Liénard–Chipart criterion

In control system theory, the Liénard–Chipart criterion is a stability criterion modified from Routh–Hurwitz stability criterion, proposed by A. Liénard and M. H. Chipart.^[1] This criterion has a computational advantage over Routh–Hurwitz criterion because they involve only about half the number of determinant computations.^[2]

Algorithm

Recalling the Routh–Hurwitz stability criterion, it 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 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. $a_{n}>0,a_{{n-1}}>0,a_{{n-3}}>0,\ldots ;\,\Delta _{2}>0,\Delta _{4}>0,\ldots$

Henceforth, one can see that by choosing one of these conditions, the determinants required to be evaluated are thus reduced.

References

1. Liénard, A.; Chipart, M. H. (1914). "Sur le signe de la partie réelle des racines d'une équation algébrique". J. Math. Pures Appl. 10 (6): 291–346.
2. Feliks R. Gantmacher (2000). The Theory of Matrices. American Mathematical Society. pp. 221–225. ISBN 0-8218-2664-6.

External links

• Hazewinkel, Michiel, ed. (2001), "Liénard–Chipart criterion", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4