Page 242 - Electrical Engineering Dictionary

dynamical systems with delays a system dynamo a term used to describe any of
described by the linear state equation a variety of rotating machines that convert
mechanical to electrical energy, or less com-
0
x (t) = A 0 x(t) + A 1 x(t − h) + Bu(t) (1) monly, electrical to mechanical energy. Dy-
namos typically consist of a stationary struc-
where x(t) is n-dimensional vector, u(t) is ture, called the stator, supporting a rotating
m-dimensional control vector, and A 0 , A 1 , element called the rotor. Energy conversion
and B are constant matrices of appropriate di- occurs via Faraday induction. A field wind-
mensions and h> 0 is a constant delay. For ing (or in some smaller machines, permanent
a given admissible control and initial data, magnets) is mounted on one of the mechan-
the above differential equation (1) with devi- ical structures and produces a magnetic flux.
ating argument has a unique solution derived An armature winding is mounted on the other
by the method of steps. structure, and rotation of the rotor produces
The state space for dynamical sys- relative motion between the field flux and
(2) n
tem (1), X = W ([−h, 0],R ), is infinite- the coils of an armature winding, inducing
1
dimensional Sobolev space of absolutely a Faraday voltage in the armature coil. This
continuous functions defined on [−h, 0] with Faraday induced voltage is the source of elec-
values in R n and with square integrable trical energy at the dynamo output.
derivatives.
Linear unbounded operator connected dynamometer a rotating device used to
with the dynamical system (1) generates the measure the steady-state torque and power
solution in the state space X and has in- output of rotating machines. Dynamometers
finite number of eigenvalues each of finite generally provide precise control of the load
multiplicity. The corresponding eigenfunc- torque applied to a test machine, and power
tions may form the basis in the infinite- output is determined through precise speed
dimensional state space X. measurements.
It should be stressed that it is possible
to consider another type of linear dynamical
systems with delays, namely systems with
multiple delays, systems with delays in the
controlorneutraldynamicalsystemswithde-
layed derivative.

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2000 by CRC Press LLC