The authors provide a systematic way for the use of polynomial parameter-dependent Lyapunov functions in the robust disturbance attenuation and the robust asymptotic stability problems of finite-dimensional LTIPD systems with constant time-delay parameters in the state. The system under consideration is of the form: \[ \begin{aligned} \dot x(t)&= A_1(\rho)x(t)+A_2(\rho)x(t-h)+B_u u(t)+B_w w(t),\\ x(t)&=\varphi(t), \qquad t\in[-h,0],\\ z(t)&= C_1x(t)+C_2u(t), \end{aligned} \] where \(C_1^TC_2=0\), \(h\) is the constant time-delay parameter, \(\varphi(t)\) is the continuous vector valued initial function, \(z(t)\) is the controlled output, \(x(t), u(t)\) and \(w(t)\) are the state vector, the control input and the disturbance vector, respectively. The affine parameter dependence of the system matrices means that \[ A_1(\rho)=A_0+\rho_1A_1+\rho_2A_2+\dots+\rho_mA_m \]
\[ A_2(\rho)=A_{d0}+\rho_1A_{d1}+\rho_2A_{d2}+\dots+\rho_mA_{dm}. \] Sufficient conditions of increasing precision for the existence of polynomial parameter-dependent quadratic Lyapunov functions are given using LMI formulations. It is shown that the state feedback control can be determined to guarantee the stability of the closed -loop system independently of the time-delay. Some simulation results show that the obtained control law can achieve robust stability and disturbance attenuation, simultaneously.