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