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A Unified KKL-based Interval Observer for Nonlinear Discrete-time Systems

Thach Ngoc Dinh, Gia Quoc Bao Tran

TL;DR

This work addresses the challenge of bounding the state of a generic nonlinear discrete-time system under measurement disturbances without assuming a specific structure on the dynamics. It adopts a Kazantzis–Kravaris–Luenberger (KKL) approach by mapping the state through a transformation $T$ into a higher-dimensional target form that admits an interval observer in the $z$-coordinates; the original state bounds are then recovered via a Lipschitz left-inverse $T^*$, ensuring guaranteed interval bounds. Key contributions include: (i) a Sylvester-equation-based construction of $T$ with Lipschitz injectivity and an explicit left-inverse; (ii) a concrete interval observer design in the transformed domain with nonnegative Schur dynamics ensuring exponential convergence in the disturbance-free setting; and (iii) an illustrative example validating the method. The framework provides robust, guaranteed state-bound estimates for nonlinear systems without requiring prior knowledge of the dynamics’ structure, with potential extensions to continuous-time systems and adaptive observer tuning.

Abstract

This work proposes an interval observer design for nonlinear discrete-time systems based on the Kazantzis-Kravaris/Luenberger (KKL) paradigm. Our design extends to generic nonlinear systems without any assumption on the structure of its dynamics and output maps. Relying on a transformation putting the system into a target form where an interval observer can be directly designed, we then propose a method to reconstruct the bounds in the original coordinates using the bounds in the target coordinates, thanks to the Lipschitz injectivity of this transformation achieved under Lipschitz distinguishability when the target dynamics have a high enough dimension and are pushed sufficiently fast. An academic example serves to illustrate our methods.

A Unified KKL-based Interval Observer for Nonlinear Discrete-time Systems

TL;DR

This work addresses the challenge of bounding the state of a generic nonlinear discrete-time system under measurement disturbances without assuming a specific structure on the dynamics. It adopts a Kazantzis–Kravaris–Luenberger (KKL) approach by mapping the state through a transformation into a higher-dimensional target form that admits an interval observer in the -coordinates; the original state bounds are then recovered via a Lipschitz left-inverse , ensuring guaranteed interval bounds. Key contributions include: (i) a Sylvester-equation-based construction of with Lipschitz injectivity and an explicit left-inverse; (ii) a concrete interval observer design in the transformed domain with nonnegative Schur dynamics ensuring exponential convergence in the disturbance-free setting; and (iii) an illustrative example validating the method. The framework provides robust, guaranteed state-bound estimates for nonlinear systems without requiring prior knowledge of the dynamics’ structure, with potential extensions to continuous-time systems and adaptive observer tuning.

Abstract

This work proposes an interval observer design for nonlinear discrete-time systems based on the Kazantzis-Kravaris/Luenberger (KKL) paradigm. Our design extends to generic nonlinear systems without any assumption on the structure of its dynamics and output maps. Relying on a transformation putting the system into a target form where an interval observer can be directly designed, we then propose a method to reconstruct the bounds in the original coordinates using the bounds in the target coordinates, thanks to the Lipschitz injectivity of this transformation achieved under Lipschitz distinguishability when the target dynamics have a high enough dimension and are pushed sufficiently fast. An academic example serves to illustrate our methods.
Paper Structure (7 sections, 4 theorems, 43 equations, 1 figure)

This paper contains 7 sections, 4 theorems, 43 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main Results
  3. Problem Statement
  4. Properties of $T$
  5. Interval Observer Design
  6. An Illustrative Example
  7. Conclusion

Key Result

Lemma 1

Efimov-12 Consider vectors $a$, $a^+$, $a^-$ in $\mathbb{R}^{n}$ such that $a^- \leq a \leq a^+$. For any $A \in \mathbb{R}^{m\times n}$,

Figures (1)

  • Figure 1: Results for $x_{k,1}$. Left: Convergence in the absence of noise; Right: Comparison between different observer parameters in the presence of noise.

Theorems & Definitions (10)

  • Definition 1
  • Lemma 1
  • Lemma 2
  • Definition 2
  • Remark 1
  • Lemma 3
  • proof
  • Remark 2
  • Theorem 1
  • proof