Differentiation Of Non-integer Order Research Articles

Analysis and Applications of the Chaotic Hyperbolic Memristor Model with Fractional Order Derivative

This work presents a new five-dimensional fractional-order chaotic system that includes a feedback memristor. A Lorenz-Stenflo-based fractional-order chaotic system with five dimensions that contains feedback memristor dynamics is used to do this. This work focuses on chaotic models through the ABC fractional derivative, a concept we rigorously examine. We designed and usedsophisticated numerical algorithms to model fractional-order dynamics with the utmost precision to understand this system’s complex characteristics. These numerical approaches excel at non-integer order differentiation, which standard numerical methods struggle with. Our research uses fractional calculus to increase the system’s complexity and robustness, revealing its hyperchaotic nature. We demonstrate that the system is chaotic and stable over fractional orders using eigenvalues, Lyapunov exponents, Kaplan-Yorke dimensions, maximal exponents, phase portraits, and equilibrium points. Our conclusions are strengthened by using these numerical systems, intended for this work, and analytical tools. We improve our understanding of fractional-order chaotic systems with our research. It illuminates how to improve electrical integrations and create more sophisticated nonlinear dynamical systems. This work establishes the approach and insights for future research, guiding the development of new systems with enhanced dynamical features

Non-Markovian dynamics of open quantum system with memory

In this paper, non-Markovian generalization of master equation for open quantum system is proposed. Non-Markovian dynamics of two-level quantum system with memory and interaction with environment is described. To describe this system, the Gorini–Kossakowski–Sudarshan equation for open quantum states is generalized by taking into account power-law fading memory. The non-Markovian quantum processes with power-law memory are described by using integration and differentiation of non-integer orders. Complete positivity and bi-positivity in non-Markovian quantum dynamics with memory are described. An example of two-level quantum systems with power-law memory is suggested. Exact solution of the non-Markovian master equations of two-level open quantum systems with memory is derived.

Quantum Maps with Memory from Generalized Lindblad Equation.

In this paper, we proposed the exactly solvable model of non-Markovian dynamics of open quantum systems. This model describes open quantum systems with memory and periodic sequence of kicks by environment. To describe these systems, the Lindblad equation for quantum observable is generalized by taking into account power-law fading memory. Dynamics of open quantum systems with power-law memory are considered. The proposed generalized Lindblad equations describe non-Markovian quantum dynamics. The quantum dynamics with power-law memory are described by using integrations and differentiation of non-integer orders, as well as fractional calculus. An example of a quantum oscillator with linear friction and power-law memory is considered. In this paper, discrete-time quantum maps with memory, which are derived from generalized Lindblad equations without any approximations, are suggested. These maps exactly correspond to the generalized Lindblad equations, which are fractional differential equations with the Caputo derivatives of non-integer orders and periodic sequence of kicks that are represented by the Dirac delta-functions. The solution of these equations for coordinates and momenta are derived. The solutions of the generalized Lindblad equations for coordinate and momentum operators are obtained for open quantum systems with memory and kicks. Using these solutions, linear and nonlinear quantum discrete-time maps are derived.

On the evolution of fractional diffusive waves

In physics, phenomena of diffusion and wave propagation have great relevance; these physical processes are governed in the simplest cases by partial differential equations of order 1 and 2 in time, respectively. By replacing the time derivatives in the above standard equations with pseudo-differential operators interpreted as derivatives of non integer order (nowadays misnamed as of fractional order) we are lead to generalized processes of diffusion that may be interpreted as slow diffusion and interpolating between diffusion and wave propagation. In mathematical physics we may refer these interpolating processes to as fractional diffusion-wave phenomena. In this work we analyze and simulate both the situations in which the input function is a Dirac delta generalized function and a box function, restricting ourselves to the Cauchy problem. In the firsrst case we get the fundamental solutions (or Green functions) of the problem whereas in the latter case the solutions are obtained by a space convolution of the Green function with the input function. In order to clarify the matter for the non-specialist readers, we briefly recall the basic and essential notions of the fractional calculus (the mathematical theory that regards the integration and differentiation of noninteger order) with a look at the history of this discipline.

Discrete Non-Integer Order Differentiator Models Using Moth-Flame Optimization Algorithm

Discrete rational approximation models to the non-integer order differentiator sλ, where λ ε (0, 1), using Moth-Flame Optimization (MFO) algorithm is proposed in this paper. The proposed metaheuristic optimization approach used to design the discrete non-integer order differentiators (DNODs) does not employ any s-to-z domain mapping function to perform the discretization operation. Frequency domain characteristics of DNODs, solution reliability, and algorithm convergence performances are investigated among MFO and an advanced evolutionary algorithm called Particle Swarm Optimization with adaptive inertia weight (PSO-w). Results demonstrate the effectiveness of MFO in outperforming PSO-w in solving this non-linear and multimodal optimization problem. The proposed DNODs also exhibit better performance in comparison with the designs based on techniques such as Nelder-Mead Simplex algorithm and Cuckoo Search Algorithm published in recent literature.

