\( \DeclareMathOperator{\abs}{abs} \newcommand{\ensuremath}[1]{\mbox{$#1$}} \)
| (%i1) |
LieScalar
(
[
l
]
)
:
=
block
(
[
f
,
h
,
x
,
k
,
Lfh
]
,
if ( length ( l ) < 3 ) or ( length ( l ) > 4 ) then error ( "Aufruf mit 3 oder 4 Argumenten." ) , f : l [ 1 ] , h : l [ 2 ] , x : l [ 3 ] , if length ( l ) = 3 then k : 1 else k : l [ 4 ] , if not ( nonnegintegerp ( k ) ) then error ( "Ordnung k muss natürliche Zahl sein." ) , if k = 0 then return ( h ) else Lfh : jacobian ( [ h ] , x ) . f , return ( LieScalar ( f , Lfh , x , k − 1 ) ) ) $ |
| (%i2) |
ObservabilityMap
(
f
,
h
,
x
)
:
=
block
(
[
i
,
n
]
,
n : length ( x ) , makelist ( LieScalar ( f , h , x , i ) , i , 0 , n − 1 ) ) $ |
| (%i3) | ObservabilityMatrix ( f , h , x ) : = jacobian ( ObservabilityMap ( f , h , x ) , x ) $ |
| (%i4) |
LieBracket
(
[
l
]
)
:
=
block
(
[
f
,
g
,
x
,
k
,
Df
,
Dg
,
ad
]
,
if ( length ( l ) < 3 ) or ( length ( l ) > 4 ) then error ( "Aufruf mit 3 oder 4 Argumenten." ) , f : l [ 1 ] , g : l [ 2 ] , x : l [ 3 ] , if length ( l ) = 3 then k : 1 else k : l [ 4 ] , if not ( nonnegintegerp ( k ) ) then error ( "Ordnung k muss natürliche Zahl sein." ) , if k = 0 then return ( g ) else Df : jacobian ( f , x ) , Dg : jacobian ( g , x ) , ad : list_matrix_entries ( Dg . f − Df . g ) , return ( LieBracket ( f , ad , x , k − 1 ) ) ) $ |
| (%i8) |
f
:
[
−
x2
−
x3
,
x1
+
a
·
x2
,
c
+
x3
·
(
x1
−
b
)
]
;
h : log ( x3 ) ; x : [ x1 , x2 , x3 ] $ n : length ( x ) $ |
\[\operatorname{(f) }\left[ -\ensuremath{\mathrm{x3}}-\ensuremath{\mathrm{x2}}\operatorname{,}a\, \ensuremath{\mathrm{x2}}+\ensuremath{\mathrm{x1}}\operatorname{,}\left( \ensuremath{\mathrm{x1}}-b\right) \, \ensuremath{\mathrm{x3}}+c\right] \]
\[\operatorname{(h) }\log{\left( \ensuremath{\mathrm{x3}}\right) }\]
| (%i9) | Q : ratsimp ( ObservabilityMatrix ( f , h , x ) ) ; |
\[\operatorname{(Q) }\begin{pmatrix}0 & 0 & \frac{1}{\ensuremath{\mathrm{x3}}}\\ 1 & 0 & -\frac{c}{{{\ensuremath{\mathrm{x3}}}^{2}}}\\ -\frac{c}{\ensuremath{\mathrm{x3}}} & -1 & -\frac{{{\ensuremath{\mathrm{x3}}}^{3}}+\left( b c-c\, \ensuremath{\mathrm{x1}}\right) \, \ensuremath{\mathrm{x3}}-2 {{c}^{2}}}{{{\ensuremath{\mathrm{x3}}}^{3}}}\end{pmatrix}\]
| (%i11) |
v
:
col
(
invert
(
Q
)
,
n
)
$
v : list_matrix_entries ( v ) ; |
\[\operatorname{(v) }\left[ 0\operatorname{,}-1\operatorname{,}0\right] \]
| (%i14) |
cp
:
a0
+
a1
·
s
+
a2
·
s
^
2
+
s
^
3
$
k1 : sum ( coeff ( cp , s , i ) · LieBracket ( − f , v , x , i ) , i , 0 , n ) $ transpose ( ratsimp ( k1 ) ) ; |
\[\operatorname{ }\begin{pmatrix}-\ensuremath{\mathrm{x3}}+a\, \ensuremath{\mathrm{a2}}+\ensuremath{\mathrm{a1}}+{{a}^{2}}-1\\ \left( 1-{{a}^{2}}\right) \, \ensuremath{\mathrm{a2}}-a\, \ensuremath{\mathrm{a1}}-\ensuremath{\mathrm{a0}}-{{a}^{3}}+2 a\\ \left( \ensuremath{\mathrm{a2}}+a\right) \, \ensuremath{\mathrm{x3}}-c\end{pmatrix}\]
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