(e) construct the Root Loci for -∞ ≤k ≤∞.

(a) Find the transfer function of the system.

(b) Is the system of minimum phase, and why?

2. Design a feedback controller to regulate the output of the plantto any prescribed constant level without any steady-state error, and make the resulting closed-loop polesat s =-70,-100.

3. Given a controller C(s)=(s-1)/(s+1) and a plant P(s)=s/[(s-I1)(s+2)] in negative feedback, determine the stability of the feedback system.

(a) open loop pole(s) and zero(s)；

(b) intersection of the asymptotes (Centroid)；

(c) angles of asymptotes；

(d)intersection of the Root Loci with the imaginary axis and the value of k, then

(a) A Nyquist path is shown in Fig. 1.Sketch the Nyquist plot of this system for K ＞0 in accordance with fig. 1, and indicate clearly I. theintersection(s) of the Nyquist plot with real and imaginary axes if any II. each mapping of pointa:(0+jω,ω→∞⁺),6:(0+jω,ω→0⁺),c:(,φ→0⁺)and d:(0-jω,ω→∞⁺) in yourplot.

(b) Suppose that the Nyquist plot ( K ＞0) you obtained in part (a) is depicted in Fig. 2, where thesolid line is part of the Nyquist plot when ω is from ∞ ⁺ to 0⁺, and it has an intersection with axis-Re(L(s)) at -K.Using Nyquist criterion, determine the range of K ＞ 0 such that the closed-loop system can be stabilized.If the system is unstable due to the range of K, find the number of closed-loop pole(s) in the right-half ofs-plane.

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