## 3 Control System Design
### LQI Control Law Design
\begin{bmatrix}\mathbf{A}& \mathbf{O}_{10\times 3}\\ -\mathbf{C}&\mathbf{O}_{3}\end{bmatrix}\begin{bmatrix}\mathbf{X}_{\mathbf{b}}\\ \mathbf{X}_{i}\end{bmatrix} }_{\dot{\mathbf{X}}}+\underbrace{ \begin{bmatrix}\mathbf{B}\\ \mathbf{O}_{3}\end{bmatrix} \mathbf{U}+\underbrace{ \begin{bmatrix}\mathbf{D}\\ \mathbf{r_{b}}\end{bmatrix} }_{\dot{\mathbf{D}}}\end{bmatrix}_{\dot{\mathbf{D}}} \tag{16}\]  
A state-feedback control law is designed in the form of:  
\[\mathbf{U}=- \begin{bmatrix}\mathbf{K}_{\mathbf{f}}&\mathbf{K}_{i}\end{bmatrix} \widetilde{\mathbf{ X}} \tag{17}\]  
where the optimal matrix gains are computed by the LQI control method:  
\[\underbrace{ \begin{bmatrix}\mathbf{K}_{\mathbf{f}}&\mathbf{K}_{i}\end{bmatrix} }_{\mathbf{K}}= \mathbf{R}^{-1}\overline{\mathbf{B}}^{T}\mathbf{P} \tag{18}\]  
such that the control law (17) minimizes the following quadratic cost function:  
\[\min_{\mathbf{U}}\mathbf{U}=\int\limits_{0}^{\infty}\Big{(}\widetilde{\mathbf{X}}^{T}\bm {Q}\widetilde{\mathbf{X}}+\mathbf{U}^{T}\mathbf{R}\mathbf{U}\Big{)}dt \tag{19}\]  
\(\mathbf{Q}\) and \(\mathbf{R}\) are positive-definite weighting matrices of the state and input, respectively. \(\mathbf{P}\) is the positive definite and symmetric matrix, which satisfies the algebraic Riccati Equation (20):  
\[\widetilde{\mathbf{A}}^{T}\mathbf{P}+\mathbf{P}\overline{\mathbf{A}}-\mathbf{P}\overline{\mathbf{B}} \mathbf{R}^{-1}\overline{\mathbf{B}}^{\mathrm{T}}\mathbf{P}+\mathbf{Q}=0 \tag{20}\]  
The schematic drawing of the designed control system is depicted in Figure 4.  
Assume that the vector of disturbances and unmodelled dynamics \(\widetilde{\mathbf{D}}\) is bounded by a non-negative value \(\gamma\) such that \(\left\|\widetilde{\mathbf{D}}\right\|_{2}\leq\gamma\). To analyze the robustness of the closed-loop system, a Lyapunov function candidate is chosen as follows:  
\[V=\widetilde{\mathbf{X}}^{\mathrm{T}}\mathbf{P}\widetilde{\mathbf{X}} \tag{21}\]  
Taking the time-derivative of \(V\) and applying the control law in (17) and the algebraic Riccati equation in (20) results in the following:  
\[ \begin{array}{l}\dot{V}=\widetilde{\mathbf{X}}^{T}\Big{(}\big{(}\overline{\mathbf{A} }-\overline{\mathbf{B}}\mathbf{K}\big{)}^{T}\mathbf{P}+\mathbf{P}\big{(}\overline{\mathbf{A}}- \overline{\mathbf{B}}\mathbf{K}\big{)}\Big{)}\widetilde{\mathbf{X}}+\widetilde{\mathbf{D}}^{T }\mathbf{P}\widetilde{\mathbf{X}}+\widetilde{\mathbf{X}}^{T}\mathbf{P}\widetilde{\mathbf{D}}\\ =-\widetilde{\mathbf{X}}^{T}\Big{(}\mathbf{P}\overline{\mathbf{B}}\mathbf{R}^{-1}\overline\mathbf{{B}}^{\mathrm{T}}\mathbf{P}+\mathbf{Q}\Big{)}\widetilde{\mathbf{X}}+\widetilde{\mathbf{D}}^{T }\mathbf{P}\widetilde{\mathbf{X}}+\widetilde{\mathbf{X}}^{T}\mathbf{P}\widetilde{\mathbf{D}}\\ \leq\Big{[}-\lambda_{\min}\Big{(}\mathbf{P}\overline{\mathbf{B}}\mathbf{R}^{-1}\overline{ \mathbf{B}}^{\mathrm{T}}\mathbf{P}+\mathbf{Q}\Big{)}\left\|\widetilde{\mathbf{X}}\right\|_{2}+ 2\gamma\lambda_{\max}(\mathbf{P})\Big{]}\left\|\widetilde{\mathbf{X}}\right\|_{2}\end{array}  \tag{22}\]  
where \(\lambda_{\min}(. )\) and \(\lambda_{\max}(. )\) indicate the minimum and maximum eigenvalues of the corresponding matrix. In order to preserve the system stability, \(\dot{V}\) must be negative-definite. Figure 4: The proposed LQI control system. From Equation (22), one can see that \(\dot{V}<0\) whenever the state satisfies the following condition:  
\[\left\|\widetilde{\mathbf{X}}\right\|_{2}>\frac{2\lambda_{\max}(\mathbf{P})}{\lambda _{\min}\left(\mathbf{P}\widetilde{\mathbf{B}}\mathbf{R}^{-1}\widetilde{\mathbf{B}}^{\mathrm{T}} \mathbf{P}+\mathbf{Q}\right)}\gamma \tag{23}\]  
which implies the boundedness of \(\widetilde{\mathbf{X}}\).