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Nonlinear Discrete-Time Observers with Physics-Informed Neural Networks

Hector Vargas Alvarez, Gianluca Fabiani, Ioannis G. Kevrekidis, Nikolaos Kazantzis, Constantinos Siettos

arXiv:2402.12360v1math.NAcs.AIcs.NAmath.DS

TL;DR

Nonlinear discrete-time observer design is challenging, and conventional approaches can be locally valid or rely on restrictive conditions. The paper uses PINNs to learn a state transformation within a single-step exact observer linearization framework. Greedy-wise training substantially improves approximation near steep gradients, with reported error of order 10^-2 near an edge point.

  • Problem

    Nonlinear discrete-time observer design is challenging, and conventional approaches can be locally valid or rely on restrictive conditions.

    Source: §1 · p. 2

  • Method

    PINNs learn a nonlinear state transformation satisfying first-order inhomogeneous functional equations within the single-step exact observer linearization framework.

    Source: §1 · p. 3, §2 · p. 4

  • Results

    10^-2 error near (-0.91, -0.91) with greedy-wise training, which substantially improves approximation over single-step full-domain training near steep gradients.

    Source: §4.1 · p. 11, §4.2 · p. 16

  • Takeaways & Limitations

    Greedy-wise training is used to address steep gradients that transformation maps may exhibit near inherent singularities.

    Source: §5 · p. 20

Abstract

from arXiv · show

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.

1 Introduction

Reliable state estimates matter when critical states cannot be measured directly, but nonlinear discrete-time observer design remains challenging. The paper applies PINNs within a single-step exact observer linearization framework to learn the required state transformation map.

1 Introduction

  • Observers use system models and available measurements to reconstruct state variables that cannot be measured directly.The motivation includes control-algorithm implementation and system-condition monitoring.Source: §1 · p. 1
  • Conventional nonlinear observer approaches can be locally valid or require restrictive conditions, motivating alternatives for nonlinear discrete-time systems.The introduction notes that local linearizations may ignore dominant nonlinearities, while exact observer methods rely on restrictive conditions.Source: §1 · p. 2
  • Prior learning-based approaches include supervised learning that may not explore the full state space and an unsupervised method for the discrete-time framework.The paper also identifies an ANN architecture proposed to improve numerical map approximation and its extension in the transient phase.Source: §1 · pp. 2–3
  • The proposed PINN learns a state transformation map satisfying first-order inhomogeneous functional equations within the single-step exact observer linearization framework.The resulting observer has linearizable error dynamics and assignable convergence rates under rather mild assumptions.Source: §1 · p. 3

2 Preliminaries and Methodological Framework

The framework designs a discrete-time observer by finding a locally invertible transformation that yields linear observer dynamics up to an output injection term. A PINN approximates this transformation, and Newton iterations recover state estimates in the original coordinates.

2 Preliminaries and Methodological Framework

  • The framework assumes an analytic discrete-time system and local observability around the equilibrium.The stated setup places the origin at an equilibrium and imposes rank conditions on the system Jacobian and output derivative.Source: §2 · p. 3
  • The method seeks a locally invertible transformation T that makes the transformed observer dynamics linear up to an output injection term.The transformation must satisfy first-order non-homogeneous functional equations; under the stated theorem assumptions, an analytic, locally invertible solution exists.Source: §2 · p. 4
  • Choosing A with eigenvalues inside the unit disc makes the transformed estimation error decay, with eigenvalue magnitudes regulating its convergence rate.Local invertibility of T then implies that the original-state estimate approaches the actual state asymptotically.Source: §2 · pp. 4–5
  • The single-step design avoids the transformed-output linearity requirement used in a two-step approach.The paper identifies that requirement as the main mathematical source of restrictive conditions in the two-step approach.Source: §2 · p. 5
  • A two-hidden-layer PINN with a linear output layer approximates T by minimizing a loss over a grid in a domain around equilibrium.The loss includes functional-equation, equilibrium, and Jacobian conditions, weighted equally in the illustrative examples.Source: §2 · pp. 5–6
  • Greedy zero-th-order continuation trains progressively over nested subdomains to address steep gradients resembling singularities.Subdomain sizes are selected to provide sufficient accuracy.Source: §2 · p. 6

2.1 Inverse of the Nonlinear Transformation

  • Newton iterations solve T(x)=z at each time step to recover the state estimate in the original coordinates.The paper sets relative and absolute tolerances to 1E−06 for its computations.Source: §2.1 · p. 6

3 Illustrative Benchmark Problems

The section introduces two benchmark problems for evaluating PINN-based observer design, each with an analytically derived transformation map. The examples include a transformation singularity and, in the second problem, an unmeasured state to reconstruct.

