Abstract
The article applies nonlinear optimal (H-infinity) control to underactuated robotic systems using as a case study Furuta’s pendulum. The pendulum’s dynamic model is first transformed to an equivalent form after applying partial feedback linearization. The later description of the pendulum’s dynamics undergoes approximate linearization which takes place round a temporary operating point (equilibrium) recomputed at each iteration of the control algorithm. The linearization makes use of Taylor series expansion of the state-space model of the system and needs computation of the associated Jacobian matrices. For the approximately linearized model of the pendulum an H-infinity feedback controller is developed. The controller’s gain is computed through the repetitive solution of an algebraic Riccati equation which is also performed at each step of the control method. The stability analysis is based on Lyapunov’s method. First it is shown that the control loop satisfies the H-infinity tracking performance condition. Next, under moderate conditions it is also shown that the global asymptotic stability of the control loop can be assured.
Similar content being viewed by others
References
Shiriaev A, Freidovich L, Robertsson A, Johansson R, Sandberg A (2007) Virtual-holonomic-constraints-based design of stable oscillations of Furuta pendulum: theory and experiments. IEEE Trans Robot 23(4):827–832
Freidovich L, Shiriaev A, Gordillo F, Gmez-Estern F, Aracil J (2009) Partial-energy-shaping control for orbital stabilization of high-frequency oscillations of the furuta pendulum. IEEE Trans Control Syst Technol 17(4):853–858
Morena-Valenzuela J, Aguilar-Avelar C, Puga-Guzman S, Santibanex V (2016) Adaptive neural network control for the trajecotry tracking of the Furuta pendulum. IEEE Trans Cybern 46:1–14
Park MS, Chwa D (2009) Swing-up stabilization control of inverted pendulum systems with coupled sliding-mode contorl method. IEEE Trans Ind Electron 56(9):3541–3555
Siriaev A, Perman JW, Canudas-de-Wit C (2005) Constructuve tool for orbital stabilization of underactuated nonlinear systems: virtual constraints approach. IEEE Trans Autom Control 50(8):1166–1176
Liu S (2014) Stable synchronization of two Furuta pendulums network based on the model of controlled Lagrangian. In: Proceedings of the 33rd Chinese control conference, Niajing, China
Tanaka S, Xiu X, Yamasaki T (2011) New results on energy-based swing-up control for rotational pendulum. In: Proceedings of the 18th IFAC world congress, Milan, Italy
Mathew NJ, Rao KK, Sivakumaran N (2013) Swing-up and stabilization control of a rotary inverted pendulum. In: 10th IFAC symposium on dynamics and control of process systems, Mumbai, India
Zhang J, Zhang Y (2011) Optimal linear modelling and its applications on swing-up and stabilization control for rotary inverted pendulum. In: Proceedings of the 30th Chinese control conference, Yantai, China
Stamens ON, Aamo OM, Kaasa GO (2011) A constructive speed observer design for optimal Euler–Lagrange systems. Automatica 47:2233–2238
Aracil J, Acosta J, Gordillo E (2013) A nonlinear hybrid controller for swinging-up and stabilizing the Furuta pendulum. Control Eng Pract 21:989–993
Fabbri T, Fenucci D, Falasca S, Gamba M, Bicchi A (2013) Packet-based dynamic control of a Furuta pendulum over Ethernet. In: IEEE MED 2013, 21st Mediterranean conference on control and automation, Crete, Greece
La Hera P, Freidovich L, Shiriaev A, Mettin U (2009) New approach for swinging-up the Furuta pendulum: theory and experiments. Mechatronics 19:1240–1250
Chang DE (2010) Stabilizability of controlled Lagrangian systems of two degrees of freedom and one degree of under-actuation by the energy-shaping method. IEEE Trans Autom Control 55(8):1888–1893
Zhao X, Zhang Z, Huang J (2016) Energy-based swing-up control of rotary parallel inverted pendulum. In: IEEE WCICA 2016, Proceedings of 12th world congress on intelligent control and automation, Guilin, China
Aguilar-Ibanez C, Sira-Ramirez H (2002) Control of the Furuta pendulum based on a linear differential flatness approach. In: IEEE ACC 2002, Proceedings of the American control conference, Anchorage, Alaska
