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Strong nonlinear detectability and moving horizon estimation for nonlinear systems with unknown inputs

Yang Guo, Jaime A. Moreno, Stefan Streif

TL;DR

The paper tackles state estimation for nonlinear discrete-time systems with unknown inputs and bounded measurement noise, introducing strong nonlinear detectability as a necessary and sufficient condition for the existence of unknown input state estimators (UISEs). It develops UISE designs based on moving horizon estimation (MHE) using full-order models, with Lyapunov-based guarantees that extend robust MHE to unbounded disturbances. To reduce online complexity and latency, it then derives a tighter detectability notion and a reduced-order model, enabling a two-stage MHE-based UISE that first estimates a reduced-state and then reconstructs the full state. A crop-growth simulation demonstrates improved accuracy of UISE variants over conventional MHE and showcases substantial computational savings with the two-stage approach. Overall, the work provides a rigorous framework for UISE design in nonlinear systems with unknown inputs, with practical implications for robust estimation, fault detection, and security applications.

Abstract

This paper considers state estimation for general nonlinear discrete-time systems subject to measurement noise and possibly unbounded unknown inputs. To approach this problem, we first propose the concept of strong nonlinear detectability. This condition is sufficient and necessary for the existence of unknown input state estimators (UISEs), which reconstruct states from noisy sampled measurements and yield bounded estimation error even for unbounded unknown inputs. Based on the proposed detectability notion, a UISE is designed via a moving horizon estimation strategy using a full-order model as well as past and current measurements. Next, we tighten this detectability notion to design a two-stage MHE-based UISE, which is computationally more efficient than the MHE-based UISE using full-order models. In a simulation example with a plant growth process, both variants of MHE-based UISEs are compared with a conventional MHE to illustrate the merits of the developed methods.

Strong nonlinear detectability and moving horizon estimation for nonlinear systems with unknown inputs

TL;DR

The paper tackles state estimation for nonlinear discrete-time systems with unknown inputs and bounded measurement noise, introducing strong nonlinear detectability as a necessary and sufficient condition for the existence of unknown input state estimators (UISEs). It develops UISE designs based on moving horizon estimation (MHE) using full-order models, with Lyapunov-based guarantees that extend robust MHE to unbounded disturbances. To reduce online complexity and latency, it then derives a tighter detectability notion and a reduced-order model, enabling a two-stage MHE-based UISE that first estimates a reduced-state and then reconstructs the full state. A crop-growth simulation demonstrates improved accuracy of UISE variants over conventional MHE and showcases substantial computational savings with the two-stage approach. Overall, the work provides a rigorous framework for UISE design in nonlinear systems with unknown inputs, with practical implications for robust estimation, fault detection, and security applications.

Abstract

This paper considers state estimation for general nonlinear discrete-time systems subject to measurement noise and possibly unbounded unknown inputs. To approach this problem, we first propose the concept of strong nonlinear detectability. This condition is sufficient and necessary for the existence of unknown input state estimators (UISEs), which reconstruct states from noisy sampled measurements and yield bounded estimation error even for unbounded unknown inputs. Based on the proposed detectability notion, a UISE is designed via a moving horizon estimation strategy using a full-order model as well as past and current measurements. Next, we tighten this detectability notion to design a two-stage MHE-based UISE, which is computationally more efficient than the MHE-based UISE using full-order models. In a simulation example with a plant growth process, both variants of MHE-based UISEs are compared with a conventional MHE to illustrate the merits of the developed methods.

Paper Structure

This paper contains 10 sections, 11 theorems, 68 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem setup
  3. Unknown input state estimators with full-order models
  4. Strong nonlinear detectability
  5. MHE-based UISE
  6. Unknown input state estimators with reduced-order models
  7. A tighter notion for strong nonlinear detectability
  8. Two-stage MHE-based UISE
  9. Numerical example
  10. Conclusions

Key Result

Theorem III.2

The system eq:sys is strong nonlinearly detectable according to Definition def:strong_detect if and only if there exist $\alpha_1, \alpha_2 \in \mathcal{K}_{\infty}$, $\sigma_v, \sigma_y, \in \mathcal{K}$, $\mu \in (0,1)$ and an incremental storage function $V(x, \widetilde{x}): \mathbb{X} \times for all $(x,u,w, v, y)$, $(\widetilde{x},u,\widetilde{w}, \widetilde{v}, \widetilde{y})\in \mathbb{

Figures (3)

  • Figure 1: True $x^{d,1}$ (black) and estimated $x^{d,1}$ of UISE in Theorem \ref{['theo:MHE_full']} (green), two-stage UISE in Theorem \ref{['theo:stab']} (red) and standard MHE from schiller2023 (blue).
  • Figure 2: Estimation errors of UISE in Theorem \ref{['theo:MHE_full']} (green), two-stage UISE in Theorem \ref{['theo:stab']} (red) and standard MHE from schiller2023 (blue)
  • Figure 3: Unknown input $w$.

Theorems & Definitions (27)

  • Definition II.1: State estimator
  • Definition II.2: Unknown input state estimator
  • Definition III.1: Strong nonlinear detectability
  • Theorem III.2
  • Lemma III.3
  • proof
  • Theorem III.4
  • proof
  • Lemma III.5
  • proof
  • ...and 17 more