Skip to main content
SpringerLink
Log in

Stabilization and regulation of infinite-dimensional systems using coprime factorizations

  • 1. Tutorial Lectures
  • Conference paper
  • First Online:
Analysis and Optimization of Systems: State and Frequency Domain Approaches for Infinite-Dimensional Systems

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 185))

Abstract

This paper surveys some of the results on stabilization and regulation of infinite-dimensional systems which have been obtained within the so-called fractional representation approach to feedback system analysis and synthesis. The relationship with state-space concepts is carefully discussed. The following topics are addressed: Rings of transfer functions and coprime factorizations, Pritchard-Salamon systems, External and internal closed-loop stability, Closed-loop stability and pole-zero cancellations, The Nyquist stability criterion, Closed-loop stability and the existence of coprime factorizations, Parametrization of all stabilizing controllers for a given plant, Existence of finite-dimensional stabilizing compensators, Strong stabilization by finite-dimensional controllers, The internal model principle, PI-control of uncertain infinite-dimensional systems.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. V. Anatharam. “On stabilization and the existence of coprime factorizations,” IEEE Trans. Auto. Control, vol. 30, pp. 1030–1031, 1985.

    Article  Google Scholar 

  2. B. D. O. Anderson and M. R. Gevers. “On multivarable pole-zero cancellations and the stability of feedback systems,” IEEE Trans. Circ. and Sys., vol. 28, pp. 830–833, 1981.

    Article  MathSciNet  Google Scholar 

  3. M. J. Balas. “Linear distributed parameter systems: Closed-loop exponential stability with a finite-dimensionsal controller,” Automatica, vol. 20, pp. 371–377, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  4. M. J. Balas. “Finite-dimensional control of distributed parameter systems by Galerkin approximation of infinite-dimensional controllers,” J. Math. Anal. & Appl., vol. 114, pp. 17–36, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  5. S. P. Banks. State-Space and Frequency-Domain Methods in the Control of Distributed Parameter Systems, London: P. Peregrinus, 1983.

    MATH  Google Scholar 

  6. J. Bontsema. Dynamic Stabilization of Large Flexible Structures. PhD Thesis, Mathematics Institute, University of Groningen, 1989.

    Google Scholar 

  7. J. Bontsema and R. F. Curtain, “Perturbation properties of a class of infinite-dimensional systems with unbounded control and observation,” IMA J. Math. Control & Information, vol. 5, pp. 333–352, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  8. J. Bontsema, R. F. Curtain, and J. M. Schumacher. “Robust control of flexible structures: A case study,” Automatica, vol. 24, pp.177–186, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  9. F. M. Callier and C. A. Desoer. “A graphical test for checking the stability of a linear time-invariant feedback system,” IEEE Trans. Auto. Control, vol. 17, pp.773–780, 1972.

    Article  MATH  MathSciNet  Google Scholar 

  10. F. M. Callier and C. A. Desoer. “On symplifying a graphical stability criterion for linear distributed feedback systems,” IEEE Trans. Auto. Control, vol. 21, pp. 128–129, 1976.

    Article  MATH  MathSciNet  Google Scholar 

  11. F. M. Callier and C. A. Desoer. “Open-loop unstable convolution feedback systems with dynamical feedback,” Automatica, vol. 12, pp. 507–518, 1976.

    Article  MATH  MathSciNet  Google Scholar 

  12. F. M. Callier and C. A. Desoer. “An algebra of transfer functions for distributed linear time-invariant systems,” IEEE Trans. Circ. and Sys., vol. 25, pp. 651–662, 1978. (Correction in vol. 26, p. 360).

    Article  MATH  MathSciNet  Google Scholar 

  13. F. M. Callier and C. A. Desoer. “Simplifications and clarifications on the paper ‘An algebra of transfer functions for distributed linear time-invariant systems’,” IEEE Trans. Circ. and Sys., vol. 27, pp. 320–323, 1980.

    Article  MATH  MathSciNet  Google Scholar 

  14. F. M. Callier and C. A. Desoer, “Stabilization, tracking and disturbance rejection in multivariable convolution systems,” Annales de la Societé Scientifique de Bruxelles, vol. 94, pp.7–51, 1980.

    MathSciNet  Google Scholar 

  15. F. M. Callier and J. Winkin. “Distributed system transfer functions of exponential order,” Int. J. Control, vol. 43, pp. 1353–1373, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  16. M. J. Chen and C. A. Desoer. “Necessary and sufficient conditions for robust stability of linear distributed feedback systems,” Int. J. Control, vol. 35, pp. 255–267, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  17. C. Corduneanu. Almost periodic functions. New York: J. Wiley, 1968.

