Inverted pendulum stabilization: Characterization of codimension-three triple zero bifurcation via multiple delayed proportional gains |
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| Affiliation: | 1. Laboratoire des Signaux et Systèmes, CNRS-Supélec-Université Paris Sud, 3 rue Joliot-Curie, 91192 Gif-sur-Yvette cedex, France;2. Institut Polytechnique des Sciences Avancées, 7 rue Maurice Grandcoing, 94200 Ivry-sur-Seine, France;3. Université de Lorraine, CRAN, UMR 7039, 2 avenue de la forêt de Haye, 54516 Vandoeuvre-lès-Nancy Cedex, France;4. CNRS, CRAN, UMR 7039, 2 avenue de la forêt de Haye, 54516 Vandoeuvre-lès-Nancy Cedex, France;1. Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany;2. Institut Denis Poisson, CNRS, Université de Tours, Université d''Orléans, Parc de Grandmont, 37200 Tours, France;1. Grenoble INP, Gipsa-Lab (UMR5216), F-38400 Saint Martin d?Hères, France;2. Université de Lyon, INSA Lyon, Ampère (UMR5005), F-69621 Villeurbanne, France;1. Department of General Surgery, Chiba University Graduate School of Medicine, 1-8-1 Inohana, Chuoku, Chiba 260-8677, Japan;2. Department of Plastic, Reconstructive & Aesthetic Surgery, Chiba University Graduate School of Medicine, 1-8-1 Inohana, Chuoku, Chiba 260-8677, Japan;3. Department of Radiology, Chiba University Graduate School of Medicine, 1-8-1 Inohana, Chuoku, Chiba 260-8677, Japan;1. Department of Mathematics, Payame Noor University (PNU), P.O. Box 19395-3697, Tehran, Iran;2. Center of Excellence on Soft Computing and Intelligent Information Processing (SCIIP), Ferdowsi University of Mashhad, Iran;3. Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Iran;1. Department of Mathematical Sciences, Huzhou University, Zhejiang 313000, China;2. School of Risk and Actuarial Studies and CEPAR, UNSW Business School, The University of New South Wales, Sydney, NSW 2052, Australia;1. Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian 116023, China;2. State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116023, China |
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| Abstract: | The paper considers the problem of stabilization of systems possessing a multiple zero eigenvalue at the origin. The controller that we propose, uses multiple delayed measurements instead of derivative terms. Doing so, we increase the performances of the closed loop in presence of system uncertainties and/or noisy measurements. The problem formulation and the analysis is presented through a classical engineering problem which is the stabilization of an inverted pendulum on a cart moving horizontally. On one hand, we perform a nonlinear analysis of the center dynamics described by a three dimensional system of ordinary differential equations with a codimension-three triple zero bifurcation. On the other hand, we present the complementary stability analysis of the corresponding linear time invariant system with two delays describing the behavior around the equilibrium. The aim of this analysis is to characterize the possible local bifurcations. Finally, the proposed control scheme is numerically illustrated and discussed. |
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| Keywords: | Time-delay Stability Delayed feedback Nonhyperbolic dynamics Center manifold Normal forms Local bifurcation analysis Inverted pendulum |
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