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

$$f(z)=a_0{z}^n+a_1{z}^{n-1}+\cdots +a_n,a_0>0$$

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

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

where Template:Math is the Template:Mvar-th leading principal minor of the Hurwitz matrix associated with Template:Mvar.

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

$$\begin{array}{cc}1)& a_n>0,a_{n-2}>0,a_{n-4}>0,\dots \\ & {\mathrm{\Delta }}_1>0,{\mathrm{\Delta }}_3>0,\dots \\ 2)& a_n>0,a_{n-2}>0,a_{n-4}>0,\dots \\ & {\mathrm{\Delta }}_2>0,{\mathrm{\Delta }}_4>0,\dots \\ 3)& a_n>0,a_{n-1}>0,a_{n-3}>0,\dots \\ & {\mathrm{\Delta }}_1>0,{\mathrm{\Delta }}_3>0,\dots \\ 4)& a_n>0,a_{n-1}>0,a_{n-3}>0,\dots \\ & {\mathrm{\Delta }}_2>0,{\mathrm{\Delta }}_4>0,\dots \end{array}$$

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 Template:Math is never needed to be checked):

$$\begin{array}{cc}& a_n>0,a_1>0,a_3>0,a_5>0,\dots ;\\ & {\mathrm{\Delta }}_{n-1}>0,{\mathrm{\Delta }}_{n-3}>0,{\mathrm{\Delta }}_{n-5}>0,\dots ,\{\begin{array}{cc}n\text{ even}& {\mathrm{\Delta }}_3>0\\ n\text{ odd}& {\mathrm{\Delta }}_2>0\end{array}\end{array}$$

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

External links

• Template:SpringerEOM

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