This thesis addresses robust stability, robust H¡Û and H2 performance and design of discrete-time polytopic systems with an LFT uncertainty assumed at each vertex. A sequence of relaxed sufficient analysis results based on the HPPDL matrix approach has been extended to cope with such more general uncertainty structure. The state feedback gain matrix to achieve robust H¡Û and H2 performance can be easily computed from the derived sufficient LMIs. The larger the degree of homogeneous polynomial is, the lower H¡Û and H2 norm are achieved. Numerical examples are included to illustrate the derived results.