## 5 Simulation Analysis
### Simulation Results Using FTC Strategies  
In this section, different FTC strategies are presented and respective simulation results are used to analyze the system response. The system model with moderate failure is used in the following analysis, where the corresponding matrices are presented in Table 6. Different FTC methods are used to form a closed-loop system and compare the control effects. Since the gradual faults are considered in the NT-DRAS, the FTC target should be maintaining the system robust performance under gradual fault conditions. Meanwhile, the FTC controller should be with a fixed structure as well as an efficient and effective control gain, which can not only save the control resources but also ensure the system reliability during the FTC process. Based on the above consideration, two fault-tolerant control methods analyzed in this section are guaranteed cost control (GCC) method [33] and CO-FTC method proposed in this paper. **Remark 4**. : _GCC method is robust control design method for system with considered uncertainty [33]. The method is suitable for the following uncertain system:_  
\[\dot{\mathbf{x}}(t)=(\mathbf{A}+\mathbf{\Delta}\mathbf{A})\mathbf{x}(t)+(\mathbf{B}+\mathbf{\Delta}\mathbf{B}) \mathbf{u}(t) \tag{26}\]  
_A cost function in the following form_  
\[\mathbf{J}=\int_{0}^{t}\Bigl{(}\mathbf{x}^{\mathrm{T}}(\tau)\mathbf{Q}\mathbf{x}(\tau)+\mathbf{u} (\tau)^{\mathrm{T}}\mathbf{R}\mathbf{u}(\tau)\Bigr{)}d\tau \tag{27}\]  
_was defined to deal with the uncertain matrices, \([\mathbf{\Delta}\mathbf{A}\ \mathbf{\Delta}\mathbf{B}]=\mathbf{D}\mathbf{F}[\mathbf{E_{1}}\mathbf{E_{2}}]\), where matrices \(\mathbf{D}\) and \(\mathbf{E}\) describe the structure information. The corresponding control gain form is given as an LMI condition (see (19)), where \(\mathbf{X}\) and \(\mathbf{W}\) are positive matrices, then for a given positive scalar \(\varepsilon\), and if the LMI holds, then, the state feedback guaranteed cost control law can be designed as \(\mathbf{K}=\mathbf{W}\mathbf{X}^{-1}\) where the cost has upper bound \(\mathbf{J}\leq\mathbf{x}_{0}^{-1}\mathbf{X}^{-1}\mathbf{x}_{0}\)._  
From Remark 4, it is shown that GCC is a fault-tolerant control gain design method based on state feedback, where weighting matrices \(\mathbf{Q}\) and \(\mathbf{R}\) are chosen to decrease defined cost functional. As opposed to the GCC, the proposed OP-FTC uses not only state feedback but also the output error integration feedback to design the control law. In order to remove the system uncertainty, new conditions, based on convex analysis, are deduced and the control gain matrix determined from the corresponding LMIs. The comparison between the two methods applied to the system under the same moderate failure is shown in Figure 11. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Fault Conditions** & \(\mathbf{e_{\mathrm{perf}}}(\mathbf{t})\) & \(\mathbf{-}\) & \(\mathbf{e_{\mathrm{perf}}}(\mathbf{t})\) & \(\mathbf{e_{\mathrm{perf}-max}}\) \\ \hline Normal & 0.4307 & 0.0000771 & 0.03 \\ Minor fault & 0.8721 & 0.0003205 & 0.03 \\ Moderate fault & 1.0098 & 0.0004153 & 0.03 \\ \hline \hline \end{tabular}  
* Square-wave signal is used to evaluate the tracking performance here.  
\end{table} 
Table 7: System performance evaluation values under integrated fault conditions without FTC. The system response to a square-wave and to a compound sine-wave input is compared for the cases when the GCC method and the proposed OP-FTC method are applied (see Figure 11). The results indicate that both GCC and OP-FTC methods can improve the tracking performance of the system with moderate fault with respect to both the square-wave and the compound sine-wave inputs. The Figure 11(a1,b1) show the local details of tracking performance for both control methods.