# Some Practical Acoustic Design and Typical Control Strategies for Multichannel Active Noise Control
## 3 Three Active Control Strategies  
Given a certain acoustic design, the control algorithm determines the final performance of an ANC system. A lot of ANC algorithms have been developed. For example, in [PERSON]'s papers [17, 18], each of the 16 controllers use the filtered-x least mean squares (FxLMS) algorithm [27] to generate proper anti-sound. In [PERSON]'s 2017 paper [13], a commercial ANC system, namely Antsyound Tiger ANC-II, was used for noise control. To make the multichannel controllers cooperate with each other, different strategies could be employed. In this subsection, three frequently used control strategies, i.e., the centralized, decentralized, and distributed strategies, are discussed. The properties of the three strategies are briefly summarized in Table 3. First, we compare these strategies in terms of the structure (see Figure 6) and the computational complexity (see Table 4). Then, we report recent progress in the decentralized and distributed control strategies that consume less hardware resources and are much easier to be implemented in real applications. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Centralized FxLMS** & **Decentralized FxLMS** & **Distributed FxLMS** \\ \hline Secondary path modeling & \(K^{2}\) & \(K\) & \(K\) \\ Adaptive filter & \(K^{2}L\) & \(KL\) & \(KL\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Arithmetic Complexities of Various Control Methods. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Centralized FxLMS** & **Decentralized FxLMS** & **Distributed FxLMS** \\ \hline Cost function & \(J_{c}=E[\sum_{k=1}^{K}\epsilon_{k}^{2}(n)]\) & \(J_{d,k}=E[\epsilon_{k}^{2}(n)]\) & \(J_{dis,k}=E[\epsilon_{k}^{2}(n)]\) \\  & & for \(k=1,...,K\) & \(s.t.\)\(w_{k}=w_{l},l\in N_{k}/\{k\}\) \\  & & & for \(k=1,...,K\) \\ Performance in general cases & High & Low & Close to centralized FxLMS \\ Complexity & High & Low & Low \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Brief Summary of Various Control Strategies. Figure 6: Cost functions and block diagrams of ANC systems with different control strategies: (**a**) centralized; (**b**) decentralized; and (**c**) distributed. The centralized control was first proposed as the multiple error (ME) ANC algorithm in [PERSON]'s 1987 paper [28]. The MEANC algorithm could also be implemented by FxLMS or its variants [29; 30; 31; 32] if the noise signal, known as reference signal, is collected. MEANC optimizes the residue over all channels such that each controller updates the coefficients by using data from all error microphones (see the cost function in Table 3 and diagram for the centralized ANC system with \(K\) loudspeakers/error microphones in Figure 6a). Different from the centralized control, the decentralized control [33] lets each controller minimize the error power at one error microphone only (see the cost function in Table 3 and diagram in Figure 6b). Comparing Figure 6a,b, it can be seen that, the centralized controller needs to process signals from all the error microphones and model \(K^{2}\) transfer functions while the decentralized controller only processes signals from one error microphone. Therefore, the decentralized structure could decrease the computational burden from order \(K^{2}\) to \(K\). The computational complexity of these two structures are compared in Table 4.  
To balance between the complexity and overall performance, a third structure, called the distributed control, has been developed in the recent 5 years. From the cost function of the diffusion control shown in Table 3 and its diagram in Figure 6c, it can be seen that the distributed strategy also processes signals from one error microphone such that it has a complexity comparable to the decentralized strategy (see Table 4). It further combines the controller parameters along a linking path or within a neighborhood \(N_{k}\) for \(k=1,...,K\) according to the incremental [34] or diffusion [35] rule, which could increase the system robustness without involving any extra multiplication.