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State representations of linear systems with output constraints

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Abstract

We derive state space representations for linear systems that are described by input/state/output equations and that are subjected to a number of constant linear constraints on the outputs. In the case of a general linear system, the state representation of the constrained system is shown to be essentially nonunique. For linear Hamiltonian systems satisfying a nondegeneracy condition, there is a natural and unique choice of the representation which preserves the Hamiltonian structure. In the linear systems setting we give an algebraic proof that a system withn degrees of freedom underk constraints becomes a system withn−k degree of freedom. Similar results are obtained for linear gradient systems.

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References

  1. R. A. Abraham and J. E. Marsden,Foundations of Mechanics, 2nd edn., Addison-Wesley, Reading, MA, 1978.

    MATH  Google Scholar 

  2. B. D. O. Anderson, Output-nulling invariant and controllability subspaces,Proceedings of the Sixth World Congress of the International Federation of Automatic Control, Cambridge, MA, 1975, paper 43.6.

  3. V. I. Arnold,Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1978.

    MATH  Google Scholar 

  4. H. Blomberg and R. Ylinen,Algebraic Theory for Multivariable Linear Systems, Academic Press, London, 1983.

    MATH  Google Scholar 

  5. R. K. Brayton and J. K. Moser, A theory of nonlinear networks,Quart. Appl. Math.,22 (1964), 1–33 (Part I), 81–104 (Part II).

    MathSciNet  Google Scholar 

  6. J. P. Corfmat and A. S. Morse, Control of linear systems through specified input channels,SIAM J. Control Optim.,14 (1976), 163–175.

    Article  MathSciNet  Google Scholar 

  7. J. P. Den Hartog,Mechanical Vibrations, Dover, New York, 1985; reprint of original 4th edn., 1956.

    Google Scholar 

  8. M. L. J. Hautus, The formal Laplace transform for smooth linear systems, inMathematical Systems Theory (E. G. Marchesini and S. K. Mitter, eds.), pp. 29–47, Lecture Notes in Economics and Mathematical Systems, Vol. 131, Springer-Verlag, New York, 1976.

    Google Scholar 

  9. J. Hoekstra, Hamiltonse systemen met beperkingen op de uitgang (Hamiltonian systems with output restrictions, M.Sc. thesis, Department of Applied Mathematics, Twente University, 1988 (in Dutch).

  10. S. Lang,Algebra, Addison-Wesley, Reading, MA, 1965.

    MATH  Google Scholar 

  11. D. G. Luenberger, Dynamic equations in descriptor form,IEEE Trans. Automat. Control,22 (1977), 312–321.

    Article  MathSciNet  Google Scholar 

  12. C. C. MacDuffee,The Theory of Matrices, Chelsea, New York, 1956; reprint of original, 1933.

    Google Scholar 

  13. M. Malabre, Structure à l’infini des triplets invariants: Application à la poursuite parfaite de modèle, inAnalysis and Optimization of Systems (A. Bensoussan and J. L. Lions, eds.), pp. 43–53, Lecture Notes in Control and Information Sciences, Vol. 44, Springer-Verlag, Berlin, 1982.

    Chapter  Google Scholar 

  14. H. Nijmeijer and J. M. Schumacher, On the inherent integration structure of nonlinear systems,IMA J. Math. Control Inform. 2 (1985), 87–107.

    Article  Google Scholar 

  15. J. W. S. Rayleigh,The Theory of Sound, Vol. 1, Dover, New York, 1945; reprint of original, 1894.

    MATH  Google Scholar 

  16. H. H. Rosenbrock,State Space and Multivariable Theory, Wiley, New York, 1970.

    MATH  Google Scholar 

  17. A. J. van der Schaft, System Theoretic Descriptions of Physical Systems, CWI Tract 3, Centre for Mathematics and Computer Science, Amsterdam, 1984.

    Google Scholar 

  18. A. J. van der Schaft, On realization of nonlinear systems described by higher-order differential equations,Math. Systems Theory,19 (1987), 239–275.

    Article  MathSciNet  Google Scholar 

  19. A. J. van der Schaft, Equations of motion for Hamiltonian systems with constraints,J. Phys. A,20 (1987), 3271–3277.

    Article  MathSciNet  Google Scholar 

  20. J. M. Schumacher, Transformations of linear systems under external equivalence,Linear Algebra Appl.,102 (1988), 1–34.

    Article  MathSciNet  Google Scholar 

  21. J. C. Willems, Input-output and state space representations of finite-dimensional linear time-invariant systems,Linear Algebra Appl.,50 (1983), 581–608.

    Article  MathSciNet  Google Scholar 

  22. J. C. Willems, From time series to linear system, Part I: Finite-dimensional linear time-invariant systems,Automatica,22 (1986), 561–580.

    Article  MathSciNet  Google Scholar 

  23. W. M. Wonham,Linear Multivariable: Control: a Geometric Approach, 2nd edn., Springer-Verlag, New York, 1979.

    MATH  Google Scholar 

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  1. Centre for Mathematics and Computer Science, P.O. Box 4079, 1009 AB, Amsterdam, The Netherlands

    J. M. Schumacher

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  1. J. M. Schumacher
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Schumacher, J.M. State representations of linear systems with output constraints. Math. Control Signal Systems 3, 61–80 (1990). https://doi.org/10.1007/BF02551356

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  • DOI: https://doi.org/10.1007/BF02551356

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