Parabolic trough collector. Consider the fluid temperature

Chapter 8, Problem 72

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Parabolic trough collector. Consider the fluid temperature control of a parabolic trough collector (Camocho, 2012) embedded in the unity feedback structure as shown in Figure P8.3, where the open-loop plant transfer function is given by

G(s) = \(\frac{137.2 \times 10^{-6} K}{s^2+0.0224 s+196 \times 10^{-6}} e^{-39 s}\)

Approximating the time-delay term with \(e^{-s T} \approx \frac{1-\frac{T}{2} s}{1+\frac{T}{2} s}\), make a sketch of the resulting root locus (Note: After substituting the approximation, \(G(\infty)\) < 0, the positive feedback rules of Section 8.9 must be used). Mark where appropriate in the plot and find:

a. The asymptotes and their intersection with the real axis;

b. The break-in and breakaway points. (The procedures presented in Section 8.5 are also valid for positive feedback systems);

c. The range of K for closed-loop stability;

d. The value of K that will make the system oscillate and the oscillation frequency.

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