The derivation of Eq. (4.42) to calculate the settling time for a

The derivation of Eq. (4.42) to calculate the settling time for a second order system assumed an underdamped system (ζ < 1). In this problem you will calculate a similar result for a critically damped system (ζ = 1).

a. Show that the unit-step response for a system with transfer function C(s)/R(s) = a²/(s+a)² is c(t) = 1 – e^-at(1 + at).

b. Show that the settling time can be found by solving for T_s in e^-aTs (1 + aT_s) = 0.02.

c. Use MATLAB to plot e^-x(1 + x) = 0.02 vs. x. Use the plot to show that Ts = 5.834/a.