## 3 The MISP Structure  
### Proposed Improvement on RLS Algorithm  
Parameter identification is a theory in which the unknown parameters of the system are identified through an algorithm combined with input and output data. The RLS algorithm is one of the methods of parameter identification, and is given by  
\[y=\left( \begin{array}{ccc}x_{1}&\cdots&x_{n}\end{array} \right)\left( \begin{array} []{c}\theta_{1}\\ \vdots\\ \theta_{n}\end{array} \right) \tag{15}\]  
where \(y\) is the output data and \(\left( \begin{array}{ccc}x_{1}&\cdots&x_{n}\end{array} \right)\) is the input data. \(n\) is the length of unknown parameters. \(\left( \begin{array}{ccc}\theta_{1}&\cdots&\theta_{n}\end{array} \right)^{T}\) is a system parameter to be identified. The matrices of the input data, output data, and system parameters were recorded as  
\[\mathbf{Y}_{k}=\left[ \begin{array}{ccc}y_{1}&\cdots&y_{k}\end{array} \right]^{T} \tag{16}\]  
\[\mathbf{\Phi}_{k}=\left[ \begin{array}{ccc}\mathbf{\phi}_{1}^{T}&\cdots&\mathbf{\phi}_{k }^{T}\end{array} \right]^{T} \tag{17}\]  
\[\mathbf{\Theta}=\left[ \begin{array}{ccc}\theta_{1}&\cdots&\theta_{n}\end{array}  \right]^{T} \tag{18}\]  
\[\mathbf{\phi}_{i}^{T}=\left( \begin{array}{ccc}x_{1}^{i}&\cdots&x_{n}^{i}\end{array} \right) \tag{19}\]  
where \(k\) is the length of observation data. The matrix form can be expressed as  
\[\mathbf{Y}_{k}=\mathbf{\Phi}_{k}\mathbf{\Theta} \tag{20}\]  
The solution of the parameter in the RLS algorithm with the forgetting factor in [18] can be expressed as  
\[\widehat{\mathbf{\Theta}}_{k}=\widehat{\mathbf{\Theta}}_{k-1}+\mathbf{K}_{k}e_{k} \tag{21}\]  
where \(\mathbf{K}_{k}\) is the Kalman gain vector and \(e_{k}\) is the estimation error, which is given by  
\[e_{k}=y_{k}-\mathbf{\phi}_{k}^{T}\widehat{\mathbf{\Theta}}_{k-1} \tag{22}\]  
When \(\sum\limits_{i=1}^{k}\lambda^{k-i}e_{k}^{2}\) gets the minimum, \(\widehat{\mathbf{\Theta}}_{k}\) obtain the optimal solution where \(\lambda\) is the forgetting factor. \(\mathbf{K}_{k}\) can be obtained by  
\[\mathbf{K}_{k}=\frac{\mathbf{P}_{k-1}\mathbf{\phi}_{k}}{\lambda+\mathbf{\phi}_{k}^{T}\mathbf{P}_{k -1}\mathbf{\phi}_{k}} \tag{23}\]where \(P_{k}\) is the inverse of the input correlation matrix, obtained by  
\[\mathbf{P}_{k}=\mathbf{P}_{k-1}-\frac{\mathbf{P}_{k-1}\mathbf{\phi}_{k}\mathbf{\phi}_{k}^{\top}\mathbf{P} _{k-1}}{\lambda+\mathbf{\phi}_{k}^{\top}\mathbf{P}_{k}\mathbf{\phi}_{k}} \tag{24}\]  
\(\lambda\) ranges from 0 to 1 and affects the tracking speed and stability of the estimated parameters. When the forgetting factor is close to 1, the estimation parameters become stable and the misadjustment is lower; however, the tracking ability deteriorates. On the contrary, when the forgetting factor decreases, the tracking speed becomes faster, but this will increase the imbalance. To balance the relationship between the convergence rate and the tracking stability for the estimation parameters, a specific function between \(\lambda\) and \(e_{k}\) is constructed to adjust the value of \(\lambda\) according to \(e_{k}\sigma_{e}\) is the power of the a priori error signal computed with a short exponential window [17] and takes the form  
\[\sigma_{e}^{2}(k)=\alpha\sigma_{e}^{2}(k-1)+(1-\alpha)e_{k}{}^{2} \tag{25}\]  
where \(\alpha\) is the weighting factor, given by  
\[\alpha=1-1/(K_{\alpha}n) \tag{26}\]  
with \(K_{\alpha}\geq 2\). Considering the noise, the power of noise \(\sigma_{v}^{2}\) can be estimated with a longer exponential window taking the form  
\[\sigma_{v}^{2}(k)=\beta\sigma_{v}^{2}(k-1)+(1-\beta)e_{k}{}^{2} \tag{27}\]  
where \(\beta\) is the weighting factor, given by  
\[\beta=1-1/(K_{\beta}n) \tag{28}\]  
where \(K_{\beta}\geq K_{\alpha}\). Theoretically, \(\sigma_{e}\geq\sigma_{v}\) and an RLS algorithm with \(\sigma_{e}\cong\sigma_{v}\) will result in \(\lambda\cong 1\). The forgetting factor of the proposed RLS algorithm is designed in the following forms  
\[\lambda(k)=\frac{1}{1+\mu|\sigma_{e}^{2}(k)-\sigma_{v}^{2}(k)|} \tag{29}\]  
where \(\mu\) is a positive constant parameter.