Stability criteria with less LMI variables for linear system with additive time-delays

In this technical note, equivalent stability criterion with minimal number of variables for three recently reported stability criteria is proposed for a class of linear systems with additive time-varying delays. The existing delay-dependent stability criteria for additive time-delay systems have more number of matrix variables in the LMI; and hence, they require more computational cost. The proposed equivalent criteria, unlike the original ones, encompass only the matrix variables that are associated in the Lyapunov-Krasovskii functional, making the criteria mathematically less complex and computationally more attractive. The complexity involved in the existing stability criteria is attributed to the fact the cross-terms that emanate from the time-derivative of the Lyapunov-Krasovskii functional are dealt with free-weighting matrices. Hence, apart from the matrix variables that are associated in the corresponding Lyapunov-Krasovskii functional, the existing criteria also have additional matrix variables in them. In this paper, we have devised techniques to eliminate the free-weighting matrices in the existing stability criteria without sacrificing the conservatism. The resulting equivalent stability criteria, therefore, have least possible number of variables in the LMI; and hence, have minimum computational burden. The effectiveness of the proposed equivalent criteria is validated on a numerical example.