The Silver Ratio and its Relation to Controllability

Michael Muehlebach

REVIEW 2 major objections 4 minor 12 references

The Silver Ratio and its Relation to Controllability

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For two coupled pendulums, the silver ratio gives the widest reachable state space.

desk verdict A correct short calculation whose headline claim is coordinate-dependent: the silver-ratio optimum in the paper's coordinates becomes (√13−2)/3 in modal coordinates, so the rule needs a strong qualifier. read the letter →

arxiv 1908.07109 v1 pith:JWK3M4DH submitted 2019-08-19 eess.SY cs.SY

Michael Muehlebach This is my paper · ORCID

classification eess.SYcs.SY MSC 93B0593C15

keywords silverratiocontrollabilityinvertedpendulumreachablesetunitenergytimeconstantsreactionwheellinearsystems

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 note asks which ratio of time constants makes two unstable second-order systems, coupled through one common input, easiest to control. Controllability is measured as the volume of states that can be reached and returned to the origin with unit control energy, a definition that extends ordinary reachability to unstable dynamics. The author derives that volume in closed form in the chosen coordinates and finds that it is maximized when the time constants are in the silver ratio, the larger being $1+\sqrt{2}$ times the smaller. That gives a concrete design target for inverted-pendulum systems of this type, such as reaction-wheel pendula. The result is conditional on the input coefficients being independent of the time constants and on the coordinate choice.

What carries the argument

The load-bearing construction is the set $X$ in (3), the unit-energy reachable set for an unstable system; the author computes its volume by the coordinate change $T_i$ in (10), which decouples each unstable second-order equation into a stable and an unstable first-order mode, reducing the problem to two ellipsoidal constraints described by the matrix $P$ in (13). The determinant of the resulting quadratic form yields the scalar function $\epsilon(1-\epsilon)/(1+\epsilon)$, and the maximizer of that concave function is $1/\delta_s=\sqrt{2}-1$. The definition of $X$ itself, following the balanced-realization treatment of unstable systems, is what makes the volume finite and computable.

What would settle it

For a reaction-wheel or dual-pendulum setup with inertia ratio $(1+\sqrt{2})^2$, measure the unit-energy reachable volume (or the minimal energy needed to reach a fixed state) and compare with nearby ratios; if the maximum is not at the silver-ratio value, the coordinate or input-coefficient assumptions, not the algebra, would be at fault.

Watch

Extended reading notes

Core claim

The paper derives, in closed form, the volume of the set of states from which two unstable second-order systems (1)-(2) can be driven to the origin and from which the origin can be reached with a single unit-energy input. Using the transformation (10) that splits each second-order mode into stable and unstable first-order modes, the author shows the volume is proportional to $\left(\frac{\pi_1\pi_2(\pi_1-\pi_2)}{4(\pi_1+\pi_2)}\right)^2$ for $0<\pi_1\le \pi_2$, i.e. $\left(\frac{\pi_2^2}{4}\cdot\frac{\epsilon(1-\epsilon)}{1+\epsilon}\right)^2$ with $\epsilon=\pi_1/\pi_2$. The factor $\epsilon(1-\epsilon)/(1+\epsilon)$ is concave on $(0,1]$, so its unique maximizer is $\epsilon^*=\sqrt{2}-1$. The claim is therefore that the silver ratio $\delta_s=1+\sqrt{2}$ maximizes this measure of controllability; in the example rigid body, this corresponds to an inertia ratio $(1+\sqrt{2})^2$.

Load-bearing premise

The silver-ratio optimum assumes the input coefficients $v_1$ and $v_2$ stay fixed while the time constants $\pi_1$ and $\pi_2$ are varied, and it is expressed in the coordinate system chosen in the paper.

Editorial extensions

If this is right

For two unstable second-order modes driven by a common scalar input, unit-energy controllability is maximized when the time constants are in the silver ratio $\pi_2/\pi_1 = 1+\sqrt{2}$.
In the balanced rigid-body example, the corresponding optimal inertia ratio is $I_1/I_2 = (1+\sqrt{2})^2$.
The optimal ratio is independent of the absolute time-constant scale and of the input gains $v_1, v_2$ as long as those gains are constant.
The reachable volume scales as $v_1^2 v_2^2 \pi_2^4$, so stronger constant input coupling and a larger overall time constant enlarge the reachable set without changing the optimal ratio.

