Dynamics of Simple Balancing Models with Time-Delayed Switching Feedback Control

Time-delayed control in a balancing problem may be a nonsmooth function for a variety of reasons. In this paper we study a simple model of the control of an inverted pendulum by either a connected movable cart or an applied torque for which the control is turned off when the state of the pendulum is located within certain regions of phase space. Without applying a small angle approximation for deviations about the vertical position, we see structurally stable periodic orbits which may be attracting or repelling. Due to the nonsmooth nature of the control, these periodic orbits originate in various discontinuity-induced bifurcations. We also show that a coincidence of switching events can produce complicated periodic and aperiodic solutions.

[1]  T. Ohira,et al.  The time-delayed inverted pendulum: implications for human balance control. , 2009, Chaos.

[2]  Sue Ann Campbell,et al.  Calculating Centre Manifolds for Delay Differential Equations Using Maple , 2009 .

[3]  Tamás Kalmár-Nagy,et al.  Delay differential equations : recent advances and new directions , 2009 .

[4]  Mary E. Landry,et al.  Dynamics of an Inverted Pendulum with Delayed Feedback Control , 2005, SIAM J. Appl. Dyn. Syst..

[5]  Ian David Loram,et al.  Direct measurement of human ankle stiffness during quiet standing: the intrinsic mechanical stiffness is insufficient for stability , 2002, The Journal of physiology.

[6]  S. Lunel,et al.  Delay Equations. Functional-, Complex-, and Nonlinear Analysis , 1995 .

[7]  Stability , 1973 .

[8]  Juan Luis Cabrera,et al.  Controlling instability with delayed antagonistic stochastic dynamics , 2005 .

[9]  Thomas Bräunl.  Embedded robotics - mobile robot design and applications with embedded systems (2. ed.) , 2003 .

[10]  Gábor Stépán,et al.  SAMPLING DELAY AND BACKLASH IN BALANCING SYSTEMS , 2000 .

[11]  P. Glendinning.  Stability, Instability and Chaos: An Introduction to the Theory of Nonlinear Differential Equations , 1994 .

[12]  Mario di Bernardo,et al.  Complex Dynamics in a Hysteretic Relay Feedback System with Delay , 2007, J. Nonlinear Sci..

[13]  Mario di Bernardo,et al.  Piecewise smooth dynamical systems , 2008, Scholarpedia.

[14]  Enrique Ponce,et al.  The continuous matching of two stable linear systems can be unstable , 2006 .

[16]  James U. Korein,et al.  Robotics , 2018, IBM Syst. J..

[17]  B. Day,et al.  Effect of vision and stance width on human body motion when standing: implications for afferent control of lateral sway. , 1993, The Journal of physiology.

[18]  A. D. Lewis,et al.  Geometric control of mechanical systems : modeling, analysis, and design for simple mechanical control systems , 2005 .

[19]  Edwin Kreuzer,et al.  Iutam Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty , 2007 .

[20]  R. Leine,et al.  Bifurcations in Nonlinear Discontinuous Systems , 2000 .

[21]  John G. Milton,et al.  Unstable dynamical systems: Delays, noise and control , 2008 .

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

[23]  Bernd Krauskopf,et al.  Extending the permissible control loop latency for the controlled inverted pendulum , 2005 .

[24]  Bernd Krauskopf,et al.  Complex balancing motions of an inverted pendulum subject to delayed feedback control , 2004 .

[25]  Max Donath,et al.  American Control Conference , 1993 .

[26]  Gábor Stépán,et al.  Stability of time-periodic and delayed systems - a route to act-and-wait control , 2006, Annu. Rev. Control..

[27]  Houman Dallali,et al.  Modelling human balance using switched systems with linear feedback control , 2012, Journal of The Royal Society Interface.

[28]  Motoki Kouzaki,et al.  Importance of body sway velocity information in controlling ankle extensor activities during quiet stance. , 2003, Journal of neurophysiology.

[29]  T. Erneux.  Applied Delay Differential Equations , 2009 .

[30]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[31]  C. Budd,et al.  Review of ”Piecewise-Smooth Dynamical Systems: Theory and Applications by M. di Bernardo, C. Budd, A. Champneys and P. 2008” , 2020 .

[32]  Gábor Stépán,et al.  Robust Time-Periodic Control of Time-Delayed Systems , 2007 .

[33]  J. Craggs.  Applied Mathematical Sciences , 1973 .

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

[35]  H. D. Jong,et al.  Piecewise-linear Models of Genetic Regulatory Networks: Equilibria and their Stability , 2006, Journal of mathematical biology.

[36]  Alex M. Andrew.  Embedded Robotics: Mobile Robot Design and Applications with Embedded Systems , 2004 .

[37]  A. D. Lewis,et al.  Geometric Control of Mechanical Systems , 2004, IEEE Transactions on Automatic Control.

[38]  John G. Milton,et al.  Balancing with Vibration: A Prelude for “Drift and Act” Balance Control , 2009, PloS one.

[39]  Ian David Loram,et al.  The frequency of human, manual adjustments in balancing an inverted pendulum is constrained by intrinsic physiological factors , 2006, The Journal of physiology.

[40]  T. Insperger,et al.  Increasing the Accuracy of Digital Force Control Process Using the Act-and-Wait Concept , 2010, IEEE/ASME Transactions on Mechatronics.

[41]  Thomas Braunl.  Embedded Robotics , 2003 .

[42]  Haoping Wang,et al.  Hybrid control for vision based Cart-Inverted Pendulum system , 2008, 2008 American Control Conference.

[43]  Hai Lin,et al.  Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results , 2009, IEEE Transactions on Automatic Control.

[44]  Jan Sieber,et al.  Dynamics of delayed relay systems , 2006 .

[45]  Toru Ohira,et al.  Balancing with positive feedback: the case for discontinuous control , 2009, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[46]  Aleksej F. Filippov,et al.  Differential Equations with Discontinuous Righthand Sides , 1988, Mathematics and Its Applications.

[47]  Shinji Hara,et al.  Stability tests and stabilization for piecewise linear systems based on poles and zeros of subsystems , 2006, Autom..

[48]  Gábor Stépán,et al.  Balancing with Reflex Delay , 2000 .

[49]  John G Milton,et al.  On-off intermittency in a human balancing task. , 2002, Physical review letters.

[50]  R. Stengel,et al.  CONTROL systems. , 1952, Hospitals.

[51]  Yoshiyuki Asai,et al.  A Model of Postural Control in Quiet Standing: Robust Compensation of Delay-Induced Instability Using Intermittent Activation of Feedback Control , 2009, PloS one.

[52]  J. Milton,et al.  Noise-induced transitions in human postural sway. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[53]  Bernd Krauskopf,et al.  Bifurcation analysis of an inverted pendulum with delayed feedback control near a triple-zero eigenvalue singularity , 2003 .

[54]  J. Sieber,et al.  Dynamics of Symmetric Dynamical Systems with Delayed Switching , 2010 .

[55]  Constantinos N Maganaris,et al.  Active, non‐spring‐like muscle movements in human postural sway: how might paradoxical changes in muscle length be produced? , 2005, The Journal of physiology.

[56]  Enrique Ponce,et al.  Bifurcation Sets of Continuous Piecewise Linear Systems with Two Zones , 1998 .

[57]  P. Morasso,et al.  Direct measurement of ankle stiffness during quiet standing: implications for control modelling and clinical application. , 2005, Gait & posture.