As derived in Book II [452, Appendix G], the impulse response of the state-space model can be summarized in the single-input, single-output (SISO) case as
In our force-driven-mass example, we have
,
, and
. For a position output we have
while for a velocity
output we would set
. Choosing
simply feeds
the whole state vector to the output, which allows us to look at both
simultaneously:
Thus, when the input force is a unit pulse, which corresponds
physically to imparting momentum
at time 0 (because the
time-integral of force is momentum and the physical area under a unit
sample is the sampling interval
), we see that the velocity after
time 0 is a constant
, or
, as expected from
conservation of momentum. If the velocity is constant, then the
position must grow linearly, as we see that it does:
. The finite difference approximation to the time-derivative
of
now gives
, for
, which
is consistent.