Nonlinear Discrete-Time Observers with Physics-Informed Neural Networks

TL;DR

This work uses Physics-Informed Neural Networks (PINNs) to solve the discrete-time nonlinear observer state estimation problem and performs an uncertainty quantification analysis for the proposed PINN scheme and compares it with conventional power-series numerical implementations.

Abstract

We use Physics-Informed Neural Networks (PINNs) to solve the discrete-time nonlinear observer state estimation problem. Integrated within a single-step exact observer linearization framework, the proposed PINN approach aims at learning a nonlinear state transformation map by solving a system of inhomogeneous functional equations. The performance of the proposed PINN approach is assessed via two illustrative case studies for which the observer linearizing transformation map can be derived analytically. We also perform an uncertainty quantification analysis for the proposed PINN scheme and we compare it with conventional power-series numerical implementations, which rely on the computation of a power series solution.

Figures (9)

• Figure 1: Analytical solution of the NFEs (\ref{['FEsys']}) in $[-0.495,0]\times [-0.495,0]$. (a) $T_1(x_1,x_2)=\ln(1+x_1+x_2)$. A steep-gradient at $(-0.495,-0.495)$ is due to the presence of a singular point at $(x_1,x_2)=(-0.5,-0.5)$. (b) $T_2(x_1,x_2)=x_2$.
• Figure 2: Analytical solution of the functional equations (\ref{['FEsys']}). (a) $T_1(x_1,x_2)=\frac{x_1(t)}{1+x_1(t)}+0.9x_2$. (b) $T_2(x_1,x_2)=\frac{5}{2}\biggl(\frac{x_1(t)}{1+x_1(t)} + x_2\biggl)$. A steep-gradient at $(-0.91,-0.91)$ is due to the presence of a singular point at $x_1=-1$.
• Figure 3: Benchmark problem 1. Training sets (grids of $15 \times 15$ equispaced distributed points). Numerical approximation accuracy (difference between the computed and analytical solution) of $T_1(x_1,x_2)$ (left column) and $T_2(x_1,x_2)$ (right column) using the various schemes. (a),(b) $6th$ order power-series expansion of $T_1(x_1,x_2)$ and $T_2(x_1,x_2)$ and the right-hand side of the model (\ref{['eq:benchmark1']}) in $[-0.495,0]\times[-0.495,0]$. (c),(d) PINN trained in the entire domain $[-0.495,0]\times[-0.495,0]$. (e),(f) PINN trained via the greedy-wise approach.
• Figure 4: Benchmark problem 1. Test sets (grids of $20 \times 20$ Chebyshev-distributed points). Numerical approximation accuracy (difference between the computed and analytical solution) of $T_1(x_1,x_2)$ (left column) and $T_2(x_1,x_2)$ (right column) using the various schemes. (a),(b) $6th$ order power-series expansion of $T_1(x_1,x_2)$ and $T_2(x_1,x_2)$ and the right-hand side of the model (\ref{['eq:benchmark1']}) in $[-0.495,0]\times[-0.495,0]$. (c),(d) PINN trained in the entire domain $[-0.495,0]\times[-0.495,0]$. (e),(f) PINN trained via the greedy-wise approach.
• Figure 5: Benchmark Problem 1: (a)-(c) Panels involve a comprehensive comparison between the performance of Physics-Informed Neural Networks (PINN) trained using two distinct approaches: one trained on the entire domain at once (Black line) and another trained using a greedy approach (blue line). The evaluation is conducted on a test set comprising $20\times20$ Chebyshev nodes of the second type. The reported performance metrics include the $L_1$, $L_2$, and $L_\infty$ error norms, presented in terms of the median and the 5th to 95th percentiles. The evaluation is performed over 100 independent runs for the transformation $T_1(x_1, x_2)$. Panel (d) depicts a comparison between observation errors. The red line represents the difference between the analytical inverse $\Bar{x}_1$ of the transformation $T_1(x_1, x_2)$ and the real state $x_1$, whereas the blue line depicts the difference between the true state $x_1$ and its approximation obtained through Newton's method, denoted as $\hat{x}_1$.

Theorems & Definitions (2)

• Definition 2.1
• Theorem 2.1