Mikhailov criterion

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)

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.