Control of Uncertain Systems pp 1-17 | Cite as

# Robust Stabilization of a Flexible Beam Model Using a Normalized Coprime Factorization Approach

Chapter

## Abstract

The problem of robustly stabilizing a linear system subject to *H* _{ ∞ }-bounded perturbations in the numerator and the denominator of its normalized left coprime factorizations is considered for a class of infinite-dimensional systems. This class has possibly unbounded, finite-rank input and output operators which includes many delay and distributed systems. The optimal stability margin is expressed in terms of the solutions of the control and filter algebraic Riccati equations. The applicability of this theory is demonstrated by a controller design for a flexible beam with uncertain parameters.

## Keywords

Robust Stabilization Robust Controller Rigid Body Mode Coprime Factorization Close Loop Pole
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

- [1]J. Bontsema,
*Dynamic stabilization of large flexible space structures*, Ph.D. thesis University of Groningen, 1989.Google Scholar - [2]J. Bontsema, R.F. Curtain, and J.M. Schumacher,
*Robust control of flexible systems: a case study*, Automatica, vol. 24, p. 177–186, 1988.CrossRefGoogle Scholar - [3]J.A. Ball, and J.W. Helton,
*A Beurling-Lax theorem for the Lie group U(m*,*n) which contains most classical interpolation theory*, J. Operator Theory, 8, p. 107–142, 1983.Google Scholar - [4]J.A. Ball, and A. Ran,
*Optimal Hankel Norm Model Reductions and Wiener-Hopf Factorization II: the Non-Canonical Case*, Integral Equations and Operator Theory (1987),**10**, p. 416–436.CrossRefGoogle Scholar - [5]F.M. Cahier, and C.A. Desoer,
*Simplifications and New Connections on an Algebra of Transfer Functions of Distributed Linear*, Time-Invariant Systems“, IEEE Trans. Circuits of Systems, CAS-27, p. 320323, 1980.Google Scholar - [6]F.M. Cahier, and C.A. Desoer,
*An Algebra of Transfer Functions for Distributed linear Time-Invariant Systems*, IEEE Trans. Circuits and Systems 25, p. 651–663, 1978.CrossRefGoogle Scholar - [7]F.M. Cahier and C.A. Desoer,
*Stabilization*,*Tracking and Distributed Rejection in Multivariable Convolution Systems*. Ann. Soc. Sci. Bruxelles, 94, p. 7–51, 1980.Google Scholar - [8]C.C. Chu, J.C. Doyle and E.B. Lee,
*The general distance in H**c*,*optimal control theory*, Int. J. Control, vol. 44, p. 565–596, 1986.CrossRefGoogle Scholar - [9]F.M. Callier and J. Winkin,
*On Spectral Factorization and LQ-optimal Regulation of Multivariable Distributed Systems*,Int. J. Control (to appear).Google Scholar - [10]R.F. Curtain,
*Equivalence of Input-Output Stability and Exponential Stability for Infinite-Dimensional Systems*, Math. System Theory, 1988,**21**, p. 19–48.CrossRefGoogle Scholar - [11]R.F. Curtain,
*Robust Stabilizability of Normalized Coprime Factors: The Infinite-Dimensional Case*,1990, International Journal of Control (to appear).Google Scholar - [12]R.F. Curtain and K. Glover,
*Robust stabilization of infinite-dimensional systems by finite-dimensional controllers*, Systems and Control Letters, vol. 7, pp. 41–47, 1986.CrossRefGoogle Scholar - [13]R.F. Curtain and A. Ran,
*Explicit Formulas for Hankel Norm Approximations of Infinite-dimensional Systems*, Integral Equations & Operator Theory, 1989, 12, p. 455–469.CrossRefGoogle Scholar - [14]T.T. Georgiou,
*On the computation of the gap metric*, System and Control Letters, vol. 11, pp. 253–257, 1988.CrossRefGoogle Scholar - [15]T.T. Georgiou, and M.C. Smith,
*Optimal robustness in the gap metric*, 1990, IEEE Trans. Autom. Control (to appear).Google Scholar - [16]K. Glover, R.F. Curtain, and J.R. Partington,
*Realization and approximation of linear infinite-dimensional systems with error bounds*, SIAM J. Control and Optim., vol. 26, pp. 863–899, 1989.CrossRefGoogle Scholar - [17]K. Glover and D. McFarlane, Robust stabilization of normalized co-prime factors: An explicit Hoo-solution, Proc. of the ACC, pp. 842–847, 1988.Google Scholar
- [18]D. McFarlane, and K. Glover,
*Robust controller design using normalized coprime factor plant descriptions*,1990, Springer Verlag LNCIS, no 138, Berlin.Google Scholar - [19]K. Glover and D. McFarlane,
*Robust Stabilization of Normalized Co-prime Factor Plant Descriptions with H**oo**-bounded Uncertainty*, submitted to IEEE Trans. on Automatic Control, 1988.Google Scholar - [20]K.Glover and J.R. Partington,
*Robust Stabilization of Delay Systems*,Proc. MTNS, 1989 (to appear).Google Scholar - [21]P.P. Khargonekar and E. Sontag,
*On the Relation between Stable Matrix Fraction Factorizations and Regulable Realizations of Linear Systems over Rings*, IEEE Trans. AC-27, 1982, p. 627–638.Google Scholar - [22]C.N. Nett, C.A. Jacobson and M.J. Balas,
*Fractional Representation Theory: Robustness with Applications to Finite-Dimensional Control of a Class of Linear Distributed Systems*, Proc. IEEE Conf. on Decision and Control, p. 269–280, 1983.Google Scholar - [23]C.N. Nett, C.A. Jacobson and M J Balas,
*A Connection Between State Space and Doubly Co-prime Fractional Representations*, Trans. IEEE, Vol. AC-21, 9, p. 831–832, 1984.Google Scholar - [24]A.J. Pritchard and D. Salamon,
*The Linear Quadratic Optimal Control Problem for Infinite Dimensional Systems with Unbounded Input and Output Operators*, SIAM J. Control and Optimiz, 25, p. 121–144, 1987.CrossRefGoogle Scholar - [25]M. Vidyasagar,
*Control System Synthesis: A Coprime Factorization Approach*, MIT Press, 1985.Google Scholar - [26]M. Vidyasagar and H. Kimura,
*Robust controllers for uncertain linear multivariable systems*, Automatica, vol. 22, pp. 85–94, 1986.CrossRefGoogle Scholar - [27]G. Weiss,
*Admissibility of Unbounded Control Operator*, SIAM J. Control and Optim. 27, p. 527–545, 1989.CrossRefGoogle Scholar - [28]G. Weiss,
*Admissible Observation Operators for Linear Semigroups*, Israel J. Math., 65, p. 17–43, 1989.Google Scholar - [29]G. Weiss,
*The Representation of Regular Linear Systems on Hilbert Spaces*,Proceedings of the Conference on Distributed Parameter Systems, Vorau, Austria, July 1988, to appear.Google Scholar

## Copyright information

© Springer Science+Business Media New York 1990