Neural Luenberger state observer for nonautonomous nonlinear systems

This work proposes a method for model-free synthesis of a state observer for nonlinear systems with manipulated inputs, where the observer is trained offline using a historical or simulation dataset of state measurements. We use the structure of the Kazantzis-Kravaris/Luenberger (KKL) observer, extended to nonautonomous systems by adding an additional input-affine term to the linear time-invariant (LTI) observer-state dynamics, which determines a nonlinear injective mapping of the true states. Both this input-affine term and the nonlinear mapping from the observer states to the system states are learned from data using fully connected feedforward multi-layer perceptron neural networks. Furthermore, we theoretically prove that trained neural networks, when given new input-output data, can be used to observe the states with a guaranteed error bound. To validate the proposed observer synthesis method, case studies are performed on a bioreactor and a Williams-Otto reactor.