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Low-Rank Second-Order Splitting of Large-Scale Differential Riccati Equations

Stillfjord, Tony LU (2015) In IEEE Transactions on Automatic Control 60(10). p.2791-2796
Abstract
We apply first- and second-order splitting schemes to the differential Riccati equation. Such equations are very important in e.g. linear quadratic regulator (LQR) problems, where they provide a link between the state of the system and the optimal input. The methods can also be extended to generalized Riccati equations, e.g. arising from LQR problems given in implicit form. In contrast to previously proposed schemes such as BDF or Rosenbrock methods, the splitting schemes exploit the fact that the nonlinear and affine parts of the problem, when considered in isolation, have closed-form solutions. We show that if the solution possesses low-rank structure, which is frequently the case, then this is preserved by the method. This feature is... (More)
We apply first- and second-order splitting schemes to the differential Riccati equation. Such equations are very important in e.g. linear quadratic regulator (LQR) problems, where they provide a link between the state of the system and the optimal input. The methods can also be extended to generalized Riccati equations, e.g. arising from LQR problems given in implicit form. In contrast to previously proposed schemes such as BDF or Rosenbrock methods, the splitting schemes exploit the fact that the nonlinear and affine parts of the problem, when considered in isolation, have closed-form solutions. We show that if the solution possesses low-rank structure, which is frequently the case, then this is preserved by the method. This feature is used to implement the methods efficiently for large-scale problems. The proposed methods are expected to be competitive, as they at most require the solution of a small number of linear equation systems per time step. Finally, we apply our low-rank implementations to the Riccati equations arising from two LQR problems. The results show that the rank of the solutions stay low, and the expected orders of convergence are observed. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Differential Riccati equation, large-scale, low-rank, Riccati differential equation, splitting methods
in
IEEE Transactions on Automatic Control
volume
60
issue
10
pages
2791 - 2796
publisher
IEEE--Institute of Electrical and Electronics Engineers Inc.
external identifiers
  • wos:000367284100027
  • scopus:84942872964
ISSN
0018-9286
DOI
10.1109/TAC.2015.2398889
language
English
LU publication?
yes
id
98982f50-6239-4500-b0db-1ead943cea2c (old id 5265732)
alternative location
http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=7029027
date added to LUP
2015-05-05 16:37:37
date last changed
2017-10-01 03:14:18
@article{98982f50-6239-4500-b0db-1ead943cea2c,
  abstract     = {We apply first- and second-order splitting schemes to the differential Riccati equation. Such equations are very important in e.g. linear quadratic regulator (LQR) problems, where they provide a link between the state of the system and the optimal input. The methods can also be extended to generalized Riccati equations, e.g. arising from LQR problems given in implicit form. In contrast to previously proposed schemes such as BDF or Rosenbrock methods, the splitting schemes exploit the fact that the nonlinear and affine parts of the problem, when considered in isolation, have closed-form solutions. We show that if the solution possesses low-rank structure, which is frequently the case, then this is preserved by the method. This feature is used to implement the methods efficiently for large-scale problems. The proposed methods are expected to be competitive, as they at most require the solution of a small number of linear equation systems per time step. Finally, we apply our low-rank implementations to the Riccati equations arising from two LQR problems. The results show that the rank of the solutions stay low, and the expected orders of convergence are observed.},
  author       = {Stillfjord, Tony},
  issn         = {0018-9286},
  keyword      = {Differential Riccati equation,large-scale,low-rank,Riccati differential equation,splitting methods},
  language     = {eng},
  number       = {10},
  pages        = {2791--2796},
  publisher    = {IEEE--Institute of Electrical and Electronics Engineers Inc.},
  series       = {IEEE Transactions on Automatic Control},
  title        = {Low-Rank Second-Order Splitting of Large-Scale Differential Riccati Equations},
  url          = {http://dx.doi.org/10.1109/TAC.2015.2398889},
  volume       = {60},
  year         = {2015},
}