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

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

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. $a_{n}>0,a_{n-2}>0,\ldots ,\,\Delta _{3}>0,\Delta _{1}>0$
2. $a_{n}>0,a_{n-2}>0,\ldots ,\,\Delta _{4}>0,\Delta _{2}>0$
3. $a_{n}>0,a_{n-1}>0,a_{n-3}>0,\ldots ,\,\Delta _{4}>0,\Delta _{2}>0$
4. $a_{n}>0,a_{n-1}>0,a_{n-3}>0,\ldots ,\,\Delta _{3}>0,\Delta _{1}>0$

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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