The Inverted Pendulum Benchmark in Nonlinear Control Theory: A Survey

Abstract For at least fifty years, the inverted pendulum has been the most popular benchmark, among others, in nonlinear control theory. The fundamental focus of this work is to enhance the wealth of this robotic benchmark and provide an overall picture of historical and current trend developments in nonlinear control theory, based on its simple structure and its rich nonlinear model. In this review, we will try to explain the high popularity of such a robotic benchmark, which is frequently used to realize experimental models, validate the efficiency of emerging control techniques and verify their implementation. We also attempt to provide details on how many standard techniques in control theory fail when tested on such a benchmark. More than 100 references in the open literature, dating back to 1960, are compiled to provide a survey of emerging ideas and challenging problems in nonlinear control theory accomplished and verified using this robotic system. Possible future trends that we can envision based on the review of this area are also presented.

[1]  Oscar Castillo,et al.  Generation of walking periodic motions for a biped robot via genetic algorithms , 2011, Appl. Soft Comput..

[2]  Chuen-Chien Lee,et al.  Fuzzy logic in control systems: fuzzy logic controller. II , 1990, IEEE Trans. Syst. Man Cybern..

[3]  F. Andreev,et al.  Matching, linear systems, and the ball and beam , 2000, Autom..

[4]  Olfa Boubaker,et al.  Position/force control optimized by particle swarm intelligence for constrained robotic manipulators , 2011, 2011 11th International Conference on Intelligent Systems Design and Applications.

[5]  José Ángel Acosta,et al.  Furuta's Pendulum: A Conservative Nonlinear Model for Theory and Practise , 2010 .

[6]  Thomas Kailath,et al.  Linear Systems , 1980 .

[7]  Shuzhi Sam Ge,et al.  Analysis and synthesis of switched linear control systems , 2005, Autom..

[8]  Mark W. Spong,et al.  The swing up control problem for the Acrobot , 1995 .

[9]  Olfa Boubaker,et al.  Impedance Controller Tuned by Particle Swarm Optimization for Robotic Arms , 2011 .

[10]  Chung Choo Chung,et al.  Nonlinear control of a swinging pendulum , 1995, Autom..

[11]  Patrizio Colaneri,et al.  Trends in Theory of Control System Design - Status Report by the IFAC Coordinating Committee on Desi , 2008 .

[12]  Riccardo Marino,et al.  Nonlinear control design , 1995 .

[13]  Karl Johan Åström,et al.  Control systems engineering education , 1996, Autom..

[14]  Frédéric Mazenc,et al.  Asymptotic tracking of a reference state for systems with a feedforward structure , 2000, Autom..

[15]  Stephen P. Boyd,et al.  Future directions in control in an information-rich world , 2003 .

[16]  Patrizio Colaneri,et al.  Theory, algorithms and technology in the design of control systems , 2005, Annu. Rev. Control..

[17]  Jianqiang Yi,et al.  Upswing and stabilization control of inverted pendulum system based on the SIRMs dynamically connected fuzzy inference model , 2001, Fuzzy Sets Syst..

[18]  Fuhong Min,et al.  The chaotic synchronization of a controlled pendulum with a periodically forced, damped Duffing oscillator , 2011 .

[19]  R. Decarlo,et al.  Perspectives and results on the stability and stabilizability of hybrid systems , 2000, Proceedings of the IEEE.

[20]  Alfred C. Rufer,et al.  JOE: a mobile, inverted pendulum , 2002, IEEE Trans. Ind. Electron..

[21]  V. Maties,et al.  A Solution of the Inverse Pendulum on a Cart Problem Using Predictive Control , 2005, Proceedings of the IEEE International Symposium on Industrial Electronics, 2005. ISIE 2005..

[22]  Chunqing Lu.  An alternative approach for chaos: A plane pendulum with oscillating torque , 2011 .

[23]  Saeed Pezeshki,et al.  Performance Analysis of a Neuro-PID Controller Applied to a Robot Manipulator , 2012 .

[24]  Oscar Castillo,et al.  Path planning for autonomous mobile robot navigation with ant colony optimization and fuzzy cost function evaluation , 2009, Appl. Soft Comput..

