Hurwitz Matrix

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
:$p(z)=a_z^n+a_z^+\cdots+a_z+a_n$
the $n\times n$ square matrix
:$H= \begin a_1 & a_3 & a_5 & \dots & \dots & \dots & 0 & 0 & 0 \\ a_0 & a_2 & a_4 & & & & \vdots & \vdots & \vdots \\ 0 & a_1 & a_3 & & & & \vdots & \vdots & \vdots \\ \vdots & a_0 & a_2 & \ddots & & & 0 & \vdots & \vdots \\ \vdots & 0 & a_1 & & \ddots & & a_n & \vdots & \vdots \\ \vdots & \vdots & a_0 & & & \ddots & a_ & 0 & \vdots \\ \vdots & \vdots & 0 & & & & a_ & a_n & \vdots \\ \vdots & \vdots & \vdots & & & & a_ & a_ & 0 \\ 0 & 0 & 0 & \dots & \dots & \dots & a_ & a_ & a_n \end.$
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
\Delta_1(p) &= \begin a_ \end &&=a_ > 0 \\ mm\Delta_2(p) &= \begin
a_ & a_ \\
a_ & a_ \\
\end &&= a_2 a_1 - a_0 a_3 > 0\\ mm\Delta_3(p) &= \begin
a_ & a_ & a_ \\
a_ & a_ & a_ \\
0 & a_ & a_ \\
\end &&= a_3 \Delta_2 - a_1 (a_1 a_4 - a_0 a_5 ) > 0
\end

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

In engineering and stability theory, a square matrix $A$ is called a stable matrix (or sometimes a Hurwitz matrix) if every eigenvalue of $A$ has strictly negative real part, that is,
:\operatorname lambda_i< 0\,
for each eigenvalue $\lambda_i$. $A$ is also called a stability matrix, because then the differential equation
:$\dot x = A x$
is asymptotically stable, that is, $x(t)\to 0$ as $t\to\infty.$

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
:$\dot x(t)=A x(t) + B u(t)$
:$y(t)=C x(t) + D u(t)\,$
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

References

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External links

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{{Matrix classes

Matrices
Differential equations