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

Algorithm

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

$f(z) = a_0 z^n + a_1 z^ + \cdots + a_n, \quad a_0 > 0$

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

$\Delta_1 > 0,\, \Delta_2 > 0, \ \ldots, \ \Delta_n > 0,$

where is the -th leading 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:

\begin
1) \quad& a_n > 0,\ a_ > 0,\ a_ > 0,\ \ldots \\
&\Delta_1 > 0,\ \Delta_3 > 0,\ \ldots \\ pt 2) \quad& a_n > 0,\ a_ > 0,\ a_ > 0,\ \ldots \\
& \Delta_2 > 0,\ \Delta_4 > 0,\ \ldots \\ pt 3) \quad& a_n > 0,\ a_ > 0,\ a_ > 0,\ \ldots \\
&\Delta_1 > 0,\ \Delta_3 > 0,\ \ldots \\ pt 4) \quad& a_n > 0,\ a_ > 0,\ a_ > 0,\ \ldots \\
&\Delta_2 > 0,\ \Delta_4 > 0,\ \ldots
\end

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 is never needed to be checked):

$\begin & a_n > 0,\ a_1 > 0,\ a_3 > 0,\ a_5 > 0,\ \ldots; \\ & \Delta_ > 0,\ \Delta_ > 0,\ \Delta_ > 0,\ \ldots,\ \begin n \text & \Delta_3 > 0 \\ n \text & \Delta_2 > 0 \end \end$

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

References

External links

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Stability theory

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