root/library/bdm/design/arx2statespace_notes.lyx @ 1469

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\begin_body

\begin_layout Standard
System
\begin_inset Formula 
\[
y_{t}=y_{t-1}+bu_{t}+e_{t}
\]

\end_inset

 is assumed to fullfill causality, i.e.
 
\begin_inset Formula $y_{t-1}\rightarrow u_{t}\rightarrow y_{t}$
\end_inset

 
\end_layout

\begin_layout Standard
with change in notation, i.e.
 
\begin_inset Formula $y_{t-1}\rightarrow u_{t-1}\rightarrow y_{t}$
\end_inset

 the same system is: 
\begin_inset Formula 
\[
y_{t}=y_{t-1}+bu_{t-1}+e_{t}
\]

\end_inset

and can be transformed as
\begin_inset Formula 
\begin{eqnarray*}
x_{t+1} & = & x_{t}+bu_{t}+e_{t}\\
y_{t} & = & x_{t}
\end{eqnarray*}

\end_inset

Morale:
\end_layout

\begin_layout Itemize
for transcription from ARX to State-Space, the state is one time-step shorter
 than the regressor in ARX!
\end_layout

\begin_layout Subsection*
Delays in 
\begin_inset Formula $u$
\end_inset

:
\end_layout

\begin_layout Standard
System
\begin_inset Formula 
\[
y_{t}=y_{t-1}+bu_{t}+cu_{t-1}+e_{t}
\]

\end_inset

is
\begin_inset Formula 
\begin{eqnarray*}
\left[\begin{array}{c}
x_{t+1}\\
u_{t}
\end{array}\right] & =\left[\begin{array}{cc}
1 & c\\
0 & 0
\end{array}\right] & \left[\begin{array}{c}
x_{t}\\
u_{t-1}
\end{array}\right]+\left[\begin{array}{c}
b\\
1
\end{array}\right]u_{t}+e_{t}\\
y_{t} & = & x_{t}
\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
System #2:
\begin_inset Formula 
\[
y_{t}=ay_{t-1}+a_{2}y_{t-2}+bu_{t-1}+cu_{t-2}+e_{t}
\]

\end_inset

is
\begin_inset Formula 
\begin{eqnarray*}
\left[\begin{array}{c}
x_{t+1}\\
x_{t}\\
u_{t}
\end{array}\right] & =\left[\begin{array}{ccc}
a_{1} & a_{2} & c\\
1 & 0 & 0\\
0 & 0 & 0
\end{array}\right] & \left[\begin{array}{c}
x_{t}\\
x_{t-1}\\
u_{t-1}
\end{array}\right]+\left[\begin{array}{c}
b\\
\\
1
\end{array}\right]u_{t}+e_{t}\\
y_{t} & = & x_{t}
\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
Notation in BDM:
\end_layout

\begin_layout Standard
xrv = [y{0}, u{-1}]
\end_layout

\begin_layout Subsection*
Constant term
\end_layout

\begin_layout Standard
System
\begin_inset Formula 
\[
y_{t}=y_{t-1}+bu_{t-1}+g+e_{t}
\]

\end_inset

must be modelled as 
\begin_inset Formula 
\[
\left[\begin{array}{c}
x_{t+1}\\
one
\end{array}\right]=\left[\begin{array}{cc}
1 & c\\
0 & 0
\end{array}\right]\left[\begin{array}{c}
x_{t}\\
one
\end{array}\right]+\left[\begin{array}{c}
b\\
1
\end{array}\right]u_{t}+e_{t}
\]

\end_inset

where one is an extra state variable.
\end_layout

\end_body
\end_document