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Robust Moving-horizon Estimation for Nonlinear Systems: From Perfect to Imperfect Optimization

Angelo Alessandri

TL;DR

The paper addresses robust stability of moving-horizon estimators for nonlinear discrete-time systems subjected to disturbances, leveraging incremental input/output-to-state stability (i-IOSS). It develops moving-horizon estimators with a discounted quadratic cost and proves two main results: robust stability under perfect optimization when the plant is i-UIOSS (and i-UEIOSS under discounting), and $\varepsilon$-practically exponentially robust stability under imperfect optimization using a relaxed gradient stopping criterion. Numerical examples—one i-UEIOSS and one not satisfying i-UEIOSS—validate the theory and illustrate the trade-off between horizon length, discounting, and termination thresholds. Together, these results provide practical guidelines for designing robust MHE schemes that balance estimation accuracy and computational effort, with explicit conditions and bounds that can be checked in practice.

Abstract

Robust stability of moving-horizon estimators is investigated for nonlinear discrete-time systems that are detectable in the sense of incremental input/output-to-state stability and are affected by disturbances. The estimate of a moving-horizon estimator stems from the on-line solution of a least-squares minimization problem at each time instant. The resulting stability guarantees depend on the optimization tolerance in solving such minimization problems. Specifically, two main contributions are established: (i) the robust stability of the estimation error, while supposing to solve exactly the on-line minimization problem; (ii) the practical robust stability of the estimation error with state estimates obtained by an imperfect minimization. Finally, the construction of such robust moving-horizon estimators and the performances resulting from the design based on the theoretical findings are showcased with two numerical examples.

Robust Moving-horizon Estimation for Nonlinear Systems: From Perfect to Imperfect Optimization

TL;DR

The paper addresses robust stability of moving-horizon estimators for nonlinear discrete-time systems subjected to disturbances, leveraging incremental input/output-to-state stability (i-IOSS). It develops moving-horizon estimators with a discounted quadratic cost and proves two main results: robust stability under perfect optimization when the plant is i-UIOSS (and i-UEIOSS under discounting), and -practically exponentially robust stability under imperfect optimization using a relaxed gradient stopping criterion. Numerical examples—one i-UEIOSS and one not satisfying i-UEIOSS—validate the theory and illustrate the trade-off between horizon length, discounting, and termination thresholds. Together, these results provide practical guidelines for designing robust MHE schemes that balance estimation accuracy and computational effort, with explicit conditions and bounds that can be checked in practice.

Abstract

Robust stability of moving-horizon estimators is investigated for nonlinear discrete-time systems that are detectable in the sense of incremental input/output-to-state stability and are affected by disturbances. The estimate of a moving-horizon estimator stems from the on-line solution of a least-squares minimization problem at each time instant. The resulting stability guarantees depend on the optimization tolerance in solving such minimization problems. Specifically, two main contributions are established: (i) the robust stability of the estimation error, while supposing to solve exactly the on-line minimization problem; (ii) the practical robust stability of the estimation error with state estimates obtained by an imperfect minimization. Finally, the construction of such robust moving-horizon estimators and the performances resulting from the design based on the theoretical findings are showcased with two numerical examples.
Paper Structure (7 sections, 3 theorems, 107 equations, 2 figures, 2 algorithms)

This paper contains 7 sections, 3 theorems, 107 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Moving-horizon Estimation with Perfect Optimization
  3. Moving-horizon Estimation with Imperfect Optimization
  4. Numerical Results
  5. i-UEIOSS Example
  6. Not Exponential i-UIOSS Example
  7. Conclusion

Key Result

Theorem 1

If system (eq:system:general) is i-UIOSS and there exist ${\mathcal{K}L}$ functions $\alpha_x(\cdot,\cdot), \alpha_v(\cdot,\cdot)$, $\alpha_w(\cdot,\cdot)$ and $N \in {\mathop{\Bbb N}}_{\ge 1}$ such that for all $(s,k) \in [0,+\infty) \times {\mathop{\Bbb N}}_{\ge 0}$, the MHE$_N$ resulting from (eq:mhe:general) is robustly stable.

Figures (2)

  • Figure 1: State and estimated state variables of MHE$_3$ (with fmincon and $\varepsilon=0.01$) and MHE$_5$ (with fminimax and $\varepsilon=0.01$) in noise-free (on the left) and noisy (on the right, zero-mean Gaussian noises with dispersions equal to $0.1$ and $0.5$ for system and measurement disturbances, respectively) simulation runs with $x_0 = ( 2 \,,\, 2 \,,\, 2 )$, $\hat{x}_0 = ( -2 \,,\, -2 \,,\, -2 )$, and $u_t=\sin (0.2 \, t)$.
  • Figure 2: State and estimated state variables of MHE$_3$ and MHE$_4$ with multi-start search fminimax in noise-free (on the left) and noisy simulation runs (on the right, with uniformly distributed random noises in $[-0.01,0.01]$ and $[-0.5,0.5]$ for system and measurement disturbances, respectively) with $x_0 = ( 1 \,,\, 1 )$ and $\hat{x}_0 = ( -1 \,,\, -1 )$.

Theorems & Definitions (10)

  • Definition 1
  • Remark 1
  • Definition 2
  • Theorem 1
  • Remark 2
  • Definition 3
  • Definition 4
  • Theorem 2
  • Remark 3
  • Theorem 3