Technical Brief

On the Linear Control of Underactuated Nonlinear Systems Via Tangent Flatness and Active Disturbance Rejection Control: The Case of the Ball and Beam System

[+] Author and Article Information
Mario Ramírez-Neria

Department of Mechatronics,
Blvd. Calacoaya No. 7 Col. La Ermita Atizapán,
de Zaragoza Estado de México 52999, Mexico
e-mail: mramirezn@ctrl.cinvestav.mx

Hebertt Sira-Ramírez

Department of Electrical Engineering,
Mechatronics Section,
Av. IPN 2580 Col. San Pedro Zacatenco,
D.F. México, C.P. 07360, Mexico
e-mail: hsira@cinvestav.mx

Rubén Garrido-Moctezuma

Department of Automatic Control,
Av. IPN 2580 Col. San Pedro Zacatenco,
D.F. México, C.P. 07360, Mexico
e-mail: garrido@ctrl.cinvestav.mx

Alberto Luviano-Juárez

Instituto Politécnico Nacional,
Av. IPN 2580, Col. Barrio La Laguna Ticomán,
D.F. México, C.P. 07340, Mexico
e-mail: aluvianoj@ipn.mx

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received September 17, 2015; final manuscript received March 23, 2016; published online June 8, 2016. Assoc. Editor: Evangelos Papadopoulos.

J. Dyn. Sys., Meas., Control 138(10), 104501 (Jun 08, 2016) (5 pages) Paper No: DS-15-1452; doi: 10.1115/1.4033313 History: Received September 17, 2015; Revised March 23, 2016

In this paper, a systematic procedure for controller design is proposed for a class of nonlinear underactuated systems (UAS), which are non-feedback linearizable but exhibit a controllable (flat) tangent linearization around an equilibrium point. Linear extended state observer (LESO)-based active disturbance rejection control (ADRC) is shown to allow for trajectory tracking tasks involving significantly far excursions from the equilibrium point. This is due to local approximate estimation and compensation of the nonlinearities neglected by the linearization process. The approach is typically robust with respect to other endogenous and exogenous uncertainties and disturbances. The flatness of the tangent model provides a unique structural property that results in an advantageous low-order cascade decomposition of the LESO design, vastly improving the attenuation of noisy and peaking components found in the traditional full order, high gain, observer design. The popular ball and beam system (BBS) is taken as an application example. Experimental results show the effectiveness of the proposed approach in stabilization, as well as in perturbed trajectory tracking tasks.

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Fig. 1

A schematic of the BBS

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Fig. 2

Cascade structure of BBS

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Fig. 3

Block diagram of the ball and beam control system

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Fig. 4

Performance of closed loop reference trajectory tracking ball position and acceleration

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Fig. 5

Tracking error, voltage input control signal, and lumped online disturbance estimation

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Fig. 6

Tracking trajectory performance

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Fig. 7

Control inputs in the comparison test



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