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An Algebraic Interpretation to the Operator-Theoretic Approach to Stabilizability. Part I: SISO Systems

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Abstract

The purpose of this paper is to show that a duality exists between the fractional ideal approach [23, 26] and the operator-theoretic approach [4, 6, 8, 9, 33, 34] to stabilization problems. In particular, this duality helps us to understand how the algebraic properties of systems are reflected by the operator-theoretic approach and conversely. In terms of modules, we characterize the domain and the graph of an internally stabilizable plant or that of a plant which admits a (weakly) coprime factorization. Moreover, we prove that internal stabilizability implies that the graph of the plant and the graph of a stabilizing controller are direct summands of the global signal space. These results generalize those obtained in [6, 8, 9, 33, 34]. Finally, we exhibit a class of signal spaces over which internal stabilizability is equivalent to the existence of a bounded inverse for the linear operator mapping the errors e1 and e2 of the closed-loop system to the inputs u1 and u2.

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References

  1. Bourbaki, N.: Algèbre Homologique, Masson, Paris, 1980, Chapter 10.

    Google Scholar 

  2. Bourbaki, N.: Commutative Algebra, Springer-Verlag, Berlin, 1986, Chapters 1–7.

    Google Scholar 

  3. Curtain, R. F. and Zwart, H. J.: An Introduction to Infinite-Dimensional Linear Systems Theory, TAM 21, Springer-Verlag, Berlin, 1991.

    Google Scholar 

  4. Desoer, C. A., Liu, R. W., Murray, J. and Saeks, R.: Feedback system design: the fractional representation approach to analysis and synthesis, IEEE Trans. Automat. Control 25 (1980), 399–412.

    Article  Google Scholar 

  5. Desoer, C. A. and Vidyasagar, M.: Feedback Systems: Input-Output Properties, Academic Press, New York, 1975.

    Google Scholar 

  6. Doyle, J. C., Georgiou, T. T. and Smith, M. C.: The parallel projection operators of a nonlinear feedback system, Systems Control Lett. 20 (1993), 79–85.

    Article  Google Scholar 

  7. Fuchs, L. and Salce, L.: Modules over Non-Noetherian Domains, Mathematical Survey and Monographs 84, Amer. Math. Soc., Providence, RI, 2000.

    Google Scholar 

  8. Georgiou, T. T. and Smith, M. C.: Optimal robustness in the gap metric, IEEE Trans. Automat. Control 35 (1990), 673–685.

    Article  Google Scholar 

  9. Georgiou, T. T. and Smith, M. C.: Graphs, causality, and stabilizabity: Linear, shift-invariant systems on ℒ2[0,∞), Math. Control Signals Systems 6 (1993), 195–223.

    Article  Google Scholar 

  10. Glover, K. and McFarlane, D.: Robust stabilization of normalized coprime factor plant descriptions with H∞-bounded uncertainty, IEEE Trans. Automat. Control 34 (1989), 821–830.

    Article  Google Scholar 

  11. Gluesing-Luerssen, H.: Linear Delay-Differential Systems with Commensurate Delays: An Algebraic Approach, Lecture Notes in Math. 1770, Springer, New York, 2002.

    Google Scholar 

  12. Hoffman, K.: Banach Spaces of Analytic Functions, Dover, New York, 1962.

    Google Scholar 

  13. Inouye, Y.: Linear systems with transfer functions of bounded type: Factorization approach, In: N. Nagai (ed.), Linear Circuits, Systems and Signal Processing, Advanced Theory and Applications, Marcel Dekker, New York, 1990, pp. 141–173.

    Google Scholar 

  14. Malgrange, B.: Systèmes à coefficients constants, Séminaire Bourbaki 246 (1962/63), 1–11.

    Google Scholar 

  15. Nikolskii, N. K.: Treatise on the Shift Operator, Springer-Verlag, Berlin, 1986.

    Google Scholar 

  16. Oberst, U.: Multidimensional constant linear systems, Acta Appl. Math. 20 (1990), 1–175.

    Article  Google Scholar 

  17. Pillai, H. K. and Shankar, S.: A behavioral approach to control of distributed systems, SIAM J. Control Optim. 37 (1998), 388–408.

    Article  Google Scholar 

  18. Polderman, J. W. and Willems, J. C.: Introduction to Mathematical Systems Theory. A Behavioral Approach, TMA 26, Springer, Berlin, 1998.

    Google Scholar 

  19. Pommaret, J.-F.: Partial Differential Control Theory, Kluwer, Dordrecht, 2001.

    Google Scholar 

  20. Pommaret, J. F. and Quadrat, A.: Localization and parametrization of linear multidimensional control systems, Systems Control Lett. 37 (1999), 247–260.