    MATH  Google Scholar 

  18. R. F. Curtain. “Tracking and regulation for distributed parameter systems,” Matematica Applicada E Computational, vol. 2, pp. 199–218, 1983.

    MATH  MathSciNet  Google Scholar 

  19. R. F. Curtain. “Finite-dimensional compensators for parabolic distributed systems with unbounded control and observation,” SIAM J. Contr. & Opt., vol.22, pp.255–276, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  20. R. F. Curtain. “Equivalence of internal and input-output stability for infinite-dimensional systems,” Math. Systems Theory, vol. 21, pp. 19–48, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  21. R. F. Curtain. “A synthesis of time and frequency domain methods for the control of infinite-dimensional systems: A system theoretic approach,” Preprint 1989. To appear in SIAM Frontiers in Applied Mathematics.

    Google Scholar 

  22. R. F. Curtain. “Robust stabilizability of normalized coprime factors: The infinite-dimensional case,” Int. J. Control, vol. 51, pp. 1173–1190, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  23. 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.

    Article  MATH  MathSciNet  Google Scholar 

  24. R. F. Curtain, H. Logemann, S. Townley, and H. Zwart. “Well-posedness, stabilizability and admissibility for Pritchard-Salamon systems,” Report 260, Institut für Dynamische Systeme, Universität Bremen, 1992. Submitted for publication.

    Google Scholar 

  25. R. F. Curtain and A. J. Pritchard. Infinite-Dimensional Linear Systems Theory. Lecture Notes in Control and Information Sciences, vol. 8. New York: Springer-Verlag, 1978.

    MATH  Google Scholar 

  26. R. F. Curtain and D. Salamon. “Finite-dimensional compensators for infinite-dimensional systems with unbounded input operators,” SIAM J. Contr. & Opt., vol. 24, pp.797–816, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  27. R. Datko, J. Lagnese, and M. P. Polis, “An example of the effect of time delays in boundary feedback stabilization of wave equations,” SIAM J. Contr. & Opt., vol. 24, pp. 152–156, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  28. J. H. Davis. “Fredholm operators, encirclements, and stability criteria,” SIAM J. Control, vol. 10, pp. 608–622, 1972.

    Article  MATH  MathSciNet  Google Scholar 

  29. E. J. Davison. “Multivariable tuning regulators: The feedforward and robust control of a general servomechanism problem,” IEEE Trans. Auto. Control, vol. 21, pp. 35–47, 1976.

    Article  MATH  MathSciNet  Google Scholar 

  30. C. A. Desoer. “A general formulation of the Nyquist citerion,” IEEE Trans. Circuit Theory, vol. 12, pp. 230–234, 1965.

    MathSciNet  Google Scholar 

  31. C. A. Desoer and F. M. Callier. “Convolution feedback systems,” SIAM J. Control, vol. 10, pp. 737–746, 1972.

    Article  MATH  MathSciNet  Google Scholar 

  32. C. A. Desoer and W. S. Chan. “The feedback interconnection of linear time-invariant systems,” J. Franklin Inst., vol. 300, pp. 335–351, 1975.

    Article  MATH  MathSciNet  Google Scholar 

  33. C. A. Desoer and M. G. Kabuli. “Stabilization and robustness of nonlinear unity-feedback systems: Factorization approach,” Int. J. Control, vol. 47, pp.1133–1148, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  34. C. A. Desoer, R. W. Liu, J. Murray, and R. Saeks. “Feedback system design: The fractional representation approach to analysis and synthesis,” IEEE Trans. Auto. Control, vol. 25, pp. 399–412, 1980.

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  36. C. A. Desoer and Y. T. Wang. “The robust nonlinear servomechanism problem,” Int. J. Control, vol. 29, pp. 803–828, 1979.

    Article  MATH  MathSciNet  Google Scholar 

  37. C. A. Desoer and Y. T. Wang. “On the generalized Nyquist stability criterion,” IEEE Trans. Auto. Control, vol. 25, pp. 187–196, 1980. (Correction in vol. 26, p. 804).

