scholarly journals The Extended Hamiltonian Algorithm for the Solution of the Algebraic Riccati Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Zhikun Luo ◽  
Huafei Sun ◽  
Xiaomin Duan
Keyword(s):  
Riccati Equation ◽  
Iteration Method ◽  
Learning Algorithm ◽  
Positive Definite ◽  
Gradient Algorithm ◽  
Algebraic Riccati Equation ◽  
Symmetric Matrices ◽  
Algebraic Riccati Equations ◽  
Subspace Iteration ◽  
Hamiltonian Algorithm

We use a second-order learning algorithm for numerically solving a class of the algebraic Riccati equations. Specifically, the extended Hamiltonian algorithm based on manifold of positive definite symmetric matrices is provided. Furthermore, this algorithm is compared with the Euclidean gradient algorithm, the Riemannian gradient algorithm, and the new subspace iteration method. Simulation examples show that the convergence speed of the extended Hamiltonian algorithm is the fastest one among these algorithms.

PARALLEL SUBSPACE ITERATION METHOD FOR THE SPARSE SYMMETRIC EIGENVALUE PROBLEM

2006 ◽  
Vol 05 (04) ◽  
pp. 801-818 ◽  
Author(s):  
RICHARD LOMBARDINI ◽  
BILL POIRIER
Keyword(s):  
Eigenvalue Problem ◽  
Iteration Method ◽  
Degrees Of Freedom ◽  
Sparse Matrices ◽  
Model Systems ◽  
Symmetric Matrices ◽  
Six Degrees Of Freedom ◽  
Subspace Iteration ◽  
The Matrix ◽  
Parallel Iterative

A new parallel iterative algorithm for the diagonalization of real sparse symmetric matrices is introduced, which uses a modified subspace iteration method. A novel feature is the preprocessing of the matrix prior to iteration, which allows for a natural parallelization resulting in a great speedup and scalability of the method with respect to the number of compute nodes. The method is applied to Hamiltonian matrices of model systems up to six degrees of freedom, represented in a truncated Weyl–Heisenberg wavelet (or "weylet") basis developed by one of the authors (Poirier). It is shown to accurately determine many thousands of eigenvalues for sparse matrices of the order N ≈ 106, though much larger matrices may also be considered.

Filtering for Linear Stochastic Systems With Small Measurement Noise

1995 ◽  
Vol 117 (3) ◽  
pp. 425-429 ◽  
Author(s):  
Z. Aganovic ◽  
Z. Gajic ◽  
X. Shen
Keyword(s):  
Kalman Filter ◽  
Riccati Equation ◽  
Stochastic Systems ◽  
Measurement Noise ◽  
Algebraic Riccati Equation ◽  
Reduced Order ◽  
Complete Decomposition ◽  
Algebraic Riccati Equations ◽  
Optimal Filters ◽  
Linear Stochastic Systems

In this paper we present a method which produces complete decomposition of the optimal global Kalman filter for linear stochastic systems with small measurement noise into exact pure-slow and pure-fast reduced-order optimal filters both driven by the system measurements. The method is based on the exact decomposition of the global small measurement noise algebraic Riccati equation into exact pure-slow and pure-fast algebraic Riccati equations. An example is included in order to demonstrate the proposed method.

scholarly journals Analysis of an Iteration Method for the Algebraic Riccati Equation

2016 ◽  
Vol 37 (2) ◽  
pp. 624-648 ◽  
Author(s):  
Arash Massoudi ◽  
Mark R. Opmeer ◽  
Timo Reis
Keyword(s):  
Riccati Equation ◽  
Iteration Method ◽  
Algebraic Riccati Equation

scholarly journals On the square-root method for continuous-time algebraic Riccati equations

1999 ◽  
Vol 40 (4) ◽  
pp. 459-468 ◽  
Author(s):  
Linzhang Lu ◽  
C. E. M. Pearce
Keyword(s):  
Riccati Equation ◽  
Continuous Time ◽  
Riccati Equations ◽  
Algebraic Riccati Equation ◽  
Square Root ◽  
Numerical Examples ◽  
Algebraic Riccati Equations ◽  
Solution Methods ◽  
Square Root Method

AbstractWe give a simple and transparent proof for the square-root method of solving the continuous-time algebraic Riccati equation. We examine some benefits of combining the square-root method with other solution methods. The iterative square-root method is also discussed. Finally, paradigm numerical examples are given to compare the square-root method with the Schur method.

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