Robustly l∞-Stable Implementation of the Adaptive Inverse Control Scheme for Noise Cancelation

This paper addresses the issues relating to the enforcement of robust stability when implementing the Adaptive Inverse Control (AIC) scheme. In this scheme, an adaptive FIR filter is added to a closed-loop system in order to reduce the output error caused by external disturbances, enhancing the performance achieved by linear time-invariant controllers alone. A Small-Gain-Theorem-based sufficient stability condition, which accounts for the feedback interaction between the time-varying adaptive filter and the unmodeled dynamics existing in the closed-loop plant, is derived. This condition suggests that system stability can be imposed by reducing the feasible region where the FIR filter gains can lie. Thus, a constrained convex optimization problem is formulated and solved for the FIR filter form. A relation between this optimization problem and the recursive least-squares (RLS) algorithm is established. Also, a suboptimal solution, which is implementable recursively, is proposed. Finally, to demonstrate their effectiveness, these algorithms are implemented on a laser beam steering experiment.