    Article  MATH  MathSciNet  Google Scholar 

  38. P. M. G. Ferreira and F. M. Callier. “The tracking and disturbance rejection problem for distributed parameter systems,” IEEE Trans. Auto. Control, vol. 27, pp. 478–481, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  39. A. Feintuch. “Graphs of time varying linear systems and coprime factorizations,” J. Math. Anal. & Appl., vol. 163, pp. 79–85, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  40. A. Feintuch and R. Saeks. System Theory: A Hilbert Space Approach. New York: Academic Press, 1982.

    MATH  Google Scholar 

  41. B. A. Francis. “The multivariable servomechanism problem from the input-output viewpoint,” IEEE Trans. Auto. Control, vol. 22, pp. 322–328, 1977.

    Article  MATH  MathSciNet  Google Scholar 

  42. B. A. Francis. A Course in H ∞ Control Theory. Lecture Notes in Control and Information Sciences, vol. 88. New York: Springer-Verlag, 1987.

    MATH  Google Scholar 

  43. B. A. Francis and M. Vidyasagar. “Algebraic and topological aspects of the regulator problem for lumped linear systems,” Automatica, vol. 19, pp. 87–90, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  44. P. A. Fuhrmann. Linear Systems and Operators in Hilbert Space. New York: McGraw-Hill, 1981.

    MATH  Google Scholar 

  45. K. Glover, R. F. Curtain, and J. R. Partington. “Realisation and approximation of linear infinite-dimensional systems with error bounds,” SIAM J. Contr. & Opt., vol. 26, pp. 863–898, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  46. K. Glover, J. Lam, and J. R. Partington. “Rational approximation of a class of infinite-dimensional systems I: Singular values of Hankel operators,” Mathematics of Control, Signals, and Systems, vol. 3, pp. 325–344, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  47. K. Glover, J. Lam, and J. R. Partington. “Rational approximation of a class of infinite-dimensional systems II: Optimal convergence rates of L ∞-approximants,” Mathematics of Control, Signals, and Systems, vol. 4, pp. 233–246, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  48. G. Gu, E. B. Lee, and P. P. Khargonekar. “Approximation of infinite-dimensional systems,” IEEE Trans. Auto. Control, vol. 34, pp. 610–618, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  49. Y.-P. Harn and E. Polak. “Proportional-plus-multiintegral stabilizing compensators for a class of MIMO feedback systems with infinite-dimensional plants,” IEEE Trans Auto. Control, vol. 36, pp. 207–213, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  50. A. J. Helmicki, C. A. Jacobson, and C. N. Nett. “Ill-posed distributed parameter systems: A control viewpoint,” IEEE Trans. Auto. Control, vol. 36, pp. 1053–1057, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  51. B. Herz. Das Nyquist-Kriterium für Regelkreise, deren Verhalten durch lineare Funktional-Differential-Gleichungen beschrieben wird. PhD Thesis, Technische Universität Berlin, 1968.

    Google Scholar 

  52. K. Hoffman. Banach Spaces of Analytic Functions. Englewood Cliffs, N. J.: Prentice Hall, 1962.

    MATH  Google Scholar 

  53. Y. Inouye. “Parametrization of compensators for linear systems with transfer functions of bounded type,” in Proceedings of the 27th IEEE Conference on Decision and Control, (Austin, TX), pp. 2083–2088, 1988.

    Google Scholar 

  54. K. Ito. “Finite-dimensional compensators for infinite-dimensional systems via Galerkin-type approximation,” SIAM J. Contr. & Opt., vol. 28, pp. 1251–1269, 1990.

    Article  MATH  Google Scholar 

  55. C. A. Jacobson and C. N. Nett. “Linear state-space systems in infinite-dimensional space: The role and characterization of joint stabilizability/detectability,” IEEE Trans. Auto. Control, vol. 33, pp. 541–549, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  56. T. T. Jussila and H. N. Koivo. “Tuning of multivariable PI-controllers for unknown delay-differential systems,” IEEE Trans. Auto. Control, vol. 32, pp. 364–368, 1987.

    Article  MATH  Google Scholar 

  57. E. W. Kamen, P. P. Khargonekar, and A. Tannenbaum. “Stabilization of time-delay systems using finite-dimensional compensators,” IEEE Trans. Auto. Control, vol. 30, pp. 75–78, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  58. E. W. Kamen, P. P. Khargonekar, and A. Tannenbaum. “Proper stable Bezout factorizations and feedback control of linear time-delay systems,” Int. J. Control, vol. 43, pp. 837–857, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  59. T. Kobayashi. “Regulator problem for infinite-dimensional systems,” Systems and Control Letters, vol. 3, pp. 31–39, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  60. H. N. Koivo and S. A. Pohjolainen. “Tuning of multivariable PI-controllers for unknown systems with input delay,” Automatica, vol. 21, pp. 81–91, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  61. S. Lang. Algebra. Reading, MA: Addison-Wesley, 1965.

