# Nonlinear Control of Multibody Flexible Mechanisms: A Model-Free Approach
## 4 Control Tuning and Closed-Loop Response
At the same time, the \(k_{p}\) gain of the PD controller needs to be set to an high value to compensate the speed reduction introduced by the derivative term, with the consequence of leading also to a pronounced overshoot (see Figure 4). Figure 8: Motor torque, comparison with PD and PD\({}^{\mu}\) controllers. Figure 7: Curvature of the crank and follower, detail, comparison with PD\({}^{\mu}\) controller. The proposed nonlinear control is capable of avoiding this occurrence: the nonlinear action governed by the gain \(k_{n}\), which is set to a negative value, basically reduces the effects of derivative action when the tracking error is high, i.e., during the first part of the transient. This momentary reduction of derivative action does not need to be compensated by an aggressive proportional gain, which would lead to pronounced vibrations at the beginning of the transient. Figure 8 actually shows that the nonlinear control produces a peak torque that is reduced by 47% in comparison with PD control and by 56% in comparison with PD\({}^{\mu}\) control. The maximum values of the motor torque developed by the three control systems is reported in Table 3.  
In order to quantify the performance level of the three closed-loop controller tested numerically, some metrics of the response of the closed-loop performance are provided in Table 3. The performance comparison can be made in terms of settling time, which is here measured as the time needed for the tracking error to be stabilized below \(\pm\)5%. As reported in Table 3, the fastest settling time is achieved by the nonlinear controller, while the slowest is the one obtained using the PD controller. Using a relative measurement of the settling time, the PD\({}^{\mu}\) controller is 82% slower than the PD controller, while the nonlinear control is 28% faster than the PD controller. The time evolution of the percentage tracking error is shown in Figure 9.  
Another measure of the closed-loop performances can be provided by the time needed to reduce the amplitude of the follower elastic vibration. In particular, Table 3 lists the time needed to reduce the amplitude of the elastic vibration below \(1\times 10^{-4}\) m\({}^{-1}\). With regards to this index, the best performance is obtained, again, by the nonlinear controller, which requires an 88% smaller time to reduce the vibration level below the chosen threshold in comparison with the PD controller. The PD\({}^{\mu}\) controller allows a reduction of the same time over the PD controller equal to 42%. The best performance is obtained by the nonlinear \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Performance Measurement** & \(PD\) & \(PD^{\mu}\) & **Nonlinear Control** \\ \hline Settling time [\(\pm\)5\%] & 2.022 s & 3.685 & 1.455 \\ Vibration settling time [\(\pm 1\times 10^{-4}\) m\({}^{-1}\)] & 0.2775 s & 0.1616 s & 0.0329 s \\ Overshoot [\%] & 19.8\% & 2.8\% & 0.006\% \\ Peak torque [Nm] & 0.157 Nm & 0.2 Nm & 0.088 Nm \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance measurement and comparison between PD, PD\({}^{\mu}\) and nonlinear controller. Figure 9: Tracking error, percentage. controller also in terms of overshoot, which is equal to 0.006% for the nonlinear controller, 2.8% for the PD\({}^{\mu}\) controller and 19.84% for the PD controller. The limited overshoot obtained by the use of the nonlinear controller can be explained by looking at the torque profile generated during the transient, which has a limited peak value. This feature is the main advantage brought by the derivative action included in the nonlinear formulation of the controller, which allows to set the proportional gain to a limited value. In this way, the overshoot is limited, since the bandwidth of the controller is kept high by a rather pronounced value of the \(k_{d}\) gain, which in turn is compensated during the initial part of the transient by the negative value of the \(k_{n}\) gain. Table 3 lists also the peak value of the motor torque, obtained with the use of each control system.