Next  |  Prev  |  Top  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

State Space Modal Representation

Diagonal state transition matrix = modal representation:

\begin{eqnarray*}
\left[\begin{array}{c} x_1(n+1) \\ [2pt] x_2(n+1) \\ [2pt] \vdots \\ [2pt] x_{N-1}(n+1)\\ [2pt] x_N(n+1)\end{array}\right] & = &
\left[
\begin{array}{ccccc}
\lambda _1 & 0 & 0 & \cdots & 0 \\ [2pt]
0 & \lambda _2 & 0 & \cdots & 0 \\ [2pt]
\vdots & \vdots & \ddots & \vdots & \vdots \\ [2pt]
0 & 0 & 0 & \lambda _{N-1} & 0 \\ [2pt]
0 & 0 & 0 & 0 & \lambda _N \\ [2pt]
\end{array}\right]
\left[\begin{array}{c} x_1(n) \\ [2pt] x_2(n) \\ [2pt] \vdots \\ [2pt] x_{N-1}(n)\\ [2pt] x_N(n)\end{array}\right]
+
\left[\begin{array}{c} b_1 \\ [2pt] b_2 \\ [2pt] \vdots \\ [2pt] b_{N-1}\\ [2pt] b_N\end{array}\right] u(n)\\ [10pt]
y(n) & = & C \underline{x}(n) + Du(n)
\end{eqnarray*}

The $ N$ complex modes are decoupled:

\begin{eqnarray*}
x_1(n+1) &=& \lambda _1 x_1(n) + b_1 u(n)\\
x_2(n+1) &=& \lambda _2 x_2(n) + b_2 u(n)\\
&\vdots& \\
x_N(n+1) &=& \lambda _N x_N(n) + b_N u(n)\\
y(n) & = & c_1 x_1(n) + c_2 x_2(n) + \dots + c_N x_N(n) + D u(n)
\end{eqnarray*}

That is, diagonal state-space system consists of $ N$ parallel one-pole systems:

\begin{eqnarray*}
H(z) &=& C(zI-A)^{-1}B+D\\ [10pt]
&=& \zbox{D + \sum_{i=1}^N \frac{c_i b_iz^{-1}}{1-\lambda _iz^{-1}}}
\end{eqnarray*}


Next  |  Prev  |  Top  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

Download StateSpace.pdf
Download StateSpace_2up.pdf
Download StateSpace_4up.pdf

``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
CCRMA  [Automatic-links disclaimer]