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Introducing Modelling, Analysis and Control of Three-Phase Electrical Systems Using Geometric Algebra

Manel Velasco, Isiah Zaplana, Arnau Dòria-Cerezo, Josué Duarte, Pau Martí

TL;DR

This paper introduces geometric algebra (GA) as a unified framework for modeling, analysis, and control of three-phase electrical systems. By mapping traditional real-valued MIMO models to GA-valued linear SISO representations, it reduces model order and restores linearity in a GA domain, even for unbalanced configurations. Stability analysis in GA mirrors the real-valued SISO approach, via the roots of a real polynomial, and the Youla--Kučera parametrization is extended to GA to synthesize stabilizing, decoupling controllers. An experimental validation on an inverter with unbalanced load demonstrates practical viability, with GA-based decoupling yielding balanced currents and accurate tracking. This GA-centric approach opens new research directions in GA-based systems theory, robustness analysis, and Nyquist-type criteria tailored to GA dynamics.

Abstract

State-of-the-art techniques for modeling, analysis and control of three-phase electrical systems belong to the real-valued multi-input/multi-output (MIMO) domain, or to the complex-valued nonlinear single-input/single-output (SISO) domain. In order to complement both domains while simplifying complexity and offering new analysis and design perspectives, this paper introduces the application of geometric algebra (GA) principles to the modeling, analysis and control of three-phase electrical systems. The key contribution for the modeling part is the identification of the transformation that allows transferring real-valued linear MIMO systems into GA-valued linear SISO representations (with independence of having a balanced or unbalanced system). Closed-loop stability analysis in the new space is addressed by using intrinsic properties of GA. In addition, a recipe for designing stabilizing and decoupling GA-valued controllers is provided. Numerical examples illustrate key developments and experiments corroborate the main findings.

Introducing Modelling, Analysis and Control of Three-Phase Electrical Systems Using Geometric Algebra

TL;DR

This paper introduces geometric algebra (GA) as a unified framework for modeling, analysis, and control of three-phase electrical systems. By mapping traditional real-valued MIMO models to GA-valued linear SISO representations, it reduces model order and restores linearity in a GA domain, even for unbalanced configurations. Stability analysis in GA mirrors the real-valued SISO approach, via the roots of a real polynomial, and the Youla--Kučera parametrization is extended to GA to synthesize stabilizing, decoupling controllers. An experimental validation on an inverter with unbalanced load demonstrates practical viability, with GA-based decoupling yielding balanced currents and accurate tracking. This GA-centric approach opens new research directions in GA-based systems theory, robustness analysis, and Nyquist-type criteria tailored to GA dynamics.

Abstract

State-of-the-art techniques for modeling, analysis and control of three-phase electrical systems belong to the real-valued multi-input/multi-output (MIMO) domain, or to the complex-valued nonlinear single-input/single-output (SISO) domain. In order to complement both domains while simplifying complexity and offering new analysis and design perspectives, this paper introduces the application of geometric algebra (GA) principles to the modeling, analysis and control of three-phase electrical systems. The key contribution for the modeling part is the identification of the transformation that allows transferring real-valued linear MIMO systems into GA-valued linear SISO representations (with independence of having a balanced or unbalanced system). Closed-loop stability analysis in the new space is addressed by using intrinsic properties of GA. In addition, a recipe for designing stabilizing and decoupling GA-valued controllers is provided. Numerical examples illustrate key developments and experiments corroborate the main findings.
Paper Structure (14 sections, 2 theorems, 40 equations, 5 figures)

This paper contains 14 sections, 2 theorems, 40 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Motivating Reduced-Order Representation
  3. Related Work
  4. Summary of Contributions and Paper Structure
  5. Real-valued and Complex-valued Models
  6. Real-valued Representation
  7. Complex-valued Representation
  8. Geometric Algebra Representation
  9. Discussion
  10. Migrating Analisis and design tools from RV into GA
  11. GA Stability Analysis
  12. Migrating Youla--Kučera parametrization from RV to GA
  13. Experiments
  14. Conclusions

Key Result

Proposition 1

The closed-loop scheme shown in Fig.fig:gmimo, whose GA-TF is given in (eq:closedloop), and where the controller and plant are denoted by $C_{ \mathbb{G}}(\mathrm{p}), G_{ \mathbb{G}}(\mathrm{p})\in \mathcal{F}_{2,0}$, and decomposed as in (eq:gplantcontroller), is asymptotically stable if the root with $d_{pc}(\mathrm{p})=d_p(\mathrm{p})d_c(\mathrm{p})+n_p(\mathrm{p})n_c(\mathrm{p})\in \mathcal

Figures (5)

  • Figure 1: Control of three-phase electrical systems in different spaces
  • Figure 2: Scheme for the three-phase electrical system used in the examples.
  • Figure 3: Simulation results: Decoupling tracking controller ($\alpha$ and $\beta$ channels). The system model corresponds to the one depicted in Fig. \ref{['fig:scheme3']}, using eq. (\ref{['eq:gmimoexunbalanced']}). The controller is described by eq. (\ref{['eq:decontroller']}). The inputs and outputs are transformed here into the $\alpha$ and $\beta$ channels.
  • Figure 4: Laboratory set-up.
  • Figure 5: Experimental results: Geometric controller experiment.

Theorems & Definitions (10)

  • Example 1: Illustrative three-phase electrical system
  • Example 2: Real-valued MIMO model
  • Example 3: Complex-valued MIMO and SISO
  • Example 4: GA-valued MIMO and SISO
  • Proposition 1
  • proof
  • Example 5: Stability analysis with a proportional controller
  • Proposition 2
  • proof
  • Example 6: Decoupling GA-valued controller