Robust Aperiodic Sampled-Data Washout Control for Uncertain Affine Systems

Folco Giorgetti; Francesco Crocetti; Francesco Ferrante; Mario Luca Fravolini

REVIEW 3 major objections 5 minor 27 references

Robust Aperiodic Sampled-Data Washout Control for Uncertain Affine Systems

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that stabilizing the unknown equilibrium of an uncertain affine plant under aperiodic sampling is exactly equivalent to globally exponentially stabilizing the zero-disturbance hybrid system, provided a washout rank…

desk verdict The paper's main theorem has a sign error in the sufficiency proof that, as printed, leaves the if-direction unproven — but the error is likely repairable and the core idea is a legitimate new combination. read the letter →

arxiv 2505.23534 v1 pith:IPBX2IMW submitted 2025-05-29 eess.SY cs.SYmath.OC

Folco Giorgetti , Francesco Crocetti , Mario Luca Fravolini , Francesco Ferrante This is my paper

classification eess.SYcs.SYmath.OC MSC 93C5793C3093D2193D30

keywords aperiodicsamplingsampled-datacontrolwashoutfilteruncertainaffinesystemsunknownequilibriumhybriddynamicalglobalexponentialstabilitysum-of-squaresprogramming

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

An affine plant with an unknown constant disturbance has an unknown equilibrium, and aperiodic sampling defeats straightforward discretization. This paper proves that a sampled-data washout controller stabilizes that unknown equilibrium exactly when a rank condition holds and the corresponding zero-disturbance hybrid system is globally exponentially stable. The rank condition is what lets a coordinate shift erase the disturbance, so the unknown-equilibrium problem becomes a zero-equilibrium problem. The authors turn the conditions into a sum-of-squares design algorithm and verify convergence in simulation.

What carries the argument

The device that carries the argument is the sampled-data washout controller (3), with parameters $(L,G,K,R)$, studied through the hybrid model (5) whose state is $x=(z,\tau)=(x_p,\xi,q,\tau)$, where $q$ stores the last controller output, $\xi$ the last controller state, and the timer $\tau$ triggers sampling when it reaches $[T_1,T_2]$. The named central condition is the rank identity $$\mathrm{rank}\begin{bmatrix} R & K \\ G & L-I \end{bmatrix} = \mathrm{rank}\begin{bmatrix} K \\ L-I \end{bmatrix},$$ which is exactly what guarantees that for every disturbance $d$ one can find a controller-state shift $\bar{\xi}$ satisfying $(L-I)\bar{\xi}=GA^{-1}d$ and $K\bar{\xi}=RA^{-1}d$. Under that shift the disturbed closed loop becomes identical to the zero-disturbance system $H_0$, so global exponential stability of the set $\mathcal{A}_0=\{0\}\times[0,T_2]$ for $H_0$ is the only remaining requirement. For design, the paper uses a clock-dependent Lyapunov function $V(\tilde{x})=\tilde{z}^\top P(\tau)\tilde{z}$ and converts stability and rank requirements into matrix inequalities solved by sum-of-squares programming.

What would settle it

Search for a counterexample in the scalar affine case: choose an observable controller pair and matrices $G,R$ satisfying condition (6), verify that the zero-disturbance system has $\mathcal{A}_0$ globally exponentially stable, and simulate the closed loop with several nonzero disturbances $d$ and sampling gaps inside $[T_1,T_2]$; any trajectory that fails to converge to $-A^{-1}d$ contradicts the sufficiency direction of Theorem 1.

Watch

Extended reading notes

Core claim

Theorem 1 is the central discovery: for an observable controller pair $(L,K)$, the sampled-data washout controller $(L,G,K,R)$ solves the unknown-equilibrium stabilization problem if and only if the rank condition (6) holds and the zero-disturbance hybrid system $H_0$ has $\mathcal{A}_0=\{0\}\times[0,T_2]$ globally exponentially stable. In plain terms, the unknown constant disturbance does not introduce a genuinely new stabilization problem; once the controller state can be shifted to cancel the disturbance, the disturbed system and the zero-disturbance system are the same up to a coordinate change. The paper also shows the rank condition implies the plant matrix $A$ is nonsingular, so the uniqueness of the equilibrium does not need to be assumed separately.

Load-bearing premise

The load-bearing premise is that the rank condition (6) is sufficient to solve the two fixed-point equations that shift the controller state so the unknown disturbance disappears; if for some admissible disturbance and uncertainty those equations had no solution, the sufficiency proof would have no way to reduce the problem to the zero-disturbance system.

