Nonlinear Control Sy.. - Free

8.1. POWER AND ENERGY: PASSIVE SYSTEMS 203
e(t)
R
Figure 8.2: Passive network.
sense. It is not straightforward to capture the notion of good behavior within the context
of a theory of networks, or systems in a more general sense. Stability, in its many forms,
is a concept that has been used to describe a desirable property of a physical system, and
it is intended to capture precisely the notion of a system that is well behaved, in a certain
precise sense. If the notion of passivity in networks is to be of any productive use, then we
should be able to infer some general statements about the behavior of a passive network.
To study this proposition, we consider the circuit shown in Figure 8.2, where we
assume that the black box contains a passive (linear or not) circuit element. Assuming that
the network is initially relaxed, and using Kirchhoff voltage law, we have
e(t) = i(t)R + v(t)
Assume now that the electromotive force (emf) source is such that
we have,
I T
T
i
v
e2(t)dt 0. It follows that
J
T T
e2(t) dt > R2 J i2(t) dt + f T v2(t) dt.
0 0 0