Establishing improved convergence properties for the adaptive learning control

The adaptive learning approach proposed in Del Vecchio, Marino, and Tomei (2003) and Liuzzo, Marino, and Tomei (2007a), which guarantees output tracking of sufficiently smooth periodic reference signals (of known period) for classes of time-invariant nonlinear systems with sufficiently smooth unstructured uncertainties, is considered. By using a slightly different analysis (and in particular less conservative bounds), we show how improved convergence properties can be established while maintaining the same learning control structures. The delayed first arithmetic mean of the Fourier series of the uncertain periodic time function f (of known period) can be used instead of the partial sums of the Fourier series of f to further improve the result: a trigonometric polynomial, whose approximation to f is almost as good as the best approximation of f , is obtained.