Editorial extensions

If this is right

Designers can split the problem: robustly stabilize the zero-disturbance sampled-data loop, then verify the algebraic rank condition; no dedicated disturbance-compensation step is needed.
Any controller satisfying the theorem automatically preserves open-loop equilibria, so the closed loop converges to the true unknown equilibrium rather than to a shifted artificial one.
The necessary-and-sufficient character means the controller structure (3) is not the source of conservatism; conservatism is confined to the sum-of-squares feasibility certificates.
The same Lyapunov framework extends to exponential decay-rate and $H_\infty$ or $L_2$-gain performance objectives, as the paper notes.
The design tolerates any sampling sequence with first sample within $[0,T_2]$ and subsequent gaps between $T_1$ and $T_2$, covering strictly aperiodic as well as periodic sampling.

Reading between the lines

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

If the equivalence is correct, it implies a separation principle the paper leaves implicit: existing robust sampled-data stabilizers for the zero equilibrium can be retrofitted with a washout filter meeting the rank condition, without re-solving the full uncertain problem.
The rank condition is a purely algebraic certificate about the controller matrices, so a similar condition should survive in output-feedback extensions, which the paper lists as future work.
The numerical table shows the maximal allowable sampling bound $T_2$ increasing with the polynomial degree of $W$, which suggests feasibility is limited by the SOS certificate rather than by the underlying stabilizability.
Washout control originated in the stabilization of chaotic systems, so an apt test outside the paper's affine setting would be to emulate aperiodic sampling on a chaotic oscillator with unknown equilibrium; the theorem predicts the same robustness guarantees.

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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 reader's take was close, but the stress-test note is the one to pay attention to: Theorem 1's sufficiency proof is not just typo-heavy, it is wrong as written. The shift z̃ = (xp − A⁻¹d, ξ − ξ̄, u) should use xp + A⁻¹d, since the plant equilibrium is −A⁻¹d. As printed, the shifted flow leaves an extra 2d term, so the reduction to H0 fails. And the claimed GES set Ad omits the controller-state equilibrium ξ̄; if ξ̄ ≠ 0, the actual closed-loop equilibrium isn't even contained in Ad. The earlier fixed-point equations are also misprinted. The stress-test is right: the if-direction of Theorem 1 is not established as printed.

That said, the flaw looks repairable. With the correct shift (xp + A⁻¹d) and the correct target set {−A⁻¹d} × {ξ̄} × {0} × [0,T2], the argument does go through, and the rank condition (6) supplies the needed ξ̄. So the damage is to the written proof, not necessarily to the underlying claim.

What is genuinely new: this is the first combination of washout control with aperiodic sampled-data and norm-bounded uncertainty, as far as the cited literature goes. The necessary-and-sufficient reduction to zero-equilibrium stability plus a rank condition is a clean and useful reformulation, and the SOS-based design procedure in Proposition 3 is a natural and reasonably careful extension of Briat's framework. The numerical example is consistent with the claims, though it is only one scenario and no code or data are shipped, so the computational claims rest on SOSTOOLS/SeDuMi output the reader can't reproduce.

Other soft spots are minor in comparison: the polynomial degree g is hand-tuned, the LMIs are sufficient rather than necessary (which the authors acknowledge), and the prose is a bit rough in places. None of these would stop me from engaging.

Bottom line: a serious referee should look at this. The paper deserves a chance after the proof is fixed. I would not cite it in its current form, but I would bring it to a reading group precisely because the proof error is instructive and the repair is a good exercise.

Referee Report

3 major / 5 minor

Summary. The manuscript studies global exponential stabilization of the unknown zero-input equilibrium of an uncertain affine plant xp_dot = A xp + B up + d under aperiodic sampled-data dynamic state feedback of washout type. The closed loop is modeled as a hybrid system with a timer variable τ and two memory states ξ and q storing the controller state and output between samples. The main result, Theorem 1, states that, under observability of (L,K), a controller (L,G,K,R) solves Problem 1 if and only if the rank condition (6) holds and the set A0 = {0} × [0,T2] is globally exponentially stable for the zero-disturbance hybrid system H0. The paper then derives clock-dependent matrix-inequality conditions and an SOS-based design procedure, and illustrates the method on a numerical example. The central reduction is conceptually attractive, but the sufficiency proof of Theorem 1 as printed contains sign and set-definition errors that must be corrected before the main claim can be accepted.

