Adaptive Fuzzy Sliding Mode Control

• Adaptive Fuzzy Sliding Mode Control (AFSMC) is a control approach that combines sliding mode control’s robustness with fuzzy systems’ online adaptation to handle nonlinear uncertainties.
• It decomposes the control law into nominal, robust, and adaptive fuzzy components to reduce steady-state error, mitigate chattering, and compensate for unknown dynamics.
• AFSMC is applied in domains such as robotics, aerial vehicles, and power electronics, demonstrating faster convergence and improved performance under uncertainty.

Searching arXiv for recent and foundational AFSMC papers to ground the article. Adaptive Fuzzy Sliding Mode Control (AFSMC) denotes a class of nonlinear control architectures that combine the robustness of sliding mode control (SMC) with the approximation and online tuning capabilities of fuzzy or neuro-fuzzy systems. Across the arXiv literature, AFSMC is used for uncertain nonlinear plants, underactuated mechanical systems, dead-zone actuators, chaotic systems, quadrotors, flapping-wing vehicles, rolling robots, container cranes, and robotic manipulators. Its common purpose is to preserve SMC’s robustness to disturbances and model mismatch while reducing steady-state error, relaxing reliance on exact models, and attenuating chattering through adaptive fuzzy compensation, higher-order sliding modes, or boundary-layer smoothing (Azimi et al., 2017).

1. Conceptual structure and major variants

In the literature, AFSMC is not a single canonical controller but a family of designs built around a recurring decomposition: a sliding variable is defined from tracking errors; a robust SMC term enforces reachability; and a fuzzy, interval-valued fuzzy, or neuro-fuzzy component approximates unknown nonlinearities, disturbances, dead-zones, or gain variations. This general pattern appears in zero-order Takagi–Sugeno disturbance compensation for a chaotic pendulum, in adaptive fuzzy inference for an experimental overhead container crane, in interval type-2 fuzzy higher-order sliding mode control for chaotic systems, and in interval-valued fuzzy modeling for quadrotor control (Bessa et al., 2022).

Several distinct architectural lineages are represented in the sources. One lineage uses direct fuzzy compensation embedded in the control law, typically with a form such as $u=u_{eq}+u_{sw}+u_{fuzzy}$, where the fuzzy term estimates residual uncertainty online; this is explicit in the crane formulation of Bessa et al. (Bessa et al., 2022). A second lineage uses higher-order sliding mode control, especially the Super Twisting algorithm, with adaptive type-2 fuzzy systems approximating both the unknown plant term and the super-twisting signals; this structure is central to the chaotic-system controller in (Hendel et al., 2015). A third lineage is indirect adaptive control, in which a Takagi–Sugeno interval-valued fuzzy model first estimates unknown plant terms $f(x)$ and $g(x)$, after which a sliding-mode law is designed from the estimated dynamics (Bouhentala et al., 2022).

A further variant replaces conventional premise-parameterized fuzzy rules with evolving neuro-fuzzy structures. The Parsimonious Controller (PAC) removes rule premise parameters entirely by representing each rule with a hyperplane, adapts consequent parameters using SMC theory in single-pass form, and adds bias-variance-based rule growing and pruning without user-defined thresholds (Ferdaus et al., 2018). In the spherical rolling robot of Kayacan et al., AFSMC is organized as a two-degree-of-freedom feedback-error-learning scheme in which a conventional PD or PID branch stabilizes the plant while a zero-order TSK neuro-fuzzy network progressively learns the uncertain inverse dynamics and disturbances (Kayacan et al., 2021).

This diversity implies that AFSMC should be understood as a design paradigm rather than a fixed algorithm. In some papers the adaptive fuzzy component tunes rule consequents only; in others it tunes antecedent and consequent parameters; in still others it adjusts sliding-surface coefficients rather than approximating the plant directly. The flapping-wing MAV altitude controller, for example, is described as tuning both antecedent and consequent parameters of the fuzzy system, with SMC theory used to derive the adaptation laws and confirm closed-loop stability (Ferdaus et al., 2018).

