This survey aims to present a comprehensive and systematic synthesis of concepts and results on the minimal state space realization problem for positive, linear, time-invariant systems. Positive systems are systems for which the state and the output are always non-negative for any non-negative initial state and input. They are used to model phenomena in which the variables must take non-negative values due to the nature of the underlying physical system. Restricting the state–space realization to positive systems makes the problem extremely different and much more difficult than that for ordinary systems. Indeed, a minimal positive realization may have a dimension even much greater than the order of the transfer function it realizes. Although the problem of finding a finite-dimensional positive state–space realization of a given transfer function has been solved, the characterization of minimality for positive systems is still an open problem. This survey introduces the reader to different aspects of the problem and presents the mathematical approaches used to tackle it as well as some relevant related problems. Moreover, some partial results are presented. Finally, a comprehensive bibliography on positive systems, organized by topics, is provided.

本综述旨在对正、线性、时不变系统的最小状态空间实现问题的概念和结果进行全面系统的综合。正系统是状态和输出对于任何非负初始状态和输入总是非负的系统。它们用于对由于基础物理系统的性质而变量必须取非负值的现象进行建模。将状态空间的实现限制在正系统中会使问题变得非常不同，并且比普通系统要困难得多。实际上，最小正实现的维度可能比它实现的传递函数的阶数还要大得多。尽管找到给定传递函数的有限维正状态空间实现的问题已经解决，但正系统的极小性表征仍然是一个悬而未决的问题。本调查向读者介绍了该问题的不同方面，并介绍了用于解决该问题的数学方法以及一些相关的相关问题。此外，还给出了一些部分结果。最后，提供了按主题组织的关于积极系统的综合参考书目。