Regulating Stability Margins in Symbiotic Control: A Low-Pass Filter Approach

Emre Yildirim; John T. Hrynuk; Tansel Yucelen

REVIEW 3 major objections 4 minor 11 references

Regulating Stability Margins in Symbiotic Control: A Low-Pass Filter Approach

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper claims that a low-pass-filtered fixed-gain control law regulates stability margins in symbiotic control, nearly tripling the delay margin at high gain with only a minor tracking cost.

desk verdict Solid Lyapunov work but the margin numbers in Table I rest on an unspecified loop transfer function; the practical claim is not reproducible as written. read the letter →

arxiv 2411.09881 v1 pith:J5AJIOMG submitted 2024-11-15 eess.SY cs.SY

Emre Yildirim , Tansel Yucelen , John T. Hrynuk This is my paper · ORCID

classification eess.SYcs.SY MSC 93C8093D0593C40

keywords symbioticcontrolstabilitymarginslow-passfilterfixed-gainadaptivedelaymargingainexogenousdisturbances

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

Symbiotic control couples a fixed-gain control with an adaptive learning law so that raising the fixed-gain parameter improves tracking without requiring prior knowledge of uncertainty bounds. The practical obstacle this paper targets is that a large fixed-gain parameter also shrinks the stability margins, leaving the closed loop vulnerable to time delays and gain variations. The paper proposes a new fixed-gain control law in which the control signal is drawn toward a low-pass filtered copy of itself, which curbs the aggressive behavior that erodes the margins. It proves that the resulting closed loop has bounded signals for bounded disturbances (Theorem 1) and, for constant disturbances with zero leakage, that the tracking error and fixed-gain control converge to zero (Theorem 2). Numerical examples show the delay margin at $\alpha=10$ rising from $0.0294$ to $0.0814$ with the filter, nearly three times the unfiltered value, while tracking performance changes only slightly.

What carries the argument

The load-bearing mechanism is the low-pass filter (11), $\dot{u}_{fl}=-\epsilon_2(u_{fl}-u_f)-\mu_2 u_{fl}$, with time constant $\tau=1/(\epsilon_2+\mu_2)$ and gain $K=\epsilon_2/(\epsilon_2+\mu_2)$, together with the extra integral term added in (10). The pair makes the fixed-gain control $u_f$ converge toward its own smoothed signal $u_{fl}$, limiting how aggressively $u_f$ can act. Lemma 1's equivalence between (10) and the dynamic form (12) is what lets the stability proof go through, because (12) puts $u_f$ into a form where the quadratic energy function (16) and the matrix conditions on (17)-(18) apply; Theorem 1 then yields boundedness, and Theorem 2 uses the same structure to get convergence. In the examples, the delay and gain margins are obtained from the loop transfer function broken at the control input, and the filter parameters $\epsilon_1,\epsilon_2$ are the tuning knobs that trade margins against the quadratic tracking cost.

What would settle it

Simulate the full nonlinear closed loop (1)-(3), (7), (10)-(11) with a time delay inserted at the control input and record the smallest delay that destabilizes it; if this delay disagrees with the linear delay-margin values in Table I, then the paper's robustness claim does not hold for the actual nonlinear system.

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Extended reading notes

Core claim

The paper's central claim is that replacing the standard fixed-gain law (5) with the filtered law (10)-(11) regulates the stability margins of symbiotic control without substantially degrading tracking. The new law adds the term $-\alpha\epsilon_1\int_0^t (u_f(s)-u_{fl}(s))\,ds$ to the standard fixed-gain control and defines $u_{fl}$ by the low-pass filter (11), so that $u_f$ is encouraged to track its filtered version rather than reacting aggressively. Lemma 1 rewrites the law in the equivalent dynamic form (12), which makes the stability analysis possible: with the quadratic energy function (16), Theorem 1 gives an exponential bound on the closed-loop signals, and Theorem 2 shows $\lim_{t\to\infty}(e(t),u_f(t))=(0,0)$ for constant disturbances when both leakage parameters are zero. The numerical study compares gain and delay margins read from the loop transfer function, reporting that the filtered law keeps better margins at all tested $\alpha$ and, at $\alpha=10$, nearly triples the delay margin ($0.0814$ versus $0.0294$) with a modest increase in the quadratic tracking cost. The fixed-gain core is linear, so the paper states the same architecture can be dropped into the nonlinear symbiotic control framework unchanged.

