The minimal positive realization problem for linear systems is to find positive systems with a state space of minimal dimensions while keeping their transfer functions unchanged. Here the necessary and suffi-cient condition is established for the linear systems with third-order two-different-pole transfer functions to have a three-dimensional minimal positive realization by developing a new approach. The approach is to identify a canonical form of positive realizations with the help of polyhedral cone theory, and the iden-tification consists of a sequential edge transformations with some invariant feature. The analyses are then successfully extended to the linear systems with general nth-order two-different-pole transf er functions. At the same time, the approach can also be used to generate infinitely many distinct minimal positive realizations for some linear systems.(c) 2022 European Control Association. Published by Elsevier Ltd. All rights reserved.
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