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A Model-Based Extended State Observer for Discrete-Time Linear Multivariable Systems

Jinfeng Chen, Zhiqiang Gao, Qin Lin

TL;DR

The paper develops a model-based extended state observer (MB-ESO) for discrete-time, multivariable linear systems by augmenting the state with disturbances and demonstrates a rigorous link to unknown input observers (UIO). It establishes a necessary and sufficient condition—absence of invariant zeros between disturbances and outputs—for MB-ESO and its variant (with non-diagonal disturbance gains), and shows that MB-ESO can emulate a delayed UIO when conditions on the disturbance gain matrix and observer structure are met. An exact transfer function for disturbance estimation error is derived, enabling explicit error bounds that depend on the disturbance relative degree and the observer bandwidth; the error decreases monotonically with time and bandwidth under appropriate assumptions. Numerical simulations corroborate the theoretical connection and error properties, illustrating the advantages of MB-ESO in disturbance decoupling and smoothing in noisy measurements, and highlighting the impact of invariant zeros on disturbance estimation for alternative estimators. Overall, the work provides a solid theoretical foundation and practical tools for disturbance estimation in multivariable discrete-time systems, with implications for safety-critical control and robust design.

Abstract

A model-based extended state observer (MB-ESO) and its variant are proposed for discrete-time linear multivariable systems, where multiple disturbances are defined as an extended state vector in the same manner as in the original formulation of ESO. The variant MB-ESO extends the MB-ESO to address cases where the disturbance gain matrix is non-diagonal. Leveraging the connection between the variant MB-ESO and the well-known unknown input observer (UIO), the condition for the existence of a MB-ESO and its variant in multivariable systems is established, for the first time, i.e., no invariant zeros exist between the disturbances and the plant outputs. It is shown that, with the observer eigenvalues all placed at the origin and the subsystems decoupled, the variant MB-ESO produces the identical disturbance estimation as that of UIO. Moreover, the error characteristics of MB-ESO and its variant are analyzed and the transfer functions associated with the disturbance estimation errors are derived. It is demonstrated both mathematically and in simulations that the disturbance estimation error of MB-ESO decreases monotonically with respect to both the observer eigenvalues and time.

A Model-Based Extended State Observer for Discrete-Time Linear Multivariable Systems

TL;DR

The paper develops a model-based extended state observer (MB-ESO) for discrete-time, multivariable linear systems by augmenting the state with disturbances and demonstrates a rigorous link to unknown input observers (UIO). It establishes a necessary and sufficient condition—absence of invariant zeros between disturbances and outputs—for MB-ESO and its variant (with non-diagonal disturbance gains), and shows that MB-ESO can emulate a delayed UIO when conditions on the disturbance gain matrix and observer structure are met. An exact transfer function for disturbance estimation error is derived, enabling explicit error bounds that depend on the disturbance relative degree and the observer bandwidth; the error decreases monotonically with time and bandwidth under appropriate assumptions. Numerical simulations corroborate the theoretical connection and error properties, illustrating the advantages of MB-ESO in disturbance decoupling and smoothing in noisy measurements, and highlighting the impact of invariant zeros on disturbance estimation for alternative estimators. Overall, the work provides a solid theoretical foundation and practical tools for disturbance estimation in multivariable discrete-time systems, with implications for safety-critical control and robust design.

Abstract

A model-based extended state observer (MB-ESO) and its variant are proposed for discrete-time linear multivariable systems, where multiple disturbances are defined as an extended state vector in the same manner as in the original formulation of ESO. The variant MB-ESO extends the MB-ESO to address cases where the disturbance gain matrix is non-diagonal. Leveraging the connection between the variant MB-ESO and the well-known unknown input observer (UIO), the condition for the existence of a MB-ESO and its variant in multivariable systems is established, for the first time, i.e., no invariant zeros exist between the disturbances and the plant outputs. It is shown that, with the observer eigenvalues all placed at the origin and the subsystems decoupled, the variant MB-ESO produces the identical disturbance estimation as that of UIO. Moreover, the error characteristics of MB-ESO and its variant are analyzed and the transfer functions associated with the disturbance estimation errors are derived. It is demonstrated both mathematically and in simulations that the disturbance estimation error of MB-ESO decreases monotonically with respect to both the observer eigenvalues and time.

Paper Structure

This paper contains 18 sections, 12 theorems, 100 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Contributions
  4. Organization and Notation
  5. A Model-Based Extended State Observer and Its Variant
  6. The Connection between UIO and MB-ESO
  7. Unknown Input Observer
  8. Finite-Step Convergence of Estimated Disturbance in MB-ESO
  9. Error Characteristics of the Estimated Disturbance in MB-ESO
  10. Numerical Simulations and Discussion
  11. Validations of the Connection: UIO & MB-ESO
  12. Example 1
  13. Example 2
  14. Validations of the Characteristics of MB-ESO
  15. Example 3
  16. ...and 3 more sections

Key Result

Lemma 1

If $(A_0, E_0, C_0)$ has no invariant zeros, then $(A_0, C_0)$ is observable.

Figures (11)

  • Figure 1: Hierarchical relationship among the geometric approach, the UIO, and the disturbance estimator.
  • Figure 2: The estimated disturbances using the variant MB-ESO.
  • Figure 3: The estimated disturbances using UIO.
  • Figure 4: The actual disturbances and their estimates by MB-ESO and UIO.
  • Figure 5: The estimated disturbances using UIO.
  • ...and 6 more figures

Theorems & Definitions (30)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 1
  • proof
  • Lemma 4
  • proof
  • ...and 20 more