Relationship Between Integer Order Systems and Fractional Order Systems and Its Two Applications

Existence of periodic solutions and stability of fractional order dynamic systems are two important and difficult issues in fractional order systems(FOS) field. In this paper, the relationship between integer order systems(IOS) and fractional order systems is discussed. A new proof method based on the above involved relationship for the non existence of periodic solutions of rational fractional order linear time invariant systems is derived. Rational fractional order linear time invariant autonomous system is proved to be equivalent to an integer order linear time invariant non-autonomous system. It is further proved that stability of a fractional order linear time invariant autonomous system is equivalent to the stability of another corresponding integer order linear time invariant autonomous system. The examples and state figures are given to illustrate the effects of conclusion derived.

Guest Editorial for Special Issue on Fractional Order Systems and Controls

I.INTRODUCTION FRACTIONAL calculus has been applied in all MAD(modeling,analysis and design)aspects of control systems engineering since Shunji Manabe’s pioneering work in early 1960s.The 2016 International Conference on Fractional Differentiation and Its Applications(ICFDA)was held in Novi Sad,Serbia,July 18-20.Quoting from the

CLASSIFICATION OF SOLUTIONS TO HIGHER FRACTIONAL ORDER SYSTEMS

Let 0<α,β<n and f,g∈ C([0,∞)×[0,∞))be two nonnegative functions.We study nonnegative classical solutions of the system{(-△)^(α/2)u=f(u,v)in R^(n),(-△)^(β/2)v=g(u,v)in R^(n),and the corresponding equivalent integral system.We classify all such solutions when f(s,t)is nondecreasing in s and increasing in t,g(s,t)is increasing in s and nondecreasing in i,and f(μ^(n-α)s,μ^(n-β)t)/μ^(n-α),g(μ^(n-α)s,μ^(n-β)t)/μ^(n-β)are nonincreasing in μ>0 for all s,t≥0.The main technique we use is the method of moving spheres in integral forms.Since our assumptions are more general than those in the previous literature,some new ideas are introduced to overcome this difficulty.

Criteria for Response Monotonicity Preserving in Approximation of Fractional Order Systems

In approximation of fractional order systems,a significant objective is to preserve the important properties of the original system.The monotonicity of time/frequency responses is one of these properties whose preservation is of great importance in approximation process.Considering this importance,the issues of monotonicity preservation of the step response and monotonicity preservation of the magnitude-frequency response are independently investigated in this paper.In these investigations,some conditions on approximating filters of fractional operators are found to guarantee the preservation of step/magnitude-frequency response monotonicity in approximation process.These conditions are also simplified in some special cases.In addition,numerical simulation results are presented to show the usefulness of the obtained conditions.

Guest Editorial for Special Issue on Fractional Order Systems and Controls

I.INTRODUCTION FRACTIONAL calculus is about differentiation and integration of non-integer orders.Using integer-order models and controllers for complex natural or man-made systems is simply for our own convenience while the nature runs in a fractional order dynamical way.Using integer order traditiona tools for modelling and control of dynamic systems may resul in suboptimum performance,that is,using fractional order calculus tools,we could be'more optimal'as already doc-

An Exploration on Adaptive Iterative Learning Control for a Class of Commensurate High-order Uncertain Nonlinear Fractional Order Systems

This paper explores the adaptive iterative learning control method in the control of fractional order systems for the first time. An adaptive iterative learning control(AILC) scheme is presented for a class of commensurate high-order uncertain nonlinear fractional order systems in the presence of disturbance.To facilitate the controller design, a sliding mode surface of tracking errors is designed by using sufficient conditions of linear fractional order systems. To relax the assumption of the identical initial condition in iterative learning control(ILC), a new boundary layer function is proposed by employing MittagLeffler function. The uncertainty in the system is compensated for by utilizing radial basis function neural network. Fractional order differential type updating laws and difference type learning law are designed to estimate unknown constant parameters and time-varying parameter, respectively. The hyperbolic tangent function and a convergent series sequence are used to design robust control term for neural network approximation error and bounded disturbance, simultaneously guaranteeing the learning convergence along iteration. The system output is proved to converge to a small neighborhood of the desired trajectory by constructing Lyapnov-like composite energy function(CEF)containing new integral type Lyapunov function, while keeping all the closed-loop signals bounded. Finally, a simulation example is presented to verify the effectiveness of the proposed approach.

The Ellipsoidal Invariant Set of Fractional Order Systems Subject to Actuator Saturation:The Convex Combination Form

The domain of attraction of a class of fractional order systems subject to saturating actuators is investigated in this paper. We show the domain of attraction is the convex hull of a set of ellipsoids. In this paper, the Lyapunov direct approach and fractional order inequality are applied to estimating the domain of attraction for fractional order systems subject to actuator saturation. We demonstrate that the convex hull of ellipsoids can be made invariant for saturating actuators if each ellipsoid with a bounded control of the saturating actuators is invariant. The estimation on the contractively invariant ellipsoid and construction of the continuous feedback law are derived in terms of linear matrix inequalities(LMIs). Two numerical examples illustrate the effectiveness of the developed method.

A New Technique for Solving Fractional Order Systems: Hermite Collocation Method

In this study, we establish an approximate method which produces an approximate Hermite polynomial solution to a system of fractional order differential equations with variable coefficients. At collocation points, this method converts the mentioned system into a matrix equation which corresponds to a system of linear equations with unknown Hermite polynomial coefficients. Construction of the method on the aforementioned type of equations has been presented and tested on some numerical examples. Results related to the effectiveness and reliability of the method have been illustrated.

Stabilizing controller design for nonlinear fractional order systems with time varying delays

To deal with stabilizing of nonlinear affine fractional order systems subject to time varying delays,two methods for finding an appropriate pseudo state feedback controller are discussed.In the first method,using the Mittag-Lefler function,Laplace transform and Gronwall inequality,a linear stabilizing controller is derived,which uses the fractional order of the delayed system and the upper bound of system nonlinear functions.In the second method,at first a sufficient stability condition for the delayed system is given in the form of a simple linear matrix inequality(LMI)which can easily be solved.Then,on the basis of this result,a stabilizing pseudo-state feedback controller is designed in which the controller gain matrix is easily computed by solving an LMI in terms of delay bounds.Simulation results show the effectiveness of the proposed methods.

Lyapunov Stability Analysis for Incommensurate Nabla Fractional Order Systems

This paper investigates the problem of stability analysis for a class of incommensurate nabla fractional order systems.In particular,both Caputo definition and Riemann-Liouville definition are under consideration.With the convex assumption,several elementary fractional difference inequalities on Lyapunov functions are developed.According to the essential features of nabla fractional calculus,the sufficient conditions are given first to guarantee the asymptotic stability for the incommensurate system by using the direct Lyapunov method.To substantiate the efficacy and effectiveness of the theoretical results,four examples are elaborated.

Fractional-order heterogeneous memristive Rulkov neuronal network and its medical image watermarking application

This article proposes a novel fractional heterogeneous neural network by coupling a Rulkov neuron with a Hopfield neural network(FRHNN),utilizing memristors for emulating neural synapses.The study firstly demonstrates the coexistence of multiple firing patterns through phase diagrams,Lyapunov exponents(LEs),and bifurcation diagrams.Secondly,the parameter related firing behaviors are described through two-parameter bifurcation diagrams.Subsequently,local attraction basins reveal multi-stability phenomena related to initial values.Moreover,the proposed model is implemented on a microcomputer-based ARM platform,and the experimental results correspond to the numerical simulations.Finally,the article explores the application of digital watermarking for medical images,illustrating its features of excellent imperceptibility,extensive key space,and robustness against attacks including noise and cropping.

Video Transmission Secrecy Improvement Based on Fractional Order Hyper Chaotic System

In the Digital World scenario,the confidentiality of information in video transmission plays an important role.Chaotic systems have been shown to be effective for video signal encryption.To improve video transmission secrecy,compressive encryption method is proposed to accomplish compression and encryption based on fractional order hyper chaotic system that incorporates Compressive Sensing(CS),pixel level,bit level scrambling and nucleotide Sequences operations.The measurement matrix generates by the fractional order hyper chaotic system strengthens the efficiency of the encryption process.To avoid plain text attack,the CS measurement is scrambled to its pixel level,bit level scrambling decreases the similarity between the adjacent measurements and the nucleotide sequence operations are done on the scrambled bits,increasing the encryption.Two stages are comprised in the reconstruction technique,the first stage uses the intra-frame similarity and offers robust preliminary retrieval for each frame,and the second stage iteratively improves the efficiency of reconstruction by integrating inter frame Multi Hypothesis(MH)estimation and weighted residual sparsity modeling.In each iteration,the residual coefficient weights are modified using a mathematical approach based on the MH predictions,and the Split Bregman iteration algorithm is defined to resolve weighted l1 regularization.Experimental findings show that the proposed algorithm provides good compression of video coupled with an efficient encryption method that is resistant to multiple attacks.

An Adaptive Neuro-Fuzzy Inference System to Improve Fractional Order Controller Performance

The design and analysis of a fractional order proportional integral deri-vate(FOPID)controller integrated with an adaptive neuro-fuzzy inference system(ANFIS)is proposed in this study.Afirst order plus delay time plant model has been used to validate the ANFIS combined FOPID control scheme.In the pro-posed adaptive control structure,the intelligent ANFIS was designed such that it will dynamically adjust the fractional order factors(λandµ)of the FOPID(also known as PIλDµ)controller to achieve better control performance.When the plant experiences uncertainties like external load disturbances or sudden changes in the input parameters,the stability and robustness of the system can be achieved effec-tively with the proposed control scheme.Also,a modified structure of the FOPID controller has been used in the present system to enhance the dynamic perfor-mance of the controller.An extensive MATLAB software simulation study was made to verify the usefulness of the proposed control scheme.The study has been carried out under different operating conditions such as external disturbances and sudden changes in input parameters.The results obtained using the ANFIS-FOPID control scheme are also compared to the classical fractional order PIλDµand conventional PID control schemes to validate the advantages of the control-lers.The simulation results confirm the effectiveness of the ANFIS combined FOPID controller for the chosen plant model.Also,the proposed control scheme outperformed traditional control methods in various performance metrics such as rise time,settling time and error criteria.

