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Quantized Dissipative Uncertain Model for Fractional T_S Fuzzy systems with Time_Varying Delays Under Networked Control System

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper claims that one set of LMI feasibility conditions guarantees a filter for delayed Takagi-Sugeno fuzzy singular systems that keeps the filtering error impulse-free and asymptotically stable under sensor faults, event-triggered…

desk verdict The advertised fractional, quantized, event-triggered framework never appears in the equations, and the central LMI theorems are stated without proof; this paper is not ready for peer review. read the letter →

arxiv 2506.02788 v1 pith:66O5Y5V7 submitted 2025-06-03 eess.SY cs.SY

Muhammad Shamrooz Aslam , Hazrat Bilal , Sumeera Shamrooz This is my paper
classification eess.SYcs.SY MSC 93C4293C2393B3693D30
keywords T-Sfuzzysystemssingulartime-varyingdelaysH-infinityfilteringevent-triggeredcontrolquantizationsensorfaultslinearmatrixinequalities
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

This paper tries to establish a complete filter-design recipe for a nonlinear plant modeled as a Takagi-Sugeno fuzzy singular system with time-varying delay, parameter uncertainty, sensor faults, and a network that sends measurements only when an event-triggered condition fires and then quantizes them logarithmically. The recipe is a set of linear matrix inequalities (LMIs): if a computer can find matrices satisfying inequalities (24)-(28) and (31), then the paper claims the filtering error system is impulse-free and asymptotically stable with prescribed noise-rejection level $\gamma$, and equations (30) hand back the filter matrices. The reason to care is practical: in networked control, delays, data reduction, and faulty sensors are the norm, so a filter that provably tolerates all three at once is directly usable. The paper demonstrates the design on a nonlinear DC motor model and reports smaller minimal $\gamma$ than four earlier T-S fuzzy filter designs for the same delay bounds.

What carries the argument

The engine of the proof is a fuzzy Lyapunov-Krasovskii functional: a Lyapunov function built from fuzzy-membership-dependent matrices plus single and double integral terms over the delay intervals, which converts the stability and dissipativity requirement into matrix inequalities. Randomness of the delays is encoded by a Bernoulli variable that splits the delay into two distributions (Assumptions 1-2), the sensor faults enter through a diagonal fault matrix $\beta$ in (16)-(18), and the quantized event-triggered channel is folded into the error dynamics as communication-induced delay. Theorems 2 and 3 then linearize the general conditions by assuming bounds $\rho_i$ on the membership-function derivatives $h_i$, producing strict LMIs; equations (30) recover the filter gains.

What would settle it

Simulate the DC motor example with the given disturbances and uncertainties, record the maximum of $|\dot h_i(t)|$ for the membership functions in (37) along the trajectory, and compare it with the chosen $\rho_i$ (the simulations set $\rho_1=\rho_2=100$). If any realized derivative exceeds the assumed bound, Assumption 4 is violated for that run and the LMI-certified guarantee does not apply; if the bound holds, the numerical demonstration is consistent with the theorem.

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Extended reading notes

Core claim

The paper claims that Theorems 2 and 3 give sufficient conditions for the existence of a resilient delayed singular fuzzy filter for system (8): under Assumptions 3 and 4, feasibility of the LMIs (24)-(28) and (31) guarantees that the filtering error system (21) is impulse-free and asymptotically stable with $H_\infty$ performance level $\gamma$, for any time-varying delays satisfying Assumption 2 and uncertainties of the linear-fractional form (7). The filter matrices are then recoverable from equations (30). This extends earlier T-S fuzzy $H_\infty$ filter designs to a setting that simultaneously includes sensor faults, event-triggered transmission, logarithmic quantization, random time-varying delays, and a unified dissipativity requirement covering $H_\infty$, passivity, $L_2$-$L_\infty$, and purely dissipative performance.

Load-bearing premise

The guarantee rests on the assumption that the fuzzy membership functions never change faster than the chosen rates $\rho_i$, yet the paper states those rates for the example without computing them from the dynamics; if the actual rates are larger, the LMI conditions are not sufficient and the filter is not guaranteed.

