Polynomial Stability

A polynomial is said to be stable if all of its roots have negative real parts.

The pole of a transfer function $G(s)$ are the roots of its denominator polynomial. Thus, to determine the stability of an LTI system, we typically need to:

• Check the root positions of a polynomial
• Determine the stability of a polynomial

Consider a polynomial:

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

If $a(s)$ is stable, then $a_i>0$ for $i=1,\dots ,n$. Note that this is only a necessary condition for stability! We can have all positive coefficients but still have an unstable polynomial.