2. Sliding manifolds, reachability, and control decomposition

The sliding manifold is the structural core of AFSMC. For many SISO tracking problems, the basic choice is the first-order surface

$s(e,\dot e)=\dot e+\lambda e,\qquad \lambda>0,$

which appears in the chaotic pendulum controller and in the general uncertain second-order formulation underlying fuzzy supertwisting control (Bessa et al., 2022). On the manifold $s=0$, the reduced error dynamics become exponentially stable. For MIMO trajectory tracking, the same idea is written in vector form. In the cylindrical manipulator controller,

$s(t)=\dot e(t)+\Lambda e(t),$

with $$\Lambda=\operatorname{diag}(\lambda_1,\lambda_2,\lambda_3)>0$$, and $\Lambda(t)$ is produced online by fuzzy inference rather than kept fixed (Pham et al., 7 Aug 2025).

Underactuated systems motivate coupled or hierarchical manifolds. Azimi et al. define

$$s_2(x,t)=C_2x_3+x_4,\qquad s_1(x,t)=C_1(x_1-z)+x_2,$$

with a virtual reference

$z=z_u\,\sat(s_2/\phi_2),$

so that the unactuated coordinate influences the actuated channel through the manifold design itself (Azimi et al., 2017). The crane controller adopts a two-input switching vector

$f(x)$0

that explicitly couples the unactuated swing angle $f(x)$1 into the manifold, allowing the actuated trolley position $f(x)$2 and cable length $f(x)$3 to be regulated while load swing is damped (Bessa et al., 2022).

The control law is then split into nominal, robust, and adaptive components. The most explicit decomposition in the sources is

$f(x)$4

where $f(x)$5 cancels the nominal model or enforces manifold dynamics, $f(x)$6 is the switching or saturation-based robust term, and $f(x)$7 estimates the uncertainty (Bessa et al., 2022). Closely related forms appear in the chaotic pendulum,

$f(x)$8

and in the dead-zone-compensating electro-hydraulic controller,

$f(x)$9

with the discontinuity often smoothed by $g(x)$0 or $g(x)$1 to reduce chattering (Bessa, 2022).

Higher-order AFSMC replaces first-order switching by continuous second-order action. In the adaptive type-2 fuzzy second-order sliding mode controller, the super-twisting terms are

$g(x)$2

and the full law is $g(x)$3, with finite-time convergence of $g(x)$4 and $g(x)$5 when the gains dominate the uncertainty bound (Hendel et al., 2015). The fuzzy supertwisting buck-converter controller likewise uses

$g(x)$6

while a fuzzy inference system adapts the sliding-surface slope $g(x)$7 online (Kareem et al., 2012).

3. Fuzzy modeling, approximation targets, and online adaptation

The fuzzy component in AFSMC serves different approximation targets depending on the plant and uncertainty structure. In the chaotic pendulum, a zero-order Takagi–Sugeno fuzzy system estimates the disturbance term $g(x)$8 as

$g(x)$9

where the input is the sliding variable $s(e,\dot e)=\dot e+\lambda e,\qquad \lambda>0,$0, the rule consequents are $s(e,\dot e)=\dot e+\lambda e,\qquad \lambda>0,$1, and the normalized weights $s(e,\dot e)=\dot e+\lambda e,\qquad \lambda>0,$2 satisfy $s(e,\dot e)=\dot e+\lambda e,\qquad \lambda>0,$3 (Bessa et al., 2022). In the crane, each switching variable $s(e,\dot e)=\dot e+\lambda e,\qquad \lambda>0,$4, $s(e,\dot e)=\dot e+\lambda e,\qquad \lambda>0,$5, drives its own T-S network with rule base

$s(e,\dot e)=\dot e+\lambda e,\qquad \lambda>0,$6

and the approximation is

$s(e,\dot e)=\dot e+\lambda e,\qquad \lambda>0,$7

This fuzzy term compensates residual model uncertainties beyond the nominal crane dynamics (Bessa et al., 2022).

