Question8. Problem 9. For a similar PD controlled positioning system given in the previous problem, where Ge(s) = K(s+ 4) and Gp(s) = 1(»+2}(s*+10s+41) sketch the root-locus of the system for K = 1 by means of the computations requested below. A tyipical cascaded control system is presented in the figure below. (a) When sketching the root locus, if necessary, make use of the asymptotes finding o, and 0, that are the intersecting point and angles with the real axis, respectively with the following formula. Efinite poles-E finite zeros and e,= efinite poles-finite pzeros (2k+1)m efinite poles-efinite pzeros -where k = 0,11, t2, . (b) If the root locus intersects the jes-axis, find the values of poles at crossing points, the value of gain Kat the crossings points and write the range of gain K making the system stable. (c) If there are complex poles find the angle of departure. (d) Find the breakaway and break-in points if they exist. (e) Plot the root-locus of the system for the positive values of gain in MATLAB. () Repeat the same steps for a PID controlled positioning system that has the following transfer functions. Ge(s) = KG+X+0.5) and Gp(s) =; (s+2)(s²+18s+41) R(s) as) Ge(s) G,(s)

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