On the gain of entrainment in stable linear control systems with a nonlinear output

Michael Margaliot; Ram Massas

arxiv: 2606.23149 · v1 · pith:VIYNSKVUnew · submitted 2026-06-22 · 🧮 math.OC

On the gain of entrainment in stable linear control systems with a nonlinear output

Ram Massas , Michael Margaliot This is my paper

Pith reviewed 2026-06-26 07:42 UTC · model grok-4.3

classification 🧮 math.OC

keywords gain of entrainmentnonlinear outputlinear control systemsperiodic inputsconvexityBregman divergenceentrainmentasymptotic stability

0 comments

The pith

Convex nonlinear outputs produce nonnegative gain of entrainment for any periodic input in stable linear systems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Stable linear control systems with nonlinear output maps can exhibit a nonzero gain of entrainment, where the time-average output under periodic forcing exceeds the output produced by the constant input that has the same mean value. This difference arises solely from the nonlinearity, since linear outputs always give zero gain. The paper shows that the sign of the gain is controlled by the curvature of the output map: convexity on the controllable subspace forces the gain to be nonnegative for every periodic input, while concavity forces it to be nonpositive. The gain also equals the average Bregman divergence between the entrained periodic orbit and the equilibrium point associated with the averaged input.

Core claim

In an asymptotically stable linear system the gain of entrainment for a static nonlinear output equals the difference between the time-average of the output along the unique periodic solution entrained by a T-periodic input and the value of the output at the equilibrium produced by the constant input equal to the average of that periodic input. For twice-differentiable outputs the leading-order term is determined by the Hessian of the output evaluated along the equilibrium trajectory. When the output is convex on the controllable subspace this difference is nonnegative for every periodic input and equals the average Bregman divergence between the entrained orbit and the averaged equilibrium.

What carries the argument

The average Bregman divergence between the entrained periodic orbit and the equilibrium for the averaged input, which encodes the mismatch created by nonlinearity and determines the sign of GOE.

If this is right

GOE is nonnegative for every periodic input whenever the output is convex on the controllable subspace.
For quadratic output maps, GOE admits explicit frequency-domain formulas that isolate the contribution of each input harmonic.
These formulas yield necessary and sufficient conditions on the output coefficients that fix the sign of GOE and identify the periodic input that maximizes GOE under an energy constraint.
The same conclusions apply to concrete models such as an RLC circuit and a compartmental pharmacodynamic system with nonlinear drug-effect map.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Engineers could deliberately shape the nonlinearity of a sensor or actuator map to amplify or suppress the average response to known periodic disturbances.
The Bregman-divergence representation may link entrainment analysis to convex optimization or information-geometric methods applied to dynamical systems.
Analogous sign results might hold for discrete-time linear systems or for systems whose state equations are themselves mildly nonlinear, though these cases lie outside the present analysis.

Load-bearing premise

The linear system is asymptotically stable, guaranteeing a unique entrained periodic orbit for any periodic input.

What would settle it

An asymptotically stable linear system together with a convex output map on its controllable subspace and a periodic input for which the numerically computed gain of entrainment is negative.

Figures

Figures reproduced from arXiv: 2606.23149 by Michael Margaliot, Ram Massas.

**Figure 1.** Figure 1: Gain of entrainment (GOE) in a system with a scalar output y = h(x, u). We compare the effect of two controls with the same average value. Left: u(t) = v(t) is a T-periodic control, γ v (t) is the corresponding globally attracting periodic trajectory, and y¯ is the average value of the output along γ v . Right: u(t) ≡ v¯, where v¯ := 1 T ∫ T 0 v(t) dt, is the corresponding constant control and e v¯ is the … view at source ↗

**Figure 2.** Figure 2: A series RLC circuit driven by a voltage source u(t). and h(z) = z ⊤Qz. Note that A is Hurwitz. Since Q is positive-definite, h is strictly convex and we conclude that GOE(v) > 0 for any T-periodic control v that generates a nonconstant periodic orbit γ v . In other words, if our goal is to maximize the average energy stored in the circuit then a nonconstant T-periodic voltage input is better than a consta… view at source ↗

