Undermodeled adaptive filtering: an a priori error bound for the Steiglitz-McBride method

Practical applications of adaptive IIR filtering are confronted with undermodeled (or reduced-order) cases: the order chosen for the adaptive identifier is inferior to the true degree of the unknown system. Most known results for adaptive IIR filters concern only the sufficient order case, and rarely admit direct extensions to the undermodeled case. As exact matching is excluded by undermodeling, critical to the acceptance of any algorithm are the approximation properties which result in the undermodeled ease. In this direction, we establish an a priori error bound for the Steiglitz-McBride algorithm. In particular, if the input and disturbance are both white noise processes, and if M is the chosen order for the identifier, we show that the L₂-norm of the error function at any stationary point can be no larger than the M+1st Hankel singular value of the unknown system. This gives a meaningful bound, and yields the first formal result which affirms that the Steiglitz-McBride method is capable of satisfactory approximation properties for the undermodeled case. Conditions under which the Steiglitz-McBride model is close to an optimal L₂-norm or Hankel-norm approximant are obtained as an elementary consequence of our bound. Our result also provides the first bound on the “bias” introduced by a colored disturbance