    Article  Google Scholar 

  21. Pommaret, J. F. and Quadrat, A.: A functorial approach to the behaviour of multidimensional control systems, Appl. Math. Computer Sci. 13 (2003), 7–13.

    Google Scholar 

  22. Quadrat, A.: The fractional representation approach to synthesis problems: An algebraic analysis viewpoint I. (Weakly) doubly coprime factorizations, II. Internal stabilization, SIAM J. Control Optim. 42 (2003), 266–299, 300–320.

    Article  Google Scholar 

  23. Quadrat, A.: On a generalization of the Youla–Kučera paramatrization. Part I: The fractional ideal approach to SISO systems, Systems Control Lett. 50 (2003), 135–148.

    Article  Google Scholar 

  24. Quadrat, A.: On a generalization of the Youla–Kučera parametrization. Part II: The lattice approach to MIMO systems, Math. Control Signals Systems (2005), to appear.

  25. Quadrat, A.: A lattice approach to analysis and synthesis problems, Math. Control Signals Systems (2005), to appear.

  26. Quadrat, A.: Every internally stabilizable multidimensional system admits a doubly coprime factorization, In: Proceedings of MTNS04, Leuven, Belgium (05-09/07/04).

  27. Quadrat, A.: “Stabilizing” the stabilizing controllers, In: Proceedings of MTNS04, Leuven, Belgium (05-09/07/04).

  28. Quadrat, A.: An elementary proof of the general Q-parametrization of all stabilizing controllers, In: Proceedings of the World IFAC Congress, Prague, Czeck Republic (04-08/07/05), to appear.

  29. Rotman, J. J.: An Introduction to Homological Algebra, Academic Press, New York, 1979.

    Google Scholar 

  30. Shankar, S.: Geometric completeness of distribution spaces, Acta Appl. Math. 77 (2003), 168–180.

    Article  Google Scholar 

  31. Shankar, S. and Sule, V. R.: Algebraic geometry aspects of feedback stabilization, SIAM J. Control Optim. 30 (1992), 11–30.

    Article  Google Scholar 

  32. Smith, M. C.: On the stabilization and the existence of coprime factorizations, IEEE Trans. Automat. Control 34 (1989), 1005–1007.

    Article  Google Scholar 

  33. Sule, V. R.: Feedback stabilization over commutative rings: The matrix case, SIAM J. Control Optim. 32 (1994), 1675–1695; Corrigendum, SIAM J. Control Optim. 36 (1998), 2194–2195.

    Article  Google Scholar 

  34. Vidyasagar, M.: On the use of right-coprime factorizations in distributed feedback systems containing unstable systems, IEEE Trans. Circuits Systems 25 (1978), 916–921.

    Article  Google Scholar 

  35. Vidyasagar, M.: Control System Synthesis: A Factorization Approach, MIT Press, Cambridge, MA, 1985.

    Google Scholar 

  36. Wood, J.: Modules and behaviours in nD systems theory, Multidimens. Systems Signal Process. 11 (2000), 11–48.

    Article  Google Scholar 

  37. Zames, G. and Francis, B.: Feedback, minimax sensitivity, and optimal robustness, IEEE Trans. Automat. Control 28 (1983), 585–601.

    Article  Google Scholar 

  38. Zerz, E.: Topics in Multidimensional Linear Systems Theory, Lecture Notes in Control and Inform. Sci. 256, Springer, New York, 2000.

    Google Scholar 

  39. Zhu, S. Q.: Graph topology and gap topology for unstable systems, IEEE Trans. Automat. Control 34 (1989), 848–855.

    Article  Google Scholar 

  40. Zhu, S. Q.: On normalized Bezout fractions of distributed LTI systems, IEEE Trans. Automat. Control 36 (1991), 489–491.

    Article  Google Scholar 

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  1. CAFE, INRIA Sophia Antipolis, 2004 route des Lucioles, BP 93, 06902, Sophia Antipolis Cedex, France

    A. Quadrat

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  1. A. Quadrat
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Correspondence to A. Quadrat.

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Mathematics Subject Classifications (2000)

93C05, 93D25, 93B25, 93B28, 16D40, 30D55, 47A05.

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Quadrat, A. An Algebraic Interpretation to the Operator-Theoretic Approach to Stabilizability. Part I: SISO Systems. Acta Appl Math 88, 1–45 (2005). https://doi.org/10.1007/s10440-005-6697-2

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