Routh–Hurwitz stability criterion

An algebraic test that determines whether every root of a real polynomial lies in the open left half of the complex plane, without explicitly calculating the roots.

Routh array: A tabular arrangement of polynomial coefficients used to count roots in the right half-plane. Its first-column sign changes give the criterion’s root count.
Characteristic polynomial: A polynomial whose roots encode the eigenvalues or modes of a linear system. For linear systems, this is the polynomial to which the criterion is applied.
Linear time-invariant system: A linear system whose governing parameters do not change with time. Its characteristic polynomial translates dynamic stability into a root-location question.
Edward John Routh: A British mathematician and engineer known for contributions to mechanics and stability theory. His 1877 method supplied the tabular test now bearing his name.
Hurwitz matrix: A structured matrix formed from polynomial coefficients whose leading principal minors encode stability conditions. Its determinants provide an equivalent coefficient-based test.
Complex plane: The plane whose points represent complex numbers through real and imaginary coordinates. Stability is decided by which side of its imaginary axis contains each root.
Control system stability: The property that a control system’s response remains bounded or returns to equilibrium after disturbances. The criterion tests asymptotic stability through the closed-loop characteristic polynomial.
Adolf Hurwitz: A German mathematician whose work included criteria for locating polynomial roots. His determinant conditions gave a second algebraic formulation of the stability test.
Cauchy index: An integer that counts signed jumps of a real rational function across a specified interval. The Routh procedure can be understood through sign variations associated with this index.
Left half-plane: The set of complex numbers with strictly negative real part. The criterion’s target region is precisely this half of the complex plane.

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