In [13]:

    

#greek symbols (must be imported first)
from sympy.abc import*
from sympy import *
from sympy import MatrixSymbol, Identity
init_printing(use_latex=True) #For good look equations
x1 = Symbol('x_1')
x2 = Symbol('x_2')


    

In [14]:

    

A1 = [x1**3, 0]
A2 = [sin (x1), x2]


    

In [15]:

    

A = Matrix((A1,A2))


    

In [16]:

    

A


    

    
Out[16]:





$$\left[\begin{matrix}x_{1}^{3} & 0\\\sin{\left (x_{1} \right )} & x_{2}\end{matrix}\right]$$

In [17]:

    

A**-1


    

    
Out[17]:





$$\left[\begin{matrix}\frac{1}{x_{1}^{3}} & 0\\- \frac{\sin{\left (x_{1} \right )}}{x_{1}^{3} x_{2}} & \frac{1}{x_{2}}\end{matrix}\right]$$

In [32]:

    

A1 = x1**3
A2 = sin (x1)+  x2
X = Matrix([A1,A2])
X


    

    
Out[32]:





$$\left[\begin{matrix}x_{1}^{3}\\x_{2} + \sin{\left (x_{1} \right )}\end{matrix}\right]$$

In [34]:

    

Y = Matrix([x1,x2])
J = X.jacobian(Y)
J


    

    
Out[34]:





$$\left[\begin{matrix}3 x_{1}^{2} & 0\\\cos{\left (x_{1} \right )} & 1\end{matrix}\right]$$

In [35]:

    

J**-1


    

    
Out[35]:





$$\left[\begin{matrix}\frac{1}{3 x_{1}^{2}} & 0\\- \frac{\cos{\left (x_{1} \right )}}{3 x_{1}^{2}} & 1\end{matrix}\right]$$

In [40]:

    

y = Symbol('y')
function = y


    

In [41]:

    

#(diferencial dy, desde 0, a y)
integrate(function, (y,0,y))


    

    
Out[41]:





$$\frac{y^{2}}{2}$$

In [51]:

    

f, g = symbols('f g', cls=Function)


    

In [52]:

    

f(x).diff(x)


    

    
Out[52]:





$$\frac{d}{d x} f{\left (x \right )}$$

In [53]:

    

cos(f(t)).diff(t,t)


    

    
Out[53]:





$$- \sin{\left (f{\left (t \right )} \right )} \frac{d^{2}}{d t^{2}}  f{\left (t \right )} + \cos{\left (f{\left (t \right )} \right )} \left(\frac{d}{d t} f{\left (t \right )}\right)^{2}$$

In [135]:

    

y1d,y1dd,y1ddd = symbols('\dot{y}_1 \ddot{y}_1 \dddot{y}_1 ',cls=Function)
y2d,y3dd,y3ddd = symbols('\dot{y}_2 \ddot{y}_2 \dddot{y}_2 ',cls=Function)
x1,x2,x3,x4,x5,x6 = symbols('x_1 x_2 x_3 x_4 x_5 x_6',cls=Function)
v1,v2,v3,v4 = symbols('v_1 v_2 v_3 v_4',cls=Function)
z1,z2,z3,z4 = symbols('z_1 z_2 z_3 z_4',cls=Function)


    

In [136]:

    

u1= y1d(t)*tan(x3(t))
u1


    

    
Out[136]:





$$\dot{y}_{1}{\left (t \right )} \tan{\left (\operatorname{x_{3}}{\left (t \right )} \right )}$$

In [137]:

    

simplify(u1.diff(t))


    

    
Out[137]:





$$\frac{\dot{y}_{1}{\left (t \right )} \frac{d}{d t} \operatorname{x_{3}}{\left (t \right )}}{\cos^{2}{\left (\operatorname{x_{3}}{\left (t \right )} \right )}} + \tan{\left (\operatorname{x_{3}}{\left (t \right )} \right )} \frac{d}{d t} \dot{y}_{1}{\left (t \right )}$$

In [138]:

    

bx = Matrix(([0],[v1(t)*tan(x3(t))]))
bx


    

    
Out[138]:





$$\left[\begin{matrix}0\\\operatorname{v_{1}}{\left (t \right )} \tan{\left (\operatorname{x_{3}}{\left (t \right )} \right )}\end{matrix}\right]$$

In [142]:

    

Dx = Matrix(([cos(x3(t)), 0],[0, z1(t)/cos(x3(t))**2]))
Dx


    

    
Out[142]:





$$\left[\begin{matrix}\cos{\left (\operatorname{x_{3}}{\left (t \right )} \right )} & 0\\0 & \frac{\operatorname{z_{1}}{\left (t \right )}}{\cos^{2}{\left (\operatorname{x_{3}}{\left (t \right )} \right )}}\end{matrix}\right]$$

In [174]:

    

V = Matrix(([z1(t)],[v2(t)]))
V


    

    
Out[174]:





$$\left[\begin{matrix}\operatorname{z_{1}}{\left (t \right )}\\\operatorname{v_{2}}{\left (t \right )}\end{matrix}\right]$$

In [177]:

    

def sfeed(Dx, bx, V):
    betax = Dx**-1*V
    alfax = Dx**-1*bx
    return simplify(alfax + betax)


    

In [180]:

    

sfeed(Dx, bx, V)


    

    
Out[180]:





$$\left[\begin{matrix}\frac{\operatorname{z_{1}}{\left (t \right )}}{\cos{\left (\operatorname{x_{3}}{\left (t \right )} \right )}}\\\frac{\cos^{2}{\left (\operatorname{x_{3}}{\left (t \right )} \right )}}{\operatorname{z_{1}}{\left (t \right )}} \left(\operatorname{v_{1}}{\left (t \right )} \tan{\left (\operatorname{x_{3}}{\left (t \right )} \right )} + \operatorname{v_{2}}{\left (t \right )}\right)\end{matrix}\right]$$

In [179]:

    




    

    
Out[179]:





\dot{y}_1

In [ ]: