## 2 System Description and Reduced Characteristic Function  
A linear second-order controlled system with time-delay is described by:  
\[\mathbf{M}\ddot{\mathbf{x}}(t)+\mathbf{C}\dot{\mathbf{x}}(t)+\mathbf{K} \mathbf{x}(t)=\mathbf{B}\mathbf{u}(t,\tau_{1},\tau_{2})+\mathbf{f}(t) \tag{1}\]  
\[\mathbf{u}(t,\tau_{1},\tau_{2})=-\mathbf{G}_{1}\mathbf{x}(t-\tau_{1})-\mathbf{ G}_{2}\dot{\mathbf{x}}(t-\tau_{2}) \tag{2}\]where **M**, **C**, and **K** are known as the \(n\times n\) mass, damping, and stiffness matrices, respectively; **u** is a control force vector and **f** is an external applied force vector; **B** is the \(n\times p\) control input distribution matrix; **G**\({}_{2}\) and **G**\({}_{1}\) are the \(p\times n\) velocity and displacement feedback gain matrices respectively, and \(p<n\); \(\tau_{1}\) and \(\tau_{2}\) are the displacement and velocity feedback time-delays, respectively. Substituting (2) into (1) gives  
\[\textbf{M}\ddot{\textbf{x}}(t)+\textbf{C}\dot{\textbf{x}}(t)+\textbf{K} \textbf{x}(t)=\textbf{B}\Big{(}-\textbf{G}_{1}\textbf{x}(t-\tau_{1})-\textbf{ G}_{2}\dot{\textbf{x}}(t-\tau_{2})\Big{)}+\textbf{f}(t) \tag{3}\]  
Laplace transform of (3) gives  
\[\Big{(}s^{2}\textbf{M}+s\textbf{C}+\textbf{K}\Big{)}\textbf{x}(s)=-\textbf{B} (\textbf{G}_{1}\textrm{e}^{-s\tau_{1}}+s\textbf{G}_{2}\textrm{e}^{-s\tau_{2}} )\textbf{x}(s)+\textbf{f}(s) \tag{4}\]  
\[\Big{[}s^{2}\textbf{M}+s\textbf{C}+\textbf{K}+\textbf{B}(\textbf{G}_{1} \textrm{e}^{-s\tau_{1}}+s\textbf{G}_{2}\textrm{e}^{-s\tau_{2}})\Big{]}\textbf{ x}(s)=\textbf{f}(s) \tag{5}\]  
Then, the full \(n\times n\) receptance matrices of the open-loop and closed-loop systems are represented by  
\[\textbf{H}_{0}(s)=\Big{(}s^{2}\textbf{M}+s\textbf{C}+\textbf{K}\Big{)}^{-1} \tag{6}\]  
\[\textbf{H}_{\textrm{c}}(s)=\Big{[}\textbf{H}_{0}^{-1}(s)+\textbf{B}(\textbf{G }_{1}\textrm{e}^{-s\tau_{1}}+s\textbf{G}_{2}\textrm{e}^{-s\tau_{2}})\Big{]}^{ -1} \tag{7}\]  
The closed-loop receptance matrix \(\textbf{H}_{\textrm{c}}(s)\) for the system with delay can be directly related to the open-loop receptance matrix \(\textbf{H}_{0}(s)\) by using the Sherman-Morrison-Woodbury formula as follows:  
\[\textbf{H}_{\textrm{c}}(s)=\textbf{H}_{0}(s)-\textbf{H}_{0}(s)\textbf{ B}\Big{[}\textbf{I}_{\textrm{p}}+(\textbf{G}_{1}\textrm{e}^{-s\tau_{1}}+s \textbf{G}_{2}\textrm{e}^{-s\tau_{2}})\textbf{H}_{0}(s)\textbf{ B}\Big{]}^{-1}(\textbf{G}_{1}\textrm{e}^{-s\tau_{1}}+s\textbf{G}_{2}\textrm{e}^{-s \tau_{2}})\textbf{H}_{0}(s) \tag{8}\]  
The characteristic function of the second-order linear time-delay system (5) is given in the form  
\[f(\lambda)=\textrm{det}(\textbf{Z}(\lambda))=\textrm{det}\Big{[}\lambda^{2} \textbf{M}+\lambda\textbf{C}+\textbf{K}+\textbf{B}\Big{(}\textbf{G}_{1} \textrm{e}^{-\lambda\tau_{1}}+\lambda\textbf{G}_{2}\textrm{e}^{-\lambda\tau_{2} }\Big{)}\Big{]} \tag{9}\]  
where \(\textbf{Z}(\lambda)=\lambda^{2}\textbf{M}+\lambda\textbf{C}+\textbf{K}+ \textbf{B}\Big{(}\textbf{G}_{1}\textrm{e}^{-\lambda\tau_{1}}+\lambda\textbf{ G}_{2}\textrm{e}^{-\lambda\tau_{2}}\Big{)}\) is the so-called dynamic stiffness matrix of the closed-loop system (5), and \(\textbf{Z}(\lambda)=\textbf{H}_{\textrm{c}}^{-1}(\lambda)\) from (7). \(f(\lambda)\) is a transcendental function containing some exponential terms, also called quasi-polynomial, which has an infinite number of roots. The roots of \(f(\lambda)\) are also known as eigenvalues (poles or characteristic roots) of (5), whose distribution on the complex plane determines the stability and dynamic behaviour of (5). Now the reduced form of characteristic function (9) and the corresponding characteristic equation are presented. Without loss of generality, **assume that the eigenvalues of the closed-loop system with delay are distinct from those of the open-loop system**. Suppose that **A** and **Q** are nonsingular matrices of appropriate orders.