In this paper we investigate some analysis and control problems with performance for discrete-time hybrid systems in the Piece-Wise Affine form. By using arguments from the dissipativity theory for nonlinear systems, we show that $H_infty$ analysis and synthesis problems can be formulated and solved via Linear Matrix Inequalities that take into account the switching structure of the systems. Moreover, such procedures can be generalized in order to solve both robust control problems and analysis/synthesis with other performance indices like the Generalized $H_2$ norm.