Performance analysis and control of fractional-order positive systems

Unlike a general integer-order linear system, the $L_1$L1 and $L_2$L2 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 $L_1$L1 and $L_2$L2 performances in a finite time interval for a class of fractional-order positive systems. The $L_\infty $L infinity performance problem in the infinite time interval is also concerned since the boundness of the $L_\infty $L infinity 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 $L_1$L1 and $L_\infty $L infinity performance characterisations are expressed by linear programming and $L_2$L2 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 $L_p\ (p\in \{1,2,\infty \})$Lp(p is an element of{1,2,infinity})-gain guaranteed are solved for fractional-order positive systems. An example is presented to show the effectiveness of the proposed methods in this study.