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

Define the transformed system matrices by

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

We can now write

\begin{eqnarray*}
\tilde{\underline{x}}(n+1) & = &\tilde{\mathbf{A}}\tilde{\underline{x}}(n) + \tilde{\mathbf{B}}\underline{u}(n) \nonumber \\
\underline{y}(n) & = & \tilde{\mathbf{C}}\tilde{\underline{x}}(n) + \mathbf{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{\mathbf{H}}}(z) &=& \tilde{\mathbf{D}}+ \tilde{\mathbf{C}}(z\mathbf{I}- \tilde{\mathbf{A}})^{-1}\tilde{\mathbf{B}}\\
&=& \mathbf{D}+ (\mathbf{C}\mathbf{E}) \left(z\mathbf{I}- \mathbf{E}^{-1}\mathbf{A}\mathbf{E}\right)^{-1}(\mathbf{E}^{-1}\mathbf{B})\\
&=& \mathbf{D}+ \mathbf{C}\left[\mathbf{E}\left(z\mathbf{I}- \mathbf{E}^{-1}\mathbf{A}\mathbf{E}\right)\mathbf{E}^{-1}\right]^{-1} \mathbf{B}\\
&=& \mathbf{D}+ \mathbf{C}\left(z\mathbf{I}- \mathbf{A}\right)^{-1} \mathbf{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 © 2015-04-18 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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