Recent works have proposed solving the PDE for the unknown injection in a data-driven manner (e.g., via neural networks [3]–[5]), but these approaches remain largely restricted to autonomous systems. As pointed out in Bernard et al. [6], extending the KKL framework to non-autonomous systems requires incorporating an additional term to account for external inputs. For a data-driven observer synthesis, this input-related term should also be parameterized and learned. Therefore, we propose a supervised learning algorithm that learns such an additional term from input–output data through a neural network approach, as well as the nonlinear injection from the states to the observer states. We refer to the learned observer as a “neural Luenberger observer for nonlinear systems with external inputs” (Neural LOX).
Our approach comprises of two steps. First, we collect data from the autonomous system without external inputs and train the output mapping of the KKL observer using polynomial regression. We choose polynomial regression for simplicity and convexity of the formulation. Subsequently, we incorporate the effects of manipulated inputs by utilizing a residual neural network (ResNet) to parameterize and learn the new input-related, state-dependent terms in the observer. We quantify the loss metric of ResNet based on the prediction error between the true states and the observed states. Thus, our neural LOX provides a model-free solution to synthesize a KKL observer with external inputs in a data-driven manner.
To validate the proposed approach, we present two case studies. With a two-state bioreactor (used in [6]), the neural LOX algorithm successfully reconstructs state trajectories with comparable performance to the actual KKL observer in analytical form and significantly outcompetes the extended Kalman filter. We also demonstrate the scalability of the proposed approach to a more complex Williams-Otto reactor with a six-dimensional state space, where it maintains estimation accuracy.
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