Complicated dynamics of prototype continuous-line adaptive control system

The authors investigate the possible dynamics that a prototype model reference adaptive control system can experience when using the so-called σ-modification adaption law described by P.I. Ioannou and P.V. Kokotovic (1984). Limiting their attention to first-order plants with a single unknown pole and an external disturbance, they verify analytically the bifurcations that an example adaptive system has previously been shown to exhibit based on computer simulations. The parameter-space-to-bifurcation diagram is constructed summarizing all possible behaviors of the class of prototype systems under consideration. It is shown that a certain range of constant-plus-sinusoidal disturbances generate chaos. For a certain range of constant disturbances, it is shown that a sinusoidal reference input signal generates chaos as well. In both cases, chaos arises form transversal homoclinic orbits. The use of the parameter space as a design tool for the proper implementation of the scheme is demonstrated