Routh–Hurwitz theorem

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In mathematics, the Routh–Hurwitz theorem gives a test to determine whether all roots of a given polynomial lie in the left-half complex plane. Polynomials with this property are called Hurwitz stable polynomials. The Routh–Hurwitz theorem is important in dynamical systems and control theory, because the characteristic polynomial of the differential equations of a stable, linear system has roots limited to the left half plane (negative eigenvalues). Thus the theorem provides a mathematical test, the Routh–Hurwitz stability criterion, to determine whether a linear dynamical system is stable without solving the system. The Routh–Hurwitz theorem was proved in 1895, and it was named after Edward John Routh and Adolf Hurwitz.

Notations

Let $f (z)$ be a polynomial (with complex coefficients) of degree $n$ with no roots on the imaginary axis (i.e. the line $z = ic$ where $i$ is the imaginary unit and $c$ is a real number). Let us define real polynomials $P 0 (y)$ and $P 1 (y)$ by $f (iy) = P 0 (y) + iP 1 (y)$ , respectively the real and imaginary parts of $f$ on the imaginary line.

Furthermore, let us denote by:

$p$ the number of roots of $f$ in the left half-plane (taking into account multiplicities);
$q$ the number of roots of $f$ in the right half-plane (taking into account multiplicities);
$Δ arg f (iy)$ the variation of the argument of $f (iy)$ when $y$ runs from $-\infty$ to $+\infty$ ;
$w (x)$ is the number of variations of the generalized Sturm chain obtained from $P 0 (y)$ and $P 1 (y)$ by applying the Euclidean algorithm;
$I +\infty -\infty r$ is the Cauchy index of the rational function $r$ over the real line.

Statement

With the notations introduced above, the Routh–Hurwitz theorem states that:

p-q={\frac {1}{\pi }}\Delta \arg f(iy)=\left.{\begin{cases}+I_{-\infty }^{+\infty }{\frac {P_{0}(y)}{P_{1}(y)}}&{\text{for odd degree}}\\[10pt]-I_{-\infty }^{+\infty }{\frac {P_{1}(y)}{P_{0}(y)}}&{\text{for even degree}}\end{cases}}\right\}=w(+\infty )-w(-\infty ).

From the first equality we can for instance conclude that when the variation of the argument of $f (iy)$ is positive, then $f (z)$ will have more roots to the left of the imaginary axis than to its right. The equality $p - q = w (+\infty) - w (-\infty)$ can be viewed as the complex counterpart of Sturm's theorem. Note the differences: in Sturm's theorem, the left member is $p + q$ and the $w$ from the right member is the number of variations of a Sturm chain (while $w$ refers to a generalized Sturm chain in the present theorem).

Routh–Hurwitz stability criterion

We can easily determine a stability criterion using this theorem as it is trivial that $f (z)$ is Hurwitz-stable if and only if $p - q = n$ . We thus obtain conditions on the coefficients of $f (z)$ by imposing $w (+\infty) = n$ and $w (-\infty) = 0$ .

References

Routh, E. J. (1877). A Treatise on the Stability of a Given State of Motion, Particularly Steady Motion. Macmillan and co.
Hurwitz, A. (1964). "On The Conditions Under Which An Equation Has Only Roots With Negative Real Parts". In Bellman, Richard; Kalaba, Robert E. (eds.). Selected Papers on Mathematical Trends in Control Theory. New York: Dover.
Gantmacher, F. R. (2005) [1959]. Applications of the Theory of Matrices. New York: Dover. pp. 226–233. ISBN 0-486-44554-2.
Rahman, Q. I.; Schmeisser, G. (2002). Analytic theory of polynomials. London Mathematical Society Monographs. New Series. Vol. 26. Oxford: Oxford University Press. ISBN 0-19-853493-0. Zbl 1072.30006.
Explaining the Routh–Hurwitz Criterion (2020)^[1]

External links

Mathworld entry

^ Bodson, Marc (February 2020). "Explaining the Routh–Hurwitz Criterion: A Tutorial Presentation [Focus on Education]". IEEE Control Systems Magazine. 40 (1): 45–51. doi:10.1109/MCS.2019.2949974. ISSN 1941-000X. S2CID 210692106.

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