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Analysis of Linear Time-Varying (LTV) Systems

10 pagesPublished: November 2, 2021


Consider a linear time-varying (LTV) system described by the state-space equation dx(t)/dt = A(t)x(t)+B(t)u(t). The main objectives of this paper include: (i) determination of the analytical (closed-form) solutions for the fundamental matrix X(t) and the state transition matrix P(t,t0) of the LTV system, and (ii) design of feedback control, such that the closed-loop system matrix Acl(t) = A(t)-B(t)K(t), where K(t) is a gain matrix, has desirable characteristics, namely, Acl(t), is commutative and triangular. It follows that commutativity of Acl(t) will facilitate the analytical solutions of Xcl(t) and Pcl(t,t0), including Matlab solutions; while triangularization of Acl(t) will allow manual calculations of these matrices easily, especially for low-dimensional systems. This is different from a traditional pole-placement design problem where the goal is to place the poles of Acl(t) to acquire desired closed-loop stability properties. Examples are given to demonstrate the design objectives. Solutions in Matlab are given as well.

Keyphrases: analytical solution, Commutative and triangular matrix, fundamental matrix, Linear time-varying (LTV) systems, state transition matrix

In: Yan Shi, Gongzhu Hu, Quan Yuan and Takaaki Goto (editors). Proceedings of ISCA 34th International Conference on Computer Applications in Industry and Engineering, vol 79, pages 130--139

BibTeX entry
  author    = {Robert Loh and K. C. Cheok},
  title     = {Analysis of  Linear Time-Varying (LTV) Systems},
  booktitle = {Proceedings of ISCA 34th International Conference on Computer Applications in Industry and Engineering},
  editor    = {Yan Shi and Gongzhu Hu and Quan Yuan and Takaaki Goto},
  series    = {EPiC Series in Computing},
  volume    = {79},
  pages     = {130--139},
  year      = {2021},
  publisher = {EasyChair},
  bibsource = {EasyChair,},
  issn      = {2398-7340},
  url       = {},
  doi       = {10.29007/lv7b}}
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