Load-bearing premise

The margin numbers come from a linear frequency-response analysis of the control loop, and the paper assumes this analysis also describes the robustness of the full closed loop, which contains the nonlinear adaptive law (7).

Editorial extensions

If this is right

At a given $\alpha$, the filtered law yields larger gain and delay margins than the standard law, with the gap growing as $\alpha$ increases; the reported delay margin at $\alpha=10$ is $0.0814$ versus $0.0294$.
Operators can recover robustness lost by turning up $\alpha$: for a fixed quadratic tracking cost, different $(\epsilon_1,\epsilon_2)$ pairs give markedly different margins, so the filter parameters can be chosen to meet a margin target.
The stability guarantees hold for bounded exogenous disturbances with bounded rate of change, and for constant disturbances with no leakage the tracking error and fixed-gain control go to zero.
Because the fixed-gain core is linear, the new law transfers unchanged into the nonlinear symbiotic control setting of [5], so the margin regulation benefits are not confined to the linear disturbance problem.

Reading between the lines

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

Inference: the loop transfer function used to compute the margins is never written out explicitly; deriving it and testing the margin predictions on plants beyond the double integrator would tell whether the near-tripling in delay margin is a general property of the filter or an artifact of the example.
Inference: a testable extension is an automatic tuning rule for $(\epsilon_1,\epsilon_2)$ that maximizes delay margin subject to a tracking-cost budget, since Figure 6 suggests the margins vary non-monotonically with the parameters at fixed cost.
Inference: the filter's smoothing of $u_f$ may also reduce actuator slew and chatter during the adaptive transient, a side effect the paper does not quantify but that would matter in physical systems.
Inference: the assumption that linear frequency-response margins predict the robustness of the full nonlinear closed loop is the paper's least tested point; a time-delay simulation of the complete system (1)-(3), (7), (10)-(11) would directly check it.

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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

Here's my take on arXiv:2411.09881. The stability theory is fine, but the paper's main practical claim—that the low-pass filter raises delay margins nearly threefold at α=10—is not verifiable from the text because the loop transfer function is never written down. That is a real problem, not a nitpick.

What is actually new: the control law (10)-(11) is a genuinely new modification to the symbiotic control framework. Prior low-pass filtering in [6] and [7] targeted high-frequency oscillations, not stability margins, so the architecture is incremental but not derivative. The Lyapunov proofs in Theorems 1 and 2 are standard, and I checked the algebra: the boundedness and convergence conclusions follow from the stated assumptions. The numerical examples illustrate the expected tradeoff between tracking performance and margins, and the figures look consistent with the theory. The paper is also honest that the margin analysis is carried out in a linear disturbance setting rather than the full nonlinear framework.

Where it gets soft. The paper never specifies the loop transfer function used to compute the gain and delay margins in Tables I-III. It just says "broken at the control input." But the control input includes the adaptive term (6)-(7), which is nonlinear. A meaningful linear margin analysis requires a precise definition of what is in the loop: is the adaptive dynamics linearized about the equilibrium? Is it excluded? The paper gives no L(s), no state-space realization of the loop, nothing. I tried to derive a plausible loop from (1)-(5) and (10)-(11), and the numbers don't match the tabulated values; for the fixed-gain-only loop at α=10 I get a delay margin around 0.14 s, not 0.0294 s. The stress-test note says the same. So the central quantitative claim is not reproducible, and the paper offers no way to tell which of several possible loops produces Table I. This matters because the theorems prove stability but not any margin property; the practical contribution rests entirely on this unverifiable linear analysis.

A second, smaller issue: footnote 2 claims the filter carries over unchanged to the nonlinear symbiotic framework of [5], but gives no argument beyond "the linear form" of the fixed-gain law. That may be true, but it is asserted, not shown.

Overall, the idea is plausible and the stability backbone is solid. The missing loop transfer function is a fixable but real defect. If the authors provide the derivation and specify how the adaptive loop is treated, the paper would be a useful contribution to the adaptive control subfield. As it stands, the margin numbers are not something I would rely on or cite.

Recommendation: send it to peer review—the stability results alone justify a referee's time—but the review must require the explicit loop transfer function and a reproducible margin computation before acceptance.

