Optimal LQ-feedback control for a class of first-order hyperbolic distributed parameter systems

Ilyasse Aksikas; Joseph J. Winkin; Denis Dochain

ESAIM: Control, Optimisation and Calculus of Variations (2008)

  • Volume: 14, Issue: 4, page 897-908
  • ISSN: 1292-8119

Abstract

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The Linear-Quadratic (LQ) optimal control problem is studied for a class of first-order hyperbolic partial differential equation models by using a nonlinear infinite-dimensional (distributed parameter) Hilbert state-space description. First the dynamical properties of the linearized model around some equilibrium profile are studied. Next the LQ-feedback operator is computed by using the corresponding operator Riccati algebraic equation whose solution is obtained via a related matrix Riccati differential equation in the space variable. Then the latter is applied to the nonlinear model, and the resulting closed-loop system dynamical performances are analyzed.

How to cite

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Aksikas, Ilyasse, Winkin, Joseph J., and Dochain, Denis. "Optimal LQ-feedback control for a class of first-order hyperbolic distributed parameter systems." ESAIM: Control, Optimisation and Calculus of Variations 14.4 (2008): 897-908. <http://eudml.org/doc/250316>.

@article{Aksikas2008,
abstract = { The Linear-Quadratic (LQ) optimal control problem is studied for a class of first-order hyperbolic partial differential equation models by using a nonlinear infinite-dimensional (distributed parameter) Hilbert state-space description. First the dynamical properties of the linearized model around some equilibrium profile are studied. Next the LQ-feedback operator is computed by using the corresponding operator Riccati algebraic equation whose solution is obtained via a related matrix Riccati differential equation in the space variable. Then the latter is applied to the nonlinear model, and the resulting closed-loop system dynamical performances are analyzed. },
author = {Aksikas, Ilyasse, Winkin, Joseph J., Dochain, Denis},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {First-order hyperbolic PDE's; infinite-dimensional systems; LQ-optimal control; stability; optimality; first-order hyperbolic PDE's},
language = {eng},
month = {2},
number = {4},
pages = {897-908},
publisher = {EDP Sciences},
title = {Optimal LQ-feedback control for a class of first-order hyperbolic distributed parameter systems},
url = {http://eudml.org/doc/250316},
volume = {14},
year = {2008},
}

TY - JOUR
AU - Aksikas, Ilyasse
AU - Winkin, Joseph J.
AU - Dochain, Denis
TI - Optimal LQ-feedback control for a class of first-order hyperbolic distributed parameter systems
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2008/2//
PB - EDP Sciences
VL - 14
IS - 4
SP - 897
EP - 908
AB - The Linear-Quadratic (LQ) optimal control problem is studied for a class of first-order hyperbolic partial differential equation models by using a nonlinear infinite-dimensional (distributed parameter) Hilbert state-space description. First the dynamical properties of the linearized model around some equilibrium profile are studied. Next the LQ-feedback operator is computed by using the corresponding operator Riccati algebraic equation whose solution is obtained via a related matrix Riccati differential equation in the space variable. Then the latter is applied to the nonlinear model, and the resulting closed-loop system dynamical performances are analyzed.
LA - eng
KW - First-order hyperbolic PDE's; infinite-dimensional systems; LQ-optimal control; stability; optimality; first-order hyperbolic PDE's
UR - http://eudml.org/doc/250316
ER -