Editorial extensions

If this is right

  • Feasible LMIs certify a filter for the full networked setting, so a designer can check stability and noise rejection with one numerical test before implementation.
  • The recovered filter matrices give an explicit controller (19) whose error system rejects disturbances with level $\gamma$ despite sensor faults and packet-saving communication.
  • The unified dissipativity setup means one theorem covers $H_\infty$, passivity, $L_2$-$L_\infty$, and dissipative performance as special cases, so a single feasibility run serves multiple design specs.
  • For the DC motor example, the reported minimal $\gamma$ drops from the 0.31-2.22 range of four earlier T-S fuzzy filters to $10^{-3}$-$10^{-2}$, indicating less conservative delay-dependent conditions.
  • The method extends to any plant that admits a T-S fuzzy singular model with bounded membership-derivative rates, not just the DC motor and truck-trailer examples used for simulation.

Reading between the lines

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

  • The simulation uses $\rho_1=\rho_2=100$ without a computed bound on $\dot h_i$ along the actual trajectories; a reader who wants a theorem-guaranteed filter would need to verify these bounds from the operating region or the state equations.
  • The comparison in Table 2 is against delay-free $H_\infty$ T-S filter designs without sensor faults, quantization, or event triggering; the reported $\gamma$ improvements therefore measure the new delay-dependent machinery, not the cost or benefit of the extra networked features.
  • The title calls the systems fractional, but the state equations (6), (8), (15), and (21) use ordinary integer-order derivatives; applying the design to a genuinely fractional-order plant would require a separate derivation.
  • A natural next experiment is to run the recovered filter on a networked DC motor emulator that drops and quantizes measurements, and compare the measured $L_2$ gain of the estimation error with the predicted $\gamma$.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper claims to solve a quantized dissipative filtering problem for delayed fractional Takagi-Sugeno fuzzy singular systems with time-varying delays under event-triggered networked control, sensor faults, and uncertainties. The plant is modeled as a T-S fuzzy singular system (Eqs. (6), (8)), a filtering error system is formed (Eq. (21)), and Theorems 1-3 assert membership-function-dependent and strict LMI conditions guaranteeing impulse-freeness, asymptotic stability, and an H-infinity performance bound gamma. Controller matrices are to be recovered from Eq. (30). A DC motor/truck-trailer example is used to demonstrate the method. The abstract also promises a unified extended dissipativity framework combining H-infinity, dissipativity, L2-Linfinity, and passivity, although no such combined performance index appears in the main text.

Significance. If the LMI conditions were rigorously proved and the numerical study were an independent validation, the paper would provide a complete reliable H-infinity filter design procedure for a useful class of T-S fuzzy singular systems with random delays. That contribution would be of interest to researchers working on networked fuzzy filtering. However, the central claims are not supported by the text: no proofs are given for Theorems 1-3, the advertised fractional-order and event-triggered/quantization features are not present in the mathematical model, and the membership-derivative bounds on which the LMIs rest are not verified. No reproducible code, machine-checked proofs, or falsifiable parameter-free predictions are supplied. The significance can only be assessed after a full rewrite that supplies the missing derivations and definitions.

