  Optimal sensor placement is an important yet unsolved problem in control
theory. In biological organisms, genetic activity is often highly nonlinear,
making it difficult to design libraries of promoters to act as reporters of the
cell state. We make use of the Koopman observability gramian to develop an
algorithm for optimal sensor (or reporter) placement for discrete time
nonlinear dynamical systems to ease the difficulty of design of the promoter
library. This ease is enabled due to the fact that the Koopman operator
represents the evolution of a nonlinear system linearly by lifting the states
to an infinite-dimensional space of observables. The Koopman framework ideally
demands high temporal resolution, but data in biology are often sampled
sparsely in time. Therefore we compute what we call the temporally fine-grained
Koopman operator from the temporally coarse-grained Koopman operator, the
latter of which is identified from the sparse data. The optimal placement of
sensors then corresponds to maximizing the observability of the fine-grained
system. We demonstrate the algorithm on a simulation example of a circadian
oscillator.