Significance. The proposed equivalence is valuable: it separates the algebraic washout condition from the dynamical stability condition and reduces the unknown-equilibrium stabilization problem to a zero-equilibrium stabilization problem for the same hybrid data. The design procedure is constructive, uses standard SOS tools, and the numerical example demonstrates the approach. The paper also correctly observes that the conditions of Theorem 1 imply nonsingularity of A, which removes the need to verify Assumption 3 separately. If the proof issues noted below are repaired, the contribution would be a useful addition to the sampled-data washout-control literature.

major comments (3)

[§3, Theorem 1, sufficiency proof] The coordinate shift is printed as z̃ = (xp − A^{-1}d, ξ − ξbar, u), but the plant equilibrium is xp = −A^{-1}d, so the correct shift is xp + A^{-1}d. With the printed minus sign, the shifted flow satisfies \dot{\tilde{x}}_p = A \tilde{x}_p + B q + 2d, which is not the H0 dynamics. As a result, the claimed reduction of the closed-loop system to H0 is invalid and the if-direction of Theorem 1 is not established as written.
[§3, Theorem 1, same proof] The target set Ad is printed as {−A^{-1}d} × {0} × {0} × [0,T2], which omits the controller-state equilibrium ξbar. Unless ξbar = 0, the actual closed-loop equilibrium (xp,ξ,q) = (−A^{-1}d, ξbar, 0) is not contained in Ad, so global exponential convergence to Ad cannot hold. The set should be {−A^{-1}d} × {ξbar} × {0} × [0,T2]. With the corrected coordinate shift, this set is exactly the image of A0 under the shift, and the rest of the sufficiency argument goes through.
[§3, Theorem 1, first paragraph of sufficiency proof] The fixed-point selection used to prove that A is nonsingular contains a typo: the equation 'Gξ = (I_nc −L)ξ' should read 'Gx = (I_nc −L)ξ', equivalently (L−I)ξ = −Gx together with Kξ = −Rx. As printed, the first equation does not couple x and ξ in the needed way and cannot be justified from the rank condition (6). This is a load-bearing step because it is used to construct a nonzero point in ker(F) ∩ ker(I−J) that contradicts the GES of A0.

minor comments (5)

[§5, numerical example] The phrase 'initial condition 2xp(0,0) = (10,1)' appears to contain a stray factor 2; presumably it should read xp(0,0) = (10,1).
[§3, after Eq. (8)] The word 'combinining' should be 'combining'.
[§3, Theorem 1, sufficiency proof] The symbol u in z̃ = (xp − A^{-1}d, ξ − ξbar, u) denotes the controller output q, not the control input u; using u here is confusing and should be corrected to q.
[§4, Proposition 2 and Remark 3] There are typographical errors such as 'Theroem 1' in Remark 3, and the displayed equation (11) contains malformed LaTeX artifacts that make the matrix hard to read; the display should be cleaned up.
[§4, Proposition 3 proof] The proof would be easier to follow if it explicitly stated that observability of (L,K) follows from nonsingularity of Λ − I in the same way as in the proof of Proposition 2, since Theorem 1 requires this assumption.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main theorem is a genuine reduction to the zero-disturbance problem plus an algebraic rank condition.

full rationale

The derivation chain is self-contained. Theorem 1 states an equivalence between (i) the rank condition (6) and (ii) global exponential stability of A0 for the zero-disturbance system H0, on one hand, and solvability of Problem 1 for the uncertain affine plant with unknown d, on the other. These are distinct mathematical objects: the rank condition is an algebraic solvability condition ensuring that, for every d, there exists a controller-state shift xi-bar satisfying (L-I)xi-bar = G A^{-1} d and K xi-bar = R A^{-1} d, while condition (ii) concerns a different hybrid system (d = 0) and a different target set. Neither condition is defined in terms of the target result, and the proof actually constructs the required set Ad from the data. The subsequent design results (Propositions 2 and 3) verify conditions (i) and (ii) using SOS/LMI feasibility and then invoke Theorem 1, so the controller guarantee is not assumed as an input. All cited external results (Goebel et al. 2012 for hybrid systems, Teel et al. 2012 for Lyapunov conditions, Briat 2013 for sampled-data LMIs, and Takimoto/Yamamoto 2007 for the washout controller structure) are standard and do not overlap with the present authors' prior work; none is used as a self-supporting uniqueness or existence theorem. The apparent sign typos in the sufficiency proof of Theorem 1 (the coordinate shift x_p - A^{-1}d rather than x_p + A^{-1}d, and the omission of xi-bar from the displayed set Ad) are proof-correctness issues, not circularity: the intended coordinate change is constructed from the problem data, and the target set is not assumed in the hypothesis. Therefore no step in the paper reduces, by definition or by self-citation, to its own inputs.

