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Path: Dynamics-and-Control-master/0_Getting_Started/Cheatsheet.ipynb

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Kernel: Python 3 (system-wide)

The Jupyter notebook cheat sheet

This document will be available to you during tests and exams

Numeric

Basic plotting functions

Symbolic manipulation

Equation solving

Matrix math

In [1]:

import tbcontrol
tbcontrol.expectversion('0.1.2')

Numeric

In [2]:

import numpy
import scipy

In [3]:

a = numpy.array([1, 2, 3])

In [4]:

t = numpy.linspace(0, 10)

Basic plotting functions

In [5]:

import matplotlib.pyplot as plt
%matplotlib inline

In [6]:

plotfuncs = [plt.plot, 
             plt.stem,
             plt.scatter, 
             plt.semilogx, 
             plt.semilogy, 
             plt.loglog]

for i, func in enumerate(plotfuncs, 1):
    plt.subplot(2, 3, i)
    func([1, 2, 3], [2, 1, 2])
    plt.title(func.__name__)
plt.tight_layout()

Symbolic manipulation

Imports

In [7]:

import sympy
sympy.init_printing()

Symbol definitions

In [8]:

s = sympy.Symbol('s')  # A single symbol
tau, K_c = sympy.symbols('tau K_c', positive=True) # we can use real=True or complex=True for other kinds of variables

Example controller and system

In [9]:

Gc = K_c*((tau*s + 1) / (tau*s))
GvGpGm = 5 / ((10*s + 1)**2)

Working with rational functions and polynomials

We often want nice rational functions, but sympy doesn't make expressions rational by default

In [10]:

chareq = GvGpGm*Gc + 1
chareq

$\displaystyle \frac{5 K_{c} \left(s \tau + 1\right)}{s \tau \left(10 s + 1\right)^{2}} + 1$

The cancel function forces this to be a fraction. collect collects terms.

In [11]:

chareq = chareq.cancel().collect(s)
chareq

$\displaystyle \frac{5 K_{c} + 100 s^{3} \tau + 20 s^{2} \tau + s \left(5 K_{c} \tau + \tau\right)}{100 s^{3} \tau + 20 s^{2} \tau + s \tau}$

In some cases we can factor equations:

In [12]:

chareq.factor(s)

$\displaystyle \frac{5 K_{c} + 100 s^{3} \tau + 20 s^{2} \tau + s \left(5 K_{c} \tau + \tau\right)}{s \tau \left(10 s + 1\right)^{2}}$

Obtain the numerator and denominator:

In [13]:

sympy.numer(chareq), sympy.denom(chareq)

$\displaystyle \left( 5 K_{c} + 100 s^{3} \tau + 20 s^{2} \tau + s \left(5 K_{c} \tau + \tau\right), \ 100 s^{3} \tau + 20 s^{2} \tau + s \tau\right)$

If you want them both, you can use

In [14]:

chareq.as_numer_denom()

$\displaystyle \left( 5 K_{c} + 100 s^{3} \tau + 20 s^{2} \tau + s \left(5 K_{c} \tau + \tau\right), \ 100 s^{3} \tau + 20 s^{2} \tau + s \tau\right)$

Convert to polynomial in s

In [15]:

numer = sympy.poly(sympy.numer(chareq), s)

Once we have a polynomial, it is easy to obtain coefficients:

In [16]:

numer.all_coeffs()

$\displaystyle \left[ 100 \tau, \ 20 \tau, \ 5 K_{c} \tau + \tau, \ 5 K_{c}\right]$

Calculate the Routh Array

In [17]:

from tbcontrol.symbolic import routh

In [18]:

routh(numer)

$\displaystyle \left[\begin{matrix}100 \tau & 5 K_{c} \tau + \tau\\20 \tau & 5 K_{c}\\5 K_{c} \tau - 25 K_{c} + \tau & 0\\5 K_{c} & 0\end{matrix}\right]$

To get a function which can be used numerically, use lambdify:

In [19]:

f = sympy.lambdify((K_c, tau), K_c + tau)

In [20]:

f(1, 2)

$\displaystyle 3$

Functions useful for discrete systems

In [21]:

z, q = sympy.symbols('z, q')

In [22]:

Gz = z**-1/(1 - z**-1)
Gz

$\displaystyle \frac{1}{z \left(1 - \frac{1}{z}\right)}$

Write in terms of positive powers of $z$ :

In [23]:

Gz.cancel()

$\displaystyle \frac{1}{z - 1}$

Write in terms of negative powers of $z$ :

In [24]:

Gz.subs({z: q**-1}).cancel()

$\displaystyle - \frac{q}{q - 1}$

Inversion of the $z$ transform

In [25]:

from tbcontrol.symbolic import sampledvalues

In [26]:

sampledvalues(Gz, z, 10)

$\displaystyle \left[ 0, \ 1, \ 1, \ 1, \ 1, \ 1, \ 1, \ 1, \ 1, \ 1\right]$

Equation solving

Symbolic

In [27]:

x, y, z, a = sympy.symbols('x, y, z, a')
residuals = [x + y - 2, y + z - a, x + y + z]
unknowns = [x, y, z]
sympy.solve(residuals, unknowns)

$\displaystyle \left\{ x : - a, \ y : a + 2, \ z : -2\right\}$

Numeric sympy

In [28]:

residuals = [2*x**2 - 2*y**2, sympy.sin(x) + sympy.log(y)]
unknowns = [x, y]
sympy.nsolve(residuals, unknowns, [1, 3])

$\displaystyle \left[\begin{matrix}-2.21910714891375\\2.21910714891375\end{matrix}\right]$

Numeric

In [29]:

import scipy.optimize

In [30]:

def residuals(unknowns):
    x, y = unknowns
    return [2*x**2 - 2*y**2, numpy.sin(x) + numpy.log(y)]

In [31]:

starting_point = [1, 3]

In [32]:

residuals(starting_point)

$\displaystyle \left[ -16, \ 1.9400832734760063\right]$

In [33]:

scipy.optimize.fsolve(residuals, starting_point)

array([-2.21910715,  2.21910715])

Matrix math

Symbolic

In [34]:

G11, G12, G21, G22 = sympy.symbols('G11, G12, G21, G22')

Creation

In [35]:

G = sympy.Matrix([[G11, G12], [G21, G22]])
G

$\displaystyle \left[\begin{matrix}G_{11} & G_{12}\\G_{21} & G_{22}\end{matrix}\right]$

Determinant, inverse, transpose

In [36]:

G.det(), G.inv(), G.T

$\displaystyle \left( G_{11} G_{22} - G_{12} G_{21}, \ \left[\begin{matrix}\frac{G_{22}}{G_{11} G_{22} - G_{12} G_{21}} & - \frac{G_{12}}{G_{11} G_{22} - G_{12} G_{21}}\\- \frac{G_{21}}{G_{11} G_{22} - G_{12} G_{21}} & \frac{G_{11}}{G_{11} G_{22} - G_{12} G_{21}}\end{matrix}\right], \ \left[\begin{matrix}G_{11} & G_{21}\\G_{12} & G_{22}\end{matrix}\right]\right)$

Math operations: Multiplication, addition, elementwise multiplication:

In [37]:

G*G, G+G, G.multiply_elementwise(G)

$\displaystyle \left( \left[\begin{matrix}G_{11}^{2} + G_{12} G_{21} & G_{11} G_{12} + G_{12} G_{22}\\G_{11} G_{21} + G_{21} G_{22} & G_{12} G_{21} + G_{22}^{2}\end{matrix}\right], \ \left[\begin{matrix}2 G_{11} & 2 G_{12}\\2 G_{21} & 2 G_{22}\end{matrix}\right], \ \left[\begin{matrix}G_{11}^{2} & G_{12}^{2}\\G_{21}^{2} & G_{22}^{2}\end{matrix}\right]\right)$

Numeric

Creation

In [38]:

G = numpy.matrix([[1, 2], [3, 4]])

Determinant, inverse, transpose

In [39]:

numpy.linalg.det(G), G.I, G.T

(-2.0000000000000004,
 matrix([[-2. ,  1. ],
         [ 1.5, -0.5]]),
 matrix([[1, 3],
         [2, 4]]))

Math operations: Multiplication, addition, elementwise multiplication:

In [40]:

G*G, G+G, G.A*G.A

(matrix([[ 7, 10],
         [15, 22]]),
 matrix([[2, 4],
         [6, 8]]),
 array([[ 1,  4],
        [ 9, 16]]))

In [0]:

The Jupyter notebook cheat sheet

Table of Contents

Numeric

Basic plotting functions

Symbolic manipulation

Imports

Working with rational functions and polynomials

Functions useful for discrete systems

Equation solving

Symbolic

Numeric sympy

Numeric

Matrix math

Symbolic

Numeric