabian R.
T1  - An Alternative Converse Lyapunov Theorem for Discrete-Time Systems
N2  - 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 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.
KW  - Ljapunov-Funktion
KW  - stability analysis
KW  - conewise linear systems
KW  - discrete-time systems
KW  - converse Lyapunov theorems
Y1  - 2013
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-78512
ER  - 
TY  - THES
A1  - Geiselhart, Roman
T1  - Advances in the stability analysis of large-scale discrete-time systems
T1  - Fortschritte in der Stabilitätsanalyse großskaliger zeitdiskreter Systeme
N2  - Several aspects of the stability analysis of large-scale discrete-time systems are considered. An important feature is that the right-hand side does not have have to be continuous.
In particular, constructive approaches to compute Lyapunov functions are derived and applied to several system classes.
For large-scale systems, which are considered as an interconnection of smaller subsystems, we derive a new class of small-gain results, which do not require the subsystems to be robust in some sense. Moreover, we do not only study sufficiency of the conditions, but rather state an assumption under which these conditions are also necessary.
Moreover, gain construction methods are derived for several types of aggregation, quantifying how large a prescribed set of interconnection gains can be in order that a small-gain condition holds.
N2  - Es werden großskalige zeitdiskrete Systeme betrachtet, deren rechte Seite nicht als stetig angenommen wird.
Konstruktive Ansätze um Lyapunovfunktionen zu berechnen werden hergeleitet und auf mehrere Systemklassen angewandt. Für großskalige Systeme, die beschrieben sind durch die Kopplung kleinerer Systeme, wird eine neue Klasse von sogenannten Small-Gain Resultaten vorgestellt, die nicht verlangt, dass die Subsysteme robust sein müssen. Zudem untersuchen wir die Notwendigkeit der geforderten Bedingungen. Zusätzlich werden Gainkonstruktionsmethoden für verschiedene Typen von Verknüpfung hergeleitet, welche quantifizieren, wie groß eine vorgegebene Menge von Kopplungsgains sein kann, so dass eine Small-Gain-Bedingung erfüllt ist.
KW  - Ljapunov-Funktion
KW  - Konstruktionsmethoden
KW  - Ljapunov-Stabilitätstheorie
KW  - Nichtlineare Funktionalgleichung
Y1  - 2015
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-112963
ER  -