## 4 Design of the Sensor Fault Reconstruction  
The ASMO-based state estimation of the dynamic system is the purpose of reconstruction. In fact, the state variable will reach the sliding mode surface in limited time and stay there. There exists the following transformation matrix:  
\[T\in R^{n+1}=\left[ \begin{array}{c}(\overline{\dot{C}}_{i})_{\bot}^{T}P_{i} \\ \overline{\dot{C}}_{i}\end{array} \right] \tag{22}\]  
where \(\big{(}\overline{\dot{C}}_{i}\big{)}_{\bot}\in R^{(n+1)\times n}\) is a nonsingular matrix and the null space of \(\overline{\dot{C}}_{i}\). Therefore, \(\big{(}\overline{\dot{C}}_{i}\big{)}_{\bot}\) is the orthogonal complement space of \(\overline{\dot{C}}_{i}^{T}\). Then,  
\[T\overline{\dot{e}}_{i}=\left( \begin{array}{c}e_{1i}\\ e_{yi}\end{array} \right)=\overline{\dot{e}}_{i} \tag{23}\]  
The state estimation error under the new coordinates is  
\[\dot{\overline{\dot{e}}}_{i}=T(\overline{A}_{i})T^{-1}\overline{\dot{e}}_{i}+T \overline{\dot{F}}_{i}(f_{si}-v_{i})+T\overline{\dot{H}}_{i}\theta(t) \tag{24}\]  
According to transformation matrix \(T\) and \(\overline{\dot{C}}_{i}(\big{(}\overline{\dot{C}}_{i}\big{)}_{\bot})=0\), the following is true:  
\[T(\overline{A}_{i})T=\left[ \begin{array}{cc}\overline{A}_{11i}&\overline{A }_{12i}\\ \overline{A}_{21i}&\overline{A}_{22i}\end{array} \right] \tag{25}\]  
\[T\overline{\dot{F}}_{i}=\left[ \begin{array}{c}0\\ \overline{\dot{C}}_{i}P_{i}^{-1}\overline{\dot{C}}_{i}^{T}D_{i}^{T}\end{array} \right] \tag{26}\]  
\[T\overline{\dot{F}}_{i}v_{i}=\left[ \begin{array}{c}0\\ v_{2i}\end{array} \right] \tag{27}\]  
\[v_{2i} =\rho_{i}\overline{\dot{C}}_{i}P_{i}^{-1}\overline{\dot{C}}_{i}^{T}D_{i}^{T }\overline{\dot{D}_{i}^{T}\overline{\dot{e}}_{i}\|D_{i}^{T}\overline{\dot{e}}_{ i}\|} \tag{28}\] \[=\overline{\dot{C}}_{i}\overline{\dot{F}}_{i}v(t)\]\[T\overline{H}_{i}=\left[ \begin{array}{c}0\\ \overline{C}_{i}P_{i}^{-1}\overline{C}_{i}^{T}K_{i}^{T}\end{array} \right] \tag{29}\]  
Under the new coordinates, (24) can be divided as follows:  
\[\dot{\overline{e}}_{1i}=\widetilde{A}_{11i}e_{1i}+\widetilde{A}_{12i}e_{yi} \tag{30}\]  
\[\dot{\overline{e}}_{yi} =\widetilde{A}_{21i}e_{1i}+\widetilde{A}_{22i}e_{yi}+\widetilde{ C}_{i}P_{i}^{-1}\overline{C}_{i}^{T}K_{i}^{T}d(t) \tag{31}\] \[+\overline{C}_{i}P_{i}^{-1}\overline{C}_{i}^{T}D_{i}^{T}f_{i}(t)- \overline{C}_{i}P_{i}^{-1}\overline{C}_{i}^{T}D_{i}^{T}v_{i}(t)\]  
If the states of the error system arrive at the sliding mode surface in the finite time, which means \(\dot{\overline{e}}_{yi}=e_{yi}=0\), (30) and (31) can be transformed to the following:  
\[\dot{\overline{e}}_{1i}=\widetilde{A}_{11i}e_{1i} \tag{32}\]  
\[0 =\widetilde{A}_{21i}e_{1i}+\overline{C}_{i}P_{i}^{-1}\overline{C}_{i}^{T}D_ {i}^{T}f_{i}(t) \tag{33}\] \[-\overline{C}_{i}P_{i}^{-1}\overline{C}_{i}^{T}D_{i}^{T}v_{i}(t)+ \overline{C}_{i}P_{i}^{-1}\overline{C}_{i}^{T}K_{i}^{T}d(t)\]  
According to \(P_{i}\overline{F}_{i}=\overline{C}_{i}^{T}D_{i}^{T}\), \(e_{1i}\to 0\), we obtain the following:  
\[\overline{C}_{i}\overline{F}_{i}f_{i}(t)\approx\overline{C}_{i}\overline{F}_ {i}v_{i}(t)-\overline{C}_{i}\overline{H}_{i}d(t) \tag{34}\]  
As \(rank(\overline{C}_{i}\overline{F}_{i})=1\), and there exists the following pseudo inverse,  
\[(\overline{C}_{i}\overline{F}_{i})^{+}=\left[(\overline{C}_{i}\overline{F}_{ i})^{T}(\overline{C}_{i}\overline{F}_{i})\right]^{-1}(\overline{C}_{i} \overline{F}_{i})^{T} \tag{35}\]  
\[W=-(\overline{C}_{i}\overline{F}_{i})^{+}\overline{C}_{i}\overline{H}_{i} \tag{36}\]  
\(f_{si}(t)\) can be reconstructed as  
\[f_{i}(t)\rightarrow\left[(\overline{C}_{i}\overline{F}_{i})^{+}(\overline{C }_{i}\overline{F}_{i})\right]v_{i}(t)+Wd(t) \tag{37}\]  
In order to reduce chattering of the sliding mode, the sliding mode control input vector can be adjusted as follows:  
\[v_{i}=\left\{ \begin{array}{ccc}\rho_{i}(t)\frac{D_{i}\overline{e}_{yi}}{|D_ {i}\overline{e}_{yi}|+\delta_{i}}&;&if&\overline{e}_{yi}\
eq 0\\ 0&;&if&\overline{e}_{yi}=0\end{array} \right. \tag{38}\]  
where \(\delta_{i}\) is a small positive constant.