2015 |
Hmed, A. Ben; Amairi, M.; Aoun, M. Stability and resonance conditions of the non-commensurate elementary fractional transfer functions of the second kind Article de journal Dans: Communications in Nonlinear Science and Numerical Simulation, vol. 22, no. 1-3, p. 842 – 865, 2015, (Cited by: 12). Résumé | Liens | BibTeX | Étiquettes: Convergence of numerical methods, Damping, Fractional order, Fractional systems, Frequency domain analysis, Frequency domain curves, Frequency domains, Functions of the second kind, Resonance, Resonance analysis, Resonance condition, Stability, Time domain, Transfer functions @article{BenHmed2015842b,This paper deals with stability and resonance conditions of the non-commensurate elementary fractional transfer function of the second kind. This transfer function is a generalization of the elementary fractional transfer function of the second kind to an arbitrary order. It is written in the canonical form and characterized by a non-commensurate order, a pseudo-damping factor and a natural frequency. Stability and resonance analysis is done in terms of the pseudo-damping factor and the non-commensurate order. Also, an overall study of frequency-domain and time-domain performances of the considered system is done. Therefore many time-domain and frequency-domain curves are presented to help obtaining system parameters for a specified fractional order. Many illustrative examples show the efficiency of this study. Also, an application to the control of a spherical tank is also presented to show the usefulness of this study. © 2014 Elsevier B.V. |
Hmed, A. Ben; Amairi, M.; Aoun, M. Stability and resonance conditions of the non-commensurate elementary fractional transfer functions of the second kind Article de journal Dans: Communications in Nonlinear Science and Numerical Simulation, vol. 22, no. 1-3, p. 842 – 865, 2015, (Cited by: 12). Résumé | Liens | BibTeX | Étiquettes: Convergence of numerical methods, Damping, Fractional order, Fractional systems, Frequency domain analysis, Frequency domain curves, Frequency domains, Functions of the second kind, Resonance, Resonance analysis, Resonance condition, Stability, Time domain, Transfer functions @article{BenHmed2015842c,This paper deals with stability and resonance conditions of the non-commensurate elementary fractional transfer function of the second kind. This transfer function is a generalization of the elementary fractional transfer function of the second kind to an arbitrary order. It is written in the canonical form and characterized by a non-commensurate order, a pseudo-damping factor and a natural frequency. Stability and resonance analysis is done in terms of the pseudo-damping factor and the non-commensurate order. Also, an overall study of frequency-domain and time-domain performances of the considered system is done. Therefore many time-domain and frequency-domain curves are presented to help obtaining system parameters for a specified fractional order. Many illustrative examples show the efficiency of this study. Also, an application to the control of a spherical tank is also presented to show the usefulness of this study. © 2014 Elsevier B.V. |
2014 |
Hmed, A. Ben; Amairi, M.; Najar, S.; Aoun, M. Resonance study of an elementary fractional transfer function of the third kind Conférence 2014, (Cited by: 3). Résumé | Liens | BibTeX | Étiquettes: Calculations, Canonical form, Damping, Damping factors, Differentiation (calculus), Fractional calculus, Frequency domain analysis, Frequency domain curves, Frequency domains, Natural frequencies, Normalized gains, Resonance, Stability condition, Systems analysis, Transfer functions @conference{BenHmed2014c,This work is devoted to the stability and resonance study of the elementary fractional transfer function of the third kind. Some basic properties of this transfer function which is written in the canonical form and characterized by a non commensurate order, a pseudo-damping factor and a natural frequency, are presented. A resonance and stability condition is established numerically in terms of the non commensurate order and the pseudo-damping factor. Many frequency-domain curves are given to determine graphically the pseudo-damping factor and the non commensurate order for a desired normalized gain and normalized resonant frequency. Illustrative examples are presented to show the correctness and the usefulness of these curves. © 2014 IEEE. |