When the uncertainty enters through actuator nonlinearities, the fuzzy system approximates the nonlinearity rather than the plant drift. For a non-symmetric dead-zone,

$s(e,\dot e)=\dot e+\lambda e,\qquad \lambda>0,$8

the unknown offset function $s(e,\dot e)=\dot e+\lambda e,\qquad \lambda>0,$9 is estimated by a zero-order TSK system,

$s=0$0

with online update

$s=0$1

so the fuzzy term cancels the dead-zone effect without adding another discontinuous action (Bessa, 2022).

Type-2 and interval-valued variants generalize this approximation layer to explicitly represent uncertainty in the membership functions. In the adaptive type-2 fuzzy second-order sliding mode controller, interval Type-2 fuzzy systems approximate the unknown nonlinear function $s=0$2 and also generate the super-twisting signals. The paper defines three parameter vectors,

$s=0$3

for approximating $s=0$4, $s=0$5, and $s=0$6, respectively, using interval-valued Gaussian antecedents and center-of-sets type reduction (Hendel et al., 2015). In the quadrotor controller, the interval-valued fuzzy model is constructed offline from input-output data using Gustafson–Kessel clustering and the Envelope Detection Algorithm, then embedded into adaptive approximators

$s=0$7

with projection-based adaptation to keep $s=0$8 and $s=0$9 inside known bounds (Bouhentala et al., 2022).

The adaptation laws are usually chosen to cancel cross terms in a Lyapunov derivative. Representative examples include

$s(t)=\dot e(t)+\Lambda e(t),$0

for the chaotic pendulum (Bessa et al., 2022),

$s(t)=\dot e(t)+\Lambda e(t),$1

for the crane (Bessa et al., 2022), and

$s(t)=\dot e(t)+\Lambda e(t),$2

for underactuated systems (Azimi et al., 2017). The rolling-robot neuro-fuzzy controller goes further and updates Gaussian centers, widths, and consequents via SMC-theory-based rules chosen so that the derivative of $s(t)=\dot e(t)+\Lambda e(t),$3 becomes negative whenever the learning gain $s(t)=\dot e(t)+\Lambda e(t),$4 exceeds $s(t)=\dot e(t)+\Lambda e(t),$5 (Kayacan et al., 2021). By contrast, the cylindrical manipulator paper states explicitly that the “adaptation” of $s(t)=\dot e(t)+\Lambda e(t),$6 occurs implicitly through fuzzy inference and that no continuous update laws for fuzzy weights are derived (Pham et al., 7 Aug 2025).

4. Stability guarantees and convergence claims

AFSMC stability analyses in the sources are predominantly Lyapunov-based, often supplemented by Barbalat’s lemma, finite-time reachability arguments, or the LaSalle-Yoshizawa theorem. The typical Lyapunov candidate augments the sliding energy with parameter-estimation errors. For the chaotic pendulum,

$s(t)=\dot e(t)+\Lambda e(t),$7

and, under model bounds and a sufficiently large discontinuous gain $s(t)=\dot e(t)+\Lambda e(t),$8, the derivative satisfies

$s(t)=\dot e(t)+\Lambda e(t),$9

This yields boundedness of all signals, $$\Lambda=\operatorname{diag}(\lambda_1,\lambda_2,\lambda_3)>0$$0, and $$\Lambda=\operatorname{diag}(\lambda_1,\lambda_2,\lambda_3)>0$$1 by Barbalat’s lemma; consequently $$\Lambda=\operatorname{diag}(\lambda_1,\lambda_2,\lambda_3)>0$$2 and $$\Lambda=\operatorname{diag}(\lambda_1,\lambda_2,\lambda_3)>0$$3 (Bessa et al., 2022).

