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Modelling via Convex Structures

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Analysis and Synthesis of Nonlinear Control Systems

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 408))

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Abstract

This chapter presents a variety of nonlinear models related to the Takagi–Sugeno form (TS): some of them are directly inspired from the fuzzy approach in which they first appeared; others are generalisations or derivative structures of the original one. Their common characteristic is the presence of convex structures of the state, time, or exogenous parameters. Since this book is focussed on model-based approaches, different methods are presented to construct the convex models from a given nonlinear one.

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Notes

  1. 1.

    Existence and uniqueness of solutions for systems of the form \(E\dot{\boldsymbol{x}}(t)=A\boldsymbol{x}(t)\), with A and E square matrices of the same size, are guaranteed if \(\det (sE-A)\ne 0\) for some s; such systems are called regular [52]. Since this condition is only sufficient, the problem remains open for systems of the form (3.71).

  2. 2.

    If \(T_1\) were not full rank, additional variable elimination and algebraic manipulations involving differentiation would be needed in order to set up a minimal representation (3.78) with invertible \(T_1\); these are the so-called high-index differential-algebraic equations. Alternatively, a descriptor system framework might be pursued, see Sect. 3.4; in fact, all LFT representations admit a descriptor form, see Sect. 3.5.3.

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Authors and Affiliations

  1. Department of Electrical and Electronics Engineering, Sonora Institute of Technology, Obregon, Sonora, Mexico

    Miguel Bernal

  2. Instituto Universitario de Automática e Informática Industrial, Universitat Politècnica de València, Valencia, Spain

    Antonio Sala

  3. Department of Automation, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

    Zsófia Lendek

  4. Department of Automation and Control, LAMI HUMR CNRS8201, Université Polytechnique Hauts-de-France, Valenciennes, France

    Thierry Marie Guerra

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  1. Miguel Bernal
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  2. Antonio Sala
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  4. Thierry Marie Guerra
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Bernal, M., Sala, A., Lendek, Z., Guerra, T.M. (2022). Modelling via Convex Structures. In: Analysis and Synthesis of Nonlinear Control Systems. Studies in Systems, Decision and Control, vol 408. Springer, Cham. https://doi.org/10.1007/978-3-030-90773-0_3

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