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2001 | 11 | 6 | 1311-1330
Tytuł artykułu

Reduced order controllers for Burgers' equation with a nonlinear observer

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
A method for reducing controllers for systems described by partial differential equations (PDEs) is applied to Burgers' equation with periodic boundary conditions. This approach differs from the typical approach of reducing the model and then designing the controller, and has developed over the past several years into its current form. In earlier work it was shown that functional gains for the feedback control law served well as a dataset for reduced order basis generation via the proper orthogonal decomposition (POD)@. However, the test problem was the two-dimensional heat equation, a problem in which the physics dominates the system in such a way that controller efficacy is difficult to generalize. Here, we additionally incorporate a nonlinear observer by including the nonlinear terms of the state equation in the differential equation for the compensator.
Rocznik
Tom
11
Numer
6
Strony
1311-1330
Opis fizyczny
Daty
wydano
2001
Twórcy
  • Ball Aerospace Technologies Corp., Boulder, CO 80836i, U.S.A.
  • Interdisciplinary Center for Applied Mathematics, Virginia Tech, Blacksburg, VA 24061-0531, U.S.A.
  • Interdisciplinary Center for Applied Mathematics, Virginia Tech, Blacksburg, VA 24061-0531, U.S.A.
Bibliografia
  • Atwell J.A. and King B.B. (1999): Computational aspects of reduced basis feedback controllers for spatially distributed systems. — Proc. 38th IEEE CDC, Phoenix, AZ, pp.4301–4306.
  • Atwell J.A. and King B.B. (2000): Stabilized finite element codes for control of Burgers’ equation. — Proc. American Control Conf., Chicago, IL, pp.2745–2749.
  • Atwell J.A. and King B.B. (2001): Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations. — Math. Comput. Model., Vol.33, No.1–3, pp.1–19.
  • Atwell J.A. and King B.B. (2002): Reduced order controllers for spatially distributed systems via proper orthogonal decomposition. — SIAM J. Sci. Comput., (under revision).
  • Banks H.T., del Rosario R.C.H. and Smith R.C. (2000): Reduced order model feedback control design: Numerical implementation in a thin shell model. — IEEE Trans. Automat. Contr., Vol.45, No.7, pp.1312–1324.
  • Berkooz G. (1991): Observations on the proper orthogonal decomposition, In: Studies in Turbulence (T.B. Gatski, S. Sarkar and C.G. Speziale, Eds.). — New York: Springer, pp.229–247.
  • Berkooz G., Holmes P.J. and Lumley J.L. (1993): The proper orthogonal decomposition in the analysis of turbulent flows. — Annu. Rev. Fluid Mech., Vol.25, pp.539–575.
  • Burns J.A. and Kang S. (1991): A control problem for Burgers’ equation with bounded input/output. — Nonlin. Dyn., Vol.2, pp.235–262.
  • Burns J.A. and King B.B. (1995): Representation of feedback operators for hyperbolic systems, In: Computation and Control IV (K.L. Bowers and J. Lund, Eds.). — Boston: Birkhäuser, pp.57–74.
  • Burns J.A. and King B.B. (1998): A reduced basis approach to the design of low order compensators for nonlinear partial differential equation systems. — J. Vibr. Contr., Vol.4, No.3, pp.297–323.
  • Burns J.A. and Rubio D. (1997): A distributed parameter control approach to sensor location for optimal feedback control of thermal processes. — Proc. 36th IEEE CDC, San Diego, CA, pp.2243–2247.
  • Chambers D.H., Adrian R.J., Moin P., Stewart D.S. and Sung H.J. (1988): Karhunen-Loève expansion of Burgers’ model of turbulence. — Phys. Fluids, Vol.31, No.9, pp.2573–2582.
  • Demmel J.W. (1987): On condition numbers and the distance to the nearest ill-posed problem. — Numer. Math., Vol.51, No.3, pp.251–289.
  • Fahl M. (1999): Computation of PODs for fluid flows with Lanczos method. — Tech. Rep. 99–13, Universität Trier.
  • Franca L.P. and Frey S.L. (1992a): Stabilized finite element methods: I. Application to the advective-diffusive model. — Comput. Meth. Appl. Mech. Eng., Vol.95, pp.253–276.
  • Franca L.P. and Frey S.L. (1992b): Stabilized finite element methods: II. The incompressible Navier-Stokes equations. — Comput. Meth. Appl. Mech. Eng., Vol.99, pp.209–233.
  • Faulds A.L. and King B.B. (2000): Centroidal Voronoi tesselations for sensor placement. — Proc. IEEE CCA/CACSD 2000, Anchorage AK, pp.536–541.
  • Holmes P.J. (1991): Can dynamical systems approach turbulence? In: Whither Turbulence? Turbulence at the Crossroads (J.L. Lumley, Ed.). — Berlin: Verlag, pp.195–249.
  • Ito K. and Morris K.A. (1998): An approximation theory for solutions to operator Riccati equation for H ∞ control. — SIAM J. Contr. Optim., Vol.36, No.1, pp.82–99.
  • King B.B. (1995): Existence of functional gains for parabolic control systems, In: Computation and Control IV (K.L. Bowers and J. Lund, Eds.). — Boston: Birkhäuser, pp.203– 218.
  • King B.B. (1998): Nonuniform grids for reduced basis design of low order feedback controllers for nonlinear continuous systems. — Math. Models Meth. Appl. Sci., Vol., No.7, pp.1223–1241.
  • King B.B. and Sachs E.W. (2000): Semidefinite programming techniques for reduced order systems with guaranteed stability margins. — Comput. Optim. Appl., pp.37–59.
  • Kirby M., Boris J.P. and Sirovich L. (1990): A proper orthogonal decomposition of a simulated supersonic shear layer. — Int. J. Num. Meth. Fluids, Vol.10, pp.411–428.
  • Kunisch K. and Volkwein S. (1999): Control of Burgers’ equation by a reduced order approach using proper orthogonal decomposition. — JOTA, Vol.102, No.2, pp.345–371.
  • Ly H.V. and Tran H.T. (2001): Modeling and control of physical processes using proper orthogonal decomposition. — Comp. Math. Apps., Vol.33, No.1–3, pp.233–236.
  • Ly H.V. and Tran H.T. (2002): Proper orthogonal decomposition for flow calculations and optimal control in a horizontal CVD reactor. — Quart. Appl. Math, (in press).
  • Marrekchi H. (1993): Dynamic compensators for a nonlinear conservation law. — Ph.D. Dissertation, Virginia Polytechnic Institute and State University.
  • Moore B.C. (1981): Principal component analysis in linear systems: controllability, observability and model reduction. — IEEE Trans. Automat. Contr., Vol.26, No.1, pp.18–32.
  • Mustafa D. and Glover K. (1991): Controller reduction by H ∞ -balanced truncation. — IEEE Trans. Automat. Contr., Vol.36, No.6, pp.668–682.
  • Sirovich L. (1987): Turbulence and the dynamics of coherent structures, parts I–III. — Quart. Appl. Math., Vol.XLV, No.3, pp.561–590.
  • Theodoropoulou A., Adomaitis R.A. and Zafiriou E. (1998): Model reduction for optimization of rapid thermal chemical vapor deposition systems. — IEEE Trans. Semicond. Manuf., Vol.11, No.1, pp.85–98.
Typ dokumentu
Bibliografia
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