SYSTEM DYNAMICS AND CONTROL THEORY HOMEWORK #$2$

$($Due date: Dec $14$th$,2024-23$:$59-$ on MS Teams group of SysDyn&Cont. Course$)$

A closed$-$loop control system is to be designed such that its response is underdamped. Performance specifications for step response are as follows:

$10\%P.O.30\%$

$t_{s}0\mathrm{.}4s$

a$)$ Find the region of system's roots in the complex plane.

b$)$ What should be the minimum value of a third root $(p_3),$ if complex conjugate roots are dominant?

$2.$ A system is described by the following transfer function:

$G(s)=\frac{{}_n^2}{s^2+2{}_{n}s+{}_n^2}$

a$)$ Determine the region of poles in the complex plane if:

i$)$ Percent overshoot is to be less than $4\mathrm{.}32\%,$

ii$)$ Settling time based on $2\%$ criterion is to be less than $2$ seconds.

b$)$ Now, in addition to the above requirements, the following conditions are also required:

i$){}_n=4ra\frac{d}{s}$

ii$)t_p=3\mathrm{.}14s$

Is it possible to satisfy all four conditions together? If it is possible, where would the poles be located in the complex plane.