# 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
Thus, there exists \(T_{1}>0\), which can lead to  
\[\|\varepsilon_{i}(t)\|\leq\gamma_{2}=2\gamma_{1}\|P_{1}\|,\forall t\geq T_{1} \tag{52}\]  
For tracking error system Equation (46), one can find that \(\|\theta_{i}\|\leq\|K_{i}\|\gamma_{2}\) with \(\|K_{i}\|\) based on \(\|\varepsilon_{i}(t)\|\leq\gamma_{2}\). Consider the Lyapunov function \(V_{i2}(e_{i})=e_{i}^{\mathrm{T}}P_{2}e_{i}\). Similarly, one can prove that there exists an instant, \(T_{1}\), the following holds  
\[\|e_{i}(t)\|\leq 2\|K_{i}\|\gamma_{2}\|P_{2}\|\leq 4\gamma_{1}\|K_{i}\|\|P_{ 1}\|\|P_{2}\|,\forall t\geq\tilde{T}_{1} \tag{53}\]  
Using Equations (52) and (53) and setting \(T=\max\{T_{1},\tilde{T}_{1}\}\) lead to Equation (49). Moreover, if \(\Psi_{i}(x,t)\) and \(\dot{\Psi}_{i}(x,t)\) are locally Lipschitz in their arguments, it will guarantee the exponential convergence of the observation error [19] and closed-loop tracking error into  
\[\lim_{t\to\infty}\varepsilon_{i}(t)=0\quad\text{and}\quad\lim_{t\to\infty}e_{ i}(t)=0 \tag{54}\]  
After the states \(i_{\mathrm{d}}\) and \(i_{\mathrm{q}}\) and their derivatives are stable that controlled by NAC. The parameter variation is considered in the error system in Equations (43) and (46), and the error system is proved as converged to zero in Equation (54). This guarantees that the estimated perturbations track the extended states defined in Equation (36), which includes the uncertainties affected by the parameter variations and disturbances, and compensates for the control input in Equation (40). Then the linearized subsystems in Equation (37) are independent of the parameters and disturbances. The perturbation and its derivative are assumed to locally bounded as described in **Assumption 2**. The existence of these bounds can be shown in the following analysis. The perturbation and its derivative can be represented as  
\[\begin{array}{rcl}\Psi_{2}&=&F_{2}(x)+\frac{B_{2}(x)-B_{2}(0)}{B_{2}(x)}[k_{2 1}(z_{21r}-z_{21})+z_{22}-\hat{z}_{22}]\\ &=&F_{2}(x)+\frac{B_{2  
Considering Assumption 1, we have  
\[ \begin{array}{rcl}\mid\Psi_{2}\mid&\leq&\frac{1}{1-\theta_{2}}\mid F_{2}(x) \mid+\frac{\theta_{2}}{1+\theta_{2}}(\parallel k_{21}\parallel\parallel e_{21} \parallel+\mid\varepsilon_{22}\mid)\\ \mid\dot{\Psi}_{2}\mid&\leq&\mid\dot{F}_{2}(x)\mid+\mid B_{2}(x)\mid\mid u_{2} \mid+\theta_{2}(\parallel k_{21}\parallel\parallel\varepsilon_{21}\parallel+ l_{22}\mid\varepsilon_{21}\mid)\\ \mid\Psi_{3}\mid&\leq&\frac{1}{1-\theta_{3}}\mid F_{3}(x)\mid+ \frac{\theta_{3}}{1+\theta_{3}}(\parallel k_{31}\parallel\parallel e_{31} \parallel+\mid\varepsilon_{32}\mid)\\ \mid\dot{\Psi}_{3}\mid&\leq&\mid\dot{F}_{3}(x)\mid+\mid B_{3}(x)\mid\mid u_{3} \mid+\theta_{3}(\parallel k_{31}\parallel\parallel\dot{e}_{31}\parallel+l_{32} \mid\varepsilon_{31}\mid)\end{array} \]  
From the above equations, with consideration of the perturbation assumed as a smooth function of time, it can be concluded that the bound of perturbation and its derivative exist. As a result, with both the **Assumptions 1** and 2, the effectiveness of such perturbation observer-based control can be guaranteed.