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Robust stability of moving horizon estimation for continuous-time systems

Julian D. Schiller, Matthias A. Müller

TL;DR

This work develops a continuous-time moving horizon estimation framework with a time-discounted least-squares objective for nonlinear systems that are detectable in the i-iIOSS sense. By tying the horizon length to a Lyapunov-based dissipation inequality, it proves robust global exponential stability of the estimation error in a time-discounted $L^2$ to $L^∞$ sense and provides LMIs to efficiently construct i-iIOSS Lyapunov functions. The proposed approach offers practical advantages over discrete-time MHE, such as arbitrary online sampling and horizon design, and validates the theory on a nonlinear chemical reactor benchmark. The combination of RGES guarantees and easily verifiable LMIs makes the framework appealing for robust nonlinear state estimation in continuous time.

Abstract

We consider a moving horizon estimation (MHE) scheme involving a discounted least squares objective for general nonlinear continuous-time systems. Provided that the system is detectable (incrementally integral input/output-to-state stable, i-iIOSS), we show that there exists a sufficiently long estimation horizon that guarantees robust global exponential stability of the estimation error in a time-discounted $L^2$-to-$L^\infty$ sense. In addition, we show that i-iIOSS Lyapunov functions can be efficiently constructed by verifying certain linear matrix inequality conditions. In combination, we propose a flexible Lyapunov-based MHE framework in continuous time, which particularly offers more tuning possibilities than its discrete-time analog, and provide sufficient conditions for stability that can be easily verified in practice. Our results are illustrated by a numerical example.

Robust stability of moving horizon estimation for continuous-time systems

TL;DR

This work develops a continuous-time moving horizon estimation framework with a time-discounted least-squares objective for nonlinear systems that are detectable in the i-iIOSS sense. By tying the horizon length to a Lyapunov-based dissipation inequality, it proves robust global exponential stability of the estimation error in a time-discounted to sense and provides LMIs to efficiently construct i-iIOSS Lyapunov functions. The proposed approach offers practical advantages over discrete-time MHE, such as arbitrary online sampling and horizon design, and validates the theory on a nonlinear chemical reactor benchmark. The combination of RGES guarantees and easily verifiable LMIs makes the framework appealing for robust nonlinear state estimation in continuous time.

Abstract

We consider a moving horizon estimation (MHE) scheme involving a discounted least squares objective for general nonlinear continuous-time systems. Provided that the system is detectable (incrementally integral input/output-to-state stable, i-iIOSS), we show that there exists a sufficiently long estimation horizon that guarantees robust global exponential stability of the estimation error in a time-discounted -to- sense. In addition, we show that i-iIOSS Lyapunov functions can be efficiently constructed by verifying certain linear matrix inequality conditions. In combination, we propose a flexible Lyapunov-based MHE framework in continuous time, which particularly offers more tuning possibilities than its discrete-time analog, and provide sufficient conditions for stability that can be easily verified in practice. Our results are illustrated by a numerical example.
Paper Structure (14 sections, 4 theorems, 57 equations, 1 figure)

This paper contains 14 sections, 4 theorems, 57 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Problem Setup
  5. Moving horizon estimation
  6. Setup
  7. Robust stability analysis
  8. Constructing smooth i-iIOSS Lyapunov functions
  9. Numerical example
  10. Conclusion
  11. Proofs
  12. Proof of Proposition \ref{['prop:RGES']}
  13. Proof of Proposition \ref{['prop:MHE']}
  14. Proof of Theorem \ref{['thm:IOSS']}

Key Result

Proposition 1

Suppose there exists a state estimator for system eq:sys. If it is RGAS in the sense of Definition def:RGAS, then there exist functions $\psi\in\mathcal{KL}$ and $\gamma\in\mathcal{K}_\infty$ such that for all $t_i\in\mathcal{T}$, $\chi,\hat{\chi}\in\mathcal{X}$, $u\in\mathcal{M}_{\mathcal{U}}$, and $w\in\mathcal{M}_{\mathcal{W}}$. If it is RGES, then there exist $C>0$ and $\rho\in[0,1)$ such tha

Figures (1)

  • Figure 1: Top left: disturbance signal $w$; top right: sampling times $t_i$ contained in the set $\mathcal{T}$; bottom left: comparison of the estimated trajectory $\hat{x}$\ref{['eq:x_hat_CT']}, the true system trajectory $x$, and measurements $y$; bottom right: corresponding estimation error. The circles in the bottom plots correspond to the estimates $\hat{x}(t_i)$ at the sampling times $t_i\in\mathcal{T}$.

Theorems & Definitions (16)

  • Definition 1: i-iIOSS Lyapunov function, Schiller2023a
  • Definition 2: RGAS, RGES
  • Remark 1: Point-wise error bound
  • Proposition 1
  • Remark 2: Discounting
  • Remark 3: Tuning
  • Remark 4: Estimated trajectory
  • Proposition 2
  • Theorem 1: RGES of MHE
  • proof
  • ...and 6 more