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


 * $$f(z) = a_0 z^n + a_1 z^{n-1} + \cdots + a_n \, (a_0 > 0) $$

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:
 * 1) $$ a_n>0,a_{n-2}>0, \ldots;\, \Delta_{1}>0,\Delta_3>0,\ldots$$
 * 2) $$ a_n>0,a_{n-2}>0, \ldots;\, \Delta_{2}>0,\Delta_4>0,\ldots$$
 * 3) $$ a_n>0,a_{n-1}>0,a_{n-3} >0, \ldots;\, \Delta_1>0,\Delta_3>0,\ldots$$
 * 4) $$ a_n>0,a_{n-1}>0,a_{n-3} >0, \ldots;\, \Delta_2>0,\Delta_4>0,\ldots$$

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):

$$ a_n>0,a_{1}>0, a_{3}>0, a_{5}>0, \ldots;$$

$$ \Delta_{n-1}>0,\Delta_{n-3}>0,\Delta_{n-5}>0,\ldots,\{\Delta_3>0 \ (n \ even)\, \Delta_2>0 \ (n \ odd)\}.$$

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 $$ a_{n-1}>0$$ is also needed for even n.