Identification fréquentielle de systèmes par modèle non entier en utilisant la méthode des sous-espaces

Identification fréquentielle de systèmes par modèle non entier en utilisant la méthode des sous-espaces

Elena Ivanova Rachid Malti Xavier Moreau 

Laboratoire de l’Intégration du Matériau au Système, IMS UMR 5218 Université de Bordeaux, 351 Cours de la Libération F-33405 Talence cedex, France

Corresponding Author Email: 
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In this paper, an algorithm based on subspace method from the MOESP (MIMO Ouput-Erreur State sPace) family is presented, for state-space identification of continuous-time fractional commensurate models, using samples of frequency data. As compared to the rational state-space representation, the commensurate differentiation order must be estimated besides the state-space matrices, estimated with conventional subspace-based techniques using QR and singular value decomposition. This is the first method developed for multi-input multi-output system identification of fractional models in the frequency domain.


fractional state-space representation, subspace method, identification in frequency domain: deterministic and stochastic contexts

1. Introduction
2. Systèmes non entiers
3. Identification fréquentielle des systèmes non entiers par la méthode des sous-espaces
4. Exemples d’illustration
5. Conclusion

Akçay H., Turkay S. (2004). Frequency domain subspace-based identification of discretetime power spectra from nonuniformly spaced measurements. Automatica, vol. 40, no 8, p. 1333-1347.

Akçay H., Turkay S. (2011). Frequency domain subspace-based identification of discrete-time power spectra from uniformly spaced measurements. Automatica, vol. 47, no 2, p. 363-367.

Battaglia J.-L., Cois O., Puigsegur L., Oustaloup A. (2001). Solving an inverse heat conduction problem using a non-integer identified model. Int. J. of Heat and Mass Transfer, vol. 44, no 14, p. 2671-2680.

Battaglia J.-L., Maachou A., Malti R., Melchior P. (2013). Nonlinear heat diffusion simulation using volterra series expansion. International journal of thermal sciences, vol. 71, p. 80-87.

Datsko B. (2012). Mathematical modeling of nonlinear dynamics in bistable reaction-diffusion systems with fractional derivatives. Journal of Mathematical Sciences, vol. 184, no 2, p. 196-207.

Gabano J., Poinot T. (2011). Estimation of thermal parameters using fractional modelling. Signal Processing, vol. 91, no 4, p. 938-948.

Haverkamp L. R. J. (2001). State space identification. theory and practice. PhD thesis.

Hotzel R., FliessM. (1998). On linear systems with a fractional derivation: Introductory theory and examples. Mathematics and Computers in Simulation, vol. 45, p. 385-395.

Katayama T. (2005). Subspace methods for system identification. Springer.

Khemane F., Malti R., Raïssi T., Moreau X. (2012). Robust estimation of fractional models in the frequency domain using set membership methods. Signal Processing, vol. 92, p. 1591-1601.

Lawrence P., G.J. R. (1979). Recursive identification for system models of transfer function type. Proc. Instn Elec. Engrs., vol. 126, p. 283-288.

Levy E. (1959). Complex curve fitting. IRE Trans. Autom. Control, vol. 4, p. 37-43.

Liu K., Jacques R. N., Miller D. W. (1994). Frequency domain structural system identification by observability range space extraction. Proc. Amer. Control Conf., vol. 1, p. 107-111.

Maachou A., Malti R., Melchior P., Battaglia J.-L., Oustaloup A., Hay B. (2014). Nonlinear thermal systemidentification using fractional Volterra series. Control Engineering Practice, vol. 29, p. 50 - 60.

Malti R., Raïssi T., ThomassinM., Khemane F. (2010). Setmembership parameter estimation of fractional models based on bounded frequency domain data. Communications in Nonlinear Science and Numerical Simulation, vol. 15, no 4, p. 927 - 938. (IF: 2.697, SNIP: 2.055)

Malti R., Thomassin M. (2013). Differentiation similarities in fractional pseudo-state space representations and the subspace-based methods. Fractional Calculus and Applied Analysis, vol. 16, p. 273-287. Consulté sur

Malti R., Victor S., Oustaloup A. (2008). Advances in system identification using fractional models. Journal of Computational and Nonlinear Dynamics, vol. 3, no 2, p. 021401,1-7. (IF: 0.571, SNIP: 1.500)

Mathieu B., Oustaloup A., Levron F. (1995). Transfer function parameter estimation by interpolation in the frequency domain. In European control conference (ecc). Rome, Italie.

