Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-09T14:58:11.456Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  07 February 2008

Ilyasse Aksikas ,
Joseph J. Winkin  and
Denis Dochain
Ilyasse Aksikas
Affiliation:
Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta, T6G 2G6, Canada; [email protected]
Joseph J. Winkin
Affiliation:
Department of Mathematics, University of Namur (FUNDP), 8 Rempart de la Vierge, 5000 Namur, Belgium; [email protected]
Denis Dochain
Affiliation:
CESAME, Université Catholique de Louvain, 4-6 avenue G. Lemaître, 1348 Louvain-la-Neuve, Belgium; [email protected]
Get access

Abstract

The Linear-Quadratic (LQ) optimal control problem is studied for aclass of first-order hyperbolic partial differential equation modelsby using a nonlinear infinite-dimensional (distributed parameter) Hilbert state-spacedescription. First the dynamical properties of the linearized modelaround some equilibrium profile are studied. Next the LQ-feedbackoperator is computed by using the corresponding operator Riccatialgebraic equation whose solution is obtained via a relatedmatrix Riccati differential equation in the space variable. Then thelatter is applied to the nonlinear model, and the resultingclosed-loop system dynamical performances are analyzed.

Keywords

First-order hyperbolic PDE'sinfinite-dimensional systemsLQ-optimal controlstabilityoptimality
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 14 , Issue 4 , October 2008 , pp. 897 - 908
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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).
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).
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.
Aksikas, I., Winkin, J. and Dochain, D., LQ-optimal feedback regulation of a nonisothermal plug flow reactor infinite-dimensional model. Int. J. Tomography & Statistics 5 (2007) 73–78.
Aksikas, I., Winkin, J. and Dochain, D., Optimal LQ-feedback regulation of a nonisothermal plug flow reactor model by spectral factorization. IEEE Trans. Automat. Control 52 (2007) 1179–1193. CrossRef
Aksikas, I., Winkin, J. and Dochain, D., Asymptotic stability of infinite-dimensional semilinear systems: application to a nonisothermal reactor. Systems Control Lett. 56 (2007) 122–132. CrossRef
V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems. Boston: Academic Press (1993).
A. Bressan, Hyperbolic Systems of Conservation Laws: The One-Dimensional Cauchy Problem. Oxford University Press (2000).
H. Brezis, Opéateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, Mathematics Studies. North-Holland (1973).
F.M. Callier and C.A. Desoer, Linear System Theory. Springer-Verlag, New York (1991).
Callier, F.M. and Winkin, J., LQ-optimal control of infinite-dimensional systems by spectral factorization. Automatica 28 (1992) 757–770. CrossRef
P.D. Christofides, Nonlinear and Robust Control of Partial Differential Equation Systems: Methods and Application to Transport-Reaction Processes. Birkhauser, Boston (2001).
R.F. Curtain and H.J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory. Springer-Verlag, New York (1995).
Dafermos, C.M. and Slemrod, M., Asymptotic behavior of nonlinear contraction semigroups. J. Funct. Anal. 13 (1973) 97–106. CrossRef
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).
G.F. Froment and K.B. Bischoff, Chemical Reactor Analysis and Design. 2nd edition, John Wiley, New York (1990).
Ikeda, M. and Siljak, D.D., Optimality and robustness of linear quadratic control for nonlinear systems. Automatica 26 (1990) 499–511. CrossRef
Laabissi, M., Achhab, M.E., Winkin, J. and Dochain, D., Trajectory analysis of nonisothermal tubular reactor nonlinear models. Systems Control Lett. 42 (2001) 169–184. CrossRef
V. Lakshmikantham and S. Leela, Nonlinear Differential Equations in Abstract Spaces. Pergamon, Oxford (1981).
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).
Z. Luo, B. Guo and O. Morgül, Stability and Stabilization of Infinite Dimensional Systems with Applications. Springer-Verlag, London (1999).
R.H. Martin, Nonlinear Operators and Differential Equations in Banach spaces. John Wiley & Sons, New York (1976).
A. Pazy, Semigroups of Linear Operators and Application to Partial Differential Equations, Appl. Math. Sci. 44. Springer-Verlag, New York (1983).
W.H. Ray, Advanced Process Control, Series in Chemical Engineering. Butterworth, Boston (1981).
Silverman, L.M. and Meadows, H.E., Controllability and observability in time-variable linear systems. J. SIAM Control 5 (1967) 64–73. CrossRef