On the characterization of the Duhem hysteresis operator with clockwise input-output dynamics

Abstract In this paper we investigate the dissipativity property of a certain class of Duhem hysteresis operator, which has clockwise (CW) input–output (I/O) behavior. In particular, we provide sufficient conditions on the Duhem operator such that it is CW and propose an explicit construction of the corresponding function satisfying dissipation inequality of CW systems. The result is used to analyze the stability of a second order system with hysteretic friction which is described by a Dahl model.

[1]  Bayu Jayawardhana,et al.  Stability of systems with the Duhem hysteresis operator: The dissipativity approach , 2012, Autom..

[2]  P. S. Bauer.  Dissipative Dynamical Systems: I. , 1931, Proceedings of the National Academy of Sciences of the United States of America.

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

[4]  Bayu Jayawardhana,et al.  Sufficient conditions for dissipativity on Duhem hysteresis model , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[5]  Carlos Canudas de Wit,et al.  A new model for control of systems with friction , 1995, IEEE Trans. Autom. Control..

[6]  Dennis S. Bernstein,et al.  Semilinear Duhem model for rate-independent and rate-dependent hysteresis , 2005, IEEE Transactions on Automatic Control.

[7]  Mayergoyz,et al.  Mathematical models of hysteresis. , 1986, Physical review letters.

[8]  J. Willems,et al.  Every storage function is a state function , 1997 .

[9]  J. S. Wang.  Statistical Theory of Superlattices with Long-Range Interaction. I. General Theory , 1938 .

[10]  A. Schaft.  L2-Gain and Passivity Techniques in Nonlinear Control. Lecture Notes in Control and Information Sciences 218 , 1996 .

[11]  Hartmut Logemann,et al.  Systems with hysteresis in the feedback loop: existence, regularity and asymptotic behaviour of solutions , 2003 .

[12]  A. Visintin.  Differential models of hysteresis , 1994 .

[13]  Philip Hartman,et al.  Ordinary differential equations, Second Edition , 2002, Classics in applied mathematics.

[14]  Hartmut Logemann,et al.  Asymptotic Behaviour of Nonlinear Systems , 2004, Am. Math. Mon..

[15]  M. Brokate,et al.  Hysteresis and Phase Transitions , 1996 .

[16]  David Angeli,et al.  Multistability in Systems With Counter-Clockwise Input–Output Dynamics , 2007, IEEE Transactions on Automatic Control.

[17]  Billie F. Spencer,et al.  Modeling and Control of Magnetorheological Dampers for Seismic Response Reduction , 1996 .

[18]  Billie F. Spencer,et al.  Models for hysteresis and application to structural control , 1997, Proceedings of the 1997 American Control Conference (Cat. No.97CH36041).

[19]  Ian R. Petersen,et al.  Feedback Control of Negative-Imaginary Systems: Large Flexible structures with colocated actuators and sensors , 2014, ArXiv.

[20]  Peter C. Breedveld,et al.  Port-based modeling of mechatronic systems , 2004, Math. Comput. Simul..

[21]  N. G. Parke,et al.  Ordinary Differential Equations. , 1958 .

[22]  P. Dahl.  Solid Friction Damping of Mechanical Vibrations , 1976 .

[23]  Alexander Lanzon,et al.  Feedback Control of Negative-Imaginary Systems , 2010, IEEE Control Systems.

[24]  Hartmut Logemann,et al.  Extending hysteresis operators to spaces of piecewise continuous functions , 2003 .

[25]  S. Fassois,et al.  Duhem modeling of friction-induced hysteresis , 2008, IEEE Control Systems.

[26]  JinHyoung Oh,et al.  Counterclockwise Dynamics of a Rate-Independent Semilinear Duhem Model , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[27]  Bayu Jayawardhana,et al.  Dissipativity of general Duhem hysteresis models , 2011, IEEE Conference on Decision and Control and European Control Conference.

[28]  J. Willems.  Dissipative dynamical systems part I: General theory , 1972 .

[29]  I. Mayergoyz.  Mathematical models of hysteresis and their applications , 2003 .

[30]  Isaak D. Mayergoyz,et al.  The science of hysteresis , 2005 .

[31]  Kirsten Morris,et al.  Generalized dissipation in hysteretic systems , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[32]  Jonathan P. How,et al.  A KYP lemma and invariance principle for systems with multiple hysteresis non-linearities , 2001 .

[33]  D. Jordan,et al.  Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers , 1979 .

[34]  David Angeli,et al.  Systems with counterclockwise input-output dynamics , 2006, IEEE Transactions on Automatic Control.

[35]  Y. Wen.  Method for Random Vibration of Hysteretic Systems , 1976 .