Matignon D. (1998). Stability properties for generalized fractional differential systems. In Esaim : Proceedings, fractional differential systems: Models, methods and applications, vol. 5, p. 145–158.

Matignon D., Novel B. d’Andréa. (1996b, July). Some results on controllability and observability of finite-dimensional fractional differential systems. In Imacs, vol. 2, p. 952-956. Lille, France.

Matignon D., Novel B. d’Andréa. (1997, December). Observer-based controllers for fractional differential systems. In 36th ieee conference on decision and control, p. 4967-4972.

McKelvey T., Ljung L. (1996). Subspace-based multivariable system identification from frequency response data. IEEE Transactions on automatic control, vol. 41, no 7.

Mensler M., Sanada K.W. (2000). Subspace method for continuous-time system identifibation. Proc ISCIE Int Symp Stoch Syst Theory Appl (Inst Syst Control Inf Eng), vol. 32, p. 21-26.

Neumayer R., Stelzer A., Weigel R. (2003). A comparison of complex curve fitting and subspace identification algorithm for circuit modeling using frequency domain data. 33rd European Microwave Conference, Munich, vol. 3, p. 1007-1010.

Oustaloup A. (1995). La dérivation non-entière: théorie, synthèse et applications. Hermès-Paris.

Overschee P. V., Moor B. D. (1996). Continuous-time frequency domain subspace system identification. Signal Processing, vol. 52, p. 179-194.

Pintelon R. (2002). Frequency domain subspace system identification using non-parametric noise models. Automatica, vol. 38, p. 1295-1311.

Podlubny I. (1999). Fractional differential equations. San Diego, Academic Press. Sabatier J., Aoun M., Oustaloup A., Grégoire G., Ragot F., Roy P. (2006). Fractional system identification for lead acid battery sate charge estimation. Signal Processing, vol. 86, no 10, p. 2645-2657.

Sabatier J., Farges C., Merveillaut M., Feneteau L. (2012). On observability and pseudo state estimation of fractional order systems. European Journal of Control, vol. 18, no 3, p. 260-271.

Sanathanan C., Koerner J. (1963). Transfer function synthesis as a ratio of two complex polynomials. IEEE Trans. Autom. Control, vol. AC-8, p. 56-58.

Sierociuk D., Skovranek T., Macias M., Podlubny I., Petras I., Dzielinski A., Ziubinski P. (2015). Diffusion process modeling by using fractional-order models. Applied Mathematics and Computation, vol. 257, p. 2-11.

Thomassin M., Malti R. (2009). Multivariable identification of continuous-time fractional system. In ASME IDETC/CIE Conferences, p. DETC2009-86984. San Diego CA USA.

Thomassin M., Malti R. (2009). Subspace method for continuous-time fractional system identification. In Proc. of the 15th ifac symp. on system identification (sysid 2009). Saint-Malo, France.

Valério D., Sabatier J., Agrawal O. P., Tenreiro Machado J. A. (2007). Advances in fractional calculus theoretical developments and applications in physics and engineering. Springer.

Van Overschee P., De Moor B. (1996). Subspace identification for linear systems: theory, implementation, applications. Springer.

Victor S., Malti R., Garnier H., Oustaloup A. (2013). Parameter and differentiation order estimation in fractional models. Automatica, vol. 49, no 4, p. 926-935.

Wang B., Li S.-E., Peng H., Liu Z. (2015). Fractional-order modeling and parameter identification for lithium-ion batteries. Journal of Power Sources, vol. 293, p. 151 - 161.

Wang L., Cluet W. R. (1997). Frequency sampling filters: an improuved model structure for step-response identification. Automatica, vol. 33, p. 939-944.

Wang L., Mokhtar R. M. (2007). Continuous-time system identification using subspace methods. Australian and New Zealand Industrial and Applied Mathematics Journal (ANZIAM), vol. 47, p. 712-732.

Wang L., Mokhtar R. M. (2009). 2-stage identification based on frequency sampling filters and subspace frequency response. Elektrika, vol. 11, p. 27-33.

Yang Z.-J., Sanada S. (2000). Frequency domain subspace identification with the aid of the w-operator. Electrical Engineering in Japan, vol. 132, no 1, p. 326-334.