## 5 Results
### Test-Case 4: Four-Bar Flexible-Link Multibody System
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Open-Loop** & **Desired** & **Closed-Loop Poles** & **Damping Ratio** & **Closed-Loop Poles** & **Damping ratio** \\
**Poles** & **Closed-Loop Poles** & **with**\(\mathbf{k}_{0}\) & **with**\(\mathbf{k}_{0}\) & **with**\(\mathbf{k}\) & **with**\(\mathbf{k}\) \\ \hline
3.17 & \(-15\) & \(-15\) & \(1\) & \(-15\) & \(1\) \\ -3.17 & \(-20\) & \(-20\) & \(1\) & \(-20\) & \(1\) \\ \(-1.20\pm 226.2\)i & \(-10\pm 220\)i & \(-10\pm 220\)i & \(0.045\) & \(-10\pm 220\)i & \(0.045\) \\ \(-2.40\pm 322.3\)i & - & \(28.6\pm 244.7\)i & \(-0.116\) & \(-74.0\pm 215.7\)i & \(0.324\) \\ \(-5.33\pm 483.1\)i & \(-15\pm 320\)i & \(-15\pm 320\)i & \(0.047\) & \(-15\pm 320\)i & \(0.047\) \\ \(-20.86\pm 958.4\)i & \(-50\pm 500\)i & \(-50\pm 500\)i & \(0.100\) & \(-50\pm 500\)i & \(0.100\) \\ \(-46.24\pm 1427.4\)i & - & \(-22.19\pm 966.96\)i & \(0.023\) & \(-66.59\pm 923.49\)i & \(0.072\) \\ \(-58.49\pm 1605.3\)i & - & \(-46.41\pm 1429.3\)i & \(0.033\) & \(-85.13\pm 1426.8\)i & \(0.060\) \\ \(-121.1\pm 2308.9\)i & - & \(-121.1\pm 2308.5\)i & \(0.052\) & \(-121.1\pm 2308.5\)i & \(0.052\) \\ \(-287.6\pm 3551.7\)i & - & \(-287.6\pm 3551.7\)i & \(0.080\) & \(-287.6\pm 3551.7\)i & \(0.081\) \\ \(-390.8\pm 4134.8\)i & - & \(-390.7\pm 4134.9\)i & \(0.094\) & \(-390.7\pm 4134.9\)i & \(0.094\) \\ \(-947.0\pm 6396.2\)i & - & \(-947.0\pm 6396\)i & \(0.147\) & \(-947.0\pm 6396.2\)i & \(0.147\) \\ \(-1820\pm 8777.4\)i & - & \(-1820.0\pm 8777\)i & \(0.203\) & \(-1819.8\pm 8777.3\)i & \(0.203\) \\ \(-3440.7\pm 11835\)i & - & \(-3440.1\pm 11834\)i & \(0.279\) & \(-3439.9\pm 11834\)i & \(0.279\) \\ \(-6348.2\pm 15491\)i & - & \(-6348.2\pm 15490\)i & \(0.379\) & \(-6348.3\pm 15490\)i & \(0.379\) \\ \(-23417\pm 22033\)i & - & \(-23417\pm 22033\)i & \(0.728\) & \(-23417\pm 22033\)i & \(0.728\) \\ \(-40747\pm 11774\)i & - & \(-40747\pm 11773\)i & \(0.961\) & \(-40747\pm 11774\)i & \(0.961\) \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Open-loop and closed-loop poles for the test-case 4. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Stage 1: \(k_{0}\)**} & \multicolumn{2}{c}{**Stage 2: \(k\)=\(k_{0}\)+V\(k_{r}\)**} \\ \hline \(f_{0}\) & \(g_{0}\) & \(f\) & \(g\) \\ \hline \(-0.00359\) & \(-639.62\) & \(0.0738\) & \(-639.26\) \\ \(0.00000\) & -0.01 & \(-9.8664\) & \(-4850.30\) \\ \(0.00000\) & \(0.13\) & \(-5.5614\) & \(-3456.7\) \\ \(0.00163\) & \(-951.75\) & \(0.1526\) & \(-950.73\) \\ \(-0.02214\) & \(15.34\) & \(0.5697\) & \(302.04\) \\ \(0.00277\) & \(-0.34\) & \(-1.8326\) & \(-618.71\) \\ \(0.00245\) & \(-0.23\) & \(2.1344\) & \(719.62\) \\ \(-0.000001\) & \(-10.26\) & \(-1.1898\) & \(-587.82\) \\ \(0.00000\) & \(0.02\) & \(8.7912\) & \(5210.00\) \\ \(0.02216\) & \(8.43\) & \(0.6184\) & \(313.22\) \\ \(0.00011\) & \(29.19\) & \(0.2044\) & \(-68.49\) \\ \(0.00000\) & \(-2.00\) & \(0.3313\) & \(53.01\) \\ \(0.00000\) & \(1.39\) & \(-0.0977\) & \(95.78\) \\ \(-0.00008\) & \(-27.40\) & \(-0.4665\) & \(158.70\) \\ \(0.00006\) & \(34.45\) & \(0.2638\) & \(-62.60\) \\ \(0.04228\) & \(0.36\) & \(0.0343\) & \(0.30\) \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Gains for the test-case 4. Figure 11: Open-loop and closed-loop poles of the four-bar linkage (test-case 4).