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In this paper the goal is to identify the parameters of the Lorenz chaotic system, based on synchronization of identical systems using fractional calculus. The method which is used for synchronization is come from Lyapunov stability theorem and then by using fractional dynamics, control laws are improved. To this end, a Lyapunov function is presented and based on the Lyapunov stability theory and asymptotic stability criteria, some adaptation laws to estimate unknown parameters of the system are proposed. The introduced method is applied to the Lorenz chaotic system and since the goal is identification, all the parameters of the system are taken unknown. Using invariant set theory, it is proved that the parameter estimation errors converge to zero. Then the results of numerical simulations are shown and discussed and it is observed that fractional calculus has an essential effect on reducing fluctuations and settling time of the parameters convergence. At the end, the stability of the system by using fractional adaptation law is discussed. It is shown that the asymptotic stability of the synchronization error dynamics is proved using the fractional adaptation law, and this is confirmed through simulation.

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Fractional calculus is a branch of mathematics which in recent decades has been of great interest to scientists in various disciplines, including engineering. One of the applications of this branch in engineering, is in modeling the viscoelastic materials using fractional differentiation. In this article, by inserting fractional calculus as a viscoelastic material compatibility equations in nonlocal beam theory, a viscoelastic Euler-Bernoulli nano-beam with different boundary conditions at two ends, has been modeled. Material properties of a carbon nanotube is considered and two methods, pure numerical and numerical-analytical have been used for solving obtained equations in time domain. Main method is completely numerical and operator matrices used in it to discrete equations in time and spatial domain. Second method is introduced for validation of pervious method’s answers. In this method equation of system reduced to an ordinary differential equation using Galerkin and obtained equation solved using a numerical direct integrator method. Finally, in a case study, the effects of fractional order, viscoelasticity coefficient and nanlocal theory coefficient on the time response of the viscoelastic Euler-Bernoulli nano-beam with different boundary conditions have been studied.

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Energy saving, low robot mass to carried mass ratio, more ability to work in various environments, easier delivery of parts and lower production costs in flexible robots make these robots more attractive than rigid robots to many researchers and industries. But due to nonlinearities in flexible robot system and high vibration in operation points and also more sensitivity against external disturbances, control of these robots is more difficult and complex. In this paper a controller for a flexible link manipulator based on fractional calculus is practically implemented. At first the dynamic model of a single flexible-link robot is introduced. Then various controllers such as fuzzy control, PID control, and fractional order PID torque control are practically implemented on a single flexible-link robot made in laboratory, and then the performance of each controllers in decreasing of arm vibration in final desired point and tracking error reduction are investigated. Further, to compare the robustness of the designed controllers, a same constant disturbance is applied to all controllers and their performance are compared. Finally, the simulation results and experimental results show that the fractional order PID torque controller has the best results among the implemented controllers.

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Quadrotor is one the most popular models of unmanned aerial vehicles with four actuated propellers which has a simple, light weight, small mechanical structure and high maneuverability. However, its nonlinear under-actuated dynamics needs more advanced controllers for rejection of external disturbances, balancing and precise trajectory tracking. In particular, the under-actuated subsystem of the quadrotor's dynamics needs a fast response without overshoot and steady state error. In this paper, fuzzy fractional-order proportional-integral derivative (FOFPID) controller is designed for quadrotor control system using fuzzy and fractional order systems to improve response speed, tracking accuracy and system robustness respect to the conventional PID controller. Controller architecture of the under-actuated subsystem of the quadrotor's dynamics is designed based on the inner-outer loop control theory which is employed explicit and analytical inverse kinematic of system to connect the inner and outer loops. Also, dynamics of the motors and actuators saturation are considered in the quadrotor’s dynamics model and their effects are studied on the controllers' performance. In order to evaluate tracking performance of controllers, trajectory of an eight aerial maneuver is designed and controllers’ performance is assessed in the absence and presence of wind disturbance. Trajectory tracking accuracy of the controllers is studied according to the maximum absolute error and integral of absolute error criterions and is compared that shows the proposed FOFPID controller has successfully improved performance of the quadrotor system.