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Fixed-time sliding mode control with disturbance observer and variable exponent coefficient for nonlinear
systems

Figure 13. Control inputs for the proposed
fixed-time trajectory tracking sliding mode control

Figure 11. Position tracking errors for (A) joint 1
and (B) joint 2
Figure 14. Control inputs for Boukattaya et al.’s
controller

5. Conclusion

The study introduces a robust trajectory track-
ing control approach for uncertain nonlinear dy-
namic systems by leveraging a fixed-time DO.
First, a novel FVECDO was designed using fixed-
time stability theory to accurately estimate the
combined uncertainties arising from modeling in-
accuracies and external disturbances. Utilizing
the information from this DO, a sliding mode
control strategy was proposed that ensures fixed-
time convergence for trajectory tracking. This
approach incorporated a reaching law with a
state-dependent exponent coefficient in conjunc-
tion with a sliding variable designed for fixed-
time convergence. The developed controller ex-
hibited key advantages, including high robust-
ness, reduced chattering, and avoidance of singu-
larities. Lyapunov theory was employed to com-
prehensively analyze the system’s fixed-time sta-
bility and control precision. Finally, simulations
comparing this method with other control strate-
gies demonstrated its effectiveness and low sensi-
Figure 12. Estimation of lumped disturbances for tivity to the system’s initial conditions. Future
(A) joint 1 and (B) joint 2
work will explore the controller’s performance in
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