4. (30 pts.) For the unity feedback system shown in figure (a) below, the root locus sketch is shown in figure (c). Find the following: (a) Asymptotes (b) Breakaway points (c) The range of K for stability (d) The value of K to yield a 1/√2 damping ratio for the dominant second order pairs (e) The settling time T, and overshoot OS % when (-1/√√2 10.0 To improve stability, we desire the root locus to cross the jw-axis at j5.5. To accomplish this, the open-loop function is cascaded with a zero, as shown in figure (b). (e) Find the value of a and sketch the new root locus (f) Find the new range of K for stability. R(s). #2.0 are.o K C(s) re.o (s+1)(s+2)(s+3)(s+4) 85.0 SA.0 R(s). 0 K(s+a) (b) (a) C (s+1)(s+2)(s+3)(s+4) £8.0 C(s) 0.82 0.91 2- 0.975 y de 20 0.7 0 7 -2- 0.975 Y 0.91 6 0.42 0.28 0.14 0.56 Root Locus 10 0.82 0.7 40 -6 фф -8 -7 -6 3 0.56 0.42 0.28 0.14 -4 -3 -2 -1 0 Real Axis (c)