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We consider the problem of zeroing an error output of a nonlineardiscrete-time system in the presence of constant exogenous disturbances,subject to hard convex constraints on the input signal. The designspecification is formulated as a variational inequality, and we adapt aforward-backward splitting algorithm to act as an integral controller whichensures that the input constraints are met at each time step. We establish alow-gain stability result for the closed-loop system when the plant isexponentially stable, generalizing previously known results for integralcontrol of discrete-time systems. Specifically, it is shown that if thecomposition of the plant equilibrium input-output map and the integral feedbackgain is strongly monotone, then the closed-loop system is exponentially stablefor all sufficiently small integral gains. The method is illustrated viaapplication to a four-tank process.

Low-Gain Stability of Projected Integral Control for Input-Constrained Discrete-Time Nonlinear Systems