Stability of Nonlinear Differential-Algebraic Systems Via Additive Identity

Pierluigi Di Franco; Giordano Scarciotti; Alessandro Astolfi

doi:10.1109/JAS.2020.1003219

Volume 7 Issue 4

Jun. 2020

IEEE/CAA Journal of Automatica Sinica

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Article Navigation > IEEE/CAA Journal of Automatica Sinica > 2020 > 7(4): 929-941

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

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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

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Stability of Nonlinear Differential-Algebraic Systems Via Additive Identity

doi: 10.1109/JAS.2020.1003219

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Author Bio:
Pierluigi Di Franco received the bachelor and master degrees in automation engineering from the University of Rome Tor Vergata, Italy, in 2012 and 2015, respectively. He was a Visiting Student at the Electrical and Electronic Engineering Department, Imperial College of London, UK, in 2013. In 2015 he joined the Control and Power Group, Imperial College of London, UK, where he obtained the Ph.D. degree in 2019 with a thesis on analysis and control of nonlinear differential-algebraic systems and applications

Giordano Scarciotti (S’10–M’17) received the B.Sc. and M.Sc. degrees in automation engineering from the University of Rome Tor Vergata, Italy, in 2010 and 2012, respectively. In 2012 he joined the Control and Power Group, Imperial College London, London, UK, where he obtained the Ph.D. degree in 2016 with a thesis on approximation, analysis and control of large-scale systems. He is currently a Lecturer (Assistant Professor) in the Control and Power Group. His current research interests are focused on analysis and control of uncertain systems (modeled by stochastic equations and set-valued mappings), model reduction, and optimal control. He was a Visiting Scholar at New York University, USA in 2015, and at University of California Santa Barbara, USA in 2016. He is the Recipient of an Imperial College Junior Research Fellowship (2016), of the IET Control & Automation PhD Award (2016), the Eryl Cadwaladr Davies Prize (2017), and of an ItalyMadeMe award (2017). He is a Member of the IEEE CSS Conference Editorial Board, of the IFAC Technical Committee in Nonlinear Control Systems, and of the IPC of the IFAC World Congress 2020. He is the Local Organisation Chair for the EUCA European Control Conference 2022

Alessandro Astolfi (S’90–M’91–SM’00–F’09) graduated in electrical engineering from the University of Rome, Italy in 1991. In 1992 he joined ETH-Zurich where he obtained the M.Sc. degree in information theory in 1995 and the Ph.D. degree with Medal of Honor in 1995 with a thesis on discontinuous stabilization of nonholonomic systems. In 1996 he was awarded a Ph.D. from the University of Rome La Sapienza, Italy for his work on nonlinear robust control. Since 1996 he has been with the Electrical and Electronic Engineering Department of Imperial College London, UK, where he is currently Professor of nonlinear control theory and Head of the Control and Power Group. From 1998 to 2003 he was also an Associate Professor at the Dept. of Electronics and Information of the Politecnico of Milano. Since 2005 he has also been a Professor at Dipartimento di Ingegneria Civile e Ingegneria Informatica, University of Rome Tor Vergata. He has been a Visiting Lecturer in nonlinear control in several universities, including ETH-Zurich (1995–1996); Terza University of Rome (1996); Rice University, Houston (1999); Kepler University, Linz (2000); SUPELEC, Paris (2001), Northeastern University (2013). His research interests are focused on mathematical control theory and control applications, with special emphasis for the problems of discontinuous stabilization, robust and adaptive control, observer design and model reduction. He is the author of more than 150 journal papers, of 30 book chapters, and of over 240 papers in refereed conference proceedings. He is the author (with D. Karagiannis and R. Ortega) of the monograph ”Nonlinear and Adaptive Control with Applications” (Springer-Verlag). He is the Recipient of the IEEE CSS A. Ruberti Young Researcher Prize (2007), the IEEE RAS Googol Best New Application Paper Award (2009), the IEEE CSS George S. Axelby Outstanding Paper Award (2012), and the Automatica Best Paper Award (2017). He is a ”Distinguished Member” of the IEEE CSS, IEEE Fellow, and IFAC Fellow. He served as Associate Editor for Automatica, Systems and Control Letters, the IEEE Trans. on Automatic Control, the International Journal of Control, the European Journal of Control, and the Journal of the Franklin Institute; as Area Editor for the Int. J. of Adaptive Control and Signal Processing; as Senior Editor for the IEEE Trans. on Automatic Control; and as Editor-in-Chief for the European Journal of Control. He is currently Editor-in-Chief of the IEEE Trans. on Automatic Control. He served as Chair of the IEEE CSS Conference Editorial Board (2010–2017) and in the IPC of several international conferences
Corresponding author: The authors are with the Electrical and Electronic Engineering Department, Imperial College London, London SW7 2AZ, UK (e-mail: pierluigi.di-franco13@ic.ac.uk; g.scarciotti@ic.ac.uk; a.astolfi@ic.ac.uk)
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 \begin{document}$ h $\end{document} on the algebraic variable \begin{document}$ w $\end{document} is explicit only when the index is one. However, with some abuse of notation, we use \begin{document}$ h(x, w) $\end{document} for any index.
⁴Throughout the paper all mappings are assumed to be smooth.
⁵We consider the notion of “classical” solution as formulated in [27].
⁶See Hadamard’s Lemma [33].
⁷The matrices \begin{document}$ A_{i j} $\end{document} , for \begin{document}$ i = 1, 2 $\end{document} and \begin{document}$ j = 1, 2 $\end{document} , are not uniquely defined.
⁸Again, see Hadamard’s Lemma [33].
⁹See [31] for the concept of generalized \begin{document}$ {\cal{L}}_2 $\end{document} -gain.
¹⁰In [37] \begin{document}$ a $\end{document} is constant.
Received Date: 2020-02-04
Revised Date: 2020-03-24
Accepted Date: 2020-06-03

Available Online: 2020-06-11

Abstract

Abstract

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.
- Differential-algebraic systems,
- Lyapunov method,
- small-gain theorem,
- stability analysis

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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.

References(46)

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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.

Stability of Nonlinear Differential-Algebraic Systems Via Additive Identity

doi: 10.1109/JAS.2020.1003219

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