IFAC-PapersOnline

A formulation and a numerical scheme for fractional optimal control problems

04:00 PM Wednesday 31, 1969

This paper presents a general formulation and a general numerical scheme for a class of Fractional Optimal Control Problems (FOCPs). The fractional derivative is described in the Caputo sense. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of Fractional Differential Equations (FDEs). The Calculus of Variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler-Lagrange equations for the FOCP. The formulation presented and the resulting equations are very similar to those that appear in the classical optimal control theory. Thus, the present formulation essentially extends the classical control theory to fractional dynamic systems. The formulation is used to derive the control equations for a quadratic linear fractional control problem. An iterative numerical scheme is presented to find the approximate numerical solution of the resulting equations. For a linear system, this method results into a set of linear simultaneous equations, which can be solved directly. Numerical results for a FOCP are presented to demonstrate the feasibility of the method. It is shown that the solutions converge as the number of grid points increases, and the solutions approach to classical solutions as the order of the fractional derivatives approach to 1. The formulation presented is simple and can be extended to other FOCPs.

A fractional adaptation scheme for lateral control of an AGV

04:00 PM Wednesday 31, 1969

The lateral control of an autonomous guided vehicle (AGV) is highly influenced both for the longitudinal speed and the position input command (magnitude of the reference signal) of the vehicle. For that reason, a suitable strategy to govern the vehicle would be to use an adaptive and robust controller. In this paper an adaptive scheme is proposed which combines a model reference approach and a fractional order adjustment rule for a feedforward gain adjustment. Two parameters can be tuned to obtain robustness against speed and magnitude of the reference signal variations: adaptation gain, and derivative order of the adjustment rule. A model is developed for the vehicle, the design procedure is exposed, and simulation results are obtained to show the advantages of using the proposed fractional adaptation scheme.

A general fractional finite element formulation

04:00 PM Wednesday 31, 1969

Differential equations governing the behaviors of systems can be obtained through minimizations of energy functionals. In many applications, finite element formulations for a system can be developed directly using energy or some other types of functionals. One of the advantages of these formulations is that they do not require the natural boundary conditions to be imposed explicitly. In recent years, there has been a growing interest in the area of fractional variational calculus. Recently developed fractional variational formulations allow one to obtain the necessary Euler-Lagrange equations and the natural boundary conditions. However, closed form solutions for these problems may not be available, and a numerical technique may be necessary. In this paper, we present a general fractional finite element formulation for a class of fractional variational problems defined in terms of the Riemann-Liouville fractional derivatives. Specifically, we consider a functional quadratic in function y and its fractional derivative aDtαy, where α is the order of the derivative. In this formulation, the domain of the integral is divided into several elements, and y and aDtαy over the elements are written in terms of nodal variables and some shape functions. Various choices of shape functions are being considered. Here we approximate y and aDtαy over each element using a linear function and a constant function, respectively. The constant function considered here represents aDtαy at the center of the element. These approximations are used to reduce the original functional into a discretized quadratic form. Two approaches are presented to approximate aDtαy at the center of an element. In the first scheme, 1) the number of node points is doubled; 2) the function values at the new node points are computed using the linear function; and 3) aDtαy is expressed using Grunwald-Letnikov formula. In the second scheme, the Grunwald-Letnikov formula is modified to write aDtαy at the center of an element in terms of nodal functional values. This modified Grunwald-Letnikov formula, when written for α = 1, gives the central difference formula for the derivative. These approximations are substituted into the above quadratic equation, the resulting equation is minimized, and the terminal kinematic conditions are imposed to obtain a set of algebraic equations. These algebraic equations can be solved using a direct or an iterative scheme. Here we use a direct scheme. Solution of these equations provides the response of the system. To demonstrate the applications of the formulations developed here, two fractional variational problems are solved using the scheme developed here, and the results are compared with analytical solutions and the solutions obtained using other schemes. Results show that 1) solutions converge as the number of node points is increased. However, in some cases, this convergence may be slow. 2) As α approaches 1, the classical solutions are recovered as expected. For integer order systems, finite element is a well established technique, and it is widely being used in academia and industries. It is hoped that this research will initiate a similar course for fractional order system.

A note on the controllability and the observability of fractional dynamical systems

04:00 PM Wednesday 31, 1969

In this paper, we give some new results on the controllability and the observability of linear dynamical systems with a fractional derivative of order α, where α is a non integer number. We show that the observability and the controllability Gramians, recently introduced for a fractional order system, are solutions of fractional differential Lyapunov equations, thus generalizing the classical result for the integer case (α = 1). Our results can be considered as a generalization of the known corresponding results in the integer order case to the fractional order one since for α = 1, the results for the integer case are recovered.

