Positive real lemmas for fractional order systems

This paper is concerned with the positive realness of fractional order linear time-invariant systems with the commensurate order 0 <; α <; 2. Sufficient and necessary condition for a fractional order system to be positive real is derived in terms of linear matrix inequalities. Furthermore, the sufficient condition for positive real fractional order systems with pseudo state feedback is investigated. Numerical examples are given to illustrate the effectiveness of the method proposed in this paper.