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Volume 7 Issue 4
Jun.  2020

IEEE/CAA Journal of Automatica Sinica

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Pierluigi Di Franco, Giordano Scarciotti and Alessandro Astolfi, "Stability of Nonlinear Differential-Algebraic Systems Via Additive Identity," IEEE/CAA J. Autom. Sinica, vol. 7, no. 4, pp. 929-941, July 2020. doi: 10.1109/JAS.2020.1003219
Citation: Pierluigi Di Franco, Giordano Scarciotti and Alessandro Astolfi, "Stability of Nonlinear Differential-Algebraic Systems Via Additive Identity," IEEE/CAA J. Autom. Sinica, vol. 7, no. 4, pp. 929-941, July 2020. doi: 10.1109/JAS.2020.1003219

Stability of Nonlinear Differential-Algebraic Systems Via Additive Identity

doi: 10.1109/JAS.2020.1003219
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  • The stability analysis for nonlinear differential-algebraic systems is addressed using tools from classical control theory. Sufficient stability conditions relying on matrix inequalities are established via Lyapunov Direct Method. In addition, a novel interpretation of differential-algebraic systems as feedback interconnection of a purely differential system and an algebraic system allows reducing the stability analysis to a small-gain-like condition. The study of stability properties for constrained mechanical systems, for a class of Lipschitz differential-algebraic systems and for an academic example is used to illustrate the theory.

     

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  • 1Loosely speaking, the index indicates the number of time-differentiations required to reduce a DAE system to a system of ordinary differential equations, see [11] for a precise definition.
    2See [24] for detail on the transformation of fully-implicit DAE systems to the semi-explicit form and vice versa.
    3The dependence of $ h $ on the algebraic variable $ w $ is explicit only when the index is one. However, with some abuse of notation, we use $ h(x, w) $ for any index.
    4Throughout the paper all mappings are assumed to be smooth.
    5We consider the notion of “classical” solution as formulated in [27].
    6See Hadamard’s Lemma [33].
    7The matrices $ A_{i j} $ , for $ i = 1, 2 $ and $ j = 1, 2 $ , are not uniquely defined.
    8Again, see Hadamard’s Lemma [33].
    9See [31] for the concept of generalized $ {\cal{L}}_2 $ -gain.
    10In [37] $ a $ is constant.
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    Highlights

    • Representation of DAE systems as feedback interconnection.
    • Stability analysis forDAE systems via Lyapunov Method and Small Gain-like arguments.
    • Stability analysis for nonlinear mechanical systems with holonomic constraints.
    • Stability analysis of Lipschitz DAE systems.

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