## 2 Preliminaries and Motivation  
### Model Conversion for Singular Systems  
Consider a known linear continuous-time singular system described by  
\[E_{r}\dot{x}_{c}(t)=A_{r}x_{c}(t)+B_{r}u_{c}(t), \tag{1a}\] \[y_{c}(t)=C_{r}x_{c}(t), \tag{1b}\]  
where \(x_{c}(t)\in\Re^{n}\), \(u(t)\in\Re^{m}\), and \(y_{c}(t)\in\Re^{p}\) are the state, control input, and measured output vectors of the system, respectively; \(A_{r}\in\Re^{n\times m}\) and \(B_{r}\in\Re^{n\times m}\) are real constant matrices; and \(E_{r}\in\Re^{n\times m}\) is a singular matrix. For simplicity, it is assumed here that the system has no impulsive modes in the fast state. For a singular system with impulsive modes in the fast state, the methodology for eliminating the impulsive modes can be found in [2]. Through a series of coordinate transforms, the impulsive mode-free singular system can be decomposed into a reduced-order regular subsystem and a non-dynamic subsystem [2], as  
\[\left[ \begin{array}{cc}I_{\xi}&0\\ 0&0\end{array} \right[ \begin{array}{c}\dot{\hat{x}}_{c,s}(t)\\ \dot{\hat{x}}_{c,f}(t)\end{array} ]=\left[ \begin{array}{cc}\hat{A}_{s}&0\\ 0&I_{n-\xi}\end{array} \right[ \begin{array}{c}\dot{\hat{x}}_{c,s}(t)\\ \dot{\hat{x}}_{c,f}(t)\end{array} ]+\left[ \begin{array}{c}\dot{\hat{B}}_{s}\\ \dot{\hat{B}}_{f}\end{array} \right]\mu_{c}(t), \tag{2a}\] \[y_{c}(t)=\left[\hat{C}_{s}\;\hat{C}_{f}\right]\left[ \begin{array}{c}\dot{ \hat{x}}_{c,s}(t)\\ \dot{\hat{x}}_{c,f}(t)\end{array} \right], \tag{2b}\]  
where \(\hat{x}_{c,f}(t)=-\hat{B}_{f}u_{c}(t)\). Then, the corresponding reduced-order regular system with an input-output feedthrough term can be described by  
\[\dot{\hat{x}}_{c,s}(t)=\hat{A}_{s}\hat{x}_{c,s}(t)+\hat{B}_{s}u_{c}(t), \tag{3a}\] \[y_{c}(t)=\hat{C}_{s}\hat{x}_{c,s}(t)+(-\hat{C}_{f}\hat{B}_{f})u_{c}(t)=\hat{C} _{s}\hat{x}_{c,s}(t)+\hat{D}_{s}u_{c}(t), \tag{3b}\]  
where \(\hat{D}_{s}=-\hat{C}_{f}\hat{B}_{f}\), \(\hat{A}_{s}\in\Re^{\times\xi,\xi}\),\(\hat{B}_{s}\in\Re^{\times m}\), \(\hat{C}_{s}\in\Re^{\times p\times\xi}\), and \(\hat{D}_{s}\in\Re^{p\times m}\) are the state, input, output, and direct feedthrough matrices, respectively; \(\hat{x}_{c,s}(t)\in\Re^{\xi}\) is the state vector, \(u_{c}(t)\in\Re^{m}\) is the control input, and \(y_{c}(t)\in\Re^{p}\) is the measurable output of the system at time \(t\). For this equivalent model, let the associated cost function to be minimized be  
\[J\big{(}y_{c}(t),u_{c}(t)\big{)}=\frac{1}{2}\int_{0}^{t_{f}}\Big{\{}\big{[}y_{ c}(t)-r(t)\big{]}^{T}Q_{c}\big{[}y_{c}(t)-r(t)\big{]}+u_{c}(t)^{T}R_{c}u_{c}(t) \Big{\}}dt, \tag{4}\]  
where \(Q_{c}\) is a \(p\times p\) positive definite or positive semi-definite real symmetric matrix, \(R_{c}\) is an \(m\times m\) positive definite real symmetric matrix, the reference input \(r(t)\) denotes the pre-specified output trajectory, and the final index is finite, i.e., \(t_{f}<\infty\). Solving Equation (4) yields the continuous-time state feedback control law [18]  
\[ \begin{array}{rl}u_{c}(t)&=-K_{c}\hat{x}_{c,s}(t)-\overline{R_{c}^{-1}} \bigg{[}\big{(}\hat{C}_{s}-\hat{D}_{s}K_{c}\big{)}\big{(}\hat{A}_{s}-\hat{B}_ {s}K_{c}\big{)}^{-1}\hat{B}_{s}+\hat{D}_{s}\bigg{]}^{T}Q_{c}r(t)\\ &=-K_{c}\hat{x}_{c,s}(t)+E_{c}r(t),\end{array}  \tag{5}\]  
where  
\[ \begin{array}{rl}K_{c}&=\overline{R_{c}^{-1}}\Big{(}\hat{B}_{s}^{T}P_{s}+N _{s}^{T}\Big{)},\;E_{c}=-\overline{R_{c}^{-1}}\bigg{[}\big{(}\hat{C}_{s}-\hat {D}_{s}K_{c}\big{)}\big{(}\hat{A}_{s}-\hat{B}_{s}K_{c}\big{)}^{-1}\hat{B}_{s}+ \hat{D}_{s}\bigg{]}^{T}Q_{c},\\ \overline{R}_{c}&=R_{c}+\hat{D}_{s}^{T}Q_{c}\hat{D}_{s},\;N_{s}=\hat{C}_{s}^{T }Q_{c}\hat{D}_{s},\end{array}