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Stabilization of Nonlinear Singular Systems via Takagi–Sugeno Models and Robust Differentiators

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Abstract

Concerned with descriptor systems, this paper extends recent results on nonquadratic controller design. The proposal employs an exact Takagi–Sugeno model of the descriptor redundancy form and a fully nonquadratic Lyapunov function. Based on these elements, a switching control law is proposed that achieves stabilization up to the modelling area by feeding back the time derivatives of the membership functions: these are obtained from a Levant’s robust differentiator. Conditions thus obtained turned up to be linear matrix inequalities. Via suitable examples, the methodology is shown to outperform previous approaches on the subject.

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Notes

  1. It will be assumed that \(\text{rank}\begin{bmatrix} E_v&B_h\end{bmatrix}=2n-\text{rank}(\bar{E})\) \(\forall x\in {\mathbb {R}}^n\); this condition reduces to impulse controllability in the linear case [25].

  2. The differential index is the minimum number of differentiation steps required to transform a DAE system into an ordinary differential equation (ODE). If the differential index is 0, then the matrix E(x) is invertible and an ODE system can be directly obtained; if the differential index is 1, then by simple substitution the descriptor system is transformed into an ODE one. For indexes greater than 1, the Pantelides algorithm must be performed; it is already implemented in the Symbolic Math Toolbox of MATLAB [27, 29].

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Acknowledgements

This work has been supported by the CONACYT scholarship 583472, the postdoctoral fellowship for CVU 366627, the PROFAPI Projects 2016-0081 and 2016-0091, and the ITSON PFCE 2016-17.

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Authors and Affiliations

  1. Department of Electrical and Electronics Engineering, Sonora Institute of Technology, 5 de febrero 818 Sur, Ciudad Obregon, Mexico

    Juan Carlos Arceo, Raymundo Márquez, Víctor Estrada-Manzo & Miguel Bernal

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  1. Juan Carlos Arceo
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  2. Raymundo Márquez
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  3. Víctor Estrada-Manzo
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  4. Miguel Bernal
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Correspondence to Miguel Bernal.

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Arceo, J.C., Márquez, R., Estrada-Manzo, V. et al. Stabilization of Nonlinear Singular Systems via Takagi–Sugeno Models and Robust Differentiators. Int. J. Fuzzy Syst. 20, 1451–1459 (2018). https://doi.org/10.1007/s40815-018-0463-x

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  • DOI: https://doi.org/10.1007/s40815-018-0463-x

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