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Data-driven Dissipativity Analysis of Linear Parameter-Varying Systems

Chris Verhoek, Julian Berberich, Sofie Haesaert, Frank Allgöwer, Roland Tóth

TL;DR

This paper develops a direct data-driven framework for dissipativity analysis of linear parameter-varying (LPV) systems represented in discrete-time shifted-affine IO form. By leveraging a data-driven LPV representation derived from the Extended LPV Fundamental Lemma and a finite-horizon notion of dissipativity, it derives necessary and sufficient conditions that can be checked via semidefinite programs and LMIs. The authors present three computational approaches—polytopic convex-hull LMIs, S-procedure (ellipsoidal) multipliers, and sampling-based scheduling-dependent multipliers—to verify L-dissipativity from a single data set while discussing the trade-offs between conservatism and computational complexity. Through three simulation examples, including a nonlinear LPV embedding of an unbalanced disc, they demonstrate that finite-horizon dissipativity tests from data can closely approximate infinite-horizon, model-based results, with robustness to moderate noise and scalability considerations. The work provides a practical, theoretically grounded pathway for data-driven performance certification of LPV systems and motivates future extensions to storage functions and stochastic noise handling.

Abstract

We derive direct data-driven dissipativity analysis methods for Linear Parameter-Varying (LPV) systems using a single sequence of input-scheduling-output data. By means of constructing a semi-definite program subject to linear matrix inequality constraints based on this data-dictionary, direct data-driven verification of $(Q,S,R)$-type of dissipativity properties of the data-generating LPV system is achieved. Multiple implementation methods are proposed to achieve efficient computational properties and to even exploit structural information on the scheduling, e.g., rate bounds. The effectiveness and trade-offs of the proposed methodologies are shown in simulation studies of academic and physically realistic examples.

Data-driven Dissipativity Analysis of Linear Parameter-Varying Systems

TL;DR

This paper develops a direct data-driven framework for dissipativity analysis of linear parameter-varying (LPV) systems represented in discrete-time shifted-affine IO form. By leveraging a data-driven LPV representation derived from the Extended LPV Fundamental Lemma and a finite-horizon notion of dissipativity, it derives necessary and sufficient conditions that can be checked via semidefinite programs and LMIs. The authors present three computational approaches—polytopic convex-hull LMIs, S-procedure (ellipsoidal) multipliers, and sampling-based scheduling-dependent multipliers—to verify L-dissipativity from a single data set while discussing the trade-offs between conservatism and computational complexity. Through three simulation examples, including a nonlinear LPV embedding of an unbalanced disc, they demonstrate that finite-horizon dissipativity tests from data can closely approximate infinite-horizon, model-based results, with robustness to moderate noise and scalability considerations. The work provides a practical, theoretically grounded pathway for data-driven performance certification of LPV systems and motivates future extensions to storage functions and stochastic noise handling.

Abstract

We derive direct data-driven dissipativity analysis methods for Linear Parameter-Varying (LPV) systems using a single sequence of input-scheduling-output data. By means of constructing a semi-definite program subject to linear matrix inequality constraints based on this data-dictionary, direct data-driven verification of -type of dissipativity properties of the data-generating LPV system is achieved. Multiple implementation methods are proposed to achieve efficient computational properties and to even exploit structural information on the scheduling, e.g., rate bounds. The effectiveness and trade-offs of the proposed methodologies are shown in simulation studies of academic and physically realistic examples.
Paper Structure (22 sections, 6 theorems, 50 equations, 6 figures, 2 tables)

This paper contains 22 sections, 6 theorems, 50 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. Problem setting
  4. Data-driven LPV representations
  5. Data-driven dissipativity
  6. Finite-horizon dissipativity
  7. Data-driven dissipativity of LPV systems
  8. Computational approaches for verifying data-driven dissipativity
  9. Main concept
  10. Dissipativity verification via a convex hull argument
  11. Core concept
  12. Including rate bounds on $\mathit{p}$
  13. Dissipativity verification via the S-procedure
  14. Examples
  15. Example I: Model-based vs. data-driven analysis
  16. ...and 7 more sections

Key Result

Proposition 1

Given a data set ${\mathcal{D}}_{N}$ from an LPV system represented by eq:sys. For a $\bar{p}_{[1,L]}\in{\mathscr{P}}_{[1,L]}$, define the spaces For ${L}\ge n_{\mathrm{r}}$, the data set ${\mathcal{D}}_{N}$ satisfies for all $\bar{p}_{[1,L]}\in{\mathscr{P}}_{[1,L]}$, if and only if for all $\bar{p}_{[1,L]}\in{\mathscr{P}}_{[1,L]}$. This is equivalent to the existence of a vector $g\in{\mathbb{

Figures (6)

  • Figure 1: Data-dictionary for Example I.
  • Figure 2: Influence of noisy data on the $(L-\tau)$-dissipativity analysis.
  • Figure 3: Data-dictionary for Example II.
  • Figure 4: Model-based and indirect data-driven dissipativity analysis versus direct data-driven $(L-\tau)$-dissipativity analysis of an LPV system for an increasing horizon $L$. Plot (a) shows the results when the rate bounds on $p$ are not included in the analysis, while (b) shows the results when the rate bounds are included.
  • Figure 5: Control scheme considered in Example III.
  • ...and 1 more figures

Theorems & Definitions (20)

  • Definition 1: Dissipativity, HillMoylan1980
  • Definition 2: $L$-dissipativity, maupong2017lyapunov
  • Proposition 1: Extended LPV Fundamental Lemma
  • proof
  • Proposition 2: $L$-dissipativity of LPV systems
  • proof
  • Remark 1: Finite- vs. infinite-horizon dissipativity
  • Remark 2
  • Remark 3: Model-based $L$-dissipativity
  • Remark 4: Dissipativity analysis of generalized plants
  • ...and 10 more