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Time-Invariant Polytopic and Interval Observers for Uncertain Linear Systems via Non-Square Transformation

Feiya Zhu, Tarun Pati, Sze Zheng Yong

TL;DR

This work proposes a non-square, time-invariant coordinate transformation to enable stable polytopic and interval observers for all detectable linear systems with bounded disturbances. It leverages a lifted state via z = P x and a mixed-monotone embedding to guarantee enclosure of the true state and input-to-state stability of the estimate volumes, without positivity constraints. The approach applies to both continuous- and discrete-time systems and demonstrates superior performance over time-varying interval observers across several CT/DT examples. It also provides existence results for time-invariant interval observers and discusses extensions to nonlinear and hybrid systems. Overall, the method offers a robust, invariant framework for set-valued state estimation in uncertain linear dynamics.

Abstract

This paper presents novel polytopic and interval observer designs for uncertain linear continuous-time (CT) and discrete-time (DT) systems subjected to bounded disturbances and noise. Our approach guarantees enclosure of the true state and input-to-state stability (ISS) of the polytopic and interval set estimates. Notably, our approach applies to all detectable systems that are stabilized by any optimal observer design, utilizing a potentially non-square (lifted) time-invariant coordinate transformation based on polyhedral Lyapunov functions and mixed-monotone embedding systems that do not impose any positivity constraints, enabling feasible and optimal observer designs, even in cases where previous methods fail. The effectiveness of our approach is demonstrated through several examples of uncertain linear CT and DT systems.

Time-Invariant Polytopic and Interval Observers for Uncertain Linear Systems via Non-Square Transformation

TL;DR

This work proposes a non-square, time-invariant coordinate transformation to enable stable polytopic and interval observers for all detectable linear systems with bounded disturbances. It leverages a lifted state via z = P x and a mixed-monotone embedding to guarantee enclosure of the true state and input-to-state stability of the estimate volumes, without positivity constraints. The approach applies to both continuous- and discrete-time systems and demonstrates superior performance over time-varying interval observers across several CT/DT examples. It also provides existence results for time-invariant interval observers and discusses extensions to nonlinear and hybrid systems. Overall, the method offers a robust, invariant framework for set-valued state estimation in uncertain linear dynamics.

Abstract

This paper presents novel polytopic and interval observer designs for uncertain linear continuous-time (CT) and discrete-time (DT) systems subjected to bounded disturbances and noise. Our approach guarantees enclosure of the true state and input-to-state stability (ISS) of the polytopic and interval set estimates. Notably, our approach applies to all detectable systems that are stabilized by any optimal observer design, utilizing a potentially non-square (lifted) time-invariant coordinate transformation based on polyhedral Lyapunov functions and mixed-monotone embedding systems that do not impose any positivity constraints, enabling feasible and optimal observer designs, even in cases where previous methods fail. The effectiveness of our approach is demonstrated through several examples of uncertain linear CT and DT systems.

Paper Structure

This paper contains 12 sections, 4 theorems, 30 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Formulation
  4. Proposed Polytopic and Interval Observers
  5. Non-Square Time-Invariant Transformation
  6. Polytopic and Interval Observer Design
  7. Illustrative Examples
  8. CT System Examples
  9. DT System Example
  10. Conclusion and Future Work
  11. CT Systems
  12. DT Systems

Key Result

Lemma 1

Consider any $Q \in \mathbb{R}^{m \times m}$ and its Metzlerized and absolute value matrices, $Q^\textnormal{m}$ and $|Q|$, respectively. Then,

Figures (1)

  • Figure 1: Left: Representative state and its upper and lower framers obtained as projections of proposed polytopic estimates, and from time-varying interval observers based on reachability in meslem_Hinf_2020 (CT) and meslem2021reachability (DT), and on time-varying coordinate transformations in mazenc2011interval (CT) and mazenc2014interval (DT); Middle and right: Proposed polytopic sets in blue, interval estimates of meslem_Hinf_2020 (CT) and meslem2021reachability (DT) in red and interval estimates of mazenc2011interval (CT) and mazenc2014interval (DT) in green, and their respective volumes. Note that the (large) interval volumes from the approach in mazenc2014interval (DT) are plotted with a 1/100th factor such that they are more comparable with the other methods.

Theorems & Definitions (8)

  • Definition 1: Embedding System khajenejad2023tight
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2: Correctness
  • Theorem 3: Input-to-State Stability
  • proof