Robust design and robust stability analysis of active noise control systems

The problem of designing robust active control systems is addressed in this paper. A variety of active control design problems are formulated as semid efinite programming (SDP) problems. An SDP problem is a convex optimization problem, consisting of a linear objective function subject to linear matri x inequality (LMI) constraints. First, an SDP formulation is presented for the design of multichannel LMS algorithms with limited-capacity secondary s ources. Simulations show that this SDP formulation is an order of magnitude more computationally efficient than the usual non-linear constrained optim ization formulations. Secondly, the design of robust LMS algorithms is pres ented as an SDP problem. These algorithms minimize the worst-case control e rror in the presence of unknown but norm-bounded perturbations on the secon dary path model and on the primary field. Both the unstructured and structu red perturbations cases are considered. The resulting controllers are exact solutions to the robust control design problem, except in the most general case of structured perturbations when they only minimize an upper bound on the worst-case residual control error. Thirdly, SDP formulations are propo sed to compute guaranteed stability limits for the adaptive multiple-channe l leaky LMS algorithm in the presence of both unstructured and structured p erturbations on the secondary path. Monte Carlo simulations show that the o btained stability limits are much more reliable than previously used limits , based, for example, on the Gershgorin circle theorem. (C) 2001 Academic P ress.