Liénard–Chipart criterion

In control theory, the Liénard–Chipart criterion is a stability criterion modified from the Routh–Hurwitz stability criterion, proposed in 1914 by French physicists 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

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

to have negative real parts (i.e. f is Hurwitz stable) is that

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

where Δ_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:

$${\displaystyle {\begin{aligned}1)\quad &a_{n}>0,\ a_{n-2}>0,\ a_{n-4}>0,\ \ldots \\&\Delta _{1}>0,\ \Delta _{3}>0,\ \ldots \\[4pt]2)\quad &a_{n}>0,\ a_{n-2}>0,\ a_{n-4}>0,\ \ldots \\&\Delta _{2}>0,\ \Delta _{4}>0,\ \ldots \\[4pt]3)\quad &a_{n}>0,\ a_{n-1}>0,\ a_{n-3}>0,\ \ldots \\&\Delta _{1}>0,\ \Delta _{3}>0,\ \ldots \\[4pt]4)\quad &a_{n}>0,\ a_{n-1}>0,\ a_{n-3}>0,\ \ldots \\&\Delta _{2}>0,\ \Delta _{4}>0,\ \ldots \end{aligned}}}$$

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 Δ₁ > 0 is never needed to be checked):

$${\displaystyle {\begin{aligned}&a_{n}>0,\ a_{1}>0,\ a_{3}>0,\ a_{5}>0,\ \ldots$$ ;\\&\Delta _{n-1}>0,\ \Delta _{n-3}>0,\ \Delta _{n-5}>0,\ \ldots ,\ {\begin{cases}n{\text{ even}}&\Delta _{3}>0\\n{\text{ odd}}&\Delta _{2}>0\end{cases}}\end{aligned}}}

This means if n is even, the second line ends in Δ₃ > 0 and if n is odd, it ends in Δ₂ > 0 and so this is just condition (1) for odd n and condition (4) for even n from above. The first line always ends in a_n, but a_n-1 > 0 is also needed for even n.

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. Vol. 2. American Mathematical Society. pp. 221–225. ISBN 0-8218-2664-6.

External links

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