Applet below show how the closed loop poles can be approximately found by locating the zeros of the characteristic eqn. In the applet the magnitude of the characteristic equation is plotted as a contour plot with the lowest magnitudes being white and black. The resolution of the minimum can be changed by changing rs.

To find the closed loop for K = 7 for a system with open loop poles at -0.5, -1.0, 1.5 and open loop zero at -3.0, set the following parameters.

(0.5, 1.0, 1.5, 3.0, 7.0, sr, si, 6) then adjust sr, si to find where the angle condition is satisfied. For sr = -0.37, si = 1.82; K = 7.06 and f = -180.3°. For sr = -2.266, si = 0.0; K = ~ 7 and f = -540°.

The CL poles are ~-2.266, ~-0.37 ± ~j1.82. In this plot the location of the minimum is limited but it is in general agreement with that obtained from sr, si.

Gif image showing how applet should appear is shown below.

COPYRIGHT © 2006 Cuthbert Nyack.