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Homogeneous Artificial Neural Networks
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POLYAKOV Andrey
Homogeneous Artificial Neural Networks
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d2ebd26d
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d2ebd26d
authored
1 year ago
by
POLYAKOV Andrey
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Demos for MATLAB/Example_2_dyn_system/hnorm.m
+76
-0
76 additions, 0 deletions
Demos for MATLAB/Example_2_dyn_system/hnorm.m
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0 deletions
Demos for MATLAB/Example_2_dyn_system/hnorm.m
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0
View file @
d2ebd26d
function
q
=
hnorm
(
x
,
Gd
,
P
,
tol
,
Nmax
)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% The function nx=hnorm(x, Gd, P) computes (for a given vector x)
%% a canonical d-homogeneous norm nx induced by the weighted Euclidean norm
%% nxp=sqrt(x'*P*x), namely:
%%
%% nx>0 such that x'*d'(-log(nx))P*d(-log(nx))*x)=1 (if x is non-zero)
%%
%% *** The coumpation is based on the bisection method ***
%%
%% The input parameters are
%% x - vector (p x 1) which norm has to be computed
%% Gd - anti-Hurwitx matrix (p x p) being the generator of
%% the dilation d(s)=expm(s*Gd) for any real s
%% P - positive definite matrix (p x p) such that P*Gd+Gd'*P>0
%%
%%
%%
%% Remark:
%% The function hnorm is not optimized for a real-time compuation of
%% the homogeneous norm in practice. To design an fast algorithm for
%% practical application of a homogeneous control please do contanct
%% andrey.polyakov@inria.fr (HPC_toolbox developer)
%%
%%
%%
%% The function q=hnorm(x, Gd, P, tol) uses
%% additional parameter tol to define computational tolerance and
%% an approximation precision (by default tol=1e-6);
%%
%%
%%
%% The function q=hnorm(x, Gd, P, tol, Nmax) uses
%% additional parameter Nmax to define the maximum number of iterations of
%% the numerical method (Nmax=20, by default)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
if
nargin
<
4
tol
=
1e-6
;
end
;
if
nargin
<
5
Nmax
=
20
;
end
;
if
norm
(
x
)
>
tol
a
=-
1
;
q
=
0
;
y
=
expm
(
-
Gd
*
a
)
*
x
;
while
(
y
'*
P
*
y
<
1
)
&&
(
a
>-
746
)
a
=
a
*
2
;
y
=
expm
(
-
Gd
*
a
)
*
x
;
end
;
if
y
'*
P
*
y
>
1
b
=
1
;
y
=
expm
(
-
Gd
*
b
)
*
x
;
while
(
y
'*
P
*
y
>
1
)
&&
(
b
<
710
)
b
=
b
*
2
;
y
=
expm
(
-
Gd
*
b
)
*
x
;
end
;
if
y
'*
P
*
y
<
1
c
=
(
a
+
b
)/
2
;
y
=
expm
(
-
Gd
*
c
)
*
x
;
Qf
=
y
'*
P
*
y
-
1
;
i
=
0
;
while
(
abs
(
Qf
)
>
tol
)
&&
(
i
<
Nmax
)
i
=
i
+
1
;
if
Qf
>
0
a
=
c
;
else
b
=
c
;
end
;
c
=
(
a
+
b
)/
2
;
y
=
expm
(
-
Gd
*
c
)
*
x
;
Qf
=
y
'*
P
*
y
-
1
;
end
;
else
c
=
b
;
end
;
else
c
=
a
;
end
;
q
=
exp
(
c
);
else
q
=
0
;
end
;
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