This paper derives linear quadratic regulator (LQR) results for boundary-controlled parabolic partial differential equations (PDEs) via weak variations. Research on optimal control of PDEs has a rich 40-year history. This body of knowledge relies heavily on operator and semigroup theory. Our research distinguishes itself by deriving existing LQR results from a more accessible set of mathematics, namely weak-variational concepts. Ultimately, the LQR controller is computed from a Riccati PDE that must be derived for each PDE model under consideration. Nonetheless, a Riccati PDE is a significantly simpler object to solve than an operator Riccati equation, which characterizes most existing results. To this end, our research provides an elegant and accessible method for practicing engineers who study physical systems described by PDEs. Simulation examples, closed-loop stability analyses, comparisons to alternative control methods, and extensions to other models are also included.
References
1.
Avalos
, G.
, and Lasiecka
, I.
, 1996
, “Differential Riccati Equation for the Active Control of a Problem in Structural Acoustics
,” J. Optim. Theory Appl.
, 91
(3
), pp. 695
–728
.10.1007/BF021901282.
Aksikas
, I.
, Fuxman
, A.
, Forbes
, J. F.
, and Winkin
, J. J.
, 2009
, “LQ Control Design of a Class of Hyperbolic PDE Systems: Application to Fixed-Bed Reactor
,” Automatica
, 45
(6
), pp. 1542
–1548
.10.1016/j.automatica.2009.02.0173.
Ferrari-Trecate
, G.
, Buffa
, A.
, and Gati
, M.
, 2006
, “Analysis of Coordination in Multi-Agent Systems Through Partial Difference Equations
,” IEEE Trans. Autom. Control
, 51
(6
), pp. 1058
–1063
.10.1109/TAC.2006.8768054.
Zariphopoulou
, T.
, 1999
, “Optimal Investment and Consumption Models With Non-Linear Stock Dynamics
,” Math. Methods Oper. Res.
, 50
(2
), pp. 271
–296
.10.1007/s0018600500985.
Aamo
, O. M.
, Krstic
, M.
, and Bewley
, T. R.
, 2003
, “Control of Mixing by Boundary Feedback in 2D Channel Flow
,” Automatica
, 39
(9
), pp. 1597
–1606
.10.1016/S0005-1098(03)00140-76.
Vazquez
, R.
, and Krstic
, M.
, 2007
, “Explicit Output Feedback Stabilization of a Thermal Convection Loop by Continuous Backstepping and Singular Perturbations
,” 2007 American Control Conference
(ACC
), pp. 2177
–2182
.10.1109/ACC.2007.42827937.
Del Vecchio
, D.
, and Petit
, N.
, 2005
, “Boundary Control for an Industrial Under-Actuated Tubular Chemical Reactor
,” J. Process Control
, 15
(7
), pp. 771
–784
.10.1016/j.jprocont.2005.03.0048.
Doyle
, M.
, Fuller
, T.
, and Newman
, J.
, 1993
, “Modeling of Galvanostatic Charge and Discharge of the Lithium/Polymer/Insertion Cell
,” J. Electrochem. Soc.
, 140
(6
), pp. 1526
–1533
.10.1149/1.22215979.
Fuller
, T.
, Doyle
, M.
, and Newman
, J.
, 1994
, “Simulation and Optimization of the Dual Lithium Ion Insertion Cell
,” J. Electrochem. Soc.
, 141
(1
), pp. 1
–10
.10.1149/1.205468410.
Forman
, J.
, Bashash
, S.
, Stein
, J.
, and Fathy
, H.
, 2011
, “Reduction of an Electrochemistry-Based Li-Ion Battery Model via Quasi-Linearization and Padé Approximation
,” J. Electrochem. Soc.
, 158
(2
), pp. A93
–A101
.10.1149/1.351905911.
Lions
, J.
, and Mitter
, S.
, 1971
, Optimal Control of Systems Governed by Partial Differential Equations
, Vol. 396
, Springer
, Berlin
.12.
Lasiecka
, I.
, and Triggiani
, R.
