Robust stability analysis of linear time-delay systems by Lambert WW function: Some extreme point results

Robust stability conditions are derived for linear time-delay systems using Lambert WW function. The characteristic quasi-polynomials of the systems are assumed to be factorized. It is proven that if uncertainties in the coefficients of the quasi-polynomial are set in appropriate regions in the complex plane, we can enjoy extreme point results: finite number of stability checks at some points of the boarder of the regions suffice. The strength of Lambert WW function approach lies in the fact that the function is implemented on some standard software packages such as Mathematica, Maple or Matlab which afford to compute the function value very easily. The above two points make the stability test for the class of uncertain time-delay systems quite practical.