## 3 Actuator Fault Detection and Diagnosis Scheme
### Fault Diagnosis Scheme Design
When the unknown actuator fault signal is detected successfully by the designed fault detection strategy, the fault alarm triggers the fault estimation observer to start working. In the following description, a fault estimation observer is designed using an adaptive neural network technique. For the design of a fault estimation observer, two new variables are defined as: \(\Phi=[e_{1},\,\ldots,\,e_{n}]^{T}\) and \(\Xi=\text{diag}\{\tau_{1},\,\ldots,\,\tau_{n}\}\), and Equation (11) can be transformed into the following  
\[ \begin{array}{l}\dot{\omega}=-(J+J_{VS})^{-1}\Lambda(J+J_{VS})\omega+(J+J_{ VS})^{-1}D\Xi\Phi\\ +(J+J_{VS})^{-1}DF+W^{T}\varphi(t)+\varepsilon_{0}\end{array}  \tag{32}\]  
A RBF neural network-based fault estimation observer is given by  
\[ \begin{array}{l}\dot{\omega}=A_{2}\overline{\omega}-(J+J_{VS})^{-1}\hat{ \Lambda}(J+J_{VS})\dot{\omega}+(J+J_{VS})^{-1}D\Xi\dot{\Phi}\\ +(J+J_{VS})^{-1}D\tilde{F}+\tilde{W}^{T}\varphi(t)+\hat{\varepsilon}_{0}\end{array}  \tag{33}\]  
where \(\overline{\omega}\) is the estimation error which satisfies  
\[\overline{\omega}=\hat{\omega}-\omega \tag{34}\]  
\(A_{2}\) is a known Hurwitz matrix. \(\dot{\Phi}\), \(\hat{F}\), and \(\hat{W}^{T}\varphi(t)+\hat{\varepsilon}_{0}\) denote the estimated values of LOE fault factor, bias fault factor, and the generalized perturbation, respectively. They are determined by the following adaptive updated estimation algorithms  
\[\dot{\Phi}=-\dot{\Phi}-\frac{1}{c_{1}+c_{h,1}(J_{VS})}[(J+J_{VS})^{-1}D\Xi]^{ T}P_{2}\overline{\omega} \tag{35}\]  
\[\dot{\hat{F}}=-\hat{F}-\frac{1}{c_{2}+c_{h,2}(J_{VS})}[(J+J_{VS})^{-1}D]^{T}P_ {2}\overline{\omega} \tag{36}\]  
\[\dot{\hat{W}}=-\hat{W}-\frac{1}{c_{3}+c_{h,3}(J_{VS})}\varphi(t)\overline{ \omega}^{T}P_{2} \tag{37}\]  
\[\dot{\hat{\varepsilon}}_{0}=-\hat{\varepsilon}_{0}-\frac{c_{0}+c_{h,0}(J_{VS}) }P_{2}\overline{\omega} \tag{38}\]  
where \(c_{i}>0\) (\(i=0,\,1,\,2,\,3\)) are known constants, and \(c_{h,i}(J_{VS})\) are the variable structure harmonic functions of their respective updated algorithms, which change continuously with the change of \(J_{VS}\) and have known expressions. \(P_{2}\) is an unknown positive definite matrix to be determined later. Let  
\[\widetilde{\Phi}=\dot{\Phi}-\Phi \tag{39}\]  
\[\widetilde{F}=\hat{F}-F \tag{40}\]\[\widetilde{W} =\hat{W}-W \tag{41}\] \[\widetilde{\epsilon}_{0} =\hat{\epsilon}_{0}-\varepsilon_{0} \tag{42}\]  
the following error dynamics equation is obtained by subtracting Equation (32) from Equation (33):  
\[ \begin{array}{l}\ddot{\omega}=A_{2}\overline{\omega}+(J+J_{VS})^{-1}[\Lambda (J+J_{VS})\omega-\hat{\Lambda}(J+J_{VS})\hat{\omega}]\\ +(J+J_{VS})^{-1}D\[PERSON]\tilde{\Phi}+(J+J_{VS})^{-1}D\widetilde{F}+\widehat{W}^{T} \varphi(t)+\widetilde{\epsilon}_{0}\\ =A_{2}\overline{\omega}+(J+J_{VS})^{-1}[\Lambda(J+J_{VS})\omega-\hat{\Lambda} (J+J_{VS})\hat{\omega}]\\ +(J+J_{VS})^{-1}D\[PERSON]\tilde{\Phi}+(J+J_{VS})^{-1}D\widetilde{F}+\widetilde{ \rho}(t)\end{array}  \tag{43}\]  
where  
\[\widetilde{\rho}(t)=\widetilde{W}^{T}\varphi(t)+\widetilde{\epsilon}_{0} \tag{44}\]  
In this position, the second result of this study is given in the form of Theorem 2. \(\square\)  
**Theorem 2**. : _For the faulty dynamics Equation (11), a RBF neural network-based fault estimation observer is designed in Equation (33) with the adaptive parameter updated algorithms (35)-(38), and for a given Hurwitz matrix \(A_{2}\), if exists a positive definite matrix \(P_{2}\) with a known positive scalar \(\xi\), such that inequality (45) holds_  
\[Q=P_{2}[A_{2}+\xi(J+J_{VS})^{-1}]+[A_{2}^{T}+\xi((J+J_{VS})^{-1})^{T}]P_{2}<0 \tag{45}\]  
_then, the error dynamics Equation (43) is ultimately uniformly bounded (UIUB)._  
**Proof of Theorem 2. ** Considering the following Lyapunov candidate function  
\[