## 3 Enhanced LMS Algorithm
### Sliding Mode LMS (SM-LMS)  
A different structure of the adaptive filtering system is introduced using SMC for active vibration isolation. This new system prevents misadjustment from improper initial states and excessive error problems with respect to the amplitude of the desired values for the sake of the SMC. The sliding mode control is improved by the LMS in terms of smooth convergence and stability. With these properties, it is expected that the technique will enable multi-spectral control of disturbances to manage noise signals and a number of frequency elements simultaneously with optimized tracking schemes. The SMC technique is combined with the LMS to track or control \(x(t)\) due to its intrinsic benefits of insensitivity to the plant dynamics, robustness, and ability to decouple systems with high dimension [49]. The proposed SM-LMS technique is illustrated in Figure 4.  
Figure 3: Phase trajectory in sliding mode: (**a**) ideal sliding mode; and (**b**) real sliding mode. \(f[k]\) in Figure 2 is defined as follows:  
\[f[k]=\sum_{j=0}^{N}w_{j}[k]\cdot u[k-j], \tag{18}\]  
where \(w_{j}[k]\) indicates the \(j^{th}\) filter coefficient at sampling index \([k]\). \(e[k]\) is defined with the assumption that \(P_{s}(z)\) can be disregarded:  
\[e[k]=d[k]-f[k] \tag{19}\]  
Next, the sliding mode \(s[k]\) in the error state plane is designated as follows. Here, \(\alpha\) determines the slope of the sliding surface. \[s[k]=\alpha\cdot e[k]+e[k-1] \tag{20}\]  
Using a similar procedure to the derivation of the LMS algorithm, the cost function \(\xi\) for the SM-LMS is now defined as Equation (21), where \(E\) means the expected value. Note that the cost function \(\xi=E[e[k]^{2}]\) is applied in the LMS algorithm. \[\xi=E[s[k]^{2}] \tag{21}\]  
A unique minimum value is guaranteed since \(\xi\) is a quadratic function. By expanding \(\xi\) using Equations (4) and (20), the following results can be obtained:  
\[ \begin{split}\xi&=\alpha^{2}\Big{\{}E[d[k]^{2}]-2 \mathbf{w}[k]^{T}\cdot\mathbf{P}_{1}[k]+\mathbf{w}[k]^{T}\cdot\mathbf{R}_{1}[k ]\cdot\mathbf{w}[k]\Big{\}}\\ &\quad+\Big{\{}E[d[k-1]^{2}]-2\mathbf{w}[k-1]^{T}\cdot\mathbf{P }_{2}[k]+\mathbf{w}[k-1]^{T}\cdot\mathbf{R}_{2}[k]\cdot\mathbf{w}[k-1]\Big{\}} \\ &\quad+2\alpha\Big{\{}E[d[k]\cdot d[k-1]]-\mathbf{w}[k-1]^{T} \cdot\mathbf{P}_{3}[k]-\mathbf{w}[k]^{T}\cdot\mathbf{P}_{4}[k]+\mathbf{w}[k]^ {T}\cdot\mathbf{R}_{3}[k]\cdot\mathbf{w}[k-1]\Big{\}}\end{split}  \tag{22}\]  
Figure 4: Proposed sliding mode least mean squares control (SM-LMS) method. The vector **P** indicates the expected cross correlation between the reference and unwanted signals, and the matrix **R** is the expected autocorrelation of the reference:  
\[\textbf{P}_{1}[k]=E[\textbf{u}[k]\cdot d[k]] \tag{23}\]  
\[\textbf{P}_{2}[k]=E[\textbf{u}[k-1]\cdot d[k]] \tag{24}\]  
\[\textbf{P}_{3}[k]=E[\textbf{u}[k-1]\cdot d[k-1]] \tag{25}\]  
\[\textbf{P}_{4}[k]=E[\textbf{u}[k]\cdot d[k-1]] \tag{26}\]  
\[\textbf{R}_{1}[k]=E[\textbf{u}[k]\cdot\textbf{u}[k]^{T}] \tag{27}\]  
\[\textbf{R}_{2}[k]=E[\textbf{u}[k-1]\cdot\textbf{u}[k]^{T}] \tag{28}\]  
\[\textbf{R}_{3}[k]=E[\textbf{u}[k]\cdot\textbf{u}[k-1]^{T}] \tag{29}\]  
The next step is using the steepest descent scheme to obtain an algorithm for regulating the weights \(\textbf{w}[k]\). \[\textbf{w}[k+1]=\textbf{w}[k]-0.5\mu\
abla\xi \tag{30}\]  
Equation (21) is partially differentiated with respect to \(\textbf{w}[k]\) in order to derive the gradient of the cost function \(\
abla\xi\):  
\[\
abla\xi=-2\alpha^{2}\cdot\textbf{P}_{1}[k]+2\alpha^{2}\cdot\textbf{R}_{1}[k ]\cdot\textbf{w}[k] \tag{31}\]  
\[\textbf{P}_{1}[k]=E[\textbf{u}[k]\cdot d[k]] \tag{23}\]0  
For this control algorithm, the gradient \(\
abla\xi\) along with **P** and **R**, have to be determined in every time instant \(k\), as mentioned in the previous section. This is a clumsy process, and to simplify the calculating operations, instantaneous estimates of \(\xi\), **P**, and **R** should be obtained at a specific sampling instant in the LMS algorithm [34].