en>ArthurOgawa: correct markup error

2014-02-21T06:46:54Z

correct markup error

← Older revision		Revision as of 08:46, 21 February 2014
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	~~A [[polynomial]] is said to be '''stable''' if either:~~		Hello and welcome. My name is Figures Wunder. Hiring is her working day occupation now and she will not alter it anytime quickly. South Dakota is exactly where I've always been living. The preferred pastime for my children and me is to play baseball but I haven't made a dime with it.<br><br>My blog post - [http://Vcup.in/fooddeliveryservices36486 weight loss food delivery]
	* all its roots lie in the [[open set\|open]] left [[half-plane]], or
	* all its roots lie in the [[open set\|open]] [[unit disk]].

	~~The first condition provides stability for (or [[continuous-time]]) linear systems, and the second case relates to stability~~
	of [[discrete-time]] linear systems. A polynomial with the first property is called at times a [[Hurwitz polynomial]] and with the second property a [[Schur polynomial]]. Stable polynomials arise in [[control theory]] and in mathematical theory
	~~of differential~~ and ~~difference equations. A linear, [[time-invariant system]] (see [[LTI system theory]]) is said to be [[BIBO stability\|BIBO stable]] if every bounded input produces bounded output~~. ~~A linear system is BIBO stable if its characteristic polynomial~~ is ~~stable~~. ~~The denominator is required to be Hurwitz stable if the system~~ is ~~in continuous-time~~ and ~~Schur stable if~~ it ~~is in discrete-time~~. ~~In practice, stability~~ is ~~determined by applying any one of several [[stability criterion\|stability criteria]]~~.

	~~==Properties==~~
	* The ~~[[Routh-Hurwitz theorem]] provides an algorithm~~ for ~~determining if a given polynomial is Hurwitz stable.~~

	* To test if a given polynomial ''P'' (of degree ''d'') is ~~Schur stable, it suffices~~ to ~~apply this theorem to the transformed polynomial~~

	~~:<math> Q(z)=(z-1)^d P\left({{z+1}\over{z-1}}\right)~~
	~~</math>~~

	~~obtained after the [[Möbius transformation]] <math>z \mapsto {{z+1}\over{z-1}}</math> which maps the left half-plane to the open unit disc: ''P'' is Schur stable if and only if ''Q'~~' is Hurwitz stable. For higher degree polynomials the extra computation involved in this mapping can be avoided by testing the Schur stability by the Schur-Cohn test, the [[Jury stability criterion\| Jury test]] or the [[Bistritz stability criterion\| Bistritz test]].

	* Necessary condition: a Hurwitz stable polynomial (with real coefficients) has coefficients of the same sign (either all positive or all negative).

	* Sufficient condition: a ~~polynomial <math> f(z)=a_0+a_1 z+\cdots+a_n z^n</math>~~ with ~~(real) coefficients such that:~~
	~~:<math> a_n>a_{n-1}>\cdots>a_0>0,</math>~~
	~~is Schur stable.~~

	* Product rule: Two polynomials ''f'' and ''g'' are stable (of the same type) if and only if the product ''fg'' is stable.

	~~==Examples==~~
	* <math> 4z^3+3z^2+2z+1 </math> is Schur stable because it satisfies the sufficient condition;
	* <math> z^{10}</math> is Schur stable (because all its roots equal 0) but it ~~does not satisfy the sufficient condition;~~
	* <math> z^2-z-2</math> is not Hurwitz stable (its roots are -1,2) because it violates the necessary condition;
	* <math> z^2+3z+2 </math> is Hurwitz stable (its roots are -1,-2).
	* The polynomial <math> z^4+z^3+z^2+z+1 </math> (with positive coefficients) is neither Hurwitz stable nor Schur stable. Its roots are the four primitive fifth [[root of unity\|roots of unity]]

	~~::<math> z_k=\cos\left({{2\pi k}\over 5}\right)+i \sin\left({{2\pi k}\over 5}\right), \, k=1, \ldots, 4 \~~ .<~~/math~~>

	~~:Note here that~~

	::<~~math~~> ~~\cos({{2\pi}/5})={{\sqrt{5}~~-~~1}\over 4}>0.~~
	~~</math>~~

	:It is a "boundary case" for Schur stability because its roots lie on the unit circle. The example also shows that the necessary (positivity) conditions stated above for Hurwitz stability are not sufficient.

	~~==See also==~~
	* [[Stability criterion]]
	* [[Stability radius]]

	~~==External links==~~
	* [http://~~mathworld~~.~~wolfram.com~~/~~StablePolynomial.html Mathworld page]~~

	~~[[Category:Stability theory]]~~
	~~[[Category:Polynomials]]~~

	~~[[fr:Polynôme de Hurwitz]~~]

en>SchreiberBike: Correct ordinal 4rth→ 4th. Also some copy editing.

2013-11-21T05:27:56Z

Correct ordinal 4rth→ 4th. Also some copy editing.

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en>Helpful Pixie Bot: ISBNs (Build KC)

2012-05-06T06:06:58Z

ISBNs (Build KC)

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en>ArthurOgawa: correct markup error

en>SchreiberBike: Correct ordinal 4rth→ 4th. Also some copy editing.

en>Helpful Pixie Bot: ISBNs (Build KC)