Abstract: This paper shows the robust stablility of closed-loop systems with rational tranfer function, and gives a necessary and sufficient condition for robust stability of polynomial family δ(s) =" Qo(s)Po(s) +… + Qm(s)Pm(s). It states a 'necessary and sufficient finite checking for robust stability of polynomial familyδ(s) when Qi(s) satisfies some assumptions, and when Qi(s) is a general polynomial, a so-called edge-checking for robust stability ofδ(s) is obtained.