An Adaptive Gradient Law with Projection for Non-smooth Convex Boundaries

The parameter projection method, which has been designed for solving on-line parameter identification problems under smooth convex restrictions, is extended here to arbitrary convex parameter domains. Such problems arise naturally in cases where an admissible solution set results from the intersection of several (possibly smooth) convex constraints. The corresponding adaptive gradient law has the form of a special evolution projected dynamical system with a discontinuous right-hand side. The paper develops an alternative formulation of this projected dynamical system based on the multidimensional stop operator. The advantage of this approach is that the new right-hand side is continuous and the problem is thus accessible to conventional analysis methods, which easily give results on existence, uniqueness, and convergence properties of the corresponding solution trajectories. The method is tested on the parameter identification problem for complex hysteresis nonlinearities.