Hmed, A. Ben; Amairi, M.; Najar, S.; Aoun, M. Resonance study of an elementary fractional transfer function of the third kind Conférence 2014, (Cited by: 3). Résumé | Liens | BibTeX | Étiquettes: Calculations, Canonical form, Damping, Damping factors, Differentiation (calculus), Fractional calculus, Frequency domain analysis, Frequency domain curves, Frequency domains, Natural frequencies, Normalized gains, Resonance, Stability condition, Systems analysis, Transfer functions @conference{BenHmed2014e,This work is devoted to the stability and resonance study of the elementary fractional transfer function of the third kind. Some basic properties of this transfer function which is written in the canonical form and characterized by a non commensurate order, a pseudo-damping factor and a natural frequency, are presented. A resonance and stability condition is established numerically in terms of the non commensurate order and the pseudo-damping factor. Many frequency-domain curves are given to determine graphically the pseudo-damping factor and the non commensurate order for a desired normalized gain and normalized resonant frequency. Illustrative examples are presented to show the correctness and the usefulness of these curves. © 2014 IEEE. |
2011 |
Malti, Rachid; Aoun, Mohamed; Levron, Franois; Oustaloup, Alain Analytical computation of the ℋ 2-norm of fractional commensurate transfer functions Article de journal Dans: Automatica, vol. 47, no. 11, p. 2425 – 2432, 2011, (Cited by: 50). Résumé | Liens | BibTeX | Étiquettes: Analytical computations, Closed loop control systems, Differentiation (calculus), Finiteness condition, Fractional calculus, Function coefficients, General expression, Impulse response, Impulse response energy, Integral squared error, Performance indices, Rational functions, Rational transfer functions, Relative degrees, Three term control systems, Transfer functions @article{Malti20112425b,ℋ2-norm, or impulse response energy, of any fractional commensurate transfer function is computed analytically. A general expression depending on transfer function coefficients and differentiation orders is established. Then, more concise expressions are given for elementary fractional transfer functions. Unlike stable rational transfer functions, it is proven that the ℋ2-norm of stable fractional transfer functions may be infinite. Finiteness conditions are established in terms of transfer function relative degree. Moreover, it is proven that the ℋ2-norm of a fractional transfer function with a proper integrator of order less than 0.5 may be finite. The obtained results are used to evaluate the integral squared error of closed-loop control systems. © 2011 Elsevier Ltd. All rights reserved. |
Malti, Rachid; Aoun, Mohamed; Levron, Franois; Oustaloup, Alain Analytical computation of the ℋ 2-norm of fractional commensurate transfer functions Article de journal Dans: Automatica, vol. 47, no. 11, p. 2425 – 2432, 2011, (Cited by: 50). Résumé | Liens | BibTeX | Étiquettes: Analytical computations, Closed loop control systems, Differentiation (calculus), Finiteness condition, Fractional calculus, Function coefficients, General expression, Impulse response, Impulse response energy, Integral squared error, Performance indices, Rational functions, Rational transfer functions, Relative degrees, Three term control systems, Transfer functions @article{Malti20112425,ℋ2-norm, or impulse response energy, of any fractional commensurate transfer function is computed analytically. A general expression depending on transfer function coefficients and differentiation orders is established. Then, more concise expressions are given for elementary fractional transfer functions. Unlike stable rational transfer functions, it is proven that the ℋ2-norm of stable fractional transfer functions may be infinite. Finiteness conditions are established in terms of transfer function relative degree. Moreover, it is proven that the ℋ2-norm of a fractional transfer function with a proper integrator of order less than 0.5 may be finite. The obtained results are used to evaluate the integral squared error of closed-loop control systems. © 2011 Elsevier Ltd. All rights reserved. |
2004 |
Malti, Rachid; Aoun, Mohamed; Oustaloup, Alain Synthesis of fractional Kautz-like basis with two periodically repeating complex conjugate modes Conférence 2004, (Cited by: 19). Résumé | Liens | BibTeX | Étiquettes: Approximation theory, Differential equations, Dynamical systems, Extrapolation, Fractional calculus, Fractional derivatives, Generalized orthogonal basis (GOB), Identification (control systems), Laplace transforms, Linear control systems, Mathematical models, Model reduction, Orthogonal functions, Polynomials, Set theory, System stability, Transfer functions @conference{Malti2004835b,Fractional Kautz-like functions are synthesized extrapolating the definition of classical Kautz functions to fractional derivatives. The synthesized bases has two periodically repeating complex conjugate modes. The new basis extends the definition of the fractional Laguerre basis, by allowing the modes to be complex. An example is then provided in system identification context. |
