| 英文摘要 | Positive systems theory can be widely applied in various fields, such as optical filtering, chemical engineering, biology and medicine, economics, social science etc, and positive realization problem, which arose from identification of hidden Markov model, is a basic problem of positive systems theory, and draws great attention from researchers. This dissertation focuses on minimal positive realizations of nth-order rational transfer functions of discrete-time and continuous-time linear time-invariant single-input-single-output (SISO) systems. For discrete-time systems, with the help of constructive methods, conditions are provided for nth-order systems to have nth-order positive realizations, which can be applied to reducing the order of realizations in positive decomposition problem. For continuous-time systems, with the method of transformation, the results are deduced from those of discrete-time systems, which can be applied to deciding the minimal compartment number of a compartmental system and to finding a representation for a phase-type (PH) distribution. In each case, numerical examples are employed to validating the effectiveness of the obtained results. The main contributions of this dissertation include following issues: 1. Conditions are provided for third order discrete-time and continuous-time systems with complex poles to have third order positive realizations. For discrete-time systems, according to the location of the complex poles, transfer functions are divided into two groups. For one group, necessary and sufficient conditions are provided for existence of third order positive realizations; for the other, sufficient conditions are provided. With the method of shifting transformation, sufficient conditions are obtained for continuous-time transfer functions to have third order positive realizations. These results improve the existing ones in literatures. 2. With constructive methods, necessary and sufficient conditions are obtained for nth order transfer functions H(z) with (n-1) negative poles to have nth-order positive realizations; and sufficient conditions are presented for a class of nth-order transfer functions H(z) with complex poles to have nth-order positive realizations. However, the obtained conditions are necessary, in case that the sum of poles is equal to zero. 3. For nth-order continuous-time transfer functions H(s) of distinct, or multiple real poles, conditions are provided for H(s) to have nth-order positive realizations, respectively. For nth-order transfer functions with complex poles, with the help of index of positivity, necessary and sufficient conditions are provided for H(s) to have nth-order positive realizations with certain index of positivity. |
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