Abstract
An adaptive control system has to be built on top of a robust digital control system. Therefore robustness issues for the underlying controller and the shaping of the sensitivity functions for various possible values of the plant parameters are very important. After a review of some basic robustness concepts, a methodology for shaping the sensitivity functions is presented. Its application is illustrated in the context of adaptive control of a flexible transmission.
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Notes
- 1.
Robustness with respect to plant model uncertainties is not a god given property for some control strategies. It is the wise choice of some design parameters which can assure the robustness of a given control strategy in a specific context.
- 2.
At a given frequency the point belonging to the Nyquist plot of the true plant model lies in a disc of a given radius centered on the corresponding point belonging to the Nyquist plot of the nominal model.
- 3.
The criterion remains valid in the case of poles zeros cancellations. The number of encirclements should be equal to the number of unstable open-loop poles without taking into account the possible cancellations.
- 4.
There are some “pathological” systems \(\frac{B(z^{-1})}{A(z^{-1})}\), with unstable poles and zeros which can be stabilized only with open-loop unstable controllers.
- 5.
See Sung and Hara (1988) for a proof. In the case of unstable open-loop systems but stable in closed loop, this integral is positive.
- 6.
The bilinear transformation assures a better approximation of a continuous-time model by a discrete-time model in the frequency domain than the replacement of differentiation by a difference, i.e. s=(1−z −1)T s .
- 7.
The factor γ has no effect on the final result (coefficients of R and S). It is possible, however, to implement the filter without normalizing the numerator coefficients.
- 8.
To be download from the web site (http://landau-bookic.lag.ensieg.inpg.fr).
- 9.
Available on the website (http://landau-bookic.lag.ensieg.inpg.fr).
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Landau, I.D., Lozano, R., M’Saad, M., Karimi, A. (2011). Robust Digital Control Design. In: Adaptive Control. Communications and Control Engineering. Springer, London. https://doi.org/10.1007/978-0-85729-664-1_8
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DOI: https://doi.org/10.1007/978-0-85729-664-1_8
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