A Note on Robust Stability Analysis of Fractional Order Interval Systems by Minimum Argument Vertex and Edge Polynomials

By using power mapping(s =v^m),stability analysis of fractional order polynomials was simplified to the stability analysis of expanded degree integer order polynomials in the first Riemann sheet.However,more investigation is needed for revealing properties of power mapping and demonstration of conformity of Hurwitz stability under power mapping of fractional order characteristic polynomials.Contributions of this study have two folds: Firstly,this paper demonstrates conservation of root argument and magnitude relations under power mapping of characteristic polynomials and thus substantiates validity of Hurwitz stability under power mapping of fractional order characteristic polynomials.This also ensures implications of edge theorem for fractional order interval systems.Secondly,in control engineering point of view,numerical robust stability analysis approaches based on the consideration of minimum argument roots of edge and vertex polynomials are presented.For the computer-aided design of fractional order interval control systems,the minimum argument root principle is applied for a finite set of edge and vertex polynomials,which are sampled from parametric uncertainty box.Several illustrative examples are presented to discuss effectiveness of these approaches.

State Feedback Control for a Class of Fractional Order Nonlinear Systems

Using the Lyapunov function method,this paper investigates the design of state feedback stabilization controllers for fractional order nonlinear systems in triangular form,and presents a number of new results.First,some new properties of Caputo fractional derivative are presented,and a sufficient condition of asymptotical stability for fractional order nonlinear systems is obtained based on the new properties.Then,by introducing appropriate transformations of coordinates,the problem of controller design is converted into the problem of finding some parameters,which can be certainly obtained by solving the Lyapunov equation and relevant matrix inequalities.Finally,based on the Lyapunov function method,state feedback stabilization controllers making the closed-loop system asymptotically stable are explicitly constructed.A simulation example is given to demonstrate the effectiveness of the proposed design procedure.

Constrained Swarm Stabilization of Fractional Order Linear Time Invariant Swarm Systems

This paper deals with asymptotic swarm stabilization of fractional order linear time invariant swarm systems in the presence of two constraints: the input saturation constraint and the restriction on distance of the agents from final destination which should be less than a desired value. A feedback control law is proposed for asymptotic swarm stabilization of fractional order swarm systems which guarantees satisfying the above-mentioned constraints. Numerical simulation results are given to confirm the efficiency of the proposed control method.

Chaos and Combination Synchronization of a New Fractional-order System with Two Stable Node-foci

A new fractional-order Lorenz-like system with two stable node-foci has been thoroughly studied in this paper.Some sufficient conditions for the local stability of equilibria considering both commensurate and incommensurate cases are given. In addition, with the effective dimension less than three,the minimum effective dimension of the system is approximated as 2.8485 and is verified numerically. It should be affirmed that the linear differential equation in fractional-order Lorenzlike system appears to be less sensitive to the damping, represented by a fractional derivative, than the two other nonlinear equations. Furthermore, combination synchronization of this system is analyzed with the help of nonlinear feedback control method. Theoretical results are verified by performing numerical simulations.

Robust Output Feedback Control for Fractional Order Nonlinear Systems with Time-varying Delays

Robust controller design problem is investigated for a class of fractional order nonlinear systems with time varying delays.Firstly,a reduced-order observer is designed.Then,an output feedback controller is designed.Both the designed observer and controller are independent of time delays.By choosing appropriate Lyapunov functions,we prove the designed controller can render the fractional order system asymptotically stable.A simulation example is given to verify the effectiveness of the proposed approach.

Leader-following consensus of discrete-time fractional-order multi-agent systems

Leader-following consensus of fractional order multi-agent systems is investigated. The agents are considered as discrete-time fractional order integrators or fractional order double-integrators. Moreover, the interaction between the agents is described with an undirected communication graph with a fixed topology. It is shown that the leader-following consensus problem for the considered agents could be converted to the asymptotic stability analysis of a discrete-time fractional order system. Based on this idea, sufficient conditions to reach the leader-following consensus in terms of the controller parameters are extracted. This leads to an appropriate region in the controller parameters space. Numerical simulations are provided to show the performance of the proposed leader-following consensus approach.

Stabilization of Uncertain Fractional Memristor Chaotic Time⁃Delay System Based on Fractional Order Sliding Mode Control

Based on Lyapunov theorem and sliding mode control scheme,the chaos control of fractional memristor chaotic time⁃delay system was studied.In order to stabilize the system,a fractional sliding mode control method for fractional time⁃delay system was proposed.In addition,Lyapunov stability theorem was used to analyze the control scheme theoretically,which guaranteed the stability of commensurate and non⁃commensurate order systems with or without uncertainties and disturbances.Furthermore,to illustrate the feasibility of controller,the conditions for designing the controller parameters were derived.Finally,the simulation results presented the effectiveness of the designed strategy.