# Adaptive Pitch Control of Variable-Pitch PMSG Based Wind Turbine
## 3 Perturbation Observer-Based Nonlinear Adaptive Controller Design
### NAC Design of PMSG
#### 3.2.5 Stability Analysis of Closed-Loop System  
This subsection analyzes the stability of the closed-loop system equipped with the NAC designed in the previous section. At first, both the estimation error system and the tracking error system are obtained. On one hand, by defining estimation errors \(\varepsilon_{21}=z_{21}-\hat{z}_{21}\), \(\varepsilon_{22}=z_{22}-\hat{z}_{22}\), \(\varepsilon_{31}=z_{31}-\hat{z}_{31}\), \(\varepsilon_{32}=z_{32}-\hat{z}_{32}\)  
Figure 4: Block diagram of nonlinear adaptive controller. subtracting Equation (38) from Equation (37) and subtracting Equation (39) from Equation (37), the following estimation error system yields:  
\[\dot{\epsilon}_{i}=A_{i}\epsilon_{i}+\eta_{i} \tag{43}\]  
where  
\[\epsilon_{i}=\left[ \begin{array}{c}\varepsilon_{21}\\ \varepsilon_{22}\\ \varepsilon_{31}\\ \varepsilon_{32}\end{array} \right],\qquad\eta_{i}=\left[ \begin{array}{c} 0\\ \dot{\Psi}_{2}\\ 0\\ \dot{\Psi}_{3}\end{array} \right],\]  
\[A_{i}=\left[ \begin{array}{cccc}-l_{21}&1&0&0\\ -l_{22}&0&0&0\\ 0&0&-l_{31}&1\\ 0&0&-l_{32}&0\end{array} \right] \tag{44}\]  
On the other hand, define the tracking errors as \(e_{21}=y_{2\text{r}}-z_{21}\) and \(e_{31}=y_{3\text{r}}-z_{31}\). It follows from Equations (24), (26), (36), (40) and (41) that  
\[\left[ \begin{array}{c}\dot{e}_{21}\\ \dot{e}_{31}\end{array} \right]=-\left[ \begin{array}{c}k_{21}(e_{21}+ \varepsilon_{21})+\varepsilon_{22}\\ k_{31}(e_{31}+\varepsilon_{31})+\varepsilon_{32}\end{array} \right] \tag{45}\]  
Thus, the tracking error system can be summarized as  
\[\dot{e}_{i}=M_{i}e_{i}+\vartheta_{i} \tag{46}\]  
where  
\[e_{i}=\left[ \begin{array}{c}e_{21}\\ e_{22}\end{array} \right],\qquad\vartheta_{i}=\left[ \begin{array}{c}-\zeta_{ 1}\\ -\zeta_{2}\end{array} \right],\]  
\[M_{i}=\left[ \begin{array}{cc}-k_{21}&0\\ 0&-k_{31}\end{array} \right] \tag{47}\]  
with \(\zeta_{1}=\varepsilon_{22}+k_{21}\varepsilon_{21}\) and \(\zeta_{2}=\varepsilon_{32}+k_{31}\varepsilon_{31}\) being the lumped estimation error. The stability analysis of the closed-loop control system is transformed into globally uniformly ultimately bounded summarized. **Theorem 1**. : _Consider the PMSM system Equation (24) equipped the proposed NAC Equation (42) with two POs Equations (38) and (39). If the real perturbation \(\Psi_{i}(x,t)\) defined in Equation (36) satisfies_  
\[\|\Psi_{i}(x,t)\|\leq\gamma_{1} \tag{48}\]  
_then both the estimation error system Equation (43) and the tracking error system Equation (46) are, i.e.,_  
\[\|\epsilon_{i}(t)\|\leq 2\gamma_{1}\|P_{1}\|,\|e_{i}(t)\|\leq 4\gamma_{1}\|K_{i }\|\|P_{1}\|\|P_{2}\|,\forall t\geq T \tag{49}\]  
_where \(P_{i}\), \(i=1,2\) are respectively the feasible solutions of Riccati equations \(A_{i}^{\text{T}}P_{1}+P_{1}A_{i}=-I\) and \(M_{i}^{\text{T}}P_{2}+P_{2}M_{i}=-I\); and \(\|K_{i}\|\) is a constant related to \(k_{11},k_{21}\) and \(k_{22}\)._  
Proof. : For the estimation error system Equation (43), consider the following Lyapunov function:  
\[V_{i1}(\epsilon_{i})=\epsilon_{i}^{\text{T}}P_{1}\epsilon_{i} \tag{50}\]The high gains of POs Equations (38) and (39) are determined by requiring Equation (21) holds, which means \(A_{i}\) is Hurwitz. One can find a feasible positive definite solution, \(P_{1}\), of Riccati equation \(A_{i}^{\mathrm{T}}P_{1}+P_{1}A_{i}=-I\). Calculating the derivative of \(V_{i1}(\varepsilon_{i})\) along the solution of system Equation (43) and using Equation (48) to yield  
\[ \begin{array}{rcl}\dot{V}_{i1}(\varepsilon_{i})&=&\varepsilon_{i}^{\mathrm{T }}(A_{i}^{\mathrm{T}}P_{1}+P_{1}A_{i})\varepsilon_{i}+\eta_{i}^{\mathrm{T}}P_{ 1}\varepsilon_{i}+\varepsilon_{i}^{\mathrm{T}}P_{1}\eta_{i}\\ &\leq&-\|\varepsilon_{i}\|^{2}+2\|\varepsilon_{i}\|\cdot\|\eta_{i}\|\cdot\|P_{ 1}\|\\ &\leq&-\|\varepsilon_{i}\|(\|\varepsilon_{i}\|-2\gamma_{1}\|P_{1}\|)\end{array}  \tag{51}\]  
Then \(\dot{V}_{i1}(\varepsilon_{i})\leq 0\) when \(\|\varepsilon_{i}\|\geq 2\gamma_{1}\|P_{1}\|\).