This paper explores use of the Kalman filter to estimate modal states in Modal Filtered-X LMS algorithm. Modal Filtered-x LMS (M FxLMS) algorithm is an adaptive algorithm, recently proposed by the authors, to minimize global acoustic potential energy of a vibro-acoustic cavity for reducing global level of noise in the cavity. For a weakly coupled vibro-acoustic cavity, global acoustic potential energy can be expressed as sum of squares of modal amplitudes of rigid walled acoustic modes of the cavity. Therefore, the algorithm was formulated to minimize sum of squares of modal amplitudes of the rigid walled acoustic modes.
The M FxLMS algorithm turned out to be modal counterpart of the conventional FxLMS algorithm. In the conventional FxLMS algorithm, sum of squares of pressures at the location of microphones is minimized and hence, values of acoustic pressure at the location of error microphones and physical secondary path impulse responses are used. A physical secondary path impulse response represents the variation of acoustic pressure at location of error microphone when secondary source is driven by an impulse function. In the M FxLMS algorithm, sum of squares of modal amplitudes of some chosen acoustic modes is minimized and hence, values of modal amplitudes of the chosen acoustic modes and modal secondary path impulse responses are used.
A modal secondary path impulse response represents the variation of modal amplitude of a particular acoustic mode when secondary source is driven by an impulse function. In the earlier study, modal amplitude of the acoustic modes are obtained using information of acoustic pressure at some discrete locations inside the cavity and information of the chosen acoustic mode shapes at those locations. In that approach, number of the acoustic pressure sensors must be adequately more than number of the acoustic modes contributing in desired frequency range. In a view of reducing number of the acoustic pressure sensors, this paper explores use of Kalman filter to estimate modal amplitudes of the acoustic modes. A modal model of the vibro-acoustic cavity is used in the formulation of Kalman filter for estimation of modal amplitudes of acoustic modes.
A numerical study is carried out on the same irregular vibro-acoustic cavity which was used for the earlier study. An acoustic disturbance acting inside the cavity and a structural disturbance acting on the flexible surface of the cavity are considered. An acoustic control source is used as a secondary source. The study is carried out for three frequencies: a cavity controlled resonance, a panel controlled resonance and an off-resonance frequency. It is found that with the use of Kalman filter; only 6 microphones are required as against the 18 microphones required in the previous approach for the accurate estimation and then, control of modal amplitudes of first 12 acoustic modes.
Dr. Amrita Puri, Indian Institute of Technology, Jodhpur, INDIA; Prof. Subodh V. Modak, IIT Delhi, INDIA; Prof. Kshitij Gupta, IIT Delhi, INDIA