Hurwitz matrix

Short description: Algebraic matrix element to analyze a polynomial by its coefficients

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

$p(z)=a_0{z}^n+a_1{z}^{n-1}+\cdots +a_{n-1}z+a_n$

the $n\times n$ square matrix

$H=\left(\begin{array}{ccccccccc}{a}_1& a_3& a_5& \dots & \dots & \dots & 0& 0& 0\\ a_0& a_2& a_4& & & & ⋮& ⋮& ⋮\\ 0& a_1& a_3& & & & ⋮& ⋮& ⋮\\ ⋮& a_0& a_2& \ddots & & & 0& ⋮& ⋮\\ ⋮& 0& a_1& & \ddots & & a_n& ⋮& ⋮\\ ⋮& ⋮& a_0& & & \ddots & a_{n-1}& 0& ⋮\\ ⋮& ⋮& 0& & & & a_{n-2}& a_n& ⋮\\ ⋮& ⋮& ⋮& & & & a_{n-3}& a_{n-1}& 0\\ 0& 0& 0& \dots & \dots & \dots & a_{n-4}& a_{n-2}& a_n\end{array}\right).$

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:

$\begin{array}{rlrl}{\mathrm{\Delta }}_1(p)& =\left|\begin{array}{c}{a}_1\end{array}\right|& & =a_1>0\\ {\mathrm{\Delta }}_2(p)& =\left|\begin{array}{cc}{a}_1& a_3\\ a_0& a_2\\ \end{array}\right|& & =a_2{a}_1-a_0{a}_3>0\\ {\mathrm{\Delta }}_3(p)& =\left|\begin{array}{ccc}{a}_1& a_3& a_5\\ a_0& a_2& a_4\\ 0& a_1& a_3\\ \end{array}\right|& & =a_3{\mathrm{\Delta }}_2-a_1(a_1{a}_4-a_0{a}_5)>0\end{array}$

and so on. The minors ${\mathrm{\Delta }}_k(p)$ 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 Hurwitz matrix if every eigenvalue of $A$ has strictly negative real part, that is,

$\mathrm{Re}[{\lambda }_i]<0$

for each eigenvalue ${\lambda }_i$. $A$ is also called a stable matrix, because then the differential equation

$\dot{x}=Ax$

is asymptotically stable, that is, $x(t)\to 0$ as $t\to \mathrm{\infty }.$

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

$\dot{x}(t)=Ax(t)+Bu(t)$

$y(t)=Cx(t)+Du(t)$

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". SIAM Journal on Applied Mathematics 18 (2): 407–414. doi:10.1137/0118035. 
• Dimitrov, Dimitar K.; Peña, Juan Manuel (2005). "Almost strict total positivity and a class of Hurwitz polynomials". Journal of Approximation Theory 132 (2): 212–223. doi:10.1016/j.jat.2004.10.010. 
• Gantmacher, F. R. (1959). Applications of the Theory of Matrices. New York: Interscience. 
• Hurwitz, A. (1895). "Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Teilen besitzt". Mathematische Annalen 46 (2): 273–284. doi:10.1007/BF01446812. http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002255472. 
• Khalil, Hassan K. (2002). Nonlinear Systems. Prentice Hall. 
• Lehnigk, Siegfried H. (1970). "On the Hurwitz matrix". Zeitschrift für Angewandte Mathematik und Physik 21 (3): 498–500. doi:10.1007/BF01627957. Bibcode: 1970ZaMP...21..498L. 

External links

• "Hurwitz matrix". http://planetmath.org/?op=getobj&from=objects&id={{{id}}}.