[25]  Frank L. Lewis,et al.  Neural Network Control Of Robot Manipulators And Non-Linear Systems , 1998 .

[26]  Mitsuji Sampei,et al.  An integrated software environment for the design and real-time implementation of control systems , 1997 .

[27]  Shin'ichi Yuta,et al.  Baggage Transportation and Navigation by a Wheeled Inverted Pendulum Mobile Robot , 2009, IEEE Transactions on Industrial Electronics.

[28]  V. Utkin.  Variable structure systems with sliding modes , 1977 .

[29]  Riccardo Scattolini,et al.  Stabilizing Model Predictive Control of Nonlinear Continuous Time Systems , 2003 .

[30]  Masami Iwase,et al.  Time Optimal Swing-Up Control of Single Pendulum , 2001 .

[31]  Olfa Boubaker,et al.  Minimum Jerk-Based Control for a Three Dimensional Bipedal Robot , 2011, ICIRA.

[32]  Chuen-Chien Lee.  FUZZY LOGIC CONTROL SYSTEMS: FUZZY LOGIC CONTROLLER - PART I , 1990 .

[33]  Kazuhito Yokoi,et al.  Biped walking pattern generation by a simple three-dimensional inverted pendulum model , 2003, Adv. Robotics.

[34]  Dominique Bonvin,et al.  Global stabilization of an inverted pendulum-Control strategy and experimental verification , 2009, Autom..

[35]  Boris Tovornik,et al.  Swinging up and stabilization of a real inverted pendulum , 2006, IEEE Transactions on Industrial Electronics.

[36]  Arthur D Kuo,et al.  The six determinants of gait and the inverted pendulum analogy: A dynamic walking perspective. , 2007, Human movement science.

[37]  Brett Ninness,et al.  Nonlinear model predictive control of an inverted pendulum , 2009, 2009 American Control Conference.

[38]  M. Mariton,et al.  Jump Linear Systems in Automatic Control , 1992 .

[39]  Richard B. Vinter,et al.  Optimal Control , 2000 .

[40]  Rogelio Lozano,et al.  Energy based control of the Pendubot , 2000, IEEE Trans. Autom. Control..

[41]  K. Ohnishi,et al.  Trajectory Planning of Biped Robot Using Linear Pendulum Mode for Double Support Phase , 2006, IECON 2006 - 32nd Annual Conference on IEEE Industrial Electronics.

[42]  Frank L. Lewis,et al.  Optimal Control , 1986 .

[43]  S. Dormido Bencomo.  Control learning: present and future , 2002 .

[44]  Romeo Ortega,et al.  Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment , 2002, IEEE Trans. Autom. Control..

[45]  Thomas Holzhüter,et al.  Optimal regulator for the inverted pendulum via Euler-Lagrange backward integration , 2004, Autom..

[46]  Yang Liu,et al.  On tracking control of a pendulum-driven cart-pole underactuated system , 2008, Int. J. Model. Identif. Control..

[47]  Frank Allgöwer,et al.  Assessment and Future Directions of Nonlinear Model Predictive Control , 2007 .

[48]  Daniel Liberzon,et al.  Switching in Systems and Control , 2003, Systems & Control: Foundations & Applications.

[49]  Jan M. Maciejowski,et al.  Predictive control : with constraints , 2002 .

[50]  K.J. Astrom,et al.  Bicycle dynamics and control: adapted bicycles for education and research , 2005, IEEE Control Systems.

[51]  B. Piccoli,et al.  Optimal Syntheses for Control Systems on 2-D Manifolds , 2004 .

[52]  Frédéric Mazenc,et al.  Tracking trajectories of the cart-pendulum system , 2003, Autom..

[53]  Oscar Castillo,et al.  Optimization of type-2 fuzzy systems based on bio-inspired methods: A concise review , 2012, Inf. Sci..

[54]  U. Huh,et al.  Stability experiment of a biped walking robot with inverted pendulum , 2004, 30th Annual Conference of IEEE Industrial Electronics Society, 2004. IECON 2004.

