Iterative Learning Control Problem For Riemann-Liouville Type Fractional Differential Systems

Fractional order differential equation is an important branch of differential equations,which is extended to any order differential equation.It is especially suitable for describing the physical and mechanical processes with memory and genetic phenomena.As a branch of intelligent control and data driven control,iterative learning control(ILC)is able to achieve the control technology of tracking in finite time interval.In this paper,we study the iterative learning control for Riemann-Liouville fractional differential systems.For a given target,a suitable iterative learning control law is designed,which can correct the deviation between the output signal and the given target.First,we study the nonlinear Riemann-Liouville fractional differential systems with initial state shifts,and design the type learning law(0 <?< 1)and type learning law(1 <?< 2)with initial state learning.By using the monotonicity and boundedness of the Mittag-Leffler function,the fractional Gronwall inequality and the classical method of contractive mapping,some sufficient conditions are given to ensure that the tracking error converges to zero in finite time in the sense of C?,? norm.And the validity of the theoretical results is verified by numerical examples.Secondly,we study the iterative learning control problem for Riemann-Liouville fractional differential systems with impulsive perturbation,and give a reasonable definition of piecewise continuous solutions.We design initial learning type learning law.By using of pulse fractional Gronwall inequality,some sufficient conditions are given to ensure that the tracking error of the initial state offset fractional pulse system converges to zero in the sense of PC?,? norm.And the validity of the theoretical results is verified by numerical examples.Finally,we give the exact expression for the solution of the nonhomogeneous problem of linear time-delay systems by using the delay matrix exponential notation.By designing the type and type learning law,the finite time tracking problem of linear time-delay control system is realized in the sense of C? norm.