Authors: Andrzej Dzielinski; Dominik Sierociuk

Abstract: This paper presents simple stability condition for the fractional order discrete state-space system. This is one of very few attempts to give the condition of the stability of this type of systems. It is established for the state space model introduced by the Authors.

Keywords: discrete fractional order state-space system,stability,infinite dimensional systems

Identifier: 10.3182/20060719-3-PT-4902.00084

Conference: Second IFAC Workshop on Fractional Differentiation and its Applications, 2006

Location: The Institute of Engineering of Porto (ISEP), Portugal

Start Date: Wed Jul 19 2006 - End Date: Fri Jul 21 2006

Authors: Patrick Lanusse; Alain Oustaloup

Abstract: The stability criterion proposed by Popov is commonly used to prove the stability of systems with interconnected memoryless nonlinearity and rational linear system. Fractional linear systems are tackled in this paper. Then it is shown how this criterion can be used to take into account nonlinearity lying in a sector with a corresponding complex gain. Then, for the design of a fractional order robust controller, this gain is interpreted as an unstructured uncertainty added to the structured uncertainty of the linear part of a plant model.

Keywords: Popov criterion,nonlinear system,fractional order system,perturbed system,CRONE control system design

Identifier: 10.3182/20060719-3-PT-4902.00026

Conference: Second IFAC Workshop on Fractional Differentiation and its Applications, 2006

Location: The Institute of Engineering of Porto (ISEP), Portugal

Start Date: Wed Jul 19 2006 - End Date: Fri Jul 21 2006

Authors: Alain Oustaloup; Olivier Cois; Patrick Lanusse; Pierre Melchior; Xavier Moreau; Jocelyn Sabatier

Abstract: This article deals with fractal robustness. Of physical origin, this robustness represents the insensitivity of damping in nature, resulting from the combination of fractality and non integer differentiation. The study case is a natural robust phenomenon: the relaxation of water on a porous dyke whose damping ratio is independent of the mass of moving water. The dynamic model which governs this phenomenon is a non integer order linear differential equation for which the natural frequency and the damping ratio of the oscillatory mode of the solution are determined. A remarkable result is that damping ratio is exclusively linked to differential equation non integer order which is dictated by the fractal dimension of the dyke. Damping robustness is illustrated by two isodamping half-straight lines in the operational plane, and by a frequency template in the Nyquist plane. Then, the principle of the CRONE suspension, the synthesis method and the performances are developed. This suspension system is called the CRONE suspension because of the link with the second-generation CRONE control. The transposition to automatic control of the two isodamping half-straight lines, defines the robust strategy used by the second generation CRONE control. Defining a generalization of the second generation CRONE control, the great principles of the third generation CRONE control are also given. Finally, the problem of time domain identification by non integer model is studied. Among the whole of the methods developed in the CRONE team, a single one is dealt with in this paper. This one is based on the sampling of a non integer differential equation through the Grünwald numeric approximation of the non integer derivative of a time function. An application is carried out within the context of the identification of the dynamic behaviour of a thermal system. As far as the CRONE approach applications are concerned, a first part deals with the first ones showing the performance improvement. A second part is limited to a non exhaustive list of the other applications.

Keywords: fractional (or non integer) differentiation,porous dyke,fractal robustness,CRONE suspension,CRONE control,identification by non integer model,CRONE applications

Identifier: 10.3182/20060719-3-PT-4902.00059

Conference: Second IFAC Workshop on Fractional Differentiation and its Applications, 2006

Location: The Institute of Engineering of Porto (ISEP), Portugal

Start Date: Wed Jul 19 2006 - End Date: Fri Jul 21 2006

Authors: Ana Cristina Galucio; Jean-Francois Deu; Francois Dubois

Abstract: The Gear scheme is a three-level step algorithm, backward in time and second order accurate for the approximation of classical time derivatives. In this contribution, the formal power of this scheme is proposed to approximate fractional derivative operators in the context of finite difference methods. Some numerical examples are presented and analysed in order to show the effectiveness of the present Gear scheme at the power α (Gα-scheme) when compared to the classical Grünwald-Letnikov approximation. In particular, for a fractional damped oscillator problem, the combined Gα-Newmark scheme is shown to be second-order accurate.