A numerical algorithm for differential equations with nonlinear fractional derivatives

04:00 PM Wednesday 31, 1969

A numerical integration algorithm to solve single-degree of freedom (1- DOF) nonlinear fractional differential equations (NFDEs) is developed by the use of the one stepsc scheme; Newmark-β method. The NFDE involving the fractional derivatives of the displacement and the displacement squared is numerically solved by the proposed numerical integration algorithm. In this paper, the derivation of the numerical integration algorithm, the error analysis of the algorithm, and the contribution of the displacement squared fractional derivative term on the solution of the NFDE are given.

A robust tuning method for fractional order PI controllers

04:00 PM Wednesday 31, 1969

The application of fractional controller attracts more attention in the recent years. In this paper, a new tuning method for PIα controller design is proposed for a class of unknown, stable, and minimum phase plants. We are able to design a PIα controller to ensure that the phase Bode plot is flat, i.e., the phase derivative w.r.t. the frequency is zero, at a given gain crossover frequency so that the closed-loop system is robust to gain variations and the step responses exhibit an iso-damping property. Several relay feedback tests can be used to identify the plant gain and phase at the given frequency in an iterative way. The identified plant gain and phase at the desired tangent frequency are used to estimate the derivatives of amplitude and phase of the plant with respect to frequency at the same frequency point by Bode's integral relationship. Then, these derivatives are used to design a PIα controller for slope adjustment of the Nyquist plot to achieve the robustness of the system to gain variations. No plant model is assumed during the PIα controller design. Only several relay tests are needed.

A speculative study of fractional Laplacian modeling of turbulence

04:00 PM Wednesday 31, 1969

This study makes the first attempt to use the 2/3-order fractional Laplacian modeling of Kolmogorov -5/3 scaling of fully developed turbulence and enhanced diffusing movements of random turbulent particles. Nonlinear inertial interactions and the molecular Brownian diffusivity are considered the bi-fractal mechanism behind multifractal scaling of moderate Reynolds number turbulence. Accordingly, a stochastic equation is proposed to describe turbulence intermittency. The 2/3-order fractional Laplacian representation is also used to model nonlinear interactions of fluctuating velocity components, and then we conjecture a fractional Reynolds equation, underlying fractal spacetime structures of Lévy 2/3 stable distribution and the Kolmogorov scaling at inertial scales. The new perspective of this study is that the fractional calculus is an effective approach modeling of chaotic fractal phenomena induced by nonlinear interactions.

Adaptive discretization of an integro-differential equation modeling quasi-static fractional order v

04:00 PM Wednesday 31, 1969

We study a quasi-static model for viscoelastic materials based on a constitutive equation of fractional order. In the quasi-static case this results in a Volterra integral equation of the second kind with a weakly singular kernel in the time variable involving also partial derivatives of second order in the spatial variables. We discretize by means of a discontinuous Galerkin finite element method in time and a standard continuous Galerkin finite element method in space. To overcome the problem of the growing amount of data that has to be stored and used in each time step, we introduce sparse quadrature in the convolution integral. We prove a priori and a posteriori error estimates, and develop an adaptive strategy based on the a posteriori error estimate.

Analog fractional order controller in a temperature control application

04:00 PM Wednesday 31, 1969

An analog fractional order PIα controller using a fractional order impedance device, a FractorTM (patent pending), is demonstrated in a temperature control application. The performance improvement over a standard PI controller was notable in reduction of overshoot and decreased time to stable temperature while retaining the long term stability and set point accuracy of the standard controller. The modification of the standard controller to a fractional order controller was as simple as replacing the integrator capacitor with a FractorTM.

Analog implementation of non integer order integrator via Field Programmable Analog Array

04:00 PM Wednesday 31, 1969

Recently a renewed interest has been devoted to non integer, or fractional, order systems. This is due to the fact that they well model a lot of physical systems and can be usefully applied in the area of automatic control. On drawback, mainly faced in the area of control systems, is related to their practical realization, essentially due to their infinite dimension nature. In this paper it is proposed an analog implementation of a non integer order integrator by using Field Programmable Analog Array. The reported example shows the feasibility and reliability of the proposed approach.

Analysis of fractional-order robot axis dynamics

04:00 PM Wednesday 31, 1969

Robots are complex mechatronics systems where several electric drives are employed to control the movement of articulated structures. In industrial environments they must perform tasks with rapidity and accuracy in order to produce goods and services with minimal production time. These procedures require the use of flexible robots which can act in a large workspace, thus subjected to important parameters variations and nonlinear dynamics effects. This paper investigates the fractional order dynamics during the evolution of trajectories of three robotic joints, considering the complete system dynamics.

