Generalized Super-Twisting Observer for a class of interconnected nonlinear systems with uncertainties

Jaime A. Moreno; Rania Tafat; Stefan Streif

REVIEW 3 major objections 4 minor 29 references

Generalized Super-Twisting Observer for a class of interconnected nonlinear systems with uncertainties

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Observer reaches exact finite-time estimates despite uncertain dynamics

desk verdict A solid extension of the GSTO to a useful class of interconnected systems, with a proof gap in the boundedness step and a few index typos that should be fixed. read the letter →

arxiv 2506.01754 v1 pith:IYZUAUDK submitted 2025-06-02 eess.SY cs.SY

Rania Tafat , Jaime A. Moreno , Stefan Streif This is my paper

classification eess.SYcs.SY

keywords generalizedsuper-twistingobserverfinite-timeconvergenceinterconnectednonlinearsystemsboundeduncertaintiesstrongobservabilitydiscontinuousLyapunovfunctionhigh-gaincomparison

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 letter claims that the Generalized Super-Twisting Observer (GSTO) can be extended from single two-dimensional systems to a class of strongly observable nonlinear interconnected systems with bounded uncertainties, and that the extended observer converges to the true state exactly in finite time. The class allows unknown, possibly discontinuous nonlinearities in the unmeasured channels, provided the uncertainty and interconnection terms grow no faster than linearly in the estimation errors and the measured channels are interconnected in a strict cascade. If the claim is correct, exact state estimation becomes possible in settings where continuous observers such as high-gain observers can only guarantee boundedness or input-to-state stability. The authors support the claim with a nonsmooth strong Lyapunov function and demonstrate it on a food-production system in which CO2 measurements are used to estimate dry biomass of larvae.

What carries the argument

The load-bearing object is the nonsmooth strong Lyapunov function $V_i(\xi_i) = \xi_i^\top P_i \xi_i$, where $P_i$ solves the algebraic Lyapunov equation $(A_0 - L_i C_0)^\top P_i + P_i(A_0 - L_i C_0) = -Q_i$ and $\xi_i$ is the scaled error vector $[\varphi_{i1}(e_{i1}), e_{i2}/\gamma]^\top$. This change of variables, borrowed from high-gain observer analysis, turns the super-twisting gains into eigenvalues scaled by $\gamma$ and produces the inequalities that dominate the interconnection terms. The functions $\varphi_{i1}(z) = \mu_{i1}|z|^{1/2}\operatorname{sign}(z) + \mu_{i2}z$ and $\varphi_{i2}(z) = \varphi'_{i1}(z)\varphi_{i1}(z)$ encode the discontinuous and fractional-power structure that makes exact finite-time convergence possible. Assumption 1 supplies the linear growth bounds on the interconnection residuals $\rho_{i1}$, $\rho_{i2}$, and the cascade condition (6) is what turns the ultimately bounded error result into a sequential finite-time convergence proof.

What would settle it

Take a two-subsystem plant in the required form with N=2 and choose the second-channel nonlinearity to include a term like the estimation error raised to the power 3/2, so that the residual violates the linear-growth bound; simulate the observer with increasing gain and observe whether the error reaches zero in finite time or only shrinks to a neighborhood of zero. Alternatively, to test the cascade condition, let the first measured channel of subsystem 2 depend on the first measured state of subsystem 1 as well as its second state, making the coupling residual nonzero, and check whether the error remains bounded rather than vanishing.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is a finite-time convergence theorem. For the plant (1), under Assumption 1's linear-growth bounds and the cascade condition (6), the observer (2) with positive gains $l_{ij}$ has a gain threshold $\gamma_0$ such that for every $\gamma \geq \gamma_0$ the estimation error reaches zero in finite time, despite nonvanishing bounded uncertainties $\delta_i$. The proof shows that the error dynamics form a network of nonlinearly interconnected Generalized Super-Twisting Algorithms, and that a nonsmooth quadratic Lyapunov function, quadratic in the transformed vector $\xi_i = \gamma \Gamma^{-1}[\varphi_{i1}(e_{i1}), e_{i2}]^\top$, decreases along solutions with both linear and quadratic terms. The cascade structure makes $\rho_{i1}$ vanish once the preceding subsystem has converged, so finite-time convergence propagates from subsystem 1 through subsystem $N$. The same analysis shows that without the cascade condition the observer still yields uniformly bounded errors, that is, finite-time input-to-state stability rather than exact convergence.

Load-bearing premise

The proof's finite-time conclusion rests on Assumption 1: every uncertain or unmodeled term in the second channel must grow no faster than linearly in the estimation errors, and the measured-channel interconnections must be strictly cascaded; if an uncertainty grows superlinearly or a measured channel couples non-cascadedly, the same argument only establishes boundedness, not exact convergence.