The same pattern appears in more elaborate settings. In the underactuated-system controller of Azimi et al., the Lyapunov function includes the sliding variable $$\Lambda=\operatorname{diag}(\lambda_1,\lambda_2,\lambda_3)>0$$4 and the fuzzy parameter errors $$\Lambda=\operatorname{diag}(\lambda_1,\lambda_2,\lambda_3)>0$$5, leading to

$$\Lambda=\operatorname{diag}(\lambda_1,\lambda_2,\lambda_3)>0$$6

when the switching gain $$\Lambda=\operatorname{diag}(\lambda_1,\lambda_2,\lambda_3)>0$$7 exceeds the disturbance and approximation bounds. The result is boundedness of all closed-loop signals and asymptotic convergence of both coupled surfaces $$\Lambda=\operatorname{diag}(\lambda_1,\lambda_2,\lambda_3)>0$$8 and $$\Lambda=\operatorname{diag}(\lambda_1,\lambda_2,\lambda_3)>0$$9 (Azimi et al., 2017). In the crane controller, the adaptive law cancels the uncertain term in the Lyapunov derivative so that

$\Lambda(t)$0

from which the paper concludes existence of the sliding manifold in finite time and asymptotic convergence of $\Lambda(t)$1 to zero (Bessa et al., 2022).

Higher-order AFSMC emphasizes finite-time convergence with reduced chattering. In the adaptive type-2 fuzzy second-order sliding mode controller,

$\Lambda(t)$2

which guarantees finite-time convergence of $\Lambda(t)$3; once $\Lambda(t)$4 and $\Lambda(t)$5, the original error dynamics reduce to a stable linear polynomial in $\Lambda(t)$6, implying asymptotic tracking (Hendel et al., 2015). The fuzzy supertwisting controller likewise derives a Lyapunov inequality of the form

$\Lambda(t)$7

and attributes chattering attenuation to the continuity of the supertwisting control law itself (Kareem et al., 2012).

Other proofs reflect architectural differences. The quadrotor indirect adaptive controller uses

$\Lambda(t)$8

and shows

$\Lambda(t)$9

after the switching term compensates the worst-case model errors $$s_2(x,t)=C_2x_3+x_4,\qquad s_1(x,t)=C_1(x_1-z)+x_2,$$0 and $$s_2(x,t)=C_2x_3+x_4,\qquad s_1(x,t)=C_1(x_1-z)+x_2,$$1 (Bouhentala et al., 2022). PAC establishes boundedness and convergence of the tracking error and consequent parameters by the LaSalle-Yoshizawa theorem, while the spherical rolling robot uses a baseline conventional controller to guarantee global asymptotic stability in a compact set and an SMC-theory-based learning law to drive the control surface to zero in finite time (Ferdaus et al., 2018).

Taken together, these results show that AFSMC stability claims are usually hybrid in character: finite-time reachability of the sliding manifold or control surface, combined with asymptotic convergence of tracking errors once sliding is established.

5. Application domains and reported performance

The reported applications span aerial vehicles, underactuated transport systems, chaotic oscillators, hydraulic actuators, power electronics, and robotic manipulators. In the flapping-wing micro air vehicle altitude problem, the proposed adaptive controller is designed for a four-wing FW MAV and is said to adapt to environmental disturbances by tuning antecedent and consequent fuzzy parameters, with SMC theory used to develop the adaptation laws and confirm closed-loop stability (Ferdaus et al., 2018).

For the experimental overhead container crane, the intelligent AFSMC scheme was implemented on a $$s_2(x,t)=C_2x_3+x_4,\qquad s_1(x,t)=C_1(x_1-z)+x_2,$$2 scale crane at the Institute of Mechanics and Ocean Engineering at Hamburg University of Technology. The reported highlights state that AFSMC yields faster convergence with settling time $$s_2(x,t)=C_2x_3+x_4,\qquad s_1(x,t)=C_1(x_1-z)+x_2,$$3 s versus 4 s, reduced steady-state error by 95%, suppressed chattering with smooth actuator commands, and near-ideal path following in obstacle-avoidance tests because the fuzzy term adapts online to compensate unmodeled cable friction and trolley drive nonlinearity (Bessa et al., 2022).

