# Difference between revisions of "Mikhailov criterion"

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All roots of a polynomial | All roots of a polynomial | ||

− | $$P(z)=z^n+a_{n-1}z^{n-1}+\ | + | $$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 $n\pi/2$.) | 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 | + | 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 $n$, |

− | + | \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 | ||

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

− | with a constant matrix | + | 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>< | + | <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> |

====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 | + | 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>< | + | <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 15:30, 1 July 2020

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-lineaire" , 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=32812