## 3 Design of the Adaptive Sliding Mode Observer  
To design the corresponding ASMO, the following assumptions are proposed:  
**Hypothesis 1**. : _(A, C) is observable._  
**Hypothesis 2**. : _Sensor faults are bounded matrices. It means there exists \(\gamma_{i}>0\), \(\|f_{i}\|\leq\gamma_{i}\) and the value of \(\gamma_{i}\) is unknown._  
**Hypothesis 3**. : _The external disturbance is a bounded matrix. It means there exists \(\varsigma>0\), \(\|\theta(t)\|\leq\gamma_{i}\)._  
**Hypothesis 4**. : _(\(\overline{A}_{i},\overline{C}_{i}\)) is observable, and there exists appropriate matrix \(L_{i},D_{i},K_{i}\), positive definite matrix \(P_{i}\) and \(Q_{i}\), such that \((\overline{A}_{i}-L_{i}\overline{C}_{i})\) remains stable, \(P_{i}\overline{F}_{i}=\overline{C}_{i}^{T}D_{i}^{T}\), \(P_{i}\overline{H}_{i}=\overline{C}_{i}^{T}K_{i}^{T}\) and \(\left(\overline{A}_{i}-L_{i}\overline{C}_{i}\right)^{T}P_{i}+P_{i}\overline{ \left(\overline{A}_{i}-L_{i}\overline{C}_{i}\right)}=-Q_{i}\)._  
**Hypothesis 5**. : \(\text{rank}(\overline{C}_{i}\overline{F}_{i})=\text{rank}(\overline{F}_{i})=1\)_. System (11) is a minimum phase system. It means the invariant zeros of the system are located on the left side of the complex plane. For all complex numbers satisfying \(R(s)\geq 0\), (12) holds true._  
\[ \begin{array}{l}\text{rank}\left[\begin{array}{ccc}sI-\overline{A}_{i}& 0&0\\ -A_{si}\overline{C}_{i}&sI+A_{si}&-A_{si}\\ 0&I_{1}&0\\ \overline{y}_{i}(t)=\overline{C}_{i}\overline{x}_{i}(t)\end{array}\right]+2 \end{array}  \tag{12}\]  
Therefore, the invariant zeros of the system are the characteristic values of \(\overline{A}_{i}\). According to sensor fault model (11), the SMO can be described as [22]:  
\[\left\{ \begin{array}{l}\dot{\overline{x}}_{i}(t)=\overline{A}_{i}\dot{ \overline{x}}_{i}(t)+\overline{B}u(t)+L_{i}(\overline{y}_{i}-\dot{\overline{ y}}_{i})+\overline{F}_{i}v_{i}\\ \dot{\overline{y}}_{i}(t)=\overline{C}_{i}\dot{\overline{x}}_{i}(t)\end{array} \right. \tag{13}\]where \(\widehat{\widehat{x}}_{i}(t)\) is the state observation vector of \(\widehat{x}_{i}(t)\), \(L_{i}\) is the gain matrix of the observer to be designed, \(\varepsilon_{yi}=\overline{y}_{i}-\widehat{\overline{y}}_{i}\) is the output estimation error, \(v_{i}\) is the discontinuous sliding mode control input vector which is used to cut off the effect of fault \(f_{i}\), and it is expressed as follows:  
\[v_{i}=\left\{ \begin{array}{ll}\rho_{i}(t)\frac{D_{i}\widetilde{\varepsilon}_ {yi}}{\|D_{i}\widetilde{\varepsilon}_{yi}\|}&\widetilde{\varepsilon}_{yi} \
eq 0\\ 0&\widetilde{\varepsilon}_{yi}=0\end{array} \right. \tag{14}\]  
where \(\rho_{i}(t)\) denotes the adjustable gain parameter with the advantage of adaptive adjustment; it is designed using the corresponding adaptive algorithm without knowing the upper limit of the unknown sensor fault. The adaptive algorithm is expressed as follows:  
\[\frac{\mathrm{d}\rho_{i}(t)}{\mathrm{d}t}=\eta_{i}\|D_{i}\widetilde{ \varepsilon}_{yi}\|\mathrm{sgn}(\|D_{i}\widetilde{\varepsilon}_{yi}\|- \lambda_{i}),i=1,\cdots m \tag{15}\]  
\[\rho_{i}(0)>0,\eta_{i}<2,0<\lambda_{i}<1,i=1,\cdots m \tag{16}\]  
According to (14), when \(\mathrm{sgn}(\|D_{i}\widetilde{\varepsilon}_{yi}\|-\lambda_{i})=1\), adjustable gain parameter \(\rho_{i}(t)\) increases, and the increase rate is proportional to \(\|D_{i}\widetilde{\varepsilon}_{yi}\|\). Furthermore, when \(\mathrm{sgn}(\|D_{i}\widetilde{\varepsilon}_{yi}\|-\lambda_{i})=-1\), \(\rho_{i}(t)\) decreases. It means that once a sliding mode motion occurs, which means \(\|D_{i}\widetilde{\varepsilon}_{yi}\|\) is close to zero, \(\|D_{i}\widetilde{\varepsilon}_{yi}\|\leq\lambda_{i}\) when \(\lambda_{i}\) takes a minor constant. Then, the adjustable gain parameter changes a little, which ensures that the sliding mode adjustable parameter is not too large, so that the chattering phenomenon caused by the excessive gain parameter can be weakened to some extent. **Lemma 1.