Towards a General Theory of Hybrid Dynamical Systems
Dr Andrey Savkin (University of New South Wales)
Hybrid dynamical systems (HDS) have attracted considerable
attention in recent years. In general,
HDS are those that consist of a logical discrete event
decision-making system interacting with a continuous time process.
A simple example is a climate control system in a typical home.
The on-off nature of the thermostat is modelled as a discrete event
system, whereas the furnace or air-conditioner are modelled as continuous
time systems.
Some other examples include transmissions and stepper motors,
computer disk drives, robotic systems, high-level flexible
manufacturing systems
intelligent vehicle/highway systems, communication networks,
interconnected power systems, air/sea traffic management.
In fact, many problems facing control engineers, computer
scientists, and mathematicians as they seek to use
computers to control complex physical systems, naturally
fit into the HDS framework. Study of HDS represents a difficult
and exciting challenge to control engineering and is referred
to as ``The Control Theory of Tomorrow'' by
SIAM News.
Most of the seminar will concentrate on the classes of
discretely controlled continuous-time systems.
First, we would consider a quite general model of HDS called
a differential automaton (DA). We will present results on
algebraic reducibility (reduction to a finite-state automaton) of DA,
existence of limit cycles, their global asymptotic stability.
An analog of the classic Poincare-Bendixon theorem will be given
for planar DA. It will be shown, that under some assumptions,
any DA can be represented as a product of a finite number of
eventually periodic DA.
Our main results will describe a broad class of HDS satisfying
the following properties:
(i) This HDS has a finite number of limit cycles.
(ii) Any trajectory converges to one of these cycles.
Also, we will describe a broad class of HDS with chaotic
behavior.
We will apply these results to analysis and synthesis
of complex switched flow server/arrival networks and
dynamically routed queueing networks.
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