Keywords: fractional derivatives,viscoelasticity,multi-step scheme,linear dynamics

Identifier: 10.3182/20060719-3-PT-4902.00064

Conference: Second IFAC Workshop on Fractional Differentiation and its Applications, 2006

Location: The Institute of Engineering of Porto (ISEP), Portugal

Start Date: Wed Jul 19 2006 - End Date: Fri Jul 21 2006

Authors: Mikael Enelund; Peter Olsson

Abstract: Here an integro-differential formulation for time domain poroelasticity is derived and included into a finite element framework for structural dynamics. Further, an algorithm for the time integration of the spatially discretized finite element equations is developed. The Biot poroelastic theory (formulated in the frequency domain) is taken as a starting point. In this theory, the dissipative force is proportional to the relative motion between the solid phase and the viscous fluid in the pores with a constant of proportionality, drag force coefficient, that is frequency dependent. The dissipative force is described in the time domain by a convolution term with a kernel that is the Fourier inverse of the drag force coefficient. As the exact analytical inversion of the drag force coefficient seems highly intractable, we have combined an asymptotic expansion for high frequencies with a low frequency correction term, yielding an analytically invertible drag force coefficient. The convolution with the fourier inverse of the asymptotic expansion, can be identified as a semi-derivative.

Keywords:

Identifier: 10.3182/20060719-3-PT-4902.00066

Conference: Second IFAC Workshop on Fractional Differentiation and its Applications, 2006

Location: The Institute of Engineering of Porto (ISEP), Portugal

Start Date: Wed Jul 19 2006 - End Date: Fri Jul 21 2006

Authors: Katica Hedrih

Abstract: Based on the knowledge from literature on free transversal vibrations of the one axially moving string, in this paper, coupled partial fractional order differential equations of the transversal vibrations of the axially moving sandwich double belt system with creep layer are derived and analytically solved.

Keywords: moving sandwich double belt system,partial fractional order differential coupled equations,analytical solutions,multi-frequency vibrations

Identifier: 10.3182/20060719-3-PT-4902.00030

Conference: Second IFAC Workshop on Fractional Differentiation and its Applications, 2006

Location: The Institute of Engineering of Porto (ISEP), Portugal

Start Date: Wed Jul 19 2006 - End Date: Fri Jul 21 2006

Authors: Duarte Valerio; Jose Sa da Costa

Abstract: This paper presents several tuning rules for fractional PID controllers, similar to the first and the second sets of tuning rules proposed by Ziegler and Nichols for integer PIDs. Fractional PIDs so tuned perform better than integer PIDs; in particular, step-responses have roughly constant overshoots even when the gain of the plant varies.

Keywords: fractional PIDs,tuning rules

Identifier: 10.3182/20060719-3-PT-4902.00004

Conference: Second IFAC Workshop on Fractional Differentiation and its Applications, 2006

Location: The Institute of Engineering of Porto (ISEP), Portugal

Start Date: Wed Jul 19 2006 - End Date: Fri Jul 21 2006

Authors: YangQuan Chen

Abstract: There is an increasing interest in dynamic systems and controls of noninteger orders or fractional orders. Clearly, for closed-loop control systems, there are four situations. They are 1) IO (integer order) plant with IO controller; 2) IO plant with FO (fractional order) controller; 3) FO plant with IO controller and 4) FO plant with FO controller. However, from engineering point of view, doing something better is the major concern. This review article will first show two examples that the best fractional order controller outperforms the best integer order controller. Then, we try to argue why consider fractional order control even when integer (high) order control works comparatively well. We will also address issues in fractional order PID controller tuning. Using several real world examples, we further argue that, fractional order control is ubiquitous when the dynamic system is of distributed parameter nature.

Keywords: non-integer order calculus,fractional order dynamic systems,fractional order control,fractional order PID control,distributed parameter systems,smart materials

Identifier: 10.3182/20060719-3-PT-4902.00081

Conference: Second IFAC Workshop on Fractional Differentiation and its Applications, 2006

Location: The Institute of Engineering of Porto (ISEP), Portugal

Start Date: Wed Jul 19 2006 - End Date: Fri Jul 21 2006

Authors: Tomasz Rybicki; Piotr Ostalczyk

Abstract: In this paper a simple dead-beat control of an electromagnetic servo is proposed. An electromagnetic servo is identified as a third-order linear continuous-time plant. A method of a dead-beat control of its discrete equivalence is proposed. In a control algorithm a non-integer variable-order backward difference is used.