Editorial extensions

If this is right

For this class of systems, the observer achieves exact state estimates in finite time even when the uncertainties are never zero, so the discontinuous term buys something continuous observers cannot.
Any continuous observer, including the high-gain observer obtained by setting the linear fraction coefficient to zero, can at best render the estimation error input-to-state stable with respect to the unknown input; the theorem marks the boundary of that limitation.
If the cascade condition on the measured channels is dropped, the same Lyapunov machinery yields finite-time input-to-state stability: the error remains uniformly bounded around zero rather than converging exactly.
The convergence proof gives a constructive design rule: choose any positive gains, pick a positive definite matrix, solve the Lyapunov equation for the corresponding matrix, and then take the gain scale large enough to dominate the interconnection constants.
In the presence of bounded measurement noise, the error remains bounded, and the gain scale must balance convergence speed against noise amplification.

Reading between the lines

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

The cascade-propagation argument suggests the GSTO should work for any directed acyclic interconnection graph on the measured channels, not only the strict chain in (1), as long as each residual points from an earlier-converging subsystem to a later one.
A natural test of the theorem's boundary is to violate only the linear-growth bound, for example by letting an uncertainty grow like a power higher than one in the estimation error; the constants then cease to exist, and simulations should show finite-time convergence failing while mere boundedness persists.
The food-production example points to a broader application in controlled-environment agriculture, where biological rate functions are inherently bounded and the key assumption may hold for process models without additional tuning.
The paper's own open question, when a strongly observable system can be transformed into the required class, is the practical bottleneck: the theorem applies after such a transformation exists, and finding those transformations for specific plants is where the real-world value is won.

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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

Rania, Jaime, and Stefan extend the Generalized Super-Twisting Observer to a broader class than [17]: they drop the observable linear part and global Lipschitz continuity, allow discontinuous or multivalued f_i2, and handle cascaded first-channel interconnections with arbitrary second-channel coupling. The core idea is sensible: put the error dynamics into interconnected GSTA form, use a nonsmooth quadratic Lyapunov function per subsystem, and exploit the strict cascade in the first channel to get sequential finite-time convergence. The numerical example on interconnected larvae production units is relevant and shows the GSTO clearly outperforming a high-gain observer.

The main soft spot is the boundedness step. Inequality (22) is only established on the set Omega defined by (19). The paper says this 'implies' ultimate boundedness, but the argument only protects the smallest level set contained in Omega. It does not show that trajectories starting outside that level set ever enter Omega, or that the complement of Omega cannot host an escape. Since the second-channel interconnections are not cascaded, the premise 'the terms ||xi_j|| remain bounded' used around Eq. (23) has no rigorous justification. This is a genuine gap in the proof as written, though not obviously a fatal one; a more careful Lyapunov argument on a forward-invariant set would probably fix it.

There is also an index error in Eq. (21): the second term in the bracket should use alpha_ji and beta_ji, not alpha_ii and beta_ii, to properly symmetrize the cross terms. The gamma-scaling is fine once you remember the outer gamma, but as written the negativity condition is not verifiable. The numerical section would benefit from full parameter listings and code, but for a letters paper the demonstration is adequate.

The citation pattern is honest; the authors build on their own earlier work and on Moreno's Lyapunov functions, which is appropriate here. The central claim is plausible and the class of systems is meaningful for applications like food production. I'd send this to a serious referee: the extension is real, and the proof issues are addressable, but they need to be fixed before the result can be taken as proven.

Referee Report

3 major / 4 minor

Summary. The paper extends the Generalized Super-Twisting Observer (GSTO) to a class of N interconnected nonlinear second-order subsystems with bounded uncertainties. The proposed observer (2) uses discontinuous injection terms in both channels of each subsystem. Under Assumption 1, which imposes a global linear-growth bound on the uncertainties/interconnections and a cascaded structure on the measured-channel couplings, Theorem 1 claims finite-time convergence of the estimation error to zero. The proof introduces a nonsmooth quadratic Lyapunov function per subsystem, derives a derivative bound, and then uses a cascade argument to establish sequential finite-time convergence. A numerical case study on two interconnected insect larvae production units compares the GSTO with a high-gain observer, and the authors emphasize that the discontinuous term is essential for exact estimation under nonzero uncertainties.

