A Necessary and Sufficient Condition for the Stabilization of Decentralized Time-Delay Systems by Time-Delay Controllers

It has recently been shown that a necessary and sufficient condition for the μ-stabilization of a linear time-invariant (LTI) retarded time-delay system by LTI decentralized finite-dimensional controllers is that the system should not have any μ-decentralized fixed modes with respect to decentralized static feedback. In the present work, we consider the stabilization of such systems by decentralized time-delay controllers and show that the same condition is also necessary and sufficient in this case too. This extends a previously known fact for centralized control to decentralized control. Namely, a LTI retarded time-delay system can be stabilized by decentralized LTI time-delay controllers if and only if it can be stabilized by decentralized LTI finite-dimensional controllers. We then consider the extension of these results to LTI neutral time-delay systems. We show that the necessity continues to hold for this case as well. Furthermore, sufficiency also continues to hold provided that the system has finitely many modes to the right of a vertical line passing through μ – ε for some positive ε. A number of examples are also presented to demonstrate the applications of the results obtained.