# The Hurwitz Stability Criterion

The Hurwitz Stability Criterion By: Song Yoong Siang (PhD Student) degree polynomial, of the form shown by Equation (1) is stable if all roots of this polynomial lie in the left half of the complex plane. In this situation, any solution to the linear, homogeneous differential equation will converge to zero. However, it is difficult to determine all the roots if is large. The Hurwitz test provides a necessary and sufficient condition for stability without solving the Equation (2). (1) (2)

The Hurwitz matrix for degree polynomial is describe in Equation (3). is stable if and only if the leading principal minors of are all positive.The leading principal minors, are the determinants of the upper left (1×1), (2×2), . . ., (n×n) submatrices of . Example of leading principal minors are shown in Equation (4) – (6). (3) (4) (5) Reference(s):

1. http://www.systems.caltech.edu/EE/Courses/EE32b/handouts/Hurwitz.pdf
2. http://www.cems.uvm.edu/~tlakoba/08_fall/EE_295/Routh_Hurwitz_Criterion.pdf
3. http://planetmath.org/sites/default/files/texpdf/35395.pdf