Keywords: difference equations,discrete-time systems,dead-beat control,electromagnetic servo

Identifier: 10.3182/20060719-3-PT-4902.00016

Conference: Second IFAC Workshop on Fractional Differentiation and its Applications, 2006

Location: The Institute of Engineering of Porto (ISEP), Portugal

Start Date: Wed Jul 19 2006 - End Date: Fri Jul 21 2006

Authors: Piotr W. Ostalczyk; Tomasz Rybicki

Abstract: In this paper a simple dead-beat control of an electromagnetic servo is proposed. An electromagnetic servo is identified as a third-order linear continuous-time plant. A method of a dead-beat control of its discrete equivalent is proposed. As a control algorithm a variable- fractional-order backward difference is applied.

Keywords: difference equations,discrete-time systems,dead-beat control

Identifier: 10.3182/20060719-3-PT-4902.00017

Conference: Second IFAC Workshop on Fractional Differentiation and its Applications, 2006

Location: The Institute of Engineering of Porto (ISEP), Portugal

Start Date: Wed Jul 19 2006 - End Date: Fri Jul 21 2006

Authors: H. T. C. Pedro; M. H. Kobayashi; J. M. C. Pereira; C. F. M. Coimbra

Abstract: This work advances our understanding of the drag force acting on a particle due to the oscillatory flow of a viscous fluid at finite Reynolds and Strouhal numbers. The drag force is determined by using the novel concept of Variable-Order (VO) Calculus, where the order of derivative can vary with the parameters and variables according to the dynamics of the flow. With the VO formulation we determine: (i) the region of validity of Tchen's equation for oscillatory flow, (ii) the region where the order of the derivative is fractional, but constant and (iii) the region where the strong non-linearity of the flow requires a variable order derivative to account for the increased complexity of the flow.

Keywords: variable-order calculus,diffusive-convective flows,non-linar drag

Identifier: 10.3182/20060719-3-PT-4902.00077

Conference: Second IFAC Workshop on Fractional Differentiation and its Applications, 2006

Location: The Institute of Engineering of Porto (ISEP), Portugal

Start Date: Wed Jul 19 2006 - End Date: Fri Jul 21 2006

Authors: Reyad EL-Khazali; Wajdi H. Ahmad

Abstract: In this paper, a variable structure control design is presented for a class of fractional time-delay systems of incommensurate order. First, a stable switching manifold, independent of the delay, is designed for a virtual integer-order system. Then, a switched controller that guarantees convergence to the sliding surface is developed. The premise of the design methodology is based on the fact that the stability region of the integer system is a subset of the stability region of the corresponding fractional-order system. An illustrative example is given to illustrate the main idea of this work.

Keywords: fractional systems,sliding mode control,and time-delay systems

Identifier: 10.3182/20060719-3-PT-4902.00039

Conference: Second IFAC Workshop on Fractional Differentiation and its Applications, 2006

Location: The Institute of Engineering of Porto (ISEP), Portugal

Start Date: Wed Jul 19 2006 - End Date: Fri Jul 21 2006

Authors: Pascal Serrier; Xavier Moreau; Alain Oustaloup

Abstract: The fractional differentiator is a linear operator but it can be very convenient to achieve it with nonlinear components. This paper presents an analysis method for components nonlinearities of a suspension whose force-deflection linearized transfer function is a limited-bandwidth fractional differentiator (CRONE suspension). The analysis method is based on Volterra series decomposition and on nonlinear systems association with George algebra.

Keywords: automatic control,fractional differentiation,fractional systems,nonlinear systems,Volterra series

Identifier: 10.3182/20060719-3-PT-4902.00080

Conference: Second IFAC Workshop on Fractional Differentiation and its Applications, 2006

Location: The Institute of Engineering of Porto (ISEP), Portugal

Start Date: Wed Jul 19 2006 - End Date: Fri Jul 21 2006