Nonlinear adaptive fractional order fuzzy PID control of a 2-link planar rigid manipulator with payload

The main objective of this paper is to demonstrate that fractional order operators (differentiator/integrator) can increased the robustness of an intelligent controller. In this paper, a nonlinear adaptive fractional order fuzzy proportional integral derivative (NLA – FOFPID) controller is presented to control a nonlinear, coupled, multi-input and multi-output and uncertain system i.e. a 2-link planar rigid robotic manipulator with payload. The NLA – FOFPID controller is realized using non-integer order differentiator and integrator operators in nonlinear adaptive fuzzy proportional integral derivative (NLA – FPID) controller. The control objective is to track a reference trajectory in virtual industrial environment and under model uncertainties by minimizing the combination of integral of time weighted absolute error and integral of time weighted absolute change in controller output. The industrial environment was simulated by incorporating measurement noise, sinusoidal disturbance and model uncertainties. The gains of NLA – FPID and NLA – FOFPID controllers are optimized using recently reported Backtracking Search Algorithm. Intensive simulations were performed to assess the performances of NLA – FPID and NLA – FOFPID controllers for servo and regulatory mode. It has been observed that NLA – FOFPID controller outperforms NLA – FPID controller by offering much better performance. Particularly in uncertain environment it offered very robust behavior as compared to NLA – FPID controller. The sufficient stability conditions are also established for bounded-input and bounded-output stability of overall closed loop system by using small gain theorem.

Lattice fractional calculus

Integration and differentiation of non-integer orders for N-dimensional physical lattices with long-range particle interactions are suggested. The proposed lattice fractional derivatives and integrals are represented by kernels of lattice long-range interactions, such that their Fourier series transformations have a power-law form with respect to components of wave vector. Continuous limits for these lattice fractional derivatives and integrals give the continuum derivatives and integrals of non-integer orders with respect to coordinates. Lattice analogs of fractional differential equations that include suggested lattice differential and integral operators can serve as an important element of microscopic approach to nonlocal continuum models in mechanics and physics.

REVIEW OF SOME PROMISING FRACTIONAL PHYSICAL MODELS

Fractional dynamics is a field of study in physics and mechanics investigating the behavior of objects and systems that are characterized by power-law nonlocality, power-law long-term memory or fractal properties by using integrations and differentiation of non-integer orders, i.e., by methods in the fractional calculus. This paper is a review of physical models that look very promising for future development of fractional dynamics. We suggest a short introduction to fractional calculus as a theory of integration and differentiation of noninteger order. Some applications of integro-differentiations of fractional orders in physics are discussed. Models of discrete systems with memory, lattice with long-range inter-particle interaction, dynamics of fractal media are presented. Quantum analogs of fractional derivatives and model of open nano-system systems with memory are also discussed.

Quantum dissipation from power-law memory

A new quantum dissipation model based on memory mechanism is suggested. Dynamics of open and closed quantum systems with power-law memory is considered. The processes with power-law memory are described by using integration and differentiation of non-integer orders, by methods of fractional calculus. An example of quantum oscillator with linear friction and power-law memory is considered.

Discussion on: “Simple Fractional Order Model Structures and their Applications in Control System Design”

Fractional calculus is about the integration or differentiation of non-integer orders. The use of “fractional” is purely due to historical reasons [1]. Using fractional order differential equations is believed to be able to better characterize the nature around us. Using an integer order model is only for our own convenience. Depending on the scale onwhich we characterize the dynamics of a system, more and more evidences are found that using fractional order model is ubiquitous and unavoidable [1, 2, 3]. For dynamic systems and controls, there are four possibilities 1) Integer-order plant model with integer order controller; 2) Integer-order plant model with fractional order controller; 3) Fractional-order plant with integerorder controller; and 4) Fractional-order plant model with fractional-order controller [2]. Case-1 is our traditional (integer-order) control. An example of Case 2) can be found in [3] and an example of Case-4 can be found in [4]. Examples for Case-3 are abundant such as conventional integer-order controls for flexible structures whose transfer functions are irrational or of fractional order. It is interesting to note that, reaction curve based rough modeling technique, although widely used in process industry to get a first order plus delay time (FOPDT) for initial tuning of a working PID controller with techniques such as Ziegler-Nichols tuning rule, has not been put enough emphasis in control textbooks.Webelieve that, every practicing control engineer and control engineering educator should pay attention to the importance of reaction

Non-Integer Motion Control: Application to an XY Cutting Table

A new approach to path tracking using non integer order differentiation applied to non-time-varying plants is proposed with its application on a XY cutting table. It pennits the generation of optimal movement reference-input leading to a minimum path completion time, taking into account both the physical constraints of the actuators (maximum velocity, acceleration and torque) and the bandwidth of the closedloop system. This reference-input results from the step response of a non integer order filter whose frequency response is that of an implicit generalized derivative filter. This new approach is complementary to CRONE control which synthesizes a robust control law, and which is based on real or complex non-integer order differentiation. The reference-input obtained is compared with spline function. Both are applied to an XY cutting table and actuator outputs compared.

Frequency-band complex noninteger differentiator: characterization and synthesis

The state-of-the-art on generalized (or any order) derivatives in physics and engineering sciences, is outlined for justifying the interest of the noninteger differentiation. The problems subsequent to its use in real-time operations are then set out so as to motivate the idea of synthesizing it by a recursive distribution of zeros and poles. An analysis of the existing work is also proposed to support this idea. A comprehensive study is given of the synthesis of differentiators with integer, noninteger, real or complex orders, and whose action is limited to any given frequency bandwidth. First, a definition, in the operational and frequency domains, of a frequency-band complex noninteger order differentiator, is given in a mathematical space with four dimensions which is a Banach algebra. Then, the determination of its synthesized form, by a recursive distribution of complex zeros and poles characterized by complex recursive factors, is presented. The complex noninteger differentiation order is expressed as a function of these recursive factors. The number of zeros and poles is calculated to be as low as possible while still ensuring the stability of the synthesized differentiator to be synthesized. A time validation is presented. Finally, guidelines are proposed for the conception of the synthesized differentiator.