⚡ electrical engineering

Fractional-Order Linear-Quadratic-Regulator-Based Sliding Mode Control with Half- Derivative Sliding Motion Condition for Magnetic Levitation Systems

This paper proposes a fractional-order sliding mode control strategy for magnetic levitation systems that integrates fractional-order linear-quadratic-regulator theory with a half-derivative sliding condition and sigmoid-based smoothing to achieve robust, chattering-free target tracking with enhanced stability and performance compared to conventional integer-order methods.

Original authors: Yoshihiro Maruo, Masaharu Kuroda, Natsuki Kawaguchi

Published 2026-07-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yoshihiro Maruo, Masaharu Kuroda, Natsuki Kawaguchi

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ 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

The Big Picture: Keeping a Ball Floating

Imagine you are trying to balance a steel ball in mid-air using only a magnet. This is the "magnetic levitation system" the paper studies. It's a tricky job because the ball wants to fall down due to gravity, but if you push the magnet too hard, the ball snaps up and sticks to it. The system is naturally unstable and very sensitive.

The researchers wanted to build a "smart controller" (a computer program) that keeps this ball floating perfectly, even if the wind blows or the ball gets bumped. They compared two types of controllers:

The Old Way (Integer-Order): The standard, traditional method used for decades.
The New Way (Fractional-Order): A newer, more advanced method that uses a special kind of math called "fractional calculus."

The Problem with the Old Way: The "Jittery" Driver

Think of the traditional controller like a nervous driver trying to keep a car in the center of a lane.

The Issue: To stay in the lane, the driver constantly jerks the steering wheel left and right. This is called chattering. In a magnetic system, this "jerking" causes the voltage to spike up and down rapidly. It makes the system noisy, wears out the equipment, and can actually make the ball unstable.
The Goal: The researchers wanted a controller that steers smoothly, like a professional driver, without all that jitter.

The Solution: A "Fractional" Approach

The researchers introduced a new type of math called Fractional Calculus.

The Analogy: Imagine normal math (integer calculus) as taking steps of 1 meter. You go 1 meter, then another 1 meter.
Fractional Math: This allows you to take steps of "half a meter" or "0.5 meters." It's like having a finer ruler.
Why it helps: By using these "half-steps," the controller can see the future behavior of the ball more accurately. It understands the "memory" of the system (how the ball moved a split second ago) better than the old method. This allows for a much smoother ride.

How They Built the New Controller

The paper describes a three-step recipe for building this new controller:

The Blueprint (Linear Quadratic Regulator):
First, they created a perfect "blueprint" for how the ball should move. They used a mathematical tool called LQR (Linear Quadratic Regulator) to calculate the ideal path. Think of this as drawing the perfect line on the road that the car should follow.
The "Half-Derivative" Rule:
Instead of just looking at the current position, they used a "half-derivative" rule.
- Analogy: If a normal rule says, "If you are off-center, turn the wheel," this new rule says, "If you are off-center and your momentum is carrying you there, turn the wheel gently."
- They set a condition where the "half-step" change in the error is zero. This ensures the ball stays on the path smoothly.
The Smooth Switch (Sigmoid Function):
The old controllers used a "Sign Function," which is like a light switch: it's either ON or OFF. This causes the "jerking" (chattering).
- The new controller uses a Sigmoid Function.
- Analogy: Instead of a light switch, imagine a dimmer switch. You can turn the power up or down gradually. This removes the sudden "jumps" in voltage, eliminating the jitter while still keeping the ball stable.

The Results: A Smoother Ride

The researchers tested this on a real machine with a steel ball and a magnet. Here is what happened:

Better Tracking: The new controller kept the ball exactly where it was told to go, even when the target moved. The old controller failed when the target got too close to the magnet, causing the ball to crash into it.
No More Jitter: The "voltage" (the power sent to the magnet) was incredibly smooth with the new method. With the old method, the voltage was spiking wildly.
Less Error: The new method made fewer mistakes in keeping the ball at the right height.

The "Goldilocks" Setting

One interesting finding was about the "smoothing parameter" (the dimmer switch setting).

If the setting is too low, the system is too sensitive to noise (like a microphone picking up a whisper).
If the setting is too high, the system becomes sluggish.
The researchers found a "sweet spot" (an optimal value) where the system is stable but not too sensitive. They proved mathematically that this balance is necessary to get the best performance.

Summary

In short, this paper shows that by using a more advanced type of math (fractional calculus) and a smoother control method (sigmoid function), we can make magnetic levitation systems much more stable and less noisy than before. It's like upgrading from a bumpy, jerky ride in an old car to a smooth, high-speed train ride. The researchers proved this works not just on a computer screen, but on a real physical machine.

Technical Summary: Fractional-Order Linear-Quadratic-Regulator-Based Sliding Mode Control with Half-Derivative Sliding Motion Condition for Magnetic Levitation Systems

Problem Statement
Magnetic levitation systems, particularly attractive-force types, exhibit strong nonlinearity and inherent instability, posing significant challenges for robust target tracking control. While Sliding Mode Control (SMC) is a practical nonlinear control method, conventional integer-order SMC (IOSMC) often suffers from "chattering"—high-frequency oscillations in the control input caused by the discontinuous sign function. This phenomenon can degrade system performance and damage hardware. Although Fractional-Order SMC (FOSMC) has been proposed to address these issues, existing research is largely confined to theoretical analysis and numerical simulation, with a lack of systematic design frameworks that ensure structural consistency between the controller and the linearized plant model. Furthermore, many existing FOSMC designs rely on empirical or heuristic sliding surface selection rather than optimization.

