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Volume 7 Issue 4
Jun.  2020

IEEE/CAA Journal of Automatica Sinica

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Article Contents
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.

# Stability of Nonlinear Differential-Algebraic Systems Via Additive Identity

##### doi: 10.1109/JAS.2020.1003219
• 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.

• 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].
7The matrices $A_{i j}$ , for $i = 1, 2$ and $j = 1, 2$ , are not uniquely defined.
9See [31] for the concept of generalized ${\cal{L}}_2$ -gain.
10In [37] $a$ is constant.
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