Learning-Based Robust Fixed-Time Terminal Sliding Mode Control

Adnane Saoudc; Chaimae El Mortajinea; Mostafa Bouzia; Moussa Labbadib

REVIEW 5 major objections 6 minor 28 references

Learning-Based Robust Fixed-Time Terminal Sliding Mode Control

T0 review · 5 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Adding Gaussian-process drift estimates to a terminal sliding-mode controller preserves global fixed-time stability, with a settling-time bound that grows with the GP error.

desk verdict The known-dynamics half is a reasonable adaptation, but the GP-based fixed-time theorem is broken: it rests on a missing lemma and a state-dependent settling time, so the paper is not ready for review. read the letter →

arxiv 2505.22827 v1 pith:X2DJD6HX submitted 2025-05-28 eess.SY cs.SY

Chaimae El Mortajinea , Moussa Labbadib , Adnane Saoudc , Mostafa Bouzia This is my paper

classification eess.SYcs.SY MSC 93B1293D0593C10

keywords fixed-timestabilityintegralslidingmodecontrolterminalGaussianprocessregressiondata-drivenmatchedperturbationsLyapunovnonlinearsystems

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to extend integral fixed-time terminal sliding-mode control to systems whose drift is only partly known, using Gaussian-process regression to learn the unknown part. Its central claim is that if the GP approximation error $|\Delta f_i(x)|$ is bounded and the control gain $\alpha_{i2}$ is chosen above $\bar d_i + |\Delta f_i(x)|$, the closed-loop system converges in a fixed time whose worst case is explicitly bounded and independent of initial conditions. The motivation is practical: many systems such as robot manipulators, motors, and vehicles have uncertain but learnable dynamics, and time-critical tasks need a guaranteed settling time rather than an asymptotic one. The paper provides two theorems, one for fully known dynamics and one for GP-estimated dynamics, both proved with a quadratic Lyapunov function, and illustrates the approach on a permanent-magnet synchronous motor.

What carries the argument

The load-bearing object is the exponential integral sliding variable $s_i(z_i)=z_i(t)+\alpha_{i1}\int_0^t e^{z_i(\tau)^2}\lfloor z_i(\tau)\rceil^{p_i/q_i}d\tau$, where $\lfloor \cdot \rceil^{p/q}$ is the signed fractional power $|\cdot|^{p/q}\operatorname{sign}(\cdot)$, combined with the reaching control (19). The exponentials $e^{z_i^2}$ and $e^{s_i^2}$ convert textbook finite-time inequalities into fixed-time ones: for $|z|>1$ the exponential dominates the polynomial and activates the $h>1$ branch of Lemma 1, while for $|z|\le 1$ it is bounded below by 1 and gives the $0<h<1$ branch. The GP posterior mean replaces the known drift $f_i(x)$, and the scalar gain $\alpha_{i2}$ is chosen to dominate the worst-case sum of the matched-perturbation bound $\bar d_i$ and the GP error $|\Delta f_i(x)|$, which makes the sliding surface attractive.

What would settle it

Run the PMSM simulation with a drift term that grows outside the region covered by training data and record $\max_t |\Delta f_i(x(t))|$; if it ever exceeds $\alpha_{i2} - \bar d_i$, the right-hand side of (29) goes positive and the claimed fixed-time bound cannot be validated. The paper does not report the trajectory-wise GP error, so checking whether $\sup_t |\Delta f_i(x(t))| < \alpha_{i2} - \bar d_i$ along the simulated states is the direct experimental test.

Watch

Extended reading notes

Core claim

The paper's central claim is that the closed-loop system formed by the perturbed plant (4) and the GP-based control law (19) is globally fixed-time stable whenever $\alpha_{i2} > \bar d_i + |\Delta f_i(x)|$, with settling time bounded by $T_{\max} = T(s) + T(z)$ as stated in (20)-(24). The proof first establishes fixed-time convergence of the sliding variable $s_i$ using the inequality $\dot V(s_i) \le -(\alpha_{i2} - |\Delta f_i(x)| - \bar d_i)e^{s_i^2}|s_i|$ and Lemma 4, then establishes fixed-time convergence of the tracking error $z_i$ on the sliding manifold using the exponential terminal sliding variable. In the limit where the GP error vanishes, the settling time reduces to that of the known-dynamics result, making the learned controller a plug-in replacement for the exact model.

