## 4 FTC Structure and Controller Design for NT-DRAS
### Controller Design for NT-DRAS Subjected to Gradual Faults  
The FTC problem considered in this paper is to design a controller for NT-DRAS with gradual faults, similar to the ones discussed in [23; 27], such that  
Figure 4: FTC structure for all fault processes. Under the gradual fault conditions, the NT-DRAS output \(\mathcal{S}_{\text{out}}\mathbf{y}(t)\) tracks the reference signal \(r(t)\) without steady-state error, that is \(\underset{t\rightarrow\infty}{\lim}e(t)=0\), where \(e(t)=r(t)-\mathcal{S}_{\text{out}}\mathbf{y}(t)\), \(\mathcal{S}_{\text{out}}= \begin{bmatrix}1&0&0&0\end{bmatrix} \)
2. The optimized performance to be defined later can be achieved by the designed controller. The corresponding augmented system can be described in the following form. \[\left\{ \begin{array}{l}\dot{\mathbf{x}}_{\text{aug}}(t)=\mathbf{A}_{\text{aug}}\bm {x}_{\text{aug}}(t)+\mathbf{B}_{\text{aug}}\mathbf{u}(t)+\mathbf{G}_{\text{aug}}\mathbf{w}_{ \text{aug}}(t)\\ \mathbf{y}_{\text{aug}}(t)=\mathbf{C}_{\text{aug}}\mathbf{x}_{\text{aug}}(t)\end{array} \right.\] (20) where \(\mathbf{x}_{\text{aug}}(t)= \begin{bmatrix}\int e(t)dt&\mathbf{x}(t)\end{bmatrix} ^{ \text{T}}\) is the augmented state vector, \(\mathbf{y}_{\text{aug}}(t)= \begin{bmatrix}\int e(t)dt&\mathbf{y}(t)\end{bmatrix} ^{ \text{T}}\) is the augmented output vector, and the corresponding augmented matrices are as follows \(\mathbf{A}_{\text{aug}}= \begin{bmatrix}0&-\mathcal{S}_{\text{out}}\mathbf{C}\\ 0&\mathbf{A}_{\text{F}}\end{bmatrix} \), \(\mathbf{B}_{\text{aug}}= \begin{bmatrix}0\\ \mathbf{B}_{\text{F}}\end{bmatrix} \), \(\mathbf{C}_{\text{aug}}= \begin{bmatrix}\mathbf{I}&0\\ 0&\mathbf{C}\end{bmatrix} \), and \(\mathbf{G}_{\text{aug}}= \begin{bmatrix}\mathbf{I}&0\\ 0&\mathbf{G}\end{bmatrix} \). The output feedback method is used to design a fault-tolerant controller when it is sufficient to realize some but not all control performance factors [28; 29]. The closed-loop controller is designed using state and output error integration feedback form as follows, which is essentially a PI controller: \[\mathbf{u}(t)=\mathbf{K}\mathbf{x}_{\text{aug}}(t)=\mathbf{K}_{e}\int_{0}^{t}e(\tau)d\tau+\mathbf{ K}_{\mathbf{x}}\mathbf{x}(t)\] (21) where \(\mathbf{K}= \begin{bmatrix}\mathbf{K}_{e}&\mathbf{K}_{\mathbf{x}}\end{bmatrix} \) is the fixed gain matrix to be solved by the LMI technique. Define the linear-quadratic (LQ) cost function as  
\[\mathbf{J}=\int_{0}^{t}\Bigl{(}\mathbf{x}_{\text{aug}}^{\text{T}}(\tau)\mathbf{Q}\mathbf{x}_{ \text{aug}}(\tau)+\mathbf{u}(\tau)^{\text{T}}\mathbf{R}\mathbf{u}(\tau)\Bigr{)}d\tau \tag{22}\]  
where \(\mathbf{Q}\) and \(\mathbf{R}\) are symmetric positive semi-definite and positive definite weighting matrices, respectively. Then the optimization objective is to design the control input form (see Equation (21)), so as to minimize the cost function. The following lemmas are introduced before the control gain matrix theorem is presented in further text. **Lemma 2** (Bounded Real Lemma). : _For the linear system form derived from Equation (16), and represented in the following form_  
\[\left\{ \begin{array}{l}\dot{\mathbf{x}}(t)=(\mathbf{A}_{\text{L}}+\mathbf{B}_{\text{L}} \mathbf{K}_{\text{x}})\mathbf{x}(t)+\mathbf{G}\mathbf{w}(t)\\ \mathbf{y}(t)=\mathbf{C}\mathbf{x}(t)\end{array} \right. \tag{23}\]  
_the transfer function form \(\mathbf{G}(s)=\mathbf{C}(s\mathbf{I}-\mathbf{A}_{\text{L}}-\mathbf{B}_{\text{L}}\mathbf{K}_{\text{x}})\) can be obtained. For a given performance index \(\gamma:\|y(t)\|\leq\gamma\|w(t)\|\), if a real matrix \(\mathbf{P}=\mathbf{P}^{\text{T}}>0\) exists and satisfies the following matrix inequality_  
\[ \begin{bmatrix}(\mathbf{A}_{\text{L}}+\mathbf{B}_{\text{L}}\mathbf{K}_{\text{x}})\mathbf{P}+ \mathbf{P}(\mathbf{A}_{\text{L}}+\mathbf{B}_{\text{L}}\mathbf{K}_{\text{x}})^{\text{T}}&\mathbf{I} &(\mathbf{C}\mathbf{P})^{\text{T}}\\ \ast&-\gamma^{2}\mathbf{I}&\mathbf{I}\\ \ast&-\mathbf{I}\end{bmatrix}