Neural Luenberger state observer for nonautonomous nonlinear systems

Abstract

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.

Figures (11)

• Figure 1: Neural Leunberger state observer for systems with exogenous inputs.
• Figure 2: State trajectories for $\tilde{x}_1$ and $\tilde{x}_2$ for the bioreactor system.
• Figure 3: Varying the number of neurons per layer for the bioreactor system, evaluated on a test trajectory.
• Figure 4: Varying the number training samples for the bioreactor system, evaluated on a test trajectory.
• Figure 5: Comparison of state estimates for the bioreactor system: NLOX (red) versus the analytical solution (light blue), EKF (yellow), and SMO (green).

Theorems & Definitions (14)

• Remark 1
• Definition 1: Differential observability of the drift system
• Definition 2: Instantaneous uniform observability
• Remark 2
• Lemma 1
• Lemma 2
• proof
• Lemma 3
• Lemma 4
• proof