Load-bearing premise

The proof depends on the GP approximation error $|\Delta f_i(x)|$ being uniformly bounded over the whole state space and on the designer choosing $\alpha_{i2}$ larger than $\bar d_i + |\Delta f_i(x)|$; the paper supports this with a probabilistic bound on a compact set and refers to a 'Lemma 5' that does not appear in the manuscript.

Editorial extensions

If this is right

If Theorem 5 is correct, a robot or vehicle with unknown drift can be guaranteed to settle within $T_{\max}$ regardless of where it starts, as long as the GP error bound is respected.
More training data shrinks the GP variance and therefore the settling-time bound (24); the paper's N=5 versus N=50 simulations show faster settling and smoother control with more data.
The controller retains closed-loop robustness against bounded matched perturbations, and with only three tuning parameters it is simpler to commission than the adaptive fixed-time baseline [15].
When the GP estimate becomes exact ($\Delta f_i \to 0$), all guarantees reduce to the known-dynamics Theorem 3, so the learning layer adds capability without degrading the nominal design.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A general template follows: any estimator that provides a uniform error bound on a compact set, not only Gaussian processes, could be substituted into control law (19) and inherit the same fixed-time argument.
Because the cited GP error bound is probabilistic and stated on a compact set, the 'global' fixed-time statement is best interpreted as holding on the region where training data gives confidence; outside that region the guarantee is not established.
A natural next experiment is to measure worst-case settling time on a physical platform as the training set grows, testing whether it follows the predicted dependence on $\alpha_{i2} - \bar d_i - \max|\Delta f_i|$.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper combines integral fixed-time terminal sliding mode control with Gaussian process drift estimation, and the known-dynamics controller (Theorem 3) is a sensible adaptation of the authors' earlier work and Moulay et al. The tuning is genuinely simple, and the simulations suggest the approach can work, especially with more training data. I'll give credit for the idea and for the clear writing, but the theoretical claims do not hold up as stated.

The problems are concentrated in the learning-based part, and they are load-bearing, not typos. Theorem 5's proof cites a 'Lemma 5' that does not exist in the manuscript. The GP error bound actually available is probabilistic and holds only on the compact set X from Assumption 1, but the theorem needs a deterministic global bound on |Δf_i(x)| over all of R^n. For a squared-exponential kernel, the posterior mean decays to the prior mean away from training data, while the true drift need not, so the error is typically unbounded outside X. No finite α_i2 can dominate it globally. That alone breaks the global fixed-time claim.

Even if we grant a global bound, the settling time in (24) contains |Δf_i(x)|, which is evaluated along the trajectory and varies with x(t). Definition 2 requires a single T_max independent of initial conditions, so the expression is not a fixed-time bound. The differential inequality in (29) applies Lemma 4 with A state-dependent, but Lemma 4 requires constant A<0; the argument does not fit. This is an internal inconsistency, not a minor gap.

The known-dynamics theorem also has smaller issues: equation (13) duplicates the α_i1 term, and the gain condition α_i2 > d̄_i is weaker than the α_i2 > (2/√π) d̄_i required by the proof and settling time. These are likely fixable, but they show the proofreading is not careful.

Who is this for? Readers interested in sliding mode control with learning-based uncertainty compensation might find the controller structure appealing, and the simulations are encouraging. But the paper is not formally reliable: the central theorem for the GP case is invalid as written. A serious referee would have to send it back for major revision, and the required changes are substantial. I would not send it out as is; I'd desk reject and invite resubmission if the authors can either prove a global bound under stronger assumptions or recast the result as practical fixed-time stability on a compact set. The idea may be salvageable, but this version does not deserve referee time.

