Liénard–Chipart Criterion

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

::$f(z) = a_0 z^n + a_1 z^ + \cdots + a_n \, (a_0 > 0)$

to have negative real parts (i.e. $f$ is Hurwitz stable) is that

::$\Delta_1 > 0,\, \Delta_2 > 0, \ldots, \Delta_n > 0,$

where $\Delta_i$ is the ''i''-th leading principal minor of the Hurwitz matrix associated with $f$.

Using the same notation as above, the Liénard–Chipart criterion is that $f$ is Hurwitz stable if and only if any one of the four conditions is satisfied:
# $a_n>0,a_>0, \ldots;\, \Delta_>0,\Delta_3>0,\ldots$
# $a_n>0,a_>0, \ldots;\, \Delta_>0,\Delta_4>0,\ldots$
# $a_n>0,a_>0,a_ >0, \ldots;\, \Delta_1>0,\Delta_3>0,\ldots$
# $a_n>0,a_>0,a_ >0, \ldots;\, \Delta_2>0,\Delta_4>0,\ldots$

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 $\Delta_1>0$ is never needed to be checked):

$a_n>0,a_>0, a_>0, a_>0, \ldots;$

$\Delta_>0,\Delta_>0,\Delta_>0,\ldots,\.$

This means if n is even, the second line ends in $\Delta_3>0$ and if n is odd, it ends in $\Delta_2>0$ and so this is just 1. condition for odd n and 4. condition for even n from above. The first line always ends in $a_n$, but $a_>0$ is also needed for even n.

References

External links

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Stability theory

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