ABSTRACT
In the areas of identification and adaptive control of nonlinear systems, the persistent excitation condition is normally difficult to be verified a priori. Approximation theory has undergone major advances. Fundamental approximation theory includes interpolation, least squares, and Chebyshev approximation by polynomials, splines, and orthogonal-polynomials. From the 1980s, neural networks were constructed and empirically demonstrated to approximate quite well nearly all functions encountered in practical applications. Persistence of excitation is of great importance in adaptive systems. The concept was first introduced in the context of system identification by K. J. Astrom and T. Bohlin to express the idea that the input signal to the plant should be sufficiently rich such that all the modes of the plant are excited, and convergence of the model parameters is achieved. The instability of recurrent chaotic trajectories leads to the properties of divergence of nearby trajectories and sensitivity to initial conditions.
