## 4 Wind System Control at Variable Wind Speeds from \(V_{max}\) [m/s] up to 21 [m/s]
### A Nonlinear Sliding Mode Controller  
For a sliding mode control algorithm design [50; 51; 52], the dynamical system (18), (19) is transformed to input affine form. First, let  
\[\dot{y}(t)=\
abla_{n}h\cdot\dot{n}+\
abla_{V}h\cdot\dot{V}+\
abla_{\beta}\dot{ \beta}, \tag{21}\]with \(\
abla_{n}h=\frac{dh(n,V,\beta)}{dt}\), \(\
abla_{V}h=\frac{dh(n,V,\beta)}{dV}\), \(\
abla_{\beta}h=\frac{dh(n,V,\beta)}{d\beta}\) and let us augment the dynamical system with the integrator \(\hat{\beta}=u\). Then, use the MAS equation \(\dot{n}=\frac{1}{Jn}(y-P_{GE}(n))\) to further arrive at  
\[ \begin{split}\dot{y}&=\frac{\
abla_{n}h}{Jn}y+ \
abla_{\beta}h\cdot u+\
abla_{V}h\cdot V-\frac{\
abla_{n}h}{Jn}P_{GE}\\ &\qquad\qquad\qquad\qquad:=\varphi(n,V,\beta)\cdot y+B(n,V,\beta )\cdot u+\sigma(n,V,\beta),\end{split}  \tag{22}\]  
where \(\varphi(n,V,\beta):=\frac{\
abla_{n}h}{Jn}\) is the output multiplicative factor in the output differential equation, \(B(n,V,\beta):=\
abla_{\beta}h\) is the input multiplicative factor and \(\sigma(n,V,\beta):=\
abla_{V}h\cdot V-\frac{\
abla_{n}h}{Jn}\cdot P_{GE}(n)\) is the uncertainty factor. **Assumption 1**. : _The input multiplicative factor \(B(n,V,\beta)\) and its reciprocal \(1/B(n,V,\beta)\) are well defined within the entire operating region OR2._  
**Assumption 2**. : _The uncertainty factor \(\sigma(n,V,\beta)\) is bounded within the operating region OR2. Let this bound \(\overline{\sigma}\) satisfy \(|\sigma|<\overline{\sigma}\)._  
**Remark 6**. : _Normally, the uncertainty term \(\sigma\) is partially measurable since its expression is well defined and all arguments \(n,V,\beta\) are measurable. However, original model uncertainty, parameter aging and unmodeled dynamics will be lumped inside this term altogether. Take, e.g., \(\varphi+\hat{\varphi}\) and \(B+\hat{B}\) as additive uncertainties about nominal values \(\varphi\), \(B\), then \(\hat{\varphi}y+\hat{B}u\) could easily be lumped inside \(\sigma\). Both Assumption 1 and Assumption 2 are very reasonable and hold for quite wide operating conditions._  
Let the following controller be used:  
\[u=B^{-1}\Big{(}\dot{y}^{*}-\varphi y\Big{)}-B^{-1}Ksgn(g), \tag{23}\]  
where \(g:=e\) is the switching variable, which is also the control error \(e=y-y^{*}\) for \(y^{*}=P_{TV}^{*}\), a reference input. In the proposed control law, \(K>0\) is a design gain. The factor \(B^{-1}\Big{(}\dot{y}^{*}-\varphi y\Big{)}\) is the equivalent control component and \(B^{-1}Ksgn(g)\) is called the switching component. The most direct stability analysis can be formalized with the next theorem. **Theorem 1**. : _The control system obtained by the output Equation (22) coupled with controller (23) is stable for \(K>\overline{\sigma}\)._  
Proof of Theorem 1.: Substitute \(u\) from (23) in (22) and we obtain \(\dot{y}=\dot{y}^{*}-Ksgn(g)+\sigma\), which is equivalent to \(\dot{g}=-Ksgn(g)+\sigma\). Define the Lyapunov function \(V(g)=0.5g^{2}\) and calculate \(\dot{V}=g\dot{g}\). Case (1) When \(g>0\), \(sgn(g)=1\) and \(\dot{g}=-K+\sigma\leq-K+\overline{\sigma}<0\), which makes \(\dot{V}=g\dot{g}<0\). Case (2) When \(g<0\), \(sgn(g)=-1\) and \(\dot{g}=K+\sigma\geq K-\overline{\sigma}>0\), which makes \(\dot{V}=g\dot{g}<0\). Based on the two cases above, from the fact that \(g:=e\) and \(e=y-y^{*}\) depends on bounded \(y^{*}\) and on \(y\), the control system is Lyapunov-stable with bounded \(y\). \(\square\)  
**Remark 7**. : _The upper bound \(\overline{\sigma}\) may be estimated more or less accurately in practice. It is customary to choose a large enough \(K>\overline{\sigma}\) to allow for a safe robustness margin. The advantage with respect to the linear double integral controller is the stability certification._  
**Remark 8**: _Although the proposed control draws inspiration from the sliding mode method, it is seen as a particular type of sliding mode, as explained next. For one, the equivalent control component \(B^{-1}(\dot{y}^{*}-\varrho y)\) found within the control law, does not compensate for some nominal value \(\sigma\) of the uncertainty.