Bounded Real Lemmas for Fractional Order Systems

This paper derives the bounded real lemmas corresponding to L_∞ norm and H_∞ norm (L-BR and H-BR) of fractional order systems. The lemmas reduce the original computations of norms into linear matrix inequality (LMI) problems, which can be performed in a computationally efficient fashion. This convex relaxation is enlightened from the generalized Kalman-Yakubovich-Popov (KYP) lemma and brings no conservatism to the L-BR. Meanwhile, an H-BR is developed similarly but with some conservatism. However, it can test the system stability automatically in addition to the norm computation, which is of fundamental importance for system analysis. From this advantage, we further address the synthesis problem of H_∞ control for fractional order systems in the form of LMI. Three illustrative examples are given to show the effectiveness of our methods.