  Engineered cyberphysical systems are growing increasingly large and complex.
These systems require scalable controllers that robustly satisfy state and
input constraints in the presence of additive noise -- such controllers should
also be accompanied by theoretical guarantees on feasibility and stability. In
our companion paper, we introduced Distributed and Localized Model Predictive
Control (DLMPC) for large-scale linear systems; DLMPC is a scalable closed-loop
MPC scheme in which subsystems need only exchange local information in order to
synthesize and implement local controllers. In this paper, we provide recursive
feasibility and asymptotic stability guarantees for DLMPC. We leverage the
System Level Synthesis framework to express the maximal positive robust
invariant set for the closed-loop system and its corresponding Lyapunov
function, both in terms of the closed-loop system responses. We use the
invariant set as the terminal set for DLMPC, and show that this guarantees
feasibility with minimal conservatism. We use the Lyapunov function as the
terminal cost, and show that this guarantees stability. We provide fully
distributed and localized algorithms to compute the terminal set offline, and
also provide necessary additions to the online DLMPC algorithm to accommodate
coupled terminal constraint and cost. In all algorithms, only local information
exchanges are necessary, and computational complexity is independent of the
global system size -- we demonstrate this analytically and experimentally. This
is the first distributed MPC approach that provides minimally conservative yet
fully distributed guarantees for recursive feasibility and asymptotic
stability, for both nominal and robust settings.
