In this paper we consider discrete-time piecewise affine hybrid systems with Boolean inputs, outputs and states and we show that they can be represented in a canonical form where the logic variables influence the switching between different submodels but not the continuous-valued dynamics. We exploit this representation for studying Lagrange stability and developing performance analysis procedures based on linear matrix inequalities. Moreover, by using arguments from dissipativity theory for nonlinear systems, we generalize our approach to solve the $H_infty$ analysis problem.