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Numerical solution of large‐scale Lyapunov equations, Riccati equations, and linear‐quadratic optimal control problems
Author(s) -
Benner Peter,
Li JingRebecca,
Penzl Thilo
Publication year - 2008
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.622
Subject(s) - algebraic riccati equation , cholesky decomposition , linear quadratic regulator , mathematics , riccati equation , linear quadratic gaussian control , lyapunov equation , linear system , optimal control , lyapunov function , mathematical optimization , differential equation , mathematical analysis , nonlinear system , eigenvalues and eigenvectors , physics , quantum mechanics
We study large‐scale, continuous‐time linear time‐invariant control systems with a sparse or structured state matrix and a relatively small number of inputs and outputs. The main contributions of this paper are numerical algorithms for the solution of large algebraic Lyapunov and Riccati equations and linear‐quadratic optimal control problems, which arise from such systems. First, we review an alternating direction implicit iteration‐based method to compute approximate low‐rank Cholesky factors of the solution matrix of large‐scale Lyapunov equations, and we propose a refined version of this algorithm. Second, a combination of this method with a variant of Newton's method (in this context also called Kleinman iteration) results in an algorithm for the solution of large‐scale Riccati equations. Third, we describe an implicit version of this algorithm for the solution of linear‐quadratic optimal control problems, which computes the feedback directly without solving the underlying algebraic Riccati equation explicitly. Our algorithms are efficient with respect to both memory and computation. In particular, they can be applied to problems of very large scale, where square, dense matrices of the system order cannot be stored in the computer memory. We study the performance of our algorithms in numerical experiments. Copyright © 2008 John Wiley & Sons, Ltd.