# Routh Hurwitz criterion and closed loop poles

Cuthbert Nyack
This Applet shows the Routh Hurwitz criterion applied to a system with 3 real poles at -a, -b and -c and how the closed loop poles can be determined by moving the imaginary s axis by the parameter r in the applet.

eg Parameters in table applet
(8.0, 1.0, 1.0, 1.0, r = 0.0), this system is oscillatory for r = 0.0 with poles 0.0 ± 1.732j on the imaginary axis.
changing r to -0.01 shows b1 going -ve and the imaginary s axis is now to the left of the complex conjugate poles( 2 sign changes). When r changes from -3 to -3.01 c1 goes -ve, the number of sign changes increase to 3 and the imaginary s axis is now to the left of the real pole. The closed loop poles of this system are therefore -3.0, 0.0 ± 1.732j.
In this case after the complex conjugate poles r1, r2 have been determined, then the real pole can be calculated as r0 = (abc + K)/(r1r2).
In the first applet the same conclusions are reached with the eg parameters (16.0, 1.0, 1.0, 1.0, r = 0.0). In this case K = 8 corresponds to the middle of the plot. With r = 0.0, b1(green) crosses the zero line at K = 8. With r = -3.0, a1(yellow) is -ve and c1(red) crosses the zero line at K = 8.

eg (4.0, 1.0, 1.0, 1.0, r = 0.0), this system has closed loop poles at ~ -0.21 ± 1.376j, and ~ -2.59.

eg (2.0, 0.1, 2.5, 2.5, r = 0.0), this system has 3 real closed loop poles at ~ -0.77, ~ -1.05, ~ -3.3.

Gif images below shows how applet should appear.  