Efficient algorithms for solving modified matrix equations of linear systems

1982 ◽  
Vol 70 (4) ◽  
pp. 406-407 ◽  
Author(s):  
S.B. Haley
Keyword(s):  
Linear Systems ◽  
Efficient Algorithms ◽  
Matrix Equations

Model reference control in quasi-linear systems with a parametric feed-forward compensator and state-feedback stabilization controller

2021 ◽  
pp. 014233122110564
Author(s):  
Weizhen Liu ◽  
Guangren Duan ◽  
Dake Gu
Keyword(s):  
Linear Systems ◽  
State Feedback ◽  
Closed Loop ◽  
Feedback Stabilization ◽  
Matrix Equations ◽  
Parametric Solution ◽  
Model Reference ◽  
Feed Forward ◽  
Model Reference Control ◽  
Sylvester Matrix Equations

In this paper, a parametric feed-forward compensator and a parametric state-feedback stabilization controller are proposed for the model reference control to a class of quasi-linear systems. Quasi-linear systems are a special type of nonlinear systems whose coefficient matrices contain the state variables and also a time-varying parameter vector. The parametric state-feedback stabilization controller guarantees the stability of the closed-loop system and the parametric feed-forward compensator compensates the effect of the reference model state to the tracking error. The complete parametrization of the parametric feed-forward compensator is established based on a complete parametric solution to a class of generalized Sylvester matrix equations and solution of a coefficient matrix such that two matrix equations are satisfied. The established parametric state-feedback stabilization controller only needs a complete parametric solution to the same generalized Sylvester matrix equations but with different sets of freely designed parameters that represent the degrees of design freedom and may be further utilized to improve the system performance. A linear closed-loop form with the desired eigenstructure can be derived with the proposed parametric feed-forward compensator and parametric state-feedback stabilization controller, and a constant linear can even be obtained in certain cases. A numerical example and the application in spacecraft rendezvous are provided to illustrate the effectiveness of the proposed approach.

scholarly journals Investigation of High-Efficiency Iterative ILU Preconditioner Algorithm for Partial-Differential Equation Systems

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1461 ◽  
Author(s):  
Yan-Hong Fan ◽  
Ling-Hui Wang ◽  
You Jia ◽  
Xing-Guo Li ◽  
Xue-Xia Yang ◽  
...  
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Linear Systems ◽  
High Efficiency ◽  
Iterative Scheme ◽  
Processing Unit ◽  
Matrix Equations ◽  
Partial Differential ◽  
Central Processing ◽  
Ilu Factorization

In this paper, we investigate an iterative incomplete lower and upper (ILU) factorization preconditioner for partial-differential equation systems. We discretize the partial-differential equations into linear equation systems. An iterative scheme of linear systems is used. The ILU preconditioners of linear systems are performed on the different computation nodes of multi-central processing unit (CPU) cores. Firstly, the preconditioner of general tridiagonal matrix equations is tested on supercomputers. Then, the effects of partial-differential equation systems on the speedup of parallel multiprocessors are examined. The numerical results estimate that the parallel efficiency is higher than in other algorithms.

scholarly journals A Flexible Global GCRO-DR Method for Shifted Linear Systems and General Coupled Matrix Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Jing Meng ◽  
Xian-Ming Gu ◽  
Wei-Hua Luo ◽  
Liang Fang
Keyword(s):  
Linear Systems ◽  
Numerical Experiments ◽  
Matrix Equations ◽  
The Novel ◽  
Dr Method ◽  
Shifted Linear Systems ◽  
Novel Method ◽  
General Coupled Matrix Equations ◽  
Subspace Recycling ◽  
The Right

In this paper, we mainly focus on the development and study of a new global GCRO-DR method that allows both the flexible preconditioning and the subspace recycling for sequences of shifted linear systems. The novel method presented here has two main advantages: firstly, it does not require the right-hand sides to be related, and, secondly, it can also be compatible with the general preconditioning. Meanwhile, we apply the new algorithm to solve the general coupled matrix equations. Moreover, by performing an error analysis, we deduce that a much looser tolerance can be applied to save computation by limiting the flexible preconditioned work without sacrificing the closeness of the computed and the true residuals. Finally, numerical experiments demonstrate that the proposed method illustrated can be more competitive than some other global GMRES-type methods.

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