Summary: In this article, 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 with 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 nonincreasing 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 article, 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 with state of the art approaches.
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