Rosenbrock Methods for Solving Riccati Differential Equations

Benner, Peter; Mena, Hermann

doi:10.1109/tac.2013.2258495

Cited by 47 publications

(48 citation statements)

References 18 publications

(30 reference statements)

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“…The right-hand side of (3.9) is simpler to evaluate than the one in (3.8), so the implementation of (3.9) is more efficient [15]. If we assume,…”

Section: Linearly Implicit Euler Methodsmentioning

confidence: 99%

Numerical solution of the infinite-dimensional LQR problem and the associated Riccati differential equations

Benner

Mena

2018

Journal of Numerical Mathematics

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Abstract:The numerical analysis of linear quadratic regulator design problems for parabolic partial differential equations requires solving Riccati equations. In the finite time horizon case, the Riccati differential equation (RDE) arises. The coefficient matrices of the resulting RDE often have a given structure, e.g., sparse, or low-rank. The associated RDE usually is quite stiff, so that implicit schemes should be used in this situation. In this paper, we derive efficient numerical methods for solving RDEs capable of exploiting this structure, which are based on a matrix-valued implementation of the BDF and Rosenbrock methods. We show that these methods are suitable for large-scale problems by working only on approximate low-rank factors of the solutions. We also incorporate step size and order control in our numerical algorithms for solving RDEs. In addition, we show that within a Galerkin projection framework the solutions of the finite-dimensional RDEs converge in the strong operator topology to the solutions of the infinite-dimensional RDEs. Numerical experiments show the performance of the proposed methods.

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“…The right-hand side of (3.9) is simpler to evaluate than the one in (3.8), so the implementation of (3.9) is more efficient [15]. If we assume,…”

Section: Linearly Implicit Euler Methodsmentioning

confidence: 99%

Numerical solution of the infinite-dimensional LQR problem and the associated Riccati differential equations

Benner

Mena

2018

Journal of Numerical Mathematics

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“…Using the spectral property of Kronecker products gives the positive definiteness of the 2m × 2m matrix in (8). Thus, it has a unique Cholesky factorization given bŷ…”

Section: Casementioning

confidence: 99%

“…Recent developments concerning their solution in the large-scale case use, e.g., BDF methods [7] which require the solution of an ARE in each step, or Rosenbrock type methods [8] involving the solution of a Lyapunov equation in each step. Both ways involve either implicitly or explicitly largescale Lyapunov equation, such that LRCF-ADI can be applied.…”

Section: Ab 2 Re (µ)B − B(g T B)mentioning

confidence: 99%

Efficient handling of complex shift parameters in the low-rank Cholesky factor ADI method

2012

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The solution of large-scale Lyapunov equations is a crucial problem for several fields of modern applied mathematics. The low-rank Cholesky factor version of the alternating directions implicit method (LRCF-ADI) is one iterative algorithm that computes approximate low-rank factors of the solution. In order to achieve fast convergence it requires adequate shift parameters, which can be complex if the matrices defining the Lyapunov equation are unsymmetric. This will require complex arithmetic computations as well as storage of complex data and thus, increase the overall complexity and memory requirements of the method. In this article we propose a novel reformulation of LRCF-ADI which generates real low-rank factors by carefully exploiting the dependencies of the iterates with respect to pairs of complex conjugate shift parameters. It significantly reduces the amount of complex arithmetic calculations and requirements for complex storage. It is hence often superior in terms of efficiency compared to other real formulations.

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“…The idea so far in the large-scale case has been to apply the matrix versions of common time-stepping methods, e.g. BDF methods [6], [25] or Rosenbrock methods [7], [25], and realise that in each step an ARE or a number of Lyapunov equations have to be solved. Low-rank algorithms for the solution of these equations exist and we refer to [9], [31] for recent surveys, see also [5].…”

Section: Introductionmentioning

confidence: 99%

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

Stillfjord

2015

IEEE Trans. Automat. Contr.

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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.

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Rosenbrock Methods for Solving Riccati Differential Equations

Cited by 47 publications

References 18 publications

Numerical solution of the infinite-dimensional LQR problem and the associated Riccati differential equations

Numerical solution of the infinite-dimensional LQR problem and the associated Riccati differential equations

Efficient handling of complex shift parameters in the low-rank Cholesky factor ADI method

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

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