Abstract
This study investigates the positive real lemmas for singular fractional-order linear time-invariant systems with the fractional order α ∈ ( 0 , 1 ) . Firstly, a novel condition for the stability and extended strictly positive realness of fractional-order systems is derived in terms of linear matrix inequalities. Secondly, a novel condition for the admissibility and extended strictly positive realness of singular fractional-order systems is obtained with two complex variables. Then, another novel condition for the admissibility and extended strictly positive realness of singular fractional-order systems is established with only one complex variable. Thirdly, the positive real controller synthesis problems are analysed and the corresponding positive real controllers are designed based on the positive real lemmas. Finally, four numerical examples are provided to show the effectiveness of the proposed results in this study.
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