On the circle criterion for boundary control systems in factor form: Lyapunov stability and Lur'e equations

Piotr Grabowski; Frank M. Callier

ESAIM: Control, Optimisation and Calculus of Variations (2005)

  • Volume: 12, Issue: 1, page 169-197
  • ISSN: 1292-8119

Abstract

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A Lur'e feedback control system consisting of a linear, infinite-dimensional system of boundary control in factor form and a nonlinear static sector type controller is considered. A criterion of absolute strong asymptotic stability of the null equilibrium is obtained using a quadratic form Lyapunov functional. The construction of such a functional is reduced to solving a Lur'e system of equations. A sufficient strict circle criterion of solvability of the latter is found, which is based on results by Oostveen and Curtain [Automatica34 (1998) 953–967]. All the results are illustrated in detail by an electrical transmission line example of the distortionless loaded ℜ𝔏ℭ𝔊 -type. The paper uses extensively the philosophy of reciprocal systems with bounded generating operators as recently studied and used by Curtain in (2003) [Syst. Control Lett.49 (2003) 81–89; SIAM J. Control Optim.42 (2003) 1671–1702].

How to cite

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Grabowski, Piotr, and Callier, Frank M.. "On the circle criterion for boundary control systems in factor form: Lyapunov stability and Lur'e equations." ESAIM: Control, Optimisation and Calculus of Variations 12.1 (2005): 169-197. <http://eudml.org/doc/90787>.

@article{Grabowski2005,
abstract = { A Lur'e feedback control system consisting of a linear, infinite-dimensional system of boundary control in factor form and a nonlinear static sector type controller is considered. A criterion of absolute strong asymptotic stability of the null equilibrium is obtained using a quadratic form Lyapunov functional. The construction of such a functional is reduced to solving a Lur'e system of equations. A sufficient strict circle criterion of solvability of the latter is found, which is based on results by Oostveen and Curtain [Automatica34 (1998) 953–967]. All the results are illustrated in detail by an electrical transmission line example of the distortionless loaded $\mathfrak\{RLCG\}$-type. The paper uses extensively the philosophy of reciprocal systems with bounded generating operators as recently studied and used by Curtain in (2003) [Syst. Control Lett.49 (2003) 81–89; SIAM J. Control Optim.42 (2003) 1671–1702]. },
author = {Grabowski, Piotr, Callier, Frank M.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Infinite-dimensional control systems; semigroups; Lyapunov functionals; circle criterion.; infinite-dimensional control systems; Lyapunov functionals; circle criterion},
language = {eng},
month = {12},
number = {1},
pages = {169-197},
publisher = {EDP Sciences},
title = {On the circle criterion for boundary control systems in factor form: Lyapunov stability and Lur'e equations},
url = {http://eudml.org/doc/90787},
volume = {12},
year = {2005},
}

TY - JOUR
AU - Grabowski, Piotr
AU - Callier, Frank M.
TI - On the circle criterion for boundary control systems in factor form: Lyapunov stability and Lur'e equations
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2005/12//
PB - EDP Sciences
VL - 12
IS - 1
SP - 169
EP - 197
AB - A Lur'e feedback control system consisting of a linear, infinite-dimensional system of boundary control in factor form and a nonlinear static sector type controller is considered. A criterion of absolute strong asymptotic stability of the null equilibrium is obtained using a quadratic form Lyapunov functional. The construction of such a functional is reduced to solving a Lur'e system of equations. A sufficient strict circle criterion of solvability of the latter is found, which is based on results by Oostveen and Curtain [Automatica34 (1998) 953–967]. All the results are illustrated in detail by an electrical transmission line example of the distortionless loaded $\mathfrak{RLCG}$-type. The paper uses extensively the philosophy of reciprocal systems with bounded generating operators as recently studied and used by Curtain in (2003) [Syst. Control Lett.49 (2003) 81–89; SIAM J. Control Optim.42 (2003) 1671–1702].
LA - eng
KW - Infinite-dimensional control systems; semigroups; Lyapunov functionals; circle criterion.; infinite-dimensional control systems; Lyapunov functionals; circle criterion
UR - http://eudml.org/doc/90787
ER -

