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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
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| of a real polynomial. | |
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| ==Hurwitz matrix and the Hurwitz stability criterion==
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| Namely, given a real polynomial
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| :<math>p(z)=a_{0}z^n+a_{1}z^{n-1}+\cdots+a_{n-1}z+a_n</math>
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| the <math>n\times n</math> [[square matrix]]
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| :<math> H(p) := \begin{bmatrix} | |
| a_1 & a_3 & a_5 & a_7 & \ldots & 0\\
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| a_0 & a_2 & a_4 & a_6& \ldots & 0\\
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| 0 & a_1 & a_3 & a_5& \ldots & 0\\
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| 0 & a_0 & a_2 & a_4& \ldots & 0\\
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| 0 & 0 & a_1 & a_3& \ldots & 0\\
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| \vdots & \vdots & \vdots & \vdots& \ddots& \vdots\\
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| 0 & 0 & 0 & 0& \ldots& a_n\\
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| \end{bmatrix}</math>
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| is called '''Hurwitz matrix''' corresponding to the polynomial <math>p</math>. It was established by [[Adolf Hurwitz]] in 1895 that a real polynomial is [[Stable polynomial|stable]]
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| (that is, all its roots have strictly negative real part) if and only if all the leading principal [[Minor (linear algebra)|minors]] of the matrix <math>H(p)</math> are positive:
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| :<math>
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| \begin{align}
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| \Delta_1(p) &= \begin{vmatrix} a_{1} \end{vmatrix} &&=a_{1} > 0 \\[2mm]
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| \Delta_2(p) &= \begin{vmatrix}
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| a_{1} & a_{3} \\
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| a_{0} & a_{2} \\
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| \end{vmatrix} &&= a_2 a_1 - a_0 a_3 > 0\\[2mm]
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| \Delta_3(p) &= \begin{vmatrix}
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| a_{1} & a_{3} & a_{5} \\
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| a_{0} & a_{2} & a_{4} \\
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| 0 & a_{1} & a_{3} \\
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| \end{vmatrix} &&= a_3 \Delta_2 - a_1 (a_1 a_4 - a_0 a_5 ) > 0
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| \end{align}
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| </math>
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| and so on. The minors <math>\Delta_k(p)</math> are called the [[Hurwitz determinant]]s.
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| ==Hurwitz stable matrices==
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| In [[engineering]] and [[stability theory]], a [[square matrix]] <math>A</math> is called ''' stable matrix''' (or sometimes '''Hurwitz matrix''') if every [[eigenvalue]] of <math>A</math> has [[negative number|strictly negative]] [[real part]], that is, | |
| :<math>\mathop{\mathrm{Re}}[\lambda_i] < 0\,</math>
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| for each eigenvalue <math>\lambda_i</math>. <math>A</math> is also called a '''stability matrix''', because then the [[ordinary differential equation|differential equation]]
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| :<math>\dot x = A x</math>
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| is [[stability theory|asymptotically stable]], that is, <math>x(t)\to 0</math> as <math>t\to\infty.</math>
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| If <math>G(s)</math> is a (matrix-valued) [[transfer function]], then <math>G</math> is called '''Hurwitz''' if the [[pole (complex analysis)|poles]] of all elements of <math>G</math> have negative real part. Note that it is not necessary that <math>G(s),</math> for a specific argument <math>s,</math> be a Hurwitz matrix — it need not even be square. The connection is that if <math>A</math> is a Hurwitz matrix, then the [[dynamical system]]
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| :<math>\dot x(t)=A x(t) + B u(t)</math> | |
| :<!-- This is no typo. There must be no dot on y --><math>y(t)=C x(t) + D u(t)\,</math>
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| has a Hurwitz transfer function.
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| Any hyperbolic [[Fixed point (mathematics)|fixed point]] (or [[equilibrium point]]) of a continuous [[Dynamical system (definition)|dynamical system]] is locally [[Asymptotic stability|asymptotically stable]] if and only if the [[Jacobian]] of the dynamical system is Hurwitz stable at the fixed point.
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| The Hurwitz stability matrix is in crucial part on [[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]].
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| ==See also==
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| * [[Z-matrix (mathematics)|Z-matrix]]
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| * [[M-matrix]]
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| * [[P-matrix]]
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| * [[Perron–Frobenius theorem]]
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| ==References==
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| * {{cite journal
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| | author = Hurwitz, A.
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| | year = 1895
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| | title = Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Teilen besitzt
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| | journal = [http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN235181684_0046 ''Mathematische Annalen Nr. 46''], Leipzig
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| | pages = 273–284
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| }}
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| *{{cite journal
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| | author = Gantmacher, F.R.
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| | year = 1959
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| | title = Applications of the Theory of Matrices
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| | journal = Interscience, New York
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| | volume = 641
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| | issue = 9
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| | pages = 1–8
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| }}
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| * Hassan K. Khalil (2002). ''Nonlinear Systems''. Prentice Hall.
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| * Siegfried H. Lehnigk, [http://www.springerlink.com/content/h192106tq8nl2274/ ''On the Hurwitz matrix''], ''Zeitschrift für Angewandte Mathematik und Physik (ZAMP)'', May 1970
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| * Bernard A. Asner, Jr., ''On the Total Nonnegativity of the Hurwitz Matrix'', SIAM Journal on Applied Mathematics, Vol. 18, No. 2 (Mar., 1970)
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| *Dimitar K. Dimitrov and Juan Manuel Peña, [http://portal.acm.org/citation.cfm?id=1063186.1063190 ''Almost strict total positivity and a class of Hurwitz polynomials''], Journal of Approximation Theory, Volume 132, Issue 2 (February 2005)
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| ==External links==
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| *{{planetmath reference|id=5395|title=Hurwitz matrix}}
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| {{PlanetMath attribution|id=5395|title=Hurwitz matrix}}
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| [[Category:Matrices]]
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| [[Category:Differential equations]]
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