Instability analysis and improvement of robustness of adaptive control

The effects of unmodeled high frequency dynamics and bounded disturbances on stability and performance of adaptive control schemes are analyzed. Five possible types of instability mechanisms—parameter drift, ‘linear’ instability, ‘fast adaptation’ instability, ‘high frequency’ instability, and ‘throughput’ instability—are analyzed using simple examples. A procedure is used to construct Lyapunov-like functions for a modified adaptive controller applied to a dominant plant of relative degree one, in the presence of parasitics and disturbances, and obtain sufficient conditions under which none of the five types of instability can occur. The modified scheme is robust in the sense that it guarantees the existence of a large region of attraction from which all the trajectories remain bounded and the state errors converge exponentially to a much smaller residual set. The size of the region of attraction depends on the speed of parasitics in such a way that as the parasitics become infinitely fast, the region of attraction becomes the whole space.