A geometric perspective on adaptive system performance

A geometric perspective is presented for the frequency domain performance prediction technique for adaptive systems. The geometry is simply that of an n-dimensional sphere, intersecting the origin, bisected by a hyperplane of lower dimension, which also intersects the origin. The amplitudes of all the significant frequency components of the adaptive prediction error are shown to lie on this hyperplane, and are thus bounded by the extrema of the intersection of the hyperplane and sphere. The hyperplane is determined by the original non-adaptive system, while the sphere can be determined largely by the system designer. This simple geometry allows straightforward, accurate computation of adaptive system performance bounds, and suggests simple design techniques for improving performance in desired frequency ranges. A simulation example is provided that illustrates the use of this geometry in designing adaptive systems to meet prespecified performance objectives