EE 475 Fall 2006 HW #9Due: 11/13/20061. hand sketch the root locus of 1+KG(s)=0 for the following G’s. Describe what you did regarding the X and O marks, the real axis, the asymptotes, the jw-axis crossing, and break away point. Departure/arrival angles recommended also.a. G(s) = (s+2)/{s(s+1)(s+3)(s+10)}b. G(s) = 1/{s(s+1)(s+5)(s+10)}c. G(s) = (s^2+2s+2)/{s^2(s+1)}d. G(s) = (s^2+10s+50)/{s^2(s+1)(s+5)}e. G(s) = (s^2+2s+2)/{s^2(s+5)(s+10)}f. G(s) = (s+2)/{s^2(s+10)(s^2+6s+25)}2. A feedback control system has two blocks in the forward path: C(s) and G(s), and it has a single block in the feedback path H(s), with negative feedback. For each of the following situations, set up the appropriate root locus equation, and use matlab to generate the root locus as the parameter varies from 0 to +infinity.a. C(s)=K, G(s)=(s+3)/{s(s^2+4s+5)}, H(s) = 1/(s+1)b. C(s)=1, G(s)=1/{s^2+(1+a)s +(1+a)}, H(s)=(s+5)/(5s+5)c. C(s)=1, G(s)=(s+b)/{s(s+1)(s^2+8s+52)}, H(s)=(s+5)/(0.5s+5)d. C(s)=K(s+1)/(s+13), G(s)=(s^2+81)/{s^2(s^2+100)}, H(s)=1; Is there a K so that the dominant pole has a damping ratio greater 0.5? If there is, find K so that damping ratio is 0.707.e. C(s)=10, G(s)=1/(s^2+s), H(s)=(Ks+1); find the value of K so that the dominant pole has a damping ratio 0.5.f. C(s)=K/(s+2), G(s)=1/{s(s^2+2s+2)}, H(s) = 1; Find the value of K so thatthe dominant pole has a damping ratio of 1.3. Consider a unity gain feedback control system. The plant transfer function is G(s)=1/(s^2+5s+6). Let the controller be of the form C(s) =K(s+z)/(s+p). Design the controller (ie choose K, z, p>0) so that the closed loop system has poles at -1+-j.4. Assume unity feedback system again. Plant transfer function G(s) = 1/{s(s+3)(s+6)}. The desired specifications are: step response settling time <= 5 sec for 1% tolerance; overshoot <= 17%. Is a proportional controller sufficient? If yes, design it; if not design a different controller to achieve the desired specs.5. In the previous problem, if the settling time must be <= 2 sec. Re-answer the