A method of adaptation between steepest-descent and Newton's algorithm for multichannel active control of tonal noise and vibration

Abstract

Active control methods have been applied to a number of practical problems in which it is necessary to control a tonal disturbance. For example, the control of engine noise and vibration in vehicles, propeller noise in aircraft and the vibration produced by reciprocating machinery in many industrial systems. In such applications the steepest-descent algorithm has been widely employed, in part due to its robustness to variations in the plant response. This robustness, however, comes at the expense of a potentially slow convergence speed and this may limit the performance in applications where the disturbance is non-stationary. To improve the speed of convergence, an iterative least-squares algorithm can be employed, such as Newton’s algorithm. The convergence of these algorithms is less dependent on the potentially large eigenvalue spread of a multichannel plant matrix and, therefore, can theoretically achieve more rapid convergence. However, these algorithms are significantly less robust to plant response variations and, therefore, their practical performance can be somewhat limited. Generalised algorithms have been presented which combine steepest-descent and Newton’s method in or- der to provide a fixed compromise between convergence and robustness. This paper presents a method of adaptively combining steepest-descent and Newton’s method in order to achieve both rapid convergence and robustness to plant response variations. The two algorithms are combined into a single update equation in which a single mixing parameter facilitates a trade-off between the two algorithms. A method of adapting this parameter to minimise the cost function is presented and the performance of the proposed algorithm is assessed through a series of simulations. The proposed combination algorithm is shown to improve the control performance in the presence of plant response variations compared to both the steepest-descent and Newton’s algorithms.

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ORCID for Jordan Cheer: orcid.org/0000-0002-0552-5506

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