Degenerate gradient flows: a comparison study of convergence rate estimates

Degenerate gradient flows arise in the context of adaptive control of linear systems when the usual gradient algorithm is used for the parameter update law. It is well known that in general parameter convergence is not guaranteed without further assumptions. The standard approach utilizes the notion of a persistently exciting input and different authors have derived different convergence rate estimates. In a recent paper Brockett re-examined this issue and developed a rate estimate using a property of symmetric matrices related to the condition number. In this paper we compare two well-known convergence rate estimates from the persistently exciting point of view with Brockett's estimate through a semianalytical numerical study. We establish a common footing by relating the assumptions of each theorem to the parameters specified under the persistently exciting condition. Our analysis shows that for all parameter values Anderson's result yields a tighter bound than the other two estimates. In each case the magnitude of the difference depends on the time it takes for the uniform observability condition to hold in the persistently exciting assumption. The shorter the time is, the larger the difference is.