Routh Hurwitz criterion and closed loop poles, third order poly

Cuthbert Nyack

This Applet shows the Routh Hurwitz criterion applied to a system with a cubic characteristic equation s³ + a1 s² + a2 s + a3 + K = 0 and how the closed loop poles can be determined by moving the imaginary s axis by the parameter r in the applet.

eg Parameters
(1.0, 3.0, 3.0, 1.0, r = 0.0), this system is stable for r = 0.0.
changing r to -0.5 shows b1 ~ = 0 and for r = -0.51, b1 is -ve.
When b1 goes -ve there are 2 sign changes in the first column. this means that as r changes from -0.5 to -0.51 it passes over 2 complex conjugate poles. The real part of the pole is -0.5 and the imaginary part can be devived from the auxiliary equation 1.5 w² = 1.125. with w = 0.866. w can also be calculated from the period of oscillation.

Making r more -ve shows that c1 = 0 when r = -2.0 and c1 goes -ve when r = -2.01. When c1 goes -ve there are now 3 sign changes in the first column and there are now 3 poles to the right of the moved imaginary s axis.

Summarising we get 3 closed loop poles at -2.0, -0.5 ± 0.866j

Consider the case (1.0, 3.0, 4.0, 1.0, r = 0.0), this is also stable but when r = -1, both a1 and c1 ~ 0.0 and both go -ve when r = -1.01. Using the auxiliary equation gives s³ + s = 0 with s = 0, ±j. This means that all the poles have the same real part and the 3 closed loop poles are -1.0, -1.0 ± j.

Consider the case (1.0, 3.0, 5.0, 1.0, r = 0.0), this is also stable but when r changes from -0.54 to -0.55 c1 goes -ve. When c1 goes -ve there is only one sign change in the first column and there there is only 1 real root to the right of the moved imaginary s axis. When r changes from -1.22 to -1.23 the number of sign changes moves from 1 to 3 (a1, c1 -ve) . Using the auxiliary equation gives 0.68s² + 1.47 = 0 with s = ±j1.47. This means that the 3 closed loop poles are ~-0.54, ~-1.23 ± ~j1.47.

Consider the case (1.0, 4.0, 5.0, 0.95, r = 0.0), this is also stable but when r changes from -0.79 to -0.80 c1 goes -ve. When c1 goes -ve there is only one sign change in the first column and there there is only 1 real root to the right of the moved imaginary s axis. When r changes from -1.25 to -1.26 the number of sign changes moves from 1 to 2 (b1 -ve). This means that there are now 2 poles to the right of the moved imaginary axis. When r changes from -1.94 to -1.95 the number of sign changes increases from 2 to 3 (a1, c1 -ve) and thare are now 3 poles to the right of the moved imaginary s axis.
Therefore the 3 closed loop poles are all real and are at ~-0.8, ~-1.26 and ~-1.94.

The applet below show the same thing. This time the H axis is r.
For the parameters (8.0, 3.0, 3.0, 1.0, 1.0) the H axis goes from r = -4 to r = +1. The number of sign changes change at r = 0 and r = 3. The CC pair of poles has real part 0 and the real pole is at -3.

Gif images below shows how applet should appear.

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