Distributed model predictive control via finite-step control Lyapunov functions

Abstract

As opposed to classical converse Lyapunov theorems, finite-step converse results are constructive and offer a different starting point: for sufficiently large finite step ahead, say $M$, in an explicit sense, any scaled norm can serve as a converse finite-step Lyapunov function. As for interconnected discrete-time systems, similar lines of argument lead to ``non-conservative'' small-gain conditions. Motivated by this viewpoint, this paper develops a distributed model predictive control framework for constrained interconnected nonlinear discrete-time systems. Each subsystem solves one local optimization problem at each system time step instant by setting the local stage function in form of a local control finite-step like Lyapunov function, using time-aligned neighbor predictions, optimized state-constraint tightening radii, and a finite-step small-gain terminal inequality. Since received neighbor predictions need not equal the trajectories generated by future receding-horizon optimizations, the nominal converse certificates do not alone ensure recursive feasibility or stability. We therefore develop shift-compatible constraint margins, a local one-step terminal feasibility test for networks that are affine in control, and an analytical bound for the prediction and reoptimization mismatch. The resulting analysis gives recursive feasibility, constraint satisfaction, and a practical $M$-step Lyapunov estimate, with asymptotic convergence when the prediction and reoptimization mismatch bound tends to zero. For constrained linear networks, the conditions reduce to finite-dimensional matrix, QP, and SOCP tests. The framework is specialized to current sharing and terminal bus voltage safety under DC/DC power converters' operational constraints in a two-DGU DC microgrid evaluated on a small laboratory-scale prototype.