Stability Conditions for Second Order Systems

Result:
The roots of a (strictly) second order polynomial will have negative real parts if and only if all the coefficients are of the same sign.
Proof:
For simplicity consider the polynomial
s2 + 2b s + c = 0
The roots are given by For the first case, take c >b2 . The roots are complex where the real parts are given by -b. ( If c=b2, the roots are both given by -b. )
For the second case, take b2 > c > 0. The roots are real. But since c > 0, both roots will be negative if and only if b>0 because . ( If c=0, one of the roots is given by 0 while the other is given by -2b ).

The last case is given by c<0. At least one of the roots will be positive:

1. if b>0, one of the roots is given by 1. if b<0, one of the roots is given by 1. if b=0, one of the roots is given by To summarize, we need c>0 and b>0 in order for both roots to have negative real parts. Also, if c>0 and b>0, we are guaranteed to have roots with negative real parts.
(An alternative proof will be to use the Routh-Hurwitz method.)

This page is maintained by Tomas B. Co (tbco@mtu.edu). Last revised 12/14/99.

Tomas B. Co
Associate Professor
Department of Chemical Engineering
Michigan Technological University
1400 Townsend Avenue
Houghton, MI 49931-1295