On delay-independent stability of linear systems: Generalized Lyapunov equation

Abstract

This paper presents some delay-independent stability criteria for linear systems with time delay in the form x(t) = Ax(t) + Bx(t − τ). The main result states that the system is asymptotically stable independent of delay if there are positive scalar a and positive definite matrices P and Q satisfying a generalized Lyapunov equation A <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sup> P + PA + α <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−1</sup> B <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sup> PB + αP + Q = 0. Optimization of the main result and comparison with other criteria are made through analysis and examples. It is shown that the present criteria are less conservative for a class of linear systems. The computation involves a convex optimization problem over only one positive parameter α.