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Finding the nearest bounded-real port-Hamiltonian system

Karim Cherifi, Nicolas Gillis, Punit Sharma

TL;DR

The paper tackles the problem of finding the nearest bounded-real (BR) system to a given linear time-invariant model by exploiting a close relationship between BR systems, scattering passivity, and port-Hamiltonian (PH) representations expressed in sPH form. It derives a BR-sPH equivalence via LMIs, formulates the nearest-BR problem in sPH terms, and develops an optimization framework that combines semidefinite programming with alternating optimization and Nesterov-style acceleration. The authors implement an SDP-based BR check and two AO-based algorithms (one with interior-point solves and another with fast gradient steps) and validate them on a small benchmark, a large RLC ladder, and synthetic data, demonstrating good accuracy and favorable compute times, especially with extrapolation. The work provides a practical route to obtain physically meaningful, passive BR models from data and lays groundwork for data-driven BR-PH system identification and passivity enforcement in complex networks.

Abstract

In this paper, we consider linear time-invariant continuous control systems which are bounded real, also known as scattering passive. Our main theoretical contribution is to show the equivalence between such systems and port-Hamiltonian (PH) systems whose factors satisfy certain linear matrix inequalities. Based on this result, we propose a formulation for the problem of finding the nearest bounded-real system to a given system, and design an algorithm combining alternating optimization and Nesterov's fast gradient method. This formulation also allows us to check whether a given system is bounded real by solving a semidefinite program, and provide a PH parametrization for it. We illustrate our proposed algorithms on real and synthetic data sets.

Finding the nearest bounded-real port-Hamiltonian system

TL;DR

The paper tackles the problem of finding the nearest bounded-real (BR) system to a given linear time-invariant model by exploiting a close relationship between BR systems, scattering passivity, and port-Hamiltonian (PH) representations expressed in sPH form. It derives a BR-sPH equivalence via LMIs, formulates the nearest-BR problem in sPH terms, and develops an optimization framework that combines semidefinite programming with alternating optimization and Nesterov-style acceleration. The authors implement an SDP-based BR check and two AO-based algorithms (one with interior-point solves and another with fast gradient steps) and validate them on a small benchmark, a large RLC ladder, and synthetic data, demonstrating good accuracy and favorable compute times, especially with extrapolation. The work provides a practical route to obtain physically meaningful, passive BR models from data and lays groundwork for data-driven BR-PH system identification and passivity enforcement in complex networks.

Abstract

In this paper, we consider linear time-invariant continuous control systems which are bounded real, also known as scattering passive. Our main theoretical contribution is to show the equivalence between such systems and port-Hamiltonian (PH) systems whose factors satisfy certain linear matrix inequalities. Based on this result, we propose a formulation for the problem of finding the nearest bounded-real system to a given system, and design an algorithm combining alternating optimization and Nesterov's fast gradient method. This formulation also allows us to check whether a given system is bounded real by solving a semidefinite program, and provide a PH parametrization for it. We illustrate our proposed algorithms on real and synthetic data sets.
Paper Structure (24 sections, 3 theorems, 43 equations, 5 figures, 3 tables, 2 algorithms)

This paper contains 24 sections, 3 theorems, 43 equations, 5 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contribution and outline of the paper
  3. PH systems and their relationship to BR systems
  4. Formulation of the nearest BR system problem in sPH form
  5. Checking BRness
  6. Interior-point methods
  7. Extrapolated Alternating Optimization (E-AO)
  8. Optimizing $Q^i$
  9. Optimizing $(J,Z)$
  10. Projecting onto $\mathcal{Z}$
  11. Acceleration with extrapolation
  12. E-AO algorithm for the nearest BR problem \ref{['eq:mainoptprob']}
  13. Numerical Experiments
  14. Small system from BoyBK89
  15. Checking BRness
  16. ...and 9 more sections

Key Result

Theorem 1

Every sPH system in the form eq:phsystem1 with $K_s \succ 0$ is BR.

Figures (5)

  • Figure 1: Relative error of the different variants of Algorithm \ref{['alg:eao']} with different initializations for the system \ref{['sys:boyd']}.
  • Figure 2: RLC ladder circuit with $n=200,m=1$: Relative error of Algorithm \ref{['alg:checkBRfgm']} with extrapolation ($\beta = 0.5$) and without extrapolation ($\beta = 0$).
  • Figure 3: RLC ladder circuit with $n=200,m=1$: Relative error of Algorithm \ref{['alg:eao']} with extrapolation ($\beta = 0.5$) and without extrapolation ($\beta = 0$), initialized by the CVX solution for problem \ref{['eq:mainoptprob2']}.
  • Figure 4: RLC ladder circuit with $n=200,m=1$: Relative error of several variants of Algorithm \ref{['alg:eao']}, namely EAO-IPM(Id), with and without extrapolation, and EAO-FGM(Alg1) where Alg1 is given 15 seconds to initialize EAO-FGM.
  • Figure 5: EAO-FGM vs. EAO-IPM: average relative error over time on 10 synthetic data sets with $(n=25,m=10)$ and $(n=50,m=25)$.

Theorems & Definitions (10)

  • Definition 1: BroLME20
  • Definition 2
  • Theorem 1
  • proof
  • Definition 3
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Remark 1