The design of cruise control systems in heavy vehicles, such as big rigs, is especially challenging due to the extreme variations in payload. Assume that the frequency response for the transfer function from fuel mass flow to vehicle speed is shown in Figure P10.13.

This response includes the dynamics of the engine, the gear box, the propulsion shaft, the differential, the drive shafts, the chassis, the payload, and tire dynamics. Assume that the system is controlled in a closed-loop, unity-feedback loop using a proportional compensator (van der Zalm, 2008).

a. Make a plot of the Nyquist diagram that corresponds to the Bode plot of Figure P10.13.

b. Assuming there are no open-loop poles in the right half-plane, find out if the system is closed-loop stable when the proportional gain K = 1.

c. Find the range of positive K for which the system is closed-loop stable.

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-10 -20 -30 -40 -50 -60 90 45 -45 -90 106 105 10-4 103 102 10 10 10! 10 103 1o Frequency (rad/s) FIGURE P10.13 Phase (deg) Magnitude (dB)