Robustness in adaptive filtering: How much is enough?

The issue of robustness of adaptive filtering algorithms has been investigated in the literature using the H_∞ paradigm. In particular, in the constant parameter case, the celebrated (normalized) least mean squares (LMS) algorithm has been shown to coincide with the central H_∞-filter ensuring the minimum achievable disturbance attenuation level. In this paper, the problem is re-examined by taking into account the robust performance of three classical algorithms (normalized LMS, Kalman filter, central H_∞-filter) with respect to both measurement noise and parameter drift. It turns out that normalized LMS does not guarantee any finite level of H_∞-robustness. On the other hand, it is shown that striving for the minimum achievable attenuation level leads to a trivial nondynamic estimator with poor H₂-performance. This motivates the need for a design approach balancing H₂ and H_∞ performance criteria