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This Final covers Week 5-8.  Show all required calculations, MATLAB code and MATLAB plots for full credit.

    1. A u nity feedback control system has a loop transfer function:

Gc(s)G(s) = K / ((1 + s/4)(1 + s)(1 + s/20)(1 + s/80))

where K = 10.  Use MATLAB to obtain the Bode diagram of this system.

    1. A closed-loop system has a loop transfer function:

L(s) = Gc(s)G(s) = K / (s(s + 8)(s + 12))

      1. Determine the gain K so that the phase margin is 60 degree. 
      2. For the gain K selected in part (a) determine the gain margin of the system.
    1. An uncompensated control system with unity feedback has a plant transfer function:

G(s) = K / (s(s/2 + 1)(s/6 + 1))

We want to have a velocity error constant of Kv = 20.  We also want to have a phase margin of approximately 45 degrees and a closed-loop bandwidth greater than ω = 4 rad/s.  Use two identical cascaded phase-lead networks to compensate the system.

    1. The transfer function of a plant and a zero-order system shown below is

G(z) = K(z + 0.5) / (z(z – 1))

    1. Use MATLAB to plot the root locus.
    2. Determine the rang e of gain K for a stable system.
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