New Results on Positive Realness in the Presence of Delayed Dynamics

Abstract

Positive realness is a very important tool for the achievement of hyperstability and passivity of dynamic systems. This paper is devoted to extend some positive realness results of transfer functions in the presence of point-delayed delayed dynamics. Sufficiency-type conditions which guarantee the positive realness of delayed transfer functions under point delays are given. The value of the direct input-output interconnection gain is seen to be crucial in the performed analysis. The relevance of the results rely in the importance of the hyperstability property of closed-loop systems under non-linear and time-varying controller devices. In fact if the feed-forward controlled plant has a strictly positive real transfer function then the closed-loo system is asymptotically hyperstable , that is, globally asymptotically Lyapunov´s stable for any non-linear time-varying controller which belongs to a hyperstable class defined as that which satisfies a Popov´s type inequality.