Algorithms to Compute the Largest Invariant Set Contained in an Algebraic Set for Continuous-Time and Discrete-Time Nonlinear Systems

Laura Menini; Corrado Possieri; Antonio Tornambè

doi:10.1109/JAS.2019.1911819

Volume 7 Issue 1

Jan. 2020

IEEE/CAA Journal of Automatica Sinica

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Article Navigation > IEEE/CAA Journal of Automatica Sinica > 2020 > 7(1): 57-69

Laura Menini, Corrado Possieri and Antonio Tornambè, "Algorithms to Compute the Largest Invariant Set Contained in an Algebraic Set for Continuous-Time and Discrete-Time Nonlinear Systems," IEEE/CAA J. Autom. Sinica, vol. 7, no. 1, pp. 57-69, Jan. 2020. doi: 10.1109/JAS.2019.1911819

Citation:

Laura Menini, Corrado Possieri and Antonio Tornambè, "Algorithms to Compute the Largest Invariant Set Contained in an Algebraic Set for Continuous-Time and Discrete-Time Nonlinear Systems," IEEE/CAA J. Autom. Sinica, vol. 7, no. 1, pp. 57-69, Jan. 2020. doi: 10.1109/JAS.2019.1911819

Citation:

Laura Menini, Corrado Possieri and Antonio Tornambè, "Algorithms to Compute the Largest Invariant Set Contained in an Algebraic Set for Continuous-Time and Discrete-Time Nonlinear Systems," IEEE/CAA J. Autom. Sinica, vol. 7, no. 1, pp. 57-69, Jan. 2020. doi: 10.1109/JAS.2019.1911819

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Algorithms to Compute the Largest Invariant Set Contained in an Algebraic Set for Continuous-Time and Discrete-Time Nonlinear Systems

doi: 10.1109/JAS.2019.1911819

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Author Bio:
Laura Menini received the Laurea degree in 1993 and the Ph.D. degree in 1997, from the University di Roma Tor Vergata, where she is currently Full Professor. She co-authored the books Symmetries and Semi-invariants in the Analysis of Nonlinear Systems (Springer, 2011) and Mathematical Methods for System Theory (WSP, 1998). She chairs the IFAC TC 2.1 (Control Design) and is the Associate Editor of IEEE Transactions on Automatic Control, Automatica, Systems and Control Letters. Her research interests include the use of algebraic geometry for control problems and the design of observers for nonlinear systems

Corrado Possieri received the bachelor and master degrees in medical engineering and the Ph.D. degree in computer science, control and geoinformation from the University of Roma Tor Vergata, Italy, in 2011, 2013, and 2016, respectively. In 2016, he visited the University of California, Santa Barbara (UCSB). In 2018, he joined the Dipartimento di Elettronica e Telecomunicazioni at the Politecnico di Torino, where he was an Assistant Professor. Subsequently, in 2019, he joined the Istituto di Analisi dei Sistemi ed Informatica “A. Ruberti” of the Consiglio Nazionale delle Ricerche, where he is currently a Researcher. His research interests include stability and control of hybrid systems, the application of computational algebraic geometry techniques to control problems, observers, stochastic systems, and optimization. He is Associate Editor of IEEE CSS Conference Editorial Board, of the EUCA Conference Editorial Board, and of the IFAC Conference Editorial Board

Antonio Tornambè received the Laurea degree in electronic engineering from the University of Roma La Sapienza, Rome, Italy, in 1985. He is currently Full Professor of control theory at Università di Roma Tor Vergata. His research interests include control theory for linear and nonlinear systems, observer design, and the use of algebraic geometry for control problems. He is the co-author of Discrete- Event System Theory: an Introduction (WSP, Singapore, 1995), Mathematical Methods for System Theory (WSP, Singapore, 1998), and Symmetries and Semi-invariants in the Analysis of Nonlinear Systems (Springer, London, 2011)
Corresponding author: C. Possieri is with the Istituto di Analisi dei Sistemi ed Informatica “A. Ruberti,” Consiglio Nazionale delle Ricerche (IASI-CNR), Roma 00185, Italy (e-mail: corrado.possieri@iasi.cnr.it)
Received Date: 2019-08-20
Revised Date: 2019-10-09
Accepted Date: 2019-10-27

Available Online: 2019-11-04

Abstract

Abstract

In this paper, some computational tools are proposed to determine the largest invariant set, with respect to either a continuous-time or a discrete-time system, that is contained in an algebraic set. In particular, it is shown that if the vector field governing the dynamics of the system is polynomial and the considered analytic set is a variety, then algorithms from algebraic geometry can be used to solve the considered problem. Examples of applications of the method (spanning from the characterization of the stability to the computation of the zero dynamics) are given all throughout the paper.
- Asymptotic stability,
- discrete-time systems,
- invariance,
- nonlinear systems

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References(50)

References

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Computational tools are proposed to determine the largest invariant set, with respect to either a continuous-time or a discrete-time system, that is contained in an algebraic set.
If the vector field governing the dynamics of the system is polynomial and the considered analytic set is a variety, then algorithms from algebraic geometry can be used to solve the considered problem.
Examples of applications of the method, spanning from the characterization of the stability to the computation of the zero dynamics, are given throughout the paper.

Algorithms to Compute the Largest Invariant Set Contained in an Algebraic Set for Continuous-Time and Discrete-Time Nonlinear Systems

doi: 10.1109/JAS.2019.1911819

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