References

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  1. H. Abou-Kandil, G. Freiling, V. Ionescu and G. Jank, Matrix Riccati Equations in Control and Systems Theory, Series: Systems & Control: Foundations & Applications. Birkhauser (2003).  Zbl1027.93001
  2. I. Aksikas, Analysis and LQ-Optimal Control of Infinite-Dimensional Semilinear Systems: Application to a Plug Flow Reactor. Ph.D. thesis, Université Catholique de Louvain, Belgium (2005).  
  3. I. Aksikas, J. Winkin and D. Dochain, Stability analysis of an infinite-dimensional linearized plug flow reactor model, in Proceedings of the 43rd IEEE Conference on Decision and Control, CDC (2004) 2417–2422.  
  4. I. Aksikas, J. Winkin and D. Dochain, LQ-optimal feedback regulation of a nonisothermal plug flow reactor infinite-dimensional model. Int. J. Tomography & Statistics5 (2007) 73–78.  Zbl1112.93054
  5. I. Aksikas, J. Winkin and D. Dochain, Optimal LQ-feedback regulation of a nonisothermal plug flow reactor model by spectral factorization. IEEE Trans. Automat. Control52 (2007) 1179–1193.  
  6. I. Aksikas, J. Winkin and D. Dochain, Asymptotic stability of infinite-dimensional semilinear systems: application to a nonisothermal reactor. Systems Control Lett.56 (2007) 122–132.  Zbl1112.93054
  7. V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems. Boston: Academic Press (1993).  Zbl0776.49005
  8. A. Bressan, Hyperbolic Systems of Conservation Laws: The One-Dimensional Cauchy Problem. Oxford University Press (2000).  Zbl0997.35002
  9. H. Brezis, Opéateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, Mathematics Studies. North-Holland (1973).  Zbl0252.47055
  10. F.M. Callier and C.A. Desoer, Linear System Theory. Springer-Verlag, New York (1991).  Zbl0744.93002
  11. F.M. Callier and J. Winkin, LQ-optimal control of infinite-dimensional systems by spectral factorization. Automatica28 (1992) 757–770.  Zbl0776.49023
  12. P.D. Christofides, Nonlinear and Robust Control of Partial Differential Equation Systems: Methods and Application to Transport-Reaction Processes. Birkhauser, Boston (2001).  Zbl1018.93001
  13. R.F. Curtain and H.J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory. Springer-Verlag, New York (1995).  Zbl0839.93001
  14. C.M. Dafermos and M. Slemrod, Asymptotic behavior of nonlinear contraction semigroups. J. Funct. Anal.13 (1973) 97–106.  Zbl0267.34062
  15. D. Dochain, Contribution to the Analysis and Control of Distributed Parameter Systems with Application to (Bio)chemical Processes and Robotics. Thèse d'Agrégation de l'Enseignement Supérieur, Université Catholique de Louvain, Louvain-la-Neuve, Belgium (1994).  
  16. G.F. Froment and K.B. Bischoff, Chemical Reactor Analysis and Design. 2nd edition, John Wiley, New York (1990).  
  17. M. Ikeda and D.D. Siljak, Optimality and robustness of linear quadratic control for nonlinear systems. Automatica26 (1990) 499–511.  Zbl0717.93021
  18. M. Laabissi, M.E. Achhab, J. Winkin and D. Dochain, Trajectory analysis of nonisothermal tubular reactor nonlinear models. Systems Control Lett.42 (2001) 169–184.  Zbl0985.93030
  19. V. Lakshmikantham and S. Leela, Nonlinear Differential Equations in Abstract Spaces. Pergamon, Oxford (1981).  Zbl0456.34002
  20. I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories, Volume II: Abstract Hyperbolic-like Systems over a Finite Time Horizon. Cambridge University Press (2000).  Zbl0961.93003
  21. Z. Luo, B. Guo and O. Morgül, Stability and Stabilization of Infinite Dimensional Systems with Applications. Springer-Verlag, London (1999).  Zbl0922.93001
  22. R.H. Martin, Nonlinear Operators and Differential Equations in Banach spaces. John Wiley & Sons, New York (1976).  
  23. A. Pazy, Semigroups of Linear Operators and Application to Partial Differential Equations, Appl. Math. Sci.44. Springer-Verlag, New York (1983).  Zbl0516.47023
  24. W.H. Ray, Advanced Process Control, Series in Chemical Engineering. Butterworth, Boston (1981).  
  25. L.M. Silverman and H.E. Meadows, Controllability and observability in time-variable linear systems. J. SIAM Control5 (1967) 64–73.  Zbl0163.11001

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