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Improving the performance of a chaotic nonlinear system of fractional-order...
Table 2. The average sum of absolute errors for state and rate of change of state regulation for
fractional-order sliding mode control and conventional sliding mode control in the presence of parameter
uncertainties and external disturbances

The average State Rate of change of
sum of absolute regulation state regulation
errors. error (E x ) error (E x−dot )
SMC 7.2701 9.2106
FO-SMC 3.4487 0.1598
Abbreviations: FO-SMC, fractional-order sliding mode controllers; SMC, sliding mode controller.

may lead to performance degradation and exces- Sliding mode control is widely recognized for
sive energy consumption. Conversely, FO-SMC its robustness against uncertainties and external
achieves a more stable state trajectory, effectively disturbances, making it a preferred choice for non-
suppressing oscillations and ensuring faster con- linear systems. However, a major drawback of
vergence to the desired operating point. This is conventional SMC is the chattering phenomenon,
attributed to the fractional-order terms, which in- particularly in high-speed dynamic systems. Our
troduce an additional degree of freedom, enabling analysis revealed that while SMC successfully sta-
the controller to adapt dynamically to variations bilized the system in finite time, it also introduced
in system parameters. significant fluctuations in x 1 and x 1 -dot states.
Additionally, Figures 14 and 15 further vali- To address this limitation, FO-SMC was imple-
date the enhanced stability and robustness of FO- mented, leading to a smoother system response
SMC in regulating the sliding surface and state and reduced state oscillations. Additionally, FO-
rate trajectories under uncertain conditions. The SMC improved system adaptability and enhanced
sliding surface in FO-SMC converges significantly resistance to uncertainties and disturbances.
faster than conventional SMC, ensuring improved
tracking accuracy and predictable system behav- The superiority of FO-SMC over conventional
ior. Moreover, Table 2 quantitatively confirms the SMC was demonstrated through both qualita-
advantage of FO-SMC, with a notable reduction tive (graphical) and quantitative (Tables 1 and 2)
in the state regulation error (from 7.2701 in SMC comparisons. The key advantages of the proposed
to 3.4487 in FO-SMC) and a drastic decrease in approach are summarized as follows:
the rate of change of state regulation error (from
9.2106 in SMC to 0.1598 in FO-SMC). These re- • Fractional-order integration enhances sta-
sults highlight the robustness of the FO-SMC, bility: Transforming the system into a
proving that it reduces steady-state error, and fractional-order framework inherently in-
ensure smoother transient response and lower en- troduces a fractional-order integration op-
ergy consumption in the presence of disturbances. eration, which helps reduce system errors
and improves stability.
5. Conclusion • Innovative hybrid control strategy: Un-
like conventional methods, the proposed
One of the key challenges in nonlinear control sys- FO-SMC approach integrates a linear con-
tems, particularly in BLDC motors, is their sus- tinuous term alongside relay control (as
ceptibility to parameter uncertainties and exter- described in Equation (32)), resulting in
nal disturbances. Traditional SMC techniques, smoother control behavior and signifi-
while offering strong robustness, tend to suf- cantly reducing chattering effects.
fer from high-frequency oscillations (chattering) • Improved adaptability and robustness:
and degraded performance when exposed to un- Leveraging fractional-order dynamics in
certainties. In contrast, FO-SMC leverages the the controller significantly enhances the
memory effect of fractional calculus to provide system’s adaptability while increasing its
a smoother control response while maintaining resilience to parameter uncertainties and
strong disturbance rejection capabilities. In this external disturbances.
study, a FO-SMC was designed to regulate the
chaotic behavior of a FO-BLDC system. The sys- These findings highlight the effectiveness of
tem was effectively driven toward its equilibrium fractional-order control methodologies in sup-
point by carefully selecting an appropriate sliding pressing chaotic behaviors, ensuring smooth op-
surface and a suitable control input, mitigating eration, and enhancing the robustness of nonlin-
chaotic oscillations. ear systems. The proposed FO-SMC framework
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