  In distributed model predictive control (DMPC), where a centralized
optimization problem is solved in distributed fashion using dual decomposition,
it is important to keep the number of iterations in the solution algorithm,
i.e. the amount of communication between subsystems, as small as possible. At
the same time, the number of iterations must be enough to give a feasible
solution to the optimization problem and to guarantee stability of the closed
loop system. In this paper, a stopping condition to the distributed
optimization algorithm that guarantees these properties, is presented. The
stopping condition is based on two theoretical contributions. First, since the
optimization problem is solved using dual decomposition, standard techniques to
prove stability in model predictive control (MPC), i.e. with a terminal cost
and a terminal constraint set that involve all state variables, do not apply.
For the case without a terminal cost or a terminal constraint set, we present a
new method to quantify the control horizon needed to ensure stability and a
prespecified performance. Second, the stopping condition is based on a novel
adaptive constraint tightening approach. Using this adaptive constraint
tightening approach, we guarantee that a primal feasible solution to the
optimization problem is found and that closed loop stability and performance is
obtained. Numerical examples show that the number of iterations needed to
guarantee feasibility of the optimization problem, stability and a prespecified
performance of the closed-loop system can be reduced significantly using the
proposed stopping condition.