Referee Report

3 major / 4 minor

Summary. The paper proposes a low-pass-filter modification of the fixed-gain component in the recently proposed 'symbiotic control' framework, replacing the standard fixed-gain law (5) with the new law (10)-(11), equivalently the first-order ODE (12). The authors prove boundedness of all closed-loop signals for bounded matched disturbances with leakage (Theorem 1) and asymptotic convergence of the error and fixed-gain signals to zero for constant disturbances with zero leakage (Theorem 2). The claimed contribution, regulation of stability margins, is supported by two numerical examples in which gain and delay margins of the standard and new fixed-gain laws are tabulated (Tables I-III) for a double-integrator plant. The paper argues that the filter limits the aggressive behavior of the fixed-gain control and thereby improves margins with only a minor loss of tracking performance.

Significance. If the margin data are reproducible, the contribution is practically valuable: a simple, fixed-gain modification that roughly triples the delay margin at high alpha with a modest performance penalty. The Lyapunov analysis in Theorems 1-2 is standard, self-contained, and algebraically consistent, and the paper is explicit about the design parameters (alpha, epsilon1, epsilon2) rather than fitting them to data. However, the paper's central advertised result--stability-margin regulation--is supported only by a linear loop analysis whose transfer function is never written down. The stability theorems prove boundedness and convergence but no margin property, so the practical claim currently rests on an unverifiable numerical computation. The manuscript also contains an internal inconsistency in the stated value of beta4, and the claimed extension to the nonlinear framework of [5] is asserted rather than proved.

major comments (3)

[Section IV and Tables I-III] The loop transfer function whose margins are tabulated is never defined. The phrase 'loop transfer function (broken at the control input)' is not a specification: the reader cannot determine whether the loop includes the adaptive law (7), whether (7) is linearized about the tracking equilibrium, or which signal is broken. Reproducing Tables I-III from (1)-(11) with or without the adaptive loop requires a concrete L(s), and different natural choices produce substantially different margins. Since Theorems 1-2 establish only boundedness and convergence, the paper's claim that the low-pass filter 'regulates stability margins' rests entirely on this unverifiable linear analysis. Please write out L(s) explicitly for both the standard and new fixed-gain laws, state the treatment of the adaptive law in the loop, and provide the numerical data or code used to generate Tables I-III.
[Section III, Theorem 1] Theorem 1 as stated does not explicitly require a positive leakage parameter, yet its proof needs l2 = 2*mu1 - mu1*d1 - d2 > 0, which is impossible when mu1 = 0 for any positive d1, d2. Thus the boundedness statement is not proved for the zero-leakage case. Please state the intended assumption on mu1 explicitly; if mu1 = 0 is meant to be covered for time-varying disturbances, a separate argument is needed.
[Section IV, parameter selection] The numerical example lists 'beta4 = 0.1' as a learning parameter, but beta4 is defined in (16) as beta4 = epsilon2^{-1} * alpha * beta2 * epsilon1, and the proof of Theorem 1 relies on this relation to obtain the quadratic form (24)-(25). With the example's beta2 = 3, epsilon1 = 3, epsilon2 = 10, the definition gives beta4 = 0.9*alpha, i.e., 0.9, 4.5, and 9 for alpha = 1, 5, 10. The example therefore does not implement the analyzed controller as written. Please correct the stated value of beta4 or explain why the cancellation in (24)-(25) is unaffected by the discrepancy.

minor comments (4)

[Section III, after (25)] The sentence 'where (24) can equivalently be rewritten as' is inaccurate: equation (25) is an upper bound, not an equivalent form, because the term -lambda(M2)||ufl||^2 discards the positive contribution of uf in the quadratic form -varrho^T M2 varrho.
[Section IV, Table I] Table I appears to have a dangling '1' after the Delay row; please clean up the table formatting.
[Section IV, margin remarks] The statement that 'a constant or time-varying disturbance does not affect the computation of stability margins' is only transparent if the loop transfer function excludes the adaptive law (7); if the adaptive law is included, the disturbance enters (7) and can affect the operating point about which a linearization would be taken. Please clarify how the margin computation treats this.
[Abstract/Introduction] The term 'stability margins' is used without specifying which margins (gain, phase, delay) until Section IV; a brief definition or pointer early in the paper would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the stability proof is self-contained, and the margin tables are design demonstrations rather than predictions fitted to data.

