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System Identification: Formulation

  • Qi He
  • Le Yi Wang
  • G. George Yin
Chapter
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Abstract

Consider a single-input–single-output (SISO) linear time-invariant (LTI) stable discrete-time system
$$y(t) =\displaystyle\sum\limits_{ i=0}^{\infty }a_{ i}u(t - i) + d(t),\quad t = t_{0} + 1,\ldots,$$
(2.1)
where {y(t)} is the noise corrupted observation, {d(t)} is the disturbance, {u(t)} is the input with u(t) = 0 for t < 0, and \(a =\{ a_{i},i = 0,1,\ldots \},\) satisfying
$$\|a\|_{1} =\displaystyle\sum\limits_{ i=0}^{\infty }\vert a_{ i}\vert < \infty.$$
To proceed, we define
$$\begin{array}{ll} & \theta = (a_{0},a_{1},\ldots,a_{m_{0}-1})^{\prime} \in {\mathbb{R}}^{m_{0} }, \\ & \widetilde{\theta } = (a_{m_{0}},a_{m_{0}+1},\ldots )^{\prime}, \end{array}$$
(2.2)
where z′ denotes the transpose of z.

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Copyright information

© Qi He, Le Yi Wang, and G. George Yin 2013

Authors and Affiliations

  • Qi He
    • 1
  • Le Yi Wang
    • 2
  • G. George Yin
    • 3
  1. 1.Department of MathematicsUniversity of CaliforniaIrvineUSA
  2. 2.Department of Electrical and Computer EngWayne State UniversityDetroitUSA
  3. 3.Department of MathematicsWayne State UniversityDetroitUSA

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