Reading between the lines

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

If the input coefficients inherit time-constant dependence (for example $v_i$ proportional to $1/\pi_i^2$), the objective gains new $\pi$-dependence and the optimal ratio will shift off $1+\sqrt{2}$; mapping those cases would delimit when the silver-ratio rule is valid.
The same determinant calculation for $N$ coupled unstable modes would produce a rational function of $N-1$ ratios; identifying its maximizer is a natural algebraic extension that may yield other metallic ratios.
Because the volume measure is coordinate-dependent, the practical design rule should be re-derived in the actuator and sensor coordinates of a specific robot; the silver ratio is a canonical-coordinate rule, not a physical invariant.

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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

Colleague,

The one thing to know: this is a correct, short calculation, and the silver-ratio connection is new, but the headline rule is coordinate-dependent in a way that the abstract and conclusion do not disclose. In the coordinates used in the paper (x_i, xdot_i), the reachable set volume is proportional to (π1 π2 (π1−π2)/(π1+π2))^2, which maximizes at π1/π2 = √2−1. But the same calculation in modal coordinates q_i = (xdot_i + π_i x_i, xdot_i − π_i x_i) gives a determinant proportional to π1^3 π2^3 (π1−π2)^2/(π1+π2)^2, and the optimum ratio moves to (√13−2)/3 ≈ 0.535. So the silver ratio is an artifact of the chosen state scaling. Section I does admit coordinate dependence, but the abstract, conclusion, and the inertia recommendation in (22) present the rule without the qualifier. That mismatch is the main problem.

What the paper does well: the derivation is transparent, the calculus is right, and the connection to the golden-ratio/Fibonacci literature is real and not overclaimed. There are no fitted parameters and no postdiction. The v_i-independence footnote is actually satisfied in the pendulum example—v_i = ±1/(m g0) does not depend on I_i. The stable/unstable decomposition in (12) is imported from [2] without proof; it is standard, so this is a presentation gap rather than a correctness gap. More important, the example only maps to the single-input form (1)-(2) if the two components of the applied force are linearly dependent; that condition is asserted but not verified for the systems named in the introduction, and it is not a generic property of a single force in 3D.

Who is this for? A reader interested in numerical coincidences in control might enjoy it, and it is a useful object lesson in how controllability metrics change under coordinate transformations. It deserves a serious referee because the mathematics is honest and correct, but the revision should qualify every claim of 'maximized at the silver ratio' with 'in the canonical physical coordinates used here' and either justify those coordinates as the right ones for the design objective or reduce the strength of the design rule. I would not cite the silver-ratio rule as a design principle, but I might cite the paper as a clean example of coordinate dependence.

Recommendation: send to peer review with requests for revision, not desk reject.

Referee Report

2 major / 4 minor

Summary. The note studies two unstable second-order systems coupled through a common input and measures controllability by the volume of the state-space reachable with unit energy while the trajectory starts and returns to the origin. In the normalized coordinates defined by Eq. (10), the volume is reduced to Eq. (15), whose maximizer over the time-constant ratio is the inverse silver ratio. The paper then recommends the silver-ratio inertia ratio for an inverted-pendulum example and concludes that controllability is maximized when the time-constant ratio equals the silver ratio.

Significance. The algebraic core of the paper is correct: the transformation (10), the dynamics (11), and the volume computation (14) are internally consistent, and the maximization of ϵ(1−ϵ)/(1+ϵ) is elementary and correct. The paper also has the virtue of being fully closed-form, with no fitted parameters or postdicted predictions. However, the volume criterion is not coordinate-invariant, and the paper itself concedes this in Section I. The abstract and conclusion nonetheless present the silver-ratio rule without that qualifier, and the design recommendation in Eq. (22) inherits the coordinate dependence. This is a load-bearing issue for the advertised message, not a mere presentation detail.

major comments (2)