3 Illustrative Benchmark Problems

  • The section evaluates the proposed method in two benchmark case studies.Benchmark Problem 1's transformation has a singular point at (−0.5, −0.5), producing a steep gradient near (−0.495, −0.495).Source: §3 · p. 7, Figure 1 · p. 8

3.1 Benchmark Problem 1

  • T1(x1, x2) = ln(1 + x1 + x2) and T2(x1, x2) = x2 are the closed-form transformation map for Benchmark Problem 1.The measured output y coincides with x2, and the map is considered on [−0.495, 0] × [−0.495, 0] because it is singular when x1 + x2 = −1.Source: §3.1 · pp. 7–8

3.2 Benchmark Problem 2

  • Benchmark Problem 2 treats x2 as an unidentified constant parameter or disturbance to estimate, while x1 is measured.The state transformation map is considered on [−0.91, 0] × [−0.91, 0] and has a singularity when x1 = −1.Source: §3.2 · pp. 8–9
  • Because Benchmark Problem 2 has an eigenvalue at 1, the original observer-design method cannot be applied, but the extension relaxing the Poincaré-domain assumption permits observer design.The extension replaces that assumption with a more general Siegel-type condition.Source: §3.2 · pp. 8–9
  • The functional equations for Benchmark Problem 2 admit a unique analytically derived closed-form solution.The benchmark compares the PINN-identified map with the analytical solution to assess approximation accuracy and the resulting observer's performance.Source: §3.2 · p. 9

4 Numerical results

The numerical studies compare sixth-order power-series approximations with PINNs trained over the full domain or progressively using a greedy-wise approach. Across the two benchmarks, greedy-wise training improves approximation near singular regions, while the second benchmark also tests whether PINN results are stable across repeated runs.

4 Numerical results

  • The study compares sixth-order multivariate Taylor solutions with PINNs; the latter use a two-hidden-layer, five-neuron-per-layer network, and greedy continuation reuses weights and biases across expanding subdomains.The PINNs use sigmoid activations and finite differences; the power-series order was chosen to align coefficient count with the PINN parameter count and because higher-order systems are computationally challenging.Source: §4 · pp. 10–11

4.1 Benchmark Problem 1

  • In Benchmark Problem 1, the sixth-order series gives zero error for T2 but T1 accuracy deteriorates to order 10^0 at the grid edge; the authors note this could affect unmeasured-state estimation.The reported error is concentrated beyond the linearization-relevant point [0, 0], particularly near the singular point.Source: §4.1 · p. 11, Figure 3 · p. 12
  • For Benchmark Problem 1, the difference between the true state and the state recovered using the PINN-derived inverse and Newton’s method converges to zero.The comparison uses the analytical inverse map and its numerical approximation.Source: §4.1 · p. 11
  • In Benchmark Problem 1, greedy-wise PINN training significantly outperforms full-domain training, whose error remains of order 1 near the singularity.The test-set comparison shows similar behavior to the training-set results.Source: §4.1 · p. 11, Figure 3 · p. 12

4.2 Benchmark Problem 2

  • In Benchmark Problem 2, uncertainty runs from 100 to 400 produce medians and variances of the same practical order, indicating convergence of PINN training.The analysis concerns T1 and T2 on a 20 × 20 Chebyshev-Lobatto test grid.Source: §4.2 · p. 16, Table 4 · p. 19
  • In Benchmark Problem 2, sixth-order power series are poor near the singular point, while full-domain PINNs generally have lower numerical error but remain modest near the singularity.Reported full-domain PINN errors are of order 10^0 for T1 and 10^1 for T2; the series error reaches order 10^1 at the grid edge.Source: §4.2 · pp. 15–16, Figure 6 · p. 17
  • In Benchmark Problem 2, greedy-wise training improves approximation near the domain edge to order 10^-2 and outperforms single-domain training for T2; test-set results are similar to training results.Figure 7 shows the test-set comparison on a Chebyshev-Lobatto grid.Source: §4.2 · p. 16, Figure 7 · p. 18

5 Conclusions

The paper introduces a single-step PINN framework for nonlinear discrete-time observer design, learning a transformation map through inhomogeneous functional equations. It reports numerical precision that outperforms power-series expansion methods and uses greedy training with continuation near singular regions.

  • Figure 9 compares errors for the analytical inverse of T2 and the state estimate obtained through Newton’s method.Source: Figure 9 · p. 20
  • The PINN scheme exhibits numerical precision that outperforms established power-series expansion solutions.Source: §5 · p. 20
  • The framework learns a nonlinear transformation map for observer design by solving first-order inhomogeneous functional equations in a single step.The map linearizes the observer dynamics up to an output injection term.Source: §5 · pp. 19–20
  • Greedy-wise training refines convergence near abrupt gradients, while zeroth-order continuation supplies improved parameter estimates near singular regions.Source: §5 · p. 20
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