Curtright field

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. This criterion has a computational advantage over Routh–Hurwitz criterion because they involve only about half the number of determinant computations.

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

${\mathrm{\Delta }}_1>0,{\mathrm{\Delta }}_2>0,\dots ,{\mathrm{\Delta }}_n>0,$

where ${\mathrm{\Delta }}_i$ 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. $a_n>0,a_{n-2}>0,\dots ,{\mathrm{\Delta }}_3>0,{\mathrm{\Delta }}_1>0$
2. $a_n>0,a_{n-2}>0,\dots ,{\mathrm{\Delta }}_4>0,{\mathrm{\Delta }}_2>0$
3. $a_n>0,a_{n-1}>0,a_{n-3}>0,\dots ,{\mathrm{\Delta }}_4>0,{\mathrm{\Delta }}_2>0$
4. $a_n>0,a_{n-1}>0,a_{n-3}>0,\dots ,{\mathrm{\Delta }}_3>0,{\mathrm{\Delta }}_1>0$

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

References

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