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
![$ n$](http://www.dsprelated.com/josimages_new/pasp/img146.png)
![$ C A^{n-1}
B$](http://www.dsprelated.com/josimages_new/pasp/img269.png)
![$ n\geq0$](http://www.dsprelated.com/josimages_new/pasp/img270.png)
![$ p\times q$](http://www.dsprelated.com/josimages_new/pasp/img271.png)
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