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]


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

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

where is the i-th principal minor of the Hurwitz matrix associated with .

Using the same notation as above, the Liénard–Chipart criterion is that is Hurwitz-stable if and only if any one of the four conditions is satisfied:

Hence one can see that by choosing one of these conditions, the number of determinants required to be evaluated is reduced.


  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.

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