Stabilization of fractional exponential systems including delays. (English) Zbl 1265.93211

Summary: This paper analyzes the BIBO stability of fractional exponential delay systems which are of retarded or neutral type. Conditions ensuring stability are given first. As is the case for the classical class of delay systems these conditions can be expressed in terms of the location of the poles of the system. Then, in view of constructing robust BIBO stabilizing controllers, explicit expressions of coprime and Bézout factors of these systems are determined. Moreover, nuclearity is analyzed in a particular case.


93D21 Adaptive or robust stabilization
34A08 Fractional ordinary differential equations
34K40 Neutral functional-differential equations
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[1] Bonnet C., Partington J. R.: Bézout factors and \({L}^1\)-optimal controllers for delay systems using a two-parameter compensator scheme. IEEE Trans. Automat. Control 44 (1999), 1512-1521 · Zbl 0959.93052 · doi:10.1109/9.780415
[2] Bonnet C., Partington J. R.: Analysis of fractional delay systems of retarded and neutral type. Preprint 2000 · Zbl 1007.93065 · doi:10.1016/S0005-1098(01)00306-5
[3] Bonnet C., Partington J. R.: Coprime factorizations and stability of fractional differential systems. Systems Control Lett. 41 (2000), 167-174 · Zbl 0985.93048 · doi:10.1016/S0167-6911(00)00050-5
[4] Curtain R. F., Zwart H. J.: An Introduction to Infinite Dimensional Linear Systems Theory. Springer-Verlag, Berlin 1995 · Zbl 0839.93001
[5] Glover K., Curtain R. F., Partington J. R.: Realization and approximation of linear infinite dimensional systems with error bounds. SIAM J. Control Optim. 26 (1988), 863-898 · Zbl 0654.93011 · doi:10.1137/0326049
[6] Gripenberg G., Londen S. O., Staffans O.: Volterra Integral and Functional Equations. Cambridge University Press, Cambridge, U.K. 1990 · Zbl 1159.45001
[7] Hille E., Phillips R. S.: Functional analysis and semi-groups. American Mathematical Society, Providence, R. I., 1957 · Zbl 0392.46001
[8] Hotzel R.: Some stability conditions for fractional delay systems. J. Math. Systems, Estimation, and Control 8 (1998), 1-19 · Zbl 0913.93068
[9] Loiseau J.-J., Mounier H.: Stabilisation de l’équation de la chaleur commandée en flux. Systèmes Différentiels Fractionnaires, Modèles, Méthodes et Applications. ESAIM Proceedings 5 (1998), 131-144 · Zbl 0913.73052 · doi:10.1051/proc:1998003
[10] Matignon D.: Représentations en variables d’état de modèles de guides d’ondes avec dérivation fractionnaire. Thèse de doctorat, Univ. Paris XI, 1994
[11] Partington J. R.: An Introduction to Hankel Operators. Cambridge University Press, Cambridge, U.K. 1988 · Zbl 0668.47022
[12] Weber E.: Linear Transient Analysis. Volume II. Wiley, New York 1956 · Zbl 0073.21801
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