Algebraic methods for stability in linear methods assume that the characteristic polynomial has no roots on the imaginary axis (Hermite's criterion, Cauchy index, Routh method,...).
Determining roots on the imaginary axis is necessary to decide whether a polynomial is unstable or marginally stable. Also it seems necessary for the delay stability notion.
Split p(iw), where w is real, into its real and imaginary parts. This reduces the problem to the one of finding the gcd of two real polynomials in a single real variable.
Thanks Gunther,
Obvioulsy your solution simplifies partially the problem. Evaluating p(iw) transforms the problem into searching for solutions on the real axis; this is standard in Gantmacher, Marden, et cetera. The problem is that if we assume that p(iw)=U(w)+iV(w) with U,V real polynomials in the unknown w, the greatest common divisor of U and V is not assured to include exactly all the pure imaginary roots. For example, irreducible polynomials (w^2+a) with a>0 can divide U and V.
- Badges
- Science topic
- Similar topics
- Mathematics