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    Assignment 3: Root locus of a physical system (cont'd...)

    3.5 Fig.3.2 models a pressure control system with a plant $G = \dfrac{1}{(s+1)(s+3)}$ and a controller ${G}_c = \dfrac{K(s+2)}{s}$.
    a) Sketch the loci of the close-loop system poles for varying $K$.
    b) Find, reasonably accurately, the value of K for a damping ratio 0.5 for the dominating pair of poles.

    3.6 Draw the Root loci of a robotic manipulator with a flexible joint, whose open loop TF is given by $G(s)H(s) = \dfrac{K}{s(0.1s^2+0.4s+0.5)}$.
    a) How does the nature of the transient response, in terms of time constant and damping ratios changes as K is increased?
    b) What is the limiting value of K for stability?

    3.7 For a unstable system with loop TF, $G(s)H(s) = \dfrac{K(s+1)}{s(s-1)}$
    a) sketch the loci of the system poles and find K for a system time constant of 1 sec.

    3.8 For a closed loop system shown in fig 3.1, $G(s) = \dfrac{4K}{(0.5s+3)(2s^2+8s+12)} \quad$ and $H(s) = \dfrac{0.1}{(0.05s+0.4)}$.

    a) Sketch the loci of the close loop poles for varying K.
    b) Find the approximate value of K so that the damping ratio of the dominating closed loop poles will be 0.5.

    3.9 For a unity negative feedback system with $G = \dfrac{K}{s(s^2+2s+5)}$:
    a) sketch the loci of the closed loop poles.
    b) Find the value of K for which the time constant of the dominating pair of closed loop poles is 2 sec.
    c) Determine the position of the third pole.
    d) Calculate the unit step response of the approximated system.

    3.10 Plot the loci for a system with open loop TF $G(s)H(s) = \dfrac{K}{s(s+4)}$ and use them to find the lowest value of $K$ for which the settling time of the system will be minimum. Also, comment on the change of nature of transient response with the increase of $K$.


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