Advanced Stability Analysis for Fractional-Order Chaotic DC Motors Subject to Saturation and Rate Limitations
Abstract
1. Introduction
1.1. Motivation and Background
1.2. Modeling with Input Constraints
1.3. Gap and Contribution Overview
1.4. Main Contributions
- Unified stability framework: A comprehensive Lyapunov-based stability framework is developed for FO chaotic BLDCMs under input constraints including saturation, rate limits, and their combination. To our knowledge, this is the first study that considers both rate and amplitude limitations simultaneously in FO chaotic control design.
- Novel Lyapunov candidates: Tailored Lyapunov functions are constructed for each control scenario—unconstrained, saturated, rate-limited, and jointly constrained—based on a combination of direct and indirect Lyapunov methods.
- Input-constrained controller design: Multiple controller configurations are derived from stability conditions, enabling implementations with reduced sensor requirements (single-, double-, and triple-input variants).
- Benchmark validation: The effectiveness of the proposed framework is demonstrated through extensive simulations on FOBLDCMs with varying degrees of input constraints and nonlinear behavior.
1.5. Organization
2. Preliminaries
3. Main Results
3.1. No Limitation Control
3.2. Saturation Control
3.3. Rate Limitations
3.4. Saturation and Rate Limitation Simultaneously
4. Numerical Simulations and Performance Evaluation
- Integral of squared error (ISE): Measures the accumulated squared tracking error over the simulation time for each state variable, reflecting the long-term tracking performance of the system.
- Root mean square error (RMSE): Evaluates the average deviation between the system outputs and their reference trajectories, providing a normalized metric for comparison.
- Control energy: Represents the total energy consumed by the control inputs, calculated as the integral of the squared control signal, and is used to evaluate the energy efficiency of the controller.
- Maximum control effort (): Denotes the peak value of the control input, serving as an indicator of actuator feasibility and the extent of input constraints such as saturation.
- Scenario 1: Control without any actuator limitations;
- Scenario 2: Control with rate limitation constraint only;
- Scenario 3: Control under both saturation and rate limitation constraints simultaneously.
Scenario 1: Control Without Any Actuator Limitations
- Proposed set of controllers based on available sensors: This paper introduces a comprehensive set of controllers tailored to the available sensors. In contrast, Ref. [34] proposed only a feedback linear controller, which can be considered a subset of the more general controller set presented in this work.
- Simultaneous consideration of saturation and rate limitation on the controller: This study addresses the FOBLDCM under the simultaneous constraints of saturation and rate limitation on the controller. In comparison, Ref. [34] only considered saturation as a limitation without accounting for rate limitation. Furthermore, other valuable papers did not consider any constraints [17,18,19,23,24,39].
- Robust solution approach: The proposed results in this paper are derived using a presented novel Lyapunov candidate to establish novel stability conditions in the presence of rate limitations and saturated control inputs. On the other hand, Ref. [34] relied on estimations of the solutions to the given equations, which presents a less robust approach than the one presented in this paper.
5. Conclusions
- Novel Lyapunov candidates: Dedicated Lyapunov functions are constructed for each control scenario—namely, unconstrained, saturated, rate-limited, and jointly constrained—by integrating both direct and indirect Lyapunov-based techniques tailored to fractional-order dynamics.
- Input-constrained controller design: Multiple controller configurations are derived based on the developed stability conditions, enabling practical implementation with reduced sensor dependencies through single-input, double-input, and triple-input controller variants. To the best of our knowledge, this is the first study to simultaneously address both amplitude and rate constraints in the control design of FO chaotic systems.
- Benchmark validation: The effectiveness of the proposed framework is validated through extensive numerical simulations conducted on FO chaotic BLDCMs under varying degrees of nonlinear behavior and input limitations, providing a solid benchmark for future studies.
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- For manufacturers: It is recommended to embed FO control logic and constraint-aware algorithms within motor control chips or firmware, enabling higher-fidelity performance in FO dynamic environments. Support for configurable constraint-aware control modules should be considered during motor drive design.
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- For system integrators and end users: When selecting control algorithms for high-precision or safety-critical systems, priority should be given to those that explicitly handle rate and amplitude constraints. The use of adaptable multi-input controllers can provide significant performance benefits when sensor configurations are flexible.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
FOBLDCM | Fractional-order brushless DC motor |
FC | Fractional calculus |
FO | Fractional order |
BLDCM | Brushless DC motor |
PID | Proportional–integral–derivative |
ADRC | Active disturbance rejection control |
I&I | Immersion and Invariance |
DSPs | Digital signal processors |
ISE | Integral of squared error |
RMSE | Root mean square error |
UAVs | Unmanned aerial vehicles |
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Input | Design Controller | |
---|---|---|
Single Input | ||
Double Input | ||
Triple Input | ||
Triple Input |
Performance Indices | ISE | RMSE | Control Energy | |
---|---|---|---|---|
Controller from [39] | ||||
Our proposed controller (close to [39]) | ||||
Proposed controller Scenario 1 |
Rate Limitation | Input | Design Controller |
---|---|---|
Single Input | ||
Double Input | ||
Triple Input | ||
Single Input | ||
Double Input | ||
Double Input | ||
Triple Input |
Performance Indices | ISE | RMSE | Control Energy | |
---|---|---|---|---|
Proposed controller Scenario 3 |
Feature | Proposed Work | [34] | [39] |
---|---|---|---|
Controller design based on sensor availability | Comprehensive set (single/double/triple input) | Only feedback linearization | Only feedback linearization |
Constraints considered | Both saturation and rate limitation | Only saturation | No constraints |
Stability analysis method | Lyapunov-based analytical approach | Estimation-based | Estimation-based |
Robustness | High (based on derived sufficient conditions) | Moderate lack of rate-limit handling | Moderate lack of saturation and rate-limit handling |
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Alaviyan Shahri, E.S.; Chen, Y.; Pariz, N. Advanced Stability Analysis for Fractional-Order Chaotic DC Motors Subject to Saturation and Rate Limitations. Fractal Fract. 2025, 9, 369. https://doi.org/10.3390/fractalfract9060369
Alaviyan Shahri ES, Chen Y, Pariz N. Advanced Stability Analysis for Fractional-Order Chaotic DC Motors Subject to Saturation and Rate Limitations. Fractal and Fractional. 2025; 9(6):369. https://doi.org/10.3390/fractalfract9060369
Chicago/Turabian StyleAlaviyan Shahri, Esmat Sadat, Yangquan Chen, and Naser Pariz. 2025. "Advanced Stability Analysis for Fractional-Order Chaotic DC Motors Subject to Saturation and Rate Limitations" Fractal and Fractional 9, no. 6: 369. https://doi.org/10.3390/fractalfract9060369
APA StyleAlaviyan Shahri, E. S., Chen, Y., & Pariz, N. (2025). Advanced Stability Analysis for Fractional-Order Chaotic DC Motors Subject to Saturation and Rate Limitations. Fractal and Fractional, 9(6), 369. https://doi.org/10.3390/fractalfract9060369