This paper proposes a new hierarchical identification method for FRACTIONAL-ORDER systems. In this method, a SISO (single input, single output) state space model has been considered in which parameters and also state variables should be estimated. By using a linear transformation and a shift operator, the system will be transformed into a form appropriate for identification of a FRACTIONAL ORDER system. Then, the unknown parameters will be identified through a recursive least squares method and the states will be estimated using a FRACTIONAL ORDER Kalman filter. This identification method is based on the hierarchical identification principle that reduces the computational burden and is easy to implement on computer. The promising performance of the proposed method is verified using two stable FRACTIONAL-ORDER systems.

In this paper, a novel conformable FRACTIONAL ORDER (FO) sliding mode control technique is studied for a class of FO chaotic systems in the presence of uncertainties and disturbances. First, a novel FO nonlinear surface based on conformable FO calculus is proposed to design the FO sliding mode controller. Then, asymptotic stability of the controller is derived by means of the Lyapunov direct method via conformable FO operators. The stability analysis is performed in the sliding and reaching phase. In addition, the realization of reaching phase is guaranteed in finite time and the reaching time is calculated analytically. The proposed control approach has some superiorities. Reduction of the chattering phenomenon, high robustness against the uncertainty and external disturbance, and fast convergence speed are the main advantages of the proposed control scheme. Moreover, it has simple calculations because of using conformable FO operators in the control design. The numerical simulations verify the efficiency of the proposed controller.

This paper presents a FRACTIONAL-ORDER adaptive controller based on sliding mode control for synchronization of commensurate FRACTIONAL-ORDER chaotic systems. The adaptive controller is a FRACTIONAL PID controller, which the coefficients will be tuned according to a proper adaptation mechanism. FRACTIONAL PID coefficients are updated using the gradient method when a proper sliding surface is chosen. To illustrate the effectiveness and performance of the controller, the proposed controller is implemented on chaotic FRACTIONAL-ORDER Genesio Tessi and Coullet systems. Performance of FRACTIONAL-ORDER PID adaptive controller (PaIlDm) based on synchronization error, and control signal is compared with the conventional adaptive controller (PID) and sliding mode controller (SMC). The simulation results show the efficiency of the proposed controller.

In this paper, the FRACTIONAL Sumudu decomposition method (FSDM) is employed to handle the time-FRACTIONAL PDEs and system of time-FRACTIONAL PDEs. The FRACTIONAL derivative is described in the Caputo sense. The approximate solutions are obtained by using FSDM, which is the coupling method of FRACTIONAL decomposition method and Sumudu transform. The method, in general, is easy to implement and yields good results. Illustrative examples are included to demonstrate the validity and applicability of the proposed technique.

In this study, we construct the FRACTIONAL-ORDER Bernstein wavelets for solving stochastic FRACTIONAL integro-differential equations. FRACTIONAL-ORDER Bernstein wavelets and their properties are presented for the first time.  The FRACTIONAL integral operator of FRACTIONAL-ORDER Bernstein wavelets together with the Gaussian integration method is applied to reduce stochastic FRACTIONAL integro-differential equations to the solution of algebraic equations which can be simply solved to obtain the solution of the problem.  Also, an error estimation for our approach is introduced.  The numerical results demonstrate that our scheme is simply applicable, efficient, powerful and very precise at the small number of basis functions.

In this paper, based on a FRACTIONAL ORDER Bergman minimal model, a robust strategy for regulation of blood glucose in type 1 diabetic patients is presented. Glucose/insulin concentration in the patient body is controlled through the injection under the patients skin by the pump. Many various controllers for this system have been proposed in the literature. However, most of them have consider the system as an integer ORDER system. Moreover, the majority of the presented methods su er from an important disadvantage that is long settling time of the control system. Thus, the contribution of this paper in comparison with previous related works is presenting a FRACTIONAL back-stepping sliding mode control that considerably reduces the required time for glucose to reach its desired level. Due to the sliding mode design, the proposed controller is robust against external disturbances. Due to the back-stepping design, convergence of each state variable of the system to its desired value can be guaranteed separately. Simulation results verify the satisfactory performance of the proposed controller.

This paper proposes and analyzes an applicable approach for numerically computing the solution of FRACTIONAL optimal control-affine problems. The FRACTIONAL derivative in the problem is considered in the sense of Caputo. The approach is based on a FRACTIONAL-ORDER hybrid of block-pulse functions and Jacobi polynomials. ‎First‎, ‎the corresponding Riemann-Liouville FRACTIONAL integral operator of the introduced basis functions is calculated‎. ‎ Then, an approximation of the FRACTIONAL derivative of the unknown state function is obtained by considering an approximation in terms of these basis functions‎. ‎ Next, ‎using the dynamical system and applying the FRACTIONAL integral operator‎, ‎an approximation of the unknown control function is obtained based on the given approximations of the state function and its derivatives‎. ‎ Subsequently‎, ‎all the given approximations are substituted into the performance index‎. ‎Finally‎, ‎the optimality conditions transform the problem into a system of algebraic equations‎. ‎An error upper bound of the approximation of a function based on the FRACTIONAL hybrid functions is provided‎. ‎The method is applied to several numerical examples‎, and ‎the experimental results confirm the efficiency and capability of the method.  Furthermore, they demonstrate a good agreement between the approximate and exact solutions‎. ‎