/home/travis/build/casadi/binaries/casadi/docs/tutorials/python/src/integration/temp.py
September 28, 2016
CasADi tutorial
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o p t s [ " r e l t o l " ] = 1 e -6
opts [ " grid " ] = ts
opts [ " o u t p u t _ t 0 " ] = True
s i m = i n t e g r a t o r (" sim ", " c v o d e s ", dae , o p t s )
s o l = s i m ( x0 = [ x0 , y0 ] , p= 0)
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f i g u r e ()
p l o t ( s o l [ : ,0 ] , s o l [ : ,1 ] )
t i t l e ( ’ Van d e r P o l p h a s e s p a c e ’)
x l a b e l ( ’ x ’)
y l a b e l ( ’ y ’)
show ()
s o l = s o l [ ’ x f ’ ] . f u l l ().T
Plot the trajectory
from numpy i m p o r t *
i m p o r t numpy
from c a s a d i i m p o r t *
from p y l a b i m p o r t *
Error:
Van der To
Pol phase
space
/usr/lib/pymodules/python2.7/matplotlib/__init__.py:758: UserWarning: Found matplotlib configuration
conform
with the XDG base directory standard,
2.0 in ~/.matplotlib/.
_get_xdg_config_dir())
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ODE integration
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Let’s construct a simple Van der Pol oscillator.
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u =
x =
y =
ode
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dae
SX. sym (" u ")
SX. sym (" x ")
SX. sym (" y ")
= v e r t c a t ((1- y * y )* x - y + u , x )
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y
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DAE problem formulation as expected by CasADi’s integrators:
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= { ’ x ’ : v e r t c a t ( x , y ), ’ p ’ : u , ’ ode ’ : ode }
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The whole series of sundials options are available for the user
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o p t s = {}
opts [ " f s e n s _ e r r _ c o n " ] = True
opts [ " quad_err_con " ] = True
o p t s [ " a b s t o l " ] = 1 e -6
o p t s [ " r e l t o l " ] = 1 e -6
t e n d = 10
opts [ " t0 " ] = 0
opts [ " t f " ] = tend
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F = i n t e g r a t o r (" F ", " c v o d e s ", dae , o p t s )
p r i n t "%d −> %d " % (F. n _ i n (),F. n _ o u t ())
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Setup the Integrator to integrate from 0 to t=tend, starting at [x0,y0] The output of Integrator is the state at the
end of integration. To obtain the whole trajectory of states, use Simulator:
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t s =numpy . l i n s p a c e (0, t e n d ,100)
x0 = 0; y0 = 1
o p t s = {}
opts [ " f s e n s _ e r r _ c o n " ] = True
opts [ " quad_err_con " ] = True
o p t s [ " a b s t o l " ] = 1 e -6
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Sensitivity for initial conditions
Create the Integrator
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d e f o u t ( dx0 ) :
r e s = F( x0 = [ x0 + dx0 , y0 ] )
r e t u r n r e s [ " x f " ] . f u l l ()
dx0 =numpy . l i n s p a c e (-2,2,100)
o u t = array( [ o u t ( dx ) f o r dx i n dx0 ] ). s q u e e z e ()
d x t e n d = o u t [ : ,0 ] - s o l [ -1,0 ]
f i g u r e ()
p l o t ( dx0 , d x t e n d )
g r i d ()
t i t l e ( ’ I n i t i a l p e r t u r b a t i o n map ’)
x l a b e l ( ’ dx ( 0 ) ’)
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/home/travis/build/casadi/binaries/casadi/docs/tutorials/python/src/integration/temp.py
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September 28, 2016
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y l a b e l ( ’ dx ( t e n d ) ’)
show ()
Initial perturbation map
True sensitivity
Linearised sensitivity
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Initial perturbation map
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dx(tend)
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dx(tend)
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dx(0)
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dx(0)
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#
d i n t e g r a t o r = F. d e r i v a t i v e (1,0)
r e s = d i n t e g r a t o r ( d e r _ x 0 = [ x0 , y0 ] , f w d 0 _ x 0 = [ 1,0 ] )
A = r e s [ " fwd0_xf " ] [ 0 ]
A = float(A) # FIXME
p l o t ( dx0 ,A* dx0 )
l e g e n d (( ’ T r u e s e n s i t i v i t y ’, ’ L i n e a r i s e d
p l o t (0,0, ’ o ’)
show ()
The interpetation is that a small initial circular patch of phase space evolves into ellipsoid patches at later stages.
s e n s i t i v i t y ’))
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def out( t ) :
r e s = d i n t e g r a t o r ( d e r _ x 0 = [ x0 , y0 ] , f w d 0 _ x 0 = [ 1,0 ] )
A= r e s [ " f w d 0 _ x f " ] . f u l l ()
r e s = d i n t e g r a t o r ( d e r _ x 0 = [ x0 , y0 ] , f w d 0 _ x 0 = [ 0,1 ] )
B= r e s [ " f w d 0 _ x f " ] . f u l l ()
r e t u r n array( [ A,B ] ). s q u e e z e ().T
c i r c l e = array( [ [ s i n ( x ), c o s ( x ) ]
f o r x i n numpy . l i n s p a c e (0,2* p i ,100) ] ).T
f i g u r e ()
p l o t ( s o l [ : ,0 ] , s o l [ : ,1 ] )
g r i d ()
f o r i i n range(10) :
J = o u t ( t s [ 10* i ] )
e= 0.1* numpy . d o t ( J , c i r c l e ).T+ s o l [ 10* i , : ]
p l o t ( e [ : ,0 ] , e [ : ,1 ] , c o l o r = ’ r e d ’)
show ()
/home/travis/build/casadi/binaries/casadi/docs/tutorials/python/src/integration/temp.py
September 28, 2016
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f i g u r e ()
p l o t ( uv , o u t )
g r i d ()
t i t l e ( ’ Dependence o f
x l a b e l ( ’ u ’)
y l a b e l ( ’ s t a t e ’)
l e g e n d (( ’ x ’, ’ y ’))
show ()
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Symbolic intergator results
Since Integrator is just another Function, the usual CasADi rules for symbolic evaluation are active.
We create an MX ’w’ that contains the result of a time integration with: - a fixed integration start time , t=0s
- a fixed integration end time, t=10s - a fixed initial condition (1,0) - a free symbolic input, held constant during
integration interval
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u=MX. sym (" u ")
w = F( x0 =MX( [ 1,0 ] ), p=u ) [ " x f " ]
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f = F u n c t i o n ( ’ f ’, [ u ] , [ w ] )
We construct an MXfunction and a python help function ’out’
d e f o u t (u) :
w0 = f ( u )
r e t u r n w0. f u l l ()
o u t (0)
[ [ -2.54394406 ]
[ -0.43934198 ] ]
print
o u t (1)
[ [ -0.25398164 ]
[ 1.3963845 ] ]
Let’s plot the results
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uv =numpy . l i n s p a c e (-1,1,100)
o u t = array( [ o u t ( i ) f o r
i
i n uv ] ). s q u e e z e ()
state
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The figure reveals that perturbations perpendicular to the phase space trajectory shrink.
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print
Dependence of final state on input u
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f i n a l s t a t e on i n p u t u ’)
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