Active control of flexible systems

  • M. J. Balas
Contributed Papers


Since mechanically flexible systems are distributed-parameter systems, they are infinite-dimensional in theory and, in practice, must be modelled by large-dimensional systems. The fundamental problem of actively controlling very flexible systems is to control a large-dimensional system with a much smaller dimensional controller. For example, a large number of elastic modes may be needed to describe the behavior of a flexible satellite; however, active control of all these modes would be out of the question due to onboard computer limitations and modelling error. Consequently, active control must be restricted to a few critical modes. The effect of the residual (uncontrolled) modes on the closed-loop system is often ignored.

In this paper, we consider the class of flexible systems that can be described by a generalized wave equation,utt+Au=F, which relates the displacementu(x,t) of a body Θ inn-dimensional space to the applied force distributionF(x,t). The operatorA is a time-invariant symmetric differential operator with a discrete, semibounded spectrum. This class of distributed parameter systems includes vibrating strings, membranes, thin beams, and thin plates.

The control force distribution
$$F(x,t) = \sum\limits_{i = 1}^M { \delta (x - x_i )f_i (t)} $$
is provided byM point force actuators located at pointsxi on the body. The displacements (or their velocities) are measured byP point sensorsyi(t)=u(zj,t), orut(zj,t),j=1, 2, ...,P, located at various pointszj along the body.

We obtain feedback control ofN modes of the flexible system and display the controllability and observability conditions required for successful operation. We examine the control and observation spillover due to the residual modes and show that the combined effect of spillover can lead to instabilities in the closed-loop system. We suggest some remedies for spillover, including a straightforward phase-locked loop prefilter, to remove the instability mechanism.

To illustrate the concepts of this paper, we present the results of some numerical studies on the active control of a simply supported beam. The beam dynamics are modelled by the Euler-Bernoulli partial differential equation, and the feedback controller is obtained by the above procedures. One actuator and one sensor (at different locations) are used to control three modes of the beam quite effectively. A fourth residual mode is simulated, and the destabilizing effect of control and observation spillover together on this mode is clearly illustrated. Once observation spillover is eliminated (e.g., by prefiltering the sensor outputs), the effect of control spillover alone on this system is negligible.

