ROUTH-HURWITZ METHOD

The problem of the stability of a dynamic system is often reduced to determining whether the real parts of all the roots of a polynomial are negative.

For a polynomial of fourth degree with real-valued coefficients,

alxi + a3x3 + a2x2 + axx + a0 = 0 (an > 0) (1)

Routh* shows that the real parts of all the roots are negative if and only if all the coefficients a0, alt ■ • •, a4, and the discriminant

R = ахаф з — афі — axax2 (2)

are of the same sign. For a polynomial of nth degree, Hurwitz established a criterion in terms of a series of determinants. Hurwitz’s determinants for the fourth-degree equation 1, with a0 > 0, are

These must all be positive if the real parts of all the roots are negative, and vice versa. For a fourth-degree polynomial, determinants of order 1 to 4 are involved. The diagonal from the upper left to lower right contains coefficients with increasing indices beginning with 1. The indices of the coefficients decrease from left to right in each row. Negative indices and indices greater than the degree of the polynomial involved are replaced by zero. These rules are sufficient to establish the determinants for a polynomial of any degree n.

It is easy to show that Routh and Hurwitz conditions are equivalent. The proof for Routh’s conditions for polynomials of third and fourth degrees can be found in Kdrman and Biot’s book;f that for Hurwitz can be found in Uspensky. J The literature related to this algebraic problem is very extensive. In Bateman’s review§ on this subject, more than 100 papers are quoted.

* Routh, Advanced Rigid Dynamics, Vol. II, Macmillan Co., London (1930).

f Mathematical Principles in Engineering, McGraw-Hill, New York (1940).

J Uspensky, Theory of Equations, McGraw-Hill (1948).

§ H. Bateman, The Control of Elastic Fluids, Bull. Am. Math. Soc. 51, 601-646 (1945).