⚡ electrical engineering

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

This paper establishes sufficient stability conditions for feedback interconnections between linear systems and Duhem hysteresis operators based on their clockwise or counter-clockwise input-output dynamics, and leverages these findings to propose a control design methodology that stabilizes linear plants with hysteretic actuators or sensors without requiring precise knowledge of the hysteresis model.

Original authors: Ruiyue Ouyang, Bayu Jayawardhana

Published 2026-06-03

📖 5 min read🧠 Deep dive

CC BY 3.0

Original authors: Ruiyue Ouyang, Bayu Jayawardhana

Original paper licensed under CC BY 3.0 (http://creativecommons.org/licenses/by/3.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to drive a car, but the steering wheel has a strange, sticky memory. If you turn it left, it doesn't just stay left; it "remembers" where you turned it before, and it resists moving back to the center. This is hysteresis. It's a common quirk in many machines, like the metal in a magnet or the rubber in a shock absorber.

This paper is about a team of engineers (Ouyang and Jayawardhana) who figured out how to keep a machine stable even when its "steering wheel" (the actuator or sensor) has this sticky, memory-filled behavior. They didn't try to fix the stickiness or calculate exactly how sticky it is. Instead, they looked at the direction the stickiness pushes things.

The Two Types of "Stickiness"

The authors realized that this memory effect behaves in one of two main ways, which they call Counter-Clockwise (CCW) and Clockwise (CW). Think of these as two different dance moves:

Counter-Clockwise (CCW): Imagine a dancer who, when you push them, they push back harder the more you push, but they always seem to be "leading" the movement in a specific, safe loop. In the real world, this is like a piezo-electric actuator (used in high-precision microscopes). When you give it electricity, it moves, but it has a "lag" that follows a specific, predictable loop.
Clockwise (CW): Imagine a dancer who, when you push them, they seem to lag behind and "drag" their feet, creating a loop in the opposite direction. In the real world, this is like friction in a mechanical system. When you try to slide a heavy box, the friction force fights you, creating a different kind of loop.

The Problem: The Feedback Loop

The paper looks at a system where a Linear Machine (a predictable, standard engine) is connected to a Hysteretic Part (the sticky dancer). They are connected in a loop: the machine tells the dancer what to do, and the dancer tells the machine what happened.

The big question is: Will this loop spin out of control, or will it settle down?

If the machine and the dancer are "dancing" in the wrong combination, the system might shake itself apart. If they are dancing in the right combination, the system will naturally calm down and stop moving.

The Solution: A "Stability Map"

The authors created a set of rules (mathematical conditions) to predict if the system will be safe. They didn't need to know the exact formula for the stickiness. They only needed to know:

Is the machine "Counter-Clockwise" or "Clockwise"?
Is the hysteresis "Counter-Clockwise" or "Clockwise"?

They found four possible pairings (like a 2x2 grid):

CCW Machine + CCW Hysteresis: These can work together if connected with a "positive" feedback (like two people pushing in the same direction).
CW Machine + CCW Hysteresis: These work well with "negative" feedback (like a thermostat that turns the heat off when it gets too hot).
CCW Machine + CW Hysteresis: These also work with "negative" feedback.
CW Machine + CW Hysteresis: These can work with "positive" feedback.

The "Energy" Trick

How did they prove this? They used a concept called Lyapunov functions, which is like a "Energy Meter."

Imagine the whole system (machine + sticky part) has a total amount of "energy."

If the system is unstable, this energy meter goes up forever, and the machine explodes or shakes violently.
If the system is stable, the authors showed that this energy meter always goes down (or stays the same) over time.

They built a special "Energy Meter" for each of the four cases. They proved that if you connect the machine and the hysteresis in the right way (positive or negative feedback), the "Energy Meter" will always drain down until the system stops moving.

The "No-Need-to-Know-Everything" Design

The most useful part of this paper is the Control Design Method.

Usually, to fix a machine with sticky parts, engineers have to build a complex "inverse" model to cancel out the stickiness. It's like trying to un-stick a magnet by calculating the exact magnetic field in reverse. It's hard and requires perfect information.

This paper says: You don't need to do that.
You just need to know the "direction" of the stickiness (CCW or CW). Once you know that, you can design a controller (a simple set of rules for the machine) that guarantees stability, even if you don't know the exact details of the stickiness.

Real-World Examples in the Paper

The authors tested their theory with two examples:

A Mass-Spring-Damper System: Imagine a weight on a spring with a shock absorber.
- They attached a Piezo-actuator (CCW) to it. They designed a controller, and the system settled down smoothly.
- They attached a Friction-like actuator (CW) to it. They designed a different controller, and again, the system settled down.

The Bottom Line

This paper provides a universal safety checklist for machines with "sticky" parts.

Don't try to perfectly model the stickiness.
Do identify if the stickiness is "Counter-Clockwise" or "Clockwise."
Do connect the machine and the stickiness in the matching way (Positive or Negative feedback).

