Stability of multichannel sound control systems

Sufficient conditions for the stability of decentralised, as well as fully coupled, active multichannel harmonic sound systems are derived. Such systems are being increasingly used to attenuate undesirable noise from harmonic (or narrowband) sources, at specific locations. These conditions are based on generalisations of the Gershgorin theorem. The advantage of using these generalisations is that they yield less conservative results than those based directly on the Gershgorin theorem, and more important, the conditions are applicable to systems where uncertainties in the physical parameters, and/or the frequency of the primary source within a given bandwidth, are assumed. The intuitive assertion that adding to the optimisation cost function a term which prevents using too much control effort is undesirable from the viewpoint of the steady-state error is formally shown. An analysis of the behaviour of the steady-state error as a function of the control effort is provided. A method to determine a small control effort which still guarantees stability of the optimisation algorithm is derived. A simple example is worked out analytically and simulation results for a more complicated example are provided.