Question

My friend Rodrigo from Escola de Engenharia Mauá in Brazil has been experimenting with a twin–rotor aerodynamic propulsion system. He ran a sequence of open–loop frequency response experiments and obtained the data in Table 1. For ease of data input, the measurements are given in Real–Imaginary format instead of more usual magnitude in dB–phase.

1. Plot the points and sketch an approximate Bode plot for the TWR. Hint: the process is open–loop stable and minimum–phase, that is, all its poles and zeros have negative real part.

2. Obtain a transfer function model for the system. Use any numerical fitting techniques or Matlab toolboxes you want, or just trial–and-error by visual inspection of the graph.

3. Trace the root–locus of the model and its Nyquist plot.

4. Using the root–locus, the Nyquist criterion, and simulations, discuss in qualitative terms the stability and performance of the system in closed–loop.

5. Suggest a possible control scheme to improve the performance: PID, lead–lag compensator, pole–assignment, whatever makes sense. More than one solution is possible.

6. Design and analyze the behavior of the system that incorporates your controller. If necessary change the parameters of your previous design, or pick a different controller structure. 7. You don’t really think that the model built using experimental data is very reliable, do you? In fact the real system is nonlinear, multivariable, and controlled in discrete time, as you suspect, though we are going to stick with classical techniques. Change the numbers in the response values a little bit, and check if the controller you designed still works.

Frequency (rad/s) Response (Re+Im
0.10 5.00−0.09i
0.50 4.99−0.94i
1.00 1.78−7.18i
2.00 −1.81+0.28i
3.00 −0.13+0.11i
5.00 −0.02+0.01i
6.00 −0.00−0.02i
8.00 0.00−0.02i
10.00 0.00−0.01i
20.00 0.00−0.01i
​













​

Answer 1

The student is tasked with analyzing frequency response data to design and analyze a control system, including creating Bode plots, deriving transfer functions, plotting root locus and Nyquist curves, discussing stability, designing a control scheme, and considering the effect of system nonlinearity on the controller's performance.

Step-by-step explanation:

The question asks for the analysis of an experimental data set to create a Bode plot, derive a transfer function model, sketch the root locus and Nyquist plot, discuss stability and performance in closed-loop, suggest a control improvement scheme, and analyze the effect of system nonlinearity on the designed controller. Given that the real system is known to be nonlinear, multivariable, and controlled in discrete time, classical control techniques will still be applied for educational purposes, keeping in mind that the results may not fully capture the dynamics of the actual system.

To address the tasks: one would begin by plotting the given frequency response data points and sketching a Bode plot. Next, a transfer function model can be obtained using numerical techniques such as MATLAB's System Identification Toolbox or fitting methods. After that, the root locus and Nyquist plots are sketched to analyze the system's stability and performance in closed-loop. To improve performance, a control scheme such as a PID controller or a lead-lag compensator might be suggested, and the controller would be designed accordingly. Finally, to verify the robustness of the designed controller, slight variations in the system's response values would be tested to observe the impact on the system's performance.