## 5 Robot Propulsion
### Control Design
On this basis, a non-integer integrator is also considered for comparison purposes with the form:  
\[C_{\text{I}}(s)=\frac{K}{s^{\lambda}} \tag{13}\]  
where \(\lambda\in(0,1]\) is its order. Unlike PID, the parameters of fractional integrator are calculated analytically from the design specifications, which results in  
\[C_{\text{I}}(s)=\frac{15.62}{s^{0.62}} \tag{14}\]  
Moreover, this controller was approximated by the Oustaloup method with four poles and four zeros in the frequency range \([0.001,100]\) rad/s as follows:  
\[C_{\text{I}}^{*}(s)=\frac{32.08s^{4}+1981s^{3}+6511s^{2}+1196s+11.7}{36.47s^{ 4}+372.9s^{3}+202.9s^{2}+6.175s+0.01} \tag{15}\]  
The Bode plots of the three designed controllers, namely \(C_{\text{PID}}(s)\), \(C_{\text{I}}(s)\), and \(C_{\text{I}}^{*}(s)\), are shown in \[L_{cl}(s)=\frac{1}{\left(s/\omega_{c}\right)^{\gamma}+1} \tag{9}\]0 As can be seen, the PID controller provides a phase of \(44^{\circ}\) at 5 rad/s and a positive gain to the system at relatively high frequencies, which will increase the speed of the system, but at the same time, the crossover frequency specification will not be met. In contrast, the non-integer integrator has a bit higher phase at 5 rad/s, namely \(56^{\circ}\), and a positive gain of 15 dB, so the design specifications can be better fulfilled. As for the approximation of the fractional integrator, it should be noted that it performs very close to \(C_{\text{I}}(j\omega)\), with only a small variation in the phase from the desired \(\omega_{cg}\) being observed. Figure 10 illustrates Bode plots of the open-loop system when applying the designed controllers and the approximation. From these frequency responses, the first remark that can lay bare is that the PID controller does not meet the design specifications: as it was commented, the controller increases the speed of the system (\(\omega_{cg}=38\) rad/s) and the phase margin is \(75^{\circ}\). The reason for the frequency response mismatch is due to the fact that the optimization process was approached from a time perspective, to adjust the time response of the system for a step input. On the contrary, the system fulfills the design specifications perfectly when applying a fractional order integrator. Furthermore, the phase is flat at crossover frequency and almost constant within an interval around it. In other words, the system is robust to gain variations and the overshoot of the response will be almost constant close to \(\omega_{cg}\). The issues mentioned above can be also observed from the results plotted in Figure 11, which illustrates the step response of the reference model and that corresponding to each link of the robot when applying the PID (see Figure 11a) and the fractional integrator and its approximation (see Figure 11b). As far as the PID controller is concerned, the response for the third link shows a good fit with respect to the reference model response. However, the characteristics of this behavior do not hold for the other links: the overshoot increases as the gain decreases, although the system is faster for the second and third links compared to the other control strategy. This behavior is not observed in the system for the fractional integrator, which also approximates the step response of the reference model and, in the case of applying the controller to other links, the overshoot is maintained, ensuring robustness to gain variations. Regarding the integrator approach, the responses show small differences with respect to the ideal case, with the system velocity and overshoot remaining unchanged for all three links. \[L_{cl}(s)=\frac{1}{\left(s/\omega_{c}\right)^{\gamma}+1} \tag{9}\]1 Frequency response of the reference model and the controlled system in open-loop. \[L_{cl}(s)=\frac{1}{\left(s/\omega_{c}\right)^{\gamma}+1} \tag{9}\]2 Frequency response of the designed integer and fractional order controllers.