## Appendix B  
According to [23; 27], the sufficient condition for the existence of the fault-tolerant controller with the constraints of linear quadratic (LQ) performance can be restated as  
\[\big{(}\mathbf{A}_{\rm aug}+\mathbf{B}_{\rm aug}\mathbf{K}\big{)}^{\rm T}\mathbf{P}+\mathbf{P} \big{(}\mathbf{A}_{\rm aug}+\mathbf{B}_{\rm aug}\mathbf{K}\big{)}+\mathbf{Q}+\mathbf{K}^{\rm T}\bm {R}\mathbf{K}+\left(\frac{1}{\gamma^{2}}\right)\mathbf{P}\mathbf{G}_{\rm aug}\mathbf{G}_{\rm aug }^{\rm T}\mathbf{P}<0 \tag{15}\]  
Using Lemma 3 and choosing \(\mathbf{\psi}=\mathbf{Q}+\mathbf{K}^{\rm T}\mathbf{R}\mathbf{K}+\left(\frac{1}{\gamma^{2}}\right) \mathbf{P}\mathbf{G}_{\rm aug}\mathbf{G}_{\rm aug}^{\rm T}\mathbf{P}\), \(\xi=\mathbf{P}\big{(}\mathbf{A}_{\rm aug}+\mathbf{B}_{\rm aug}\mathbf{K}\big{)}\), this sufficient condition can be transformed into the following form by using Lemma 4. \[ \begin{bmatrix}\mathbf{Q}+\mathbf{K}^{\rm T}\mathbf{R}\mathbf{K}+\left(\frac{1}{\gamma^{2}} \right)\mathbf{P}\mathbf{G}_{\rm aug}\mathbf{G}_{\rm aug}^{\rm T}\mathbf{P}+\mathbf{P}-\left( \mathbf{S}+\mathbf{S}^{-\rm T}\right)&\mathbf{S}^{\rm T}+\big{(}\mathbf{A}_{\rm aug}+\mathbf{B}_ {\rm aug}\mathbf{K}\big{)}^{\rm T}\mathbf{P}\\ \mathbf{S}+\mathbf{P}\big{(}\mathbf{A}_{\rm aug}+\mathbf{B}_{\rm aug}\mathbf{K}\big{)}&-\mathbf{P}\end{bmatrix} <0 \tag{16}\]  
By a congruence transformation matrix \( \begin{bmatrix}\mathbf{V}&0\\ 0&\mathbf{X}\end{bmatrix} \) with \(\mathbf{V}=\mathbf{S}^{-1}\) and \(\mathbf{X}=\mathbf{P}^{-1}\), the following form can be further deduced. \[ \begin{bmatrix}\mathbf{V}^{\rm T}\mathbf{Q}\mathbf{V}+\mathbf{V}^{\rm T}\mathbf{K}^{\rm T}\mathbf{R} \mathbf{K}\mathbf{V}+\left(\frac{1}{\gamma^{2}}\right)\mathbf{V}^{\rm T}\mathbf{P}\mathbf{G}_{\rm aug }\mathbf{G}_{\rm aug}^{\rm T}\mathbf{P}\mathbf{V}+\mathbf{V}^{\rm T}\mathbf{P}\mathbf{V}-\left(\mathbf{V} \mathbf{+}\mathbf{V}^{\rm T}\right)&\mathbf{X}+\mathbf{V}^{\rm T}\big{(}\mathbf{A}_{\rm aug}+\mathbf{ B}_{\rm aug}\mathbf{K}\big{)}^{\rm T}\\ \big{(}\mathbf{A}_{\rm aug}+\mathbf{B}_{\rm aug}\mathbf{K}\big{)}\mathbf{V}+\mathbf{X}&-\mathbf{X} \end{bmatrix} <0 \tag{17}\]  
By multiple applications of Lemma 4 and by defining \(\mathbf{N}=\mathbf{K}\mathbf{V}\) and \(\mathbf{M}=\mathbf{P}\mathbf{V}\), the deduced form can be transformed into the final form as follows:  
\[ \begin{bmatrix}-\big{(}\mathbf{V}\mathbf{+}\mathbf{V}^{\rm T}\big{)}&\mathbf{X}\mathbf{+}\mathbf{V}^{ \rm T}\mathbf{A}_{\rm aug}^{\rm T}\mathbf{+}\mathbf{N}^{\rm T}\mathbf{B}_{\rm aug}^{\rm T}& \mathbf{V}^{\rm T}&\mathbf{M}^{\rm T}\mathbf{G}_{\rm aug}^{\rm T}&\mathbf{V}^{\rm T}&\mathbf{N}^{ \rm T}\\ \mathbf{*}&\mathbf{*}&\mathbf{-}\mathbf{X}&0&0&0\\ \mathbf{*}&\mathbf{*}&\mathbf{*}&-\gamma^{2}\mathbf{I}&0&0\\ \mathbf{*}&\mathbf{*}&\mathbf{*}&\mathbf{*}&-\mathbf{Q}^{-1}&0\\ \mathbf{*}&\mathbf{*}&\mathbf{*}&\mathbf{*}&-\mathbf{R}^{-1}\end{bmatrix} <0 \tag{18}\]  
Since the matrices \(\mathbf{A}_{\rm aug}\) and \(\mathbf{B}_{\rm aug}\) have uncertain modules \(\mathbf{A}_{\rm F}\) and \(\mathbf{B}_{\rm F}\) respectively, therefore, the final deduced LMI form cannot be solved. Applying Lemma 1, which is intended to remove uncertainty based on the convex analysis and proof, the uncertain modules can be replaced with their boundary forms \(\mathbf{A}_{\text{L}}\) and \(\mathbf{B}_{\text{L}}\). Consequently, the solvable LMI forms can be obtained as follows:  
\[ \begin{bmatrix}-\left(\mathbf{V+}\mathbf{V}^{T}\right)&\mathbf{X+}\mathbf{V}^{T}\mathbf{A}_{\text{L }}^{T}\mathbf{+}\mathbf{N}^{T}\mathbf{B}_{\text{L}}^{T}&\mathbf{V}^{T}&\mathbf{M}^{T}\mathbf{G}_{aug}^{T }&\mathbf{V}^{T}&\mathbf{N}^{T}\\ \ast&-\mathbf{X}&0&0&0&0\\ \ast&\ast&-\mathbf{X}&0&0&0\\ \ast&\ast&\ast&-\gamma^{2}\mathbf{I}&0&0\\ \ast&\ast&\ast&\ast&-\mathbf{Q}^{-1}&0\\ \ast&\ast&\ast&\ast&-\mathbf{R}^{-1}\end{bmatrix}