American Institute of Mathematical Sciences

Advanced
2016, 6(4): 413-435. doi: 10.3934/naco.2016018

A POD projection method for large-scale algebraic Riccati equations

Boris Kramer 1, and  John R. Singler 2,

1. 

Department of Aeronautics and Astronautics, Massachusetts Institute of Technology,Cambridge, MA 02139, United States
2. 

Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409-0020, United States

Received  February 2016 Revised  September 2016 Published  December 2016

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, large-scale, proper orthogonal decomposition, control theory., reduced-order modeling.
Mathematics Subject Classification: Primary: 49; Secondary: 6.
Citation: Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018
References:
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show all references

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[25]

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[26]

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[27]

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[28]

N. Darivandi, K. Morris and A. Khajepour, An algorithm for LQ optimal actuator location, Smart Materials and Structures, 22 (2013), 035001. Google Scholar

[29]

B. T. Dickinson and J. R. Singler, Nonlinear model reduction using group proper orthogonal decomposition, Int. J. Numer. Anal. Model., 7 (2010), 356-372.  Google Scholar

[30]

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