## 4 Numerical Examples  
Consider a five-degree of freedom system, as shown in Figure 1, with the following system matrices [27],  
\[\mathbf{M}=\mathbf{I},\mathbf{C}=\left[ \begin{array}{ccccc}1&0&0&0&0\\ 0&1&0&0&-1\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&-1&0&0&1\end{array} \right]\ \mathbf{K}=\left[ \begin{array}{ccccc}10&-5&0&0&0\\ -5&10&-5&0&0\\ 0&-5&10&-5\\ 0&0&0&-5&5\end{array} \right].\]  
Other matrices involved are given as follows:  
\[\mathbf{B}=\left[ \begin{array}{ccccc}0&0\\ 0&0\\ 0&0\\ -1&0\\ 1&1\end{array} \right]\ \mathbf{G}_{1}=\left[ \begin{array}{ccccc}2&1&0&1&-1\\ 0&2&-1&0&1\end{array} \right],\ \mathbf{G}_{2}=\left[ \begin{array}{ccccc}1&0&1&0&1\\ 0&1&0&1&0\end{array} \right]\]  
The five pairs of open-loop eigenvalues are \(\{\lambda_{01}\}=\{-0.0102\pm 0.2852\mathrm{i},\ -0.0320\pm 1.6808\mathrm{i},\ -0.0441\pm 1.3137\mathrm{i},\ -0.0605\pm 1.9050\mathrm{i},\ -0.1532\pm 0.8177\mathrm{i}\}\), sorted by the real part. Its eigenvalues of the closed-loop system without time-delays from \(\mathbf{J}_{\mathrm{m1}}(\lambda)\) are \(\{\lambda_{1i}\}=\{-0.0830\pm 3.6405\mathrm{i},\ -0.2940\pm 0.8781\mathrm{i},\ -0.4687\pm 4.129\mathrm{i},\ -0.4869\pm 3.1234\mathrm{i},\ -0.6674\pm 1.4719\mathrm{i}\}\), obtained by the contour integral method with radius \(R\) = 5, \(r=p=2\), \(q=12\), and denoted by \"+\" in Figure 2. Thus, radius \(R\) = 5 is chosen to compute dominant eigenvalues of the closed-loop system with time-delays \(\tau_{1}=1.0,\ \tau_{2}=0.5\) from \(\mathbf{J}_{\mathrm{m}}(\lambda)\), which is slightly larger than the modulus of the first pair eigenvalues of \(\{\lambda_{1i}\}\), i.e., \(-0.0830\pm 3.6405\mathrm{i}\) (it is marked by a red arrow in Figure 2). Other parameters of the contour integration are \(r=p=2\) and \(q=14\). The latter parameter can be determined by counting the number of eigenvalues of \(\mathbf{J}_{\mathrm{m}}(\lambda)\) inside the contour in the next paragraph. Computed thirteen dominant eigenvalues are tabulated in Table 1, which are also denoted by \"o\" in Figure 2. The obtained eigenvalues are compared with those given by a spectral  
Figure 1: A five-degree of freedom closed-loop control system [27]. method [8], which computes all the eigenvalues in the region of the complex plane of \(\text{Re}(\lambda)>-6\). It should be noted that the spectral method requires the knowledge of system matrices, and its results are also tabulated in Table 1 and denoted by \"*\" in Figure 3. Its corresponding eigenvalues inside the contour are completely identical to those given by the contour integral method within the same contour (denoted by \"o\"), as shown in Figure 3. For the spectral method, the closed-loop time delayed control systems (i.e., (1) and (2)) with \(\mathbf{f}(t)=0\) are analysed by the following first-order state space model,  
\[\dot{\mathbf{z}}(t)=\mathbf{A}_{0}\mathbf{z}(t)+\mathbf{A}_{1}\mathbf{z}(t- \tau_{1})+\mathbf{A}_{2}\mathbf{z}(t-\tau_{2})\]  
where \(\mathbf{z}=\left(\mathbf{x}^{\text{T}}\dot{\mathbf{x}}^{\text{T}}\right)^{ \text{T}}\) and  
\[\mathbf{A}_{0}=\left[ \begin{array}{cc}0&\mathbf{I}\\ -\mathbf{M}^{-1}\mathbf{K}&-\mathbf{M}^{-1}\mathbf{C}\end{array} \right]_{ \text{T}}\,\,\mathbf{A}_{1}=\left[ \begin{array}{cc}0&0\\ -\mathbf{M}^{-1}\mathbf{B}\mathbf{G}_{1}&0\end{array} \right],\,\,\mathbf{A}_{2 }=\left[ \begin{array}{cc}0&0\\ 0&-\mathbf{M}^{-1}\mathbf{B}\mathbf{G}_{2}\end{array} \right].\] \begin{table}
\begin{tabular}{c c} \hline \hline
**The Contour Integral Method (\(R=5\))** & **A Spectral Method [8] (Re(\(\lambda\))>-6)** \\ \hline \(0.0083\pm 4.3588\)i & \(0.0083\pm 4.3588\)i \\ \(-0.1267\pm 2.8611\)i & \(-0.1267\pm 2.8611\)i \\ \(-0.1300\pm 0.9773\)i & \(-0.1300\pm 0.9773\)i \\ \(-0.2429\pm 3.8060\)i & \(-0.2429\pm 3.8060\)i \\ \(-1.2293\pm 1.1821\)i & \(-1.2293\pm 1.1821\)i \\ \(-2.6245\pm 3.2784\)i & \(-2.6245\pm 3.2784\)i \\ \(-4.2116+0.0000\)i & \(-4.2116+0.0000\)i \\  & \(-4.4613\pm 8.4646\)i \\  & \(-5.3755\pm 12.7017\)i \\  & \(-5.4304\pm 14.9364\)i \\  & \(-5.7236\pm 17.0715\)i \\ \hline \hline \end{tabular}
\end{table}