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Initial-Condition Response

Going back to the time domain, we have the linear discrete-time state-space model

\begin{eqnarray*}
\underline{y}(n) & = & C\, \underline{x}(n) + D\,\underline{u}(n)\\ [10pt]
\underline{x}(n+1) & = & A\, \underline{x}(n) + B\, \underline{u}(n)
\end{eqnarray*}

and its ``impulse response''

$\displaystyle {\mathbf{h}}(n) \eqsp \left\{\begin{array}{ll}
D, & n=0 \\ [5pt]
CA^{n-1}B, & n>0 \\
\end{array} \right.
$

Given zero inputs and initial state $ \underline{x}(0)\ne \underline{0}$ , we get

$\displaystyle \underline{y}_x(n) \eqsp C A^{n-1}\underline{x}(0), \quad n=0,1,2,\ldots\,.
$

By superposition (for LTI systems), the complete response of a linear system is given by the sum of its forced response (such as the impulse response) and its initial-condition response


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``Introduction to State Space Models'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2014-03-24 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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