[55]  Denis Gillet,et al.  A Distributed Architecture for Teleoperation over the Internet with Application to the Remote Control of an Inverted Pendulum , 2000 .

[56]  Benedetto Piccoli,et al.  Time Optimal Swing-Up of the Planar Pendulum , 2008, IEEE Trans. Autom. Control..

[57]  Olfa Boubaker,et al.  FPGA MODELLING AND REAL-TIME EMBEDDED CONTROL DESIGN VIA LABVIEW SOFTWARE: APPLICATION FOR SWINGING-UP A PENDULUM , 2012 .

[58]  J. Aracil,et al.  The inverted pendulum: a benchmark in nonlinear control , 2004, Proceedings World Automation Congress, 2004..

[59]  P. Olver.  Nonlinear Systems , 2013 .

[60]  Vadim I. Utkin,et al.  Sliding Modes in Control and Optimization , 1992, Communications and Control Engineering Series.

[61]  Karen Rudie,et al.  A survey of modeling and control of hybrid systems , 1997 .

[62]  Romeo Ortega,et al.  Robustness of adaptive controllers - A survey , 1989, Autom..

[63]  C C Lee,et al.  FUZZY LOGIC IN CONTROL SYSTEM FUZZY LOGIC CONTROLLER-PART II , 1990 .

[64]  Dongkyoung Chwa,et al.  Swing-Up and Stabilization Control of Inverted-Pendulum Systems via Coupled Sliding-Mode Control Method , 2009, IEEE Transactions on Industrial Electronics.

[65]  Javier Aracil,et al.  A family of smooth controllers for swinging up a pendulum , 2008, Autom..

[66]  John Y. Hung,et al.  Variable structure control: a survey , 1993, IEEE Trans. Ind. Electron..

[67]  K. Furuta,et al.  Computer control of a double inverted pendulum , 1978 .

[68]  Denis Gillet,et al.  Distributed architecture for teleoperation over the internet , 2001 .

[69]  Juan R. Castro,et al.  A T-S Fuzzy Logic Controller for biped robot walking based on adaptive network fuzzy inference system , 2010, The 2010 International Joint Conference on Neural Networks (IJCNN).

[70]  Kirsten Morris,et al.  Friction and the Inverted Pendulum Stabilization Problem , 2008 .

[71]  Karl Johan Åström,et al.  Adaptive Control , 1989, Embedded Digital Control with Microcontrollers.

[72]  Amit Patra,et al.  Swing-up and stabilization of a cart-pendulum system under restricted cart track length , 2002, Syst. Control. Lett..

[73]  Minoru Sasaki,et al.  Motion Control of Biped Lateral Stepping Based on Zero Moment Point Feedback for Adaptation to Slopes , 2011 .

[74]  Mark W. Spong,et al.  Underactuated mechanical systems , 1998 .

[75]  Li-Xin Wang,et al.  Adaptive fuzzy systems and control - design and stability analysis , 1994 .

[76]  Markus Schöberl,et al.  Applications of energy based control methods for the inverted pendulum on a cart , 2009, Robotics Auton. Syst..

[77]  I. D. Landau,et al.  A survey of model reference adaptive techniques - Theory and applications , 1973, Autom..

[78]  Luis T. Aguilar,et al.  Neuro-Fuzzy Based Output Feedback Controller Design for Biped Robot Walking , 2011, Soft Computing for Intelligent Control and Mobile Robotics.

[79]  Su-Yong Shim,et al.  Swing-up control for an inverted pendulum with restricted cart rail length , 2009 .

[80]  Bor-Chin Chang,et al.  A DSP microprocessor hybrid control of an inverted pendulum , 2009, 2009 IEEE International Conference on Control and Automation.

[81]  L. Praly,et al.  Adding integrations, saturated controls, and stabilization for feedforward systems , 1996, IEEE Trans. Autom. Control..

[82]  Huibert Kwakernaak,et al.  Linear Optimal Control Systems , 1972 .

[83]  Mohammed Abdelati,et al.  Design and Implementation of an Experimental Segway Model , 2009 .

