7. The Jupyter notebook cheat sheet

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7.1. Table of Contents

Numeric

Basic plotting functions

Symbolic manipulation

Equation solving

Matrix math

import tbcontrol
tbcontrol.expectversion('0.1.2')

7.2. Numeric

import numpy
import scipy

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

t = numpy.linspace(0, 10)

7.3. Basic plotting functions

import matplotlib.pyplot as plt
%matplotlib inline

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()

7.4. Symbolic manipulation

7.4.1. Imports

import sympy
sympy.init_printing()

Symbol definitions

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

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

7.4.2. Working with rational functions and polynomials

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

chareq = GvGpGm*Gc + 1
chareq

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

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

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

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

In some cases we can factor equations:

chareq.factor(s)

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

Obtain the numerator and denominator:

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

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

If you want them both, you can use

chareq.as_numer_denom()

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

Convert to polynomial in s

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

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

numer.all_coeffs()

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

Calculate the Routh Array

from tbcontrol.symbolic import routh

routh(numer)

$$\left[\begin{matrix}100 \tau & 5 K_{c} \tau + \tau\\20 \tau & 5 K_{c}\\- 25 K_{c} + \tau \left(5 K_{c} + 1\right) & 0\\5 K_{c} & 0\end{matrix}\right]$$

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

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

f(1, 2)

$$3$$

7.4.3. Functions useful for discrete systems

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

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

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

Write in terms of positive powers of \(z\):

Gz.cancel()

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

Write in terms of negative powers of \(z\):

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

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

Inversion of the \(z\) transform

from tbcontrol.symbolic import sampledvalues

sampledvalues(Gz, z, 10)

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

7.5. Equation solving

7.5.1. Symbolic

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)

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

7.5.2. Numeric sympy

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

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

7.5.3. Numeric

import scipy.optimize

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

starting_point = [1, 3]

residuals(starting_point)

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

scipy.optimize.fsolve(residuals, starting_point)

array([-2.21910715,  2.21910715])

7.6. Matrix math

7.6.1. Symbolic

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

Creation

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

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

Determinant, inverse, transpose

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

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

Math operations: Multiplication, addition, elementwise multiplication:

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

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

7.6.2. Numeric

Creation

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

Determinant, inverse, transpose

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:

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

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