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.