Optimal tuning of fractional controllers using genetic algorithms

This study addresses the optimization of fractional algorithms for the discrete-time control of linear and non-linear systems. The paper starts by analyzing the fundamentals of fractional control systems and genetic algorithms. In a second phase the paper evaluates the problem in an optimization perspective. The results demonstrate the feasibility of the evolutionary strategy and the adaptability to distinct types of systems.

Application of fractional algorithms in the control of a robotic bird

In this paper, it is studied the dynamics of the robotic bird in terms of time response and robustness. It is analyzed the wing angle of attack and the velocity of the bird, the tail influence, the gliding flight and the flapping flight. The results are positive for the construction of flying robots. The development of computational simulation based on the dynamic of the robotic bird should allow testing strategies and different algorithms of control such as integer and fractional controllers.

Effect of fractional orders in the velocity control of a servo system

The application of fractional-order PID controllers is now an active field of research. This article investigates the effect of fractional (derivative and integral) orders upon system's performance in the velocity control of a servo system. The servo system consists of a digital servomechanism and an open-architecture software environment for real-time control experiments using MATLAB/Simulink tools. Experimental responses are presented and analyzed, showing the effectiveness of fractional controllers. Comparison with classical PID controllers is also investigated.

Approximating fractional derivatives through the generalized mean

This paper addresses the calculation of fractional order expressions through rational fractions. The article starts by analyzing the techniques adopted in the continuous to discrete time conversion. The problem is re-evaluated in an optimization perspective by tacking advantage of the degree of freedom provided by the generalized mean formula. The results demonstrate the superior performance of the new algorithm.

Fractional derivatives: probability interpretation and frequency response of rational approximations

The theory of fractional calculus (FC) is a useful mathematical tool in many applied sciences. Nevertheless, only in the last decades researchers were motivated for the adoption of the FC concepts. There are several reasons for this state of affairs, namely the co-existence of different definitions and interpretations, and the necessity of approximation methods for the real time calculation of fractional derivatives (FDs). In a first part, this paper introduces a probabilistic interpretation of the fractional derivative based on the Grünwald-Letnikov definition. In a second part, the calculation of fractional derivatives through Padé fraction approximations is analyzed. It is observed that the probabilistic interpretation and the frequency response of fraction approximations of FDs reveal a clear correlation between both concepts.

A numerical analysis of generalized Boussinesq-type equation using continuos/discontinuous FEM

An improved class of Boussinesq systems of an arbitrary order using a wave surface elevation and velocity potential formulation is derived. Dissipative effects and wave generation due to a time-dependent varying seabed are included. Thus, high-order source functions are considered. For the reduction of the system order and maintenance of some dispersive characteristics of the higher-order models, an extra O(mu 2n+2) term (n ??? N) is included in the velocity potential expansion. We introduce a nonlocal continuous/discontinuous Galerkin FEM with inner penalty terms to calculate the numerical solutions of the improved fourth-order models. The discretization of the spatial variables is made using continuous P2 Lagrange elements. A predictor-corrector scheme with an initialization given by an explicit RungeKutta method is also used for the time-variable integration. Moreover, a CFL-type condition is deduced for the linear problem with a constant bathymetry. To demonstrate the applicability of the model, we considered several test cases. Improved stability is achieved.

A fractional linear system view of the fractional brownian motion

Nonlinear Dynamics, Vol. 38

Fractional signal processing: scale conversion

In Proceedings of the “ECCTD '01 - European Conference on Circuit Theory and Design, Espoo, Finland, August 2001

Approximating fractional derivatives in the perspective of system control

The theory of fractional calculus goes back to the beginning of the theory of differential calculus, but its application received attention only recently. In the area of automatic control some work was developed, but the proposed algorithms are still in a research stage. This paper discusses a novel method, with two degrees of freedom, for the design of fractional discrete-time derivatives. The performance of several approximations of fractional derivatives is investigated in the perspective of nonlinear system control.

Analysis of the Van der Pol oscillator containing derivatives of fractional order

In this paper a modified version of the classical Van der Pol oscillator is proposed, introducing fractional-order time derivatives into the state-space model. The resulting fractional-order Van der Pol oscillator is analyzed in the time and frequency domains, using phase portraits, spectral analysis and bifurcation diagrams. The fractional-order dynamics is illustrated through numerical simulations of the proposed schemes using approximations to fractional-order operators. Finally, the analysis is extended to the forced Van der Pol oscillator.

Fractional Dynamics of Computer Virus Propagation

We propose a fractional model for computer virus propagation. The model includes the interaction between computers and removable devices. We simulate numerically the model for distinct values of the order of the fractional derivative and for two sets of initial conditions adopted in the literature. We conclude that fractional order systems reveal richer dynamics than the classical integer order counterpart. Therefore, fractional dynamics leads to time responses with super-fast transients and super-slow evolutions towards the steady-state, effects not easily captured by the integer order models.

Analysis of diffusion process in fractured reservoirs using fractional derivative approach

The fractal geometry is used to model of a naturally fractured reservoir and the concept of fractional derivative is applied to the diffusion equation to incorporate the history of fluid flow in naturally fractured reservoirs. The resulting fractally fractional diffusion (FFD) equation is solved analytically in the Laplace space for three outer boundary conditions. The analytical solutions are used to analyze the response of a naturally fractured reservoir considering the anomalous behavior of oil production. Several synthetic examples are provided to illustrate the methodology proposed in this work and to explain the diffusion process in fractally fractured systems.

Reply to: Comments on “Particle Swarm Optimization with Fractional-Order Velocity”

We agree with Ling-Yun et al. [5] and Zhang and Duan comments [2] about the typing error in equation (9) of the manuscript [8]. The correct formula was initially proposed in [6, 7]. The formula adopted in our algorithms discussed in our papers [1, 3, 4, 8] is, in fact, the following: ...

Rhapsody in fractional

This paper studies several topics related with the concept of “fractional” that are not directly related with Fractional Calculus, but can help the reader in pursuit new research directions. We introduce the concept of non-integer positional number systems, fractional sums, fractional powers of a square matrix, tolerant computing and FracSets, negative probabilities, fractional delay discrete-time linear systems, and fractional Fourier transform.

Delay approximation of fractional integrals

This paper explores the calculation of fractional integrals by means of the time delay operator. The study starts by reviewing the memory properties of fractional operators and their relationship with time delay. Based on the time response of the Mittag-Leffler function an approximation of fractional integrals consisting of time delayed samples is proposed. The tuning of the approximation is optimized by means of a genetic algorithm. The results demonstrate the feasibility of the new perspective and the limits of their application.