Problem 1. For the feedback control system given, where R(s) C(s) G.(s) G,(s) 1 G.(s) = K(s + 10) and Gp(s): (s + 1)(s + 2)(s + 6) (a) Sketch the root locus. When sketching the root locus, if necessary, make use of the asymptotes finding oa and Oa that are the intersecting point and angles with the real axis, respectively, with the following formula, Efinite poles-E finite zeros #finite poles-#finite pzeros (2k+1)r and a = #finite poles-Hfinite pzeros where k = 0,1, +2, .. (b) If applies, what are the values of the gain and poles at the imaginary axis crossings? (c) Write the range of the gain making the system stable? (d) It is determined that the dominant poles are at s1,2 = -1+ j3.5 when the gain is K = 8. Find the location of the third pole for this value of gain. %3D (e) What is the system type? () Find the steady-state errors for the same gain, K = 8, when r(t) = 0.6u(t) step and 0.6tu(t) ramp inputs are applied, respectively. (g) In order to improve the steady state response of the system add a PI controller having the same gain and a zero of s = -0.2. Now find the steady-state errors for the same inputs.