major comments (4)
  1. [Section 2, Eqs. (6) and (8)] The title and abstract promise a 'fractional T-S fuzzy system', but the plant model in Eqs. (6) and (8) and the filtering error system (21) are written with ordinary integer-order derivatives only. No Caputo or Riemann-Liouville derivative, fractional order, or fractional stability notion is introduced anywhere in Section 2. The claimed fractional setting is therefore not actually modeled, and the main theorems cannot be read as results about fractional systems.
  2. [Section 3, Theorems 1-3] Theorems 1, 2, and 3 are stated as assertions but no proofs are given. There is no Lyapunov-Krasovskii functional, no computation of its derivative, and no derivation of the transition from the membership-function-dependent conditions of Theorem 1 to the strict LMIs (24)-(28) via Assumptions 3-4. In addition, condition (27) is typeset as a non-inequality with undefined blocks, and the matrices in (27)-(29) are only partially defined with mismatched dimensions. The recovery formula (30) assumes nonsingular matrices U and W satisfying congruence relations without any proof of existence or invertibility. Thus the central sufficiency claim is unverifiable as written.
  3. [Abstract and Section 2.3] The abstract and contribution list state that an event-triggered scheme and logarithmic quantization are implemented and that the quantization effect is transformed into induced communication delays. However, no event-triggering condition, no quantization map, no sector-bound representation, and no delay-induction argument appear in Section 2 or in the problem formulation. The system used in Theorems 1-3 does not contain any quantization or event-triggering variables; consequently the claimed results do not cover the advertised network control mechanism.
  4. [Section 4, Table 2 and Example 2] The numerical validation is not independent. In Example 2 the parameter gamma is fixed to 1.5 and the LMIs are reported feasible, which is only a consistency check on a chosen parameter, not a computation of minimal gamma or a falsifiable prediction. Table 2 compares minimal gamma values with [61], [63], [65], and [67], but the rows labeled 'Corollary 1' report only orders of magnitude (10^-3, 10^-2, ...) with no explicit values or problem data, so the claimed less conservatism is not checkable. Furthermore, Assumption 4 requires bounds rho_i on the membership-function derivatives h_i, and the chosen rho_1 = rho_2 = 100 are never verified against the simulated trajectory; if these bounds fail, the sufficiency of the LMIs is void.
minor comments (5)
  1. [Throughout] The manuscript contains many corrupted formulas and typos ('ABSTARCT', 'asymptomatically stable', 'associative Lyapunov'), which make it hard to parse; a full editorial cleanup is needed.
  2. [Tables 3 and 4] Table 3 and Table 4 are identical but labeled differently, and Figure captions 5-6 refer to states and the truck-trailer system without matching the example numbering.
  3. [Notations] The notation section is incomplete: several symbols used in Theorems 1-3, such as hat(P)_i, Gamma_ij, Xi_ij, and the scalar a_k, are not fully defined.
  4. [Assumptions 3 and 4] Assumption 3 is stated with an incomplete inequality and no explicit relation to the membership functions, and the relationship between Assumption 3 and Assumption 4 is unclear; this should be corrected if the manuscript is revised.
  5. [References] Some references have malformed DOIs or incomplete bibliographic data (e.g., reference [9] contains 'https://doi.org/110.1109/TAC.2011.2178629'), and references [62]-[68] are not consistently formatted.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation found; the main gaps are unproved theorems and unverified assumptions, not input-output circularity.

full rationale

I walked the derivation chain from the delayed T-S fuzzy singular model (8) through the filtering error system (21), Theorem 1, the strict LMI conversion in Theorems 2-3, and the DC motor examples. No step exhibits a claimed prediction that is equivalent, by the paper's own equations, to a fitted input or to a self-cited premise. Theorems 1-3 are asserted without proofs and with garbled typesetting, and condition (27) is not a well-formed inequality, but omitted proofs and uncheckable typesetting are correctness/verifiability gaps rather than circularity. Assumptions 3-4 postulate bounds lambda_i and rho_i on the membership derivatives, and Example 4.2 simply "found that the LMIs (24)-(28) and (31) ... are feasible" with rho_1=rho_2=100 and gamma=1.5, without verifying the derivative bounds; this makes the numerical validation unverified, but the LMI conditions are not defined in terms of the feasibility outcome. Prescribing gamma=1.5 and then testing LMI feasibility is a standard synthesis consistency check, not a prediction forced by construction, and the Table 2 comparisons to [61,63,65,67] are external benchmarks. The many self-citations appear in standard modeling citations (T-S fuzzy modeling, sensor faults, event-triggered schemes), but no load-bearing uniqueness or derivation is imported from those self-citations. I therefore find no significant circularity.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The derivation rests on a large stack of assumed structure: integer-order dynamics despite the fractional title, a Bernoulli-split delay model, ad hoc membership-derivative bounds, and standard uncertainty and singularity assumptions. The performance level and delay constants are chosen to make the LMI example work rather than predicted. No new physical entity is introduced.

