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This paper presents the analysis of the stability properties of proportional-integralderivative (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. It is also shown that adding a low-pass filter, which might be necessary to reduce sensitivity to high-frequency noise, under certain conditions on the derivative feedback gain may destabilize the closed loop even for arbitrarily large cutoff frequencies. The analysis of these potential sensitivity problems leads us to the introduction of a "robustified" notion of stability called 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 cutoff 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 for which 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. [ABSTRACT FROM AUTHOR]