Parabolic trough collector. The fluid temperature of a

QUESTION:

Parabolic trough collector. The fluid temperature of a parabolic trough collector (Camacho, 2012) will be controlled by using a unity feedback structure as shown in Figure P6.11. Assume the open-loop plant transfer function is given by

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

Use the Routh-Hurwitz criteria to find the range of gain K that will result in a closed-loop stable system. Note: Pure time-delay dynamics, such as the one in the transfer function of the parabolic trough collector, cannot be treated directly using the Routh-Hurwitz criterion because it is represented by a nonrational factor. However, a Padé approximation can be used for the nonrational component. The Padé approximation was introduced in Problem 6.61, but it can appear in different forms. Here, it is suggested you use a first-order approximation of the form

\(e^{-s T} \approx \frac{1-\frac{T}{2} s}{1+\frac{T}{2} s}\)