In control theory and the theory of differential equations, the Routh–Hurwitz stability criterion is a mathematical test that is a necessary and sufficient condition for the stability of a linear time-invariant (LTI) dynamical system or control system. A stable system is one whose output signal is bounded; the position, velocity or energy do not increase to infinity as time goes on. The Routh test is an efficient recursive algorithm that English mathematician Edward John Routh proposed in 1876 to determine whether all the roots of the characteristic polynomial of a linear system have negative real parts. German mathematician Adolf Hurwitz independently proposed in 1895 to arrange the coefficients of the polynomial into a square matrix, called the Hurwitz matrix, and showed that the polynomial is stable if and only if the sequence of determinants of its principal submatrices are all positive. The two procedures are equivalent, with the Routh test providing a more efficient way to compute the Hurwitz determinants ( Δ i {\displaystyle \Delta _{i}} ) than computing them directly. A polynomial satisfying the Routh–Hurwitz criterion is called a Hurwitz polynomial. The importance of the criterion is that the roots p of the characteristic equation of a linear system with negative real parts represent solutions ept of the system that are stable (bounded). Thus the criterion provides a way to determine if the equations of motion of a linear system have only stable solutions, without solving the system directly. For discrete systems, the corresponding stability test can be handled by the Schur–Cohn criterion, the Jury test and the Bistritz test. The Routh test can be derived through the use of the Euclidean algorithm and Sturm's theorem in evaluating Cauchy indices. Hurwitz derived his conditions differently.
Using Euclid's algorithm The criterion is related to Routh–Hurwitz theorem. From the statement of that theorem, we have p − q = w ( + ∞ ) − w ( − ∞ ) {\displaystyle p-q=w(+\infty )-w(-\infty )} where:
p {\displaystyle p} is the number of roots of the polynomial f ( z ) {\displaystyle f(z)} with negative real part;
q {\displaystyle q} is the number of roots of the polynomial f ( z ) {\displaystyle f(z)} with positive real part (according to the theorem, f {\displaystyle f} is supposed to have no roots lying on the imaginary line); w(x) is the number of variations of the generalized Sturm chain obtained from P 0 ( y ) {\displaystyle P_{0}(y)} and P 1 ( y ) {\displaystyle P_{1}(y)} (by successive Euclidean divisions) where f ( i y ) = P 0 ( y ) + i P 1 ( y ) {\displaystyle f(iy)=P_{0}(y)+iP_{1}(y)} for a real y. By the fundamental theorem of algebra, each polynomial of degree n must have n roots in the complex plane (i.e., for an ƒ with no roots on the imaginary line, p + q = n). Thus, we have the condition that ƒ is a (Hurwitz) stable polynomial if and only if p − q = n (the proof is given below). Using the Routh–Hurwitz theorem, we can replace the condition on p and q by a condition on the generalized Sturm chain, which will give in turn a condition on the coefficients of ƒ.
Using matrices Let f(z) be a complex polynomial. The process is as follows:
Compute the polynomials P 0 ( y ) {\displaystyle P_{0}(y)} and P 1 ( y ) {\displaystyle P_{1}(y)} such that f ( i y ) = P 0 ( y ) + i P 1 ( y ) {\displaystyle f(iy)=P_{0}(y)+iP_{1}(y)} where y is a real number. Compute the Sylvester matrix associated to P 0 ( y ) {\displaystyle P_{0}(y)} and P 1 ( y ) {\displaystyle P_{1}(y)} . Rearrange each row in such a way that an odd row and the following one have the same number of leading zeros. Compute each principal minor of that matrix. If at least one of the minors is negative (or zero), then the polynomial f is not stable.
Example Let f ( z ) = a z 2 + b z + c {\displaystyle f(z)=az^{2}+bz+c} (for the sake of simplicity we take real coefficients) where c ≠ 0 {\displaystyle c\neq 0} (to avoid a root in zero so that we can use the Routh–Hurwitz theorem). First, we have to calculate the real polynomials P 0 ( y ) {\displaystyle P_{0}(y)} and P 1 ( y ) {\displaystyle P_{1}(y)} :
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