The peak time of a second order system with a step input can be easily calculated as shown below. Is it possible to do the same when input is a rectangular pulse?
To explain, A second order system can be represented as: $$\frac{C(s)}{R(s)} = \frac{\omega^{2}_n}{s (s^2 + 2\zeta\omega_n s + \omega^2_n)}$$ The system is underdamped, so the time domain equation is: $$c(t) = 1- \frac{e^{-\zeta\omega_n t}}{\sqrt{1-\zeta^2}} \sin(\omega_n\sqrt{ (1-\zeta^2)} \cdot t + \phi)$$ At peak response, $$\frac{dc(t)}{dt} = 0$$ $$\frac{e^{-\zeta \omega_n t}}{\sqrt{1-\zeta^2}} \big[- \omega_n \sqrt{1-\zeta^2}\cdot \cos(\omega_n \sqrt{1-\zeta^2}t + \phi)+ \zeta \omega_n \sin (\omega_n \sqrt{1-\zeta^2}t + \phi)\big] = 0 $$ $$\tan[\omega_n \sqrt{1-\zeta^2}t + \phi] = \frac{\sqrt{1-\zeta^2}}{\zeta} = \tan\phi $$ $$[\omega_n \sqrt{1-\zeta^2}t + \phi] = n \pi$$ for n = 1,2,3,4... $$t_{peak} = \frac{\pi}{\omega_n \sqrt{1-\zeta^2}}$$
I would like to derive the same way but with rectangular pulse input, $$\frac{C(s)}{R(s)} = \bigg[\frac{e^{-as} - e^{-bs}}{s}\bigg] \frac{\omega^{2}_n}{(s^2 + 2\zeta\omega_n s + \omega^2_n)} $$ where a and b are the start and end time of the pulse. In the rectagular pulse case, it seems difficult to separate time variable from others with the exponential term staying on the left hand side. Is there a method or reference I can use?
