Page 129 - IJOCTA-15-4

Fixed-time sliding mode control with disturbance observer and variable exponent coefficient for nonlinear
systems
ensuring finite-time stabilization of system states. outlined in Zuo et al. 14,22,30 To achieve rapid
Nonlinear terminal SMC outperforms traditional convergence, 31 introduced a novel fixed-time sta-
bility theorem for neural network synchroniza-
linear SMC in several ways, including lower
steady-state tracking errors, convergence within tion, providing a more precise settling time esti-
mate.
a finite timeframe, and enhanced dynamic
response. 14–19 However, general terminal SMC is
frequently afflicted with singularity problems. 14,15 Uncertainties and disturbances significantly
Effective solutions to this issue have been affect control performance in nonlinear dynam-
proposed, broadly categorized into direct ical systems. Adaptive control-based distur-
approaches 16,17 and switching methods. 20,21 It bance rejection techniques estimate and com-
is also worth noting that terminal SMC and pensate for bounds on disturbances and uncer-
linear SMC produce similar convergence rates tainties, as detailed in Zhao and Jia 32 though
when system states are far from equilibrium. they tend to be conservative. Based on the
To improve this, the concept of fast terminal literature, disturbance observers (DOs) provide
SMC (FTSMC) was introduced, ensuring rapid a robust solution by addressing both model
transient convergence regardless of the distance uncertainties and external disturbances. Con-
from equilibrium. 17 Nonetheless, the convergence ventional DOs for nonlinear systems 33,34 typi-
time in general FTSMC may become excessively cally assume slow-varying or constant uncertain-
long as initial system values increase. In high- ties, which may not hold in practice. To en-
performance systems, a convergence time inde- sure robust fixed-time stability in nonlinear sys-
pendent of initial conditions is desirable. This tems, it is essential to observe both external
need led to the emergence of fixed-time stabil- disturbances and internal uncertainties within
ity theory, which has inspired various fixed-time a predetermined timeframe. While finite-time
control systems. 22 The formation control chal- DOs 35,36 offer a generic solution for correcting
lenge for multi-robot systems with time delay observation time, their observation time can be-
was addressed in Wang et al., 23 considering both come unbounded if the initial observer error is
undirected and directed topologies. A cooper- large. Furthermore, current observers often re-
ative control scheme combining fixed-time and quire prior knowledge of disturbance bounds or
switching strategies to ensure consensus in first- assume a zero rate of change for disturbances. 37
and second-order multi-agent systems was pro- As a result, a fixed-time DO with less strin-
posed in Du et al. 24 An adaptive fuzzy fixed-time gent constraints on both uncertainties and distur-
controller was designed for robots operating un- bances is required for nonlinear dynamical sys-
der uncertainty, based on position tracking error tems. To overcome these challenges, this study
restrictions, to ensure fast system response. 25 proposes a fixed-time variable exponent coeffi-
For the challenge of adaptive attitude control cient DO (FVECDO) that enables reliable obser-
in flexible spacecraft facing unpredictable dis- vation of lumped disturbances, comprising both
turbances and actuator failures, 26 developed a model inaccuracies and external disturbances,
continuous adaptive control technique that com- within a fixed time, under more relaxed assump-
bines the sliding mode technique with a fixed- tions.
time control strategy, enhancing both the pre-
cision and stability of the spacecraft’s attitude.
Ahmed and Azar 15 presented a fixed-time non- Building on the discussion above, this paper
singular terminal sliding variable to robustly con- explores a method for trajectory tracking control
trol second-order systems, altering the control with fixed-time convergence in nonlinear systems
law near the origin to avoid singularities. In affected by lumped disturbances. Specifically,
Zhang et al., 27 a terminal sliding surface tailored an FVECDO-based fixed-time trajectory track-
for fixed-time control of submarine-launched mis- ing sliding mode control (FTTSMC) approach
sile attitude tracking was proposed, using a si- is developed, incorporating a fixed-time sliding
nusoidal function to address singularity issues. variable and a reaching strategy with a state-
Ni et al. 28 developed a rapid fixed-time con- dependent exponent coefficient. It effectively ad-
troller for energy storage devices using sliding dresses lumped disturbances, ensuring that tra-
mode control theory and the saturation func- jectory tracking errors approach an area close
tion method to avoid singularities. Fixed-time to the origin within a time bound that is inde-
terminal SMC techniques are also described pendent of the system’s initial state. The key
in Wang et al. 29 and related works. Most of outcomes of this research are outlined as fol-
these rely on the fixed-time stability principles lows:
671