General concepts

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):

$C = \{y, y^(n),.., y^{(n-1)}; u^{(k)}, k \geq 0 \}$



In [1]:

    
from sympy.abc import*
from sympy import *

t = Symbol('t')
x = Function('x')(t)
x









    Out[1]:





x(t)

Vector space of one forms:

$dC$ is the set of one forms of the symbols in $C$, example: $dC = \{dx, dy, dz, da, ... \}$
$\xi$ is a vector space of one-forms (the differential vector space spanned over the field $\kappa$ by the elements of $dC$)
$\kappa$ is the field of meromorphic functions (Quotient fild)
$\xi = span_ {\kappa} \{dC \} $

Example of a time dependent symbols in python:



In [4]:

    
#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









    Out[5]:





$$\left[\begin{matrix}0\end{matrix}\right]$$

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}

\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 $\beta _k \in \kappa$

Operators $d$ and $d/dt$ in $\xi$:

Closed and exact forms

A one form $\omega$ is closed if $d\omega = 0$
A one form $\omega$ is exact if $\omega = d \varphi$ for some $\varphi \in \kappa$.

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.

How to check is a one-form is exact (integrable)

exact ($\omega$):

if ($d\omega == 0$):

then $\omega$ is closed, and find a function $\varphi$ such that $\omega = d \varphi$.

elif: $d\omega \wedge \omega == 0$,

The vector space $span_\kappa \{\omega\}$ is integrable. An integrating factor $\lambda$ exists, such that $\lambda \omega = d\varphi$. Find $\lambda$ .

else

$\omega$ is no exact

Forbenius theorem

Gives a condition for complete integrability
Given $\omega \in \xi$, there exists a function $\varphi$ such that $span_\kappa \{\omega \} = span_ \kappa \{d \varphi \}$ if and only if:

$d \omega \wedge \omega$ = 0

(Generalization) A subspace $\Omega \in \xi$ is closed (integrable) if $\Omega $ has a basis which consists only of closed forms.

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

Pfaff-Darboux Theorem, rank of a one-form

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









    Out[1]:





$$\left[\begin{matrix}3 dx_{2} dx_{3} + 3 dx_{3} dx_{2}\end{matrix}\right]$$



In [2]:

    
expand(dw*dw*w)









    Out[2]:





$$\left[\begin{matrix}27 \operatorname{x_{2}}{\left (t \right )} dx_{2} dx_{3} dx_{2} dx_{3}^{2} + 27 \operatorname{x_{2}}{\left (t \right )} dx_{2} dx_{3}^{2} dx_{2} dx_{3} + 27 \operatorname{x_{2}}{\left (t \right )} dx_{3} dx_{2} dx_{3} dx_{2} dx_{3} + 27 \operatorname{x_{2}}{\left (t \right )} dx_{3} dx_{2}^{2} dx_{3}^{2} + 27 \operatorname{x_{3}}{\left (t \right )} dx_{2} dx_{3} dx_{2} dx_{3} dx_{2} + 27 \operatorname{x_{3}}{\left (t \right )} dx_{2} dx_{3}^{2} dx_{2}^{2} + 27 \operatorname{x_{3}}{\left (t \right )} dx_{3} dx_{2} dx_{3} dx_{2}^{2} + 27 \operatorname{x_{3}}{\left (t \right )} dx_{3} dx_{2}^{2} dx_{3} dx_{2} + 9 dx_{2} dx_{3} dx_{2} dx_{3} dx_{1} + 9 dx_{2} dx_{3}^{2} dx_{2} dx_{1} + 9 dx_{3} dx_{2} dx_{3} dx_{2} dx_{1} + 9 dx_{3} dx_{2}^{2} dx_{3} dx_{1}\end{matrix}\right]$$

Examen



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









    Out[3]:





$$\left[\begin{matrix}0\\3 dx_{2} dx_{3}\\3 dx_{3} dx_{2}\end{matrix}\right]$$



In [ ]: