Optimal sensor placement is an important yet unsolved problem in control theory. In this paper, we make use of the Koopman observability gramian to develop an algorithm for optimal sensor placement of discrete time autonomous nonlinear dynamical systems . The Koopman operator lifts nonlinear dynamics to a higher dimensional space where the dynamics evolve approximately linearly in time. Data in biology are often sampled sparsely in time , therefore a method is developed to compute a temporally fine-grained Koopman operator from the temporally coarse-grained Koopman operator . In the case of noisy data, a closed form expression for the error in the coarse-grained Koopman operator is derived. A novel algorithm for optimal sensor placement is developed for the case of a fixed initial condition. The method is finally demonstrated on a simulation example  .
Optimal sensor placement is an important yet unsolved problem in control theory. In URLanisms, 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 .
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this paper, we
URLanisms, genetic activity is often highly nonlinear, making it difficult to design libraries of promoters to act as reporters of the cell state. We
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placement of discrete time autonomous
(or reporter) placement for discrete time
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. The Koopman operator lifts nonlinear dynamics to a higher dimensional space where the dynamics evolve approximately linearly in time. Data
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
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, therefore a method is developed to compute a
. Therefore we compute what we call the
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. In the case of noisy data, a closed form expression for the error in the coarse-grained Koopman operator is derived. A novel algorithm for optimal sensor placement is developed for the case of a fixed initial condition. The method is finally demonstrated
, 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
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of a circadian oscillator