**Figure 3.** Figure 3: Energy x⊤(t)Qx(t) in the RLC circuit as a function of time t for the control u(t) = 1 (solid line) and for the control u(t) = 1 + sin(ω0t) [dashed line]. -1 0 1 2 3 4 5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 4.** Figure 4: State variables x1(t) vs x2(t) in the RLC circuit for the control u(t) = 1 + sin(ω0t) and initial condition x(0) = 0. The value e v¯ = [ 2 0]⊤ is marked by x. the constant [periodic] control, h(x(t)) converges to the constant h(e v¯ ) = 1 [periodic orbit γ v ] and that the time average of h(γ v (t)) is larger than 1 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

read the original abstract

A control system admits a positive gain of entrainment (GOE) if entrainment to a periodic input yields a larger output, on average, than the output generated by the corresponding constant input with the same mean value. We analyze GOE in continuous-time stable linear control systems with a static nonlinear output map. Although linear systems with linear outputs have zero GOE, we show that a nonlinear output may generate a nontrivial GOE through the mismatch between the average output along the entrained periodic orbit and the output evaluated at the corresponding averaged equilibrium. We derive a second-order characterization of GOE for smooth output maps revealing that the leading-order contribution is determined by the curvature of the output map. We then show that if the output is convex (concave) on the controllable subspace, then GOE is nonnegative (nonpositive) for every periodic input. Furthermore, GOE admits a natural geometric interpretation as the average Bregman divergence between the entrained periodic orbit and the equilibrium associated with the averaged input. For the special case of quadratic output functions, we derive explicit frequency-domain formulas for GOE. These yield necessary and sufficient conditions guaranteeing the sign of GOE, characterize the contribution of individual input harmonics, and lead to an optimal periodic excitation that maximizes GOE under an energy constraint. The theoretical results are illustrated using an electrical RLC circuit and a compartmental pharmacodynamic model with a nonlinear drug-effect map.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper defines gain of entrainment for stable linear systems with nonlinear outputs and proves that convexity on the controllable subspace implies nonnegative GOE via a Bregman divergence argument.

read the letter

The key point is that this paper introduces the gain of entrainment (GOE) as a measure of how much a periodic input boosts the average output over its constant mean in stable linear systems with nonlinear readouts. It proves that if the output is convex on the controllable subspace, GOE is nonnegative for any periodic input, and interprets this as the average Bregman divergence between the periodic orbit and the averaged equilibrium.

What is new here is the GOE definition and its characterizations. The second-order curvature analysis, the convexity sign theorem, and the Bregman link are fresh. For quadratic outputs they also give frequency-domain formulas that break down the contribution of each harmonic and identify the best periodic input under power constraints. These build on standard linear system facts but apply them in a new way to nonlinear outputs.

The paper does this cleanly. The central argument rests on integrating the state equation over one period to show the average state matches the equilibrium of the mean input. Convexity then directly yields the inequality through the Bregman property. The stress test confirms this logic holds without circularity or extra assumptions beyond stability.

Soft spots are limited. The work is mostly theoretical, with the RLC circuit and pharmacodynamic model serving as quick illustrations rather than deep case studies. Readers might want more on how sensitive the results are to the stability margin or to small nonlinearities in the dynamics themselves, though the paper sticks to its linear assumption. The frequency formulas are explicit but their numerical use in optimization is not explored in depth.

This is aimed at control engineers and theorists who deal with periodic forcing in systems like electrical circuits or biological models where the output is nonlinear. Someone looking for analytic ways to compare periodic and constant excitations will find the convexity result and the quadratic formulas useful.

It deserves serious referee time because the derivations are internally consistent and the new concepts are well-motivated by applications.

I would recommend sending it out for review. The core claims check out on the given logic, and the frequency-domain part adds practical value.

