A Class of Adaptive Nonlinear Hoo Filters with guaranteed I_2 Stability

We pose an identification problem that involves a nonlinear output equation and proceed to suggest an approximate linear solution. The approximation is obtained in two stages. We first replace the nonlinear functional by a linear relation, thus reducing the problem to a standard linear Hoo-setting. We then suggest constructing an approximation that results in an overall feedback structure in order to meet the desired robustness and stability properties. By combining the linear Hoo solution with a widely-used small gain theorem we show, under suitable conditions, that the approximate solution still leads to a filter with guaranteed I_2 stability. An examaple in the context of pole-zero (or IIR) system identificaiton is discussed in details. The proposed structure is also shown to include, as special cases, several adaptive filters that have been employed earlier in the literature in the context of IIR modeling. In particular, two algorithms due to Feintuch and Landau, as well as the so-called pseudo-linear regression algorithm, are discussed within the framework proposed herein.