Lyapunov stability analysis of a mass–spring system subject to friction
This paper deals with the stability analysis of a mass–spring system subject to friction using Lyapunov-based arguments. As the described system presents a stick–slip phenomenon, the mass may then periodically stick to the ground. The objective consists of developing numerically tractable conditions...
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| Published in | Systems & control letters Vol. 150; p. 104910 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier B.V
01.04.2021
Elsevier |
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| Online Access | Get full text |
| ISSN | 0167-6911 1872-7956 1872-7956 |
| DOI | 10.1016/j.sysconle.2021.104910 |
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| Abstract | This paper deals with the stability analysis of a mass–spring system subject to friction using Lyapunov-based arguments. As the described system presents a stick–slip phenomenon, the mass may then periodically stick to the ground. The objective consists of developing numerically tractable conditions ensuring the global asymptotic stability of the unique equilibrium point. The proposed approach merges two intermediate results: The first one relies on the characterization of an attractor around the origin, to which converges the closed-loop trajectories. The second result assesses the regional asymptotic stability of the equilibrium point by estimating its basin of attraction. The main result relies on conditions allowing to ensure that the attractor issued from the first result is included in the basin of attraction of the origin computed from the second result. An illustrative example draws the interest of the approach.
•The stability analysis of mass–spring system subject to a nonlinear friction force is conducted using quadratic Lyapunov functions, leading to stability tests expressed by LMIs.•We give a precise estimation of a global attractor with emphasis on the practical consequences.•If there exists a basin of attraction, we provide an algorithm to give an inner-estimation of the latter.•We mix the two previous results to state the global exponential stability of the system. This work proves then rigorously what was experimentally already noted: for a reference speed large enough, the unique equilibrium point is globally exponentially stable. |
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| AbstractList | This paper deals with the stability analysis of a mass–spring system subject to friction using Lyapunov-based arguments. As the described system presents a stick–slip phenomenon, the mass may then periodically stick to the ground. The objective consists of developing numerically tractable conditions ensuring the global asymptotic stability of the unique equilibrium point. The proposed approach merges two intermediate results: The first one relies on the characterization of an attractor around the origin, to which converges the closed-loop trajectories. The second result assesses the regional asymptotic stability of the equilibrium point by estimating its basin of attraction. The main result relies on conditions allowing to ensure that the attractor issued from the first result is included in the basin of attraction of the origin computed from the second result. An illustrative example draws the interest of the approach.
•The stability analysis of mass–spring system subject to a nonlinear friction force is conducted using quadratic Lyapunov functions, leading to stability tests expressed by LMIs.•We give a precise estimation of a global attractor with emphasis on the practical consequences.•If there exists a basin of attraction, we provide an algorithm to give an inner-estimation of the latter.•We mix the two previous results to state the global exponential stability of the system. This work proves then rigorously what was experimentally already noted: for a reference speed large enough, the unique equilibrium point is globally exponentially stable. This paper deals with the stability analysis of a mass-spring system subject to friction using Lyapunov-based arguments. As the described system presents a stick-slip phenomenon, the mass may then periodically stick to the ground. The objective consists in developing numerically tractable conditions ensuring the global asymptotic stability of the unique equilibrium point. The approach proposed merges two intermediate results: The first one relies on the characterization of an attractor around the origin, in which converge the closed-loop trajectories. The second result assesses the regional asymptotic stability of the equilibrium point by estimating its basin of attraction. The main result relies on conditions allowing to ensure that the attractor issued from the first result is included in the basin of attraction of the origin computed form the second result. An illustrative example draws the interest of the approach. This paper deals with the stability analysis of a mass-spring system subject to friction using Lyapunov-based arguments. As the described system presents a stick-slip phenomenon, the mass may then periodically stick to the ground. The objective consists of developing numerically tractable conditions ensuring the global asymptotic stability of the unique equilibrium point. The proposed approach merges two intermediate results: The first one relies on the characterization of an attractor around the origin, to which converges the closed-loop trajectories. The second result assesses the regional asymptotic stability of the equilibrium point by estimating its basin of attraction. The main result relies on conditions allowing to ensure that the attractor issued from the first result is included in the basin of attraction of the origin computed from the second result. An illustrative example draws the interest of the approach. |
| ArticleNumber | 104910 |
| Author | Tarbouriech, Sophie Gouaisbaut, Frédéric Barreau, Matthieu |
| Author_xml | – sequence: 1 givenname: Matthieu orcidid: 0000-0002-9432-254X surname: Barreau fullname: Barreau, Matthieu email: barreau@kth.se organization: Division of Decision and Control Systems, KTH Royal Institute of Technology Stockholm, Sweden – sequence: 2 givenname: Sophie surname: Tarbouriech fullname: Tarbouriech, Sophie email: tarbour@laas.fr organization: LAAS-CNRS, Université de Toulouse, CNRS, UPS, Toulouse, France – sequence: 3 givenname: Frédéric surname: Gouaisbaut fullname: Gouaisbaut, Frédéric email: fgouaisb@laas.fr organization: LAAS-CNRS, Université de Toulouse, CNRS, UPS, Toulouse, France |
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| Cites_doi | 10.1109/MCS.2008.929425 10.1109/TCST.2008.917873 10.1016/j.automatica.2019.06.017 10.1115/1.3140698 10.1023/B:NODY.0000017482.61599.86 10.1109/9.376053 10.1109/TAC.2017.2774443 10.1016/j.automatica.2015.05.015 10.1016/j.automatica.2007.01.021 10.1016/0005-1098(94)90209-7 10.1109/TAC.2015.2506993 10.1137/18M1234795 10.1109/9.847103 10.5194/ms-6-15-2015 10.1109/TCST.2006.886434 10.1109/TAC.2013.2273279 10.1016/j.arcontrol.2020.04.010 10.1016/S0020-7462(01)00073-7 10.1006/jsvi.1997.1053 |
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| Keywords | Global asymptotic stability LMI Friction Regional asymptotic stability Attractor Lyapunov methods |
| Language | English |
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| Snippet | This paper deals with the stability analysis of a mass–spring system subject to friction using Lyapunov-based arguments. As the described system presents a... This paper deals with the stability analysis of a mass-spring system subject to friction using Lyapunov-based arguments. As the described system presents a... |
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| SubjectTerms | Analysis of PDEs Attractor Friction Global asymptotic stability LMI Lyapunov methods Mathematics Optimization and Control Regional asymptotic stability |
| Title | Lyapunov stability analysis of a mass–spring system subject to friction |
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