Bounded real lemmas for fractional order systems

Abstract

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-YakubovichPopov (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.

Robust H∞ Control for Fractional Order Systems with Order α (0 < α < 1)

“…Compare with the existing results in [23,24], two real matrices are used to replace the conjugate matrix or Hermitian matrix; Theorem 1 in the paper can be solved easily. Moreover, when α → 1, Theorem 1 can be used to solve H ∞ control for integer order systems.…”

“…The LMI-based conditions for FOS with order 0 < α < 2 have been reported in [21], which can be used to determine the stability for FOS in the uniform form. H ∞ control and analysis have been considered in [22][23][24].…”

Adaptive sliding mode fault tolerant control for interval Type-2 fuzzy singular fractional-order systems

“…In Zhang et al (2020), a unified framework of stability theorem for FOSs with 0 < α < 2 is given. When FOSs are subject to external disturbance, an H ∞ control method for FOSs is proposed in Liang et al (2015). Because state variables of FOSs often cannot represent physical variables in a natural way to provide a mathematical model, a singular fractionalorder systems (SFOSs) model is considered.…”

Positive real lemmas for fractional order systems

“…[19][20][21] sufficient conditions for internal and external positivity of FOS. Moreover, [22] and [25] concerned about the bounded real lemmas for FOS and investigated an improved bounded real lemma which is non-conservative. However, there was no discussion about the positive real lemma for FOS yet.…”