    MATH  Google Scholar 

  62. S. P. Levkov and J. Korbicz. “Convolution algebra representation of systems described by linear hyperbolic partial differential equations,” Int. J. Control, vol. 49, pp. 2029–2044, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  63. H. Logemann. “Finite-dimensional stabilization of infinite-dimensional systems: A frequency-domain approach,” Report 124, Institut für Dynamische Systeme, Universität Bremen, 1984.

    Google Scholar 

  64. H. Logemann. Funktionentheoretische Methoden in der Regelungstheorie unendlichdimensionaler Systeme. PhD Thesis, Report 156, Institut für Dynamische Systeme, Universität Bremen, 1986.

    Google Scholar 

  65. H. Logemann. “On the existence of finite-dimensional compensators for retarded and neutral systems,” Int. J. Control, vol. 43, pp. 109–121, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  66. H. Logemann. “Transfer-function conditions for the stability of neutral and Volterra integrodifferential systems,” IMA J. Math. Control & Information, vol. 3, pp. 9–19, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  67. H. Logemann. “Finitely generated ideals in certain algebras of transfer functions for infinite-dimensional systems,” Int. J. Control, vol. 45, pp. 247–250, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  68. H. Logemann. “On the transfer function of a neutral system: Characterization of exponential stability in input-output terms,” Systems and Control Letters, vol. 9, pp. 393–400, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  69. H. Logemann, J. Bontsema, and D. H. Owens. “Low-gain control of distributed parameter systems with unbounded control and observation,” Control — Theory and Advanced Technology, vol. 4, pp. 429–445, 1988.

    MathSciNet  Google Scholar 

  70. H. Logemann and D. H. Owens. “Robust high-gain feedback control of infinite-dimensional minimum-phase systems,” IMA J. Math. Control & Information, vol. 4, pp. 195–220, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  71. H. Logemann and D. H. Owens. “Low-gain control of unknown infinite-dimensional systems: A frequency-domain approach,” Dynamics and Stability of Systems, vol. 4, pp. 13–29, 1989.

    MATH  MathSciNet  Google Scholar 

  72. H. Logemann and H. J. Zwart. “On robust PI-control for infinite-dimensional systems,” SIAM J. Contr. & Opt., vol. 30, pp. 573–593, 1992

    Article  MATH  MathSciNet  Google Scholar 

  73. J. Lunze. Robust Multivariable Feedback Control. New York: Prentice Hall, 1989.

    MATH  Google Scholar 

  74. P. M. Mäkilä. “Laguerre series approximation of infinite-dimensional systems,” Automatica, vol. 26, pp. 985–995, 1990.

    Article  MATH  Google Scholar 

  75. M. Morari. “Robust stability of systems with integral control,” IEEE Trans. Auto. Control, vol. 30, pp. 574–577, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  76. S. Mossaheb. “A Nyquist type stability criterion for linear multivariable delayed systems,” Int. J. Control, vol. 32, pp. 821–847, 1980.

    Article  MATH  MathSciNet  Google Scholar 

  77. C. N. Nett. The Fractional Representation Approach to Robust Linear Feedback Design: A Self-Contained Exposition. MSc Thesis, ESCS Dept., Rensselaer Polytech. Inst., Troy, NY, 1984.

    Google Scholar 

  78. C. N. Nett. “Algebraic aspects of linear control system stability,” IEEE Trans. Auto. Control, vol. 31, pp. 941–949, 1986.

    Article  MATH  Google Scholar 

  79. 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,” in Proceedings of the 22nd IEEE Conference on Decision and Control, (San Antonio, TX), pp. 268–280, 1983.

    Google Scholar 

  80. C. N. Nett, C. A. Jacobson, and M. J. Balas. “A connection between state-space and doubly coprime fractional representations,” IEEE Trans. Auto. Control, vol. 29, pp. 831–833, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  81. H. Nyquist. “Regeneration theory,” Bell System Technical Journal, vol. 11, pp. 126–147, 1932.

    MATH  Google Scholar 

  82. D. H. Owens and A. Chotai. “High performance controllers for unknown multivariable systems,” Automatica, vol. 18, pp. 583–587, 1982.