Assumptions & free parameters 1 free parameters · 7 assumptions · 0 invented entities

No new physical entities are introduced. The clock variable tau and the memory states (xi, q) are standard hybrid-modeling variables rather than invented entities. The only hand-tuned parameter is the polynomial degree g in the SOS relaxation.

free parameters (1)

Polynomial degree g of W(tau) = 1, 2, 3, 4, 5 in Table 1; g = 4 used in the example
The user must choose the degree of the polynomial matrix W(tau); higher degrees improve feasibility for T2. This is a hand-tuned parameter of the SOS relaxation, not derived from plant data.

assumptions (7)

domain assumption Assumption 1: sampling intervals satisfy 0 <= t1 <= T2 and T1 <= t_{k+1} - t_k <= T2.
This is the aperiodic sampling model; without these uniform bounds, the hybrid timer model and the exponential stability analysis do not apply.
domain assumption Assumption 2: matrix B has full column rank.
Used in Lemma 1 to conclude from B q = 0 that the held control q is zero on the omega-limit set. The authors note redundant columns can be removed.
domain assumption Assumption 3: the matrix A is nonsingular for all admissible uncertainties, so a unique unforced equilibrium exists for every d.
The problem is otherwise unsolvable; the theorem later derives nonsingularity from its own conditions.
domain assumption Assumption 4: the uncertainty is norm-bounded, A = A0 + D Delta E, B = B0 + D Delta F with Delta^T Delta <= I.
Standard robust control parameterization, used in the LMI derivations of Propositions 2 and 3.
domain assumption The pair (L,K) is observable.
Needed for the necessity direction in Lemma 1 and Proposition 1; the authors argue it can always be enforced by reducing the controller order or is automatic in the proposed parameterization.
standard math The hybrid Lyapunov theorem of Teel, Forni and Zaccarian (2012) and the hybrid systems framework of Goebel et al. (2012).
Used to conclude global exponential stability from Lyapunov inequalities and to characterize omega-limit sets.
domain assumption SOSTOOLS and SeDuMi can verify the polynomial matrix inequalities arising from the SOS relaxation.
The design algorithm treats numerical feasibility of the SOS conditions as a stability certificate, with no formal guarantee of solver success or numerical conditioning.

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Cite this review

Pith. "Pith review of Robust Aperiodic Sampled-Data Washout Control for Uncertain Affine Systems." pith.science (2026). https://pith.science/paper/IPBX2IMW

@misc{pith2026250523534,
  author       = {Pith},
  title        = {Pith review of: Robust Aperiodic Sampled-Data Washout Control for Uncertain Affine Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IPBX2IMW}},
  note         = {Machine review of arXiv:2505.23534}
}

read the original abstract

In this paper, we address the problem of designing an aperiodic sampled-data controller stabilizing the zero-input equilibrium of an uncertain affine plant. The closed-loop system is modeled as a hybrid dynamical system incorporating a timer triggering the occurrence of the sampling events and two memory states storing the value of the controller state and controller output at each sampling time. Necessary and sufficient conditions on the controller parameters are given to establish the sought property. A constructive controller design algorithm based on sum-of-squares programming is given. A numerical example illustrates the effectiveness of the approach.

Figures

Figures reproduced from arXiv: 2505.23534 by the authors.

**Figure 2.** Evolution of the control input u [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

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Reference graph

Works this paper leans on

27 extracted references · 23 canonical work pages

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ENTRY address author booktitle chapter doi edition editor eid howpublished institution journal key month note number organization pages publisher school series title type url volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sent...
[2]

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[3]

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ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type url volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence := ...
[4]

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Pith tools

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

[1] [1]

, " * write output.state after.block = add.period write newline

ENTRY address author booktitle chapter doi edition editor eid howpublished institution journal key month note number organization pages publisher school series title type url volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sent...

[2] [2]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

[3] [3]

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ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type url volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence := ...

[4] [4]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize ":" * " " *...

[5] [5]

and Yoon, S.Y

Arthur, K.M. and Yoon, S.Y. (2020). Robust stabilization at uncertain equilibrium by output derivative feedback control. ISA transactions, 107, 40--51

work page 2020

[6] [6]

and Yoon, S.Y

Arthur, K.M. and Yoon, S.Y. (2021). Robust compensation of uncertain equilibrium in the control of dynamic systems. In 2021 American Control Conference (ACC), 1119--1124

work page 2021

[7] [7]

Bazanella, A.S., Kokotovic, P.V., and Silva, A.S. (1997). On the control of dynamic systems with unknown operating point. In Proceedings of the European Control Conference 1997, 3434--3439

work page 1997

[8] [8]

Bhasin, S., Kamalapurkar, R., Dinh, H.T., and Dixon, W.E. (2013). Robust identification-based state derivative estimation for nonlinear systems. IEEE Transactions on Automatic Control, 58(1), 187--192

work page 2013