THE DYNAMICS OF UNIVERSAL ADAPTIVE STABILIZATION: COMPUTATIONAL AND ANALYTICAL STUDIES

摘要

We consider the dynamics of closed loop systems arising from universal adaptive stabilization, with scalar feedback gain, of a class of linear systems with n-dimensional state vector. The state and gain dynamics evolve in R~n x R. We analyze qualitatively the dynamics of the closed loop nonlinear system and of an associated linear system-the limit system-obtained by implementing the linear feedback determined by the limit value of the gain. We focus on cases when this limit system can have imaginary axis eigenvalues. We give a classification of the dynamics on the corresponding center manifold, for which we derive analytical and computational results. The analysis is always made complicated by the degeneracy of the vector field defining the adaptation law. Further degenerate behavior, arising when the feedback law involves switching in the sign of the gain, is also considered.