A graphical test for checking the stability of a linear time-invariant feedback system

A graphical test is developed for checking the condition
\inf_{Re s \geq 0}|1 + k\hat{g}(s)| > 0
where
k
is a nonzero real constant and
\hat{g}
is the sum of a finite number of right-half plane poles and the Laplace transform of an integrable function plus a series of delayed impulses. As a conseqence,
1 + k\hat{g}
is, in
Re s \geq 0
, asymptotic to an almost periodic function, say
\hat{f}
, for
|s| \rightarrow \infty
. Theorem 1 gives a necessary and sufficient condition involving the curve
\{\hat{f}(j\omega)|\omega \in R\}
to ensure that
\inf_{Re s \geq 0} |\hat{f}(s)| > 0
; Corollary 1 gives a corresponding graphical test. Theorem 2 and Corollary 2 give a necessary and sufficient condition involving the curve
\{1 + K\hat{g}(j\omega)|\omega \in R\}
and a graphical test to ensure
\inf_{Re s \geq 0}| + k\hat{g}(s)| > 0
, a condition that guarantees the L
^p
stability of the feedback system for any
p \in [1,\infty]
.