Open Access

ARTICLE

New Integrating Methods for Time-Varying Linear Systems and Lie-Group Computations

Chein-Shan Liu¹

1 Department of Mechanical and Mechatronic Engineering, Taiwan Ocean University, Keelung, Taiwan. E-mail: csliu@mail.ntou.edu.tw

Computer Modeling in Engineering & Sciences 2007, 20(3), 157-176. https://doi.org/10.3970/cmes.2007.020.157

Abstract

In many engineering applications the Lie group calculation is very important. With this in mind, the subject of this paper is for an in-depth investigation of time-varying linear systems, and its accompanied Lie group calculations. In terms of system matrix A in Eq. (11) and a one-order lower fundamental solution matrix associated with the sub-state matrix function A_s^s, we propose two methods to nilpotentlize the time-varying linear systems. As a consequence, we obtain two different calculations of the general linear group. Then, the nilpotent systems are further transformed to a unique new system Ż(t) = B(t)Z(t), which having a special simple B(t) ∈ sl(n+1,R) with B_s^s and B₀⁰ vanishing. Correspondingly, we get a third calculation of the general linear group. By using the nilpotent property we can develop quite simple numerical scheme of nilpotent type to calculate the state transition matrix. We also develop a Lie-group solver in terms of the exponential mapping of B. Several numerical examples were employed to assess the performance of proposed schemes. Especially, the new Lie-group solver is very stable and highly accurate.

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Keywords

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