  In this paper, we present a tube-based framework for robust adaptive model
predictive control (RAMPC) for nonlinear systems subject to parametric
uncertainty and additive disturbances. Set-membership estimation is used to
provide accurate bounds on the parametric uncertainty, which are employed for
the construction of the tube in a robust MPC scheme. The resulting RAMPC
framework ensures robust recursive feasibility and robust constraint
satisfaction, while allowing for less conservative operation compared to robust
MPC schemes without model/parameter adaptation. Furthermore, by using an
additional mean-squared point estimate in the objective function the framework
ensures finite-gain $\mathcal{L}_2$ stability w.r.t. additive disturbances. As
a first contribution we derive suitable monotonicity and non-increasing
properties on general parameter estimation algorithms and tube/set based RAMPC
schemes that ensure robust recursive feasibility and robust constraint
satisfaction under recursive model updates. Then, as the main contribution of
this paper, we provide similar conditions for a tube based formulation that is
parametrized using an incremental Lyapunov function, a scalar contraction rate
and a function bounding the uncertainty. With this result, we can provide
simple constructive designs for different RAMPC schemes with varying
computational complexity and conservatism. As a corollary, we can demonstrate
that state of the art formulations for nonlinear RAMPC are a special case of
the proposed framework. We provide a numerical example that demonstrates the
flexibility of the proposed framework and showcase improvements compared to
state of the art approaches.