If you follow these steps, the system will naturally calm itself down, just like a spinning top that eventually finds its balance and stops. This saves engineers from doing impossible math and allows them to build stable systems even when they don't fully understand the "sticky" parts inside.

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

Problem Statement
This paper addresses the stability analysis and control design for feedback interconnections between a linear system and a Duhem hysteresis operator. Hysteresis is a prevalent nonlinearity in diverse systems, including piezo-actuators, ferromagnetic materials, and mechanical friction. While various models exist (e.g., Preisach, backlash, elastic-plastic), this work focuses on the Duhem operator due to its ability to capture complex input-output (I/O) dynamics. Specifically, the authors investigate systems where the linear plant and the hysteresis operator exhibit either counter-clockwise (CCW) or clockwise (CW) I/O dynamics. The core challenge is to establish stability conditions and design controllers for these interconnected systems without requiring precise parametric knowledge of the hysteresis operator, relying instead on its structural I/O properties.

Methodology
The authors employ a dissipativity-based approach, utilizing storage functions (Lyapunov functions) derived from the specific I/O characteristics of the subsystems.

Characterization of Dynamics:
- CCW Dynamics: Defined by the inequality $\liminf_{T\to\infty} \int_0^T \langle \dot{y}(t), u(t) \rangle dt > -\infty$ . For Duhem operators, this is characterized by a storage function $H^\circlearrowleft$ satisfying $\frac{d}{dt}H^\circlearrowleft \leq \dot{y}u$ .
- CW Dynamics: Defined by the dual inequality $\liminf_{T\to\infty} \int_0^T y(t)^T \dot{u}(t) dt > -\infty$ . For Duhem operators, this is characterized by a storage function $H^\circlearrowright$ satisfying $\frac{d}{dt}H^\circlearrowright \leq y\dot{u}$ .
- The paper reviews conditions on the Duhem functions $f_1$ and $f_2$ (specifically involving an anhysteresis function $f_{an}$ and traversing functions) that guarantee these CCW or CW properties.
Stability Analysis of Interconnections:
The paper analyzes four distinct cases of interconnection between a linear system $P$ and a Duhem operator $\Phi$ (summarized in Table I of the original text):
- Case (a): Both $P$ and $\Phi$ are CCW (Positive feedback). Stability is proven using a composite Lyapunov function involving the linear system's quadratic form and the Duhem storage function $H^\circlearrowleft$ . Conditions depend on the sector bound of the anhysteresis function $f_{an}$ .
- Case (b): $P$ is CW and $\Phi$ is CCW (Negative feedback). Stability is established using Linear Matrix Inequalities (LMIs) that are independent of the specific parameters of $\Phi$ , relying only on its CCW property.
- Case (c): $P$ is CCW and $\Phi$ is CW (Negative feedback). Stability is proven using the linear system's S-CCW property and the Duhem storage function $H^\circlearrowright$ .
- Case (d): Both $P$ and $\Phi$ are CW (Positive feedback). Stability is analyzed using extended state-space representations and LMIs, requiring bounds on the functions $f_1$ and $f_2$ .
Controller Design:
Based on the stability theorems, the authors propose a control design methodology. The controller $C$ is designed to ensure the cascaded linear system (plant + controller) satisfies the specific LMI or algebraic conditions required for the corresponding interconnection case. This allows for stabilization without inverting the hysteresis operator.

Key Contributions and Results

Sufficient Stability Conditions: The paper provides rigorous sufficient conditions for the stability of closed-loop systems involving Duhem hysteresis. These conditions are categorized by the CCW/CW nature of the plant and the hysteresis operator.
Lyapunov Construction: Explicit constructions of Lyapunov functions are provided for each interconnection case, combining the energy of the linear system with the storage functions ( $H^\circlearrowleft$ or $H^\circlearrowright$ ) of the hysteresis operator.
Controller Synthesis: A systematic algorithm is presented for designing controllers for linear plants with hysteretic actuators or sensors. The method relies on solving LMIs or algebraic inequalities, avoiding the need for precise hysteresis modeling or inverse compensation.
Numerical Validation: The theoretical results are validated through numerical examples involving a mass-damper-spring system with both CCW (piezo-like) and CW (friction-like) hysteretic actuators. Simulations demonstrate that the designed controllers successfully stabilize the systems, driving velocities to zero and positions to constants.

Significance and Claims
The authors claim that this work offers a different paradigm for controlling systems with hysteresis. Unlike traditional approaches that often require defining an inverse hysteresis operator to cancel the nonlinearity, this method exploits the inherent dissipative properties (CCW or CW) of the hysteresis. Consequently, the control design does not require precise information about the hysteresis operator's parameters, only its qualitative I/O dynamics. The paper asserts that this dissipativity approach can be extended to nonlinear plants that exhibit CCW dynamics, broadening the applicability of the results beyond linear systems. The work is motivated by and builds upon previous results regarding the interconnection of negative imaginary systems and CCW systems, extending these concepts to include CW dynamics and the specific structure of the Duhem operator.