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Model Predictive Control (MPC) can be applied to safety-critical controlproblems, providing closed-loop safety and performance guarantees.Implementation of MPC controllers requires solving an optimization problem atevery sampling instant, which is challenging to execute on embedded hardware.To address this challenge, we propose a framework that combines a tightenedsoft constrained MPC formulation with supervised learning to approximate theMPC value function. This combination enables us to obtain a correspondingoptimal control law, which can be implemented efficiently on embeddedplatforms. The framework ensures stability and constraint satisfaction forvarious nonlinear systems. While the design effort is similar to that ofnominal MPC, the proposed formulation provides input-to-state stability (ISS)with respect to the approximation error of the value function. Furthermore, weprove that the value function corresponding to the soft constrained MPC problemis Lipschitz continuous for Lipschitz continuous systems, even if the optimalcontrol law may be discontinuous. This serves two purposes: First, it allows torelate approximation errors to a sufficiently large constraint tightening toobtain constraint satisfaction guarantees. Second, it paves the way for anefficient supervised learning procedure to obtain a continuous value functionapproximation. We demonstrate the effectiveness of the method using a nonlinearnumerical example.

Learning Soft Constrained MPC Value Functions: Efficient MPC Design and Implementation providing Stability and Safety Guarantees