Advertisement

Complex Analysis and Operator Theory

, Volume 4, Issue 2, pp 319–390 | Cite as

Well-Posed State/Signal Systems in Continuous Time

  • Mikael KurulaEmail author
  • Olof J. Staffans
Article

Abstract

We introduce a new class of linear systems, the L p -well-posed state/signal systems in continuous time, we establish the foundations of their theory and we develop some tools for their study. The principal feature of a state/signal system is that the external signals of the system are not a priori divided into inputs and outputs. We relate state/signal systems to the better-known class of well-posed input/state/output systems, showing that state/signal systems are more flexible than input/state/output systems but still have enough structure to provide a meaningful theory. We also give some examples which point to possibilities for further study.

Keywords

Systems theory State/signal systems Infinite-dimensional systems Well-posed systems System nodes Input/state/output systems 

Mathematics Subject Classification (2000)

Primary 93A05 47A48 Secondary 93B28 94C05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arov D.Z., Nudelman M.A.: Passive linear stationary dynamical scattering systems with continuous time. Integr. Equ. Oper. Theory 24, 1–45 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Arov D.Z.: A survey on passive networks and scattering sytstems which are lossless or have minimal losses. Archiv Elektronik Übertragungstechnik 49, 252–265 (1995)Google Scholar
  3. 3.
    Arov, D.Z.: Passive linear systems and scattering theory. In: Dynamical Systems, Control Coding, Computer Vision, Progress in Systems and Control Theory, vol. 25, pp. 27–44. Birkhäuser Verlag, Basel (1999)Google Scholar
  4. 4.
    Arov, D.Z., Staffans, O.J.: State/signal linear time-invariant systems theory. Part I: discrete time systems. In: The State Space Method, Generalizations and Applications, Operator Theory: Advances and Applications, vol. 161, pp. 115–177. Birkhäuser-Verlag, Basel (2005)Google Scholar
  5. 5.
    Arov D.Z., Staffans O.J.: State/signal linear time-invariant systems theory, passive discrete time systems. Int. J. Robust Nonlinear Control 17, 497–548 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Arov, D.Z., Staffans, O.J.: State/signal linear time-invariant systems theory. Part III: transmission and impedance representations of discrete time systems. In: Operator Theory, Structured Matrices, and Dilations. Tiberiu Constantinescu Memorial Volume, pp. 104–140. Theta Foundation, Bucharest (2007)Google Scholar
  7. 7.
    Arov D.Z., Staffans O.J.: State/signal linear time-invariant systems theory. Part IV: affine representations of discrete time systems. Complex Anal. Oper. Theory 1, 457–521 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Arov, D.Z., Staffans, O.J.: Two canonical passive state/signal shift realizations of passive discrete time behaviors. Draft at http://web.abo.fi/~staffans/ (2008, submitted)
  9. 9.
    Åström K.J., Hägglund T.: PID Controllers: Theory, Design and Tuning. Instrument Society of America, Research Triangle Park, NC (1995)Google Scholar
  10. 10.
    Belevitch V.: Classical Network Theory. Holden-Day, San Francisco (1968)zbMATHGoogle Scholar
  11. 11.
    Curtain, R.F., Weiss, G.: Well posedness of triples of operators (in the sense of linear systems theory). In: Control and Optimization of Distributed Parameter Systems, International Series of Numerical Mathematics, vol. 91, pp. 41–59. Birkhäuser-Verlag, Basel (1989)Google Scholar
  12. 12.
    Kato, T.: Perturbation theory for linear operators. Classics in Mathematics (reprint of the 1980 edn). Springer, Berlin (1995)Google Scholar
  13. 13.
    Kurula M., Staffans O.: A complete model of a finite-dimensional impedance-passive system. Math. Control Signals Syst. 19(1), 23–63 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kurula, M.: Passive continuous-time state/signal systems (2009, in preparation)Google Scholar
  15. 15.
    Kurula, M., Zwart, H., van der Schaft, A., Behrndt, J.: Dirac structures and their composition on Hilbert spaces. Draft available at http://web.abo.fi/~mkurula/ (2008, submitted)
  16. 16.
    Malinen J., Staffans O.J.: Conservative boundary control systems. J. Differ. Equ. 231(1), 290–312 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Malinen J., Staffans O.J.: Impedance passive and conservative boundary control systems. Complex Anal. Oper. Theory 1(2), 279–300 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Malinen J., Staffans O.J., Weiss G.: When is a linear system conservative? Q. Appl. Math. 64(1), 61–91 (2006)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Pazy A.: Semi-Groups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin (1983)Google Scholar
  20. 20.
    Polderman J.W., Willems J.C.: Introduction to Mathematical Systems Theory: A Behavioral Approach. Springer, New York (1998)Google Scholar
  21. 21.
    Salamon D.: Infinite dimensional linear systems with unbounded control and observation: a functional analytic approach. Trans. Am. Math. Soc. 300, 383–431 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Salamon D.: Realization theory in Hilbert space. Math. Syst. Theory 21, 147–164 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Staffans O.J.: Passive and conservative continuous-time impedance and scattering systems. Part I: well-posed systems. Math. Control Signals Syst. 15, 291–315 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Staffans, O.J.: Passive and conservative infinite-dimensional impedance and scattering systems (from a personal point of view). In: Mathematical Systems Theory in Biology, Communication, Computation, and Finance. IMA Volumes in Mathematics and its Applications, vol. 134, pp. 375–414. Springer, New York (2002)Google Scholar
  25. 25.
    Staffans O.J.: Well-Posed Linear Systems. Cambridge University Press, Cambridge (2005)zbMATHCrossRefGoogle Scholar
  26. 26.
    Staffans O.J., Weiss G.: Transfer functions of regular linear systems. Part II: the system operator and the Lax-Phillips semigroup. Trans. Am. Math. Soc. 354, 3229–3262 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Staffans O.J., Weiss G.: Transfer functions of regular linear systems. Part III: inversions and duality. Integr. Equ. Oper. Theory 49, 517–558 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Tucsnak M., Weiss G.: How to get a conservative well-posed linear system out of thin air. Part II: controllability and stability. SIAM J. Control Optim. 42, 907–935 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Weiss G.: Admissibility of unbounded control operators. SIAM J. Control Optim. 27, 527–545 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Weiss G.: Admissible observation operators for linear semigroups. Israel J. Math. 65, 17–43 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Weiss, G.: The representation of regular linear systems on Hilbert spaces. In: Control and Optimization of Distributed Parameter Systems. International Series of Numerical Mathematics, vol. 91, pp. 401–416. Birkhäuser-Verlag, Basel (1989)Google Scholar
  32. 32.
    Weiss G.: Transfer functions of regular linear systems. Part I: characterizations of regularity. Trans. Am. Math. Soc. 342, 827–854 (1994)zbMATHCrossRefGoogle Scholar
  33. 33.
    Weiss G., Staffans O.J., Tucsnak M.: Well-posed linear systems—a survey with emphasis on conservative systems. Int. J. Appl. Math. Comput. Sci. 11, 7–34 (2001)zbMATHMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Department of MathematicsÅbo Akademi UniversityÅboFinland

Personalised recommendations