# Fault-Tolerant Control for Carrier-Based Aircraft Based on Adaptive Fuzzy Sliding-Mode Method
## 4 Control Strategy
### Design of AFSMFTC  
As mentioned above, the collected flight-path tracking errors can be transformed into attitude control commands with the assistance of the glide-path generation subsystem and the guidance subsystem. Consequently, the landing control problem can be transformed into an attitude control problem. The AFSMFTC structure is shown in Figure 4.  
Figure 4: AFSMFTC Structure. Letting the control command for the expected attitude angle be \(x_{1d}= \begin{bmatrix}\phi_{d}&\theta_{d}&\beta_{d}\end{bmatrix} ^{T}\), the state-tracking error can be defined as  
\[e_{1}(t)=x_{1}(t)-x_{1d}(t) \tag{17}\]  
The sliding-mode switching function is designed as  
\[s_{1}(t)=e_{1}(t)+c_{1}\int_{0}^{t}e_{1}(t)dt \tag{18}\]  
Meanwhile, the sliding-mode control law is designed as  
\[x_{2}=B^{-1}\big{(}-A+\dot{x}_{1d}-c_{1}e_{1}-\eta\text{sgn}(s_{1})\big{)} \tag{19}\]  
where \(\eta\geq|d(t,x_{1},x_{2})|\), the proof of the invertibility of \(B\) is provided in reference [35]. In addition, it is difficult to measure the upper limit of unknown perturbations during real-world CBA landing because of the complex and random landing environment. Therefore, this study further designs adaptive fuzzy sliding-mode control laws to offset errors induced by unknown disturbances without an upper perturbation limit; switching terms are substituted by consecutive fuzzy systems, which can effectively reduce buffeting on rudder surfaces, and \(\hat{h}(s_{1})\) is selected and used to approximate \(\eta\text{sgn}(s_{1})\). The fuzzy system is constructed in accordance with fuzzy rules in the form of an IF-THEN statement:  
\[R^{(I)}:\text{IF}\;s_{i}\;\text{is}\;A_{1}^{I}\;\text{and}\;\cdots\;\text{and} \;s_{i}^{(n)}\;\text{is}\;A_{n}^{I}\]  
\[\text{THEN}\;h_{i}\;\text{is}\;B_{i}^{I}\]  
where \(n\) is the order of the derivative \(s_{i}\). A product inference engine, single-value defuzzifier and centre-average defuzzifier are used to output the fuzzy system as  
\[h_{i}=\frac{\sum_{l=1}^{N}q_{i}^{l}\left(\prod\limits_{\mu_{A_{k}^{l}}}^{k=1}( x_{k})\right)}{\sum_{l=1}^{N}\left(\prod\limits_{\mu_{A_{k}^{l}}}^{k=1}(x_{k}) \right)} \tag{20}\]  
where \(\mu_{A_{k}^{l}}(x  
\[s_{2}(t)=e_{2}(t)+c_{2}\int_{0}^{t}e_{2}(t)dt \tag{32}\]  
and the derivative of \(s_{2}(t)\) is taken as  
\[\dot{s}_{2}=\dot{e}_{2}+c_{2}e_{2}=\dot{x}_{2}-\dot{x}_{2d}+c_{2}e_{2} \tag{33}\]  
The equation \(\dot{x}_{2}(t)=F+G\Lambda u(t)\) is substituted into Equation (33) to obtain  
\[\dot{s}_{2}=F+G\Lambda u(t)-\dot{x}_{2d}+c_{2}e_{2} \tag{34}\]  
If \(\mathrm{P}=\left(G\Lambda\right)^{-1}\), \(\lambda=F+ks_{2}+c_{2}e_{2}-\dot{x}_{2d}+\mathrm{sgn}(s_{2})\) and the estimated value of \(\mathrm{P}\) is defined as \(\hat{\mathrm{P}}\), then the error of estimate is \(\widetilde{\mathrm{P}}=\mathrm{P}-\mathrm{P}\). The control law and adaptive law are designed as  
\[u=-\hat{\mathrm{P}}\lambda \tag{35}\]  
\[s_{1}(t)=e_{1}(t)+c_{1}\int_{0}^{t}e_{1}(t)dt \tag{18}\]0  
\(\square\)  
**Theorem 2**. : _If the control law \(u=-\hat{\mathrm{P}}\lambda\) is used, then the second equation in the closed-loop system (8) is stable, and the tracking error \(e_{2}\) converges to zero in a limited time. This also means that the aircraft-fault subsystem is progressively stable._  
Proof.