Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Learning-Enhanced Observer for Linear Time-Invariant Systems with Parametric Uncertainty

Hao Shu

TL;DR

The paper tackles state estimation for discrete-time LTI systems subject to modest parametric uncertainty by introducing a Learning-Enhanced Observer (LEO) that optimizes nominal matrices $A,B,C$ using a gradient-based steady-state output discrepancy loss and then reconstructs an improved Luenberger observer. By modeling perturbations as a slowly varying LTV and leveraging local LTI approximations, the method reframes observer design as a learnable optimization while preserving interpretability. Empirical results across multiple dimensions show consistent reductions in steady-state estimation error, typically exceeding 15%, with strong statistical significance. The approach offers a scalable, data-informed enhancement to classical observers, with open-source code and clear avenues for theoretical guarantees and extensions to broader dynamical settings.

Abstract

This work introduces a learning-enhanced observer (LEO) for linear time-invariant systems with uncertain dynamics. Rather than relying solely on nominal models, the proposed framework treats the system matrices as optimizable variables and refines them through gradient-based minimization of a steady-state output discrepancy loss. The resulting data-informed surrogate model enables the construction of an improved observer that effectively compensates for moderate parameter uncertainty while preserving the structure of classical designs. Extensive Monte Carlo studies across diverse system dimensions show systematic and statistically significant reductions, typically exceeding 15\%, in normalized estimation error for both open-loop and Luenberger observers. These results demonstrate that modern learning mechanisms can serve as a powerful complement to traditional observer design, yielding more accurate and robust state estimation in uncertain systems. Codes are available at https://github.com/Hao-B-Shu/LTI_LEO.

Learning-Enhanced Observer for Linear Time-Invariant Systems with Parametric Uncertainty

TL;DR

The paper tackles state estimation for discrete-time LTI systems subject to modest parametric uncertainty by introducing a Learning-Enhanced Observer (LEO) that optimizes nominal matrices using a gradient-based steady-state output discrepancy loss and then reconstructs an improved Luenberger observer. By modeling perturbations as a slowly varying LTV and leveraging local LTI approximations, the method reframes observer design as a learnable optimization while preserving interpretability. Empirical results across multiple dimensions show consistent reductions in steady-state estimation error, typically exceeding 15%, with strong statistical significance. The approach offers a scalable, data-informed enhancement to classical observers, with open-source code and clear avenues for theoretical guarantees and extensions to broader dynamical settings.

Abstract

This work introduces a learning-enhanced observer (LEO) for linear time-invariant systems with uncertain dynamics. Rather than relying solely on nominal models, the proposed framework treats the system matrices as optimizable variables and refines them through gradient-based minimization of a steady-state output discrepancy loss. The resulting data-informed surrogate model enables the construction of an improved observer that effectively compensates for moderate parameter uncertainty while preserving the structure of classical designs. Extensive Monte Carlo studies across diverse system dimensions show systematic and statistically significant reductions, typically exceeding 15\%, in normalized estimation error for both open-loop and Luenberger observers. These results demonstrate that modern learning mechanisms can serve as a powerful complement to traditional observer design, yielding more accurate and robust state estimation in uncertain systems. Codes are available at https://github.com/Hao-B-Shu/LTI_LEO.

Paper Structure

This paper contains 16 sections, 5 theorems, 32 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Learning-Enhanced Luenberger Observer
  4. From a Noisy LTI System to a Slowly-Varying LTV Model
  5. Local LTI Approximation of an LTV System
  6. Learning-Based Refinement of Luenberger Observers
  7. Practical Considerations
  8. Invariance and Regularization Against Similarity Drift
  9. Conditions Number of the Observability Matrices.
  10. Observability During Learning
  11. Algorithm Summary
  12. Numerical Experiments
  13. Experimental Setup
  14. Statistical Results
  15. Visual Example
  16. ...and 1 more sections

Key Result

Proposition 1

Let the LTV system in Eq. (eq:LTVa) be driven by a fixed input sequence $u_k$. Define $X_{K,N} := (x_K\ x_{K+1}\ \cdots\ x_{K+N-1})$. If $X_{K,N}$ has full column rank, then there exists an LTI system: such that

Figures (1)

  • Figure 1: A visual comparison illustrating the validity of the proposed learning refinement. The curves labeled Real denote the ground truth trajectory, while Ordinary-Open and Ordinary-Luenberger correspond to nominal open-loop and Luenberger observers. Enhanced-Open and Enhanced-Luenberger show the respective observers after learning-based parameter refinement.

Theorems & Definitions (5)

  • Proposition 1
  • Proposition 2
  • Lemma 3
  • Corollary 4
  • Proposition 5