Ramirez-Neria M, Sira-Ramirez H, Garrido-Moctezuma R, Luviano-Suarez A (2014) On the linear active disturbance rejection control of the Furuta pendulum. In: IEEE ACC 2014, American control conference, Portland, Oregon, USA
Aguilar-Avelar C, Moreno-Valenzuela J (2015) A composite controller for trajectory tracking applied to the Furuta pendulum. ISA Trans 57:286–294
Aguilar-Avelar C, Moreno-Valenzuela J (2014) A feedback linearization controller for trajectory tracking of the Furuta pendulum. In: 2014 American control conference, Portland, Oregon, USA
Ramirez-Neria M, Sira-Ramirez H, Garrido-Moctezuma R, Luviano-Juarez A (2014) Linear active disturbance rejection control of underactuated systems: the case of the Furuta pendulum. ISA Trans 53:920–928
Rigatos G, Siano P (2015) A new nonlinear H-infinity feedback control approach to the problem of autonomous robot navigation. J Intell Ind Syst 1(3):179–186
Rigatos G, Siano P, Wira P, Profumo F (2015) Nonlinear H-infinity feedback control for asynchronous motors of electric trains. J Intell Ind Syst 1(2):85–98
Rigatos GG, Tzafestas SG (2007) Extended Kalman filtering for fuzzy modelling and multi-sensor fusion. Math Comput Model Dyn Syst 13:251–266
Basseville M, Nikiforov I (1993) Detection of abrupt changes: theory and applications. Prentice-Hall, Englewood Cliffs
Toussaint GJ, Basar T, Bullo F (2000) \(H_{\infty }\) optimal tracking control techniques for nonlinear underactuated systems. In: Proceedings of IEEE CDC 2000, 39th IEEE conference on decision and control, Sydney, Australia
Rigatos G, Zhang Q (2009) Fuzzy model validation using the local statistical approach. Fuzzy Sets Syst 60(7):882–904
Rigatos GG (2011) Modelling and control for intelligent industrial systems: adaptive algorithms in robotcs and industrial engineering. Springer, New York
Rigatos G (2013) Advanced models of neural networks: nonlinear dynamics and stochasticity in biological neurons. Springer, New York
Rigatos G (2015) Nonlinear control and filtering using differential flatness approaches: applications to electromechanicsl systems. Springer, New York
Rigatos G (2017) Intelligent renewable energy systems: modelling and control. Springer, New York
Rigatos G (2017) State-space appproaches for modelling and control in financial engineering: systems theory and machine learning methods. Springer, New York
Gibbs BP (2011) Advanced Kalman filtering, least squares and modelling: a practical handbook. Wiley, New York
Simon D (2006) A game theory approach to constrained minimax state estimation. IEEE Trans Signal Process 54(2):405–412
Hernandez-Guzman VM, Antonio-Cruz M, Ortigoza RS (2016) Linear state feedback regulation of the Furuta pendulum: design based on differential flatness and root locus. IEEE Access 4:8721–8736
Park MD, Chwwa D (2009) Swing-up and stabilization control of inverted pendulum systems via coupled sliding-mode control method. IEEE Trans Ind Electron 56(9):3541–3555
Suzuki S, Furuta K, Pan Y (2003) State-dependent sliding vector VS control and application to swing-up control of pendulum. In: Proceedings of the 42nd IEEE conference on decision and control, Maui, Hawai, USA
Man WS, Lin JS (2010) Nonlinear control design for a class of underactuated systems. In: 2010 IEEE internatinal conference on control applications, proceedings of the 2010 IEEE multi-conference on systems and control, Yokohama, Japan
Salgado I, Kamal S, Bandyopadhyay B, Chairez I, Fridman L (2016) Control of discrete time systems based on recurrent super-twisting-like algorithm. ISA Trans 64:47–55
Acknowledgements
Funding was provided by Unit of Industrial Automation / Industrial Systems Institute (Grant No. Ref. 5352 / Nonlinear control and filtering).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rigatos, G., Siano, P., Abbaszadeh, M. et al. Nonlinear H-infinity control for underactuated systems: the Furuta pendulum example. Int. J. Dynam. Control 6, 835–847 (2018). https://doi.org/10.1007/s40435-017-0348-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40435-017-0348-0