Controllability of linear time-invariant sampled-data systems with nonconstant shaping functions

This paper is concerned with the controllability of single-input linear time-invariant plants after introduction of sampling in such a way that, for the time
kT < t \leq (k + 1)T
, the input is
u_{k}f(t - kT)
, where u
k
is a real constant and
f(t)
is a bounded piecewise continuous shaping-function given in
0 < t \leq T
. Necessary and sufficient conditions for complete controllability of such sampled-data systems in terms of
T
and
f(t)
are given. This is a generalization of a well-known theorem for the special case
f(t) \equiv 1
due to Kalman et al. The results may prove to be important if for practical reasons the shaping function
f(t) \equiv 1
cannot be realized.