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Optimal control schemes have achieved remarkable performance in numerousengineering applications. However, they typically require high computationalcost, which has limited their use in real-world engineering systems with fastdynamics and/or limited computation power. To address this challenge,Neighboring Extremal (NE) has been developed as an efficient optimal adaptionstrategy to adapt a pre-computed nominal control solution to perturbations fromthe nominal trajectory. The resulting control law is a time-varying feedbackgain that can be pre-computed along with the original optimal control problem,and it takes negligible online computation. However, existing NE frameworksonly deal with state perturbations while in modern applications, optimalcontrollers (e.g., predictive controllers) frequently incorporate previewinformation. Therefore, a new NE framework is needed to adapt to such previewperturbations. In this work, an extended NE (ENE) framework is developed tosystematically adapt the nominal control to both state and previewperturbations. We show that the derived ENE law is two time-varying feedbackgains on the state perturbation and the preview perturbation. We also developschemes to handle nominal non-optimal solutions and large perturbations toretain optimal performance and constraint satisfaction. Case study on nonlinearmodel predictive control is presented due to its popularity but it can beeasily extended to other optimal control schemes. Promising simulation resultson the cart inverted pendulum problem demonstrate the efficacy of the ENEalgorithm.

Extended Neighboring Extremal Optimal Control with State and Preview Perturbations