Causal canonical decomposition of hysteresis systems

•Hysteresis systems are approximately rate independent for slow inputs.•Hysteresis systems are causal.•This paper presents a new decomposition of hysteresis operators into a causal rate-independent component and a causal nonhysteretic component that vanishes for slow inputs. Hysteresis is a special type of behavior found in many areas including magnetism, mechanics, biology, economics, etc. One of the characteristics of hysteresis systems is that they are approximately rate independent for slow inputs. It is possible to express this characteristic in mathematical language by decomposing hysteresis operators as the sum of a rate independent component and a nonhysteretic component which vanishes in steady state for slow inputs. This decomposition -called canonical decomposition- is possible for a class of hysteresis operators for which a continuous input leads to a continuous output and a continuous hysteresis loop. The canonical decomposition can be obtained using the concept of confluence . On the other hand, hysteresis systems are causal which means that their output depends on the current and/or previous values of the input but not on the future values of that input. Are the components of the canonical decomposition also causal? The answer is en general negative. The lack of causality of 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 other useful structures. This paper proposes a new decomposition of hysteresis operators called causal canonical decomposition in which the rate independent component and the nonhysteretic component are both causal. The main tool to obtain the causal canonical decomposition is a new mathematical equation that we call uniform confluence. Using this equation we show that the causal canonical decomposition is unique. The concepts introduced in the paper are applied to the hysteretic scalar semilinear Duhem model as a case study.