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Contraction Analysis of Nonlinear Systems on Riemannian Manifolds

Abstract : Stability of equilibrium points of nonlinear systems is one of the central issues of nonlinear control theory and applications. Stability analysis often boils down to searching for a Lyapunov candidate that adequately dissipates along the system’s solutions. The last two decades have witnessed a growing need to go beyond stability of an equilibrium, by imposing that any two solutions of a system eventually converge to one another. Such an incremental version of Lyapunov stability (contraction) indeed proves useful in observer design, synchronization and trajectory tracking. However, analysis methods to contraction are still far from being standardized, particularly for systems evolving on manifolds such as rotation dynamics in special orthonormal group, Lagrangian systems modeled in non-Euclidean configuration space and quantum systems in density matrix space. The main objective of this thesis is to provide further understanding of contractive systems on manifolds and to propose applicable methods to ensure contraction. More precisely, the contributions of the thesis are:C1 Introduce the new tool, based on the complete lift, for contraction analysis. This new tool makes it possible to carry out contraction analysis on manifolds in a coordinate-free manner and to understand the geometric essence of contractive systems.C2 Show that Finsler-Lyapunov functions play a similar role for contraction as Lyapunov functions for stability analysis. In particular, we show that a contractive system always admits a Finsler-Lyapunov function.C3 Provide new geometric characterizations of contractive systems.First, we show that contraction can be fully characterized on a tubular neighborhood of the base manifold of the tangent bundle, therefore relaxing the main results of C1. Second, we establish a connection between Lyapunov stability and contraction. The connection is made by Krasovskii’s method. It is shown that a Lyapunov function can be directly constructed using the information of contraction, in which the latter is concerned with objects in the tangent bundle while the former is an object on the base manifold.C4 Study local exponential stability of nontrivial solutions of systems on manifolds. It is found that such stability has close relationship with contraction, and is easier to use than contraction in some situations.Necessary and sufficient conditions are obtained. An illustrative example, namely, convergence analysis of observer of Lagrangian system is given to show the effectiveness of the approach.C5 Study robustness of contraction by converting contraction into transverse stability. It is shown that contraction is robust when the system flow along the horizontal manifold is hyperbolic. The method is then extended to study almost global input to state stability, which proves particularly useful for observer design on compact manifolds.
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Submitted on : Monday, May 23, 2022 - 8:24:16 AM
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  • HAL Id : tel-03675229, version 1

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Dongjun Wu. Contraction Analysis of Nonlinear Systems on Riemannian Manifolds. Automatic. Université Paris-Saclay; Harbin Institute of Technology (Chine), 2022. English. ⟨NNT : 2022UPAST037⟩. ⟨tel-03675229⟩

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