[84]  Robin J. Evans,et al.  Hybrid Dynamical Systems: Controller and Sensor Switching Problems , 2012 .

[85]  Karl Johan Åström,et al.  GLOBAL STABILIZATION OF THE INVERTED PENDULUM USING MODEL PREDICTIVE CONTROL , 2002 .

[86]  Thierry Floquet,et al.  Second‐order sliding mode control of underactuated mechanical systems I: Local stabilization with application to an inverted pendulum , 2008 .

[87]  W. Siebert.  Circuits, Signals and Systems , 1985 .

[88]  L. M. Sonneborn,et al.  The Bang-Bang Principle for Linear Control Systems , 1964 .

[89]  Shuzhi Sam Ge,et al.  Adaptive Neural Network Control of Robotic Manipulators , 1999, World Scientific Series in Robotics and Intelligent Systems.

[90]  Katsuhisa Furuta,et al.  Control of unstable mechanical system Control of pendulum , 1976 .

[91]  A. Morse,et al.  Basic problems in stability and design of switched systems , 1999 .

[92]  Oscar Castillo,et al.  Systematic design of a stable type-2 fuzzy logic controller , 2008, Appl. Soft Comput..

[93]  C.W. Anderson,et al.  Learning to control an inverted pendulum using neural networks , 1989, IEEE Control Systems Magazine.

[94]  Frank Allgöwer,et al.  Nonlinear model predictive control : towards new challenging applications , 2009 .

[95]  R. Lozano,et al.  Stabilization of the inverted pendulum around its homoclinic orbit , 2000 .

[96]  K Furuta,et al.  Swing-up Control of Inverted Pendulum Using Pseudo-State Feedback , 1992 .

[97]  Murat Arcak,et al.  Constructive nonlinear control: a historical perspective , 2001, Autom..

[98]  Scott A. Bortoff,et al.  Robust Swing-Up Control for a Rotational Double Pendulum , 1996 .

[99]  Kemalettin Erbatur,et al.  Natural ZMP Trajectories for Biped Robot Reference Generation , 2009, IEEE Transactions on Industrial Electronics.

[100]  Nohé R. Cázarez-Castro,et al.  Designing Type-1 and Type-2 Fuzzy Logic Controllers via Fuzzy Lyapunov Synthesis for nonsmooth mechanical systems , 2012, Eng. Appl. Artif. Intell..

[101]  Vadim I. Utkin,et al.  Sliding mode control in electromechanical systems , 1999 .

[102]  Christopher Edwards,et al.  Sliding mode control : theory and applications , 1998 .

[103]  Peter J. Gawthrop,et al.  Neural networks for control systems - A survey , 1992, Autom..

[104]  Manfred Morari,et al.  Model predictive control: Theory and practice - A survey , 1989, Autom..

[105]  D. Bernstein,et al.  Dynamics and control of a 3D pendulum , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[106]  Fernando Morilla,et al.  A Java/Matlab-based environment for remote control system laboratories: illustrated with an inverted pendulum , 2004, IEEE Transactions on Education.

[107]  Kouhei Ohnishi,et al.  A Bipedal Locomotion Planning Based on Virtual Linear Inverted Pendulum Mode , 2006, IEEE Transactions on Industrial Electronics.

[108]  Jay H. Lee,et al.  Model predictive control: past, present and future , 1999 .

[109]  Kazuo Tanaka,et al.  An approach to fuzzy control of nonlinear systems: stability and design issues , 1996, IEEE Trans. Fuzzy Syst..

[110]  Sung-Hee Lee,et al.  The Reaction Mass Pendulum (RMP) Model for Humanoid Robot Gait and Balance Control , 2009 .

[111]  F. L. Chernousko,et al.  Time-optimal swing-up feedback control of a pendulum , 2006 .

[112]  Mrdjan Jankovic,et al.  TORA example: cascade- and passivity-based control designs , 1996, IEEE Trans. Control. Syst. Technol..

[113]  N. Ono,et al.  Attitude control of a triple inverted pendulum , 1984 .

[114]  David Angeli.  Almost global stabilization of the inverted pendulum via continuous state feedback , 2001, Autom..

