Data-driven state observation for nonlinear systems based on online learning

For controlling nonlinear processes represented by state-space models, a state observer is needed to estimate the states from the trajectories of measured variables. While model-based observer synthesis is traditionally challenging due to the difficulty of solving pertinent partial differential equations, this article proposes an efficient model-free, data-driven approach for state observation, which is suitable for data-driven nonlinear control without accurate nonlinear models. Specifically, by using a Chen-Fliess series representation of the observer dynamics, state observation is endowed with an online least squares regression formulation that can be solved by gradient flow with performance guarantees. When the target state trajectories for regression are unavailable, by exploiting the Kazantzis-Kravaris/Luenberger observer structure, state observation is reduced to a dimensionality reduction problem amenable to an online implementation of kernel principal component analysis. The proposed approach is demonstrated by a limit cycle dynamics and a chaotic system.