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All roots of a polynomial
 
All roots of a polynomial
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063770/m0637701.png" /></td> </tr></table>
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$$P(z)=z^n+a_{n-1}z^{n-1}+\dotsb+a_0$$
  
with real coefficients have strictly negative real part if and only if the complex-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063770/m0637702.png" /> of a real variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063770/m0637703.png" /> describes a curve (the Mikhailov hodograph) in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063770/m0637704.png" />-plane which starts on the positive real semi-axis, does not hit the origin and successively generates an anti-clockwise motion through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063770/m0637705.png" /> quadrants. (An equivalent condition is: The radius vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063770/m0637706.png" />, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063770/m0637707.png" /> increases from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063770/m0637708.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063770/m0637709.png" />, never vanishes and monotonically rotates in a positive direction through an angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063770/m06377010.png" />.)
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with real coefficients have strictly negative real part if and only if the complex-valued function $z=P(i\omega)$ of a real variable $\omega\in[0,\infty)$ describes a curve (the Mikhailov hodograph) in the complex $z$-plane which starts on the positive real semi-axis, does not hit the origin and successively generates an anti-clockwise motion through $n$ quadrants. (An equivalent condition is: The radius vector $P(i\omega)$, as $\omega$ increases from $0$ to $+\infty$, never vanishes and monotonically rotates in a positive direction through an angle $n\pi/2$.)
  
This criterion was first suggested by A.V. Mikhailov [[#References|[1]]]. It is equivalent to the [[Routh–Hurwitz criterion|Routh–Hurwitz criterion]]; however, it is geometric in character and does not require the verification of determinant inequalities (see [[#References|[2]]], [[#References|[3]]]). The Mikhailov criterion gives a necessary and sufficient condition for the asymptotic stability of a linear differential equation of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063770/m06377011.png" />,
+
This criterion was first suggested by A.V. Mikhailov [[#References|[1]]]. It is equivalent to the [[Routh–Hurwitz criterion|Routh–Hurwitz criterion]]; however, it is geometric in character and does not require the verification of determinant inequalities (see [[#References|[2]]], [[#References|[3]]]). The Mikhailov criterion gives a necessary and sufficient condition for the asymptotic stability of a linear differential equation of order $n$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063770/m06377012.png" /></td> </tr></table>
+
\begin{equation*} x ^ { ( n ) } + a _ { n  - 1} z ^ { ( n - 1 ) } + \dots + a _ { 0 } x = 0, \end{equation*}
  
 
with constant coefficients, or of a linear system
 
with constant coefficients, or of a linear system
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063770/m06377013.png" /></td> </tr></table>
+
\begin{equation*} \dot { x } = A x , \quad x \in {\bf R} ^ { n }, \end{equation*}
  
with a constant matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063770/m06377014.png" />, the characteristic polynomial of which is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063770/m06377015.png" /> (see [[#References|[4]]]).
+
with a constant matrix $A$, the characteristic polynomial of which is $P ( z )$ (see [[#References|[4]]]).
  
 
Mikhailov's criterion is one of the frequency criteria for the stability of linear systems of automatic control (closely related to, for example, the [[Nyquist criterion|Nyquist criterion]]). A generalization of Mikhailov's criterion is known for systems of automatic control with delay, for impulse systems (see [[#References|[5]]]), and there is also an analogue of Mikhailov's criterion for non-linear control systems (see [[#References|[6]]]).
 
Mikhailov's criterion is one of the frequency criteria for the stability of linear systems of automatic control (closely related to, for example, the [[Nyquist criterion|Nyquist criterion]]). A generalization of Mikhailov's criterion is known for systems of automatic control with delay, for impulse systems (see [[#References|[5]]]), and there is also an analogue of Mikhailov's criterion for non-linear control systems (see [[#References|[6]]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Mikhailov,   ''Avtomat. i Telemekh.'' , '''3'''  (1938)  pp. 27–81</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.G. Chebotarev,   N.N. Meiman,   "The Routh–Hurwitz problem for polynomials and entire functions"  ''Trudy Mat. Inst. Steklov.'' , '''76'''  (1949)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.A. Lavrent'ev,   B.V. Shabat,   "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft.  (1967)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  B.P. Demidovich,   "Lectures on the mathematical theory of stability" , Moscow  (1967)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  L.S. Gnoenskii,   G.A. Kamenskii,   L.E. El'sgol'ts,   "Mathematical foundations of the theory of control systems" , Moscow  (1969)  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A. Blaquiére,   "Mécanique non-lineaire" , Gauthier-Villars  (1960)</TD></TR></table>
+
<table>
 