[115]  Nohé R. Cázarez-Castro,et al.  Fuzzy logic control with genetic membership function parameters optimization for the output regulation of a servomechanism with nonlinear backlash , 2010, Expert Syst. Appl..

[116]  Javier Aracil,et al.  A new controller for the inverted pendulum on a cart , 2008 .

[117]  Jun Zhao,et al.  Hybrid control for global stabilization of the cart-pendulum system , 2001, Autom..

[118]  Kazunobu Yoshida,et al.  Swing-up control of an inverted pendulum by energy-based methods , 1999, Proceedings of the 1999 American Control Conference (Cat. No. 99CH36251).

[119]  Yasuhisa Hasegawa,et al.  Passive Dynamic Autonomous Control for the Multi-Locomotion Robot , 2011 .

[120]  Fernando Morilla,et al.  Virtual and remote control labs using Java: a qualitative approach , 2002 .

[121]  Kazuhito Yokoi,et al.  Biped walking stabilization based on linear inverted pendulum tracking , 2010, 2010 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[122]  Mi-Ching Tsai,et al.  Synchronisation control of parallel dual inverted pendulums driven by linear servomotors , 2007 .

[123]  Thierry Floquet,et al.  Second‐order sliding mode control of underactuated mechanical systems II: Orbital stabilization of an inverted pendulum with application to swing up/balancing control , 2008 .

[124]  Karl Johan Åström,et al.  Theory and applications of adaptive control - A survey , 1983, Autom..

[125]  Frank Allgöwer,et al.  Nonlinear Model Predictive Control , 2007 .

[126]  Wilson J. Rugh,et al.  Research on gain scheduling , 2000, Autom..

[127]  Katsuhisa Furuta,et al.  Swing up control of inverted pendulum , 1991, Proceedings IECON '91: 1991 International Conference on Industrial Electronics, Control and Instrumentation.

[128]  Katsuhisa Furuta,et al.  Swinging up a pendulum by energy control , 1996, Autom..

[129]  Hongxing Li,et al.  Variable universe adaptive fuzzy control on the quadruple inverted pendulum , 2002 .

[130]  Tore Hägglund,et al.  Automatic Tuning and Adaptation for PID Controllers—A Survey , 1992 .

[131]  Hongnian Yu,et al.  Tracking control of a pendulum-driven cart-pole underactuated system , 2007, 2007 IEEE International Conference on Systems, Man and Cybernetics.

[132]  David Q. Mayne,et al.  Constrained model predictive control: Stability and optimality , 2000, Autom..

[133]  Martin Buss,et al.  Cost-oriented virtual reality and real-time control system architecture , 2003, Robotica.

[134]  Shin'ichi Yuta,et al.  Trajectory tracking control for navigation of the inverse pendulum type self-contained mobile robot , 1996, Robotics Auton. Syst..

[135]  Peter J. Gawthrop,et al.  Intermittent Predictive Control of an Inverted Pendulum , 2006 .

[136]  Shanmuganathan Rajasekar,et al.  Diffusion dynamics near critical bifurcations in a nonlinearly damped pendulum system , 2012 .

[137]  P. Horacek.  Laboratory Experiments for Control Theory Courses: A Survey , 2000 .

[138]  Rey-Chue Hwang,et al.  A self-tuning PID control for a class of nonlinear systems based on the Lyapunov approach , 2002 .

[139]  Karl Johan Åström.  Hybrid control of inverted pendulums , 1998 .

[140]  Yaping Dai,et al.  MATALB & Internet Based Remote Control Laboratory , 2009, 2009 Chinese Control and Decision Conference.

[141]  Meng Joo Er,et al.  Humanoid 3D Gait Generation Based on Inverted Pendulum Model , 2007, 2007 IEEE 22nd International Symposium on Intelligent Control.

[142]  Maki K. Rashid.  Simulation of Intelligent Single Wheel Mobile Robot , 2007 .

[143]  Karl Johan Åström,et al.  The Reaction Wheel Pendulum , 2007, The Reaction Wheel Pendulum.