Stability analysis in continuous and discrete time


Besseling, Nicolaas Cornelis (2012) Stability analysis in continuous and discrete time. thesis.

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Abstract:The relation between continuous time systems and discrete time systems is the
main topic of this research. A continuous time system can be transformed into
a discrete time system using the Cayley transform. In this process the generator
of the semigroup is mapped to a difference operator, the cogenerator. Stability
analysis plays a central role in the study of the relation between continuous and discrete time systems. Do stable continuous time systems correspond to stable discrete time systems? This is the main question of this dissertation.
For many stable continuous time systems the corresponding discrete time
systems are stable as well. In Banach spaces however, several examples are
known of stable semigroups where the corresponding cogenerators have unsta-
ble power sequences. In Hilbert spaces no such examples are known. It remains
an open problem whether for every stable continuous time system the corre-
sponding discrete time system is stable as well.
This dissertation addresses the main question in three ways.
First, a growth bound for the cogenerator is provided for exponentially stable
semigroups in Hilbert spaces. Using Lyapunov equations it is shown that for
such semigroups the power sequence of the corresponding cogenerator cannot
grow faster than ln(n).
Second, we extend the class of stable continuous time systems for which the
corresponding discrete time systems are stable as well. For this, the notion
of Bergman distance is introduced. The Bergman distance defines a metric
for semigroups and a metric for power sequences. If the Bergman distance is
finite, the two semigroups have the same stability behaviour. This holds for two
power sequences as well. Furthermore, the Bergman distance is preserved by
the Cayley transform. This enables us to extend this class.
Third, the inverse of the generator is taken into account in the stability
analysis. For exponentially stable semigroups on Banach spaces similarity is
shown between the growth of the semigroup of the inverse and the growth of the
cogenerator. For bounded semigroups on Hilbert spaces it is shown that if the
semigroup generated by the inverse is bounded, the growth of the cogenerator
is bounded as well.
Item Type:Thesis
Electrical Engineering, Mathematics and Computer Science (EEMCS)
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