A novel design of double chirped pulse amplification laser systems implementing a combination of negatively and positively chirped pulse amplification is proposed for the first time. Without utilizing any extra dispersion compensation element, this design can sufficiently cancel out the second-, third- and especially fourth-order dispersion simultaneously, just by optimizing the parameters of the stretcher and compressor in first chirped pulse amplification stage which applies negatively chirped pulse amplification. The numerical results indicate that near Fourier-transform-limited pulse duration about 20fs can be achieved in high-peak-power femtosecond laser systems up to multi-Petawatt level. This design not only provides a feasible solution for the dispersion control in high-contrast and high-peak-power femtosecond laser systems, but also can avoid the degradation of temporal contrast induced by seed energy loss in the presence of additional dispersion compensation components.