Minimal Positive Realizations for Transfer Functions with Negative Poles

In this paper, we discuss the minimal positive realizations for nth-order rational transfer function of a discrete time-invariant linear single-input-single-output (SISO) system with negative poles. By the constructive method, the necessary and sufficient conditions are provided for these transfer functions to have a positive realization of dimension n. With the benefit of these results, an improvement is made on the existing results about the positive decomposition problem in positive system theory. Numerical examples are employed to validate the results.