Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimal Interval Observers for Bounded Jacobian Nonlinear Dynamical Systems

Mohammad Khajenejad, Tarun Pati, Sze Zheng Yong

Abstract

In this chapter, we introduce two interval observer designs for discrete-time (DT) and continuous-time (CT) nonlinear systems with bounded Jacobians that are affected by bounded uncertainties. Our proposed methods utilize the concepts of mixed-monotone decomposition and embedding systems to design correct-by-construction interval framers, i.e., the interval framers inherently bound the true state of the system without needing any additional constraints. Further, our methods leverage techniques for positive/cooperative systems to guarantee global uniform ultimate boundedness of the framer error, i.e., the proposed interval observer is input-to-state stable. Specifically, our two interval observer designs minimize the $\mathcal{H}_{\infty}$ and $L_1$ gains, respectively, of the associated linear comparison system of the framer error dynamics. Moreover, our designs adopt a multiple-gain observer structure, which offers additional degrees of freedom, along with coordinate transformations that may improve the feasibility of the resulting optimization programs. We will also discuss and propose computationally tractable optimization formulations to compute the observer gains. Finally, we compare the efficacy of the proposed designs against existing DT and CT interval observers.

Optimal Interval Observers for Bounded Jacobian Nonlinear Dynamical Systems

Abstract

In this chapter, we introduce two interval observer designs for discrete-time (DT) and continuous-time (CT) nonlinear systems with bounded Jacobians that are affected by bounded uncertainties. Our proposed methods utilize the concepts of mixed-monotone decomposition and embedding systems to design correct-by-construction interval framers, i.e., the interval framers inherently bound the true state of the system without needing any additional constraints. Further, our methods leverage techniques for positive/cooperative systems to guarantee global uniform ultimate boundedness of the framer error, i.e., the proposed interval observer is input-to-state stable. Specifically, our two interval observer designs minimize the and gains, respectively, of the associated linear comparison system of the framer error dynamics. Moreover, our designs adopt a multiple-gain observer structure, which offers additional degrees of freedom, along with coordinate transformations that may improve the feasibility of the resulting optimization programs. We will also discuss and propose computationally tractable optimization formulations to compute the observer gains. Finally, we compare the efficacy of the proposed designs against existing DT and CT interval observers.

Paper Structure

This paper contains 17 sections, 9 theorems, 70 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Preliminaries
  4. Problem Formulation
  5. Proposed Interval Observer
  6. Interval Observer Structure and Framer Property (Correctness)
  7. Construction of Interval Framers
  8. $L_1$ or $\mathcal{H}_{\infty}$-Robust Observer Design
  9. Error Dynamics
  10. Observer Design
  11. Coordinate Transformation
  12. Illustrative Examples
  13. CT System Example 1
  14. CT System Example 2
  15. DT System Example 1
  16. ...and 2 more sections

Key Result

proposition 1

moh2022intervalACC For a mapping $g :{\mathcal{Z}} \subset \mathbb{R}^{n_z} \to \mathbb{R}^{p}$, if $J^g({z}) \in [\underline{J}^g,\overline{J}^g]$ for all ${z \in \mathcal{Z}}$, where $\underline{J}^g,\overline{J}^g \in \mathbb{R}^{p \times n_z}$ are known matrices, then the function $g$ can be de where the matrix $H\in\mathbb{R}^{p \times n_z}$, satisfies

Figures (4)

  • Figure 1: CT Example 1: State, $x_3$, and its upper and lower framers and error of our proposed observer, $\overline{x}^{L_1}_3,\underline{x}^{L_1}_3,\varepsilon^{L_1}_3$ for the $L_1$-robust interval observer, $\overline{x}^{\mathcal{H}^{\infty}}_3,\underline{x}^{\mathcal{H}^{\infty}}_3,\varepsilon^{\mathcal{H}^{\infty}}_3$ for the $\mathcal{H}^{\infty}$-robust interval observer, and $\overline{x}^{DMN}_3,\underline{x}^{DMN}_3,\varepsilon^{DMN}_3$ for the observer in dinh2014interval.
  • Figure 2: CT Example 2: State, $x_1$, and its upper and lower framers and error of our proposed observer, $\overline{x}^{L_1}_3,\underline{x}^{L_1}_3,\varepsilon^{L_1}_3$ for the $L_1$-robust interval observer, $\overline{x}^{\mathcal{H}^{\infty}}_3,\underline{x}^{\mathcal{H}^{\infty}}_3,\varepsilon^{\mathcal{H}^{\infty}}_3$ for the $\mathcal{H}^{\infty}$-robust interval observer.
  • Figure 3: DT Example 1: State, $x_2$, and its upper and lower framers and error of our proposed observers, $\overline{x}^{L_1}_2,\underline{x}^{L_1}_2,\varepsilon^{L_1}_2$ for the ${L_1}$-robust interval observer, $\overline{x}^{\mathcal{H}_{\infty}}_2,\underline{x}^{\mathcal{H}_{\infty}}_2,\varepsilon^{\mathcal{H}_{\infty}}_2$ for the $\mathcal{H}_{\infty}$-robust interval observer, and $\overline{x}^{TA}_2,\underline{x}^{TA}_2,\varepsilon^{TA}_2$ for the observer in tahir2021synthesis.
  • Figure 4: DT Example 2: State, $x_3$, and its upper and lower framers and error of our proposed observers, $\overline{x}^{L_1}_3,\underline{x}^{L_1}_3,\varepsilon^{L_1}_3$ for the ${L_1}$-robust interval observer, $\overline{x}^{\mathcal{H}_{\infty}}_3,\underline{x}^{\mathcal{H}_{\infty}}_3,\varepsilon^{\mathcal{H}_{\infty}}_3$ for the $\mathcal{H}_{\infty}$-robust interval observer.

Theorems & Definitions (29)

  • definition 1: Interval
  • definition 2: Jacobian Sign-Stability
  • proposition 1: Jacobian Sign-Stable Decomposition
  • remark 1
  • definition 3: Mixed-Monotonicity and Decomposition Functions
  • proposition 2: Tight Decomposition Functions for Linear Systems
  • Proof
  • proposition 3: Tight and Tractable DT Decomposition Functions for Nonlinear JSS Mappings moh2022intervalACC & moh2022intervalACC
  • definition 4: One-Sided Decomposition Functions
  • definition 5: Generalized Embedding System
  • ...and 19 more