The Routh-Hurwitz Criterion
Routh-Hurwitz decides stability from the characteristic polynomial's coefficients alone, without computing any roots.
Stability Without Root-Finding
The Routh-Hurwitz criterion tests whether all roots of a polynomial lie in the left half-plane using only arithmetic on its coefficients. This let engineers judge closed-loop stability before computers made root-finding trivial, and it still gives symbolic stability conditions on design parameters.
Necessary condition
For all roots to be in the left half-plane, every coefficient of the characteristic polynomial must be present and have the same sign. If any coefficient is zero or has an opposite sign, the system is not stable. This quick check catches many unstable cases immediately, but it is not sufficient on its own for higher orders.
The Routh array
Build a table from the coefficients: the first two rows list alternate coefficients, and each subsequent row is computed from the two above by a cross-multiplication pattern. The criterion states that the number of right-half-plane roots equals the number of sign changes in the first column of the completed array.
- No sign changes in the first column: system is stable.
- Each sign change: one root in the right half-plane.
- A zero in the first column signals a root on or near the imaginary axis and needs a special substitution.
Design use
The real power of Routh-Hurwitz is symbolic. Leave a gain K as a free parameter, build the array, and the requirement that first-column entries stay positive yields the exact range of K for which the loop is stable. This gives the stability boundary directly, which is far more informative than a single stable-or-not answer.
Limitations
The test handles only polynomials with real coefficients and gives no information about transient quality, margins, or robustness. It also does not accommodate pure time delays, whose e^(-s*T) term is not polynomial. For those, frequency-domain methods are used instead.
Even so, Routh-Hurwitz remains a compact, exact stability check and a standard classroom tool for mapping how a design parameter moves a system across the stability boundary.