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    Assignment 3: Root locus of a physical system

    3.1 Plot the loci of the closed loop system with the following open loop TF for varying K from $0$ to $\infty$. Determine the range of $K$ for which the closed loop system is stable. If any break away or break in happen, determine the value of $K$ at that point.

    a) $K/s\quad $ b) $K/(s+1)\quad$ c) $K/(s-1)\quad$ d) $K/s^2\quad$ e) $K/(s^2+4) \quad$

    f) $\dfrac{K}{s^{2}-4} \quad$ g) $\dfrac{K}{s(s^{2}+2s+2)} \quad$ h) $\dfrac{K(s+2)}{s(s+3)}\quad$

    i)$\dfrac{K}{(s^{2}+2s+2)(s^{2}+6s+10)} \quad$ j)$\dfrac{K(s+2)}{(s+1)(s^{2}+6s+11.25)}\quad$

    3.2 Sketch the general shape of the loci of the open loop system given by the TF $\dfrac{K(s+1)}{s^{2}}$. Use the angle condition to show that the locus contains a circle of unity radius centered at -1.

    3.3 For the system shown in Fig. 3.1 with $H = 1,G = \dfrac{K(s+1)}{s(s+2)}$:
    a) Draw the root loci.
    b) Use them to find $K$ for a 2 sec time constant of the dominating closed loop pole.
    c) Find the other system pole either from root locus or analytically.
    d) Calculate $c(t)$, for step input. Also find the steady-state errors for unit step and unit ramp input.

    Feedback system

    Fig 3.1 Feedback system

    3.4 Fig 3.3 shows a unity feedback system with controller. System TF $G(s) = \dfrac{1}{s(s+1)}$, sketch and compare the loci for a) ${G}_c = K \quad$ b) ${G}_c = \dfrac{K}{(s+2)}\quad$ c) ${G}_c = K(s+2)$. Try to realize the effect of adding a pole or a zero to the open loop pole-zero pattern. How these addition affects the stability?

    Feedback system with controller

    Fig 3.2 Feedback system with controller

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