Significance. If Theorem 1 is correct, this is a useful contribution: it gives finite-time exact state estimation for a nontrivial class of interconnected nonlinear systems with bounded unknown inputs, relaxing earlier global Lipschitz and linear-observability assumptions. The use of a nonsmooth quadratic Lyapunov function is appropriate for super-twisting-like dynamics, and the paper makes the structural assumptions on the interconnections explicit and falsifiable. The numerical study supports the qualitative claim that the discontinuous term avoids the peaking of a high-gain observer. However, the proof as written contains a load-bearing gap in the transition from a local Lyapunov inequality on a restricted set to global ultimate boundedness, and there are index/factor errors in the key inequality (21). These issues are substantial enough that the main theorem is not established in the present form, though they appear fixable within the paper's scope.

major comments (3)

[III, Eq. (21)] In deriving (21) from (18) via the Young inequality (20), the coefficient of ||ξ_i||^2 is written with α_ii and β_ii in the second term for every j. After exchanging summation indices, the second term should be (λ_max{P_j}/γ^2)(α_ji/μ_i2 + β_ji γ), not (λ_max{P_j}/γ^2)(α_ii/μ_i2 + β_ii γ). As written, the bracket is not the correct result of applying (20) to the last term of (18), and the claimed negativity of the quadratic coefficients for large γ is not verifiable. This is a load-bearing step because the proof of (22) depends on those coefficients being positive.
[III, Eqs. (19)–(22)] The inequality ˙V ≤ -Σ c_i V_i^{1/2} - Σ \tilde c_i V_i is only established on the set Ω defined by (19). The set Ω is not forward invariant, and its complement is unbounded: for N = 2, take ξ_1 = 0 and let ξ_2 grow; then (19) fails for i = 1 while the point is not in a compact set. The sentence that the smallest level set of V contained in Ω is positively invariant only applies to trajectories that already start inside that level set. No argument shows that arbitrary initial conditions enter Ω or that trajectories outside Ω remain bounded. Therefore the conclusion that the trajectories are 'ultimately and uniformly bounded' does not follow. This gap also undermines the boundedness premise used in the cascade argument around Eq. (23), which is critical because ρ_i2 in (5) can couple all subsystems and a later subsystem may escape before the first one is forced to zero.
[Theorem 1 and III, Eq. (23)] Theorem 1 states that for every l_ij > 0 there exists γ_0 such that convergence holds for all γ ≥ γ_0. The proof, however, requires additionally that the gains μ_i1, μ_i2 be sufficiently large (see the sentence after (23)), and the ordering of the design choices is not stated: if μ must be chosen after γ, the theorem should quantify over μ as well. In Eq. (23) the coupling terms use α_ij and β_ij with i = 1, but the correct indices should be α_1j and β_1j. As written, the displayed expression is inconsistent with the bounds in (4). The cascade conclusion needs a precise statement of the admissible gain ranges and a rigorous proof that the negative linear term in V1 dominates the coupling terms for all bounded ||ξ_j||.

minor comments (4)

[III, Eq. (7)] The term 'g1ei2' should be 'g_i e_i2' to be consistent with the subsystem index.
[II, Eq. (3)] The formula for φ_i2(z) is garbled in the typesetting; it should read φ_i2(z) = (μ_i1^2/2) sign(z) + (3/2) μ_i1 μ_i2 |z|^{1/2} sign(z) + μ_i2^2 z, which is the product φ'_i1(z) φ_i1(z).
[IV, numerical parameters] In the parameter list, 'li1 = 1.1, li1 = 3' is presumably meant to be 'l_i1 = 1.1, l_i2 = 3', and the scalars 'γ1 = 0.1, γ2 = 0.5' are not defined in the theory where γ is a single scalar gain.
[IV, Figures 3 and 4] The axes in Figures 3 and 4 are not labeled; please add labels and units so the relative-error plots are interpretable.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the finite-time convergence claim is derived from the error dynamics and explicit Assumption 1, with self-citations used only as standard proof tools.

full rationale

The paper's central claim, Theorem 1, is not presupposed by its inputs. The convergence proof starts from the explicit error dynamics (7), introduces the standard Lyapunov function of the super-twisting class from [19], [21], and then derives the Lyapunov inequality (22) under Assumption 1 and the cascade condition (6). The self-citations to Moreno's prior work are used as mathematical tools (Lyapunov functions, GSTO structure, strong observability interpretation), not as substitutes for the proof of the interconnected finite-time result. No fitted parameters are renamed as predictions; the simulation parameters are chosen and compared against a high-gain observer, not fitted to a claimed prediction. The only notable weakness is a proof gap in the boundedness argument around equations (19)-(23), but that is a correctness/detail issue, not circularity: the conclusion is not equivalent by construction to any assumed quantity. Therefore the circularity score is 0.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The central claim rests on a structural system class and boundedness assumptions rather than on fitted parameters. No new physical entities are introduced. The observer gains are design degrees of freedom, not inferred from data.

free parameters (1)

Observer gains l_ij, gamma, mu_ij = Simulation: l_i1=1.1, l_i2=3, gamma1=0.1, gamma2=0.5, mu=[[0.03,1],[0.01,1]]
Chosen by hand for the numerical study; the theorem only asserts existence of gamma0 and does not depend on the specific values. Listed for completeness as design parameters, not fitted to data.