Application of a sigma-point Kalman-filter for the online estimation of fractional order impedance m

04:00 PM Wednesday 31, 1969

To operate a solid oxide fuel cell (SOFC) safely and efficiently, it is necessary to monitor the values of characteristic model parameters of the cell impedance. This paper addresses the problem to compute online-estimates of model parameters for SOFCs, whose impedance is represented by a fractional (non-integer) order impedance model. The fractional model of the impedance is approximated by a serial connection of a finite number of RC-elements. Assuming that the time-dependent model parameters of the impedance are treated as statevariables of an augmented state vector, the linearity-property is lost and the application of the classical linear Kalman-filter for state estimation is not longer possible. Hence, the so-called sigma-point Kalman-filter is used in order to solve this online estimation problem for nonlinear system dynamics. The feasibility and quality of this online identification approach is outlined by using simulated data with additional noise.

Approximation and synthesis of non integer order systems

04:00 PM Wednesday 31, 1969

This paper presents a new method to approximate and to synthesize fractional systems represented by an explicit transfer function. We first present the distribution of relaxation times function method to approximate this type of function and hopefully to make circuit designers more aware of these advantages when designing fractal system circuits and fractal filters. Computer simulations of the circuit model by B2SPICE were used to demonstrate clearly our derivation results. An analog circuit model of the foster network can be synthesized.

Automatic loop shaping in QFT by using CRONE structures

04:00 PM Wednesday 31, 1969

This work focus on the problem of automatic loop shaping in QFT, where traditionally the search of an optimum design, a non convex and nonlinear optimization problem, is simplified by linearizing and/or convexifying the problem. In this work, the authors propose a suboptimal solution using a fixed structure in the compensator. However, in relation to previous work, the main idea consists in the study of the use of fractional compensators, which give singular properties to automatically shape the open loop gain function by using a minimum set of parameters. (Modified) CRONE controllers are considered as possible candidate structures.

Backward Euler method as a positivity preserving method for abstract integral equations of convoluti

04:00 PM Wednesday 31, 1969

It is shown that the backward Euler approximation to the solution of a wide class of linear, homogeneous equations with memory can be expressed as an average of the solution itself. This result implies that the numerical solution inherits some qualitative properties of the exact solution, such as positivity and contractivity. Numerical experiments, showing that neither the telegraph nor the fractional diffusion-wave two-dimensional equations preserve positivity, are provided.

Caputo linear fractional differential equations

04:00 PM Wednesday 31, 1969

This paper is devoted to the study of nonsequential linear fractional differential equations with constant coefficients involving the Caputo fractional derivatives. The Laplace transform is applied to obtain the general explicit solutions for the equations being studied in terms of Mittag-Leffler functions and generalized Wright functions. Conditions are given for obtaining linearly independent solutions which form a fundamental system of solutions. Some examples are presented.

Comparison of different orders Padé fractional order PD0.5 control algorithm implementations

04:00 PM Wednesday 31, 1969

This paper studies the performance of different order Padé Fractional Order (FO) PD0.5 controllers applied to the leg joint control of a hexapod robot with two dof legs and joint actuators with saturation. For simulation purposes the robot prescribed motion is characterized through several locomotion variables and for the walking performance evaluation are used two indices, one based on the mean absolute density of energy per travelled distance and the other on the hip trajectory errors. A set of simulation experiments reveals the influence of the different order Padé PD0.5 controllers tuning upon the proposed indices.

Contribution of non integer integro-differential operators (NIDO) to the geometrical understanding o

04:00 PM Wednesday 31, 1969

Advances in fractional analysis suggest a new way for the understanding of Riemann's conjecture. This analysis shows that any divisible natural number may be related to phase angles naturally associated with a certain class of non integer integro differential operators. It is shown that the subset of prime numbers is most likely related to a phase angle of ±π/ 4 to a 1/2-order differential equations and with their singularities. Riemann's conjecture asserting that, if s is a complex number, the non trivial zeros of zeta function 1/ζ(s)=Σn=1∞ µ(n)/ns in the gap [0,1], is characterized by s1/2 (1+2iθ), can be understood as a consequence of the properties of 1/2-order fractional differential equations on the prime number set. This physical interpretation suggests opportunities for revisiting flitter and cryptographic methodologies.

