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
- [math]\displaystyle{ p(z)=a_{0}z^n+a_{1}z^{n-1}+\cdots+a_{n-1}z+a_n }[/math]
the [math]\displaystyle{ n\times n }[/math] square matrix
- [math]\displaystyle{ H= \begin{pmatrix} 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_{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 & \dots & \dots & \dots & a_{n-4} & a_{n-2} & a_n \end{pmatrix}. }[/math]
is called Hurwitz matrix corresponding to the polynomial [math]\displaystyle{ p }[/math]. It was established by Adolf Hurwitz in 1895 that a real polynomial with [math]\displaystyle{ a_0 \gt 0 }[/math] is stable (that is, all its roots have strictly negative real part) if and only if all the leading principal minors of the matrix [math]\displaystyle{ H(p) }[/math] are positive:
- [math]\displaystyle{ \begin{align} \Delta_1(p) &= \begin{vmatrix} a_{1} \end{vmatrix} &&=a_{1} \gt 0 \\[2mm] \Delta_2(p) &= \begin{vmatrix} a_{1} & a_{3} \\ a_{0} & a_{2} \\ \end{vmatrix} &&= a_2 a_1 - a_0 a_3 \gt 0\\[2mm] \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 ) \gt 0 \end{align} }[/math]
and so on. The minors [math]\displaystyle{ \Delta_k(p) }[/math] are called the Hurwitz determinants. Similarly, if [math]\displaystyle{ a_0 \lt 0 }[/math] 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 [math]\displaystyle{ A }[/math] is called a Hurwitz matrix if every eigenvalue of [math]\displaystyle{ A }[/math] has strictly negative real part, that is,
- [math]\displaystyle{ \operatorname{Re}[\lambda_i] \lt 0\, }[/math]
for each eigenvalue [math]\displaystyle{ \lambda_i }[/math]. [math]\displaystyle{ A }[/math] is also called a stable matrix, because then the differential equation
- [math]\displaystyle{ \dot x = A x }[/math]
is asymptotically stable, that is, [math]\displaystyle{ x(t)\to 0 }[/math] as [math]\displaystyle{ t\to\infty. }[/math]
If [math]\displaystyle{ G(s) }[/math] is a (matrix-valued) transfer function, then [math]\displaystyle{ G }[/math] is called Hurwitz if the poles of all elements of [math]\displaystyle{ G }[/math] have negative real part. Note that it is not necessary that [math]\displaystyle{ G(s), }[/math] for a specific argument [math]\displaystyle{ s, }[/math] be a Hurwitz matrix — it need not even be square. The connection is that if [math]\displaystyle{ A }[/math] is a Hurwitz matrix, then the dynamical system
- [math]\displaystyle{ \dot x(t)=A x(t) + B u(t) }[/math]
- [math]\displaystyle{ y(t)=C x(t) + D u(t)\, }[/math]
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
 |  |
