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Digitizing State Space Models (Simplistically)

Starting with a continuous-time state-space model

\begin{eqnarray*}
\dot{\underline{x}}(t) & = & \mathbf{A}\, \underline{x}(t) + \mathbf{B}\, \underline{u}(t)\\
\underline{y}(t) &=& \mathbf{C}\, \underline{x}(t) + {\mathbf D}\, \underline{u}(t) \\ [10pt]
\longleftrightarrow\quad
s\underline{X}(s) -\underline{x}(0) & = & \mathbf{A}\, \underline{X}(s) + \mathbf{B}\, \underline{U}(s)\\
\underline{Y}(s) &=& \mathbf{C}\, \underline{X}(s) + {\mathbf D}\, \underline{U}(s)
\end{eqnarray*}

we can, e.g., apply Backward Euler, Trapezoidal Rule (Bilinear Transform), or anything in between:

$\displaystyle \zbox{s = g \frac{1-z^{-1}}{1+\alpha z^{-1}}, \quad \alpha\in[0,1]}
$

to get, letting $ g=(1+\alpha)/T$ and defining $ \underline{x}_n=\underline{x}(nT)$ ,

\begin{eqnarray*}
\frac{\underline{x}_n-\underline{x}_{n-1}}{T} & = & \mathbf{A}\, \left[\frac{\underline{x}_n+\alpha \underline{x}_{n-1}}{1+\alpha}\right]
+ \mathbf{B}\, \left[\frac{\underline{u}_n+\alpha \underline{u}_{n-1}}{1+\alpha}\right]\\ [10pt]
\underline{y}_n &=& \mathbf{C}\, \underline{x}_n + {\mathbf D}\, \underline{u}_n
\end{eqnarray*}

for zero initial conditions $ \underline{x}(0)=\underline{0}\; \Rightarrow$

\begin{eqnarray*}
\underline{x}_{n+1} & = &
\left(I-\mathbf{A}\frac{T}{1+\alpha}\right)^{-1}
\left(I+\mathbf{A}\frac{\alpha T}{1+\alpha}\right) \underline{x}_n\\
&+& \left(I-\mathbf{A}\frac{T}{1+\alpha}\right)^{-1}\mathbf{B}T\, \left(\frac{z + \alpha}{1+\alpha}\right) u_n\\ [10pt]
\end{eqnarray*}

where $ z\,u_n\isdef u_{n+1}$

More sophisticated methods will digitize in a manner that conserves energy and/or momentum


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Download DigitizingNewton.pdf
Download DigitizingNewton_2up.pdf
Download DigitizingNewton_4up.pdf

``Introduction to Physical Signal Models'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2020-06-27 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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