On the Characterization of the Duhem Hysteresis Operator with Clockwise Input-Output Dynamics

Technical Summary: On the Characterization of the Duhem Hysteresis Operator with Clockwise Input-Output Dynamics

Problem Statement
Hysteresis is a prevalent nonlinear phenomenon in physical systems (e.g., piezo-actuators, ferromagnetic materials, mechanical systems). While input-output (I/O) behaviors are often characterized as counterclockwise (CCW) or clockwise (CW), existing literature has extensively analyzed CCW dynamics (associated with passivity from input to output derivative) but lacks a rigorous framework for CW dynamics in the context of Duhem hysteresis operators. Specifically, there is a need to characterize Duhem operators that exhibit CW I/O behavior, where the supply rate is defined by the product of the output and the input derivative ($y\dot{u}$), rather than the input and output derivative ($\dot{y}u$). This is particularly relevant for mechanical systems with hysteretic friction, where the relationship between relative displacement and friction force often follows CW dynamics.

Methodology
The authors investigate a specific class of Duhem hysteresis operators defined by the differential equation:
$$\dot{y}(t) = f_1(y(t), u(t))\dot{u}_+(t) + f_2(y(t), u(t))\dot{u}_-(t)$$
where $\dot{u}_+$ and $\dot{u}_-$ represent the positive and negative parts of the input derivative, respectively.

To characterize the CW property, the paper introduces three auxiliary functions derived from the operator's defining functions $f_1$ and $f_2$:

1. Anhysteresis Function ($f_{an}$): A curve where $f_1 = f_2$, representing the locus of points where the hysteresis loop collapses.
2. Traversing Function ($\omega_\Phi$): Describes the trajectory of the operator when a monotone increasing or decreasing input is applied from a specific state.
3. Intersecting Function ($\Lambda$): Defines the unique intersection point between the traversing trajectory and the anhysteresis curve.

Using these functions, the authors construct a storage function $H_\curvearrowright: \mathbb{R}^2 \to \mathbb{R}^+$ defined by the integral of the anhysteresis function minus the integral of the traversing function. The core of the methodology involves deriving sufficient conditions on $f_1$ and $f_2$ (specifically regarding the sign of a derived function $F$ relative to the anhysteresis curve) to ensure that the time derivative of this storage function satisfies the CW dissipation inequality:
$$\frac{d}{dt}H_\curvearrowright(y(t), u(t)) \leq y(t)\dot{u}(t)$$

Key Contributions

1. Sufficient Conditions for CW Dynamics: The paper provides explicit sufficient conditions on the functions $f_1$ and $f_2$ of a Duhem operator to guarantee that the operator exhibits clockwise I/O dynamics. This is formalized in Theorem 1.
2. Explicit Construction of Storage Function: Unlike previous works that established existence, this paper offers an explicit construction of the storage function $H_\curvearrowright$ (Equation 20) that satisfies the CW dissipation inequality. This function is shown to be the "available storage function" for the supply rate $y\dot{u}$.
3. Generalization to Saturated Outputs: The results are extended in Proposition 2 to accommodate Duhem operators with saturated outputs, a common feature in physical models like the Dahl model.
4. Stability Analysis Framework: The paper demonstrates how the constructed storage function can be utilized as a Lyapunov function to analyze the stability of systems incorporating CW hysteresis.

Results

• Theoretical Validation: The authors prove that if the constructed function $H_\curvearrowright$ satisfies the dissipation inequality, the operator is CW. The proof relies on the properties of the intersecting function $\Lambda$ and the behavior of the traversing function $\omega_\Phi$.
• Application to Dahl Model: The theory is applied to the Dahl model of friction. The paper explicitly derives the storage function $H_\curvearrowright$ for the Dahl model (Equation 26) and verifies that it satisfies the required dissipation inequality.
• Stability of Second-Order Systems: Using the derived storage function, the authors analyze a second-order mechanical system with hysteretic friction described by the Dahl model. By constructing a composite Lyapunov function $V = \frac{1}{2}kx_1^2 + \frac{1}{2}mx_2^2 + H_\curvearrowright(x_3, x_1)$, they demonstrate that the system is passive from the applied force to the velocity. Consequently, a simple proportional feedback control law ($F = -dx_2$) is shown to guarantee the asymptotic convergence of the velocity to zero, without requiring precise knowledge of the friction parameters ($\rho$ and $F_C$).

Significance
The paper claims that its primary significance lies in providing a dual extension to existing CCW characterization results, specifically tailored for systems with CW I/O dynamics. By establishing a rigorous dissipativity framework for Duhem operators with CW behavior, the work enables the stability analysis of mechanical systems with hysteretic friction without needing to invert the operator (which may be physically unrealizable or non-invertible). Furthermore, the explicit construction of the storage function offers a practical tool for system identification, allowing researchers to restrict the class of Duhem operators fitted to measurement data based on their CW properties. The results bridge the gap between abstract dissipativity theory and the practical modeling of friction-induced hysteresis in control systems.