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Distributed Resilient Interval Observer Synthesis for Nonlinear Discrete-Time Systems

Mohammad Khajenejad, Scott Brown, Sonia Martinez

TL;DR

A novel recursive distributed estimation algorithm aimed at synthesizing input and state interval observers for nonlinear bounded-error discrete-time multiagent systems and proposes a distributed interval-valued observer that is guaranteed to contain the disturbance and system states.

Abstract

This paper introduces a novel recursive distributed estimation algorithm aimed at synthesizing input and state interval observers for nonlinear bounded-error discrete-time multi-agent systems. The considered systems have sensors and actuators that are susceptible to unknown or adversarial inputs. To solve this problem, we first identify conditions that allow agents to obtain nonlinear bounded-error equations characterizing the input. Then, we propose a distributed interval-valued observer that is guaranteed to contain the disturbance and system states. To do this, we first detail a gain design procedure that uses global problem data to minimize an upper bound on the $\ell_1$ norm of the observer error. We then propose a gain design approach that does not require global information, using only values that are local to each agent. The second method improves on the computational tractability of the first, at the expense of some added conservatism. Further, we discuss some possible ways of extending the results to a broader class of systems. We conclude by demonstrating our observer on two examples. The first is a unicycle system, for which we apply the first gain design method. The second is a 145-bus power system, which showcases the benefits of the second method, due to the first approach being intractable for systems with high dimensional state spaces.

Distributed Resilient Interval Observer Synthesis for Nonlinear Discrete-Time Systems

TL;DR

A novel recursive distributed estimation algorithm aimed at synthesizing input and state interval observers for nonlinear bounded-error discrete-time multiagent systems and proposes a distributed interval-valued observer that is guaranteed to contain the disturbance and system states.

Abstract

This paper introduces a novel recursive distributed estimation algorithm aimed at synthesizing input and state interval observers for nonlinear bounded-error discrete-time multi-agent systems. The considered systems have sensors and actuators that are susceptible to unknown or adversarial inputs. To solve this problem, we first identify conditions that allow agents to obtain nonlinear bounded-error equations characterizing the input. Then, we propose a distributed interval-valued observer that is guaranteed to contain the disturbance and system states. To do this, we first detail a gain design procedure that uses global problem data to minimize an upper bound on the norm of the observer error. We then propose a gain design approach that does not require global information, using only values that are local to each agent. The second method improves on the computational tractability of the first, at the expense of some added conservatism. Further, we discuss some possible ways of extending the results to a broader class of systems. We conclude by demonstrating our observer on two examples. The first is a unicycle system, for which we apply the first gain design method. The second is a 145-bus power system, which showcases the benefits of the second method, due to the first approach being intractable for systems with high dimensional state spaces.
Paper Structure (31 sections, 16 theorems, 81 equations, 6 figures, 2 algorithms)

This paper contains 31 sections, 16 theorems, 81 equations, 6 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Formulation
  4. Distributed Interval Framer Design
  5. Preliminary System Transformation
  6. Interval Framer Design
  7. Distributed Resilient Interval Observer Synthesis
  8. Stability of the Observer Design
  9. C-ISS and Error Minimizing Observer Synthesis
  10. First Approach
  11. Second Approach
  12. Nonlinear Observations & Nonlinear Attacks
  13. Illustrative Examples and Comparisons
  14. Unicycle Target
  15. Power System
  16. ...and 16 more sections

Key Result

Proposition 1

If a mapping $f: \mathcal{Z} \subset \mathbb{R}^n \to \mathbb{R}^p$ has Jacobian matrices satisfying $J^f(x) \in [\ul{J}^f,\overline{J}^f]$, $\forall z \in \mathcal{Z}$, where $\ul{J}^f,\overline{J}^f \in \mathbb{R}^{p \times n}$ are known bounds, then the mapping $f$ can be decomposed into an addit where the matrix $H\in\mathbb{R}^{p \times n}$ satisfies and the function $\mu$ is Jacobian sign-s

Figures (6)

  • Figure 1: A schematic of the intersection-based network update step.
  • Figure 2: Simple static example of "min" consensus.
  • Figure 3: Framers and estimation errors for $x_2$ and $x_3$. The framer plots (left column) show the estimates from the worst and best performing agents, with upper bounds in red and lower bounds in blue. The error plots (right column) show the error from the worst performing agent in a black dashed line and the best performing agent in solid green.
  • Figure 4: Framers and estimation errors for $\theta$ and $\dot\theta$, comparing our approach ($\textsc{DSISO}$) with the observer from XW-HS-FZ-GC:23 (labeled WSZC).
  • Figure 5: State framers (upper bound in red, lower bound in blue), as well as errors for selected state dimensions for the power system example. Only the minimum error is plotted.
  • ...and 1 more figures

Theorems & Definitions (36)

  • Definition 1: Jacobian Sign-Stable khajenejadtight23
  • Proposition 1: Jacobian Sign-Stable (JSS) Decomposition MK-FS-SZY:22a
  • Definition 2: Mixed-Monotone Decomposition Functions
  • Proposition 2: Tight and Tractable Decomposition Functions for JSS Mappings
  • Proposition 3
  • Definition 3: Correct Individual Framers
  • Definition 4: Distributed Resilient Interval Framer
  • Definition 5: Collective Framer Error
  • Definition 6: Collective Input-to-Sate Stable (C-ISS) Distributed Resilient Interval Observer
  • Remark 1
  • ...and 26 more