The parabolic trough collector (Camacho, 2012) is a Type 0 system as can be seen from its transfer function,

We want the system operating in the critically damped mode, but with reduced steady-state error. Using the root locus and a Padé approximation,

do the following:

a. Substitute the Padé approximation for the delay and find the gain necessary to have the system operating with a damping factor, ζ = 0.5. Also, find the corresponding steady-state error.

b. Design a PI compensator to obtain a zero steady-state error while maintaining ζ = 0.5.

c. Simulate the resulting design using MATLAB to verify your design.

137.2 x 10-6 K G(s) = -39s %3D s2 + 0.0224s + 196 x 10-6