Applet below show the Bode plot and step response of a system with 4 real poles and 2 complex conjugate zeros.

Magnitude(red) starts with slope 0 and ends with slope -40dB. Phase(green) starts at 0 and ends at -p. In between it may go below -p.

Step response is in pink and magnitude vs phase is in yellow.

One of the characteristics of this system is that it may be stable at low and high frequencies but unstable in between. This can happen when the phase dips below -p and bounces back to go above -p.

For the eg parameters (K, 1.5, 1.5, 1.0, 0.1, 1.2, 2.0) system is stable for K < ~1.75 and > 7.7 but unstable for intermediate values of K.

Changing the position of the zeros can give an acceptable step response eg (10.0, 1.0, 1.5, 1.0, 0.1, 0.5, 0.7) has good step response with a phase margin of ~60°.

For eg parameters (K, 1.5, 1.5, 1.0, 0.1, 2.5, 1.0), the system is stable for K < 1.75 and unstable for larger K. Placing the zeros close to the imaginary axis can produce a loop in the magnitude vs phase plot. (30.0, 1.0, 1.5, 1.0, 0.1, 0.05, 1.15).

Gif image below shows how applet should appear.

COPYRIGHT © 2006 Cuthbert Nyack.