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

    3.11 In fig 3.2 a plant transfer function $G(s) = 1/s(s-4)$ represents a tall rocket which is open loop unstable system.
    a) Plot the loci and comment whether the system can be stabilized by chosing suitable gain.
    b) If not, could the unstable pole of $G$ be canceled by zero of $G_{c}$ to stabilize the system, and if not, why not?
    c) Will the system be stable if $G_{c}$ is taken as $G_{c} = \dfrac{K(s+2)}{s+20}$.

    3.12 Motor position servo with rate feedback is shown in 3.3
    a)Sketch the loci and find $K$ for a system with damping ratio 0.5 for the dominating poles.
    b)Find the steady-state error for step and ramp input for K of part(a).

    Servo position control

    Fig 3.3 Servo position control

    3.13 Servo position control (see Fig 3.1) is described with $G(s) = 1/[s(s+3)]$, and $G_{c} = K(s+1)/(s+p)$. Choose the value of $p$ and $K$ such that:
    a) The steady-state unit ramp following error is zero.
    b) The damped natural frequency of oscillatory in the transient response is 4 rad/sec.

    3.14 For a system with a open loop TF $\dfrac{K(s+5)}{s(s+2)(s+3)}$:
    a) Sketch the loci of the closed loop poles for varying $K$. Show that for $K = 8$ the close loop poles are located at $-4, -0.5 \pm 3.12j$.
    b) Sketch the loci to show the effect of variation $\delta$ of the open loop pole at $-2$ on the close loop poles for $K = 8$. Which direction of variation is dangerous?

    3.15 Consider the feedback control system with an close loop block diagram shown in fig. 3.4 where open loop gain $K$ and coefficient $K_{t}$ both are variable. Draw the root locus of the system when both $K$ and $K_t$ vary.

    Feedback system

    Fig 3.4 Feedback control system

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