This paper studies the H∞ synthesis for discrete linear systems with state saturation nonlinearity. By introducing
a free matrix whose infinity normis less than or equal to 1, the state under saturation constraint is confined in a convex
hull. Based on this, a bounded real lemma for discrete linear systems with state saturation is obtained in terms of matrix
inequalities which can be solved by using the presented iterative linear matrix inequality algorithm. The synthesis problem
is also solved and an explicit expression for the desired state feedback control law is given. A numerical example shows the
effectiveness and correctness of the presented method.