Key Words

Distributed-parameter systems flexible systems partial differential equations 


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  1. 1.
    Kato, T.,Perturbation Theory for Linear Operators, Springer-Verlag, New York, New York, 1966.Google Scholar
  2. 2.
    Strang, G., andFix, G.,An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, New Jersey, 1973.Google Scholar
  3. 3.
    Russell, D.,Controllability and Stabilizability Theory for Linear Partial Differential Equations: Recent Progress and Open Questions, University of Wisconsin, Mathematics Research Center, MRC Technical Summary Report No. 1700, 1976.Google Scholar
  4. 4.
    Fattorini, H.,On Complete Controllability of Linear Systems, Journal of Differential Equations, Vol. 3, pp. 391–402, 1967.Google Scholar
  5. 5.
    Slemrod, M.,A Note on Complete Controllability and Stabilizability for Linear Control Systems in Hilbert Space, SIAM Journal of Control, Vol. 12, pp. 500–508, 1974.Google Scholar
  6. 6.
    Larson, V., Likins, P., andMarsh, E.,Optimal Estimation and Attitude Control of a Solar Electric Propulsion Spacecraft, IEEE Transactions on Aerospace and Electronics, Vol. AES-13, pp. 35–47, 1977.Google Scholar
  7. 7.
    Meirovitch, L., Van Landingham, H., andÖz, H.,Control of Spinning Flexible Spacecraft by Modal Synthesis, Paper Presented at the 27th IAF Congress, Anaheim, California, 1976.Google Scholar
  8. 8.
    Larson, V., andLikins, P.,An Application of Modern Control Theory to an Elastic Spacecraft, Paper Presented at the ESA Symposium on Dynamics and Control of Non-Rigid Space Vehicles, Frascati, Italy, 1976.Google Scholar
  9. 9.
    Köehne, M.,Optimal Feedback Control of Flexible Mechanical Systems, Paper Presented at the IFAC Symposium on Distributed Parameter Control Systems, Banff, Canada, 1971.Google Scholar
  10. 10.
    Skelton, R., andLikins, P.,Techniques of Modelling and Model Error Compensation in Linear Regulator Problems, Advances in Control and Dynamic Systems, Vol. 14, Edited by C. T. Leondes, Academic Press, New York, New York, 1977.Google Scholar
  11. 11.
    Kwakernaak, H., andSivan, R.,Linear Optimal Control Systems, John Wiley and Sons, New York, New York, 1972.Google Scholar
  12. 12.
    Kalman, R., andBucy, R.,New Results in Linear Filtering and Prediction Theory, Transactions of ASME, Journal of Basic Engineering, Vol. 83, pp. 95–108, 1961.Google Scholar
  13. 13.
    Luenberger, D.,An Introduction to Observers, IEEE Transactions on Automatic Control, Vol. AC-16, pp. 596–602, 1977.Google Scholar
  14. 14.
    Simon, J., andMitter, S.,A Theory of Modal Control, Information and Control, Vol. 13, pp. 316–353, 1968.Google Scholar
  15. 15.
    Gould, L., Murphy, A., andBerkman, E.,On the Simon-Mitter Pole Allocation Algorithm: Repeated Eigenvalues, IEEE Transactions on Automatic Control, Vol. AC-15, pp. 259–260, 1970.Google Scholar
  16. 16.
    Bongiorno, J., andYoula, D.,On Observers in Multi-Variable Control Systems, International Journal of Control, Vol. 8, pp. 221–243, 1968.Google Scholar
  17. 17.
    Chen, C.,Introduction to Linear System Theory, Holt, Rinehart and Winston, New York, New York, 1970.Google Scholar
  18. 18.
    Noble, B.,Applied Linear Algebra, Prentice-Hall, Englewood Cliffs, New Jersey, 1969.Google Scholar
  19. 19.
    Ginter, S.,Attitude Stabilization of Large Flexible Satellites, Massachusetts Institute of Technology, Aeronautics and Astronautics Department, MS Thesis, 1978.Google Scholar
  20. 20.
    Prado, G.,Observability, Estimation, and Control of Distributed Parameter Systems, Massachusetts Institute of Technology, Electronic Systems Laboratory, Report No. ESL-R-457, 1971.Google Scholar
  21. 21.
    Gustafson, D., andSpeyer, J.,Linear Minimum Variance Filters Applied to Carrier Tracking, IEEE Transactions on Automatic Control, Vol. AC-21, pp. 65–73, 1976.Google Scholar
  22. 22.
    Viterbi, A.,Principles of Coherent Communication, McGraw-Hill Book Company, New York, New York, 1966.Google Scholar
  23. 23.
    Balas, M.,Modal Control of Certain Flexible Dynamic Systems, SIAM Journal on Control and Optimization (to appear).Google Scholar
  24. 24.
    Potter, J.,Matrix Quadratic Solutions, SIAM Journal on Applied Mathematics, Vol. 14, pp. 496–501, 1966.Google Scholar
  25. 25.
    Balas, M., andCanavin, J.,An Active Modal Control System Philosophy for a Class of Large Space Structures, Paper Presented at the AIAA Symposium on Dynamics and Control of Large Flexible Spacecraft, Blacksburg, Virginia, 1977.Google Scholar
  26. 26.
    Balas, M.,Observer Stabilization of Singularly Perturbed Systems, AIAA Journal on Guidance and Control, Vol. 1, pp. 93–95, 1978.Google Scholar
  27. 27.
    Narendra, K., andKudva, P.,Stable Adaptive Schemes for System Identification and Control, Parts I and II, IEEE Transactions on Systems, Man, and Cybernetics, Vol. SMC-4, pp. 542–560, 1974.Google Scholar
  28. 28.
    Carroll, R., andLindorff, D.,An Adaptive Observer for Single-Input Single-Output Linear Systems, IEEE Transactions on Automatic Control, Vol. AC-18, pp. 428–435, 1973.Google Scholar
  29. 29.
    Likins, P.,The Application of Multivariable Control Theory to Spacecraft Attitude Control, Paper Presented at the 4th IFAC Symposium on Multivariable Technological Systems, Fredericton, New Brunswick, Canada, 1977.Google Scholar

Copyright information

© Plenum Publishing Corporation 1978

Authors and Affiliations

  • M. J. Balas
    • 1
  1. 1.The Charles Stark Draper LaboratoryCambridge

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