Systems are unstable if their characteristic equations have a positive root (solution). Determine whether each of the following characteristic equations represents a stable or unstable system.
(i) s3 + 6s2 + 11s + 6 = 0
(ii) s3 + s2 - 8s - 12 = 0
Systems are unstable if their characteristic equations have a positive root (solution). Determine whether each of the following characteristic equations represents a stable or unstable system.
(i) s3 + 6s2 + 11s + 6 = 0
(ii) s3 + s2 - 8s - 12 = 0
The first polynomial obviously does not have positive roots because when s > 0.
For the second polynomial, there are several ways to reason. First, Descartes' rule of signs says that it has exactly one positive root. One can also note that if , then , but, say, , so must have an x-intercept. I am sure there are other ways; what you want probably depends on what methods you have studied.
Edit: With a bit of trial-and error (or some factorization tools) one can see that . One can use the rational root theorem to check if a polynomial has rational roots.