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We complete the first step towards the resolution of several decades-oldchallenges in disturbance-robust adaptive control. For a scalar linear systemwith an unknown parameter for which no a priori bound is given, with adisturbance that is of unlimited magnitude and possibly persistent (not squareintegrable), and without a persistency of excitation necessarily verified bythe state, we consider the problems of (practical) gain assignment relative tothe disturbance. We provide a solution to these heretofore unsolved feedbackdesign problems with the aid of infinite-dimensional nonlinear feedbackemploying distributed delay of the state and input itself. Specifically, inaddition to (0) the global boundedness of the infinite-dimensional state of theclosed-loop system when the disturbance is present, we establish (1) practicalinput-to-output stability with assignable asymptotic gain from the disturbanceto the plant state; (2) assignable exponential convergence rate; and (3)assignable radius of the residual set. The accompanying identifier in ouradaptive controller guarantees (4) boundedness of the parameter estimate evenwhen disturbances are present; (5) an ultimate estimation error which isproportional to the magnitude of the disturbance with assignable gain whenthere exists sufficient excitation of the state; and (6) exact parameterestimation in finite-time when the disturbance is absent and there issufficient excitation. Among our results, one reveals a tradeoff between"learning capacity" and "disturbance robustness:" the less sensitive theidentifier is to the disturbance, the less likely it is to learn the parameter.

On Disturbance-to-State Adaptive Stabilization without Parameter Bound by Nonlinear Feedback of Delayed State and Input