Reduced Order Controllers for Spatially Distributed Systems via Proper Orthogonal Decomposition

A method for reducing controllers for systems described by partial differential equations (PDEs) is presented. This approach differs from an often used method of reducing the model and then designing the controller. The controller reduction is accomplished by projection of a large scale finite element approximation of the PDE controller onto low order bases that are computed using the proper orthogonal decomposition (POD). Two methods for constructing input collections for POD, and hence low order bases, are discussed and computational results are included. The first uses the method of snapshots found in POD literature. The second is a new idea that uses an integral representation of the feedback control law. Specifically, the kernels, or functional gains, are used as data for POD. A low order controller derived by applying the POD process to functional gains avoids subjective criteria associated with implementing a time snapshot approach and performs favorably.

[1]  Jerrold E. Marsden,et al.  Empirical model reduction of controlled nonlinear systems , 1999, IFAC Proceedings Volumes.

[2]  I. Kevrekidis,et al.  Low‐dimensional models for complex geometry flows: Application to grooved channels and circular cylinders , 1991 .

[3]  P. Holmes.  Can dynamical systems approach turbulence , 1990 .

[4]  Guangcao Ji,et al.  Partially Observed Analytic Systems with Fully Unbounded Actuators and Sensors-FEM Algorithms , 1998, Comput. Optim. Appl..

[5]  L. Sirovich,et al.  Optimal low-dimensional dynamical approximations , 1990 .

[6]  Belinda B. King,et al.  Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations , 2001 .

[7]  B. Moore.  Principal component analysis in linear systems: Controllability, observability, and model reduction , 1981 .

[8]  J. A. Burns,et al.  Feedback Control of a Thermal Fluid Using State Estimation , 1998 .

[9]  Gal Berkooz,et al.  Observations on the Proper Orthogonal Decomposition , 1992 .

[10]  Belinda B. King.  Nonuniform Grids for Reduced Basis Design of Low Order Feedback Controllers for Nonlinear Continuous Systems , 1997, Universität Trier, Mathematik/Informatik, Forschungsbericht.

[11]  D. Rubio,et al.  A distributed parameter control approach to sensor location for optimal feedback control of thermal processes , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[12]  P. Holmes,et al.  The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows , 1993 .

[13]  J. Demmel.  On condition numbers and the distance to the nearest ill-posed problem , 2015 .

[14]  Ekkehard W. Sachs,et al.  Semidefinite Programming Techniques for Reduced Order Systems with Guaranteed Stability Margins , 2000, Comput. Optim. Appl..

[15]  M. Kirby,et al.  A proper orthogonal decomposition of a simulated supersonic shear layer , 1990 .

[16]  Ioannis G. Kevrekidis,et al.  Alternative approaches to the Karhunen-Loève decomposition for model reduction and data analysis , 1996 .

[17]  K. Kunisch,et al.  Control of the Burgers Equation by a Reduced-Order Approach Using Proper Orthogonal Decomposition , 1999 .

[18]  Ruth F. Curtain,et al.  Model reduction for control design for distributed parameter systems , 2003 .

[19]  H. Tran,et al.  Proper Orthogonal Decomposition for Flow Calculations and Optimal Control in a Horizontal CVD Reactor , 2002 .

[20]  Belinda B. King,et al.  Nonlinear dynamic compensator design for flow control in a driven cavity , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[21]  E. Zafiriou,et al.  Model reduction for optimization of rapid thermal chemical vapor deposition systems , 1998 .

[22]  D. Chambers,et al.  Karhunen-Loeve expansion of Burgers' model of turbulence , 1988 .

[23]  Irena Lasiecka.  Finite Element Approximations of Compensator Design for Analytic Generators with Fully Unbounded Controls/Observations , 1995 .

[24]  J. Lumley.  Whither Turbulence? Turbulence at the Crossroads , 1990 .

[25]  Marco Fahl.  Computation of PODs for Fluid Flows with Lanczos Methods , 1999, Universität Trier, Mathematik/Informatik, Forschungsbericht.

[26]  R. H. Fabiano,et al.  On the shape of finite dimensional approximations of feedback functional gains , 1994 .

[27]  Ricardo C. H. del Rosario,et al.  Reduced-order model feedback control design: numerical implementation in a thin shell model , 2000, IEEE Trans. Autom. Control..

[28]  K. Karhunen.  Zur Spektraltheorie stochastischer prozesse , 1946 .

[29]  G. M. Kepler,et al.  Reduced order model compensator control of species transport in a CVD reactor , 2000 .

[30]  K. Hulsing,et al.  Methods of Computing Functional Gains for LQR Control of Partial Differential Equations , 1999 .

[31]  L. Sirovich.  Turbulence and the dynamics of coherent structures. I. Coherent structures , 1987 .

[32]  S. Mitter,et al.  Representation and Control of Infinite Dimensional Systems , 1992 .

[33]  Roberto Triggiani,et al.  Min-max game theory and algebraic Riccati equations for boundary control problems with continuous input-solution map. Part II: The general case , 1994 .

[34]  H. Tran,et al.  Modeling and control of physical processes using proper orthogonal decomposition , 2001 .

[35]  H. Hotelling.  Analysis of a complex of statistical variables into principal components. , 1933 .

[36]  George E. Karniadakis,et al.  Unsteady Two-Dimensional Flows in Complex Geometries: Comparative Bifurcation Studies with Global Eigenfunction Expansions , 1997, SIAM J. Sci. Comput..