Methodology
The authors propose a systematic design strategy for FOSMC applied to a voltage-controlled attractive-force-type magnetic levitation system. The methodology proceeds through the following steps:

Modeling and Linearization: The nonlinear dynamics of the magnetic levitation system are linearized around an equilibrium point. This integer-order model is then extended into a fractional-order state-space model using the Caputo definition of the derivative with an order of $\alpha = 0.5$ . To facilitate servo control, integrators are incorporated into the state vector to handle target tracking errors.
Sliding Surface Design via Fractional-Order LQR: Unlike heuristic approaches, the sliding surface coefficient vector is derived using Fractional-Order Linear Quadratic Regulator (LQR) theory. An optimal feedback gain vector is obtained by minimizing a quadratic cost function tailored for the fractional-order dynamics using a gradient-based optimization algorithm (specifically MATLAB's fminunc).
Equivalent Control and Sliding Condition: The equivalent control input is analytically derived by imposing the condition that the 0.5th-order derivative of the switching function is zero. This ensures structural consistency between the switching function and the control input.
Switching Control and Chattering Suppression: To mitigate chattering, the discontinuous sign function in the switching control input is replaced with a continuous sigmoid function. This modification allows the system to be treated as a continuous vector field, eliminating the need for the Filippov differential inclusion framework.
Stability Analysis: The stability of the closed-loop system is analyzed using Lyapunov theory. The authors demonstrate that by applying the semi-group property of the Caputo derivative, the stability analysis of the fractional-order system with a sigmoid-based switching input can be reduced to an equivalent integer-order case.
- In the absence of disturbances, the system exhibits exponential stability within the boundary layer.
- In the presence of bounded disturbances, the switching function converges to a quasi-sliding mode within a bounded region (boundary layer).
- The size of this boundary layer is shown to depend on the control gain, the smoothing parameter ( $\delta$ ), and the disturbance bound, rather than the fractional order itself.
State Observation: A fractional-order state observer is implemented using the pole placement method to estimate unmeasurable states, including fractional-order derivatives, utilizing Oustaloup filters for numerical approximation.

Key Contributions

Systematic Design Framework: The paper presents the first comprehensive framework that integrates fractional-order LQR for sliding surface design with a systematic derivation of FOSMC laws, ensuring theoretical consistency with the linearized plant model.
Half-Derivative Sliding Condition: The study introduces a specific design strategy where the 0.5th-order derivative of the switching function is set to zero, facilitating the derivation of the equivalent control input.
Rigorous Stability Analysis: The authors provide a rigorous stability proof showing that the use of a sigmoid function allows the system to avoid Filippov solutions while guaranteeing Lyapunov stability within a boundary layer. They clarify the trade-off between boundary layer thickness and noise sensitivity, deriving a cost function to determine the optimal smoothing parameter.
Experimental Verification: The study moves beyond simulation to validate the proposed method on an actual magnetic levitation experimental apparatus, comparing it directly against conventional IOSMC.

Results
Experimental and simulation results demonstrate that the proposed FOSMC outperforms conventional IOSMC in several key areas:

Tracking Accuracy: FOSMC achieved significantly lower Integral of Absolute Error (IAE) and Integral of Time-weighted Absolute Error (ITAE) compared to IOSMC. In experiments, FOSMC successfully tracked target values where IOSMC failed (e.g., at a 3.5 mm gap, IOSMC caused the ball to crash into the electromagnet).
Chattering Suppression: The Total Variation (TV) of the control voltage was drastically reduced in FOSMC (by more than 50% in experiments and orders of magnitude in simulations), indicating effective elimination of chattering.
Robustness: The system maintained stability and tracking performance under varying target positions, whereas IOSMC exhibited instability at lower gaps.
Parameter Optimization: The study identified an optimal smoothing parameter ( $\delta_{opt} \approx 1.14$ ) that balances steady-state accuracy against sensitivity to noise and unmodeled dynamics, validating the derived trade-off model.

Significance and Claims
The authors claim that this research establishes FOSMC as a practical and reliable control strategy for unstable, nonlinear plants like magnetic levitation systems. The primary significance lies in bridging the gap between theoretical fractional-order control and real-world implementation. By utilizing a linearized model at the equilibrium point as the design basis, the study ensures that the controller's structure is consistent with the plant dynamics, allowing for transparent and reproducible gain determination.

The paper asserts that the proposed method achieves both theoretical consistency and implementation effectiveness. It highlights that the use of the sigmoid function not only suppresses chattering but also simplifies the stability analysis by avoiding complex differential inclusion frameworks. Furthermore, the study provides a principled guideline for selecting the smoothing parameter, addressing a common practical challenge in SMC design. The authors conclude that while future work should explore multi-degree-of-freedom and more highly nonlinear systems, the current results validate the utility of fractional-order control in enhancing tracking accuracy and reducing control input oscillations in magnetic levitation applications.