+
<tr><td valign="top">[1]</td> <td valign="top">  A.V. Mikhailov, ''Avtomat. i Telemekh.'' , '''3'''  (1938)  pp. 27–81</td></tr>
 
+
<tr><td valign="top">[2]</td> <td valign="top">  N.G. Chebotarev, N.N. Meiman, "The Routh–Hurwitz problem for polynomials and entire functions"  ''Trudy Mat. Inst. Steklov.'' , '''76'''  (1949)  (In Russian)</td></tr>
 +
<tr><td valign="top">[3]</td> <td valign="top">  M.A. Lavrent'ev, B.V. Shabat, "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft.  (1967)  (Translated from Russian)</td></tr>
 +
<tr><td valign="top">[4]</td> <td valign="top">  B.P. Demidovich, "Lectures on the mathematical theory of stability" , Moscow  (1967)  (In Russian)</td></tr>
 +
<tr><td valign="top">[5]</td> <td valign="top">  L.S. Gnoenskii, G.A. Kamenskii, L.E. El'sgol'ts, "Mathematical foundations of the theory of control systems" , Moscow  (1969)  (In Russian)</td></tr>
 +
<tr><td valign="top">[6]</td> <td valign="top">  A. Blaquiére, "Mécanique non-linéaire" , Gauthier-Villars  (1960)</td></tr>
 +
</table>
  
 
====Comments====
 
====Comments====
Recently a generalization of stability criteria with respect to roots of polynomials has been found. It is named after V.L. Kharitonov [[#References|[a1]]]. The generalization is that the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063770/m06377016.png" /> in the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063770/m06377017.png" /> take values in given intervals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063770/m06377018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063770/m06377019.png" />. The problem addressed by Kharitonov is whether all polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063770/m06377020.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063770/m06377021.png" /> are strictly stable. It turns out that the stability of only four specific polynomials has to be investigated in order to answer this question.
+
Recently a generalization of stability criteria with respect to roots of polynomials has been found. It is named after V.L. Kharitonov [[#References|[a1]]]. The generalization is that the coefficients $a_i$ in the polynomial $p ( z ) = z ^ { n } + a _ { n  - 1} z ^ { n - 1 } + \ldots + a _ { 0 }$ take values in given intervals $[ a _ { i } ^ { - } , a _ { i } ^ { + } ]$, $i = 0 , \ldots , n - 1$. The problem addressed by Kharitonov is whether all polynomials $p ( z )$ with $a _ { i } \in [ a _ { i } ^ { - } , a _ { i } ^ { + } ]$ are strictly stable. It turns out that the stability of only four specific polynomials has to be investigated in order to answer this question.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.L. Kharitonov,  "Asymptotic stability of an equilibrium position of a family of systems of linear differential equations"  ''Differential Eq.'' , '''14''' :  11  (1978)  pp. 1483–1485  ''Differentsial'nye Uravnen.'' , '''14''' :  11  (1978)  pp. 2086–2088</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B.R. Barmish,  "New tools for robustness analysis" , ''IEEE Proc. 27th Conf. Decision and Control, Austin, Texas, December 1988'' , IEEE  (1988)  pp. 1–6</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. LaSalle,  "Stability by Liapunov's direct method" , Acad. Press  (1961)</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  V.L. Kharitonov,  "Asymptotic stability of an equilibrium position of a family of systems of linear differential equations"  ''Differential Eq.'' , '''14''' :  11  (1978)  pp. 1483–1485  ''Differentsial'nye Uravnen.'' , '''14''' :  11  (1978)  pp. 2086–2088</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  B.R. Barmish,  "New tools for robustness analysis" , ''IEEE Proc. 27th Conf. Decision and Control, Austin, Texas, December 1988'' , IEEE  (1988)  pp. 1–6</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  S. LaSalle,  "Stability by Liapunov's direct method" , Acad. Press  (1961)</td></tr></table>

