Control of Hysteretic Systems Through Inverse Compensation RAM

n an 1881 study of ferromagnetism [37], James A. Ewing coined the term hysteresis, which means “to lag behind” in Greek. In Figure 1, which is reproduced from [1], the horizontal axis is the magnitude of the average magnetic field H in a soft-iron ring, while the vertical axis is the magnitude of the average magnetic flux density B. The plot was obtained by varying the magnetic field in a quasi-static manner and recording the average magnetic flux density. The following observations can be made from this example: 1) Causality: The output B depends on past and current values of the input H only. 2) Monotonicity: The output B increases and decreases with a corresponding change in the input H. 3) Presence of a major loop: The set of points in the (H, B) plane that can be reached lie between two curves that together form the major loop. The increasing part of this loop is obtained by first decreasing the magnetic field H to a large negative value −Hmax for which the magnetic flux density saturates at −Bmax, and then increasing it to Hmax. 4) Minor loop closure: Suppose that the input H and output B at some instant are such that (H, B) is on the major loop. If the input is changed to a value H̃ and then changed back to H, then the output changes its value to B̃ and back to B. In other words, (H, B) → (H̃, B̃) → (H, B). 5) Energy Dissipation: It can be shown that the energy needed from a power source to complete a loop, say between (H1, B1) and (H2, B2), is proportional to the area enclosed by the closed loop. 6) Order Preservation: The paths for increasing inputs in the (H, B) plane are nonintersecting, as are the paths for decreasing inputs. Control of Hysteretic Systems Through Inverse Compensation