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State-space representation for fractional order controllers. (English) Zbl 0964.93024
The authors’ abstract reads as follows:
“This article proposes an infinite-dimensional state-space realization for linear filters with transfer function $C_d(s)\underset={}\Delta C_0((1+ s/\omega_b)/(1+ s/\omega_h))^d,$ where $$0< \omega_b< \omega_h$$ and $$0< d< 1$$. This exponentially stable representation is derived from the Taylor expansion at zero of the function $$(1- z)^d$$, and is made up of an infinite number of first-order ordinary differential equations. Finite-dimensional approximations obtained by truncating this representation are shown to converge towards $$C_d$$ in $$H_\infty$$. An example of feedback loop incorporating this approximation of $$C_d$$ (car suspension) is presented, for which robustness of closed-loop resonance and step response overshoot vis-à-vis a variation in the vehicle mass is achieved.”
This interesting paper finds its motivation in and is geared towards application to a real-world problem.
It is partly a marriage between an idea that dates back to Bode and modern infinite-dimensional systems theory as described and developed by Curtain and Zwart.

##### MSC:
 93B15 Realizations from input-output data 93C80 Frequency-response methods in control theory 93C95 Application models in control theory
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##### References:
 [1] Åström, K. J. (1997). Limitations on control system performance. Proceedings of the Fourth ECC. [2] Bergeon, B.; Irving, I., Commande robuste à modèle fréquentiel de référence, Apii, 24, 83-97, (1990) · Zbl 0687.93020 [3] Bode, H.W., Network analysis and feedback amplifier design, (1945), Van Nostrand New York [4] Curtain, R.F.; Zwart, H., An introduction to infinite-dimensional linear systems theory, (1995), Springer New York · Zbl 0839.93001 [5] Flajolet, P.; Odlyzko, A.M., Singularity analysis of generating functions, SIAM journal on discrete mathematics, 2, 21-240, (1990) · Zbl 0712.05004 [6] Mbodje, B.; Montseny, G., Boundary fractional derivative control of the wave equation, IEEE transactions on automatic control, 40, 2, 378-382, (1995) · Zbl 0820.93034 [7] Oustaloup, A.; Mathieu, B.; Lanusse, P., The CRONE control of resonant plants: application to a flexible transmission, European journal of control, 1, 2, (1995) [8] Oustaloup, A.; Moreau, X.; Nouillant, M., The CRONE suspension, Control engineering practice, 4, 8, 1101-1108, (1996) [9] Raynaud, H.-F.; Zergaı̈noh, A., State-space representation of fractional linear filters, Proceedings of the fourth ECC, (1997) · Zbl 0964.93024
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