Fundamental limitations of ANC in one -dimensional ducts using 2 sensors and 1 actuator
Active noise control (ANC) has remained a challenging problem in the controls community, even with the rapid development of computers and state-of-the-art digital signal processors (DSP). Despite some limited success in commercial applications, there exists very little published theoretical work that can be used for design purposes. One possible obstacle is that the area of ANC comprises a combination of unrelated backgrounds. In general, a control engineer does not have sufficient knowledge in the physics of acoustics. This may explain why much of the ANC research focuses on adaptive control solutions which does not require deep insight into the acoustics. An adaptive system, however, is nonlinear and has its own open problems. For example, closed-loop stability and achievable performance of an adaptive ANC system cannot be predicted a priori and is generally difficult to analyze. This is the main reason we choose to discuss only linear, fixed-filter ANC systems.
In this research we study in detail ANC systems designed to cancel noise in acoustic ducts. Due to the geometry of duct aspect ratio, we can simplify the description of the sound wave propagation along the duct to one-dimensional. Our emphasis is on analysis and design of fixed-controllers using at most two sensors and one actuator. The main thrust of this dissertation is to develop fundamental limitation theories tailored to such duct ANC problems. In particular, we apply the recently developed theories on single-input-two-output (SITO) feedback system. The alignment angle approach is found to be well suited to our ANC setup and has become the main machinery used in the development of stability and performance analysis, as well as in the selection of sensor and actuator locations. We present a new tradeoff between closed-loop stability margins and achievable performance resulting from constraints imposed by certain duct configurations. This inherent limitation can be hard to overcome with one-sensor design schemes such as feedforward controllers prevalently used in ANC literature. We demonstrate via analysis and simulations the advantages of using a two-input-single-output (TISO) controller.