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Journal of Applied Mathematics
Volume 2014, Article ID 134604, 10 pages
Research Article

Computation of Gram Matrix and Its Partial Derivative Using Precise Integration Method for Linear Time-Invariant Systems

1Key Laboratory of Electronic Equipment Structure Design of Ministry of Education, Xidian University, 2 Taibai Road, Xi’an 710071, China
2Wuhan Second Ship Design and Research Institute, 450 Zhongshan Road, Wuhan 430064, China

Received 14 October 2013; Revised 9 March 2014; Accepted 9 March 2014; Published 22 April 2014

Academic Editor: Alexander Timokha

Copyright © 2014 Sulan Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Gram matrix is an important tool in system analysis and design as it provides a description of the input-output behavior for system; its partial derivative matrix is often required in some numerical algorithms. It is essential to study computation of these matrices. Analytical methods only work in some special circumstances; for example, the system matrix is diagonal matrix or Jordan matrix. In most cases, numerical integration method is needed, but there are two problems when compute using traditional numerical integration method. One is low accuracy: as high accuracy requires extremely small integration step, it will result in large amount of computation; and another is stability and stiffness issues caused by the dependence on the property of system matrix. In order to overcome these problems, this paper proposes an efficient numerical method based on the key idea of precise integration method (PIM) for the Gram matrix and its partial derivative of linear time-invariant systems. Since matrix inverse operation is not required in this method, it can be used with high precision no matter the system is normal or singular. The specific calculation algorithm and block diagram are also given. Finally, numerical examples are given to demonstrate the correctness and validity of this method.