  We consider the problem of zeroing an error output of a nonlinear
discrete-time system in the presence of constant exogenous disturbances,
subject to hard convex constraints on the input signal. The design
specification is formulated as a variational inequality, and we adapt a
forward-backward splitting algorithm to act as an integral controller which
ensures that the input constraints are met at each time step. We establish a
low-gain stability result for the closed-loop system when the plant is
exponentially stable, generalizing previously known results for integral
control of discrete-time systems. Specifically, it is shown that if the
composition of the plant equilibrium input-output map and the integral feedback
gain is strongly monotone, then the closed-loop system is exponentially stable
for all sufficiently small integral gains. The method is illustrated via a
simple numerical example.
