Stability Analysis Of Fractional Order Nonlinear System And Its Design Of PI~α Controller

The fractional-order system theory has played a very important role in the promotionof modern mathematics and physics. It not only extends the classical integer-order systemtheory, but also as mathematical tools describe the research objects in nature better.Stability is a key issue to ensure the proper functioning of the system. Therefore the studyof stability problem for fractional-order nonlinear system has important theoretical andpractical significance.Firstly，the restricted conditions of double parameter Mittag-Leffler function estimatetheorem that presented by I. Podlubny are analyzed. It is proved that the constraint scopeof the theorem is not right, so the double parameter Mittag-Leffler function is redefinedand the double parameter Mittag-Leffler function estimate theorem is improved. Then anew stability theory that available for a class of fractional-order nonlinear system isproposed. In addition, the new stability theory is proved by means of improved doubleparameter Mittag-Leffler function estimate theorem and Gronwall theorem.Secondly, thePI~αcontroller for the fractional-order hyperchaotic Chen system isdesigned. The controller with the proportion and integral of error system can make Chensystems synchronization. The stability of the Chen error system is proved by thefractional-order nonlinear system stability theory. Furthermore, it concludes that theeigenvalue of system parameter matrix only depends on the diagonal element of theproportion matrix and integral matrix, and the conclusion is proved by disk theorem. Thenthe affect of the proportion parameters and integral parameters is analyzed. Numericalsimulations verify the effectiveness of the method.Finally, for the balance control problem of the fractional-order unified chaotic system,thePI~αcontroller is put forward. Subsequently, the condition of parameters selections forthe stability of the fractional-order unified chaotic system is given out, and then thestability is proved by the stability theory. Simulation results demonstrate the controller canmake the fractional-order unified chaotic system balance commendably.