Active noise control using an adaptive bacterial foraging optimization algorithm

Abstract

This paper presents an adaptive bacterial foraging optimization (ABFO) algorithm for an active noise control system. The conventional active noise control (ANC) systems often use the gradient-based filtered-X least mean square algorithms to adapt the coefficients of the adaptive controller. Hence, there is a possibility to converge to local minima. In addition, this class of algorithms needs prior identification of the secondary path. The ABFO algorithm helps the ANC system to prevent falling into local minima. The proposed ANC system is also simpler since it does not need any prior information of the secondary path. Moreover, the adaptive strategy of the algorithm results in improved search performance compared with the basic bacterial foraging optimization algorithm, as well as other conventional algorithms. Experimental studies are performed for nonlinear primary path along with linear and nonlinear secondary path. The results show the effectiveness of the proposed ABFO-based ANC system for different kinds of input noise.

Appendix: Proposed algorithm for ABFO-based ANC system

The proposed algorithm for ABFO-based ANC system has three main phases as follows:

Phase 1 Initializing the following parameters.

• $S$ is the number of bacteria to be used for searching the total region.

• $p$ is the dimension of the search space or the number of parameters to be optimized.

• $N_s$ is the swimming length of bacteria after each tumbling in a chemotactic loop.

• $N_c$ is the number of iterations to be undertaken in a chemotactic loop, $\left( {N_c >N_s } \right)$.

• $P_{ed}$ is the probability of continuing the elimination and dispersal event.

• $P$ is the location of each bacterium in the first bacteria population.

• $C^{i}\left( j \right)$ is the current run-length unit or step size taken in random direction for $i$th bacterium.

• $C_{{\text{ initial}}}$ is the initialized run-length unit. This is assumed to be constant for all the bacteria.

• $\varepsilon _{{\text{ initial}}}$ is the initialized precision goal. This is assumed to be constant for all the bacteria.

• $\varepsilon ^{i}\left( j \right)$ is the required precision fitness in the current generation of the $i$th bacterium.

• $ku$ is the user defined number of consecutive generations.

• ${\text{ flag}}^{i}$ is the number of generations during which the $i^{th}$ bacterium has not improved its own fitness.

• $\alpha$ is the run-length factor.

• $\beta$ is the cost function factor.

Phase 2 Obtaining the incoming noise samples.

In this step the input noise samples to the ANC system are taken. For each noise sample the algorithm runs separately and the corresponding residual noise is obtained.

• (a) While the input noise samples come to the ANC system — do the following

Phase 3 Running the Iterative Algorithm for Optimization.

This section models the ABFO operations (including chemotaxis, reproduction, and elimination-and-dispersal) on bacterial population.

• (b) Chemotaxis loop: for $j=1,2,\ldots ,N_c$

• 1. Filtering the input noise. In the adaptive Volterra filter, $S$ groups of the anti-noise signal $ds_i \left( n \right)$ can be generated with considering the $S$ groups of filter coefficients $W_i \left( n \right)=\left[ {w_i \left( {0,n} \right),w_i \left( {1,n} \right),\ldots ,w_i \left( {L-1,n} \right)} \right],i=1,\ldots ,S,$ given by

  $$\begin{aligned} \left[ {{\begin{array}{c} {y_1 \left( n \right)} \\ {y_2 \left( n \right)} \\ {.} \\ {.} \\ {.} \\ {y_S \left( n \right)} \\ \end{array}}} \right]&= \left[ {{\begin{array}{cccc} w_1 \left({0,n} \right)&w_1 \left( {1,n} \right)&\ldots&w_1 \left( {L-1,n} \right) \\ w_2 \left({0,n} \right)&w_2 \left( {1,n} \right)&\ldots&w_2 \left( {L-1,n} \right) \\ \vdots&\vdots&\vdots&\vdots \\ w_S \left( {0,n} \right)&w_S \left( {1,n} \right)&\ldots&w_S \left( {L-1,n} \right) \\ \end{array} }} \right]\nonumber \\&.\left[ {{\begin{array}{c} x\left( n \right) \\ x\left( {n-1} \right) \\ \vdots \\ x\left( {n-\frac{L}{2}+1} \right) \\ x^{2}\left( n \right) \\ x^{2}\left( {n-1} \right) \\ \vdots \\ x^{2}\left( {n-\frac{L}{2}+1} \right) \\ \end{array} }} \right],\end{aligned}$$

  (9)
  $$\begin{aligned} \left[ {{\begin{array}{c} ds_1 \left( n \right) \\ ds_2 \left( n \right) \\ {.} \\ {.} \\ {.} \\ {ds_S \left( n \right)} \\ \end{array} }} \right]&= \left[ {{\begin{array}{cccc} y_1&\left( n \right)y_1 \left( {n-1} \right)&\ldots&y_1 \left( {n-L_1 +1} \right) \\ y_2&\left( n \right)y_2 \left( {n-1} \right)&\ldots&y_2 \left( {n-L_1 +1} \right) \\ \vdots&\vdots&\vdots&\vdots \\ y_S&\left( n \right)y_S \left( {n-1} \right)&\ldots&y_S \left( {n-L_1 +1} \right) \\ \end{array} }} \right]\nonumber \\&.\left[ {{\begin{array}{c} {s_1 } \\ {s_2 } \\ {\vdots } \\ {s_{L_1 } } \\ \end{array} }} \right]. \end{aligned}$$

  (10)

where $\left[ {s_1,s_2,\ldots ,s_{L_1 } } \right]$ denotes the impulse response of the secondary path $S\left( z \right)$, and $W_i \left( n \right)$ denotes the coefficient vector of the $i$th bacterium of Volterra filter $W\left( z \right)$ at time $n$, with length $L$.

