Interaction Between Multiple Feedforward Active Control Systems

Centralized feedforward techniques are often used in active control systems that simultaneously adjust the inputs to multiple secondary sources to minimize the sum of the squared outputs from multiple error sensors. If the number of transducers is large, such centralized control systems require considerable computational power and cabling. In this paper, the stability and performance of decentralized adaptive feedforward systems are considered, in which a number of smaller adaptive controllers are implemented independently, each driving subsets of all the secondary sources to minimize the sum of squared outputs of subsets of the error sensors. The limit of this philosophy is when each secondary source is adjusted to minimize the output of only one error sensor. A general theoretical analysis of such decentralized feedforward controllers is developed. It is also shown that a simple condition can be derived, using the Gerschgorin circle theorem, which provides a sufficient, hut not necessary, condition for the stability of the overall system. Unfortunately, this condition appears to be rather conservative for larger systems. Examples of the application of this general analysis are presented for two and three channel active sound control systems operating in free space and a 32-channel sound control system operating in an enclosure. Provided the error sensors are close to the secondary source used to control them than the other secondary sources, the stability of the control system is seen to be quite robust to decentralization. The steady-state performance is also similar to that of a fully coupled control system, although the transient response generally involves oscillatory modes, which sometimes decay only rather slowly.