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Research Papers

Adaptive-Robust Stabilization of the Furuta's Pendulum Via Attractive Ellipsoid Method

[+] Author and Article Information
Patricio Ordaz

Research Center on Technology of
Information and Systems (CITIS),
Autonomous University of Hidalgo State,
Mineral de la Reforma Hidalgo 42074, México
e-mail: jp.ordaz.oliver@gmail.com

Alex Poznyak

Automatic Control Department,
Center for Research and
Advanced Studies (CINVESTAV),
México City 07360, México
e-mail: apoznyak@ctrl.cinvestav.mx

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received April 19, 2014; final manuscript received November 23, 2015; published online December 23, 2015. Assoc. Editor: Srinivasa M. Salapaka.

J. Dyn. Sys., Meas., Control 138(2), 021005 (Dec 23, 2015) (8 pages) Paper No: DS-14-1182; doi: 10.1115/1.4032130 History: Received April 19, 2014; Revised November 23, 2015

This paper focuses on the issue of adaptive-robust stabilization of the Furuta's pendulum around unstable equilibrium where the dynamical model is unknown. The control scheme lies at the lack of the dynamical model as well as external disturbances. The stabilization analysis is based on the attractive ellipsoid method (AEM) for a class of uncertain nonlinear systems having “quasi-Lipschitz” nonlinearities. Even more, a modification of the AEM concept that permits to use online information obtained during the process is suggested here. This adjustment (or adaptation) is made only in some fixed sample times, so that the corresponding gain matrix of the robust controller is given on time interval too. Furthermore, under a specific “regularized persistent excitation condition,” the proposed method guarantees that the controlled system trajectories remain inside an ellipsoid of a minimal size (the minimal size is refereed to as the minimal trace of the corresponding inverse ellipsoidal matrix). Finally, the adaptive process describes a region of attraction (ROA) of the considered system under adaptive-robust nonlinear control law.

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Figures

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Fig. 2

The class C(A,δ1,δ2) of quasi-Lipschitz functions (scalar case)

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Fig. 1

Furuta's pendulum system

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Fig. 3

System position x1 and x2

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Fig. 4

System velocity x3 and x4

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Fig. 5

Invariant ellipsoids Eti corresponding to trajectories x1(t) versus x3(t)

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Fig. 6

Invariant ellipsoids Eti corresponding to trajectories x2(t) versus x4(t)

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