Referee Report

5 major / 6 minor

Summary. This paper proposes an integral fixed-time terminal sliding mode control scheme for a class of nonlinear systems with matched perturbations. In the known-dynamics case, a control law with a single reaching gain is claimed to yield global fixed-time stability with an explicit settling time bound. In the partially unknown case, Gaussian process regression is used to approximate the drift, and a second theorem claims global fixed-time stability under a gain condition involving the GP estimation error. The paper reports simulations on a PMSM model with different training dataset sizes. The central claims of the learning-based part are not supported by the provided analysis, and there are inconsistencies in the known-dynamics theorem.

Significance. If the learning-based theorem were valid, the paper would offer a simple, single-parameter fixed-time sliding mode controller that leverages GP uncertainty bounds, with an explicit settling time. The constructive design and the explicit settling time formulas for the known-dynamics case are useful elements. However, the main novelty, namely the GP-based global fixed-time guarantee, rests on a nonexistent lemma and a state-dependent condition, and the error bound used is local and probabilistic. These issues are load-bearing, so the contribution as it stands is not established. The manuscript does not include machine-checked proofs or code, but the analytical claims are the core of the paper.

major comments (5)

[IV-B, Theorem 5] The global fixed-time claim requires a uniform bound on |\Delta f_i(x)| = |f_i(x) - \hat{f}_i(x)| for all x in R^n, but the only bound provided in Section IV-A is the GP regression bound |f_i(x) - \varrho_i(x)| \le \tilde{\chi}_i \sigma_i(x), which is stated for x in X and holds only with high probability. Assumption 1 bounds the RKHS norm only on the compact set X. With the squared-exponential kernel used in Section V, the posterior mean \hat{f}_i tends to the prior mean away from the training data while f_i need not decay, so |\Delta f_i(x)| is typically unbounded on R^n. Consequently, no finite \alpha_{i2} can satisfy \alpha_{i2} > \bar{d}_i + |\Delta f_i(x)| for all initial conditions, and the global fixed-time property is not established.
[IV-B, Theorem 5, Eq. (24)] The settling time bound in (24) contains |\Delta f_i(x)|, which is evaluated along the trajectory and is therefore state-dependent. Definition 2 requires a constant T_max independent of initial conditions; a bound whose denominator depends on x(t) does not qualify as a fixed-time settling time. Moreover, the proof sets A = -(\alpha_{i2} - |\Delta f_i(x)| - \bar{d}_i) and applies Lemma 4, which assumes a constant A < 0. Since A is state-dependent, the differential inequality (29) does not fit the hypotheses of Lemma 1 or Lemma 4, so the fixed-time conclusion does not follow.
[IV-B, Theorem 5 proof] The proof states that the boundedness of |\Delta f_i(x)| is 'corroborated by Lemma 5', but no Lemma 5 appears in the manuscript. The GP error bound in Section IV-A is not presented as a formal lemma and is a probabilistic statement on the compact set X, not a deterministic global bound. This missing support is load-bearing for the theorem.
[III, Theorem 3] The theorem statement says 'If \alpha_{i2} > \bar{d}_i', whereas the control law (7) and the settling time (12) require \alpha_{i2} > 2\sqrt{\pi}\,\bar{d}_i (or, equivalently, \alpha_{i2} > 2\bar{d}_i/\sqrt{\pi} if the intended condition from Lemma 2 is used) to ensure that the denominator in (12) is positive. As stated, the condition is insufficient; for instance, with \bar{d}_i = 1 and \alpha_{i2} = 1.05, the expression 1/(\alpha_{i2} - 2\sqrt{\pi}\bar{d}_i) is negative and the claimed bound is meaningless. The statement, control law, and proof must be made consistent.
[II-A, Lemma 4] The settling-time formula in Lemma 4 is incorrect. For the case |z| > 1, \dot{V} \le A z^3 = 2^{3/2} A V^{3/2}, so Lemma 1 gives T_1 \le -1/(\sqrt{2} A), not -1/(2A). The final bound T_max = -(2\sqrt{2}+1)/(2A) also does not match the two case times as computed. Since Theorem 5's settling-time estimate relies on this lemma, the numerical value in (24) is not justified.

