#### Impulse Response of State Space Models

As derived in Book II [449, Appendix G], the impulse response of the state-space model can be summarized as

Thus, the th ``sample'' of the impulse response is given by for . Each such ``sample'' is a matrix, in general.

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.

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Zero-Input Response of State Space Models

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State Definition