full rationale

The derivation chain in Section III is self-contained. Lemma 1 is an algebraic equivalence between the implementable control law (10) and the analysis form (12), illustrated by direct substitution of (13); it does not assume the margin claims. Theorems 1 and 2 prove boundedness of all closed-loop signals and asymptotic convergence using a Lyapunov function with Young's inequalities and matrix positive-definiteness conditions; neither theorem relies on the numerical margin results. The stability-margin tables in Section IV are computed after choosing the filter parameters α, ε1, and ε2, so the parameters are design choices and the margins are presented as consequences of the design, not quantities used to define the law. No fitted parameter is renamed as a prediction, and no data subset is used to force a later claim. The paper does omit the explicit loop transfer function used for Tables I-III, which is a reproducibility defect and a correctness risk, but it is not a circular reduction because the paper never defines the control law in terms of the tabulated margins. The self-citations to [5]-[8] provide background and the transferability claim in footnote 2 is an assertion about extending the linear fixed-gain structure to the nonlinear setting, not a load-bearing theorem whose conclusion is imported from the same authors without independent content. No uniqueness theorem is invoked to forbid alternatives, and the central stability result is proved within the paper. Under the hard rule that circularity must be exhibited as an equation reducing to an input or a fitted parameter renamed as a prediction, no such step is present.

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

The central claim is supported by Lyapunov proofs and simulations; the proofs rest on standard stability theorems, and the simulations use user-selected gains. No parameters are fitted to force the margin results, so the circularity burden is low.

free parameters (3)

α (fixed-gain parameter) = 1, 5, 10 in examples
User-selected design parameter; central to the gain-margin tradeoff.
ε1 (filter penalty parameter) = 3 in Example 1; 1-7 in Example 2
Chosen to balance margins and performance; not fitted.
ε2 (filter bandwidth parameter) = 10 in Example 1; 1.5-30 in Example 2
Chosen to balance margins and performance; not fitted.

assumptions (6)

standard math Lyapunov stability theorem and LaSalle-Yoshizawa theorem
Used in the proofs of Theorems 1 and 2.
standard math Young's inequality
Used to bound disturbance terms in equations (22)-(23).
domain assumption (A,B) stabilizable and B full column rank
Required for state-feedback stabilization and pseudo-inverse B_i in (5).
domain assumption Disturbance bounded with bounded time derivative
Needed for the bound l* in Theorem 1.
ad hoc to paper Sufficiently large β2 α for positive-definiteness of M3 and M4
Assumed to ensure the Lyapunov matrices are positive definite in Theorems 1 and 2.
ad hoc to paper μ2 sufficiently small to keep low-pass filter gain near unity
Stated after (11) so the filter does not force u_fl to zero.

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Cite this review

Pith. "Pith review of Regulating Stability Margins in Symbiotic Control: A Low-Pass Filter Approach." pith.science (2026). https://pith.science/paper/J5AJIOMG

@misc{pith2026241109881,
  author       = {Pith},
  title        = {Pith review of: Regulating Stability Margins in Symbiotic Control: A Low-Pass Filter Approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J5AJIOMG}},
  note         = {Machine review of arXiv:2411.09881}
}

read the original abstract

Symbiotic control synergistically integrates fixed-gain control and adaptive learning architectures to mitigate system uncertainties more predictably than adaptive learning alone and without requiring prior knowledge of uncertainty bounds as compared to fixed-gain control alone. Specifically, increasing the fixed-gain control parameter achieves a desired level of closed-loop system performance while the adaptive law simultaneously learns and suppresses the system uncertainties. However, stability margins can be reduced when this parameter is large and this paper aims to address this practical challenge. To this end, we propose a new fixed-gain control architecture predicated on a low-pass filter approach to regulate stability margins in the symbiotic control framework. In addition to the presented system-theoretical results focusing on the stability of the closed-loop system, we provide two illustrative numerical examples to demonstrate how the low-pass filter parameters are chosen for the stability margin regulation problem without significantly compromising the closed-loop system performance.

Figures

Figures reproduced from arXiv: 2411.09881 by the authors.

**Figure 2.** Closed-loop system performances with the symbiotic control [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Closed-loop system performances with the symbiotic control [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Closed-loop system performances with the symbiotic control [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: Delay margin based on ϵ1 and ϵ2 when α = 10. 0 20 40 60 80 100 0 0.5 1 1.5 2 2.5 3 106 0 5 10 15 20 25 30 0 0.5 1 1.5 2 104 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: Quadratic cost based on ϵ1 and ϵ2 when α = 10. where α = 10 is selected. The following observations are now immediate10: • Increasing ϵ1 has a diminishing impact on the delay margin when ϵ2 is fixed. Beyond a certain point, it almost does not have an effect on the dela…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

[6]

Symbiotic Control of Uncertain Dynamical Systems: Harnessing Synergy Between Fixed-Gain Control and Adaptive Learning Architectures

