Numerical Algebra, Control and Optimization (NACO)

A POD projection method for large-scale algebraic Riccati equations

Pages: 413 - 435, Volume 6, Issue 4, December 2016      doi:10.3934/naco.2016018

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Boris Kramer - Department of Aeronautics and Astronautics, Massachusetts Institute of Technology,Cambridge, MA 02139, United States (email)
John R. Singler - Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409-0020, United States (email)

Abstract: The solution of large-scale matrix algebraic Riccati equations is important for instance in control design and model reduction and remains an active area of research. We consider a class of matrix algebraic Riccati equations (AREs) arising from a linear system along with a weighted inner product. This problem class often arises from a spatial discretization of a partial differential equation system. We propose a projection method to obtain low rank solutions of AREs based on simulations of linear systems coupled with proper orthogonal decomposition. The method can take advantage of existing (black box) simulation code to generate the projection matrices. We also develop new weighted norm residual computations and error bounds. We present numerical results demonstrating that the proposed approach can produce highly accurate approximate solutions. We also briefly discuss making the proposed approach completely data-based so that one can use existing simulation codes without accessing system matrices.

Keywords:  Algebraic Riccati equations, reduced-order modeling, large-scale, proper orthogonal decomposition, control theory.
Mathematics Subject Classification:  Primary: 49; Secondary: 65.

Received: February 2016;      Revised: September 2016;      Available Online: December 2016.