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This paper presents two methods for implementing complex-valued sliding mode controllers in three-phase power converters. The paper includes the description of the algorithms and a detailed analysis of the proposed implementations. The methods, that are easy to code and have a low computational burden, retain the sliding mode properties of robustness and fast response and do not require any additional processing often used to decouple the dynamics of the three-phase system. The performance of the methods is compared in numerical simulations, and the algorithms are experimentally tested in a microcontroller using a Hardware-in-the-Loop platform.

Geometric algebra (GA) is a mathematical tool for geometric computing, providing a framework that allows a unified and compact approach to geometric relations which in other mathematical systems are typically described using different more complicated elements. This fact has led to an increasing adoption of GA in applied mathematics and engineering problems. However, the scarcity of symbolic implementations of GA and its inherent complexity, requiring a specific mathematical background, make it challenging and less intuitive for engineers to work with. This prevents wider adoption among more applied professionals. To address this challenge, this paper introduces SUGAR (Symbolic and User-friendly Geometric Algebra Routines), an open-source toolbox designed for Matlab and licensed under the MIT License. SUGAR facilitates the translation of GA concepts into Matlab and provides a collection of user-friendly functions tailored for GA computations, including support for symbolic operations. It supports both numeric and symbolic computations in high-dimensional GAs. Specifically tailored for applied mathematics and engineering applications, SUGAR has been meticulously engineered to represent geometric elements and transformations within two and three-dimensional projective and conformal geometric algebras, aligning with established computational methodologies in the literature. Furthermore, SUGAR efficiently handles functions of multivectors, such as exponential, logarithmic, sinusoidal, and cosine functions, enhancing its applicability across various engineering domains, including robotics, control systems, and power electronics. Finally, this work includes four distinct validation examples, demonstrating SUGAR's capabilities across the above-mentioned fields and its practical utility in addressing real-world applied mathematics and engineering problems., Comment: 33 pages, 6 figures, journal paper submitted to ACM TOMS

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.