minor comments (6)

[III, Eq. (13)] The derivative expression for \dot{s}_i contains the term \alpha_{i1}\exp(z_i^2)\lfloor z_i\rceil^{p_i/q_i} twice; from (6) only one such term should appear. Correcting this typo is necessary for the substitution of (7) to yield \dot{s}_i = -\frac{\sqrt{\pi}}{2}\alpha_{i2}\exp(s_i^2)\lfloor s_i\rceil^0 + d_i.
[V] The kernel described as 'squared-exponential kernels k(x,x') = exp(-l|x-x'|)' is the exponential/Laplacian kernel, not a squared-exponential kernel; the notation should be corrected.
[Abstract/Conclusion] The abstract claims that the method 'outperforms an established adaptive fixed-time sliding mode control approach', but the simulation section does not include a comparison with such a method; only results with different dataset sizes are shown.
[II-B] In (4), 'd_i \in R^n' should be 'd_i \in R' since it is a scalar perturbation component.
[IV-A] The GP error bound is stated without a probability level; the standard result (e.g., from [19]) gives a bound that holds with probability at least 1-\delta. A precise statement is needed, and it should be cited as a lemma rather than referenced as a nonexistent 'Lemma 5'.
[II-A] The notation \lfloor z \rceil^0 is not explicitly defined; it appears to denote the sign function, but this should be stated.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation; the main gap is a missing Lemma 5 and a state-dependent settling time, which are correctness issues, not circularity.

full rationale

The derivation chain is not circular. The control law (7)/(19) is constructed to cancel the known or GP-estimated drift, leaving the sliding dynamics (25). Theorem 5's condition α_i2 > d̄_i + |Δf_i(x)| is a gain-dominance condition, not a fitted quantity renamed as a prediction; the GP error bound in Section IV-A is cited from the external GP literature [19]. Lemma 2 from the authors' prior work [24] is a self-citation, but it is a parameter-free stability lemma for an auxiliary scalar system with stated assumptions that do not include the closed-loop target; it is therefore independent support rather than a circular premise. The notable weaknesses are the invocation of a nonexistent 'Lemma 5' to assert the boundedness of |Δf_i(x)| globally, and the fact that the settling time (24) contains |Δf_i(x)|, making it state-dependent and incompatible with Definition 2's constant Tmax. These are missing support and mathematical-validity gaps, not reductions of a prediction to its own input, so they do not raise the circularity score.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard Lyapunov and GP regression machinery, plus an unstated global error bound for the Gaussian process approximation. The theorem conditions are stated in terms of the unknown error |Δf_i(x)|, which makes the guarantee unverifiable in practice.

free parameters (5)

Reaching gain α_i2 = α_i2 = 4 in simulations
The theorem requires α_i2 > d̄_i + |Δf_i(x)|, but |Δf_i(x)| is unknown, so the designer cannot verify the condition. In simulation α_i2 = 4 with d̄_i = 1.
Sliding surface gain α_i1 and fractional power p_i/q_i = α_i1 = 6, p_i/q_i = 0.8
Designer-chosen parameters appearing in the settling time bounds (10) and (22); they are not fitted to data but are required for the fixed-time property.
Perturbation bound d̄_i = d̄_i = 1
The problem statement says the perturbation upper bound is unknown, but the control law and settling time use ¯d_i as a known number.
GP kernel length scale l and noise variance = l = 1; ρ_F not specified
Hyperparameters are chosen from prior knowledge; no tuning procedure is given, and the kernel is mislabeled as squared-exponential.
Training dataset size N = N = 5 and N = 50 per GP
The simulation compares two dataset sizes; the method's performance and the GP error bound depend on N.

