Since the output in Fig.E.1 is a linear combination of
the input and states , one or more poles can be
canceled by the zeros induced by this linear combination. When that
happens, the canceled modes are said to be
unobservable. Of course, since we
started with a transfer function, any pole-zero cancellations should
be dealt with at that point, so that the state space realization will
always be controllable and observable. If a mode is
uncontrollable, the input cannot affect it; if it is unobservable, it
has no effect on the output. Therefore, there is usually no reason to
include unobservable or uncontrollable modes in a state-space
model.^{E.6}

A physical example of uncontrollable and unobservable modes is
provided by the plucked vibrating string of an electric guitar
with one (very thin) magnetic pick-up. In a vibrating string,
considering only one plane of vibration, each
quasi-harmonic^{E.7}overtone corresponds to a mode
of vibration [86] which may be modeled by a pair of
complex-conjugate poles in a digital filter which models a particular
point-to-point transfer function of the string.
All modes of vibration having a node at the plucking point are
uncontrollable at that point, and all modes having a node at
the pick-up are
unobservable at that point. If an ideal string is plucked at
its midpoint, for example, all even numbered harmonics will not be
excited, because they all have vibrational nodes at the string
midpoint. Similarly, if the pick-up is located one-fourth of the
string length from the bridge, every fourth string harmonic will be
``nulled'' in the output. This is why plucked and struck strings are
generally excited near one end, and why magnetic pick-ups are located
near the end of the string.

A basic result in control theory is that a system in state-space form
is controllable from a scalar input signal if and only
if the matrix

has full rank (i.e., is invertible). Here, is . For
the general case, this test can be applied to each of the
columns of , thereby testing controllability from each input in
turn. Similarly, a state-space system is observable from a given
output if and only if

is nonsingular (i.e., invertible), where is . In the
-output case, can be considered the row corresponding to the
output for which observability is being checked.