Causal canonical decomposition of hysteresis systems
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Hysteresis is a special type of behavior found in many areas including magnetism, mechanics, bi-ology, economics, etc. One of the characteristics of hysteresis systems is that they are approximatelyrate independent for slow inputs. It is possible to express this characteristic in mathematical languageby decomposing hysteresis operators as the sum of a rate independent component and a nonhystereticcomponent which vanishes in steady state for slow inputs. This decomposition -calledcanonical decom-position- is possible for a class of hysteresis operators for which a continuous input leads to a continuousoutput and a continuous hysteresis loop. The canonical decomposition can be obtained using the conceptofcon uencewhich is an equation that continuous hysteresis operators should verify.On the other hand, hysteresis systems are causal which means that their output depends on the currentand/or previous values of the input but not on the future values of that input. Are the components ofthe canonical decomposition also causal? The answer is en general negative. The lack of causalityof these components means that they cannot be written in the form of differential equations, integro-differential equations, partial differential equations, partial integro-differential equations and many otheruseful structures.This paper proposes a new decomposition of hysteresis operators calledcausal canonical decompositionin which the rate independent component and the nonhysteretic component are both causal. The maintool to obtain the causal canonical decomposition is a new mathematical equation that we calluniformcon uence. Using this equation we show that the causal canonical decomposition is unique. The conceptsintroduced in the paper are applied to the hysteretic scalar semilinear Duhem model as a case study.
CitationIkhouane, F. Causal canonical decomposition of hysteresis systems. "Communications in nonlinear science and numerical simulation", 1 Octubre 2020, vol. 89, p. 105278:1-105278:12.