References

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  1. W. Arendt and C.J.K. Batty, Tauberian theorems and stability of one-parameter semigroups. Trans. Amer. Math. Soc.306 (1988) 837–852.  
  2. A.V. Balakrishnan, On a generalization of the Kalman-Yacubovic lemma. Appl. Math. Optim.31 (1995) 177–187.  
  3. F. Bucci, Frequency domain stability of nonlinear feedback systems with unbounded input operator. Dynam. Contin. Discrete Impuls. Syst.7 (2000) 351–368.  
  4. F.M. Callier and J. Winkin, LQ-optimal control of infinite-dimensional systems by spectral factorization. Automatica28 (1992) 757–770.  
  5. R.F. Curtain, Linear operator inequalities for strongly stable weakly regular linear systems. Math. Control Signals Syst.14 (2001) 299–337.  
  6. R.F. Curtain, Regular linear systems and their reciprocals: application to Riccati equations. Syst. Control Lett.49 (2003) 81–89.  
  7. R.F. Curtain, Riccati equations for stable well-posed linear systems: The generic case. SIAM J. Control Optim.42 (2003) 1671–1702.  
  8. R.F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory. Heidelberg, Springer (1995).  
  9. R.F. Curtain, H. Logemann and O. Staffans, Stability results of Popov-type for infinite – dimensional systems with applications to integral control. Proc. London Math. Soc.86 (2003) 779–816.  
  10. H. Górecki, S. Fuksa, P. Grabowski and A.Korytowski, Analysis and Synthesis of Time-Delay Systems. Warsaw & Chichester: PWN and J. Wiley (1989).  
  11. P. Grabowski, On the spectral – Lyapunov approach to parametric optimization of DPS. IMA J. Math. Control Inform.7 (1990) 317–338.  
  12. P. Grabowski, The LQ controller problem: an example. IMA J. Math. Control Inform.11 (1994) 355–368.  
  13. P. Grabowski, On the circle criterion for boundary control systems in factor form. Opuscula Math.23 (2003) 1–25.  
  14. P. Grabowski and F.M. Callier, Admissible observation operators. Duality of observation and control using factorizations. Dynamics Continuous, Discrete Impulsive Systems6 (1999) 87–119.  
  15. P. Grabowski and F.M. Callier, On the circle criterion for boundary control systems in factor form: Lyapunov approach. Facultés Universitaires Notre-Dame de la Paix à Namur, Publications du Département de Mathématique, Research Report 07 (2000), FUNDP, Namur, Belgium.  
  16. P. Grabowski and F.M. Callier, Boundary control systems in factor form: Transfer functions and input-output maps. Integral Equations Operator Theory41 (2001) 1–37.  
  17. P. Grabowski and F.M. Callier, Circle criterion and boundary control systems in factor form: Input-output approach. Internat. J. Appl. Math. Comput. Sci.11 (2001) 1387–1403.  
  18. P. Grabowski and F.M. Callier, On the circle criterion for boundary control systems in factor form: Lyapunov stability and Lur'e equations. Facultés Universitaires Notre-Dame de la Paix à Namur, Publications du Département de Mathématique, Research Report 05 (2002), FUNDP, Namur, Belgium.  
  19. U. Grenander and G. Szegö, Toeplitz Forms and Their Application, Berkeley: University of California Press (1958).  
  20. K. Hoffman, Banach Spaces of Analytic Functions. Englewood Cliffs: Prentice-Hall (1962).  
  21. I. Lasiecka and R. Triggiani, Differential and Algebraic Riccati Equations with Application to Boundary/Point Control Problems: Continuous Theory and Approximation Theory. Lect. Notes Control Inform. Sci.164 (1991) 1–160.  
