Bode Plot and Step Response for 4th Order Polynomial

Cuthbert Nyack
Applet shows the Bode plot and step response for a system whose open loop transfer function denominator is a fourth order polynomial.
The magnitude(red) starts with a slope of 0 and ends with a slope of -80dB/decade. The phase(green) starts from zero and ends at -2p. The step response is in pink and the magnitude vs phase plot in yellow. The step response is adjusted to give a final value of 1.
The gain and phase margins are shown on the applet.

for all poles at -1 coeffs are 4, 6, 4, 1;
Increasing K to 4 shows that both the phase and gain margins at 1rad/s and the system is unstable at this frequency.
for all poles at -1.2 coeffs are 4.8, 8.64, 6.912, 2.0736;
Setting the parameters to (8.27, 4.8, 8.65, 6.9, 2.1, 3.0) show the system is oscillatory at ~1.2rad/s. K = 2.12 gives a gain margin of 11.8dB and a phase margin of 163.8°.
for all poles at -1.5 coeffs are 6, 13.5, 13.5, 5.065;
For parameters (8.27, 6.0, 13.5, 13.5, 5.05, 3.0), System is oscillatory at 1.5rad/s with K = 20.27. With K = 5.7, Gain margin = 11dB and phase margin is 124°.

Gif image below shows how applet should appear.

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COPYRIGHT 2006 Cuthbert Nyack.