Liénard–Chipart criterion

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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 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.

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. ^ Felix Gantmacher (2000). The Theory of Matrices. American Mathematical Society. pp. 221–225. ISBN 0-8218-2664-6.