Jupyter Notebook Cheat Sheet

30-04-2021
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Jupyter Notebook Cheat Sheet This Jupyter Notebook cheat sheet will help you to find your way around the well-known Notebook App, a subproject of Project Jupyter. You'll probably know the Jupyter notebooks pretty well - it's one of the most well-known parts of the Jupyter ecosystem! Karlijn Willems Jupyter Notebook Cheat Sheet September 19th, 2017 This Jupyter Notebook cheat sheet will help you to find your way around the well-known Jupyter Notebook.

This document will be available to you during tests and exams

7.1. Table of Contents¶

7.2. Numeric¶

7.3. Basic plotting functions¶

7.4. Symbolic manipulation¶

7.4.1. Imports¶

Symbol definitions

Example controller and system

7.4.2. Working with rational functions and polynomials¶

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

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

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

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

In some cases we can factor equations:

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

Obtain the numerator and denominator:

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

If you want them both, you can use

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

Convert to polynomial in s

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

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

Calculate the Routh Array

$$left[begin{matrix}100 tau & 5 K_{c} tau + tau20 tau & 5 K_{c}- 25 K_{c} + tau left(5 K_{c} + 1right) & 05 K_{c} & 0end{matrix}right]$$

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

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7.4.3. Functions useful for discrete systems¶

Write in terms of positive powers of (z):

Write in terms of negative powers of (z):

Inversion of the (z) transform

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

7.5. Equation solving¶

7.5.1. Symbolic¶

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

7.5.2. Numeric sympy¶

$$left[begin{matrix}-2.219107148913752.21910714891375end{matrix}right]$$

7.5.3. Numeric¶

7.6. Matrix math¶

7.6.1. Symbolic¶

Creation

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

Determinant, inverse, transpose

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

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Math operations: Multiplication, addition, elementwise multiplication:

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

7.6.2. Numeric¶

Creation

Determinant, inverse, transpose

Math operations: Multiplication, addition, elementwise multiplication: