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The PhasorArray Toolbox for Harmonic Analysis and Control Design

Maxime Grosso, Pierre Riedinger, Jamal Daafouz

TL;DR

The paper addresses the challenge of practical harmonic-domain analysis for time-periodic systems by introducing the PhasorArray Toolbox, an object-oriented MATLAB framework that represents periodic matrices through harmonic coefficients in a 3D array and provides overloaded arithmetic, Toeplitz constructions, truncation strategies, visualization, and LMI-based solvers. It integrates Lyapunov, Riccati, and LMI-based synthesis within the harmonic framework and offers a PhasorSS simulator for end-to-end validation. Key contributions include a coherent architectural design, methods for constructing and reducing harmonic models, and a streamlined workflow from modeling to robust control synthesis with compatibility for YALMIP. The toolbox lowers the entry barrier to harmonic-domain methods, enabling practitioners to perform harmonic analysis, stability assessment, and controller design for LTP systems with a practical, MATLAB-based toolchain.

Abstract

We present a MATLAB package called the Pha-sorArray Toolbox that has been developed to make harmonic analysis and control methods both practical and user-friendly. The toolbox adopts an object-oriented architecture that enables intuitive manipulation of periodic matrices through overloaded operators for addition, multiplication, convolution, and automatic Toeplitz construction. Its advanced features include harmonic Sylvester, Lyapunov and Riccati equations solvers, and seamless integration with YALMIP, thereby facilitating advanced control and analysis techniques based on Linear Matrix Inequalities (LMIs) in the harmonic framework.

The PhasorArray Toolbox for Harmonic Analysis and Control Design

TL;DR

The paper addresses the challenge of practical harmonic-domain analysis for time-periodic systems by introducing the PhasorArray Toolbox, an object-oriented MATLAB framework that represents periodic matrices through harmonic coefficients in a 3D array and provides overloaded arithmetic, Toeplitz constructions, truncation strategies, visualization, and LMI-based solvers. It integrates Lyapunov, Riccati, and LMI-based synthesis within the harmonic framework and offers a PhasorSS simulator for end-to-end validation. Key contributions include a coherent architectural design, methods for constructing and reducing harmonic models, and a streamlined workflow from modeling to robust control synthesis with compatibility for YALMIP. The toolbox lowers the entry barrier to harmonic-domain methods, enabling practitioners to perform harmonic analysis, stability assessment, and controller design for LTP systems with a practical, MATLAB-based toolchain.

Abstract

We present a MATLAB package called the Pha-sorArray Toolbox that has been developed to make harmonic analysis and control methods both practical and user-friendly. The toolbox adopts an object-oriented architecture that enables intuitive manipulation of periodic matrices through overloaded operators for addition, multiplication, convolution, and automatic Toeplitz construction. Its advanced features include harmonic Sylvester, Lyapunov and Riccati equations solvers, and seamless integration with YALMIP, thereby facilitating advanced control and analysis techniques based on Linear Matrix Inequalities (LMIs) in the harmonic framework.
Paper Structure (18 sections, 1 theorem, 18 equations, 6 figures)

This paper contains 18 sections, 1 theorem, 18 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Harmonic Modeling and Implementation Challenges
  3. PhasorArray Toolbox
  4. Design Philosophy and Architecture
  5. Extended capabilities
  6. Basic construction, manipulation and visualization
  7. Building a PhasorArray from a Periodic Matrix
  8. Visualization Methods
  9. Harmonic reduction
  10. Basic PhasorArray Operations
  11. Finite dimensional Fourier or Toeplitz-Block extraction representations from a PhasorArray
  12. Control design for LTP Systems
  13. Linear Time-Periodic Systems
  14. Lyapunov Equations
  15. Riccati Equations and LQR Control
  16. ...and 3 more sections

Key Result

Theorem II.1

Consider a general dynamical system admitting Caratheodory's solution. Then, for any $T>0$, $X=\mathcal{F}(x)$ satisfies: where $\mathcal{N} = \mathrm{Id}_n \otimes \mathrm{diag}( \texttt{j} \omega \mathbb{Z})$ and conversely if $X$ belongs to $\mathcal{H}=\textrm{Im}_{\mathcal{F}}(L^2_{loc})$ (see Blin2022NecessaryTime for a characterization of $\mathcal{H}$).

Figures (6)

  • Figure 1: 3D array structure with harmonic coefficients stored as slices: $\texttt{A(:,:,h+1)}$ is the DC component, $\texttt{A(:,:,h+2)}$ is the 1rst harmonic, $\texttt{A(:,:,1)}$ is the $-h$th harmonic.
  • Figure 2: Visualization examples for periodic matrix $A(t)$ approximated by $h=31$ harmonics.
  • Figure 3: Visualization example. Left: harmonic spectrum showing full representation, threshold-based filtering (neglect), and fixed-order truncation (trunc). Right: time-domain reconstruction comparing accuracy of different reduction methods.
  • Figure 4: Barsurf of absolute value of $\mathcal{T}(A)_h$ of $((2h+1)n\times (2h+1)m)$ dimension and $\mathcal{F}(A)_h$ of $(2h+1)n\times m)$ dimension for $h=8$.
  • Figure 5: Comparison of solution of Riccati $S(t)$ and LMI $P(t)$.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Theorem II.1
  • Remark V.1