Parabolic trough collector. As discussed in Section 10.12,

QUESTION:

In Problem 71, MATLAB Chapter 8, we used MATLAB to plot the root locus for the speed control of an HEV rearranged as a unity-feedback system, as shown in Figure P7. 31 (Preitl, 2007). The plant and compensator were given by

G(s) = \(\frac{K(s+0.6)}{(s+0.5858)(s+0.0163)}\)

and we found that K = 0.78, resulted in a critically damped system.

a. Use MATLAB or any other program to plot the following:

i. The Bode magnitude and phase plots for that system, and

ii. The response of the system, c(t), to a step input, r(t) = 4 u(t). Note on the c(t) curve the rise time, \(T_r\), and settling time, \(T_s\), as well as the final value of the output.

b. Now add an integral gain to the controller, such that the plant and compensator transfer function becomes

G(s) = \(\frac{K_1\left(s+Z_c\right)(s+0.6)}{s(s+0.5858)(s+0.0163)}\)

where \(K_1=0.78\) and \(Z_{c}=\frac{K_{2}}{K_{1}}=0.4\). Use MATLAB or any other program to do the following:

i. Plot the Bode magnitude and phase plots for this case.

ii. Obtain the response of the system to a step input, r(t) = 4 u(t). Plot c(t) and note on it the rise time, \(T_r\), percent overshoot, %OS, peak time, \(T_p\), and settling time, \(T_s\).

c. Does the response obtained in Parts a or b resemble a second-order overdamped, critically damped, or underdamped response? Explain.