Stability and assignment of spectrum in systems with discrete time lags

The asymptotic stability with a prescribed degree of time delayed systems subject to multiple bounded discrete delays has received important attention in the last years. It is basically proved that the α -stability locally in the delays (i.e., all the eigenvalues have prefixed strictly negative real parts located in Re ⠡ s ≤ − α < 0 ) may be tested for a set of admissible delays including possible zero delays either through a set of Lyapunov's matrix inequalities or, equivalently, by checking that an identical number of matrices related to the delayed dynamics are all stability matrices. The result may be easily extended to check the ε -asymptotic stability independent of the delays, that is, for all the delays having any values, the eigenvalues are stable and located in Re ⠡ s ≤ ε → 0 − . The above referred number of stable matrices to be tested is 2 r for a set of distinct r point delays and includes all possible cases of alternate signs for summations for all the matrices of delayed dynamics. The manuscript is completed with a study for prescribed closed-loop spectrum assignment (or “pole placement†) under output feedback.