## 3 Overview of the Selected Techniques
### Global Adaptive Observer (GAO)
#### 3.1.2 Observer Design  
Let us the consider the following transformation for the state vector \(\xi\):  
\[\Xi=\mathcal{T}\xi, \tag{13}\]  
where \(\Xi=\bigl{[} \begin{array}{cc}\Xi_{1}&\Xi_{2}\end{array} \bigr{]}^{T}\) is the transformed state vector and the non-singular matrix \(\mathcal{T}\) is given by:  
\[\mathcal{T}=\frac{(1+\eta)^{-1}}{\omega_{n}^{2}}\biggl{[} \begin{array}{cc}1 &-\frac{1}{\omega_{n}}\\ \eta\omega_{n}&1\end{array} \biggr{]}.\]  
For the state vector \(\Xi\), the transformed state and output matrices are given by:  
\[\mathcal{A}_{\Xi}=\mathcal{T}\mathcal{A}\mathcal{T}^{-1}=\mathcal{A},\mathcal{ C}_{\Xi}=\mathcal{C}\mathcal{T}^{-1}.\]  
Then, the transformed dynamical model for the phase voltage \(U_{a}\) is given by:  
\[\dot{\Xi} =\mathcal{A}_{\Xi}\Xi, \tag{14}\] \[U_{a} =\mathcal{C}_{\Xi}\Xi. \tag{15}\]  
The following Luenberger observer can be designed for the system (14) and (15):  
\[\dot{\Xi}=\mathcal{A}_{\Xi}\dot{\Xi}+\mathcal{L}\bigl{(}U_{a}-\dot{U}_{a} \bigr{)} \tag{16}\]  
where'represents estimated value, \(\dot{U}_{a}=\mathcal{C}\dot{\Xi}\), \(\mathcal{L}\in\mathbb{R}^{2\times 1}\) is the observer gain matrix, and  
\[\dot{\mathcal{A}}_{\Xi}=\biggl{[} \begin{array}{cc}0&1\\ -\eta\omega_{n}^{2}&0\end{array} \biggr{]}.\]To study the convergence of the observer, let us consider the state and parameter estimation errors as:  
\[\hat{\Xi} = \Xi-\hat{\Xi},\] \[\hat{\eta} = \eta-\hat{\eta}.\]  
Then, the closed-loop error dynamics for the system (14) and (15) and the observer (16) are given by:  
\[\hat{\Xi} =\hat{\Xi}-\hat{\Xi},\] \[=\mathcal{A}_{\Xi}\[PERSON]-\left\{\mathcal{A}_{\Xi}\hat{\Xi}+\mathcal{ L}(C\Xi-C\hat{\[PERSON]})\right\},\] \[=\mathcal{A}_{\Xi}\[PERSON]-\left\{\mathcal{A}_{\Xi}\hat{\Xi}- \mathcal{A}_{\Xi}\hat{\Xi}+\mathcal{A}_{\Xi}\hat{\Xi}+\mathcal{L}\big{(} \mathcal{C}_{\Xi}\Xi-C_{\Xi}\hat{\Xi}\big{)}\right\},\] \[=(\mathcal{A}_{\Xi}-\mathcal{L}\mathcal{C}_{\Xi})\left(\[PERSON]-\hat {\Xi}\right)+\big{(}\mathcal{A}_{\Xi}-\mathcal{A}_{\Xi}\big{)}\hat{\[PERSON]},\] \[=(\mathcal{A}_{\Xi}-\mathcal{L}\mathcal{C}_{\Xi})\hat{\Xi}+ \mathcal{R}\omega_{n}^{2}\hat{\mu}\hat{\Xi}_{1}, \tag{17}\]  
where \(\mathcal{R}=\left[ \begin{array}{cc}0&1\end{array} \right]^{T}\). For suitably designed observer gain matrix \(\mathcal{L}\in\mathbb{R}^{2\times 1}\), the matrix, \(\mathcal{A}_{\Xi}-\mathcal{L}\mathcal{C}_{\Xi}\) can be made Hurwitz, i.e., all the eigenvalues have strictly negative real parts. This makes the closed-loop error system strictly positive real (SPR) [41] and according to the SPR lemma, there exists a matrix \(\mathcal{M}=\mathcal{M}^{T}>0\), such that  
\[(\mathcal{A}_{\Xi}-\mathcal{L}\mathcal{C}_{\Xi})^{T}\mathcal{M}+ M(\mathcal{A}_{\Xi}-\mathcal{L}\mathcal{C}_{\Xi}) =-\mathcal{Q},\mathcal{Q}=\mathcal{Q}^{T}>0, \tag{18}\] \[\mathcal{M}\mathcal{R} =\mathcal{C}_{\Xi}^{T}. \tag{19}\]  
The following Lyapunov function is considered for system (17):  
\[V(\hat{\Xi},\hat{\eta})=\hat{\Xi}^{T}\mathcal{M}\hat{\Xi}+\gamma^{-1}\hat{\eta }^{2},\gamma>0. \tag{20}\]  
Then, the derivative of the Lyapunov function becomes:  
\[\dot{V}=-\hat{\Xi}^{T}\hat{\Xi}+2\big{(}U_{a}-\hat{U}_{a}\big{)}\hat{\eta} \omega_{n}^{2}\hat{\Xi}_{1}-2\gamma^{-1}\hat{\eta}\hat{\eta}. \tag{21}\]  
If the frequency update law is chosen as:  
\[\hat{\eta}=\gamma\omega_{n}^{2}\big{(}U_{a}-\hat{U}_{a}\big{)}\hat{\Xi}_{1}. \tag{22}\]  
Then, it becomes that \(\dot{V}\leq 0\), which proves the Lyapunov stability of the estimation error dynamics. From the estimated transformed state vector \(\hat{\Xi}\), the state vector \(\xi\) can be found through the following equation:  
\[\hat{\xi}=\mathcal{T}^{-1}\hat{\Xi}, \tag{23}\]  
where  
\[\mathcal{T}^{-1}=\omega_{n}^{2}\bigg{[} \begin{array}{cc}1&\frac{1}{\omega_ {n}}\\ -\hat{\eta}\omega_{n}&1\end{array} \bigg{]}.\]  
The process described above can be applied in the same way to the grid phase voltages \(U_{b}\) and \(U_{c}\). As the frequency is same for all phases, only one frequency identification law is enough.