Adaptive control of time-varying systems with time-varying delays

This paper deals with a robustly stable adaptive pole-placement based controller for time-delay linear systems with possibly time-varying and unbounded point delays under unmodeled dynamics and bounded disturbances. Both the plant parameters and delays are allowed to be time-varying and differentiable with respect to time with bounded derivative. The mechanism used for robust adaptive stabilization consists of a relative dead zone which freezes the adaptation for small sizes of the adaptation error compared with the estimated size of the contribution of the uncertainties to the filtered output. The proposed estimation algorithm combines the use of that adaptation relative dead zone with a sigma t modification rule for updating the parameter estimates and with a modified updating rule which weights the classical least-squares type updating rule with the first and second powers of the covariance matrix as well as with the identity matrix. The dead zone adaptation mechanism prevents against potential instability caused by uncertainties, the modified updating rule guarantees positive definiteness of the covariance matrix for all time under appropriate initial conditions. Finally, the sigma-modification ride prevents against potential divergence of the parameter estimates caused by the modified covariance updating rule. A variant of the sigma- modification is proposed in order to improve the algorithm properties for the case when an absolute upper-bound for the plant parameters is known for all time.