[Section IV, Eqs. (14)–(18); Section I] The optimal ratio ϵ*=√2−1 is not invariant under changes of state coordinates, despite being presented without qualification in the abstract and conclusion. The paper states in Section I that all results depend on the choice of coordinates, but Eq. (17) and the design rule Eq. (22) are phrased as intrinsic properties. The coordinate transformation (10) is not orthogonal, so the Lebesgue measure used for the volume of X changes under reparameterization. For example, in the modal coordinates q_i=(ẋ_i+π_i x_i, ẋ_i−π_i x_i), the volume of the reachable set expressed in those coordinates is proportional to π1^3π2^3(π1−π2)^2/(π1+π2)^2; with π2 fixed, the ε-dependent factor is ε^3(1−ε)^2/(1+ε)^2, whose maximizer satisfies 3ε^2+4ε−3=0, i.e., ε=(√13−2)/3≈0.535, not √2−1. Thus the silver-ratio result is a property of the chosen state realization (10), not of the physical system, and the claims in the abstract, Section VI, and Eq. (22) must be explicitly scoped to those coordinates or supported by a physical argument for why those coordinates are the correct ones for measuring controllability volume.
[Section V, Eqs. (20)–(22)] The recommended inertia ratio I1/I2=δs² depends on two additional restrictions that the example does not establish. The first is the footnote-1 assumption that v1 and v2 do not depend on π1 and π2, which does hold for the particular example (20)–(21). The second is the statement immediately before Eq. (22) that IF1(t) and IF2(t) are linearly dependent, which reduces the two-axis dynamics to the single-input form (1)–(2). In a general inverted-pendulum system these two force components are not automatically linearly dependent, and the optimal inertia ratio would be different if they are not. The sentence presenting Eq. (22) should therefore state both conditions explicitly and should not be read as a recommendation for the unconstrained two-axis system.

minor comments (4)

[Section III, Eq. (9)] The indices in the displayed inequality appear to be swapped: the term describing the ξ1 direction should use β1², and the term describing the ξ2 direction should use β2².
[Section V] The phrase “More concrete examples of such cases cases include” contains a duplicated word “cases”.
[Sections I–IV] The quantities π1 and π2 are eigenvalues with units of inverse time, so calling them “time constants” is imprecise; the time constants are 1/π1 and 1/π2. Please adjust the terminology for consistency with the dynamics in (1)–(2).
[Section IV, Eq. (12)] Equation (12) uses T before T:=diag(T1,T2) is defined; reorder the definitions so that T is introduced before it appears in the formula for X.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the silver-ratio optimum is the maximizer of a derived closed-form volume formula, with no fitted parameters or self-referential load-bearing steps.

full rationale

The paper's central claim is that, for the system (1)-(2) with v1=v2=1, the reachable-set volume is proportional to (π2^2/4 · ε(1−ε)/(1+ε))^2 (Eqs. 14-15), and that this expression is maximized at ε = √2−1, yielding the silver ratio (Eqs. 17-18). The derivation starts from the standard controllability/reachability ellipsoid result (Eq. 6, citing Callier and Desoer) and the stable/unstable decomposition (Eqs. 9-13), then applies determinant algebra to get the volume formula. The optimization of ε(1−ε)/(1+ε) is a self-contained calculus step; no parameter is fitted to data, no quantity is postdicted, and the cited external results do not contain the silver-ratio conclusion. The footnote-1 assumption that v1 and v2 do not depend on π1 and π2 is an explicit modeling assumption, not a circular definition, and it is stated before the volume expression is maximized. The acknowledged coordinate dependence of the volume measure is a mathematical caveat about the meaningfulness of the result, not a circularity: the paper explicitly says the results depend on the choice of coordinates in Section I, and the advertised claim is qualified as holding in a canonical set of coordinates. Therefore, the derivation chain is not equivalent to its inputs by construction, and there is no self-citation chain that forces the result.

