On closed-form expressions for mean squares in discrete-continuous systems

When a system is to be optimized with respect to the mean square of some variable, a closed-form expression for that mean square is usually desired. The problem of obtaining such expressions for discrete-continuous systems-i.e., systems made up of both sampled-data and continuous subsystems-has been a difficulty in the past. The reason for this is that the spectral densities of the variables of interest often contain rational functions of
\exp (j2\pifT)
combined multiplicatively with rational functions of
f, f
being the frequency coordinate of the spectral densities, and
T
the sampling period. Presented here is a technique for finding the desired closed-form expressions. It is based on the relation
\int\min{-j\infty}\max{j\infty} P^{\ast}(e^{s^{T}})Q(s)ds = \oint P^{\ast}(z)Q^{\ast}(z)z^{-1}dz
, where
Q^{\ast} (z)
is the "
Z
-transform" of
Q (s)
, To illustrate the technique, closed-form formulas for the output and ripple of discrete-continuous systems and for the control error of sampled-data feedback systems are derived, and an application to a "track-while-scan" system is given.