    Article  MATH  Google Scholar 

  83. S. A. Pohjolainen. “Robust multivariable PI-controllers for infinite-dimensional systems,” IEEE Trans. Auto. Control, vol. 27, pp. 17–30, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  84. S. A. Pohjolainen. “Robust controller for systems with exponentially stable strongly continuous semigroups,” J. Math. Anal. & Appl., vol. 111, pp. 622–636, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  85. S. A. Pohjolainen and I. Lätti. “Robust controller for boundary control systems,” Int. J. Control, vol. 38, pp. 1189–1197, 1983.

    Article  MATH  Google Scholar 

  86. I. Postlethwaite and A. G. J. MacFarlane. A Complex Variable Approach to the Analysis of Linear Multivariable Feedback Systems. Lecture Notes in Control and Information Sciences, vol. 12. New York: Springer-Verlag, 1979.

    MATH  Google Scholar 

  87. A. J. Pritchard and D. Salamon. “The linear quadratic control problem for retarded systems with delays in control and observation,” IMA J. Math. Control & Information, vol. 2, pp. 335–362, 1985.

    Article  MATH  Google Scholar 

  88. A. J. Pritchard and D. Salamon. “The linear quadratic control problem for infinite-dimensional systems with unbounded input and output operators,” SIAM J. Contr. & Opt., vol. 25, pp. 121–144, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  89. R. Rebarber. “Conditions for the equivalence of internal and external stability for distributed parameter systems”, in Proceedings of the 30th IEEE Conference on Decision and Control, (Brighton, UK), pp. 3008–3013, 1991.

    Google Scholar 

  90. M. von Renteln. “Every subring R of N with A \(\left( {\bar D} \right)\) ⊂ R is not adequate,” Acta Sci. Math., vol. 39, pp. 139–140. 1977.

    MATH  Google Scholar 

  91. H. H. Rosenbrock. State Space and Multivariable Theory. London: Nelson, 1970.

    MATH  Google Scholar 

  92. W. Rudin. Real and Complex Analysis. New York: McGraw-Hill, 2nd edition, 1974.

    MATH  Google Scholar 

  93. R. Saeks and J. Murray. “Feedback system design: The tracking and disturbance rejection problems,” IEEE Trans. Auto. Control, vol. 26, pp. 203–217, 1981.

    Article  Google Scholar 

  94. R. Saeks, J. Murray, O. Chua, C. Karmokolias, and A. Iyer. “Feedback system design: The single-variate case — Part I,” Circ. Sys. and Sig. Proc., vol. 1, pp. 137–169, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  95. R. Saeks, J. Murray, O. Chua, C. Karmokolias, and A. Iyer. “Feedback system design: The single-variate case — Part II,” Circ. Sys. and Si. Proc., vol. 2, pp. 3–34, 1983.

    Article  MATH  Google Scholar 

  96. D. Salamon. Control and Observation for Neutral Systems, Research Notes in Mathematics, vol. 91. London: Pitman, 1984.

    Google Scholar 

  97. J. M. Schumacher. “A direct approach to compensator design for distributed parameter systems,” SIAM J. Contr. & Opt., vol. 21, pp. 823–836, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  98. J. M. Schumacher. “Finite-dimensional regulators for a class of infinite-dimensional systems,” Systems and Control Letters, vol. 3, pp. 7–12, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  99. M. C. Smith. “On the generalized Nyquist stability criterion,” Int. J. Control, vol. 34, pp. 885–920, 1981.

    Article  MATH  Google Scholar 

  100. M. C. Smith. “On stabilization and the existence of coprime factorizations,” IEEE Trans. Auto. Control, vol. 34, pp. 1005–1007, 1989.

    Article  MATH  Google Scholar 

  101. O. J. Staffans. “Feedback stabilization of a scalar functional differential equation.” J. of Integral Equations, vol. 10, pp. 319–342, 1985.

    MathSciNet  Google Scholar 

  102. O. J. Staffans. “Stabilization of a distributed system with a stable compensator,” Mathematics of Control, Signals, and Systems, vol. 5, pp. 1–22, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  103. H. Ukai and T. Iwazumi. “Design of servo systems for distributed parameter systems by finite-dimensional dynamic compensator,” Int. J. Systems Science, vol. 21, pp. 1025–1046, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  104. J. M. E. Valenca and C. J. Harris. “Nyquist criterion for input-output stability of multivariable systems,” Int. J. Control, vol. 31, pp. 917–935, 1980.

