## 3 Analysis of Fault Model by Using Convex Optimization
### Rationality Analysis of Solving Conditions Using Convex Optimization  
Since the control gain matrix is regarded as the solution of convex optimization using LMI technique, therefore, the optimization condition needs to be determined such that the optimized solution results in the stable system.  
To obtain the solving conditions for the control law, the four critical subsystems are chosen as the controlled plants instead of the original system with uncertainty. Based on these four specific subsystems, the premise of optimization is determined and then the solution is obtained using the LMI technique. The FTC law existence condition using this treatment is at least sufficient, which is stated in Lemma 1.  
There are four stable areas of the subsystems under an assumed feedback (see Figure 3). According to the convex optimization, the optimized stable area of the closed-loop system will be located at the intersection of the four areas. This corresponds to LMI technique approach to determine the control gain matrix. In this instance there are four LMIs due to the multiple solving conditions, which is similar to the conditions presented in [26], where [PERSON] et al. studied the LPV tracking problem using multiple Lyapunov functions. Although the solving condition is critical, since the stability region becomes smaller, the optimized solution will be at least sufficient once its existence is established. The method proposed in this paper deals with the problem of determining control gains for systems with uncertainty.  
**Lemma 1**.: _If a state feedback control gain matrix \(\mathbf{K}_{\mathbf{x}}\) can result in a stable system matrix \(\mathbf{A}_{\text{L}}+\mathbf{B}_{\text{L}}\mathbf{K}_{\mathbf{x}}\), then it also results in a stable uncertain controllable matrix pair \(\left(\mathbf{A}+\mathbf{\Delta}\mathbf{A}\quad\mathbf{B}+\mathbf{\Delta}\mathbf{B}\right)\), where_  
\[\left\{\begin{array}{l}\mathbf{A}_{\text{L}}\in\left\{\begin{array}{ll}\mathbf{A}+ |\mathbf{\Delta}\mathbf{A}|_{\text{bound}},&\mathbf{A}-|\mathbf{\Delta}\mathbf{A}|_{\text{bound}} \\ \mathbf{B}_{\text{L}}\in\left\{\begin{array}{ll}\mathbf{B}+|\mathbf{\Delta}\mathbf{B}|_{\text {bound}},&\mathbf{B}-|\mathbf{\Delta}\mathbf{B}|_{\text{bound}}\end{array}\right\}\end{array} \right.\end{array}\right\} \tag{17}\]  
Figure 3: Optimized stable area of the closed-loop system using convex optimization.  
in which, \(\mathbf{A}_{\rm L}\in\left\{\mathbf{A}+\left|\mathbf{\Delta A}\right|_{\rm bound},\quad\mathbf{A} -\left|\mathbf{\Delta A}\right|_{\rm bound}\right\}\) means that the matrix \(\mathbf{A}_{\rm L}\) can be chosen as \(\mathbf{A}+\left|\mathbf{\Delta A}\right|_{\rm bound}\) or \(\mathbf{A}-\left|\mathbf{\Delta A}\right|_{\rm bound}\), which represent the system's upper and lower bound matrices due to gradual faults. Similarly, \(\mathbf{B}_{\rm L}\in\left\{\mathbf{B}+\left|\mathbf{\Delta B}\right|_{\rm bound},\quad\mathbf{ B}-\left|\mathbf{\Delta B}\right|_{\rm bound}\right\}\) means that the matrix \(\mathbf{B}_{\rm L}\) can be determined as \(\mathbf{B}+\left|\mathbf{\Delta B}\right|_{\rm bound}\) or \(\mathbf{B}-\left|\mathbf{\Delta B}\right|_{\rm bound}\), which represent the control upper and lower bound matrices due to gradual faults._  
**Proof of Lemma 1.** See Appendix A. \(\square\)