Infinite set of independent variables:
The infinite set of independet variables or symbols can be represented as:
$C = \{a,b, ..., x, y, ... a_i, b_i ,... \}$
$C_{ss}$ is associated to the state-space form (finite set):
$C = \{x_i, i = 1, ..., n; u^{(k)}, k \geq 0 \}$
$C_{io}$ is associated to the input-output description (finite set):
$C = \{y, y^(n),.., y^{(n-1)}; u^{(k)}, k \geq 0 \}$
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from sympy.abc import*
from sympy import *
t = Symbol('t')
x = Function('x')(t)
x
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Vector space of one forms:
Example of a time dependent symbols in python:
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#Create a set of one-forms
from tools import*
from nlcontrol import*
var = ['x','y','z']
x,y,z = def_vars(var,True)
dx,dy,dz = def_1forms(var)
a = sin(x+y)*dx
b = (x**2 + 2*y)*dy
c = z*dz
w = a + b + c
w = 2*x*dx*dy + 2*y*dy*dz
In [5]:
r = ediff(w,var)
#Existe error en r[1], es aun a test, corregir en l el futuro por mientras no usar
r
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Any element of $\xi$ has the next form for state space representation
\begin{equation} \omega = \sum_{i = 1}^{n} {\alpha _i dx_i} + \sum_{k \geq 0} {\beta _k du^{(k)}} \end{equation}or
\begin{equation} \omega = \sum_{i = 1}^{n-1} {\alpha _i dy^{(i)}} + \sum_{k \geq 0} {\beta _k du^{(k)}} \end{equation}for i/o representation. Where $\alpha_i$ and $\beta _k \in \kappa$
The exacteness can be locally o globally.
Poincaré's Lemma: Let $\omega$ be a closed one form, then exist a function $\varphi \in \kappa$ such that locally $\omega = d\varphi$
closure = local exactness
Any exact one-form is closed. But a closed one-form can be not globally exact.
One-form colinear to an exact form:
If $\omega$ is colinear to an exact form, then there exist functions $\lambda$ and $\varphi$ in $\kappa$ such that $\lambda \omega = d\varphi$. This is equivalent to say that $span_\kappa \{\omega\} = span_\kappa \{d\varphi\}$
The function $\lambda$ is called an integrating factor.
exact ($\omega$):
if ($d\omega == 0$):
elif: $d\omega \wedge \omega == 0$,
else
$d \omega \wedge \omega$ = 0
Let $\Omega = span _{\kappa} \{\omega_1, ...,\omega_n \}$ , $\Omega$ is integrable if:
$d \omega _i \wedge \omega _1, ..., \omega _r = 0$ for any $i = 1,.., r$
Ojo: para cada i
The rank $r$ of the one-form $\omega$ is defined by
$(d\omega )^r \wedge \omega \neq 0$ and $(d\omega )^{r+1} \wedge \omega = 0$
Use to search of the minimal number of coordinates needed to express some one-form.
r+1 represent the smallest number of exact-one form $d\varphi _1,...d\varphi _r $ such that $\omega \in span\{ d\varphi _1,...d\varphi _r\}$
In [1]:
from tools import*
from nlcontrol import*
var = ['x_1','x_2','x_3']
x1,x2,x3 = def_vars(var,True)
dx1,dx2,dx3 = def_1forms(var)
w = dx1 + 3*x3*dx2 + 3*x2*dx3
dw = ediff(w,var)
dw
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In [2]:
expand(dw*dw*w)
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In [3]:
a = dx1
b = 3*x3*dx2
c = 3*x2*dx3
w = [a,b,c]
dw = ediff(w,var)
#Suma estos valores y es el resultado
dw
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