Question

For the 2nd order system given Ts =7 sec and Tp= 3 sec find the value of % OS and location of poles.

Answer 1

$T_s= \frac{4}{\xi w_n}\\ w_n= \frac{0.571}{\xi}\\ \xi= \frac{\pi}{\frac{0.571}{\xi} \sqrt{1-\xi^2}}\\ \sqrt{1- \xi^2}=(\frac{\pi}{3(0.571)}) \xi\\ 1-4363 \xi^2\\ \xi= 0.479\\ \therefore w_n= \frac{4}{7*0.479}= 1.193 rad/sec\\ s^2+(2*0.479*1.193)s+1.193)s+1.193^2=0\\ s= -\frac{1.143±\sqrt{1.143^2-4*1.423}}{2}\\ s=-0.542±j0.903$