Efficient matrix-valued algorithms for solving stiff Riccati differential equations

Choi, Chiu; Laub, Alan J.

doi:10.1109/cdc.1989.70248

Cited by 21 publications

(31 citation statements)

References 4 publications

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“…In the literature there is a large variety of approaches to compute the solution of the DRE (1) (see, e.g., [6][7][8]). However, none of these methods is suitable for large-scale control problems, since the computational effort grows at best like n 3 , where n is the dimension of the state of the control system.…”

Section: Solving Large-scale Dresmentioning

confidence: 99%

Numerical solution of differential Riccati equations arising in optimal control for parabolic PDEs

Mena

Benner

2007

Proc Appl Math and Mech

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The numerical treatment of linear-quadratic regulator problems on finite time horizons for parabolic partial differential equations requires the solution of large-scale differential Riccati equations (DREs). Typically the coefficient matrices of the resulting DRE have a given structure (e.g. sparse, symmetric or low rank). Here we discuss numerical methods for solving DREs capable of exploiting this structure. These methods are based on a matrix-valued implementation of the BDF methods. The crucial question of suitable stepsize and order selection strategies is also addressed.

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Section: Solving Large-scale Dresmentioning

confidence: 99%

Numerical solution of differential Riccati equations arising in optimal control for parabolic PDEs

Mena

Benner

2007

Proc Appl Math and Mech

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“…In this context, several approaches to solve RDEs have been proposed, [3], [4], [5]. Particularly, matrix-valued algorithms for solving RDEs based on generalizations of the BDF methods have been considered, see [6], [7], [8]. These methods are also suitable for large scale RDEs arising in optimal control problems for parabolic partial differential equations [9].…”

Section: Introduction We Consider Time-varying Riccati Differentiamentioning

confidence: 99%

Rosenbrock Methods for Solving Riccati Differential Equations

Benner¹,

Mena

2013

IEEE Trans. Automat. Contr.

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Abstract-The Riccati differential equation (RDE) arises in several fields like optimal control, optimal filtering, H∞ control of linear timevarying systems, differential games, etc. In the literature there is a large variety of approaches to compute its solution. Particularly for stiff RDEs, matrix-valued versions of the standard multi-step methods for solving ordinary differential equations have given good results. In this paper we discuss a particular class of one-step methods. These are the linearimplicit Runge-Kutta methods or Rosenbrock methods. We show that they offer a practical alternative for solving stiff RDEs. They can be implemented with good stability properties and allow for a cheap step size control. The matrix valued version of the Rosenbrock methods for RDEs requires the solution of one Sylvester equation in each stage of the method. For the case in which the coefficient matrices of the Sylvester equation are dense, the Bartels-Stewart method can be efficiently applied for solving the equations. The computational cost (computing time and memory requirements) is smaller than for the multi-step methods.

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“…Therefore, the extension of the present reduction DQ method to the transient convection-diffusion equations are also obviously applicable. For the details see reference [9].…”

Section: On Time-dependent Problemsmentioning

confidence: 99%

A Lyapunov Formulation for Efficient Solution of the Poisson and Convection–Diffusion Equations by the Differential Quadrature Method

Chen¹,

Zhong²,

Chen

1998

Journal of Computational Physics

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Efficient matrix-valued algorithms for solving stiff Riccati differential equations

Cited by 21 publications

References 4 publications

Numerical solution of differential Riccati equations arising in optimal control for parabolic PDEs

Numerical solution of differential Riccati equations arising in optimal control for parabolic PDEs

Rosenbrock Methods for Solving Riccati Differential Equations

A Lyapunov Formulation for Efficient Solution of the Poisson and Convection–Diffusion Equations by the Differential Quadrature Method

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