Hurwitz matrix explained

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.

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 x n square matrixH = \begin{pmatrix} a _{1} & a _{3} & a _{5} & ... & ... & ... & 0 & 0 & 0 \\ a _{0} & a _{2} & a _{4} & & & & \vdots & \vdots & \vdots \\ 0 & a _{1} & a _{3} & & & & \vdots & \vdots & \vdots \\ \vdots & a _{0} & a _{2} & \ddots & & & 0 & \vdots & \vdots \\ \vdots & 0 & a _{1} & & \ddots & & a _{n} & \vdots & \vdots \\ \vdots & \vdots & a _{0} & & & \ddots & a _{n-1} & 0 & \vdots \\ \vdots & \vdots & 0 & & & & a _{n-2} & a _{n} & \vdots \\ \vdots & \vdots & \vdots & & & & a _{n-3} & a _{n-1} & 0 \\ 0 & 0 & 0 & ... & ... & ... & a _{n-4} & a _{n-2} & a _{n \end{pmatrix}.} is called Hurwitz matrix corresponding to the polynomial p . It was established by Adolf Hurwitz in 1895 that a real polynomial with a _{0} > 0 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:\begin{align} \Delta _{1(p)} & = \begin{vmatrix} a _{1} \end{vmatrix} && = a _{1} > 0 \\[ 2 mm] \Delta _{2(p)} & = \begin{vmatrix} a _{1} & a _{3} \\ a _{0} & a _{2} \\ \end{vmatrix} && = a _{2} a _{1} - a _{0} a _{3} > 0 \\[ 2 mm] \Delta _{3(p)} & = \begin{vmatrix} a _{1} & a _{3} & a _{5} \\ a _{0} & a _{2} & a _{4} \\ 0 & a _{1} & a _{3} \\ \end{vmatrix} && = a _{3} \Delta _{2} - a _{1} (a _{1} a _{4} - a _{0} a _{5} ) > 0 \end{align} and so on. The minors \Delta _{k(p)} are called the Hurwitz determinants. Similarly, if a _{0} < 0 then the polynomial is stable if and only if the principal minors have alternating signs starting with a negative one.

Hurwitz stable matrices

A is called a Hurwitz matrix if every eigenvalue of A has strictly negative real part, that is,\operatorname{Re}[ λ _{i]} < 0 for each eigenvalue λ _{i} . A is also called a stable matrix, because then the differential equation• x = A x is asymptotically stable, that is, x(t)\to 0 as t\to inf ty.

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.

See also

• Liénard–Chipart criterion
• M-matrix
• P-matrix
• Perron–Frobenius theorem
• Z-matrix
• Jury stability criterion, for the analogue criterion for discrete-time systems.

References

• Asner . Bernard A. Jr. . 1970 . On the Total Nonnegativity of the Hurwitz Matrix . . 18 . 2 . 407–414 . 10.1137/0118035 . 2099475.
• Dimitrov . Dimitar K. . Peña . Juan Manuel . 2005 . Almost strict total positivity and a class of Hurwitz polynomials . . 132 . 2 . 212–223 . 10.1016/j.jat.2004.10.010. free . 11449/21728 . free .
• Book: Gantmacher, F. R. . 1959 . Applications of the Theory of Matrices . . New York.
• Hurwitz . A. . 1895 . Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Teilen besitzt . . 46 . 2 . 273–284 . 10.1007/BF01446812. 121036103 .
• Book: Khalil, Hassan K. . 2002 . Nonlinear Systems . Prentice Hall.
• Lehnigk . Siegfried H. . 1970 . On the Hurwitz matrix . . 21 . 3 . 498–500 . 1970ZaMP...21..498L . 10.1007/BF01627957. 123380473 .