Stabilizing control of an underactuated 2-dimensional TORA with only rotor angle measurement. (English) Zbl 1286.93182

Summary: One dimensional Translational Oscillation with a Rotational Actuator (TORA) system has been used as a benchmark for motivating the study of nonlinear control techniques. In this paper, a novel underactuated 2-dimensional TORA (2DTORA), which has one actuated rotor and two underactuated translational carts, is presented. The analysis of controllability around the system’s equilibriums yielded the controllable equilibriums and constraints on physical parameters. To stabilize the system to its controllable equilibriums from any initial conditions, we propose a simple linear controller containing the rotor angle and angular velocity. The controller is derived from a proper Lyapunov function, including the system’s total energy, that is used to show the passivity property of the system. In addition, a high pass filter is adopted to approximately differentiate the rotor angle so that an estimated angular velocity was used in the controller rather than measuring the actual rotor angular velocity. As a result, only the angle measurement is required for the designed feedback controller to stabilize the underactuated system. Finally, simulation results verify our control design and analysis.


93E12 Identification in stochastic control theory
93D30 Lyapunov and storage functions
93C05 Linear systems in control theory
93C10 Nonlinear systems in control theory
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[1] Wan, Global stabilization of the oscillating eccentric rotor, Nonlinear Dyn. 10 pp 49– (1996)
[2] Bupp, A benchmark problem for nonlinear control design, Int. J. Robust Nonlin. Control 8 pp 307– (1998)
[3] Jankovic, TORA example: cascade- and passivity-based control designs, IEEE Trans. Control Syst. Technol. 4 (3) pp 292– (1996)
[4] Zhao, Flexible backstepping design for tracking and disturbance attenuation, Int. J. Robust Nonlinear Control 8 pp 331– (1998) · Zbl 0925.93824
[5] Tsiotras, An L2 disturbance attenuation solution to the nonlinear benchmark problem, Int. J. Robust Nonlinear Control 8 pp 311– (1998) · Zbl 0908.93031
[6] Mracek, Control designs for the nonlinear benchmark problem via the state-dependent Riccati equation method, Int. J. Robust Nonlinear Control 8 pp 401– (1998) · Zbl 0908.93032
[7] Dussy, ”Measurement-scheduled control for the RTAC problem: an LMI approach, Int. J. Robust Nonlinear Control 8 pp 377– (1998) · Zbl 0925.93854
[8] Kolmanovsky, ”Hybrid feedback stabilization of rotational-translational actuator (RTAC) system, Int. J. Robust Nonlinear Control 8 pp 367– (1998) · Zbl 0925.93825
[9] Jiang, Global output feedback tracking for a benchmark nonlinear system, IEEE Trans. Autom. Control 45 (5) pp 1023– (2000) · Zbl 0968.93531
[10] Park, LPV controller design for the nonlinear RTAC system, Int. J. Robust Nonlinear Control 8 pp 401– (2001) · Zbl 0996.93037
[11] Hung, Design of self-tuning fuzzy sliding mode control for TORA system, Expert Syst. Appl. 32 pp 201– (2007)
[12] Petres, Approximation and complexity trade-off by TP model transformation in controller design:A case study of the TORA system, Asian J. Control 12 (5) pp 575– (2010)
[13] Bupp, Experimental implementation of integrator backstepping and passive nonlinear controllers on the RTAC testbed, Int. J. Robust Nonlinear Control 8 pp 435– (1998) · Zbl 0925.93352
[14] Pavlov , A. B. Janssen N. Wouw H. Nijmeijer Experimental output regulation for the TORA system Proceedings of 44th IEEE Conference on Decision and Control Seville, Spain 1108 1113 2005
[15] Avis, A comparison study of nonlinear control techniques for the RTAC system, Nonlinear Anal. - Real World Appl. 11 pp 2647– (2010) · Zbl 1214.93052
[16] Ortega, Passivity-based control of Euler-Lagrange systems: mechanical, electrical and electromechanical applications (1998)
[17] Alleyne, Physical insights on passivity-based TORA control designs, IEEE Trans. Control Syst. Technol. 16 (3) pp 436– (1998)
[18] Haddad, Nonlinear fixed-order dynamic compensation for passive systems, Int. J. Robust Nonlinear Control 8 pp 346– (1998) · Zbl 0921.93034
[19] Tadmor, Dissipative design, lossless dynamics, and the nonlinear TORA benchmark example, IEEE Trans. Control Syst. Technol. 9 (2) pp 391– (2001)
[20] Fantoni, Energy based control of the Pendubot, IEEE Trans. Autom. Control 45 (4) pp 725– (2000) · Zbl 1136.70329
[21] Fantoni, Stabilization of the inverted pendulum around its homoclinic orbit, Syst. Control Lett. 40 pp 197– (2000) · Zbl 0985.93026
[22] Rubi, Swing-up control problem for a self-erecting double inverted pendulum, IEE Proc. Control Theory Appl. 149 (2) pp 169– (2002)
[23] Olfati-Saber, Normal forms for underactuated mechanical systems with symmetry, IEEE Trans. Autom. Control 47 (2) pp 305– (2002) · Zbl 1364.70037
[24] Sepulchre, Constructive nonlinear control (1997)
[25] Burg, Additional notes on the TORA example: a filtering approach to eliminate velocity measurements, IEEE Trans. Control Syst. Technol. 5 (5) pp 520– (1997)
[26] Malagari, ”Globally exponential controller/obeserver for tracking in robots without velocity measurement, Asian J. Control 14 (2) pp 1– (2012) · Zbl 1286.93124
[27] Atassi, A Separation principle for the stabilization of a class of nonlinear systems, IEEE Trans. Automat. Control 44 (9) pp 1672– (1999) · Zbl 0958.93079
[28] Nazrulla , S. H. K. Khalil A novel nonlinear output feedback control applied to the TORA benchmark system Proceedings of IEEE Conference on Decision and Control Cancun, Mexico 3565 3570 2008
[29] Wit, A New model for control of systems with friction, IEEE Trans. Autom. Control 40 (3) pp 419– (1995) · Zbl 0821.93007
[30] Andersson, Friction models for sliding dry, boundary and mixed lubricated contacts, Tribol. Int. 40 pp 580– (2007)
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