# Routh Hurwitz Criterion for 4 Real Poles

Cuthbert Nyack
This Applet shows the Routh Hurwitz criterion applied to a system with 4 real poles. They allow the effect of gain and pole locations on the stability of the system to be studied. The step response of the system is also shown.

The first Applet shows the variation of the elements in the first column of the RH array as a function of gain from 0 to K
The second shows the more conventional array with the step response added. When the system is oscillatory the frequency can be calculated from the auxiliary equation or measured from the step response.

some eg parameters and their stability are shown below:-

With this system there are 2 asymptotes at ±45° so the system will always become unstable for large enough K.

eg parameters for first applet (10.0, 1.0, 1.0, 1.0, 1.0), the red line corresponding to the c1 coefficient goes -ve at K = 4 showing that the system is unstable for K > 4.0.
In the 2nd applet the eg parameters (K, 1.0, 1.0, 1.0, 1.0) show that c1 is effectively zero( greater than zero because of the small parameter added to b1) and the step response is oscillatory at K = 4.0. Increasing K beyond 4 makes c1 -ve. When c1 is zero the previous row show that when K = 4.0 the system is oscillatory with angular frequency w = 1 rad/s.

eg parameters (K, 0.5, 0.5, 1.0, 1.0) system is unstable for K > ~ 1.125.
eg parameters (K, 1.5, 1.5, 1.0, 1.0) system is unstable for K > ~ 9.375. eg parameters (20.0, 2.0, 2.0, 2.0, -0.3) system is unstable for K < ~ 2.45 and for K > 9.65.

Gif images below shows how applet should appear.