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A Comparison of Set-Based Observers for Nonlinear Systems

Nico Holzinger, Matthias Althoff

TL;DR

The paper addresses guaranteed state estimation for nonlinear discrete-time systems by comparing a broad set of set-based observers within the CORA framework. It categorizes methods into intersection-, propagation-, and interval-based approaches, and evaluates them on standardized benchmarks (e.g., Van der Pol oscillator and multi-tank systems) using objective metrics for computation time and conservatism. Key findings show that interval-based methods, particularly pDTDI and ZBKH, offer robust performance in high dimensions, while DC-based and VolMin variants struggle with scalability; constrained zonotopes balance accuracy and efficiency in low-to-mid dimensions but face scalability limits. The work provides a tool-supported, reproducible benchmark and highlights practical trade-offs guiding the selection of guaranteed state estimators for nonlinear systems.

Abstract

Set-based state estimation computes sets of states consistent with a system model given bounded sets of disturbances and noise. Bounding the set of states is crucial for safety-critical applications so that one can ensure that all specifications are met. While numerous approaches have been proposed for nonlinear discrete-time systems, a unified evaluation under comparable conditions is lacking. This paper reviews and implements a representative selection of set-based observers within the CORA framework. To provide an objective comparison, the methods are evaluated on common benchmarks, and we examine computational effort, scalability, and the conservatism of the resulting state bounds. This study highlights characteristic trade-offs between observer categories and set representations, as well as practical considerations arising in their implementation. All implementations are made publicly available to support reproducibility and future development. This paper thereby offers the first broad, tool-supported comparison of guaranteed state estimators for nonlinear discrete-time systems.

A Comparison of Set-Based Observers for Nonlinear Systems

TL;DR

The paper addresses guaranteed state estimation for nonlinear discrete-time systems by comparing a broad set of set-based observers within the CORA framework. It categorizes methods into intersection-, propagation-, and interval-based approaches, and evaluates them on standardized benchmarks (e.g., Van der Pol oscillator and multi-tank systems) using objective metrics for computation time and conservatism. Key findings show that interval-based methods, particularly pDTDI and ZBKH, offer robust performance in high dimensions, while DC-based and VolMin variants struggle with scalability; constrained zonotopes balance accuracy and efficiency in low-to-mid dimensions but face scalability limits. The work provides a tool-supported, reproducible benchmark and highlights practical trade-offs guiding the selection of guaranteed state estimators for nonlinear systems.

Abstract

Set-based state estimation computes sets of states consistent with a system model given bounded sets of disturbances and noise. Bounding the set of states is crucial for safety-critical applications so that one can ensure that all specifications are met. While numerous approaches have been proposed for nonlinear discrete-time systems, a unified evaluation under comparable conditions is lacking. This paper reviews and implements a representative selection of set-based observers within the CORA framework. To provide an objective comparison, the methods are evaluated on common benchmarks, and we examine computational effort, scalability, and the conservatism of the resulting state bounds. This study highlights characteristic trade-offs between observer categories and set representations, as well as practical considerations arising in their implementation. All implementations are made publicly available to support reproducibility and future development. This paper thereby offers the first broad, tool-supported comparison of guaranteed state estimators for nonlinear discrete-time systems.
Paper Structure (15 sections, 1 theorem, 22 equations, 4 figures, 7 tables)

This paper contains 15 sections, 1 theorem, 22 equations, 4 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Set Representations
  4. Range Bounding
  5. Problem Statement
  6. Intersection-based Methods
  7. Propagation-based Methods
  8. Interval Observers
  9. Structural Limitations of Observer Categories
  10. Comparison
  11. General Setup
  12. Special Setups
  13. Performance Metrics
  14. Results
  15. Conclusions

Key Result

Theorem 1

alamo_guaranteed_2005 Consider a continuously differentiable function $f:\mathbb{R}^n\to\mathbb{R}$, an interval vector $\mathcal{X}\in\mathbb{IR}^n$, and let $c\in \mathcal{X}$. Then where $\nabla f(\mathcal{X})$ denotes an interval enclosure of the gradient of $f$ over $\mathcal{X}$.

Figures (4)

  • Figure 1: Reachable set enclosures in the state space for the Van der Pol oscillator with $\mu = 0.1$.
  • Figure 2: Reachable set enclosures in the state space for the Van der Pol oscillator with $\mu = 5$.
  • Figure 3: Interval enclosures of state variables one and four for the 6-tank benchmark system of some selected observers.
  • Figure 4: Interval enclosures of state variable one for the 30-tank benchmark system of some selected observers.

Theorems & Definitions (3)

  • Theorem 1: Mean Value Extension
  • Remark 1
  • Remark 2