Contraposition Nyquist stability criteria for linear time-delayed systems of retarded type

This paper reports some Nyquist-type criteria for exponential/asymptotical stability in linear time-delayed systems of retarded type, by introducing contraposition separators into the return difference ratio relationships in feedbacks. The Nyquist criteria present necessary and sufficient stability conditions, delays-dependent, which can be interpreted similarly and examined graphically as we do for linear time-invariant systems. The suggested Nyquist criteria involve neither open-loop characteristic roots distribution nor encirclement/orientation of Nyquist loci, which are not available even approximately and finite-dimensionally in linear time-delayed systems. The Nyquist criteria can be easily applied to open-loop linear time-delayed systems. Examples are included to illustrate the results.