assumptions (6)

domain assumption Assumption 1: |rho_i1| and |rho_i2| satisfy linear-growth bounds (Eq. 4) with finite constants alpha, beta, and alpha_i0.
This bound on the unknown uncertainties and interconnections is required for the Lyapunov inequalities (18)-(21) to hold. It is not derived from the system class; it defines the class of admissible uncertainties.
domain assumption The first-channel interconnections are strictly cascaded: rho_i1 depends only on e_j2 for j < i (Eq. 5-6).
This structural condition is essential for the sequential finite-time proof in Section III; without it, the paper only obtains ultimate boundedness.
domain assumption g_i are known, continuous, bounded, and strictly positive.
Used in the observability map inversion and in the Lyapunov analysis (bounds g_i,m and g_i,M).
standard math Solutions of the differential equations are understood in the Filippov sense.
Required because the observer contains a discontinuous sign term; invoked in Section II.
standard math Finite-time stability criterion via Lyapunov functions (e.g., Theorem 4.2 in [3]).
Used at Eq. (15) and (22)-(23) to conclude finite-time convergence from Vdot <= -c V^(1/2).
standard math Existence and uniqueness of P_i solving the algebraic Lyapunov equation for Hurwitz A0-LiC0.
Standard linear algebra background used to define the subsystem Lyapunov functions.

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Cite this review

Pith. "Pith review of Generalized Super-Twisting Observer for a class of interconnected nonlinear systems with uncertainties." pith.science (2026). https://pith.science/paper/IYZUAUDK

@misc{pith2026250601754,
  author       = {Pith},
  title        = {Pith review of: Generalized Super-Twisting Observer for a class of interconnected nonlinear systems with uncertainties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IYZUAUDK}},
  note         = {Machine review of arXiv:2506.01754}
}

read the original abstract

The Generalized Super-Twisting Observer (GSTO) is extended for a strongly observable class of nonlinearly interconnected systems with bounded uncertainties/perturbations. A nonsmooth strong Lyapunov function is used to prove the finite-time convergence of the proposed observer to the true system's trajectories, in the presence of the uncertainties. A case study on the interaction between two food production systems is presented, comparing the proposed observer with the High Gain observer. The results emphasize the critical role of the GSTO's discontinuous term in achieving exact estimation.

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Pith tools

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

[1] [17]

L ´OPEZ -C AAMAL , F., AND MORENO , J. A. A quasicontinuous multi- variable super-twisting observer for 2 n states systems wit h lipschitz nonlinearities. International Journal of Robust and Nonlinear Control 33, 15 (2023), 8992–9017

work page 2023

[2] [1]

Centralized optimization of resource routing in interco nnected food production units with harvesting events

A L KHATIB , M., H EMPEL , A.-J., P ADMANABHA , M., AND STREIF , S. Centralized optimization of resource routing in interco nnected food production units with harvesting events. IF AC-PapersOnLine 55, 7 (2022), 322–327

work page 2022

[3] [2]

Liapunov functions and stability in control theory

B ACCIOTTI , A., AND ROSIER , L. Liapunov functions and stability in control theory. Springer Science & Business Media, 2005

work page 2005

[4] [3]

P., AND BERNSTEIN , D

B HAT, S. P., AND BERNSTEIN , D. S. Finite-time stability of continuous autonomous systems. SIAM Journal on Control and Optimization 38 , 3 (2000), 751–766

work page 2000

[5] [4]

J., B UZZARD , G

C HAKRABARTY , A., S UNDARAM , S., C ORLESS , M. J., B UZZARD , G. T., ˙ZAK , S. H., AND RUNDELL , A. E. Distributed unknown input observers for interconnected nonlinear systems. In American Control Conference (2016), pp. 101–106

work page 2016

[6] [5]

A., AND FRIDMAN , L

D ´AVILA , A., M ORENO , J. A., AND FRIDMAN , L. V ariable gains super- twisting algorithm: A Lyapunov based design. In Proc. of the 2010 American Control Conference (2010), pp. 968–973

work page 2010

[7] [6]

S., E LLOUMI , S., AND BRAIEK , N

D HBAIBI , S., T LILI , A. S., E LLOUMI , S., AND BRAIEK , N. B. hinf ty decentralized observation and control of nonlinear interc onnected sys- tems. ISA Transactions 48 , 4 (2009), 458–467

work page 2009

[8] [7]

V., AND ABOLFATHI , S

F ARAHANI , A. V., AND ABOLFATHI , S. Sliding mode observer design for decentralized multi-phase ﬂow estimation. Heliyon 8 , 2 (2022)

work page 2022