Parameter space analysis and design of an adaptive system

We investigate the so-called ??-modification scheme which has been proposed for the design of model reference adaptive control systems in the presence of unmodeled dynamics and of disturbances. Limiting attention to first order plants with a single unknown pole and an external disturbance, we verify analytically some of the bifurcations that an example adaptive system has been shown to exhibit via computer simulations. We also construct the parameter space, or bifurcation, diagram which summarizes all possible behaviors of the systems under consideration. Finally, we demonstrate the use of the parameter space as a design tool for the proper implementation of the scheme.