    Article  MATH  MathSciNet  Google Scholar 

  105. M. S. Verma. “Coprime fractional representations and stability of nonlinear feedback systems,” Int. J. Control, vol. 48, pp. 897–918, 1988.

    Article  MATH  Google Scholar 

  106. M. Vidyasagar. “Input-output stability of a broad class of linear time-invariant multivariable feedback systems,” SIAM J. Control, vol. 10, pp. 203–209, 1972.

    Article  MATH  MathSciNet  Google Scholar 

  107. M. Vidyasagar. “On the use of right-coprime factorizations in distributed feedback systems containing unstable subsystems,” IEEE Trans. Circ. and Sys., vol. 25, pp. 916–921, 1978.

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  109. M. Vidyasagar and B. D. O. Anderson. “Approximation and stabilization of distributed parameter systems by lumped systems,” Systems and Control Letters, vol. 12, pp. 95–101, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  110. M. Vidyasagar, H. Schneider, and B. A. Francis. “Algebraic and topological aspects of feedback stabilization,” IEEE Trans. Auto. Control, vol. 27, pp. 880–894, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  111. G. Weiss. “Admissibility of unbounded control operators,” SIAM J. Contr. & Opt., vol. 27, pp. 527–545, 1989.

    Article  MATH  Google Scholar 

  112. G. Weiss. “Admissible observation operators for linear semigroups,” Israel J. Math., vol. 65, pp. 12–43, 1989.

    Google Scholar 

  113. G. Weiss. “Two conjectures on the admissibility of control operators,” Preprint, 1990.

    Google Scholar 

  114. G. Weiss. “Transfer functions of regular systems. Part I: Characterizations of regularity,” Preprint, 1990.

    Google Scholar 

  115. Y. Yamamoto. “Pseudo-rational input/output maps and their realizations: A fractional representation approach to infinite-dimensional systems,” SIAM J. Contr. & Opt., vol. 26, pp. 1415–1430, 1988.

    Article  MATH  Google Scholar 

  116. Y. Yamamoto. “Equivalence of internal and external stability for a class of distributed systems,” Mathematics of Control, Signals, and Systems, vol. 4, pp. 391–409, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  117. Y. Yamamoto and S. Hara. “Relationships between internal and external stability for infinite-dimensional systems with applications to a servo problem,” IEEE Trans. Auto. Control, vol. 33, pp. 1044–1052, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  118. Y. Yamamoto and S. Hara. “Internal and external stability and robust stability for a class of infinite-dimensional systems,” Automatica, vol. 28, pp. 81–93, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  119. D. C. Youla, J. J. Bongiorno, Jr., and H. A. Jabr. “Modern Wiener-Hopf design of optimal controllers, Part I: The single-input case,” IEEE Trans. Auto. Control, vol. 21, pp. 3–14, 1976.

    Article  MATH  MathSciNet  Google Scholar 

  120. D. C. Youla, J. J. Bongiorno, Jr., and C. N. Lu. Single-loop feedback stabilization of linear multivariable dynamic plants,” Automatica, vol. 10, pp. 159–173, 1974.

    Article  MATH  MathSciNet  Google Scholar 

  121. D. C. Youla, H. A. Jabr, and J. J. Bongiorno, Jr. “Modern Wiener-Hopf design of optimal controllers, Part II: The multivariable case,” IEEE Trans. Auto. Control, vol. 21, pp. 319–338, 1976.

    Article  MATH  MathSciNet  Google Scholar 

  122. G. Zames. “Feedback and optimal sensitivity: Model reference transformations, multiplicative seminorms, and approximate inverses,” IEEE Trans. Auto. Control, vol. 26, pp. 301–320, 1981.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Dynamische Systeme, Universität Bremen, Postfach 330440, 2800, Bremen 33, Germany

    Hartmut Logemann

Authors
  1. Hartmut Logemann
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

R. F. Curtain A. Bensoussan J. L. Lions

Rights and permissions

Reprints and permissions

Copyright information

© 1993 Springer-Verlag

About this paper

Cite this paper

Logemann, H. (1993). Stabilization and regulation of infinite-dimensional systems using coprime factorizations. In: Curtain, R.F., Bensoussan, A., Lions, J.L. (eds) Analysis and Optimization of Systems: State and Frequency Domain Approaches for Infinite-Dimensional Systems. Lecture Notes in Control and Information Sciences, vol 185. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0115022

Download citation

  • DOI: https://doi.org/10.1007/BFb0115022

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-56155-2

  • Online ISBN: 978-3-540-47480-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation