Performance analysis and control of fractional‐order positive systems

Abstract

Unlike a general integer-order linear system, the L 1 and L 2 norms of its state and output variables may not be convergent in an infinite time interval even if the concerned fractional-order linear system is asymptotically stable or even Mittag-Leffler stable. Therefore, this study mainly investigates the analysis and synthesis of L 1 and L 2 performances in a finite time interval for a class of fractional-order positive systems. The L ∞ performance problem in the infinite time interval is also concerned since the boundness of the L ∞ norm of the state and output variables can be guaranteed in the infinite time interval if the concerned fractional-order positive system is asymptotical stable. The L 1 and L ∞ performance characterisations are expressed by linear programming and L 2 performance characterisation is expressed in terms of linear matrix inequality with a diagonal positive definite solution. Based on these performance conditions, the problems of stabilisation by a state feedback controller with L p (p ∈ {1, 2, ∞})-gain guaranteed are solved for fractional-order positive systems. An example is presented to show the effectiveness of the proposed methods in this study.