## 3 Control Strategies
### NAMIMO Controller Design and Stability Analysis  
Equation (18) is the state space equation of the compensated system, and the design procedure is elaborated as follows:  
Step 1: Create a _Lyapunov_ function equation that includes all error signals as follows:  
\[V=\frac{1}{2}e_{p}^{2}+\frac{1}{2}e_{q}^{2}+\frac{1}{2}\alpha e_{\omega}^{2} \tag{19}\]  
where \(\alpha\) is defined as the adaptive control rate parameter, then calculate the dynamic equation of the (19) as follows:  
\[\begin{array}{ll}\dot{V}&=e_{p}\dot{e}_{p}+e_{q}\dot{e}_{q}+\alpha e_{\omega} \dot{e}_{\omega}\\ &=e_{p}\left(-\frac{R}{L}p_{1}-\omega q_{1}-\frac{9}{2L}U_{m}^{2}+\frac{3}{2L }\sqrt{\frac{3}{2}}U_{m}M_{1}+\frac{3U_{m0}U_{d0}\cos\delta_{0}}{X}(\omega- \omega_{0})\right)\\ &+e_{q}\left(\omega p_{1}-\frac{R}{L}q_{1}+\frac{3}{2L}\sqrt{\frac{3}{2}}U_{m }M_{2}+\frac{3U_{m0}U_{d0}\sin\delta_{0}}{X}(\omega-\omega_{0})\right)-\alpha e _{\omega}\dot{\omega}\end{array} \tag{20}\]  
According to the _[PERSON]_ stability theory, (20) must be a negative definite function to ensure that the system is asymptotically stable. Therefore, the feedback linearization of (20) is performed in this step. The linearization function is as follows:  
\[M_{1}\triangleq-\frac{4\sqrt{6}}{9}L\big{(}k_{1}e_{p}+\Gamma_{1}\big{)} \tag{21}\]  
\[M_{2}\triangleq-\frac{4\sqrt{6}}{9}L\big{(}k_{2}e_{q}+\Gamma_{2}\big{)} \tag{22}\]  
where  
\[\Gamma_{1}\triangleq-\  
In order for (25) to be a negative definite function, the last term of the equation must be equal to zero. Thus, a turning function \(\phi\) is defined as follows:  
\[\phi=\frac{3U_{m0}U_{g0}\cos\delta_{0}}{X}(e_{p}+e_{q})+e_{q}p_{1}-e_{p}q_{1} \tag{26}\]  
and the adaptive law of \(\omega\) is shown below:  
\[\hat{\omega}=\alpha\int\phi dt \tag{27}\]  
Step 3: Substituting (26) and (27) into (25), the _Lyapunov_ function can be reduced to (28).  
\[\dot{V}=-k_{1}e_{p}^{2}-k_{2}e_{q}^{2} \tag{28}\]  
where \(k_{1}\) and \(k_{2}\) are controller gains. Note that (28) is a semi-negative function. Therefore, the stability of the compensation system cannot be directly derived from (28). Now, the system stability of the NAMIMO controller needs to be demonstrated.  
First, writing (28) in compact form as follows:  
\[\dot{V}=-\begin{bmatrix}e_{p}\\ e_{q}\\ e_{\omega}\end{bmatrix}^{\mathrm{T}}\begin{bmatrix}k_{1}&&\\ &k_{2}&\\ &&0\end{bmatrix}\begin{bmatrix}e_{p}\\ e_{q}\\ e_{\omega}\end{bmatrix} \tag{29}\]  
Define the error vector as:  
\[\mathbf{e}=\begin{bmatrix}e_{p}&e_{q}\end{bmatrix}^{\mathrm{T}} \tag{30}\]  
Due to:  
\[\mathbf{e}^{T}diag(k_{1},k_{2})\mathbf{e}\geq 0,\forall\mathbf{e  
Second, bringing (36) into (35) yields a new inequality as:  
\[\int_{0}^{\infty}||\mathbf{e}(t)||^{2}dt\leq\underbrace{\frac{V(0)}{\lambda_{\min} \{diag(k_{1},k_{2})\}}}_{:=\rho} \tag{37}\]  
Meanwhile, according to the analysis above, the following conditions hold:  
\[\sup_{t\geq 0}\lVert\rho\rVert,\exists \tag{38}\]  
. As a result, the following extrapolation is proven:  
\[\mathbf{e}\in L_{2}^{2} \tag{39}\]  
Finally, according to _Barhalat Lemma_[31], we have a conclusion as follows:  
\[\lim_{t\to\infty}\mathbf{e}=0\in R_{2}^{2} \tag{40}\]  
Overall, although (28) shows that the _Lyapunov_ function is semi-negative, Equation (38) proves that the whole system is asymptotically stable. This means that two power control objectives of the system can converge to the reference value.