In the chaotic pendulum, numerical results are considerably more specific. With $$s_2(x,t)=C_2x_3+x_4,\qquad s_1(x,t)=C_1(x_1-z)+x_2,$$4, $$s_2(x,t)=C_2x_3+x_4,\qquad s_1(x,t)=C_1(x_1-z)+x_2,$$5, $$s_2(x,t)=C_2x_3+x_4,\qquad s_1(x,t)=C_1(x_1-z)+x_2,$$6, and $$s_2(x,t)=C_2x_3+x_4,\qquad s_1(x,t)=C_1(x_1-z)+x_2,$$7, the tracking error remained below $$s_2(x,t)=C_2x_3+x_4,\qquad s_1(x,t)=C_1(x_1-z)+x_2,$$8 rad after a short transient $$s_2(x,t)=C_2x_3+x_4,\qquad s_1(x,t)=C_1(x_1-z)+x_2,$$9 s for a generic reference trajectory, while the control effort peaked at $z=z_u\,\sat(s_2/\phi_2),$0 mm. For stabilization of a period-1 unstable periodic orbit, the controller achieved similar final tracking accuracy with 30% less peak control amplitude, and comparison with conventional SMC indicated that AFSMC attains the same or better accuracy with reduced gain and smoother control (Bessa et al., 2022).

The electro-hydraulic dead-zone application reports a third-order actuator model controlled at 400 Hz with the model simulated at 800 Hz. In the case of perfect model knowledge but unknown dead-zone width, AFSMC achieved $z=z_u\,\sat(s_2/\phi_2),$1 m with no detectable chattering, whereas conventional SMC showed larger steady-state error. Under $z=z_u\,\sat(s_2/\phi_2),$2 valve-gain uncertainty and $z=z_u\,\sat(s_2/\phi_2),$3 variation in supply pressure $z=z_u\,\sat(s_2/\phi_2),$4, AFSMC still yielded $z=z_u\,\sat(s_2/\phi_2),$5 of order $z=z_u\,\sat(s_2/\phi_2),$6 m with smooth control voltage (Bessa, 2022).

The quadrotor study reports both modeling and closed-loop tracking metrics. For the interval-valued fuzzy model, the RMSEs on the rotational unknown terms improved from $z=z_u\,\sat(s_2/\phi_2),$7 to $z=z_u\,\sat(s_2/\phi_2),$8 for $z=z_u\,\sat(s_2/\phi_2),$9, from $f(x)$00 to $f(x)$01 for $f(x)$02, and from $f(x)$03 to $f(x)$04 for $f(x)$05 relative to a Type-1 fuzzy model. Under inertia perturbations, the interval-valued fuzzy controller recovered in $f(x)$06 s and maintained mean-squared sliding-error levels roughly half those of the Type-1 alternative in the reported tests; parameter vectors $f(x)$07 and $f(x)$08 converged within 1–2 s after step changes or disturbance injections (Bouhentala et al., 2022).

Underactuated mechanical benchmarks show the same pattern of robustness claims with explicit transient figures. For the inverted pendulum on a cart, the pole angle converged to zero in $f(x)$09 s with overshoot $f(x)$10, the cart reached 1 m in $f(x)$11 s, and $f(x)$12 N. With sinusoidal $f(x)$13 parameter variation and an additive disturbance $f(x)$14, the final error remained $f(x)$15 m and $f(x)$16. For the TORA system, position and rotor angle converged in $f(x)$17 s with error $f(x)$18 m/rad, and robustness persisted under $f(x)$19 mass variation and disturbance $f(x)$20 (Azimi et al., 2017).

