# Nonlinear Control of Multibody Flexible Mechanisms: A Model-Free Approach
## 4 Control Tuning and Closed-Loop Response  
In this section, the effectiveness of the proposed control law is investigated numerically. The plant used as the test bench is the four-bar linkage described in Section 2. All the tests presented here involve the tracking of a step signal for the angular position of the crank \(q\). The target position is \(q^{*}=\pi/2\) rad, while the initial position of the mechanism is \(q(0)=\pi/6\) rad. The effectiveness of the control system is evaluated on the basis of the precision and speed of tracking of the reference signal, but also in terms of vibration damping. The latter is accomplished by measuring the curvatures at the midspan of the crank and of the follower links. The evaluation of the performance is obtained through a comparison with two other control strategies, which share with the proposed controller the use of a single measurement, i.e., the angular position of the crank, and the model-free approach. Moreover, all three controllers are joint controllers, since they rely on measurements made to the actuated joint, and they do not explicitly take into account the elastic behavior of the mechanism. Fractional order control [41], that in its more general form is referred as \(\text{PI}^{\lambda}\text{D}^{\mu}\) control, is a control system based on fractional calculus, i.e., a generalization of standard calculus that takes into account integrals and derivatives of non-integer order [42]. In particular, a \(\text{PI}^{\lambda}\text{D}^{\mu}\) control will have a standard proportional action, an integral action of order \(\lambda\), and a derivative action of order \(\mu\). For the particular case in which \(\lambda=\mu=1\), the fraction order PID control and the standard PID control are perfectly equivalent. The \(\text{PD}^{\mu}\) control used for the numerical simulation reported in this work has been developed using FOMCON toolbox [43]. Since the exact computation of a fractional order PID control is not possible, being such controller an infinite dimension linear filter [44], a finite size approximation is needed for a real-time implementation. Several approximation techniques have been proposed in the literature [41], the one used in this work is the Oustaloup's approximation [45], which is included in the FOMCON toolbox. Figure 4 shows the behaviour of all three controllers in terms of angular position tracking. From the figure it can be clearly seen that the most accurate tracking, i.e., the one with null overshoot and minimal settling time, is achieved with the nonlinear control. The PD control produces a large overshoot, while \(\text{PD}^{\mu}\) leads to smaller overshoot with respect to the PD control, but at the same time requires a longer settling time. The tuning of the three controllers, whose parameters are reported in Table 2, has been set to achieve similar rise times. Figure 4: Angular position of the crank, comparison with PD and \(\text{PD}^{\mu}\) controllers. A more accurate comparison of the damping capability of the nonlinear controller can be deduced from the results presented in Figure 7, in which the response is plotted over an interval of 0.75 s. This comparison shows again that the nonlinear control has clearly a more pronounced damping effect, and that also the amplitude of vibration at the very beginning of the motion is less pronounced. This behavior can also be explained by looking at the torque profiles reported in Figure 8. From the figure, it can be clearly seen that the PD\({}^{\mu}\) controller produces an intense control action at the beginning of the motion, which induces pronounced vibrations. Furthermore, the PD controller has a very aggressive control effort at the beginning of motion. This is due to the fact that vibration damping in this kind of control requires to set the derivative gain \(k_{d}\) to a quite high value, with the side effect of slowing down also the position tracking.