  This paper presents the analysis of the stability properties of PID
controllers for dynamical systems with multiple state delays, focusing on the
mathematical characterization of the potential sensitivity of stability with
respect to infinitesimal parametric perturbations. These perturbations
originate for instance from neglecting feedback delay, a finite difference
approximation of the derivative action, or neglecting fast dynamics. The
analysis of these potential sensitivity problems leads us to the introduction
of a `robustified' notion of stability called \emph{strong stability}, inspired
by the corresponding notion for neutral functional differential equations. We
prove that strong stability can be achieved by adding a low-pass filter with a
sufficiently large cut-off frequency to the control loop, on the condition that
the filter itself does not destabilize the nominal closed-loop system.
Throughout the paper, the theoretical results are illustrated by examples that
can be analyzed analytically, including, among others, a third-order unstable
system where both proportional and derivative control action are necessary for
achieving stability, while the regions in the gain parameter-space for
stability and strong stability are not identical. Besides the analysis of
strong stability, a computational procedure is provided for designing strongly
stabilizing PID controllers. Computational case-studies illustrating this
design procedure complete the presentation.