Power-electronics and aerial-robotics variants add further evidence that the AFSMC label covers more than one technical route. The adaptive fuzzy supertwisting controller for a DC-DC buck converter reported, for startup with $f(x)$21 V, settling time $f(x)$22 ms, overshoot $f(x)$23, and steady-state error $f(x)$24, versus $f(x)$25 ms, $f(x)$26, and $f(x)$27 for a first-order sliding mode fuzzy controller (Kareem et al., 2012). PAC, evaluated on a bio-inspired flapping-wing MAV and a hexacopter, reported for a constant 10 m hover RMSE $f(x)$28 m with 3 rules and 12 total parameters, compared with RMSE $f(x)$29 m and about 48 parameters for G-controller, and substantially smaller parameter counts than fixed-rule TS-fuzzy and FFNN baselines (Ferdaus et al., 2018).

6. Design trade-offs, misconceptions, and methodological boundaries

A recurrent misconception is that AFSMC always means “SMC plus a fuzzy estimate of the plant dynamics.” The sources show a broader reality. In some controllers the fuzzy component estimates an additive disturbance $f(x)$30 (Bessa et al., 2022); in others it estimates the dead-zone offset $f(x)$31 (Bessa, 2022); in the quadrotor it identifies the unknown functions $f(x)$32 and $f(x)$33 through an interval-valued model (Bouhentala et al., 2022); and in the cylindrical manipulator it does not update fuzzy weights at all, but instead uses fuzzy inference to tune the sliding-surface gains $f(x)$34 online (Pham et al., 7 Aug 2025). This suggests that “adaptive fuzzy” in AFSMC refers to the adaptive fuzzy handling of uncertainty in a broad sense, not to a single estimator topology.

A second misconception is that the fuzzy layer eliminates the need for robust switching or for a stabilizing baseline controller. The opposite is standard in the cited works. The pendulum retains a discontinuous term $f(x)$35 or a saturation substitute (Bessa et al., 2022). The dead-zone controller includes $f(x)$36 (Bessa, 2022). The quadrotor uses

$f(x)$37

with $f(x)$38 sized from the known error bounds $f(x)$39 and $f(x)$40 (Bouhentala et al., 2022). The rolling robot places the neuro-fuzzy network in parallel with a conventional PD or PID branch that guarantees asymptotic stability in a compact space while the fuzzy component learns online (Kayacan et al., 2021).

Chattering reduction is another point that should be treated precisely. AFSMC does not automatically remove chattering merely by adding fuzzy logic. Different papers employ different mechanisms: boundary layers with $f(x)$41 or $f(x)$42 (Bessa et al., 2022), smooth interpolation $f(x)$43 (Bessa, 2022), continuous supertwisting laws (Kareem et al., 2012), and fuzzy approximation of the unknown dynamics to reduce the required discontinuous gain (Bessa et al., 2022). The practical implication is that smoother control is usually obtained through the joint action of approximation and switching-law modification, not from fuzzy approximation alone.

The stability guarantees are also conditional rather than unconditional. The manipulator controller requires the uncertainty bound $f(x)$44 and switching gains $f(x)$45 to obtain $f(x)$46 away from the origin (Pham et al., 7 Aug 2025). The crane formulation assumes $f(x)$47 bounded away from zero, gains satisfying reachability conditions $f(x)$48, and sufficient excitation so that fuzzy weights remain bounded and persistently adapted (Bessa et al., 2022). The underactuated-system analysis requires bounded disturbances and a fuzzy approximation residual $f(x)$49 small enough to be dominated by the robust gain $f(x)$50 (Azimi et al., 2017). These assumptions delimit the formal convergence claims.

Finally, AFSMC design involves a persistent complexity–parsimony trade-off. Richer fuzzy structures—interval type-2 systems, interval-valued models, or full antecedent-and-consequent adaptation—offer broader uncertainty representation but increase identification and tuning burden (Hendel et al., 2015). Parsimonious alternatives such as PAC explicitly remove premise parameters and use bias-variance-based rule evolution without user-defined thresholds, while maintaining SMC-theory-based adaptation of consequent parameters (Ferdaus et al., 2018). This suggests an active methodological tension in the AFSMC literature between approximation richness, real-time implementability, and provable robustness.