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. To address this challenge, Neighboring Extremal (NE) has been developed 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 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 and preview perturbations. 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.
Note to Practitioners
—Due to the vast success in predictive control and advancement in sensing, modern control applications have frequently been incorporating preview information in the control design. For example, the road profile preview obtained from vehicle crowdsourcing is exploited for simultaneous suspension control and energy harvesting, demonstrating a significant performance enhancement using the preview information despite noises in the preview (Hajidavalloo et al., 2022) Another example is thermal management for cabin and battery of hybrid electric vehicles, where traffic preview is employed in hierarchical model predictive control to improve energy efficiency (Amini et al., 2019). In Laks et al. (2011), light detection and ranging systems are used to provide wind disturbance preview to enhance the controls of turbine blades. In Yazdandoost et al. (2022), virtual water preview is employed using integrated water resources management modelling to optimize agricultural patterns and control level of water in lakes. In this work, we develop an extended neighboring extremal framework that can adapt a nominal control law to state and preview perturbations simultaneously. This setup is widely applicable as in many applications, a nominal preview is available while the preview signal can also be measured or estimated online.