Decoupled Structure-Preserving Doubling Algorithm with Truncation for Large-Scale Algebraic Riccati Equations

In \emph{Guo et al, arXiv:2005.08288}, we propose a decoupled form of the structure-preserving doubling algorithm (dSDA). The method decouples the original two to four coupled recursions, enabling it to solve large-scale algebraic Riccati equations and other related problems. In this paper, we consider the numerical computations of the novel dSDA for solving large-scale continuous-time algebraic Riccati equations with low-rank structures (thus possessing numerically low-rank solutions). With the help of a new truncation strategy, the rank of the approximate solution is controlled. Consequently, large-scale problems can be treated efficiently. Illustrative numerical examples are presented to demonstrate and confirm our claims.

[1]  Zvonimir Bujanovic,et al.  RADI: a low-rank ADI-type algorithm for large scale algebraic Riccati equations , 2015, Numerische Mathematik.

[2]  Valeria Simoncini,et al.  On two numerical methods for the solution of large-scale algebraic Riccati equations , 2014 .

[3]  Valeria Simoncini,et al.  Analysis of the Rational Krylov Subspace Projection Method for Large-Scale Algebraic Riccati Equations , 2016, SIAM J. Matrix Anal. Appl..

[4]  A. Laub.  A schur method for solving algebraic Riccati equations , 1978, 1978 IEEE Conference on Decision and Control including the 17th Symposium on Adaptive Processes.

[5]  Bruno Iannazzo,et al.  Numerical Solution of Algebraic Riccati Equations , 2012, Fundamentals of algorithms.

[6]  V. Mehrmann.  The Autonomous Linear Quadratic Control Problem , 1991 .

[7]  Khalide Jbilou,et al.  Block Krylov Subspace Methods for Large Algebraic Riccati Equations , 2003, Numerical Algorithms.

[8]  P. Benner,et al.  Numerical solution of large and sparse continuous time algebraic matrix Riccati and Lyapunov equations: a state of the art survey , 2013 .

[9]  Peter Benner,et al.  M-M.E.S.S.-1.0.1 - The Matrix Equations Sparse Solvers Library , 2016 .

[10]  W. Gawronski.  Dynamics and control of structures : a modal approach , 1998 .

[11]  Peter Benner,et al.  Stabilization of Incompressible Flow Problems by Riccati-based Feedback , 2012, Constrained Optimization and Optimal Control for Partial Differential Equations.

[12]  Wen-Wei Lin,et al.  Convergence Analysis of Structure-Preserving Doubling Algorithms for Riccati-Type Matrix Equations , 2006, SIAM J. Matrix Anal. Appl..

[13]  Wen-Wei Lin,et al.  Doubling algorithm for the discretized Bethe-Salpeter eigenvalue problem , 2018, Math. Comput..

[14]  W.-W. Lin,et al.  A structure-preserving doubling algorithm for continuous-time algebraic Riccati equations , 2004 .

[15]  Mark R. Opmeer,et al.  Analysis of an Iteration Method for the Algebraic Riccati Equation , 2016, SIAM J. Matrix Anal. Appl..

[16]  Peter Benner,et al.  Dimension Reduction of Large-Scale Systems , 2005 .

[17]  M. Heyouni,et al.  AN EXTENDED BLOCK ARNOLDI ALGORITHM FOR LARGE-SCALE SOLUTIONS OF THE CONTINUOUS-TIME ALGEBRAIC RICCATI EQUATI ON ∗ , 2008 .

[18]  Leiba Rodman,et al.  Algebraic Riccati equations , 1995 .

[19]  Peter Benner,et al.  Numerical solution of large‐scale Lyapunov equations, Riccati equations, and linear‐quadratic optimal control problems , 2008, Numer. Linear Algebra Appl..

[20]  Ji-guang Sun.  Perturbation Theory for Algebraic Riccati Equations , 1998, SIAM J. Matrix Anal. Appl..

[21]  Wen-Wei Lin,et al.  A decoupled form of the structure-preserving doubling algorithm with low-rank structures , 2020, ArXiv.

[22]  Y. Saad,et al.  Numerical solution of large Lyapunov equations , 1989 .

[23]  Biswa Nath Datta,et al.  Linear and numerical linear algebra in control theory: some research problems , 1994 .

[24]  Wen-Wei Lin,et al.  Solving large-scale continuous-time algebraic Riccati equations by doubling , 2013, J. Comput. Appl. Math..

[25]  Khalide Jbilou,et al.  An Arnoldi based algorithm for large algebraic Riccati equations , 2006, Appl. Math. Lett..

[26]  Ramon E. Moore,et al.  Computational Functional Analysis , 1986, The Mathematical Gazette.

[27]  A. J. Laub,et al.  Efficient matrix-valued algorithms for solving stiff Riccati differential equations , 1990 .

[28]  A. Laub.  Invariant Subspace Methods for the Numerical Solution of Riccati Equations , 1991 .