Advanced tools for the analysis of stability and robustness of nonlinear systems, with application to switching dynamics

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Ajouté le: 15/04/2014
Directeur : CHAILLET Antoine -
Titre : Advanced tools for the analysis of stability and robustness of nonlinear systems, with application to switching dynamics
Thèmes : Automatique, Signal, Télécoms, Systèmes embarqués
Laboratoires : L2S Laboratoire des Signaux et Systèmes UMR 8506
Description :



The L2S has a recognized expertise in the analysis and control of systems ruled by nonlinear dynamics. This is witnessed by the projects coordinated by L2S members, including the European Network of Excellence HYCON2 and the Paris-Saclay iCODE institute. The objective of this Ph.D. thesis is to develop a recently introduced notion of stability and robustness for nonlinear systems, which ensures an interesting compromise between the strengths and generalities of existing notions.

Related publications (also see this page):
  - A. Chaillet, D. Angeli, and H. Ito. Strong iISS: combination of iISS and ISS with respect to small inputs. In Proc. IEEE Conf. on Decision and Control, Hawaii, USA, Dec. 2012.
  - A. Chaillet and D. Angeli. Integral Input-to-State Stable systems in cascade. Syst. & Contr. Letters, 57(7):519–527, 2008.


Brief description

A central property tor the analysis of stability and robustness of nonlinear systems is the Input-to-State Stability (ISS), introduced by E.D. Sontag in the late eighties. This notion not only guarantees global asymptotic stability in the absence of perturbations, but also the boundedness of the responses to any bounded disturbance and the convergence to zero if disturbances vanish. ISS has contributed to numerous methodological advances in control engineering, including control under communication constraints, output feedback stabilization, optimal control, analysis of hybrid and switching systems, predictive control, neural networks, and chaotic systems. It has led to applications in domains as diverse as robotics, production lines, transportation, bio- chemical networks, or neuroscience. Despite its indisputable success, ISS remains a conservative property in practice. An interesting alternative, which still guarantees some robustness to exogenous inputs, was introduced in the late nineties: the integral input-to-state stability (iISS). Contrarily to ISS which links the steady-state error to the amplitude of the input signal, iISS measures the impact of the energy fed into the system. In contrast to ISS, iISS does not guarantee boundedness of solutions in presence of (even small, or vanishing) inputs. Recently, we have introduced an intermediary property named Strong iISS: it aims at constituting a trade-off between the strength of ISS and the generality of iISS. This notion guarantees iISS, but also boundedness of solutions in response to any input whose amplitude is below a certain threshold. The objectives of this Ph.D. thesis are:
  - To derive a necessary and sufficient characterization of Strong iISS in terms of Lyapunov functions
  - To develop control methodologies that guarantee Strong iISS of the closed-loop system
  - To study in which situations the use of saturated control laws may ensure Strong iISS
  - To exploit the notion of Strong iISS in the context of systems that switch between different dynamics.



While aiming at practical methodologies, the contribution expected from this Ph.D. thesis will mostly be of a theoretical nature. The selected candidate is expected to have strong interest and sufficient background in mathematics, especially the tools linked to the study of dynamical systems. Knowledge in the Lyapunov analysis of stability of nonlinear systems would be appreciated. Candidates with engineering, mathematical or physical background will be considered. Good level of English is required, French is not mandatory.