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“A Tutorial on Incremental Stability Analysis using Contraction Theory”

Authors: Jerome Jouffroy and Thor I. Fossen,
Affiliation: University of Southern Denmark, NTNU, Department of Engineering Cybernetics and NTNU, Centre for Ships and Ocean Structures
Reference: 2010, Vol 31, No 3, pp. 93-106.

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Keywords: Contraction theory, exponential stability, incremental stability, Lyapunov stability, methodology

Abstract: This paper introduces a methodology for differential nonlinear stability analysis using contraction theory (Lohmiller and Slotine, 1998). The methodology includes four distinct steps: the descriptions of two systems to be compared (the plant and the observer in the case of observer convergence analysis, the plant and the controller in the case of tracking controller analysis), the definition of an abstract system common to the two systems and denoted as the ´virtual system´, and the convergence study of the virtual system using its virtual dynamics representation. The approach is illustrated on several simple examples.

PDF PDF (678 Kb)        DOI: 10.4173/mic.2010.3.2



DOI forward links to this article:
  [1] (2011), doi:10.1002/9781119994138.refs
  [2] John W. Simpson-Porco and Francesco Bullo (2014), doi:10.1016/j.sysconle.2013.12.016
  [3] J.P. Maree, L. Imsland and J. Jouffroy (2014), doi:10.1080/00207721.2014.953799
  [4] Fulvio Forni and Rodolphe Sepulchre (2014), doi:10.1109/TAC.2013.2285771
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  [6] Eoin Devane and Ioannis Lestas (2017), doi:10.1016/j.automatica.2016.07.044
  [7] Qichao Ma, Ku Du, Yu Kang, Wei Xing Zheng and Jiahu Qin (2016), doi:10.1109/IECON.2016.7793665
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References:
[1] Aghannan, N. Rouchon, P. (2003). An intrisic observer for a class of Lagrangian systems, IEEE Transactions on Automatic Control, 4.6:936--945 doi:10.1109/TAC.2003.812778
[2] Angeli, D. (2002). A Lyapunov approach to incremental stability properties, IEEE Transactions on Automatic Control, 4.3:410--421 doi:10.1109/9.989067
[3] Asada, H. Slotine, J.-J.E. (1986). Robot Analysis and Control, Wiley-Interscience.
[4] Boutayeb, H.R., M. Darouach, M. (1987). Convergence analysis of the Extended Kalman Filter used as an observer for nonlinear deterministic discrete-time systems, IEEE Transactions on Automatic Control, 4.4:581--586 doi:10.1109/9.566674
[5] Egeland, O., Kristiansen, E., Nguyen, T.-D. (2001). Observer for Euler-Bernouilli beam with hydraulic drive, In Proc. IEEE Conf. on Decision and Control. Orlando, Florida, USA doi:10.1109/.2001.980864
[6] Fossen, T.I. (2002). Marine Control Systems: Guidance, Navigation and Control of Ships, Rigs and Underwater vehicles, Marine Cybernetics.
[7] Fromion, V., Scorletti, G., Ferreres, G. (1999). Nonlinear performance of a PI controlled missile: an explanation, International Journal of Robust and Nonlinear Control, 9(8):485--518 doi:10.1002/(SICI)1099-1239(19990715)9:8andlt;485::AID-RNC417andgt;3.0.CO;2-4
[8] Gajic, Z. Qureshi, M.T.J. (1995). Lyapunov matrix equation in system stability and control, Academic Press.
[9] Jouffroy, J. (2002). Stability and nonlinear systems: reflections on contraction analysis, in French. Ph.D. thesis, Universite de Savoie, Annecy, France.
[10] Jouffroy, J. (2003). A relaxed criterion for contraction theory: application to an underwater vehicle observer, In European Control Conference. Cambridge, UK.
[11] Jouffroy, J. (2003). A simple extension of contraction theory to study incremental stability properties, In European Control Conference. Cambridge, UK.
[12] Jouffroy, J. (2005). Some ancestors of contraction analysis, In Proc. Conference on Decision and Control 2005. Sevilla, Spain doi:10.1109/CDC.2005.1583029
