A Novel Active Control Strategy with Decentralized Decoupling and Wavelet Packet Transformation: Design and Verification

Active vibration control (AVC) can solve many vibration problems. However, structural vibration in underwater vehicles often involves other factors such as complex excitation and path coupling, etc. At present, the traditional algorithm (e.g., multi Filtered-x Least Mean Square, M-FxLMS) usually cannot effectively process the multi-frequency excitation and the coupling effects of the multi-secondary path, which will affect its convergence and stability to a certain extent. Consequently, a novel strategy is presented in this paper, namely, the wavelet packet transformation decentralized decoupling M-FxLMS algorithm (WPTDDM-FxLMS), which can solve the structural vibration problems mentioned above. The multi-frequency control is converted into a single-frequency line spectrum control, and the feedback compensation factor is introduced in the identification of the secondary path, both of which can simplify the multi-path control system to the parallel single-path systems. Furthermore, the WPTDDM-FxLMS algorithm is applied to the AVC in a multi-input and multi-output system (MIMO) vibration platform. Finally, the simulation and experiments show that the wavelet packet can decompose the multi-frequency excitation into a line spectrum signal, and the improvement of the decentralized decoupling and the variable step-size can effectively reduce the computation amount and increase the convergence speed and accuracy. Overall, the novel algorithm is significant for multi-path coupling vibration control. It will have certain engineering application value in underwater vehicles.

Sampling biased monotonic surfaces using exponential metrics

Monotonic surfaces spanning finite regions of ℤd arise in many contexts, including DNA-based self-assembly, card-shuffling and lozenge tilings. One method that has been used to uniformly generate these surfaces is a Markov chain that iteratively adds or removes a single cube below the surface during a step. We consider a biased version of the chain, where we are more likely to add a cube than to remove it, thereby favouring surfaces that are ‘higher’ or have more cubes below it. We prove that the chain is rapidly mixing for any uniform bias in ℤ2 and for bias λ > d in ℤd when d > 2. In ℤ2 we match the optimal mixing time achieved by Benjamini, Berger, Hoffman and Mossel in the context of biased card shuffling [2], but using much simpler arguments. The proofs use a geometric distance function and a variant of path coupling in order to handle distances that can be exponentially large. We also provide the first results in the case of fluctuating bias, where the bias can vary depending on the location of the tile. We show that the chain continues to be rapidly mixing if the biases are close to uniform, but that the chain can converge exponentially slowly in the general setting.