Deterministic and stochastic analysis of fractionally damped strings

04:00 PM Wednesday 31, 1969

Differential equations of motion of many systems moving in a fluid are described more accurately using a fractional derivative model. For example, a large plate completely immersed in a fluid medium and connected to a spring is defined by a second order differential equation containing a 3/2 order derivative term (Torvik and Bagley, 1984). A spherical particle moving in a viscous fluid experiences Basset force, which is described by a fractional derivative term (Basset, 1888; Mainardi, 1997; Tatom, 1988). It is therefore conjectured that a string moving in a fluid would experience a fractional damping force. Thus, many engineering applications containing strings and ropes such as open air high voltage transmission lines, bridges, and structures may require fractional derivatives to model them accurately. In this paper, we present a deterministic and stochastic model of a fractionally damped string. In this model, the viscous damping resulting from the string motion in a fluid is expressed by a fractional derivative. In this study, the fractional derivative is defined in the Caputo sense because it requires the normal boundary conditions to solve the problem. Using the method of separation of variables, a set of eigenfunctions is identified, and the response of the system is written as a linear combination of these functions. The properties of the eigenfunctions are used to reduce the space-time differential equation of the system into a set of infinite fractional differential equations defined in time domain only. A Laplace transform technique is used to obtain the fractional Green's function and a Duhamel integral type expression for the response of the system. For stochastic analysis, the input force is treated as a random process with specified mean and correlation functions. An expectation operator is applied on a set of expressions to obtain the stochastic behavior of the system. Several special cases are considered and closed form expressions are found for these cases. The approach is general and it can be applied to all those systems for which the existence of eigenmodes is guaranteed.

Discretization of complex-order differintegrals

04:00 PM Wednesday 31, 1969

This paper deals with the discretization of integrals and derivatives (i.e., differintegrals) of complex order. Several methods for the discretization of the operator sγ, where γ = u+jv is a complex value, are proposed. The concept of conjugated-order differintegral is also presented. The conjugated-order operator allows the use of complexorder differintegrals while still resulting in real time responses and real transfer functions. The performance of the resulting approximations is evaluated both in the time and frequency domains.

Electrochemical noise signal processing using R/S analysis and fractional Fourier transform

04:00 PM Wednesday 31, 1969

The corrosion processes of stainless steel under different solutions were examined using Electrochemical Noise (ECN). Using Rescaled Range analysis, we demonstrated that ECN signals produced by corrosion processes have non-stationary and self-similar properties. The comparison and analysis of ECN signals in both time and frequency domain showed that the conventional methods failed to give out the differences of the ECN signals obtained under different solutions. Therefore, we introduced fractional Fourier transform (FrFT) to process ECN signals, which is a powerful tool for the time-frequency analysis of self similar signals that can better describe the corrosion behaviours of the electrode in different solutions.

Enhanced diffusion in a bounded domain

04:00 PM Wednesday 31, 1969

There exit disordered media where contaminant dispersion is conveniently described by Lévy statistics. The case of enhanced diffusion corresponds to small scale motions, in the form of Continuous Time Random Walks with transition Probability densities presenting spatial diverging moments. Such CTRWs in infinite media were show to correspond, on the macroscopic scale, to diffusion equation involving Riesz-Feller derivatives of non-integer order, which are nonlocal w.r.t. space. For this reason, introducing boundary conditions sometimes results in modifying the large-scale model. We are studying here the diffusion limit of CTRWs, generalizing symmetric Lévy flights in a bounded medium, limited by two reflective barriers. The thus obtained space-fractional diffusion equations differ from the infinite domain case.

Estimation of lead acid battery state of charge with a novel fractional model

04:00 PM Wednesday 31, 1969

This paper deals with the application of fractional system identification to lead acid battery state of charge estimation. Fractional behavior of lead acid batteries is justified. A fractional system identification method recently developed is presented and a new fractional model of the battery is proposed. Based on parameter variations of this model, a state of charge estimation method is presented. Validation tests of this method on unknown state of charge are carried out. These validation tests highlights that the proposed estimation method gives a state of charge estimation with an error less than 5% whatever the operating temperature.

Finite element analysis of vibrating non-homogeneous beams with fractional derivative viscoelastic m

04:00 PM Wednesday 31, 1969

Fractional derivative rheological models are known to be very effective in describing the viscoelastic behaviour of materials, especially of polymers, and when applied to dynamic problems the resulting equations of motion, after a fractional state-space expansion, can still be studied in terms of modal analysis. However the growth in matrix dimensions carried by this expansion is in general so fast to make the calculations too cumbersome for finite element applications. This paper presents a condensation technique based on the computation of two reduced-size eigenproblems. The rheological model adopted is the Fractional Zener or Fractional Standard Linear Solid model, but the same methodology applies to problems involving different fractional derivative linear models.

Fractional calculus and the Schrödinger equation

04:00 PM Wednesday 31, 1969

In this paper, a derivation of the fractional Schrödinger equation is presented for the simple case of a pure diffusive process with dissipation. The Gaussian white noise is replaced by more general kinds of white noise and both the Markovian both the Markovian (β = 1) and non-Markovian case (0 < β > 1) are considered.