## 2 Adaptive Filtering Systems for Active Control
### Least Mean Squares Method  
The least mean squares (LMS) algorithm is the most widely used feedforward control algorithm for adaptive digital filtering systems, since it is relatively simple and easy to implement. The stochastic gradient of the squared error signal is the core of this algorithm [33], and it iteratively regulates the coefficients of transversal filters. These filters use the following two values for the updating formulation: (1) a reference input \(u[k]\) containing frequency components to be dealt with; and (2) the measured error value \(e[k]\) at the current instant, which is the difference between the filter output \(y[k]\) and the disturbance (desired value) \(d[k]\) to be tracked. The algorithm adjusts the adaptive digital filter to determine a filter output \(y[k]\) that is as close as possible to the disturbance value \(d[k]\). A formulation of the LMS algorithm for updating these coefficients can be derived by minimizing the mean squared error with a pre-defined cost function. \[y[k]=\sum_{j=0}^{N}\mathbf{w}_{j}[k]\cdot\mathbf{u}[k-j] \tag{1}\]  
The most attractive feature of the LMS method is that it does not require average correlation functions from gradient estimates and matrix inversions [34]. It starts from the steepest descent method, and the output of the FIR filter is calculated by multiplying a reference vector \(\mathbf{u}\) by a vector of filter coefficients or weights \(\mathbf{w}\). In Equation (1), \(\mathbf{u}[k-j]\) is the reference input signal at time index \(k-j\), and \(\mathbf{w}_{j}[k]\) is the \(j^{th}\) filter coefficient at time index \(k\). The desired signal \(d[k]\) is the output of the disturbance plant to be tracked and reduced, and the error \(e[k]\) is the deviation between the filter output \(y[k]\) and the desired signal \(d[k]\). \[e[k]=d[k]-y[k] \tag{2}\]  
A cost function is chosen to minimize the error. The mean squared error \(\xi\) is typically selected as a cost function since it is a quadratic function and thus guarantees a unique minimum, which is a very important advantage for the digital filtering system. \[\xi=E[e[k]^{2}] \tag{3}\]  
The cost function is expanded by substituting Equations (1) and (2) into Equation (3):  
\[\xi=E[d[k]^{2}]-2\mathbf{w}[k]^{T}\mathbf{P}+\mathbf{w}[k]^{T}\mathbf{R} \mathbf{w}[k], \tag{4}\]  
\[\mathbf{P}=E[\mathbf{u}[k]\cdot d[k]], \tag{5}\]  
\[\mathbf{R}=E[\mathbf{u}[k]\cdot\mathbf{u}[k]^{T}] \tag{6}\]  
\(\mathbf{P}\) is a cross correlation vector between the reference input and the desired signal, and \(\mathbf{R}\) is the autocorrelation matrix of the reference signal. The steepest descent concept is used to develop an algorithm that adjusts the vector of filter weights \(\mathbf{w}\). The reason why this method is applied here is that it is very straightforward and has been validated in many applications. \[\mathbf{w}[k+1]=\mathbf{w}[k]-\frac{\mu}{2}\
abla\xi \tag{7}\]  
Next, Equation (7) is partially differentiated with respect to \(\mathbf{w}[k]\), and then the gradient of the cost function \(\
abla\xi\) can be derived:  
\[\
abla\xi=2\mathbf{R}\mathbf{w}[k]-2\mathbf{P} \tag{8}\]  
From Equations (7) and (8), the steepest descent algorithm is obtained:  
\[\mathbf{w}[k+1]=\mathbf{w}[k]+\mu(\mathbf{P}-\mathbf{R}\mathbf{w}[k]) \tag{9}\]  
where \(\mu\) is a parameter that regulates the convergence speed and the stability of the adaptation procedure. Figure 1 shows a block diagram of a typical adaptive digital filtering system. The gradient of the cost function \(\
abla\xi\) is calculated in every iteration with an average of squared errors at recent sampling times. However, it is usually impossible to predict the cross correlation vector  
Figure 1: Typical adaptive digital filtering system. \(\mathbf{P}\) and the correlation matrix \(\mathbf{R}\) at every sample time. Thus, to simplify the updating equation for the filter coefficients, the LMS algorithm uses instant estimations for \(\mathbf{P}\), \(\mathbf{R}\), and the cost function \(\xi\)[35].