Then, Calculate the cost function or the residual noise function as

$$\begin{aligned} J\left( {i,j} \right)=e_i \left( n \right)=dp\left( n \right)+ds_i \left( n \right),\quad i=1,\ldots ,S. \end{aligned}$$

(11)

• 2. Let $J_{{\text{ last}}}^i =J\left( {i,j} \right)$ to save this value since we may find better value via a run.

• 3. Tumbling/Swimming loop: For $i=1,2,\ldots ,S$—take the tumbling/swimming decision

• Tumble: Generate a random vector $\Delta \left( i \right)\in R^{n}$. Each element of $\Delta \; (\Delta _m \left( i \right),m=1,2,\ldots ,p)$ is a random number on [-1, 1].

• Move (updating the position of bacteria): Let

  $$\begin{aligned} w^{i}\left( {j+1} \right)=w^{i}\left( j \right)+C^{i}\left( j \right)\frac{\Delta \left( i \right)}{\sqrt{\Delta ^{T}\left( i \right)\Delta \left( i \right)}}. \end{aligned}$$

  (12)

  This results in an adaptable step size of $C^{i}\left( j \right)$ in the direction of the tumble for $i^{th}$ bacteria. Compute $J\left( {i,j+1} \right)$ with $w^{i}\left( {j+1} \right)$.

• Swim:

• i. Let $m=0$ (counter for swim length).

• ii. While $m<N_S$

• Let $m=m+1$.

• If $J\left( {i,j+1} \right)<J_{{\text{ last}}}^i$ then let $J_{{\text{ last}}}^i =J\left( {i,j+1} \right)$, then another step of size $C^{i}\left( j \right)$ in this same direction will be taken as Eq. 12.

• Use $w^{i}\left( {j+1} \right)$ to compute the new $J\left( {i,j+1} \right)$ and then set ${\text{ flag}}^{i}=0.$

— Adaptive Chemotaxis

• Else, if $J\left( {i,j+1} \right)\ge J_{{\text{ last}}}^i$ then let ${\text{ flag}}^{i}={\text{ flag}}^{i}$ + 1 and if ${\text{ flag}}^{i}>ku$ then $C^{i}\left( {j+1} \right)=C_{{\text{ initial}}}^i$ and $\varepsilon ^{i}\left( {j+1} \right)=\varepsilon _{{\text{ initial}}}^i$.

• Else, if ${\text{ flag}}^{i}\le ku$ then let $C^{i}\left( {j+1} \right)=C^{i}\left( j \right)$ and $\varepsilon ^{i}\left( {j+1} \right)=\varepsilon ^{i}\left( j \right)$, let $m=N_S$.

• End of while statement.

• iii. If $J\left( {i,j+1} \right)<\varepsilon ^{i}\left( j \right)$ then $C^{i}\left( {j+1} \right)=C^{i}\left( j \right)/\alpha$ and $\varepsilon ^{i}\left( {j+1} \right)=\varepsilon ^{i}\left( j \right)/\beta$; Else $C^{i}\left( {j+1} \right)=C^{i}\left( j \right)$ and $\varepsilon ^{i}\left( {j+1} \right)=\varepsilon ^{i}\left( j \right)$.

— End of adaptive Chemotaxis.

• 4. End of tumbling/swimming loop. If $i\ne S$, go to (d) to process the next bacterium.

• 5. Reproduction:

• For each $i=1,2,\ldots ,S$, let

  $$\begin{aligned} J_{{\text{ health}}}^i =J\left( {i,j} \right) {\text{ for}} j=1,\ldots ,N_C +1. \end{aligned}$$

  (13)

• Sort bacteria in order of ascending values $J_{{\text{ health}}}$.

• The $S_r =S/2$ bacteria with the highest $J_{{\text{ health}}}$ values die and the other $S_r$ bacteria with the best values split and the copies that are made are placed at the same location as their parent.

• 6. Elimination-dispersal:

• For $i=1,2,\ldots ,S$, with probability$p_{ed}$, eliminate and disperse each bacterium, to keep the number of bacteria in the population constant.

• (c) End of Chemotaxis loop. — If $j<N_c$, go to Step 3.

• (d) End of while statement.

The flow chart of the described algorithm is shown in Fig. 5 .

Fig. 5

Flow chart of the proposed ABFO-based ANC algorithm