Jupyter Notebook Cheatsheet

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7.1. Table of Contents¶

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Holoviews cheatsheet. In 7: import pandas as pd import numpy as np import holoviews as hv from holoviews import opts from holoviews import streams import param hv. Extension ('bokeh', 'matplotlib') # Set dark theme from bokeh.themes import builtinthemes hv. Renderer ('bokeh'). Theme = builtin.

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¶

Jupyter

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:

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

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

Markdown

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

$$left [ 100 tau, quad 20 tau, quad 5 K_{c} tau + tau, quad 5 K_{c}right ]$$
Jupyter notebook markdown cheat sheet

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:

Jupiter Keyboard Shortcut Cheat Sheet

7.4.3. Functions useful for discrete systems¶

Write in terms of positive powers of (z):

Jupyter

Write in terms of negative powers of (z):

Inversion of the (z) transform

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

Jupyter Notebook Cheat Sheet

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¶

Jupyter Notebook Markdown Cheatsheet

Creation

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

Determinant, inverse, transpose

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

Math operations: Multiplication, addition, elementwise multiplication:

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

7.6.2. Numeric¶

Creation

Determinant, inverse, transpose

Math operations: Multiplication, addition, elementwise multiplication: