Control Engineering II, Week 5 - Note 1

Symbolic calculation and Laplace transform with python

Symbolic Calculation

Install the sympy library

!pip install sympy

Requirement already satisfied: sympy in /usr/local/lib/python3.10/dist-packages (1.12)
Requirement already satisfied: mpmath>=0.19 in /usr/local/lib/python3.10/dist-packages (from sympy) (1.3.0)

sympy library import

import sympy as sym

Call init_printing to display equations in LaTeX format

sym.init_printing(use_latex='mathjax')

Declaring symbolic variables and expressing polynomials

x, y, z = sym.symbols('x y z')
( 5*x**3 - x**2 + 3*x - 1 ) + ( -x**3 + 2*x**2 + 5 )

   3    2          
4⋅x  + x  + 3⋅x + 4

Storing a polynomial in a variable

A = -3*x*y + x**2 + 2*y**2 + x
B = ( x**2 + x - 2 ) * ( 3*x - 1 )
A, B

⎛ 2                  2            ⎛ 2        ⎞⎞
⎝x  - 3⋅x⋅y + x + 2⋅y , (3⋅x - 1)⋅⎝x  + x - 2⎠⎠

Collecting a polynomial in the symbol ’x’

sym.collect( A, x )

 2                    2
x  + x⋅(1 - 3⋅y) + 2⋅y 

Expanding a polynomial

sym.expand(B)

   3      2          
3⋅x  + 2⋅x  - 7⋅x + 2

Entering a factorable polynomial as C

C = 2*x**2 + 5*x*y + 3*y**2 + 3*x + 5*y - 2
C

   2                    2          
2⋅x  + 5⋅x⋅y + 3⋅x + 3⋅y  + 5⋅y - 2

Factoring polynomial C

sym.factor(C)

(x + y + 2)⋅(2⋅x + 3⋅y - 1)

Entering a rational polynomial expression as D

D = ( x**2 - 2*x + 1 ) / ( x**2 - 3*x + 2 )
D

 2          
x  - 2⋅x + 1
────────────
 2          
x  - 3⋅x + 2

Factoring the denominator and numerator, then canceling common factors

sym.cancel(D)

x - 1
─────
x - 2

Entering a sum of rational polynomial expressions as E

E = 1/(x-1) + 2/(x+2)
E

  2       1  
───── + ─────
x + 2   x - 1

Combining fractions and canceling common terms

sym.cancel(E)

   3⋅x    
──────────
 2        
x  + x - 2

The underscore character ’_’ >> denotes the previous output

sym.factor(_)

      3⋅x      
───────────────
(x - 1)⋅(x + 2)

Entering a very complex polynomial expression as F

F = ( x - 1 + 1/(x-1) ) / ( x - 1 - 1/(x-1) )
F

          1  
x - 1 + ─────
        x - 1
─────────────
          1  
x - 1 - ─────
        x - 1

Simplifying a complex polynomial as far as possible

sym.simplify(F)

 2          
x  - 2⋅x + 2
────────────
 x⋅(x - 2)  

Laplace Transform

Declare t and s as symbols for the Laplace transform, and define f

t, s = sym.symbols('t, s')
a = sym.symbols('a', real=True, positive=True)
f = sym.exp(-a*t)
f

exp(-a*t)

Taking the Laplace transform

F = sym.laplace_transform(f, t, s)
F

⎛  1            ⎞
⎜─────, -a, True⎟
⎝a + s          ⎠

Suppressing the conditions for existence of the Laplace transform

F = sym.laplace_transform(f, t, s, noconds=True)
F

  1  
─────
a + s

Defining a function to simplify Laplace-transform calculations

def L(f):
    return sym.laplace_transform(f, t, s, noconds=True)

def invL(F):
    return sym.inverse_laplace_transform(F, s, t)

L(f)

  1  
─────
a + s

invL(F)

 -a⋅t     
ℯ    ⋅θ(t)

$\theta(t)$ denotes the unit step function

sym.Heaviside(t)

θ(t)

sym.plot(sym.Heaviside(t), xlim=(-1, 3), ylim=(0, 1.5))

<Figure size 640x480 with 1 Axes>

<sympy.plotting.plot.Plot at 0x7b32d9367550>

Substituting a specific number for a variable in a function

invL(F).subs({a: 2})

 -2⋅t     
ℯ    ⋅θ(t)

Taking Laplace transforms of various functions

omega = sym.symbols('omega', real=True)
theta = sym.Function('theta')(t)
exp = sym.exp
sin = sym.sin
cos = sym.cos
functions = [1,
         t,
         exp(-a*t),
         t*exp(-a*t),
         t**2*exp(-a*t),
         sin(omega*t),
         cos(omega*t),
         1 - exp(-a*t),
         exp(-a*t)*sin(omega*t),
         exp(-a*t)*cos(omega*t),
         sym.diff(theta,t)
         ]
functions

[1,
 t,
 exp(-a*t),
 t*exp(-a*t),
 t**2*exp(-a*t),
 sin(omega*t),
 cos(omega*t),
 1 - exp(-a*t),
 exp(-a*t)*sin(omega*t),
 exp(-a*t)*cos(omega*t),
 Derivative(theta(t), t)]

Taking inverse Laplace transforms of various functions

L(sym.diff(theta,t)).subs({theta.subs({t:0}):0})

s*LaplaceTransform(theta(t), t, s)

Fs = [L(f) for f in functions]
Fs

[1/s,
 s**(-2),
 1/(a + s),
 (a + s)**(-2),
 2/(a + s)**3,
 omega/(omega**2 + s**2),
 s/(omega**2 + s**2),
 -1/(a + s) + 1/s,
 omega/(omega**2 + (a + s)**2),
 (a + s)/(omega**2 + (a + s)**2),
 s*LaplaceTransform(theta(t), t, s) - theta(0)]