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Similarity Transformations

A similarity transformation of a state-space system is a linear change of state variable coordinates:

$\displaystyle \underline{x}(n) \isdef \mathbf{E}\tilde{\underline{x}}(n)
$

where Substituting $ \underline{x}(n) = \mathbf{E}\tilde{\underline{x}}(n)$ gives

\begin{eqnarray*}
\mathbf{E}\tilde{\underline{x}}(n+1) & = & A \, \mathbf{E}\tilde{\underline{x}}(n) + B \underline{u}(n) \nonumber \\
\underline{y}(n) & = & C \mathbf{E}\tilde{\underline{x}}(n) + D\underline{u}(n)
\end{eqnarray*}

Premultiplying the first equation above by $ \mathbf{E}^{-1}$ gives

\begin{eqnarray*}
\tilde{\underline{x}}(n+1) & = &\left(\mathbf{E}^{-1}A \mathbf{E}\right) \tilde{\underline{x}}(n) + \left(\mathbf{E}^{-1}B\right) \underline{u}(n) \nonumber \\
\underline{y}(n) & = & \left(C \mathbf{E}\right) \tilde{\underline{x}}(n) + D\underline{u}(n)
\end{eqnarray*}

Define the transformed system matrices by

\begin{eqnarray*}
\tilde{A}&=& \mathbf{E}^{-1}A \mathbf{E}\nonumber \\
{\tilde B}&=& \mathbf{E}^{-1}B \nonumber \\
{\tilde C}&=& C \mathbf{E}\nonumber \\
{\tilde D}&=& D
\end{eqnarray*}

We can now write

\begin{eqnarray*}
\tilde{\underline{x}}(n+1) & = &\tilde{A}\tilde{\underline{x}}(n) + {\tilde B}\underline{u}(n) \nonumber \\
\underline{y}(n) & = & {\tilde C}\tilde{\underline{x}}(n) + D\underline{u}(n)
\end{eqnarray*}

The transformed system describes the same system in new state-variable coordinates

Let's verify that the transfer function has not changed:

\begin{eqnarray*}
\tilde{H}(z) &=& {\tilde D}+ {\tilde C}(zI - \tilde{A})^{-1}{\tilde B}\\
&=& D + (C \mathbf{E}) \left(zI - \mathbf{E}^{-1}A\mathbf{E}\right)^{-1}(\mathbf{E}^{-1}B)\\
&=& D + C \left[\mathbf{E}\left(zI - \mathbf{E}^{-1}A\mathbf{E}\right)\mathbf{E}^{-1}\right]^{-1} B\\
&=& D + C \left(zI - A\right)^{-1} B = \mathbf{H}(z)
\end{eqnarray*}


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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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