Liénard–Chipart criterion

Short description: Criterion on Control System Theory

In control system theory, the Liénard–Chipart criterion is a stability criterion modified from the Routh–Hurwitz stability criterion, proposed by A. Liénard and M. H. Chipart. This criterion has a computational advantage over the Routh–Hurwitz criterion because it involves only about half the number of determinant computations.

Algorithm

The Routh–Hurwitz stability criterion says that a necessary and sufficient condition for all the roots of the polynomial with real coefficients

[math]\displaystyle{ f(z) = a_0 z^n + a_1 z^{n-1} + \cdots + a_n \, (a_0 \gt 0) }[/math]

to have negative real parts (i.e. [math]\displaystyle{ f }[/math] is Hurwitz stable) is that

[math]\displaystyle{ \Delta_1 \gt 0,\, \Delta_2 \gt 0, \ldots, \Delta_n \gt 0, }[/math]

where [math]\displaystyle{ \Delta_i }[/math] is the i-th leading principal minor of the Hurwitz matrix associated with [math]\displaystyle{ f }[/math].

Using the same notation as above, the Liénard–Chipart criterion is that [math]\displaystyle{ f }[/math] is Hurwitz stable if and only if any one of the four conditions is satisfied:

1. [math]\displaystyle{ a_n\gt 0,a_{n-2}\gt 0, \ldots;\, \Delta_{1}\gt 0,\Delta_3\gt 0,\ldots }[/math]
2. [math]\displaystyle{ a_n\gt 0,a_{n-2}\gt 0, \ldots;\, \Delta_{2}\gt 0,\Delta_4\gt 0,\ldots }[/math]
3. [math]\displaystyle{ a_n\gt 0,a_{n-1}\gt 0,a_{n-3} \gt 0, \ldots;\, \Delta_1\gt 0,\Delta_3\gt 0,\ldots }[/math]
4. [math]\displaystyle{ a_n\gt 0,a_{n-1}\gt 0,a_{n-3} \gt 0, \ldots;\, \Delta_2\gt 0,\Delta_4\gt 0,\ldots }[/math]

Hence one can see that by choosing one of these conditions, the number of determinants required to be evaluated is reduced.

Alternatively Fuller formulated this as follows for (noticing that [math]\displaystyle{ \Delta_1\gt 0 }[/math] is never needed to be checked):

[math]\displaystyle{ a_n\gt 0,a_{1}\gt 0, a_{3}\gt 0, a_{5}\gt 0, \ldots; }[/math]

[math]\displaystyle{ \Delta_{n-1}\gt 0,\Delta_{n-3}\gt 0,\Delta_{n-5}\gt 0,\ldots,\{\Delta_3\gt 0 \ (n \ even)\, \Delta_2\gt 0 \ (n \ odd)\}. }[/math]

This means if n is even, the second line ends in [math]\displaystyle{ \Delta_3\gt 0 }[/math] and if n is odd, it ends in [math]\displaystyle{ \Delta_2\gt 0 }[/math] and so this is just 1. condition for odd n and 4. condition for even n from above. The first line always ends in [math]\displaystyle{ a_n }[/math], but [math]\displaystyle{ a_{n-1}\gt 0 }[/math] is also needed for even n.

External links

• Hazewinkel, Michiel, ed. (2001), "Liénard–Chipart criterion", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Li%C3%A9nard-Chipart_criterion&oldid=23382