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

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.

• The paper shows that gain of entrainment is a second-order effect determined by the output map’s curvature and the covariance of oscillatory state trajectories, while first-order effects cancel.
• The paper proves that convex outputs guarantee nonnegative gain and concave outputs guarantee nonpositive gain on the controllable subspace, with applications to RLC circuits and pharmacodynamic dosing.
• For quadratic outputs, the paper derives an exact frequency-domain decomposition, identifies necessary and sufficient sign conditions, and shows that optimal excitation concentrates energy at one frequency and input direction.

Setting and motivation

The paper studies the gain of entrainment (GOE) for a multiple-input, single-output linear control system with a static nonlinear output map—a Wiener system:

$\dot{x} = Ax + Bu, \qquad y = h(x),$

with $A$ Hurwitz, so the system entrains to any $T$-periodic input. For a $T$-periodic input $v$, the GOE is the difference between the time-averaged output along the entrained periodic orbit $\gamma^v$ and the output at the equilibrium $e^{\bar v}$ generated by the constant input $\bar v$. A positive GOE means periodic forcing with a given mean outperforms constant actuation at that mean. The authors note that for linear systems with linear outputs GOE is identically zero, and prior work on weakly contractive bilinear systems showed GOE is inherently a second-or-higher-order effect in excitation amplitude (2606.23149). The central question is therefore how nonlinearity confined to the output map can generate nonzero GOE.

Second-order characterization

A key structural fact is that for a stable linear system, the time average of the entrained orbit equals the equilibrium of the averaged input: $\overline{\gamma^v} = e^{\bar v} = -A^{-1}B\bar v$. This follows directly from averaging the dynamics over one period. Consequently, all first-order terms cancel: writing $u_\varepsilon(t) = \bar v + \varepsilon w(t)$ with zero-mean $A$0, a Taylor expansion shows

$A$1

where $A$2 is the covariance matrix of the periodic orbit about its mean. Thus the leading-order contribution is governed entirely by the curvature of $A$3 along the oscillatory component of the state trajectory; the gradient of $A$4 plays no role.

Convexity, Bregman divergence, and sign guarantees

Two consequences follow. First, if $A$5 is convex (concave) when restricted to the controllable subspace $A$6—the only region visited by periodic orbits and equilibria—then Jensen's inequality yields $A$7 ($A$8) for every periodic input, with strict inequality under strict convexity/concavity and nonconstant orbits. This is a robustness property: the sign guarantee holds regardless of the specific entries of $A$9 and $T$0, provided $T$1 is Hurwitz. Notably, convexity is sufficient but not necessary: the paper exhibits a saddle-type quadratic output ($T$2) on a fully controllable two-state system whose transfer function nevertheless produces strictly positive GOE for every periodic input.

Second, for $T$3 the paper establishes an exact identity:

$T$4

where $T$5 is the Bregman divergence. GOE is thus a geometric quantity measuring the average gap between the output graph and its tangent plane at the averaged equilibrium. When $T$6 is negative entropy on the simplex, this becomes the average Kullback–Leibler divergence between the forced orbit and the equilibrium, suggesting a possible link to entropy production in nonequilibrium statistical mechanics—though the paper does not develop this connection formally.

Quadratic outputs: frequency-domain analysis

For $T$7, the second-order expansion becomes exact and GOE equals the weighted variance $T$8 of the orbit. For multi-harmonic inputs $T$9, orthogonality of harmonics gives an additive spectral decomposition:

$T$0

where $T$1. Each harmonic contributes independently, so GOE can be estimated from frequency-response data alone without a state-space realization. The paper derives a necessary and sufficient condition for universal nonnegativity: $T$2 must be positive semidefinite for all $T$3. Under an energy constraint $T$4, the optimal excitation places all energy into a single sinusoid at the frequency–direction pair $T$5 maximizing $T$6 of this Hermitian matrix, achieving $T$7; if this quantity is nonpositive, constant input is optimal.

Applications

RLC circuit: with stored energy as the (strictly convex, positive-definite quadratic) output, GOE is provably positive for every nonconstant periodic voltage. The closed-form expression for a sinusoidal input,

$T$8

shows quadratic growth in amplitude, decay to zero at high frequency, and an optimal driving frequency strictly below the resonance frequency $T$9.

Pharmacodynamics: for a transit-compartmental model with the saturating E$v$0 dose–response map, which is concave on the positive orthant, the theory implies $v$1 for all nonnegative inputs. Periodic drug administration therefore cannot beat constant dosing with the same average dose in this model class—a practically relevant negative result, confirmed numerically across driving frequencies $v$2.

Limitations and open questions

The analysis assumes exact linearity of the dynamics; extension to weakly nonlinear systems $v$3, where nonlinearities enter both dynamics and output, remains open. Interconnected Wiener systems, where inter-subsystem coupling may introduce additional GOE mechanisms, are not treated. Stochastic forcing and its relation to noise-induced fluctuations are unaddressed. More broadly, tractable conditions for GOE in general nonlinear control systems remain an open problem, and the necessity question for sign guarantees (beyond the sufficient convexity condition) is only partially resolved by counterexample.

Conclusion

The paper establishes that in stable Wiener systems, GOE arises from the mismatch between averaging and nonlinear evaluation, is a second-order effect determined by output curvature, admits an exact Bregman-divergence representation, and—for quadratic outputs—decomposes additively over input harmonics with sharp necessary-and-sufficient sign conditions and a closed-form optimal excitation. These results connect entrainment performance to classical frequency-domain tools and yield concrete design guidance for periodic excitation protocols.