free parameters (3)
  • Delay distribution probability delta_0 = 0.15 (Example 2, Case I)
    Assumption 1 and Eq. (14) require a known probability delta_0 that the delay falls in [d_m, d_0]; the value is chosen by the authors, and LMI feasibility depends on it.
  • Performance level gamma = 1.5 (Example 2, Case I)
    The design goal gamma is fixed before solving the LMIs; the paper then reports feasibility, which is a consistency check on a chosen parameter, not an independent prediction.
  • Delay and membership derivative bounds = d_m=0.2, d_0=0.4, d_M=0.5, mu_1=0.2, mu_2=0.4, rho_1=rho_2=100
    These constants enter Assumptions 1-4 and the LMI conditions; they are hand-picked for the simulation, and no sensitivity analysis or method to determine them is given.
assumptions (6)
  • domain assumption The plant is a continuous-time integer-order singular T-S fuzzy system (System (6)-(8)), not a fractional-order system.
    All derivatives in the model are ordinary; the fractional claim in the title and abstract is never realized mathematically, so any conclusion can only apply to the integer-order model.
  • domain assumption The time-varying delay d(t) is split into two intervals with known probabilities (Assumption 1, Eq. (10)-(14)).
    The Bernoulli variable delta(t) in Eq. (14) and all subsequent delay-dependent results depend on this distributional assumption, which may not hold for arbitrary networked delays.
  • ad hoc to paper There exist constants lambda_i and rho_i bounding the membership function derivatives h_i (Assumptions 3 and 4).
    These bounds convert membership-dependent inequalities into strict LMIs in Theorems 2-3; the paper gives no method to compute them and does not verify them for the examples.
  • domain assumption Uncertainties are linear fractional: G(t)^T G(t) <= I and the matrices M_i, N_ki are known (Eq. (7), (20)).
    This is a standard structural uncertainty model, but the uncertainty matrices in the examples are selected by the authors and no identification procedure is given.
  • domain assumption The pair (E, A_i) is regular and impulse-free (Definition 1).
    Needed for the impulse-free characterization of the singular filtering error system; regularity is assumed rather than checked for the computed filter.
  • domain assumption For Example 2, state x_1(t) is bounded in [-3, 3] so the sector nonlinearity yields the T-S membership functions (37).
    The fuzzy model is only valid in this bounded sector; the paper does not discuss what happens outside it.

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Cite this review

Pith. "Pith review of Quantized Dissipative Uncertain Model for Fractional T_S Fuzzy systems with Time_Varying Delays Under Networked Control System." pith.science (2026). https://pith.science/paper/66O5Y5V7

@misc{pith2026250602788,
  author       = {Pith},
  title        = {Pith review of: Quantized Dissipative Uncertain Model for Fractional T_S Fuzzy systems with Time_Varying Delays Under Networked Control System},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/66O5Y5V7}},
  note         = {Machine review of arXiv:2506.02788}
}
read the original abstract

This paper addressed with the quantized dissipative uncertain problem for delayed fractional T_S Fuzzy system for event_triggered networked systems (E_NS), where the extended dissipativity analysis combines the H infinity, dissipativity, L2 and L infinity and passivity performance in a unified frame. To attain the high efficiency for available channel resources, measurement size decrease mechanism and event_triggered scheme (ETS) are proposed. Firstly, we present the ETS in which signal is transmitted through the channel with logical function then logarithmic quantization methodology is implemented for size reduction. Then, we transfer the original delayed fractional T_S fuzzy systems with the effect of quantization under ETS as induced communications delays. Furthermore, by employing the associative Lyapunov functional method in terms of linear matrix inequalities, adequate conditions for asymptotical stability is given. Moreover, we also construct the design fuzzy model for state space filtering system. At last, a truck_trailer model is given to show the effectiveness of the proposed strategy.

Figures

Figures reproduced from arXiv: 2506.02788 by the authors.

Figure 1
Figure 1. State Response of xt() for H Control (Example 2.) Method M d =0.5 M d =0.6 M d =0.8 M d =1 [61] 0.38 0.43 0.83 2.22 [63] 0.36 0.39 0.51 0.79 [65] 0.33 0.35 0.43 0.53 [67] 0.31 0.33 0.37 0.44 Corollary 1 3 10− 2 10− 2 10− 1 10− [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. Control input ut() for H Control (Example 2). Let 1 x t i t ( ) = ( ) and 2 x t t ( ) = ( )  be the state variable, then the system (5) with ut( ) = 0 is signed by the following state equations [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. The release instants and intervals for H Control (Example 2) [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figures from the paper (3 more)
Figure 5
Figure 5. Figure 5: State 3 xt() with and without quantization (Example 2) [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: The states and quantized state responses for Truck [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]
Figure 7
Figure 7. Figure 7: Case-II Uncertain system. To construct the reliable robust H controller design for singular T–S fuzzy system, we select the following parameters to compute the uncertain matrices [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]

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Pith tools

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