CRA based control of incommensurate fractional order systems

A proof for non existence of periodic solutions in time invariant fractional order systems

Lyapunov exponents spectrum estimation of fractional order nonlinear systems using Cloned Dynamics

In this paper, a new 4-D incommensurate fractional-order chaotic system based on adomian decomposition method is investigated. The equilibrium points of the system with incommensurate orders are analyzed. The coexisting attractors of the system with different parameters and orders are unveiled through coexisting attractors phase diagram, coexisting bifurcation model, coexisting Lyapunov exponent spectrum. The effects of the incommensurate orders of the system are also analyzed. Meanwhile, some special phenomena such as hidden attractors and chaos degradation are found. The results show that the new 4-D incommensurate fractional-order chaotic system has rich dynamic characteristics. In addition, using the DSP platform to realize the system, the results are in good agreement with those on simulation. The research results provide a theoretical basis for the application of incommensurate order chaotic systems in secure communication.

The analysis, design and circuit synthesis of a fractional order piecewise linear (PWL) system is presented in this paper. By selecting incommensurate fractional orders for the dynamical system, multi-scroll chaotic attractors are observed. The resulting fractional order PWL chaotic system satisfies the stability theorem of fractional calculus. Lyapunov exponents confirm the chaotic behavior. Moreover, a circuit synthesis is realized by considering voltage operational amplifiers. SPICE circuit simulations agree with the theoretical results.

This paper develops a fractional order adaptive fuzzy backstepping control scheme for an incommensurate fractional order uncertain nonlinear multiple-input multiple-output (MIMO) systems with external disturbance and input saturation. Combined with the adaptive and backstepping technique, the fuzzy logic system is used to approximate the unknown nonlinear uncertainties in each step of the backstepping, and the incommensurate fractional order parameters update laws for fuzzy logic system, unknown parameters as well as the external disturbances are proposed in the controller to compensate the unknown nonlinearities, unknown parameters and disturbances. With the aids of the frequency distributed model of fractional integrator for the fractional order systems in the procedure of controller design, the stability of the closed loop system is established. At last, two simulation examples are demonstrated to verify the robustness and effectiveness of the proposed controller.

This article investigates a non-equilibrium chaotic system in view of commensurate and incommensurate fractional orders and with only one signum function. By varying some values of the fractional-order derivative together with some parameter values of the proposed system, different dynamical behaviors of the system are explored and discussed via several numerical simulations. This system displays complex hidden dynamics such as inversion property, chaotic bursting oscillation, multistabilty, and coexisting attractors. Besides, by means of adapting certain controlled constants, it is shown that this system possesses a three-variable offset boosting system. In conformity with the performed simulations, it also turns out that the resultant hidden attractors can be distributively ordered in a grid of three dimensions, a lattice of two dimensions, a line of one dimension, and even arbitrariness in the phase space. Through considering the Caputo fractional-order operator in all performed simulations, phase portraits in two- and three-dimensional projections, Lyapunov exponents, and the bifurcation diagrams are numerically reported in this work as beneficial exit results.

Bifurcation control of a novel fractional-order gene regulatory network with incommensurate order and time delay

A fractional-order predator–prey biological economic system with Holling type II functional response is proposed. Local stability and Hopf bifurcation of predator–prey systems have been investigated in both commensurate and incommensurate fractional-order systems. We explore how the economic profit and fractional orders influence the local stability and Hopf bifurcation for the fractional-order predator–prey system. For an incommensurate system, we propose a new theory for the existence of Hopf bifurcation when the fractional orders are considered as bifurcation parameters. Several numerical examples are demonstrated to validate the theoretical results.

In this paper, chaos synchronization problem of the fractional incommensurate order system is investigated by using the nonlinear feedback control scheme. Synchronization sufficient conditions are analyzed based on the Lyapunov stability theory. Moreover, the control scheme can be used in secure communication via the technology of chaotic masking with complex non-periodic information signals. And numerical simulations are provided to verify the validity of the proposed method.

This manuscript, presents the fractional order disturbances observer based adaptive sliding mode hybrid projective synchronization of commensurate and incommensurate fractional order Newton–Leipnik chaotic system with disturbance. Firstly, a fractional order disturbance observer (FODO) based on adaptive sliding mode control laws for a fractional order Newton–Leipnik system with unknown bounded disturbance is studied. The appropriate control gain parameters are chosen such that the disturbance observer can approximate the unknown disturbance well. Further, we design a simple sliding mode surface based on sliding mode control technique incorporating with the designed fractional order disturbance observer for developing a bounded hybrid projective synchronization control scheme. Numerical simulation results are given to demonstrate the validity and effectiveness of the proposed sliding mode control scheme in the presence of external bounded unknown disturbance. To analyse the superiority of proposed methodology we have compared our results with earlier published results as well. Moreover, as an application, the proposed scheme is then applied to a secure communication system. Simulation results verify the proposed schemes success in the communication application.

The H∞ tracking control problem is investigated for a class of incommensurate fractional order strict feedback nonlinear systems with unknown nonlinear functions and external disturbances in this paper. The adaptive neural network H∞ tracking controller of the system is designed by combining backstepping technique, adaptive neural network technique with H∞ control theory and utilizing the frequency distributed model to construct Lyapunov functions. The H∞ tracking controller proposed in this paper can guarantee that the tracking error and all the signals in the closed-loop system are bounded, and the system has an H∞ performance for external disturbances. Finally, the simulation results verify the feasibility and effectiveness of the proposed control method.

Taking the high bypass ratio turbofan engines as the research object for the dual-rotor system under the load of blade loss, a simplified four disk dual rotor system is established. The motion differential equation of the dual rotor system is established by the finite element method, and the transient response of the dual rotor system under the blade loss load is simulated and calculated with the Newmark-β integral method, considering the effects of rotor speed, eccentricity, supporting stiffness and damping. The results show that the transient response of the system has obvious impact characteristics under the excitation of the blade loss load, and the stiffness and damping of the bearing at the rear of the fan have a significant influence on the transient response.

This paper focuses on the command-filter control of nonstrict-feedback incommensurate fractional-order systems. We utilized fuzzy systems to approximate nonlinear systems, and designed an adaptive update law to estimate the approximation errors. To overcome the dimension explosion phenomenon in the backstepping process, we designed a fractional-order filter and applied the command filter control technique. The closed-loop system was semiglobally stable, and the tracking error converged to a small neighbourhood of equilibrium points under the proposed control approach. Lastly, the validity of the developed controller is verified with simulation examples.

In this paper, discussions were made on using DFT and IDFT to calculate the transient acceleration response of a single degree system. Two key points were discussed: the sampling frequencies and the time span. Simulation were made the transient response of a single degree system's transient response, it was found that the sampling frequencies have the greatest influence on the calculation error. The calculation precision's relation with the sampling frequencies were discussed, it shows that to keep calculation error in 10%, 99% energy in the low frequency should be calculated correctly.

Linear fractional differential equations with variable coefficients and various types of fractional derivatives can be solved explicitly, using either fixed‐point theorems and successive approximations or Green's functions. In this paper, we consider a multi‐term fractional differential equation with continuous variable coefficients and incommensurate fractional orders, and we construct an explicit solution by a direct method of successive approximations, which is made rigorous by checking the solution function explicitly via substitution. The results are obtained for the most general initial value problem of an inhomogeneous equation with inhomogeneous initial conditions, the first time that an explicit analytical solution has been found for this general problem. Special cases are provided to illustrate the main results, and connections are discussed with previous results in the literature.