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Robust Fault Estimators for Nonlinear Systems: An Ultra-Local Model Design

Farhad Ghanipoor, Carlos Murguia, Peyman Mohajerin Esfahani, Nathan van de Wouw

TL;DR

The paper tackles fault estimation in uncertain nonlinear systems by augmenting the plant with an ultra-local fault model and designing a nonlinear observer to jointly estimate the state and fault dynamics. It proves stability properties: exact fault reconstruction in the absence of perturbations and input-to-state stability under model mismatch, disturbances, and measurement noise, and it provides SDP-based LMIs to bound $\mathcal{L}_2$-gain and $\mathcal{L}_2-\mathcal{L}_\infty$-induced norms. The authors formulate optimal fault-estimator design as convex programs that minimize these gains while enforcing an ISS constraint, enabling explicit performance-robustness trade-offs. The approach is demonstrated on a benchmark robotic arm, showing how learned uncertainty models and the proposed design balance fault-tracking accuracy against noise rejection. Overall, the ultra-local fault-estimation framework offers a controllable, robust mechanism for estimating time-varying process and sensor faults in nonlinear, uncertain environments.

Abstract

This paper proposes a nonlinear estimator for the robust reconstruction of process and sensor faults for a class of uncertain nonlinear systems. The proposed fault estimation method augments the system dynamics with an ultra-local (in time) internal state-space representation (a finite chain of integrators) of the fault vector. Next, a nonlinear state observer is designed based on the known parts of the augmented dynamics. This nonlinear filter (observer) reconstructs the fault signal as well as the states of the augmented system. We provide sufficient conditions that guarantee stability of the estimation error dynamics: firstly, asymptotic stability (i.e., exact fault estimation) in the absence of perturbations induced by the fault model mismatch (mismatch between internal ultra-local model for the fault and the actual fault dynamics), uncertainty, external disturbances, and measurement noise and, secondly, Input-to-State Stability (ISS) of the estimation error dynamics is guaranteed in the presence of these perturbations. In addition, to support performance-based estimator design, we provide Linear Matrix Inequality (LMI) conditions for $\mathcal{L}_2$-gain and $\mathcal{L}_2-\mathcal{L}_\infty$ induced norm and cast the synthesis of the estimator gains as a semi-definite program where the effect of model mismatch and external disturbances on the fault estimation error is minimized in the sense of $\mathcal{L}_2$-gain, for an acceptable $\mathcal{L}_2-\mathcal{L}_\infty$ induced norm with respect to measurement noise. The latter result facilitates a design that explicitly addresses the performance trade-off between noise sensitivity and robustness against model mismatch and external disturbances. Finally, numerical results for a benchmark system illustrate the performance of the proposed methodologies.

Robust Fault Estimators for Nonlinear Systems: An Ultra-Local Model Design

TL;DR

The paper tackles fault estimation in uncertain nonlinear systems by augmenting the plant with an ultra-local fault model and designing a nonlinear observer to jointly estimate the state and fault dynamics. It proves stability properties: exact fault reconstruction in the absence of perturbations and input-to-state stability under model mismatch, disturbances, and measurement noise, and it provides SDP-based LMIs to bound -gain and -induced norms. The authors formulate optimal fault-estimator design as convex programs that minimize these gains while enforcing an ISS constraint, enabling explicit performance-robustness trade-offs. The approach is demonstrated on a benchmark robotic arm, showing how learned uncertainty models and the proposed design balance fault-tracking accuracy against noise rejection. Overall, the ultra-local fault-estimation framework offers a controllable, robust mechanism for estimating time-varying process and sensor faults in nonlinear, uncertain environments.

Abstract

This paper proposes a nonlinear estimator for the robust reconstruction of process and sensor faults for a class of uncertain nonlinear systems. The proposed fault estimation method augments the system dynamics with an ultra-local (in time) internal state-space representation (a finite chain of integrators) of the fault vector. Next, a nonlinear state observer is designed based on the known parts of the augmented dynamics. This nonlinear filter (observer) reconstructs the fault signal as well as the states of the augmented system. We provide sufficient conditions that guarantee stability of the estimation error dynamics: firstly, asymptotic stability (i.e., exact fault estimation) in the absence of perturbations induced by the fault model mismatch (mismatch between internal ultra-local model for the fault and the actual fault dynamics), uncertainty, external disturbances, and measurement noise and, secondly, Input-to-State Stability (ISS) of the estimation error dynamics is guaranteed in the presence of these perturbations. In addition, to support performance-based estimator design, we provide Linear Matrix Inequality (LMI) conditions for -gain and induced norm and cast the synthesis of the estimator gains as a semi-definite program where the effect of model mismatch and external disturbances on the fault estimation error is minimized in the sense of -gain, for an acceptable induced norm with respect to measurement noise. The latter result facilitates a design that explicitly addresses the performance trade-off between noise sensitivity and robustness against model mismatch and external disturbances. Finally, numerical results for a benchmark system illustrate the performance of the proposed methodologies.
Paper Structure (19 sections, 6 theorems, 70 equations, 4 figures)

This paper contains 19 sections, 6 theorems, 70 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Ultra Local Fault Representation
  4. Augmented Dynamics
  5. Fault Estimator
  6. Fault Estimator Error Dynamics
  7. Fault Estimator Design
  8. ISS Estimation Error Dynamics
  9. $\mathcal{L}_2$ Performance Criterion
  10. $\mathcal{L}_2-\mathcal{L}_\infty$ Induced Norm Performance Criterion
  11. Optimal Fault Estimator Design
  12. Discussion on Uncertainty Models
  13. Simulation Results
  14. Model Learning for Uncertainty
  15. Robustification against Noise
  16. ...and 4 more sections

Key Result

Proposition 1

(ISS Estimation Error Dynamics) Consider the error dynamics eq:error_dynamics_combined and let Assumption assum:lipschitz holds with Lipschitz constant $\alpha$. Suppose there exist matrices $P \in {\mathbb{R}^{n_z \times n_z}},$ with $P \succ0, R \in {\mathbb{R}^{n_z \times m}}, Q \in {\mathbb{R}^{ that is a necessary and sufficient condition for ISS with respect to input $\bar{\omega}_a$.

Figures (4)

  • Figure 1: Benchmark System Schematic.
  • Figure 2: The actual actuator fault and its estimates.
  • Figure 3: The actual actuator fault and its estimates using the optimal $\mathcal{L}_2$-gain with respect to disturbance $\omega_a$ estimator.
  • Figure 4: The actual actuator fault and its estimates using different optimal criteria.

Theorems & Definitions (16)

  • Definition 1
  • Remark 1: Boundedness Assumptions
  • Proposition 1
  • Remark 2: LMI Feasibility
  • Definition 2
  • Proposition 2: $\mathcal{L}_2$-gain LMI
  • Definition 3
  • Proposition 3: $\mathcal{L}_2-\mathcal{L}_\infty$ Induced Norm LMI
  • Remark 3: Extension
  • Remark 4: Exact Estimation
  • ...and 6 more