scholarly journals An Alternating Direction Implicit Method for Solving Projected Generalized Continuous-Time Sylvester Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Yujian Zhou ◽  
Liang Bao ◽  
Yiqin Lin
Keyword(s):  
Continuous Time ◽  
Computational Cost ◽  
Implicit Method ◽  
Descriptor Systems ◽  
Low Rank ◽  
Balanced Truncation ◽  
Alternating Direction Implicit Method ◽  
Alternating Direction Implicit ◽  
Smith Method ◽  
Alternating Direction

We present the generalized low-rank alternating direction implicit method and the low-rank cyclic Smith method to solve projected generalized continuous-time Sylvester equations with low-rank right-hand sides. Such equations arise in control theory including the computation of inner products andℍ2norms and the model reduction based on balanced truncation for descriptor systems. The requirements of these methods are moderate with respect to both computational cost and memory. Numerical experiments presented in this paper show the effectiveness of the proposed methods.

scholarly journals Efficient Techniques for Solving the Periodic Projected Lyapunov Equations and Model Reduction of Periodic Systems

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Mohammad-Sahadet Hossain ◽  
M. Monir Uddin
Keyword(s):  
Model Reduction ◽  
Discrete Time ◽  
Large Scale ◽  
Computational Cost ◽  
Descriptor Systems ◽  
Low Rank ◽  
Lyapunov Equations ◽  
Adi Method ◽  
Smith Method ◽  
Alternating Direction

We have presented the efficient techniques for the solutions of large-scale sparse projected periodic discrete-time Lyapunov equations in lifted form. These types of problems arise in model reduction and state feedback problems of periodic descriptor systems. Two most popular techniques to solve such Lyapunov equations iteratively are the low-rank alternating direction implicit (LR-ADI) method and the low-rank Smith method. The main contribution of this paper is to update the LR-ADI method by exploiting the ideas of the adaptive shift parameters computation and the efficient handling of complex shift parameters. These approaches efficiently reduce the computational cost with respect to time and memory. We also apply these iterative Lyapunov solvers in balanced truncation model reduction of periodic discrete-time descriptor systems. We illustrate numerical results to show the performance and accuracy of the proposed methods.

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