A Robust Adaptive Controller for Continuous Time Systems

We prove the stability of an adaptively-controlled continuous-time plant of relative degree one in the presence of unmodeled dynamics. Specifically, we show that if the unmodeled dynamics are small in some sense, then all the signals in the closed-loop system remain bounded. Further, we show that robust performance is achieved in the sense that the mean-square tracking error converges to zero linearly with the unmodeled dynamics. In the absence of unmodeled dynamics, we show that the tracking error asymptotically goes to zero. The adaptation law considered is the usual gradient update law with projection. This shows that modifications such as normalization or relative dead-zones are not necessary to achieve stability in the presence of unmodeled dynamics.