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Sample-based Moving Horizon Estimation

Isabelle Krauss, Victor G. Lopez, Matthias A. Müller

TL;DR

The paper tackles state estimation for nonlinear discrete-time systems when measurements arrive irregularly or sparsely. It introduces a sample-based moving horizon estimation scheme with a cost tailored to missing data, and proves robust global exponential stability of the estimator under a sample-based i-IOSS detectability condition. For linear systems, it connects sample-based observability to i-IOSS, enabling design of sampling strategies that guarantee RGES. The approach is demonstrated on a biomedical HPTH-axis model, showing accurate state reconstruction under sparse measurements and robustness to disturbances.

Abstract

In this paper, we propose a sample-based moving horizon estimation (MHE) scheme for general nonlinear systems to estimate the current system state using irregularly and/or infrequently available measurements. The cost function of the MHE optimization problem is suitably designed to accommodate these irregular output sequences. We also establish that, under a suitable sample-based detectability condition known as sample-based incremental input/output-to-state stability (i-IOSS), the proposed sample-based MHE achieves robust global exponential stability (RGES). Additionally, for the case of linear systems, we draw connections between sample-based observability and sample-based i-IOSS. This demonstrates that previously established conditions for linear systems to be sample-based observable can be utilized to verify or design sampling strategies that satisfy the conditions to guarantee RGES of the sample-based MHE. Finally, the effectiveness of the proposed sample-based MHE is illustrated through a simulation example.

Sample-based Moving Horizon Estimation

TL;DR

The paper tackles state estimation for nonlinear discrete-time systems when measurements arrive irregularly or sparsely. It introduces a sample-based moving horizon estimation scheme with a cost tailored to missing data, and proves robust global exponential stability of the estimator under a sample-based i-IOSS detectability condition. For linear systems, it connects sample-based observability to i-IOSS, enabling design of sampling strategies that guarantee RGES. The approach is demonstrated on a biomedical HPTH-axis model, showing accurate state reconstruction under sparse measurements and robustness to disturbances.

Abstract

In this paper, we propose a sample-based moving horizon estimation (MHE) scheme for general nonlinear systems to estimate the current system state using irregularly and/or infrequently available measurements. The cost function of the MHE optimization problem is suitably designed to accommodate these irregular output sequences. We also establish that, under a suitable sample-based detectability condition known as sample-based incremental input/output-to-state stability (i-IOSS), the proposed sample-based MHE achieves robust global exponential stability (RGES). Additionally, for the case of linear systems, we draw connections between sample-based observability and sample-based i-IOSS. This demonstrates that previously established conditions for linear systems to be sample-based observable can be utilized to verify or design sampling strategies that satisfy the conditions to guarantee RGES of the sample-based MHE. Finally, the effectiveness of the proposed sample-based MHE is illustrated through a simulation example.

Paper Structure

This paper contains 7 sections, 5 theorems, 44 equations, 2 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES AND SETUP
  3. SAMPLE-BASED MHE
  4. STABILITY ANALYSIS
  5. LINEAR SYSTEMS AND SAMPLE-BASED i-IOSS
  6. NUMERICAL EXAMPLE
  7. CONCLUSION

Key Result

Proposition 1

The solution of the optimization problem (eq:NLP) at time $t\geq0$ is given by

Figures (2)

  • Figure 1: True states (red) and sample-based MHE results (green) when considering a measurement once a day, i.e., on average every twelfth time instant a measurement was taken. In the top-left, a zoomed-in plot of the first 10 days.
  • Figure 2: Estimation error for four different settings. We simulated the cases that a measurement was taken 12 times a day (blue), i.e., at every time instant, once a day (green), every second day (yellow), and every third day (red).

Theorems & Definitions (17)

  • Definition 1: i-IOSS
  • Definition 2: Sampling set $K$ Kra25
  • Remark 1
  • Definition 3: RGES
  • Remark 2
  • Proposition 1: Kra25a
  • Remark 3
  • Theorem 1: Sample-based MHE is RGES
  • proof
  • Remark 4
  • ...and 7 more