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.[1] This criterion has a computational advantage over the Routh–Hurwitz criterion because it involves only about half the number of determinant computations.[2]
The Routh–Hurwitz stability criterion says that a necessary and sufficient condition for all the roots of the polynomial with real coefficients
f(z)=a0zn+a1zn-1+ … +an(a0>0)
to have negative real parts (i.e.
f
\Delta1>0,\Delta2>0,\ldots,\Deltan>0,
where
\Deltai
f
Using the same notation as above, the Liénard–Chipart criterion is that
f
an>0,an-2>0,\ldots;\Delta1>0,\Delta3>0,\ldots
an>0,an-2>0,\ldots;\Delta2>0,\Delta4>0,\ldots
an>0,an-1>0,an-3>0,\ldots;\Delta1>0,\Delta3>0,\ldots
an>0,an-1>0,an-3>0,\ldots;\Delta2>0,\Delta4>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
\Delta1>0
an>0,a1>0,a3>0,a5>0,\ldots;
\Deltan-1>0,\Deltan-3>0,\Deltan-5>0,\ldots,\{\Delta3>0 (n even)\Delta2>0 (n odd)\}.
This means if n is even, the second line ends in
\Delta3>0
\Delta2>0
an
an-1>0