assumptions (5)

domain assumption Assumption 1: bounded RKHS norm of each f_i
Standard assumption in GP-based control (see [21]); invoked to bound the GP approximation error.
domain assumption Assumption 2: noisy measurements of f_i are available
Provides the data for GP training.
standard math Lemma 1, Polyakov's fixed-time stability condition
Borrowed from [6]; used to derive settling time bounds from differential inequalities.
standard math Lemma 2, fixed-time stability of the reaching dynamics
Borrowed from the authors' prior work [24]; used to bound the sliding variable reaching time.
ad hoc to paper Global uniform bound on GP error |Δf_i(x)| for all x in R^n
The cited GP bound holds only on a compact set X and in probability; the theorem requires a deterministic global bound so that α_i2 can be chosen once for all initial conditions. This is not proven and is the main unsupported premise.

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Cite this review

Pith. "Pith review of Learning-Based Robust Fixed-Time Terminal Sliding Mode Control." pith.science (2026). https://pith.science/paper/X2DJD6HX

@misc{pith2026250522827,
  author       = {Pith},
  title        = {Pith review of: Learning-Based Robust Fixed-Time Terminal Sliding Mode Control},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X2DJD6HX}},
  note         = {Machine review of arXiv:2505.22827}
}

read the original abstract

In this paper, we develop and analyze an integral fixed-time sliding mode control method for a scenario in which the system model is only partially known, utilizing Gaussian processes. We present two theorems on fixed-time convergence. The first theorem addresses the fully known system model, while the second considers situations where the system's drift is approximated utilizing Gaussian processes (GP) for approximating unknown dynamics. Both theorems establish the global fixed-time stability of the closed-loop system. The stability analysis is based on a straightforward quadratic Lyapunov function. Our proposed method outperforms an established adaptive fixed-time sliding mode control approach, especially when ample training data is available.

Figures

Figures reproduced from arXiv: 2505.22827 by the authors.

**Figure 4.** The unknown function estimations with N = 5 data. and function estimations using the online learning-based method from [17] and the FxT control learning are shown in Figs. 1–4 for the first simulation (e.g., N = 5) and in Figs. 5–6 for the second simulation (e.g., N = 50). The corresponding states in both cases are depicted in Figs. 1 and 5. As can be seen, using more datasets leads to a drastic improvement in contr… view at source ↗

**Figure 1.** State variables with N = 5 data [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

Figures from the paper (3 more)

**Figure 2.** Figure 2: Sliding variables with N = 5 data. plots depicting the states, sliding surfaces, input controls, [PITH_FULL_IMAGE:figures/full_fig_p005_2.png]

**Figure 5.** Figure 5: State variables with N = 50 data [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: The unknown function estimation with N = 50 data. VI. CONCLUSIONS This paper addresses the problem of learning-based fixedtime control for a class of nonlinear systems subjected to matched perturbations. The first part focuses on designing an integral terminal sliding…

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

V ariable structure systems with sliding mode s,

V . Utkin, “V ariable structure systems with sliding mode s,” IEEE Transactions on Automatic control , vol. 22, no. 2, pp. 212–222, 1977

work page 1977

[2] [2]

Integral sliding mode in systems ope rating under uncertainty conditions,

V . Utkin and J. Shi, “Integral sliding mode in systems ope rating under uncertainty conditions,” in Proceedings of 35th IEEE conference on decision and control , vol. 4. IEEE, 1996, pp. 4591–4596

work page 1996

[3] [3]

Finite-time stability of continuous autonomous systems,

S. P . Bhat and D. S. Bernstein, “Finite-time stability of continuous autonomous systems,” SIAM Journal on Control and Optimization , vol. 38, no. 3, pp. 751–766, 2000

work page 2000

[4] [4]

Finite time stability and stabilization of a class of continuous systems,

E. Moulay and W. Perruquetti, “Finite time stability and stabilization of a class of continuous systems,” Journal of Mathematical Analysis and Applications , vol. 323, no. 2, pp. 1430–1443, 2006

work page 2006

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