## 4 FTC Structure and Controller Design for NT-DRAS
### Controller Design for NT-DRAS Subjected to Gradual Faults
<0 \tag{24}\]  
_then the system form represented by Equation (16) is stable and the transfer function \(\mathbf{G}(s)\) satisfies \(\max\{\|\mathbf{G}(s)\|\}<\gamma\)._  
**Remark 2**. : _The parameter \(\mathbf{\gamma}\) represents the robust performance of the system and describes the quantitative relationship between the disturbance \(\mathbf{w}\) and the system output \(\mathbf{y}\). The inequality \(\max\{\|\mathbf{G}(s)\|\}<\gamma\) means that the influence of the disturbance \(\mathbf{w}\) on the system output \(\mathbf{y}\) is limited in the gain level \(\gamma\)[30]._  
**Lemma 3 (Reciprocal Projection Lemma) [31]. **_Let \(\mathbf{P}\) be any given positive definite matrix. The following statements are equivalent:_  
1. \(\mathbf{\psi}+\mathbf{\zeta}+\mathbf{\zeta}^{\mathrm{T}}<0\)_;_
2. _The LMI problem_ \( \begin{bmatrix}\mathbf{\psi}+\mathbf{P}-\left(\mathbf{S}+\mathbf{S}^{\mathrm{T}}\right)&\mathbf{ \zeta}^{\mathrm{T}}+\mathbf{S}^{\mathrm{T}}\\ \mathbf{\zeta}+\mathbf{S}&-\mathbf{P}\end{bmatrix} <0\) _is feasible with respect to_ \(\mathbf{S}\)_, where_ \(\mathbf{\psi}\) _and_ \(\mathbf{\zeta}\) _are a symmetric matrix and a general matrix, respectively._  
**Lemma 4 (Schur Complement Lemma) [32]. **_For the given symmetric matrix \(\mathbf{S}= \begin{bmatrix}\mathbf{S}_{11}&\mathbf{S}_{12}\\ \mathbf{S}_{12}^{\mathrm{T}}&\mathbf{S}_{22}^{\mathrm{T}}\end{bmatrix} \), the following three statements are equivalent:_  
1. \(\mathbf{S}<0\)_;_
2. \(\mathbf{S}_{11}<0\)_,_ \(\mathbf{S}_{22}-\mathbf{S}_{12}^{\mathrm{T}}\mathbf{S}_{11}^{-1}\mathbf{S}_{12}<0\)_;_
3. \(\mathbf{S}_{22}<0\)_,_ \(\mathbf{S}_{11}-\mathbf{S}_{12}\mathbf{S}_{22}^{-1}\mathbf{S}_{12}^{\mathrm{T}}<0\)_._  
The considered fault-tolerant controller with fixed structure and external disturbance, is in the state and output error integration feedback form (see Figure 5), realizing the pole placement and guaranteeing the robust performance. In order to solve for the controller gain, the determinate fault degree bounds and the anti-disturbance level are chosen for the proposed LMI algorithm. The existence conditions of the fault-tolerant controller gain are given in the form of the following theorem. **Theorem 1. **_Consider the closed-loop augmented system with uncertainty (see Equation (20)). For a given scalar \(\gamma>0\) in Lemma 2, and for all nonzero disturbance inputs \(w(t)\in L_{2}[0,\infty)\), choose \(\mathbf{Q}\) and \(\mathbf{R}\) as weighting matrices of linear quadratic (\(LQ\)) index, and define a symmetric positive definite matrix \(\mathbf{P}=\mathbf{X}^{-1}\), and other appropriately dimensioned common matrices \(\mathbf{V}\), \(\mathbf{N}\) and \(\mathbf{G}\). If these matrices can make the LMI conditions (see (A12)) hold, where in (A12), \(\mathbf{A}_{\mathrm{augL}}= \begin{bmatrix}0&-\mathbf{S}_{\mathrm{out}}\mathbf{C}\\ 0&\mathbf{A}_{\mathrm{L}}\end{bmatrix} \), \(\mathbf{B}_{\mathrm{augL}}= \begin{bmatrix}0\\ \mathbf{B}_{\mathrm{L}}\end{bmatrix} \), then the closed-loop system with uncertainty (see Equation (20)) can obtain the upper bound for performance indices in the following form_  
\[\mathbf{I}<\mathbf{x}_{\mathrm{aug}}^{\mathrm{T}}(0)\mathbf{P}\mathbf{x}_{\mathrm{aug}}(0)+ \gamma^{2}\int_{0}^{t}\mathbf{w}_{\mathrm{aug}}^{\mathrm{T}}(\tau)\mathbf{w}_{\mathrm{ aug}}(\tau)d\tau \tag{25}\]  
Figure 5: Controller structure for system with gradual faults in uncertain form. _The control gain matrix of the fault-tolerant controller for the closed-loop system can be determined as \(\mathbf{K}=\mathbf{N}_{\text{optimal}}\mathbf{V}_{\text{optimal}}^{-1}\), where \(\mathbf{N}_{\text{optimal}}\) and \(\mathbf{V}_{\text{optimal}}\) are the optimal solutions to the four LMIs (see (A12))._  
See Appendix B. \(\square\)