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Observer design for classes of nonlinear port-Hamiltonian systems

Filippo Ugolini, Ning Liu, Yongxin Wu, Yann Le Gorrec, Alessandro Macchelli

Abstract

This paper presents a systematic observer design methodology for a class of port-Hamiltonian (pH) systems with state-dependent input matrices. Such systems can model a wide range of electromechanical systems, including magnetic levitation systems, MEMS devices, and electro-active polymer actuators such as DEA actuators, HASEL actuators, etc. In these applications, state-dependent input matrices naturally arise when the system is modeled under quasi-static electrical assumptions. An LPV polytopic embedding framework, together with LMI-based synthesis conditions, is proposed. The nonlinear error dynamics are represented as a convex combination of linear vertex systems using an integral mean value representation, which enables systematic computation of the observer gains that ensures exponential convergence. Both constant-gain and gain-scheduled observers are derived. Numerical results demonstrate the effectiveness of the proposed observer, with the gain-scheduled design achieving a significant increase in the maximum certifiable decay rate compared with constant-gain approaches, thereby reducing conservatism.

Observer design for classes of nonlinear port-Hamiltonian systems

Abstract

This paper presents a systematic observer design methodology for a class of port-Hamiltonian (pH) systems with state-dependent input matrices. Such systems can model a wide range of electromechanical systems, including magnetic levitation systems, MEMS devices, and electro-active polymer actuators such as DEA actuators, HASEL actuators, etc. In these applications, state-dependent input matrices naturally arise when the system is modeled under quasi-static electrical assumptions. An LPV polytopic embedding framework, together with LMI-based synthesis conditions, is proposed. The nonlinear error dynamics are represented as a convex combination of linear vertex systems using an integral mean value representation, which enables systematic computation of the observer gains that ensures exponential convergence. Both constant-gain and gain-scheduled observers are derived. Numerical results demonstrate the effectiveness of the proposed observer, with the gain-scheduled design achieving a significant increase in the maximum certifiable decay rate compared with constant-gain approaches, thereby reducing conservatism.

Paper Structure

This paper contains 18 sections, 3 theorems, 40 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and problem formulation
  3. Port-Hamiltonian Systems
  4. Observer Structure
  5. Structural Assumptions
  6. Observer Design via Polytopic Embedding
  7. Integral Representation of Nonlinear Terms
  8. Polytopic Embedding
  9. Observer Synthesis via LMI
  10. Gain-Scheduled Observer
  11. Application to a Mechanical System
  12. Simplified model of DEA actuator
  13. Operating Domain and Polytopic Embedding
  14. Observer Synthesis and Performance Comparison
  15. Simulation Scenarios
  16. ...and 3 more sections

Key Result

Proposition C.1

Let $\gamma(\cdot, u) : \mathbb{R}^{2n}\times \mathbb{R}^m \to \mathbb{R}^{2n}$ be continuously differentiable in $x$ for each fixed $u \in \mathbb{R}^m$. Then the nonlinear term admits the exact representation: where $\bar{x}(s) = \hat{x} + s\tilde{x}$ for $s \in [0,1]$ is the straight-line parametrization connecting $\hat{x}$ and $x$. $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure D1: Scenario 1 (Fair comparison, $\lambda = 0.0897$): (a) Position estimation $q$ vs. $\hat{q}$. (b) Position error $\tilde{q}$ (0--0.4 s zoom). (c) Momentum estimation $p$ vs. $\hat{p}$. (d) Momentum error $\tilde{p}$ (0--0.4 s zoom). (e) Observer gains $L_1(t)$ and $L_2(t)$. (f) Observer scheduled gains $L_1(t)$ and $L_2(t)$.
  • Figure D2: Scenario 2 (Different $\lambda$ values): (a) Position estimation. (b) Position error (0--0.8 s zoom). (c) Momentum estimation. (d) Momentum error (0--0.8 s zoom).
  • Figure D3: (a) Constant Observer gains, only for $\lambda = 0.897$ since for $\lambda = 4.55$ the constant $L$ approach fails. (b) Scheduled Observer gains. (c) Scheduled Observer gains zoom.

Theorems & Definitions (8)

  • Remark B.1
  • Proposition C.1
  • proof
  • Remark C.2
  • Proposition C.3: Construction of weighting functions
  • proof
  • Theorem C.4
  • proof