Is suboptimal nonlinear MPC inherently robust?

We study the inherent robust stability properties of nonlinear discrete-time systems controlled by suboptimal model predictive control (MPC). The unique requirement to suboptimal MPC is that it does not increase the cost function with respect to a well-defined warm start, and it is therefore implementable in general nonconvex problems for which no suboptimality margins can be enforced. Because the suboptimal control law is a set-valued map, the closed-loop system is described by a difference inclusion. Under conventional assumptions on the system and cost functions, we establish nominal exponential stability of the equilibrium. If, in addition, a continuity assumption of the feasible input set holds, we prove robust exponential stability with respect to small, but otherwise arbitrary, additive process disturbances and state measurement/estimation errors. To obtain these results, we show that the suboptimal cost is a continuous exponential Lyapunov function for an appropriately augmented closed-loop system. Moreover, we show that robust recursive feasibility is implied by such (nominal) exponential cost decay. We present an illustrative example to clarify the main ideas and assumptions.