  22. I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. Part I: Abstract Parabolic Systems, Cambridge: Cambridge University Press, Encyclopedia Math. Appl.74 (2000).  
  23. I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. Part II: Abstract Hyperbolic-Like Systems over a Finite Time Horizon, Cambridge: Cambridge University Press, Encyclopedia Math. Appl.75 (2000).  
  24. A.L. Likhtarnikov and V.A. Yacubovich, The frequency domain theorem for continuous one-parameter semigroups, IZVESTIJA ANSSSR. Seria matematicheskaya.41 (1977) 895–911 (in Russian).  
  25. H. Logemann and R.F. Curtain, Absolute stability results for well-posed infinite-dimensional systems with low-gain integral control. ESAIM: COCV5 (2000) 395–424.  
  26. J.-Cl. Louis and D.Wexler, The Hilbert space regulator problem and operator Riccati equation under stabilizability. Annales de la Société Scientifique de Bruxelles105 (1991) 137–165.  
  27. Yu. Lyubich and Vû Quôc Phong, Asymptotic stability of linear differential equations in Banach spaces. Studia Math.88 (1988) 37–41.  
  28. E. Noldus, On the stability of systems having several equilibrium points. Appl. Sci. Res.21 (1969) 218–233.  
  29. E. Noldus, A. Galle and L. Jasson, The computation of stability regions for systems with many singular points. Intern. J. Control17 (1973) 641–652.  
  30. E. Noldus, New direct Lyapunov-type method for studying synchronization problems. Automatica13 (1977) 139–151.  
  31. A.A. Nudel'man and P.A. Schwartzman, On the existence of solution of some operator inequalities. Sibirsk. Mat. Zh.16 (1975) 563–571 (in Russian).  
  32. J.C. Oostveen and R.F. Curtain, Riccati equations for strongly stabilizable bounded linear systems. Automatica34 (1998) 953–967.  
  33. L. Pandolfi, Kalman-Popov-Yacubovich theorem: an overview and new results for hyperbolic control systems. Nonlinear Anal. Theor. Methods Appl.30 (1997) 735–745.  
  34. L. Pandolfi, Dissipativity and Lur'e problem for parabolic boundary control system, Research Report, Dipartamento di Matematica, Politecnico di Torino 1 (1997) 1–27; SIAM J. Control Optim.36 (1998) 2061–2081.  
  35. L. Pandolfi, The Kalman-Yacubovich-Popov theorem for stabilizable hyperbolic boundary control systems. Integral Equations Operator Theory34 (1999) 478–493.  
  36. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. New York, Springer-Verlag (1983).  
  37. D. Salamon, Realization theory in Hilbert space. Math. Systems Theory21 (1989) 147–164.  
  38. O.J. Staffans, Quadratic optimal control of stable well-posed linear systems through spectral factorization. Math. Control Signals Systems8 (1995) 167–197.  
  39. O.J. Staffans and G. Weiss, Transfer functions of regular linear systems, Part II: The system operator and the Lax-Phillips semigroup. Trans. Amer. Math. Soc.354 (2002) 3229–3262.  
  40. M. Vidyasagar, Nonlinear Systems Analysis. 2nd Edition, Englewood Cliffs NJ, Prentice-Hall (1993).  
  41. G. Weiss, Transfer functions of regular linear systems. Part I: Characterization of regularity. Trans. AMS342 (1994) 827–854.  
  42. M. Weiss, Riccati Equations in Hilbert Spaces: A Popov function approach. Ph.D. Thesis, Rijksuniversiteit Groningen, The Netherlands (1994).  
  43. M. Weiss and G. Weiss, Optimal control of stable weakly regular linear systems. Math. Control Signals Syst.10 (1997) 287–330.  
  44. R.M. Young, An Introduction to Nonharmonic Fourier Series. New York, Academic Press (1980).  

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