Latest revision as of 17:00, 15 July 2024

All roots of a polynomial

$$P(z)=z^n+a_{n-1}z^{n-1}+\dotsb+a_0$$

with real coefficients have strictly negative real part if and only if the complex-valued function $z=P(i\omega)$ of a real variable $\omega\in[0,\infty)$ describes a curve (the Mikhailov hodograph) in the complex $z$-plane which starts on the positive real semi-axis, does not hit the origin and successively generates an anti-clockwise motion through $n$ quadrants. (An equivalent condition is: The radius vector $P(i\omega)$, as $\omega$ increases from $0$ to $+\infty$, never vanishes and monotonically rotates in a positive direction through an angle $n\pi/2$.)

This criterion was first suggested by A.V. Mikhailov [1]. It is equivalent to the Routh–Hurwitz criterion; however, it is geometric in character and does not require the verification of determinant inequalities (see [2], [3]). The Mikhailov criterion gives a necessary and sufficient condition for the asymptotic stability of a linear differential equation of order $n$,

\begin{equation*} x ^ { ( n ) } + a _ { n - 1} z ^ { ( n - 1 ) } + \dots + a _ { 0 } x = 0, \end{equation*}

with constant coefficients, or of a linear system

\begin{equation*} \dot { x } = A x , \quad x \in {\bf R} ^ { n }, \end{equation*}

with a constant matrix $A$, the characteristic polynomial of which is $P ( z )$ (see [4]).

Mikhailov's criterion is one of the frequency criteria for the stability of linear systems of automatic control (closely related to, for example, the Nyquist criterion). A generalization of Mikhailov's criterion is known for systems of automatic control with delay, for impulse systems (see [5]), and there is also an analogue of Mikhailov's criterion for non-linear control systems (see [6]).

References

[1] A.V. Mikhailov, Avtomat. i Telemekh. , 3 (1938) pp. 27–81
[2] N.G. Chebotarev, N.N. Meiman, "The Routh–Hurwitz problem for polynomials and entire functions" Trudy Mat. Inst. Steklov. , 76 (1949) (In Russian)
[3] M.A. Lavrent'ev, B.V. Shabat, "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft. (1967) (Translated from Russian)
[4] B.P. Demidovich, "Lectures on the mathematical theory of stability" , Moscow (1967) (In Russian)
[5] L.S. Gnoenskii, G.A. Kamenskii, L.E. El'sgol'ts, "Mathematical foundations of the theory of control systems" , Moscow (1969) (In Russian)
[6] A. Blaquiére, "Mécanique non-linéaire" , Gauthier-Villars (1960)

Comments

Recently a generalization of stability criteria with respect to roots of polynomials has been found. It is named after V.L. Kharitonov [a1]. The generalization is that the coefficients $a_i$ in the polynomial $p ( z ) = z ^ { n } + a _ { n - 1} z ^ { n - 1 } + \ldots + a _ { 0 }$ take values in given intervals $[ a _ { i } ^ { - } , a _ { i } ^ { + } ]$, $i = 0 , \ldots , n - 1$. The problem addressed by Kharitonov is whether all polynomials $p ( z )$ with $a _ { i } \in [ a _ { i } ^ { - } , a _ { i } ^ { + } ]$ are strictly stable. It turns out that the stability of only four specific polynomials has to be investigated in order to answer this question.

References

[a1] V.L. Kharitonov, "Asymptotic stability of an equilibrium position of a family of systems of linear differential equations" Differential Eq. , 14 : 11 (1978) pp. 1483–1485 Differentsial'nye Uravnen. , 14 : 11 (1978) pp. 2086–2088
[a2] B.R. Barmish, "New tools for robustness analysis" , IEEE Proc. 27th Conf. Decision and Control, Austin, Texas, December 1988 , IEEE (1988) pp. 1–6
[a3] S. LaSalle, "Stability by Liapunov's direct method" , Acad. Press (1961)
How to Cite This Entry:
Mikhailov criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mikhailov_criterion&oldid=12540
This article was adapted from an original article by r equation','../w/w097310.htm','Whittaker equation','../w/w097840.htm','Wronskian','../w/w098180.htm')" style="background-color:yellow;">N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article