Malti, Rachid; Aoun, Mohamed; Oustaloup, Alain Synthesis of fractional Kautz-like basis with two periodically repeating complex conjugate modes Conférence 2004, (Cited by: 19). Résumé | Liens | BibTeX | Étiquettes: Approximation theory, Differential equations, Dynamical systems, Extrapolation, Fractional calculus, Fractional derivatives, Generalized orthogonal basis (GOB), Identification (control systems), Laplace transforms, Linear control systems, Mathematical models, Model reduction, Orthogonal functions, Polynomials, Set theory, System stability, Transfer functions @conference{Malti2004835,Fractional Kautz-like functions are synthesized extrapolating the definition of classical Kautz functions to fractional derivatives. The synthesized bases has two periodically repeating complex conjugate modes. The new basis extends the definition of the fractional Laguerre basis, by allowing the modes to be complex. An example is then provided in system identification context. |
2003 |
Aoun, Mohamed; Malti, Rachid; Levron, François; Oustaloup, Alain Numerical simulations of fractional systems Conférence vol. 5 A, 2003, (Cited by: 16). Résumé | Liens | BibTeX | Étiquettes: Computer simulation, Differentiation (calculus), Fractional models, Fractional systems, Integration, Laplace transforms, Mathematical models, Transfer functions, Vectors @conference{Aoun2003745b,This paper deals with the design and simulation of continuous-time models with fractional differentiation orders. Two new methods are proposed. The first is an improvement of the approximation of the fractional integration operator using recursive poles and zeros proposed by Oustaloup (1995) and Lin (2001). The second improves the simulation schema by using a modal representation. |
Malti, Rachid; Aoun, Mohamed; Cois, Olivier; Oustaloup, Alain; Levron, François H2 norm of fractional differential systems Conférence vol. 5 A, 2003, (Cited by: 19). Liens | BibTeX | Étiquettes: Algebra, Computer simulation, Differential equations, Differentiation (calculus), Explicit systems, Fractional differential systems, Impulse response, Laplace transforms, Mathematical models, Theorem proving, Transfer functions @conference{Malti2003729b, |
Aoun, Mohamed; Malti, Rachid; Levron, François; Oustaloup, Alain Numerical simulations of fractional systems Conférence vol. 5 A, 2003, (Cited by: 16). Résumé | Liens | BibTeX | Étiquettes: Computer simulation, Differentiation (calculus), Fractional models, Fractional systems, Integration, Laplace transforms, Mathematical models, Transfer functions, Vectors @conference{Aoun2003745,This paper deals with the design and simulation of continuous-time models with fractional differentiation orders. Two new methods are proposed. The first is an improvement of the approximation of the fractional integration operator using recursive poles and zeros proposed by Oustaloup (1995) and Lin (2001). The second improves the simulation schema by using a modal representation. |
Malti, Rachid; Aoun, Mohamed; Cois, Olivier; Oustaloup, Alain; Levron, François H2 norm of fractional differential systems Conférence vol. 5 A, 2003, (Cited by: 19). Liens | BibTeX | Étiquettes: Algebra, Computer simulation, Differential equations, Differentiation (calculus), Explicit systems, Fractional differential systems, Impulse response, Laplace transforms, Mathematical models, Theorem proving, Transfer functions @conference{Malti2003729, |
2002 |
Malti, Rachid; Cois, Olivier; Aoun, Mohammed; Levron, François; Oustaloup, Alain Energy of fractional order transfer functions Conférence vol. 15, no. 1, 2002, (Cited by: 5; All Open Access, Bronze Open Access). Résumé | Liens | BibTeX | Étiquettes: Automation, Calculations, Differentiation (calculus), Dynamical systems, Fractional calculus, Fractional order differentiations, Fractional order transfer function, Impulse response, Impulse response energy, Lebesgue space, Single mode, Square integrable, Strictly positive real, Transfer functions @conference{Malti2002449b,The objective of the paper is to compute the impulse response energy of a fractional order transfer function having a single mode. The differentiation order n, defined in the sense of Riemann-Liouville, is allowed to be a strictly positive real number. A necessary and sufficient condition is established on n, in order for the impulse response to belong to the Lebesgue space L2[0, ∞[ of square integrable functions on [0, ∞[. Copyright © 2002 IFAC. |
Malti, Rachid; Cois, Olivier; Aoun, Mohammed; Levron, François; Oustaloup, Alain Energy of fractional order transfer functions Conférence vol. 15, no. 1, 2002, (Cited by: 5; All Open Access, Bronze Open Access). Résumé | Liens | BibTeX | Étiquettes: Automation, Calculations, Differentiation (calculus), Dynamical systems, Fractional calculus, Fractional order differentiations, Fractional order transfer function, Impulse response, Impulse response energy, Lebesgue space, Single mode, Square integrable, Strictly positive real, Transfer functions @conference{Malti2002449,The objective of the paper is to compute the impulse response energy of a fractional order transfer function having a single mode. The differentiation order n, defined in the sense of Riemann-Liouville, is allowed to be a strictly positive real number. A necessary and sufficient condition is established on n, in order for the impulse response to belong to the Lebesgue space L2[0, ∞[ of square integrable functions on [0, ∞[. Copyright © 2002 IFAC. |