[13] Jouffroy, J. Lottin, J. (2002). On the use of contraction theory for the design of nonlinear observers for ocean vehicles, In Proc. American Control Conference 2002. Anchorage, Alaska, pp. 2647--2652 doi:10.1109/ACC.2002.1025186
[14] Jouffroy, J. Opderbecke, J. (2007). Underwater navigation using diffusion-based trajectory observers, IEEE Journal of Oceanic Engineering, 3.2:313--326 doi:10.1109/JOE.2006.880392
[15] Jouffroy, J. Slotine, J.-J.E. (2004). Methodological remarks on contraction theory, In Proc. Conference on Decision and Control 2004. Paradise Island, Bahamas doi:10.1109/CDC.2004.1428824
[16] Khalil, H. (1996). Nonlinear systems, 2nd ed.. Prentice-Hall, New-York.
[17] Krstic, M., Kanellakopoulos, I., Kokotovic, P. (1995). Nonlinear and adaptive control design, Wiley Interscience, New-York.
[18] Lanczos, C. (1970). The variational principles of mechanics, 4th ed.. Dover, New-York.
[19] Lohmiller, W. (1999). Contraction analysis for nonlinear systems, Ph.D. thesis, Dep. Mechanical Eng., M.I.T., Cambridge, Massachusetts.
[20] Lohmiller, W. Slotine, J.-J.E. (1996). Applications of metric observers for nonlinear systems, In IEEE Int. Conf. on Control Applications. Dearborn, Michigan doi:10.1109/CCA.1996.558805
[21] Lohmiller, W. Slotine, J.-J.E. (1996). On metric observers for nonlinear systems, In IEEE Int. Conf. on Control Applications. Dearborn, Michigan doi:10.1109/CCA.1996.558742
[22] Lohmiller, W. Slotine, J.-J.E. (1998). On contraction analysis for nonlinear systems, Automatica, 34(6):683--696 doi:10.1016/S0005-1098(98)00019-3
[23] Lohmiller, W. Slotine, J.-J.E. (2000). Control system design for mechanical systems using contraction theory, IEEE Transactions on Automatic Control, 4.5:984--989 doi:10.1109/9.855568
[24] Lohmiller, W. Slotine, J.-J.E. (2000). Nonlinear process control using contraction theory, A.I.Ch.E. Journal, 4.3:588--596.
[25] Luenberger, D.G. (1971). An introduction to observers, IEEE Transactions on Automatic Control, 1.6:596--602 doi:10.1109/TAC.1971.1099826
[26] Opial, Z. (1960). Sur la stabilite asymptotique des solutions d´un systeme d´ equations differentielles, Ann. Polonici Math., 7:259--267.
[27] Park, J.-K., Shin, D.-R., Chung, T.M. (2002). Dynamic observers for linear time-invariant systems, Automatica, 38:1083--1087 doi:10.1016/S0005-1098(01)00293-X
[28] Reif, K., Sonnemann, F., Unbehauen, R. (1998). An EKF-based nonlinear observer with a prescribed degree of stability, Automatica, 34(9):1119--1123 doi:10.1016/S0005-1098(98)00053-3
[29] Skjetne, R., Fossen, T.I., Kokotovic, P. (2004). Robust output maneuvering for a class of nonlinear systems, Automatica, 4.3:373--383 doi:10.1016/j.automatica.2003.10.010
[30] Slotine, J.-J.E. (2003). Modularity stability tools for distributed computation and control, Int. Journal of Adaptive Control and Signal Processing, 1.6:397--416 doi:10.1002/acs.754
[31] Slotine, J.-J.E. Li, W. (1991). Applied nonlinear control, Prentice Hall, Englewood Cliffs, New Jersey.
[32] Slotine, J.-J.E. Wang, W. (2003). A study of synchronization and group cooperation using partial contraction theory, In K.V., editor, Block Island Workshop on Cooperative Control. Springer-Verlag.
[33] Sontag, E.D. (1989). Smooth stabilization implies coprime factorization, IEEE Transactions on Automatic Control, 34:435--443 doi:10.1109/9.28018
[34] Sontag, E.D. (2000). The ISS philosophy as a unifying framework for stability-like behavior, In Nonlinear Control in the Year 2000.Vol. 2, pages 443--448. Springer-Verlag doi:10.1007/BFb0110320


BibTeX:
@article{MIC-2010-3-2,
  title={{A Tutorial on Incremental Stability Analysis using Contraction Theory}},
  author={Jouffroy, Jerome and Fossen, Thor I.},
  journal={Modeling, Identification and Control},
  volume={31},
  number={3},
  pages={93--106},
  year={2010},
  doi={10.4173/mic.2010.3.2},
  publisher={Norwegian Society of Automatic Control}
};

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