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Adaptive signal processing algorithms play an important role in many practical applications in diverse fields, such as telecommunication, radar, sonar, multimedia, biomedical engineering and noise control. Recently, a group of adaptive filtering algorithms called partial update adaptive algorithms (partial updates) has gathered considerable attention in both research and practical applications. This paper is a study of the application of PUs to very demanding, structural active noise control (ANC) systems, which are of particular interest due to their ability to provide for a global noise reduction. However, such systems are multichannel, with very high computational power requirements, which may be reduced by the application of partial updates. The paper discusses the modifications necessary to apply PUs in structural ANC systems and the potential computational power savings offered by this application. As a result, leaky versions of the PU LMS algorithms are introduced to the general public. The paper also presents two simulation examples, based on real laboratory setups, confirming high performance of the proposed algorithms.