The Applet below show the Bode Plot of

(1) a second order type 0 system.(gray blue, cyan)

G(s) = K/((T1s + 1)(T2s + 1)), C(s)/R(s) = K/(((T1s + 1)(T2s + 1)) + K))

(2) a second order type 0 system with a zero to the left of the imaginary axis.(orange, green)

G(s) = K(1 + Tns)/((T1s + 1)(T2s + 1))), C(s)/R(s) = K(1 + Tns)/(((T1s + 1)(T2s + 1)) + K(1 + Tns))

(3) a second order type 0 system with a zero to the right of the imaginary axis(nonminimum phase).(orange, red)

G(s) = K(1 - Tns)/(((T1s + 1)(T2s + 1))), C(s)/R(s) = K(1 - Tns)/(((T1s + 1)(T2s + 1)) + K(1 - Tns))

Comparison of the magnitude plots (orange, gray blue) show that a zero has the same effect regardless of whether it is to the left or right of the imaginary axis ie a break with a 20dB/decade reduction in slope at angular frequency 1/Tn.

Comparison of the phase plots (red, cyan, green) show that a zero to the left of the imaginary axis reduces the phase lag, while one to the right increases it.

Comparison of the step response plots (green, peach, pink) show that a zero to the left reduces the transient oscillations, while one to the right enhances it.

The eg parameters (5.0, 1.0, 2.0, -0.1, 1.0, 1.0) show a break in the magnitude plot, a phase advance of +45° for the zero to the left and a phase lag of 45° for the zero to the right all at 10rad/s. Increasing K to 30.0 brings the nonminimum phase system to instability. With these parameters (30.0, 1.0, 2.0, -0.1, 1.0, 1.0) the nonminimum phase system is oscillatory with period ~ 1.6s and angular frequency ~ 3.9rad/s. The Magnitude and phase plots show that the gain and phase margins for the nonminimum phase system are zero at this frequency.

Effect of the nonminimum phase zero on the stability of the system is illustrated by the following eg parameters:-

(37.5, 1.0, 2.0, -0.08, 1.0, 1.0)

(25.0, 1.0, 2.0, -0.12, 1.0, 1.0)

(18.7, 1.0, 2.0, -0.16, 1.0, 1.0)

(9.4, 1.0, 2.0, -0.32, 1.0, 1.0)

(4.7, 1.0, 2.0, -0.64, 1.0, 1.0)

(2.4, 1.0, 2.0, -1.25, 1.0, 1.0)

The second applet below show the closed loop response equivalent to the first applet. Comparison of the closed loop plots show that a zero to the left reduces the peak in the response while a zero on the right makes the peak higher.

Gif images below shows how applet should appear.

COPYRIGHT © 2006 Cuthbert Nyack.