The authors consider the perturbed nonlinear system \[ \dot x= f_0(x)+ \sum^m_1 u_if_i(x) \] with \(x_*\) as an equilibrium, i.e. \(f_j(x_*)= 0\), \(j= 0,1,\dots, m\), and also its linearization at \(x_*\): \[ \dot x= \Biggl(A_0+ \sum^m_1 u_iA_i\Biggr) x= A(u)x. \] Using the semialgebraic description it is shown that \[ r_R(A_0; (A_i))= r_{ex}(f_0; (f_i))\leq r_{as}(f_0; (f_i))\leq r_{st}(f_0; (f_i))\leq\overline r_R(A_0; (A_i)), \] where \(r_{ex}\), \(r_{as}\), \(r_{st}\) are the stability radii of the nonlinear system for exponential, asymptotic and Lyapunov stability of the equilibrium \(x_*\) while \(r_R\) and \(\overline r_R\) are the stability radius (real) and exponential instability radius for the linearized system.