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

The central derivation is a transparent linear algebra computation once the standard reachable-ellipsoid theorem and the stable/unstable decomposition are accepted. There are no fitted parameters and no new entities; the only non-derived inputs are two standard control theory theorems, the v_i independence assumption, and the coordinate system choice.

assumptions (5)

standard math The set of states reachable with unit energy for a stable linear system is the ellipsoid defined by the controllability gramian (Eq. 6).
Invoked in Section III before Eq. (6); cited to Callier and Desoer [10, Ch.8]. This is a standard linear systems result, not derived in the note.
standard math For the mixed stable/unstable system, the reachable and controllable set X is the superposition of stable and unstable subspace ellipsoids with the same gramian P (Eq. 12).
Invoked in Section IV to write Eq. (12); cited to Zhou, Salomon, Wu [2]. This external result is load-bearing for the volume formula but is not reproduced or numerically checked.
domain assumption The input coefficients v1 and v2 are independent of the time constants π1 and π2 during optimization.
Stated in footnote 1 of Section IV. If v_i depend on π_i, the optimum of Eq. (16) shifts and the silver-ratio claim fails.
ad hoc to paper The volume of X is measured in the canonical coordinates (x1, xdot1, x2, xdot2) and is not coordinate-invariant.
Acknowledged in Section I (they do, however, depend on the choice of coordinates). The definition of X and its Euclidean volume in these coordinates is a modeling choice specific to this paper.
domain assumption The attitude dynamics linearized about the upright equilibrium take the form (20)-(21) with π_i² = lmg0/I_i.
Used in Section V to connect the abstract result to an inertia ratio. It assumes small angles, a diagonal inertia tensor, and a point force at distance l.

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Cite this review

Pith. "Pith review of The Silver Ratio and its Relation to Controllability." pith.science (2026). https://pith.science/paper/JWK3M4DH

@misc{pith2026190807109,
  author       = {Pith},
  title        = {Pith review of: The Silver Ratio and its Relation to Controllability},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JWK3M4DH}},
  note         = {Machine review of arXiv:1908.07109}
}

read the original abstract

This note investigates the controllability of two unstable second-order systems that are coupled through a common input. These dynamics occur for different types of inverted-pendulum systems. Controllability is quantified by the volume of the state-space that can be reached with unit energy, provided that the system starts and ends at the origin. It is shown that controllability is maximized when the ratio between the time constants amounts to the silver ratio.

Figures

Figures reproduced from arXiv: 1908.07109 by the authors.

**Figure 1.** Shown is the function (1 − )/(1 + ) for ∈ (0, 1], where = π1/π2 is the ratio between the two time constants. The function is concave for ∈ (0, 1] and attains its maximum at 1/δs. Thus, the function (1 − )/(1 + ), as plotted in [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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T. W. Barton, ``Stabilizing the dual inverted pendulum,'' IFAC Proceedings, vol. 42, no. 24, pp. 113--118, 2010

work page 2010

[2] [1]

3 I< HJ1䘸4eV XUm 7 Ni B rR҄SkkߏR9C Rk 5 p 2!q^SZH#^Ã,>8N[Pcʈo

11em plus .33em minus .07em @technote 4000 4000 100 4000 4000 500 `\.=1000 = #1 #1 #1 0pt [0pt][0pt] #1 * \| ** #1 \@IEEEauthorblockNstyle \@IEEEauthorblockAstyle \@IEEEauthordefaulttextstyle \@IEEEauthorblockconfadjspace -0.25em \@IEEEauthorblockNtopspace 0.0ex \@IEEEauthorblockAtopspace 0.0ex \@IEEEauthorblockNinterlinespace 2.6ex \@IEEEauthorblockAinte...

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work page 1999

[4] [4]

Benavoli, L

A. Benavoli, L. Chisci, and A. Farina, ``Fibonacci sequence, golden section, K alman filter and optimal control,'' Signal Processing, vol. 80, no. 8, pp. 1483--1488, 2009

work page 2009

[5] [5]

Capponi, A

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von Brasch, J

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Donoghue, ``State estimation and control of the F ibonacci system,'' Signal Processing, vol

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S. H. Marshall and D. R. Smith, ``Feedback, control, and the distribution of prime numbers,'' Mathematics Magazine, vol. 86, no. 3, pp. 189--203, 2013

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