, 2000
, Control Theory for Partial Differential Equations: Continuous and Approximation Theories
, Cambridge University Press
, Cambridge, UK
.13.
Bensoussan
, A.
, Da Prato
, G.
, Delfour
, M. C.
, and Mitter
, S. K.
, 2007
, Representation and Control of Infinite Dimensional Systems
, 2nd ed. Birkhauser
, Boston, MA
.14.
Curtain
, R. F.
, and Zwart
, H. J.
, 1995
, An Introduction to Infinite-Dimensional Linear Systems Theory
, Springer
, New York
.15.
Hulsing
, K.
, 1999
, “Methods for Computing Functional Gains for LQR Control of Partial Differential Equations
,” Ph.D. thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA.16.
Krstic
, M.
, and Smyshlyaev
, A.
, 2008
, Boundary Control of PDEs: A Course on Backstepping Designs
(SIAM Advances in Design and Control), Society for Industrial and Applied Mathematics, Philadelphia, PA.17.
Di Domenico
, D.
, Stefanopoulou
, A.
, and Fiengo
, G.
, 2010
, “Lithium-Ion Battery State of Charge and Critical Surface Charge Estimation Using an Electrochemical Model-Based Extended Kalman Filter
,” ASME J. Dyn. Sys., Meas., Control
, 132
(6
), p. 061302
.10.1115/1.400247518.
Ravindran
, S.
, 2000
, “A Reduced-Order Approach for Optimal Control of Fluids Using Proper Orthogonal Decomposition
,” Int. J. Numer. Methods Fluids
, 34
(5
), pp. 425
–448
.10.1002/1097-0363(20001115)34:5<425::AID-FLD67>3.0.CO;2-W19.
Xu
, C.
, Ou
, Y.
, and Schuster
, E.
, 2011
, “Sequential Linear Quadratic Control of Bilinear Parabolic PDEs Based on POD Model Reduction
,” Automatica
, 47
(2
), pp. 418
–426
.10.1016/j.automatica.2010.11.00120.
Park
, H.
, and Kim
, O.
, 2000
, “A Reduction Method for the Boundary Control of the Heat Conduction Equation
,” ASME J. Dyn. Sys., Meas., Control
, 122
(3
), pp. 435
–444
.10.1115/1.128636521.
Collinger
, J.
, Wickert
, J.
, and Corr
, L.
, 2009
, “Adaptive Piezoelectric Vibration Control With Synchronized Switching
,” ASME J. Dyn. Sys., Meas., Control
, 131
(4
), p. 041006
.10.1115/1.311718922.
Burns
, J.
, and Hulsing
, K.
, 2001
, “Numerical Methods for Approximating Functional Gains in LQR Boundary Control Problems
,” Math. Comput. Modell.
, 33
(1
), pp. 89
–100
.10.1016/S0895-7177(00)00231-423.
Moura
, S.
, and Fathy
, H.
, 2011
, “Optimal Boundary Control & Estimation of Diffusion-Reaction PDEs
,” American Control Conference (ACC)
, pp. 921
–928
.24.
Athans
, M.
, and Falb
, P.
, 2007
, “Optimal Control: An Introduction to the Theory and Its Applications,
” Dover Publications
, Mineola, NY
.25.
Kuo
, B.
, and Golnaraghi
, F.
, 2003
, Automatic Control Systems
, Vol. 1, John Wiley & Sons
, Hoboken, NJ
.26.
Freudenberg
, J. S.
, 2008
, A First Graduate Course in Feedback Control
, University of Michigan
, Ann Arbor, MI
.27.
Polianin
, A. D.
, 2002
, Handbook of Linear Partial Differential Equations for Engineers and Scientists
, CRC Press LLC
, Boca Raton, FL
.28.
Smyshlyaev
, A.
, and Krstic
, M.
, 2007
, “Adaptive Boundary Control for Unstable Parabolic PDEs—Part II: Estimation-Based Designs
,” Automatica
, 43
(9
), pp. 1543
–1556
.10.1016/j.automatica.2007.02.014Copyright © 2013 by ASME
You do not currently have access to this content.
