Mechanical Engineering - Research Publications
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ItemAveraging for nonlinear systems on Riemannian manifolds(IEEE, 2013)This paper provides a derivation of the averaging methods for nonlinear time-varying dynamical systems defined on Riemannian manifolds. We extend the results on ℝ n to Riemannian manifolds by employing the language of differential geometry.
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ItemExtremum seeking control for nonlinear systems on compact Riemannian manifolds(IEEE Press, 2014)This paper formulates the extremum seeking control problem for nonlinear dynamical systems which evolve on Riemannian manifolds and presents stability results for a class of numerical algorithms defined in this context. The results are obtained based upon an extension of extremum seeking algorithms in Euclidean spaces and a generalization of Lyapunov stability theory for dynamical systems defined on Rimannian manifolds. We employ local properties of Lyapunov functions to extend the singular perturbation analysis on Riemannian manifolds. Consequently, the results of the singular perturbation on manifolds are used to obtain the convergence of extremum seeking algorithms for dynamical systems on Riemannian manifolds.
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ItemCloseness of solutions and averaging for nonlinear systems on Riemannian manifolds(IEEE, 2013)An averaging result for periodic dynamical systems evolving on Euclidean spaces is extended to those evolving on (differentiable) Riemannian manifolds. Using standard tools from differential geometry, a perturbation result for time-varying dynamical systems is developed that measures closeness of trajectories via a suitable metric on a finite time horizon. This perturbation result is then extended to bound excursions in the trajectories of periodic dynamical systems from those of their respective averages, on an infinite time horizon, yielding the specified averaging result. Some simple examples further illustrating this result are also presented.
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ItemA Local Characterization of Lyapunov Functions and Robust Stability of Perturbed Systems on Riemannian Manifolds( 2013-10-31)This paper proposes converse Lyapunov theorems for nonlinear dynamical systems defined on smooth connected Riemannian manifolds and characterizes properties of Lyapunov functions with respect to the Riemannian distance function. We extend classical Lyapunov converse theorems for dynamical systems in R n to dynamical systems evolving on Riemannian manifolds. This is performed by restricting our analysis to the so called normal neighborhoods of equilibriums on Riemannian manifolds. By employing the derived properties of Lyapunov functions, we obtain the stability of perturbed dynamical systems on Riemannian manifolds.
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ItemAveraging for nonlinear systems evolving on Riemannian manifolds( 2013-11-11)This paper presents an averaging method for nonlinear systems defined on Riemannian manifolds. We extend closeness of solutions results for ordinary differential equations on R n to dynamical systems defined on Riemannian manifolds by employing differential geometry. A generalization of closeness of solutions for periodic dynamical systems on compact time intervals is derived for dynamical systems evolving on compact Riemannian manifolds. Under local asymptotic (exponential) stability of the average vector field, we further relax the compactness of the ambient Riemannian manifold and obtain the closeness of solutions on the infinite time interval by employing the notion of uniform normal neighborhoods of an equilibrium point of a vector field. These results are also presented for time-varying dynamical systems where their averaged systems are almost globally asymptotically or exponentially stable on compact manifolds. The main results of the paper are illustrated by several examples.
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ItemOptimization Methods on Riemannian Manifolds via Extremum Seeking Algorithms( 2014-12-09)This paper formulates the problem of Extremum Seeking for optimization of cost functions defined on Riemannian manifolds. We extend the conventional extremum seeking algorithms for optimization problems in Euclidean spaces to optimization of cost functions defined on smooth Riemannian manifolds. This problem falls within the category of online optimization methods. We introduce the notion of geodesic dithers which is a perturbation of the optimizing trajectory in the tangent bundle of the ambient state manifolds and obtain the extremum seeking closed loop as a perturbation of the averaged gradient system. The main results are obtained by applying closeness of solutions and averaging theory on Riemannian manifolds. The main results are further extended for optimization on Lie groups. Numerical examples on Riemannian manifolds (Lie groups) SOp3q and SEp3q are also presented at the end of the paper.