A stability analysis of a decentralized adaptive feedback active control system of sinusoidal sound in free space

In some cases, the implementation of active control of sound in free space requires a large number of secondary sources and error sensors. In terms of control hardware, this may translate into considerable processing power requirement. A practical method to decrease processing power is to decentralize the control; that is, implement many single-input, single-output independent controllers operating simultaneously instead of a large multiple-input, multiple-output system. The main drawback of decentralized control is the risk of global instability. The purpose of this paper is to derive conditions under which globally stable control system behavior can be obtained in the case of adaptive feedback decentralized control for a sinusoidal disturbance. The main objective is to give practical conditions derived from the small gain theorem and the Nyquist criterion for the stability of the control system. These conditions only take into account the geometrical arrangement of the secondary sources and error sensors. This analysis involves a new parameter beta called "performance index," which is associated with both the convergence of the individual controllers and the global stability of the system. Simulation and experimental results are shown to illustrate the effectiveness of the developed analytical tools.