Extended Neighboring Extremal Optimal Control with State and Preview Perturbations

Optimal control schemes have achieved remarkable performance in numerous engineering applications. However, they typically require high computational cost, which has limited their use in real-world engineering systems with fast dynamics and/or limited computation power. To address this challenge, Neighboring Extremal (NE) has been developed as an efficient optimal adaption strategy to adapt a pre-computed nominal control solution to perturbations from the nominal trajectory. The resulting control law is a time-varying feedback gain that can be pre-computed along with the original optimal control problem, and it takes negligible online computation. However, existing NE frameworks only deal with state perturbations while in modern applications, optimal controllers (e.g., predictive controllers) frequently incorporate preview information. Therefore, a new NE framework is needed to adapt to such preview perturbations. In this work, an extended NE (ENE) framework is developed to systematically adapt the nominal control to both state and preview perturbations. We show that the derived ENE law is two time-varying feedback gains on the state perturbation and the preview perturbation. We also develop schemes to handle nominal non-optimal solutions and large perturbations to retain optimal performance and constraint satisfaction. Case study on nonlinear model predictive control is presented due to its popularity but it can be easily extended to other optimal control schemes. Promising simulation results on the cart inverted pendulum problem demonstrate the efficacy of the ENE algorithm.