T. Yucelen, S. B. Sarsilmaz, and E. Yildirim, ``Symbiotic control of uncertain dynamical systems: Harnessing synergy between fixed-gain control and adaptive learning architectures,'' arXiv preprint arXiv:2403.19139, 2024

work page Pith review arXiv 2024
[7]

Yucelen and E

T. Yucelen and E. Johnson, ``A new command governor architecture for transient response shaping,'' International Journal of Adaptive Control and Signal Processing, vol. 27, no. 12, pp. 1065--1085, 2013

work page 2013
[5]

T. E. Gibson, A. M. Annaswamy, and E. Lavretsky, ``On adaptive control with closed-loop reference models: T ransients, oscillations, and peaking,'' IEEE Access, vol. 1, pp. 703--717, 2013

work page 2013
[2]

R. K. Yedavalli, Robust control of uncertain dynamic systems: A linear state space approach. 1em plus 0.5em minus 0.4em New York, NY, USA: Springer, 2014

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[3]

Kurtoglu, T

D. Kurtoglu, T. Yucelen, and D. M. Tran, ``Distributed control with time transformation: User-defined finite time convergence and beyond,'' in AIAA SciTech 2022 Forum, 2022, p. 1716

work page 2022
[4]

Yucelen and W

T. Yucelen and W. M. Haddad, ``Low-frequency learning and fast adaptation in model reference adaptive control,'' IEEE Transactions on Automatic Control, vol. 58, no. 4, pp. 1080--1085, 2013

work page 2013
[8]

Gruenwald and T

B. Gruenwald and T. Yucelen, ``On transient performance improvement of adaptive control architectures,'' International Journal of Control, vol. 88, no. 11, pp. 2305--2315, 2015

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Gruenwald, T

B. Gruenwald, T. Yucelen, and J. Muse, ``Direct uncertainty minimization framework for system performance improvement in model reference adaptive control,'' Machines, vol. 5, no. 1, 2017

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[10]

Yucelen, ``Model reference adaptive control,'' in Encyclopedia of Electrical and Electronics Engineering, J

T. Yucelen, ``Model reference adaptive control,'' in Encyclopedia of Electrical and Electronics Engineering, J. G. Webster, Ed. 1em plus 0.5em minus 0.4em Wiley Online Library, 2019, pp. 1--13

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [6]

Symbiotic Control of Uncertain Dynamical Systems: Harnessing Synergy Between Fixed-Gain Control and Adaptive Learning Architectures

T. Yucelen, S. B. Sarsilmaz, and E. Yildirim, ``Symbiotic control of uncertain dynamical systems: Harnessing synergy between fixed-gain control and adaptive learning architectures,'' arXiv preprint arXiv:2403.19139, 2024

work page Pith review arXiv 2024

[2] [7]

Yucelen and E

T. Yucelen and E. Johnson, ``A new command governor architecture for transient response shaping,'' International Journal of Adaptive Control and Signal Processing, vol. 27, no. 12, pp. 1065--1085, 2013

work page 2013

[3] [5]

T. E. Gibson, A. M. Annaswamy, and E. Lavretsky, ``On adaptive control with closed-loop reference models: T ransients, oscillations, and peaking,'' IEEE Access, vol. 1, pp. 703--717, 2013

work page 2013

[4] [2]

R. K. Yedavalli, Robust control of uncertain dynamic systems: A linear state space approach. 1em plus 0.5em minus 0.4em New York, NY, USA: Springer, 2014

work page 2014

[5] [3]

Kurtoglu, T

D. Kurtoglu, T. Yucelen, and D. M. Tran, ``Distributed control with time transformation: User-defined finite time convergence and beyond,'' in AIAA SciTech 2022 Forum, 2022, p. 1716

work page 2022

[6] [4]

Yucelen and W

T. Yucelen and W. M. Haddad, ``Low-frequency learning and fast adaptation in model reference adaptive control,'' IEEE Transactions on Automatic Control, vol. 58, no. 4, pp. 1080--1085, 2013

work page 2013

[7] [8]

Gruenwald and T

B. Gruenwald and T. Yucelen, ``On transient performance improvement of adaptive control architectures,'' International Journal of Control, vol. 88, no. 11, pp. 2305--2315, 2015

work page 2015

[8] [9]

Gruenwald, T

B. Gruenwald, T. Yucelen, and J. Muse, ``Direct uncertainty minimization framework for system performance improvement in model reference adaptive control,'' Machines, vol. 5, no. 1, 2017

work page 2017