## 4 Integrated FE/FTC
### Adaptive Sliding Mode FTC Design  
A sliding surface for the System (8) is designed as follows:  
\[s_{1}=N_{1}\hat{x} \tag{14}\]  
where \(s_{1}\in R^{m}\), \(\hat{x}\in R^{n}\) is the estimate of system state \(x\), and \(N_{1}=B^{+}-Y_{1}(I_{n}-BB^{+})\) with \(B^{+}=\big{(}B^{\top}B\big{)}^{-1}B^{T}\) and an arbitrary matrix \(Y_{1}\in R^{m\times n}\). Define the state estimation error as \(e=x-\hat{x}\). Differentiating \(s_{1}\) with respect to time gives  
\[\dot{s}_{1}=N_{1}Ax+u+N_{1}Ff_{a}+N_{1}D\varepsilon-N_{1}\dot{e}_{x} \tag{15}\]  
Design the control input as  
\[u=u_{l}+u_{n} \tag{16}\]where the linear feedback component is \(u_{l}=-K\overline{x}\) with a design matrix. \(K=\left[\begin{array}{cc}K_{x}&E_{1}\end{array}\right]\), \(K_{x}\in R^{m\times n}\) is to be determined while \(E_{1}=B^{+}F\). The nonlinear component is designed as follows:  
\[u_{n}=\left\{\begin{array}{ll}-\rho_{s_{1}}(t)\frac{s_{1}}{\|s_{1}\|}\,&s_{1}\
eq 0 \\ 0,&s_{1}=0\end{array}\right.\]  
\(\|\cdot\|\) represents the Euclidean norm of a vector and the induced norm of a matrix. With \(\rho_{s_{1}}(t)=\hat{\eta}_{s_{1}}+\rho_{s_{1}}+\delta_{s1}\), \(\rho_{s_{1}}>0\), \(\delta_{s1}>0\) are design constants. The scalar \(\hat{\eta}_{s_{1}}\) is introduced to estimate \(\eta_{s_{1}}\) which is defined as \(\eta_{s_{1}}=\|N_{1}D\|\varepsilon_{0}+\|E_{1}\|\left(\overline{f}_{a}+\|\hat {f}_{a}\|\right)+\|K_{x}\|\varepsilon_{x_{0}}+N_{1}\overline{\varepsilon}_{x_ {0}}\), where \(\varepsilon_{x_{0}}\) and \(\overline{\varepsilon}_{x_{0}}\) are unknown scalars assumed to be upper bounds of \(\|\varepsilon_{x}\|\) and \(\|\overline{\varepsilon}_{x}\|\), respectively. The update law of \(\hat{\eta}_{s_{1}}\) is \(\hat{\tilde{\eta}}_{S_{1}}=\sigma_{1}\|s_{1}\|\), \(\hat{\eta}_{s_{1}}(0)>0\), with a learning rate \(\sigma_{1}>0\) to be designed.  
Define the estimation error of \(\eta_{s_{1}}\) as \(\widetilde{\eta}_{s_{1}}=\eta_{s_{1}}-\hat{\eta}_{s_{1}}\) and consider a Lyapunov function  
\[V_{s_{1}}=\frac{1}{2}\left(s_{1}^{T}s_{1}+\frac{1}{\sigma}\widetilde{\eta}_{s _{1}}^{2}\right)\]  
It follows from Equations (15) and (16) that  
\[\dot{V}_{s_{1}}=s_{1}^{T}\dot{s}_{1}-\frac{1}{\sigma_{1}}\widetilde{\eta}_{s _{1}}\hat{\eta}_{s_{1}}\leq(\omega_{s_{1}}\|x\|+\eta_{s_{1}}-\rho_{s_{1}}(t))\| s_{1}\|-\widetilde{\eta}_{s_{1}}\|s_{1}\|\leq(\omega_{s_{1}}\|x\|-\rho_{s_{1}}- \delta_{s1})\|s_{1}\|\]  
where \(\omega_{s_{1}}=\|N_{1}A-K_{x}\|+\|N_{1}M_{0}\|N_{0}\). By choosing \(\rho_{s_{1}}>\omega_{s_{1}}\phi_{s_{1}}\), with some scalar \(\phi_{s_{1}}>0\), it follows that the reaching and sliding conditions are satisfied, that is \(s_{1}^{T}\dot{s}_{1}\leq-\delta_{s1}\|s_{1}\|\) in the subset \(\Omega_{s_{1}}=\{x:\|x\|\leq\phi_{s_{1}}\}\). Thus, the controller (16) ensures that if \(x(0)\in\Omega_{s_{1}}\), then for all \(t>\|s_{1}(0)\|/\delta_{s1},s_{1}=\dot{s}_{1}=0\).  
The stability analysis of the system corresponding to the sliding mode is considered. It is assumed that the system has been controlled to remain in the sliding mode (14). Substituting the equivalent control  
\[u_{eq}=-(N_{1}Ax+N_{1}D\overline{\varepsilon})+u_{l} \tag{17}\]  
into System (8) gives the closed-loop system  
\[\dot{x}=(\Theta A-BK_{x})x+BK_{e}+\Theta D\overline{\varepsilon} \tag{18}\]  
where \(\Theta=I_{n}-BN_{1}\).  
Therefore, by designing \(K_{x}\) such that (18) is robustly stable, then the System (8) is maintained on the sliding mode with the equivalent control (17).