Strong Lyapunov Functions for Linear Time-Varying Systems Under Persistency of Excitation

Linear time-varying differential equations that arise in the study of model reference adaptive identification problems are studied in both the continuous-time and the discrete-time frameworks. The original contribution of the article is to present two new strong Lyapunov functions (e.g., Lyapunov functions with negative definite derivative), assessing global uniform asymptotic stability properties under the classical persistency of excitation condition. The first Lyapunov function, in the continuous-time framework, covers general (full-order) gradient-like adaptive observer forms—possibly taking into account the presence of projection algorithms and exhibiting piecewise continuous regressor matrices—that have so far restrictively required uniform boundedness of the (everywhere defined) derivative of the regressor. The second Lyapunov function, in the discrete-time framework, owns the advantage feature of being nonanticipating (e.g., characterized by a causal time-varying matrix that is available at runtime), which allows the designer to solve further adaptation issues.