Three typical inerter-spring-damper with negative stiffness （N-ISD） are presented to reduce the vibration of the primary system which is subjected to the displacement excitation from the base. The analytical expressions of stiffness ratio and damping ratio are derived based on the H∞ and H2 optimization design for the N-ISDs. The differential governing equations of a primary system equipped with three different N-ISDs subjected to displacement excitation are firstly developed according to D'Alembert's principle. Then the transfer function of the primary system can be further obtained. The analytical solutions for H∞ optimization of NISDs are presented based on the fixed-point theories to minimize the maximum amplitude of the primary system under harmonic displacement excitation. The analytical solutions for H2 optimization designs are presented based on the extreme value theory to minimize the mean squared displacements of the primary system under random excitation. The validity and superiority of the analytical solutions of parameter optimization for N-ISDs are verified by comparing the present results with the numerical results determined by genetic algorithms. Finally， the effects of vibration control with different types of N-ISDs are further explored. The numerical examples indicate that N-ISDs show a better vibration reduction performance than ISDs. N-SPIS-Ⅱ system shows a lower anti-resonance peak and larger resonance peak spacing under the harmonic excitations than two other N-ISDs systems. Moreover， the mean squared displacements of N-SPIS-Ⅰ and N-SPIS-Ⅱ systems are less than that of N-SIS under random excitation.