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Liénard–Chipart criterion

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Condition for control system stability

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 + ⋯ + a n , a 0 > 0 {\displaystyle f(z)=a_{0}z^{n}+a_{1}z^{n-1}+\cdots +a_{n},\quad a_{0}>0}

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

Δ 1 > 0 , Δ 2 > 0 ,   … ,   Δ n > 0 , {\displaystyle \Delta _{1}>0,\,\Delta _{2}>0,\ \ldots ,\ \Delta _{n}>0,}

where Δi is the i-th leading 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 ,   a n − 4 > 0 ,   … Δ 1 > 0 ,   Δ 3 > 0 ,   … 2 ) a n > 0 ,   a n − 2 > 0 ,   a n − 4 > 0 ,   … Δ 2 > 0 ,   Δ 4 > 0 ,   … 3 ) a n > 0 ,   a n − 1 > 0 ,   a n − 3 > 0 ,   … Δ 1 > 0 ,   Δ 3 > 0 ,   … 4 ) a n > 0 ,   a n − 1 > 0 ,   a n − 3 > 0 ,   … Δ 2 > 0 ,   Δ 4 > 0 ,   … {\displaystyle {\begin{aligned}1)\quad &a_{n}>0,\ a_{n-2}>0,\ a_{n-4}>0,\ \ldots \\&\Delta _{1}>0,\ \Delta _{3}>0,\ \ldots \\2)\quad &a_{n}>0,\ a_{n-2}>0,\ a_{n-4}>0,\ \ldots \\&\Delta _{2}>0,\ \Delta _{4}>0,\ \ldots \\3)\quad &a_{n}>0,\ a_{n-1}>0,\ a_{n-3}>0,\ \ldots \\&\Delta _{1}>0,\ \Delta _{3}>0,\ \ldots \\4)\quad &a_{n}>0,\ a_{n-1}>0,\ a_{n-3}>0,\ \ldots \\&\Delta _{2}>0,\ \Delta _{4}>0,\ \ldots \end{aligned}}}

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 Δ1 > 0 is never needed to be checked):

a n > 0 ,   a 1 > 0 ,   a 3 > 0 ,   a 5 > 0 ,   … ; Δ n − 1 > 0 ,   Δ n − 3 > 0 ,   Δ n − 5 > 0 ,   … ,   { n  even Δ 3 > 0 n  odd Δ 2 > 0 {\displaystyle {\begin{aligned}&a_{n}>0,\ a_{1}>0,\ a_{3}>0,\ a_{5}>0,\ \ldots ;\\&\Delta _{n-1}>0,\ \Delta _{n-3}>0,\ \Delta _{n-5}>0,\ \ldots ,\ {\begin{cases}n{\text{ even}}&\Delta _{3}>0\\n{\text{ odd}}&\Delta _{2}>0\end{cases}}\end{aligned}}}

This means if n is even, the second line ends in Δ3 > 0 and if n is odd, it ends in Δ2 > 0 and so this is just condition (1) for odd n and condition (4) for even n from above. The first line always ends in an, but an-1 > 0 is also needed for even n.

References

  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. Vol. 2. American Mathematical Society. pp. 221–225. ISBN 0-8218-2664-6.

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