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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- 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).
References
Adaptech (1988) WimPim + (includes, WinPIM, WinREG and WinTRAC) system identification and control software. User’s manual. 4, rue du Tour de l’Eau, 38400 St. Martin-d’Hères, France
Anderson BDO, Moore J (1971) Linear optimal control. Prentice Hall, Englewood Cliffs
Åström KJ, Wittenmark B (1984) Computer controlled systems, theory and design. Prentice-Hall, Englewood Cliffs
Doyle JC, Francis BA, Tannenbaum AR (1992) Feedback control theory. MacMillan, New York
Fenot C, Rolland F, Vigneron G, Landau ID (1993) Open loop adaptive digital control in hot-dip galvanizing. Control Eng Pract 1(5):779–790
Green M, Limebeer DJN (1995) Linear robust control. Prentice Hall, New York
Kwakernaak H (1993) Robust control and H ∞-optimization—a tutorial. Automatica 29:255–273
Kwakernaak H (1995) Symmetries in control system design. In: Isidori A (ed) Trends in control. Springer, Heidelberg, pp 17–52
Landau ID (1993b) Identification et Commande des Systèmes, 2nd edn. Série Automatique. Hermès, Paris
Landau ID (1995) Robust digital control of systems with time delay (the Smith predictor revisited). Int J Control 62(2):325–347
Landau ID, Karimi A (1996) Robust digital control using the combined pole placement sensitivity function shaping: an analytical solution. In: Proc of WAC, vol 4, Montpellier, France, pp 441–446
Landau ID, Karimi A (1998) Robust digital control using pole placement with sensitivity function shaping method. Int J Robust Nonlinear Control 8:191–210
Landau ID, Zito G (2005) Digital control systems—design, identification and implementation. Springer, London
Landau ID, Rey D, Karimi A, Voda-Besançon A, Franco A (1995a) A flexible transmission system as a benchmark for robust digital control. Eur J Control 1(2):77–96
Landau ID, Karimi A, Voda-Besançon A, Rey D (1995b) Robust digital control of flexible transmissions using the combined pole placement/sensitivity function shaping method. Eur J Control 1(2):122–133
Langer J, Constantinescu A (1999) Pole placement design using convex optimisation criteria for the flexible transmission benchmark: robust control benchmark: new results. Eur J Control 5(2–4):193–207
Langer J, Landau ID (1999) Combined pole placement/sensitivity function shaping method using convex optimization criteria. Automatica 35(6):1111–1120
Morari M, Zafiriou E (1989) Robust process control. Prentice Hall International, Englewood Cliffs
Procházka H, Landau ID (2003) Pole placement with sensitivity function shaping using 2nd order digital notch filters* 1. Automatica 39(6):1103–1107
Sung HK, Hara S (1988) Properties of sensitivity and complementary sensitivity functions in single-input single-output digital systems. Int J Control 48(6):2429–2439
Zames G (1981) Feedback and optimal sensibility: model reference transformations, multiplicative seminorms and approximate inverses. IEEE Trans Autom Control 26:301–320
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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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