Positive systems naturally arise in situations where the model tracks physical quantities. Although the linear case is well understood, analysis and controller design for nonlinear positive systems remain challenging. Model reduction methods can help tame this problem. We propose a notion of model reduction for a class of positive bilinear systems with (bounded) matrix and exogenous controls. Our reduction, called proper positive lumping, aggregates the original system such that states of the corresponding reduced model represent non-negative linear combinations of original state variables. We prove a characterization result showing that the reductions by proper positive lumping are exactly those preserving the optimality of a suitable class of value functions. Moreover, we provide an efficient polynomial-time algorithm for the computation of the minimal lumping. We numerically evaluate our approach by applying it to a number of benchmark case studies.
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