A discrete-time control design approach for periodically time-varying systems is introduced. The method employs a period-to-period (point-mapping) formulation of the system’s dynamics and a parametrization of the control input to obtain an equivalent time-invariant discrete-time representation of the system. The representation is generalized to include sampling within the period and varying sampling rates in different feedback loops. The proposed formulation allows for the design of feedback control laws using established discrete-time control methodologies. In this paper, dead-beat and optimal control laws with state- or output-feedback control are presented. An example of a multivariable control design for double inverted pendulum with periodic forcing is used to illustrate the proposed approach.
1.
Bolotin
, V. V.
, 1964, The Dynamic Stability of Elastic Systems
, Holden Day
, San Francisco
.2.
Arcara
, P.
, and Bittanti
, S.
, 2000, “Periodic Control of Helicopter Rotors for Attenuation of Vibrations in Forward Flight
,” IEEE Trans. Control Syst. Technol.
1063-6536, 8
(6
), pp. 883
–894
.3.
Calico
, R. A.
, and Wiesel
, W. E.
, 1986, “Stabilization of Helicopter Blade Flapping
,” J. Am. Helicopter Soc.
0002-8711, 31
, pp. 59
–64
.4.
Webb
, S. G.
, Calico
, R. A.
, and Wiesel
, W. E.
, 1991, “Time-Periodic Control of a Multiblade Helicopter
,” J. Guid. Control Dyn.
0731-5090, 14
(6
), pp. 1301
–1308
.5.
McKillip
, R. M.
, 1991, “Periodic Model-Following for the Control-Configured Helicopter
,” J. Am. Helicopter Soc.
0002-8711, 36
, pp. 4
–12
.6.
Tondl
, A.
, Ruijgrok
, T.
, Verhulst
, F.
, and Nabergoj
, R.
, 2000, Autoparametric Resonance in Mechanical Systems
, Cambridge University Press
, Cambridge, England
.7.
Hale
, J. K.
, 1963, Oscillations in Nonlinear Systems
McGraw-Hill
, New York
.8.
Urabe
, M.
, 1967, Nonlinear Autonomous Oscillations
, Academic
, New York
.9.
Bogoliobov
, N. N.
, and Mitropolsky
, Y. A.
, 1961, Asymptotic Methods in the Theory of Nonlinear Oscillations
, Hindustan Company
, Delhi
.10.
Thompson
, J. M. T.
, and Stewart
, H. B.
, 1986, Nonlinear Dynamics and Chaos
, Wiley, Chichester
, New York
.11.
Guckenheimer
, J.
, and Holmes
, P.
, 1983, Nonlinear Oscillations, Dynamics Systems, and Bifurcation of Vector Fields
, Springer-Verlag
, New York
.12.
Yakubovich
, V. A.
, and Starzhinski
, V. M.
, 1975, Linear Differential Equations With Periodic Coefficients
, Wiley
, New York
.13.
Sinha
, S. C.
, and Wu
, D. H.
, 1991, “An Efficient Computational Scheme for the Analysis of Periodic Systems
,” J. Sound Vib.
0022-460X, 151
, pp. 91
–117
.14.
Sinha
, S. C.
, and Butcher
, E. A.
, 1997, “Symbolic Computation of Fundamental Solution Matrices for Time-Periodic Dynamical Systems
,” J. Sound Vib.
0022-460X, 206
(1
), pp. 61
–85
.15.
Bernussou
, J.
, 1977, Point Mapping Stability
, Pergamon
, Oxford
.16.
Flashner
, H.
, 1979, “A Point Mapping Study of Dynamical Systems
,” Ph.D. thesis, University of California, Berkeley.17.
Flashner
, H.
, and Hsu
, C. S.
, 1983, “A Study of Nonlinear Periodic Systems Via The Point Mapping Method
,” Int. J. Numer. Methods Eng.
0029-5981, 19
, pp. 185
–215
.18.
Flashner
, H.
, and Guttalu
, R.
, 1990, “Analysis of Nonlinear Nonautonomous Systems by Truncated Point-Mappings
,” Int. J. Non-Linear Mech.
0020-7462, 24
(4
), pp. 327
–344
.19.
Guttalu
, R.
, and Flashner
, H.
, 1996, “Stability Analysis of Periodic Systems by Truncated Point-Mappings
,” J. Sound Vib.
0022-460X, 189
(1
), pp. 33
–54
.20.
Kwakernaak
, H.
, and Sivan
, R.
, 1972, Linear Optimal Control Systems
, Wiley Interscience
, New York
.21.
Sinha
, S. C.
, and Joseph
, P.
, 1994, “Control of General Dynamic System With Periodically Varying Parameters Via Liapunov-Floquet Transformation
,” ASME J. Dyn. Syst., Meas., Control
0022-0434, 116
, pp. 650
–658
.22.
Pandiyan
, R.
, and Sinha
, S. C.
, 1999, “Periodic Flap Control of a Helicopter Blade in Forward Flight
,” J. Vib. Control
1077-5463, 5
, pp. 761
–777
.23.
Pandiyan
, R.
, and Sinha
, S. C.
, 2001, “Time Varying Controller Synthesis for Nonlinear Systems Subjected to Periodic Parametric Loadings
,” J. Vib. Control
1077-5463, 7
, pp. 73
–90
.24.
Deshmukh
, V. S.
, and Sinha
, S. C.
, 2004, “Control of Dynamic Systems With Time-Periodic Coefficients Via the Lyapunov-Floquet Transformation and Backstepping Technique
,” J. Vib. Control
1077-5463, 10
, pp. 1517
–1533
.25.
Franklin
, G. F.
, Powell
, J. D.
, and Workman
, M. L.
, 1998, Digital Control of Dynamic Systems
, Addison-Wesley
, Reading, MA
.26.
de Boor
, C.
, 2001, A Practical Guide to Splines
, Springer-Verlag
, New York
.27.
Hohenemser
, K. H.
, and Yin
, S. K.
, 1972, “Some Applications of the Method of Multiblade Coordinates
,” J. Am. Helicopter Soc.
0002-8711, 17
(3
), pp. 3
–12
.28.
Cook
, R. D.
, Malkus
, D. S.
, Plesha
, M. E.
, and Witt
, R. J.
, 2002, Concepts and Applications of Finite Element Analysis
, Wiley
, New York
.29.
Bellman
, R. E.
, 1957, Dynamic Programming
, Princeton University Press
, NJ
.30.
Chen
, C.
, 1999, Linear System Theory and Design
, Oxford University Press
, New York
.31.
Kautsky
, J.
, and Nichols
, N. K.
, 1985, “Robust Pole Assignment in Linear State Feedback
,” Int. J. Control
0020-7179, 41
, pp. 1129
–1155
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