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Hurwitz matrix

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In mathematics, a Hurwitz matrix, or Routh–Hurwitz matrix, in engineering stability matrix, is a structured real square matrix constructed with coefficients of a real polynomial.

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Hurwitz matrix and the Hurwitz stability criterion

Namely, given a real polynomial

p ( z ) = a 0 z n + a 1 z n − 1 + ⋯ + a n − 1 z + a n

the n × n square matrix

H = ( a 1 a 3 a 5 … … … 0 0 0 a 0 a 2 a 4 ⋮ ⋮ ⋮ 0 a 1 a 3 ⋮ ⋮ ⋮ ⋮ a 0 a 2 ⋱ 0 ⋮ ⋮ ⋮ 0 a 1 ⋱ a n ⋮ ⋮ ⋮ ⋮ a 0 ⋱ a n − 1 0 ⋮ ⋮ ⋮ 0 a n − 2 a n ⋮ ⋮ ⋮ ⋮ a n − 3 a n − 1 0 0 0 0 … … … a n − 4 a n − 2 a n ) .

is called Hurwitz matrix corresponding to the polynomial p . It was established by Adolf Hurwitz in 1895 that a real polynomial is stable (that is, all its roots have strictly negative real part) if and only if all the leading principal minors of the matrix H ( p ) are positive:

Δ 1 ( p ) = | a 1 | = a 1 > 0 Δ 2 ( p ) = | a 1 a 3 a 0 a 2 | = a 2 a 1 − a 0 a 3 > 0 Δ 3 ( p ) = | a 1 a 3 a 5 a 0 a 2 a 4 0 a 1 a 3 | = a 3 Δ 2 − a 1 ( a 1 a 4 − a 0 a 5 ) > 0

and so on. The minors Δ k ( p ) are called the Hurwitz determinants.

Hurwitz stable matrices

In engineering and stability theory, a square matrix A is called stable matrix (or sometimes Hurwitz matrix) if every eigenvalue of A has strictly negative real part, that is,

R e [ λ i ] < 0

for each eigenvalue λ i . A is also called a stability matrix, because then the differential equation

x ˙ = A x

is asymptotically stable, that is, x ( t ) → 0 as t → ∞ .

If G ( s ) is a (matrix-valued) transfer function, then G is called Hurwitz if the poles of all elements of G have negative real part. Note that it is not necessary that G ( s ) , for a specific argument s , be a Hurwitz matrix — it need not even be square. The connection is that if A is a Hurwitz matrix, then the dynamical system

x ˙ ( t ) = A x ( t ) + B u ( t ) y ( t ) = C x ( t ) + D u ( t )

has a Hurwitz transfer function.

Any hyperbolic fixed point (or equilibrium point) of a continuous dynamical system is locally asymptotically stable if and only if the Jacobian of the dynamical system is Hurwitz stable at the fixed point.

The Hurwitz stability matrix is a crucial part of control theory. A system is stable if its control matrix is a Hurwitz matrix. The negative real components of the eigenvalues of the matrix represent negative feedback. Similarly, a system is inherently unstable if any of the eigenvalues have positive real components, representing positive feedback.

References

Hurwitz matrix Wikipedia
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