Skip to main content
SpringerLink
Log in

A lattice approach to analysis and synthesis problems

  • Published:
Mathematics of Control, Signals and Systems Aims and scope Submit manuscript

Abstract

Within a lattice approach, the purpose of this paper is to give general necessary and sufficient conditions for internal stabilizability and for the existence of (weakly) left-/right-/doubly coprime factorizations of multi input multi output linear systems. These results extend the ones recently obtained in [24] for single input single output systems. In particular, combining these results with the one obtained in [3, 13], we prove that every internally stabilizable multidimensional system admits doubly coprime factorizations, solving Lin's conjecture [17, 18].

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Blackadar B (1986) K-theory for operator algebras, M.S.R.I. Monographs 5. Springer, Berlin Heidelberg New York

  2. Bourbaki N (1989) Commutative algebra, Chapt 1–7, Springer, Berlin Heidelberg New York

  3. Byrnes CI, Spong MW, Tarn T-J (1984) A several complex variables approach to feedback stabilization of linear neutral delay-differential systems. Math Syst Theory 17:97–133

    Google Scholar 

  4. Curtain RF, Zwart HJ (1991) An introduction to infinite-dimensional linear systems theory. TAM 21, Springer, Berlin Heidelberg New York

  5. Desoer CA, Chan WS (1975) The feedback interconnection of lumped linear time-invariant systems. J Franklin Inst 300:335–351

    Google Scholar 

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

    Google Scholar 

  7. Dicks W, Sontag ED (1978) Sylvester domains. J Pure Appl Algebra 13:243–275

    Google Scholar 

  8. Fuhrmann PA (1976) Algebraic system theory: an analyst's point of view. J Franklin Inst 301:521–540

    Google Scholar 

  9. Georgiou TT, Smith MC (1990) Optimal robustness in the gap metric. IEEE Trans Automat Control 35:673–685

    Google Scholar 

  10. Georgiou TT, Smith MC (1993) Graphs, causality, and stabilizability: linear, shift-invariant systems on Math Control Signal Syst 6:195–223

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

    Google Scholar 

  12. Inouye Y (1990) Linear systems with transfer functions of bounded type: factorization approach. In: Linear circuits, systems and signal processing, advanced theory and applications. Nagai N, Marcel Dekker, Inc., pp. 141–173

  13. Kamen E, Khargonekar PP, Tannenbaum A (1984) Pointwise stability and feedback control of linear systems with noncommensurate time delays. Acta Applicandæ Mathematicæ 2:159–184

    Google Scholar 

  14. Khargonekar PP, Sontag ED (1982) On the relation between stable matrix fraction factorizations and regulable realizations of linear systems over rings. IEEE Trans Automat Control 27:627–638

    Google Scholar 

  15. Lam TY (1978) Serre's conjecture, lecture notes in mathematics. 635, Springer, Berlin Heidelberg New York

  16. Lam TY (1991) A first course in noncommutative rings, graduate texts in mathematics, Springer, Berlin Heidelberg New York

  17. Lin Z (1998) Feedback stabilizability of MIMO n-D linear systems. Multidimens Syst Signal Process 9:149–172

    Google Scholar 

  18. Lin Z (2001) Output feedback stabilizability and stabilization of linear n-D systems. In: Multidimensional signals, circuits and systems. Galkowski K, Wood J, Taylor and Francis

  19. Logemann H (1993) Stabilization and regulation of infinite-dimensional systems using co-prime factorizations. In: Analysis and optimization of systems: state and frequency domain approaches for infinite-dimensional systems. Lecture notes in control and information, vol 185. Springer, Berlin Heidelberg New York

  20. Mori K, Abe K (2001) Feedback stabilization over commutative rings: Further study of coordinate-free approach. SIAM J Control Optim 39:1952–1973

    Google Scholar 

  21. Mori K (2002) Parametrization of the stabilizing controllers over a commutative ring with applications to multidimensional systems. IEEE Trans Circ Sys 49:743–752

    Google Scholar 

  22. Polderman JW, Willems JC (1998) Introduction to mathematical systems theory. A Behavioral Approach, TAM 26. Springer, Berlin Heidelberg New York

  23. Quadrat A (2003) The fractional representation approach to synthesis problems: an algebraic analysis viewpoint. Part I: (weakly) doubly coprime factorizations. Part II: internal stabilization. SIAM J Control Optim 42:266–299, 300–320

    Google Scholar 

  24. Quadrat A (2003) On a generalization of the Youla-Kučera parametrization. Part I: The fractional ideal approach to SISO systems. Syst Control Lett 50:135–148

    Google Scholar 

  25. Quadrat A (2004) On a general structure of the stabilizing controllers based on stable range. SIAM J Control Optim 42:2264–2285

    Google Scholar 

  26. Quadrat A (2005) An algebraic interpretation to the operator-theoretic approach to stabilizability. Part I: SISO systems. Acta Applicandæ Mathematicæ 88:1–45

    Google Scholar 

  27. Quadrat A (2005) On a generalization of the Youla-Kučera parametrization. Part II: the lattice approach to MIMO systems. Math Control Signals Syst (to appear)

  28. Quadrat A (2003) A generalization of the Youla-Kučera parametrization for MIMO stabilizable systems. In: Proceedings of the workshop on time-delay systems (TDS03), 08-10/09/03, IFAC Workshop, INRIA Rocquencourt, France

  29. Quadrat A (2004) Every internally stabilizable multidimensional system admits a doubly coprime factorization. In: Proceedings of mathematical theory of networks and systems (MTNS04), 05-09/07/04, Leuven, Belgium

  30. Quadrat A (2004) ``Stabilizing'' the stabilizing controllers. In: Proceedings of mathematical theory of networks and systems (MTNS04), 05-09/07/04, Leuven, Belgium

  31. Rosenberg J (1986) Algebraic K-theory and its applications, graduate texts in mathematics 47. Springer, Berlin Heidelberg New York

  32. Rotman JJ (1979) An Introduction to Homological Algebra. Academic, New York

  33. Shankar S, Sule VR (1992) Algebraic geometric aspects of feedback stabilization. SIAM J Control Optim 30:11–30

    Google Scholar 

  34. Smith MC (1989) On stabilization and existence of coprime factorizations. IEEE Trans Automat Control 34:1005–1007

    Google Scholar 

  35. Sontag ED (1977) The lattice of minimal realizations of response maps over rings. Math Syst Theory 11:169–175

    Google Scholar 

  36. Sule VR (1994) Feedback stabilization over commutative rings: the matrix case. SIAM J Control Optim 32:1675–1695

    Google Scholar 

  37. Corrigendum. SIAM J Control Optim 36:2194–2195

  38. Vidyasagar M (1985) Control System Synthesis: A Factorization Approach. MIT, Cambridge

  39. Zames G (1981) Feedback and optimal sensitivity: model reference transformations, multiplicative seminorms, and approximate inverses. IEEE Trans Automat Control 26:301–320

    Google Scholar 

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

    Google Scholar 

Download references

Author information

Authors and Affiliations

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

    A. Quadrat

Authors
  1. A. Quadrat
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Quadrat.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Quadrat, A. A lattice approach to analysis and synthesis problems. Math. Control Signals Syst. 18, 147–186 (2006). https://doi.org/10.1007/s00498-005-0159-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00498-005-0159-2

Keywords

Search

Navigation