Stability Analysis and Controller Design for a Linear System with Duhem Hysteresis Nonlinearity

In this paper, we investigate the stability of a feedback interconnection between a linear system and a Duhem hysteresis operator, where the linear system and the Duhem hysteresis operator satisfy either the counter-clockwise (CCW) or clockwise (CW) input-output dynamics. More precisely, we present sufficient conditions for the stability of the interconnected system that depend on the CW or CCW properties of the linear system and the Duhem operator. Based on these results we introduce a control design methodology for stabilizing a linear plant with a hysteretic actuator or sensor without requiring precise information on the hysteresis operator.

💡 Research Summary

The paper addresses the challenging problem of guaranteeing stability for a feedback interconnection between a linear dynamical system and a hysteresis element modeled by the Duhem operator. The Duhem operator captures the characteristic rate‑dependent, input‑output hysteresis by means of a differential relationship that switches between two distinct trajectories depending on whether the input is increasing or decreasing. This non‑smooth behavior makes conventional linear analysis inadequate.

The authors first formalize the concepts of clockwise (CW) and counter‑clockwise (CCW) input‑output dynamics. A system is said to be CW if the integral (\int_0^T u^\top \dot y,dt) is bounded from below (i.e., the system dissipates more energy than it stores), while CCW corresponds to a bound on (\int_0^T y^\top \dot u,dt). These definitions are directly linked to passivity theory: CW systems behave like “output‑passive” devices, whereas CCW systems are “input‑passive”.

For the linear part, expressed in state‑space form (\dot x = A x + B u,; y = C x + D u), the paper derives linear matrix inequality (LMI) conditions that are both necessary and sufficient for the system to exhibit CW or CCW behavior. Specifically, the existence of a positive definite matrix (P) and a scalar (\lambda>0) satisfying

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