Hysteresis Dynamics in Control Systems

Hysteresis dynamics arise in control systems whenever the relationship between an input and output exhibits memory of past states, manifesting as looped or multi-valued characteristics. Such behaviour is widespread in mechanical components with frictional or magnetic effects, in thermo-mechanical devices employing shape-memory alloys, and in power-electronic converters featuring thyristor switching. The presence of hysteresis can profoundly influence stability, limit-cycle oscillations and transient response, often complicating controller design yet offering opportunities for energy harvesting or robust switching. Modelling approaches range from phenomenological operator descriptions, such as Preisach representations and relay models, to detailed gradient-flow realisations and piecewise-linear systems. Analysis techniques draw on bifurcation theory, Floquet multipliers for periodic orbits, and asymptotic methods for threshold-governed dynamics. A comprehensive understanding of hysteresis dynamics is vital for optimising performance in applications as diverse as precision positioning, vibration mitigation and renewable-energy technologies.

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Recent studies have addressed hysteresis in power-electronic control loops by examining self-oscillatory regimes in systems with thyristor converters. These works establish conditions for the emergence of periodic orbits and derive stability criteria using Lyapunov and orbital asymptotic analyses, offering design guidelines for oscillation-based sensing and regulation. In thermal-energy harvesting, simplified bang-bang hysteresis controllers coupled to nonlinear thermal reactors have been shown to support sustained periodic modes; analytical conditions on controller thresholds and system parameters guarantee the existence of limit-cycle behaviour, paving the way for compact, self-powered microsystems. Meanwhile, in flight-control and mechanical applications, novel frameworks employing Floquet theory in piecewise-linear systems enable prediction and stability assessment of limit-cycle oscillations arising from multiple nonlinear switching elements. By transforming discontinuous nonlinearities into equivalent analytic forms, these methods allow precise computation of oscillation amplitudes and stability margins, demonstrating efficacy in realistic control scenarios.

Hysteresis Dynamics in Control Systems publication trend

The graph below shows the total number of articles in hysteresis dynamics in control systems across all publications each year (not limited to Nature Index journals).

Technical terms

Hysteresis: A property of a system whereby its output depends on both its current input and its history, typically manifesting as a looped input–output characteristic and memory effect.

Preisach operator: A mathematical model that represents complex hysteresis through an aggregation of elementary relay hysteresis units, enabling quantitative description of memory and threshold behaviour.

Limit cycle: A closed, isolated trajectory in the phase space of a dynamical system, representing sustained periodic oscillations independent of initial conditions within its basin of attraction.

Bifurcation: A qualitative change in the dynamics of a system induced by variation of a parameter, often leading to the emergence or disappearance of periodic or chaotic behaviour.

Relay nonlinearity: A form of hysteretic element in control systems characterised by discrete switching between states when inputs cross predefined thresholds, commonly used to model on–off control actions.

References

1. Realization of arbitrary hysteresis by a low-dimensional gradient flow. Discrete and Continuous Dynamical Systems - B (2015).
2. Robust homoclinic orbits in planar systems with Preisach hysteresis operator. Journal of Physics Conference Series (2016).
3. Prediction of limit cycle oscillations in piecewise linear systems with multiple piecewise nonlinearities. IET Control Theory and Applications (2020).
4. Control systems with Thyristor converters. E3S Web of Conferences (2024).
5. Modelling and Investigation of a Hybrid Thermal Energy Harvester. MATEC Web of Conferences (2018).
6. Complete Asymptotic and Bifurcation Analysis for a Difference Equation with Piecewise Constant Control. Advances in Continuous and Discrete Models (2010).