# Load Frequency Control for Multi-Area Power Plants with Integrated Wind Resources
## 4 State Observer Based on Sliding Mode Control Strategies
### Stability Analysis of Whole System in Sliding Mode Dynamic  
The new SMC is designed with a better sliding surface for sliding variables to rapidly reach the surface and remain thereon. The sliding surface is therefore given below  
\[\sigma_{\_}i[z_{\_}i(t)]=L_{\_}iz_{\_}i(t)-\int\limits_{\_}0^{t}L_{\_}i(A_{\_} i-B_{\_}iJ_{\_}i)z_{\_}i(\tau)d\tau \tag{14}\]  
where \(L_{\_}i\) is a constant matrix and \(J_{\_}i\) is the design matrix. Matrix \(L_{\_}i\) is selected to guarantee that the matrix \(L_{\_}iB_{\_}i\) is invertible. The design matrix \(J_{\_}i\in R^{m_{\_}i\times n_{\_}i}\) is given, satisfying the non-linearity condition  
\[\text{Re}[\lambda_{\_}\max(A_{\_}i-B_{\_}iJ_{\_}i)]<0 \tag{15}\]For the continuous system observability rule, the estimation error defined by \(e_{i}(t)=x_{i}(t)-z_{i}(t)\) must satisfy the below equation  
\[\dot{e}_{i}=(A_{i}-T_{i}C_{i})e_{i}+\sum\limits_{ \begin{array}{c}j=1\\ j\
eq i\end{array} }^{N}H_{ij}e_{j}+F_{i}\Delta P_{di}(t) \tag{16}\]  
If we take the derivative of (13) with respect to time, we therefore obtain  
\[ \begin{array}{c}\dot{\sigma}_{t}[z_{i}(t)]=L_{i}[A_{i}z_{i}(t)\quad+\ B_{i}u_{i}(t)+ \sum\limits_{\begin{array}{c}j=1\\ j\
eq i\end{array}}^{N}H_{ij}z_{j}(t)+T_{i}(y_{t}-n_{i})]\\ -L_{i}(A_{i}-B_{i}I_{i})z_{i}(t)\end{array}  \tag{17}\]  
Setting \(\dot{\sigma}(t)=\sigma(t)=0\), we can see that the equivalent control signal is as below  
\[ \begin{array}{c}u_{i}^{eq}(t)=-(L_{i}B_{i})^{-1}[L_{i}A_{i}z_{i}(t)+\sum \limits_{\begin{array}{c}j=1\\ j\
eq i\end{array}}^{N}L_{i}H_{ij}z_{j}(t)+L_{i}T_{i}(y_{i}-n_{i})]-L_{i}(A_{ i}-B_{i}I_{i})z_{i}(t)]\\ =-(L_{i}B_{i})^{-1}[L_{i}B_{i}]iz_{i}(t)+L_{i}T_{i}(y_{i}-n_{i})+\sum\limits_{ \begin{array}{c}j=1\\ j\
eq i\end{array}}^{N}L_{i}H_{ij}z_{j}(t)]\\ \end{array}  \tag{18}\]  
Substituting \(u(t)\) into (6) yields the sliding motion:  
\[ \begin{array}{c}\dot{x}_{i}(t)=A_{i}x_{i}(t)-B_{i}I_{i}z_{i}(t)-B_{i}(L_{i} B_{i})^{-1}L_{i}T_{i}(y_{i}-n_{i})\\ -\sum\limits_{\begin{array}{c}j=1\\ j\
eq i\end{array}}^{N}B_{i}(L_{i}B_{i})^{-1}L_{i}H_{ij}z_{j}(t)+\sum\limits_{ \begin{array}{c}j=1\\ j\
eq i\end{array}}^{N}H_{ij}x_{j}(t)+F_{i}\Delta P_{di}(t)\\ =(A_{i}-B_{i}I_{i})x_{i}(t)+(B_{i}I_{i}-B_{i}(L_{i}B_{i})^{-1}L_{i}T_{i}C_{i}) e_{i}(t)\\ +\sum\limits_{\begin{array}{c}j=1\\ j\
eq i\end{array}}^{N}\left[H_{ij}-B_{i}(L_{i}B_{i})^{-1}L_{i}H_{ij}\right]x _{j}(t)+\sum\limits_{\begin{array}{c}j=1\\ j\
eq i\end{array}}^{N}B_{i}(L_{i}B_{i})^{-1}L_{i}H_{ij}e_{j}(t)+F_{i}\Delta P _{di}(t)\\ \end{array}  \tag{19}\]  
Combining system (6) and error system (18), the closed-loop system is written as  
\[\left[ \begin{array}{c}\dot{x}_{i}\\ \dot{e}_{i}\end{array} \right]=\left[ \begin{array}{cc}A_{i}-B_{i}I_{i}&\Phi_ {i}\\ 0&A_{i}-T_{i}C_{i}\end{array} \right]\left[ \begin{array}{c}x_{i}\\ e_{i}\end{array} \right]+\sum\limits_{ \begin{array}{c}j=1\\ j\
eq i\end{array} }^{N}\left[ \begin{array}{cc}H_{ij}-\Lambda_{i}H_{ij}& \Lambda_{i}H_{ij}\\ 0&H_{ij}\end{array} \right]\left[ \begin{array}{c}x_{j}\\ e_{j}\end{array} \right]+\left[ \begin{array}{c}F_{i}\Delta P_{di}(t)\\ F_{i}\Delta P_{di}(t)\end{array} \right] \tag{20}\]  
where \(\Phi_{i}=B_{i}I_{i}-B_{i}(L_{i}B_{i})^{-1}L_{i}T_{i}C_{i}\) and \(\Lambda_{i}=B_{i}(L_{i}B_{i})^{-1}L_{i}\). Equation (19) provides the condition that the MAIPS in the sliding surface is stable if the sliding motion (19) is stable and the observability condition holds following assumption 2. Therefore, the sliding motion (19) is also made Hurwitz, so the observer error \(e(t)\to 0\) when \(t\rightarrow\infty\). To prove the above condition, we postulate a theorem as follows. **Theorem 1**. : _The sliding motion (19) is asymptotically stable, if there exist symmetric positive definite matrices \(P_{i}\), \(Q_{i}\), \(i=1,\ 2,\ \ldots N\) and positive scalars \(\lambda_{i}\), \(\rho_{i}\), \(\hat{\gamma}_{i}\) and \(\overline{\gamma}_{i}\) so that the below linear matrix inequality is feasible_  
\[\left[\begin{array}{ccccc}\mathbf{x}_{i}+\sum\limits_{ \begin{subarray}{c}j=1 \\ j=1\\ j\
eq i\end{subarray}