Stability/Instability Analysis of Continuous-Time Adaptive Controllers for Pure Delay Systems

In the paper we are concerned with the problem of adaptively controlling pure delay systems. we are firstly interested in determining the conditions underwhich stability is lost due the umodeled time delay. To address this problem we consider the case of periodic signals and derive: i) Necessary and sufficient conditions for stability when the signals are of specified frequences, ii) Stability/instability conditions under slow adaptation. Secondly, we propose a globally stable adaptive regulator for systems with known, possibly time varying, time delay. All the results are derived based on the analysis of the homogeneous part of the retarded differential-difference equation (RDDE).