Referee Report

0 major / 2 minor

Summary. The paper analyzes the gain of entrainment (GOE) in asymptotically stable linear systems ˙x = Ax + Bu with static nonlinear output y = h(x). It shows that GOE vanishes for linear h but arises from the mismatch between the time-average of h along the unique entrained periodic orbit and h evaluated at the equilibrium for the averaged input. A second-order Taylor expansion characterizes the leading term via the Hessian of h. The central result is that convexity (concavity) of h on the controllable subspace implies GOE ≥ 0 (GOE ≤ 0) for every periodic input, with GOE equal to the average Bregman divergence between the orbit and the averaged equilibrium. For quadratic h, explicit frequency-domain formulas are derived that give necessary and sufficient sign conditions, decompose the contribution of input harmonics, and yield an optimal periodic excitation maximizing GOE under an energy constraint. The theory is illustrated on an RLC circuit and a compartmental pharmacodynamic model.

Significance. If the derivations hold, the manuscript supplies a matrix-based, parameter-free criterion linking output convexity to entrainment benefits, together with a geometric Bregman-divergence interpretation and computable frequency-domain expressions for the quadratic case. The restriction of convexity to the controllable subspace is sharp, and the averaging argument (integrating the dynamics over one period) is standard yet cleanly applied. These elements provide both theoretical clarity and design tools for periodic inputs in applications with nonlinear output maps.

minor comments (2)

[Abstract] Abstract: the phrase 'second-order characterization' is used without indicating the precise order of the Taylor expansion or the remainder term; a parenthetical reference to the relevant equation in §3 would clarify the leading-order claim.
[§4] §4 (quadratic case): the frequency-domain expression for GOE is stated in integral form; an explicit invocation of Parseval's theorem or the inner-product representation used to obtain the sum over harmonics would make the derivation self-contained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading, accurate summary of the contributions, and the recommendation of minor revision. We are pleased that the significance of the convexity criterion, Bregman-divergence interpretation, and frequency-domain formulas for the quadratic case was recognized.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The central claims follow directly from integrating the linear dynamics over one period (yielding avg(x) equal to the equilibrium for avg(u)) and applying the definition of Bregman divergence for a convex output map on the controllable subspace. These steps use only the system equations, the convexity assumption, and standard properties of averages and divergences; they do not reduce to fitted parameters, self-citations, or ansatzes imported from prior work by the same authors. The frequency-domain formulas for the quadratic case are likewise explicit matrix expressions derived from the same linear system. No load-bearing step matches any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard assumptions from linear control theory; no free parameters or new entities are introduced.

axioms (2)

domain assumption The linear system is asymptotically stable
Invoked to guarantee existence and uniqueness of the entrained periodic orbit and attractiveness of the averaged equilibrium.
domain assumption The output map is static and twice continuously differentiable
Required for the second-order Taylor expansion that isolates the curvature term determining leading-order GOE.

pith-pipeline@v0.9.1-grok · 5779 in / 1466 out tokens · 33615 ms · 2026-06-26T07:42:32.041781+00:00 · methodology

discussion (0)

Reference graph

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M. Margaliot, L. Grune, and T. Kriecherbauer, “Entrainment in the master equation,” Royal Society open science, vol. 5, no. 4, p. 172157, 2018

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M. Margaliot, E. D. Sontag, and T. Tuller, “Entrainment to periodic initiation and transition rates in a computational model for gene translation,” PLoS ONE, vol. 9, no. 5, p. e96039, 2014

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R. Ofir, T. Kriecherbauer, L. Grüne, and M. Margaliot, “On the gain of entrainment in the n-dimensional ribosome flow model,” J. R. Soc. Interface, vol. 20, p. 20220763, 2023

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A. Raveh, M. Margaliot, E. Sontag, and T. Tuller, “A model for competition for ribosomes in the cell,” J. R. Soc. Interface, vol. 13, no. 116, p. 20151062, 2016

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1992