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A Geometric Algebra Approach to Invariance Control in Sliding Regimes for Switched Systems

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Abstract

Within a Geometric Algebra (GA) framework, this article presents a general method for synthesis of sliding mode (SM) controllers in Single Input Single Output (SISO) switched nonlinear systems. The method, addressed as the invariance control method, rests on a reinterpretation of the necessary and sufficient conditions for the local existence of a sliding regime on a given smooth manifold. This consideration leads to a natural decomposition of the SM control scheme resulting in an invariance state feedback controller feeding a Delta–Sigma modulator that, ultimately, provides the required binary-valued switched input to the plant. As application examples, the obtained results are used to illustrate the design of an invariance controller for a switched power converter system. Using the invariance control design procedure, it is shown how well-known second order sliding regime algorithms can be obtained, via a limiting process, from traditional sliding regimes induced on linear sliding manifolds for certain nonlinear switched systems.

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Notes

  1. Recall the duality relation: \((\partial _\textbf{x}\sigma )^*=(\partial _\textbf{x}\sigma \textbf{I})=I_S(\textbf{x})\).

  2. Since, \(\epsilon >0\) and \(b > a\), the transversal condition is: \([I_S\wedge g]^*< 0\).

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Acknowledgements

M. A. Aguilar-Orduña and B. C. Gómez-León are grateful to CONACYT for its continued support during the course of their doctoral research under Scholarship Grants Nos. 702805 and 1039577 respectively.

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Authors and Affiliations

  1. Departamento de Ingeniería Eléctrica, Sección de Mecatrónica, Cinvestav IPN, Avenida IPN, No. 2803, Mexico City, Mexico

    H. Sira-Ramírez & M. A. Aguilar-Orduña

  2. Departamento de Control Automático, Cinvestav IPN, Avenida IPN, No. 2803, Mexico City, Mexico

    B. C. Gómez-León

Authors
  1. H. Sira-Ramírez
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  2. B. C. Gómez-León
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  3. M. A. Aguilar-Orduña
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Correspondence to M. A. Aguilar-Orduña.

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Communicated by Eckhard Hitzer

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This work was supported by “Consejo Nacional de Ciencia y Tecnología” (CONACYT-México) and by the “Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional” (CINVESTAV-IPN).

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Sira-Ramírez, H., Gómez-León, B.C. & Aguilar-Orduña, M.A. A Geometric Algebra Approach to Invariance Control in Sliding Regimes for Switched Systems. Adv. Appl. Clifford Algebras 33, 35 (2023). https://doi.org/10.1007/s00006-023-01281-z

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