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.^[1] This criterion has a computational advantage over the Routh–Hurwitz criterion because it involves only about half the number of determinant computations.^[2]

Algorithm

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

${\displaystyle f(z)=a_{0}z^{n}+a_{1}z^{n-1}+\cdots +a_{n}\,(a_{0}>0)}$

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

${\displaystyle \Delta _{1}>0,\,\Delta _{2}>0,\ldots ,\Delta _{n}>0,}$

where ${\displaystyle \Delta _{i}}$ is the i-th leading principal minor of the Hurwitz matrix associated with ${\displaystyle f}$.

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

1. ${\displaystyle a_{n}>0,a_{n-2}>0,\ldots ;\,\Delta _{1}>0,\Delta _{3}>0,\ldots }$
2. ${\displaystyle a_{n}>0,a_{n-2}>0,\ldots ;\,\Delta _{2}>0,\Delta _{4}>0,\ldots }$
3. ${\displaystyle a_{n}>0,a_{n-1}>0,a_{n-3}>0,\ldots ;\,\Delta _{1}>0,\Delta _{3}>0,\ldots }$
4. ${\displaystyle a_{n}>0,a_{n-1}>0,a_{n-3}>0,\ldots ;\,\Delta _{2}>0,\Delta _{4}>0,\ldots }$

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 ${\displaystyle \Delta _{1}>0}$ is never needed to be checked):

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

${\displaystyle \Delta _{n-1}>0,\Delta _{n-3}>0,\Delta _{n-5}>0,\ldots ,\{\Delta _{3}>0\ (n\ even)\,\Delta _{2}>0\ (n\ odd)\}.}$

This means if n is even, the second line ends in ${\displaystyle \Delta _{3}>0}$ and if n is odd, it ends in ${\displaystyle \Delta _{2}>0}$ and so this is just 1. condition for odd n and 4. condition for even n from above. The first line always ends in ${\displaystyle a_{n}}$, but ${\displaystyle a_{n-1}>0}$ is also needed for even n.

References

External links

• "Liénard–Chipart criterion", Encyclopedia of Mathematics, EMS Press, 2001 [1994]