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# CFSpy
CFSpy is a package to simulate the output of a control system by means of the Chen-Fliess series.
It provides:
- The list of iterated integrals indexed by words of a certain length or less.
- The list of Lie derivatives indexed by words of a certain length or less.
- A single iterated integral indexed by a given word.
- A single Lie derivative indexed by a given word.
## Overview
CFSpy is a Python library that contains the following functions:
| Function | Description |
| ---- | --- |
| [**iter_int**](https://github.com/iperezav/CFSpy/blob/main/build/lib/CFS/iter_int.py) | A function for the numerical computation of a list of iterated integrals |
| [**iter_lie**](https://github.com/iperezav/CFSpy/blob/main/build/lib/CFS/iter_lie.py) | A function for the analytical computation of a list of Lie derivatives |
| [**single_iter_int**](https://github.com/iperezav/CFSpy/blob/main/build/lib/CFS/single_iter_int.py) | A function for the numerical computation of a single iterated integral |
| [**single_iter_lie**](https://github.com/iperezav/CFSpy/blob/main/build/lib/CFS/single_iter_lie.py) | A function for the analytical computation of a single Lie derivative |
CFSpy is used for:
- Simulation of the output of a control systems.
- Reachability analysis of a control system.
# Installation
Currently, `CFSpy` supports releases of Python 3.12.4 onwards.
To install the current release:
```shell
$ pip install --upgrade CFSpy
```
# Getting Started
## Minimal Example
```python
from CFS import iter_int, iter_lie, single_iter_int, single_iter_lie
import numpy as np
from scipy.integrate import solve_ivp
import matplotlib.pyplot as plt
import sympy as sp
# Define the Lotka-Volterra system
def system(t, x, u1_func, u2_func):
x1, x2 = x
u1 = u1_func(t)
u2 = u2_func(t)
dx1 = -x1*x2 + x1 * u1
dx2 = x1*x2 - x2* u2
return [dx1, dx2]
# Input 1
def u1_func(t):
return np.sin(t)
# Input 2
def u2_func(t):
return np.cos(t)
# Initial condition
x0 = [1/3,2/3]
# Time range
t0 = 0
tf = 3
dt = 0.001
t_span = (t0, tf)
# Simulation of the system
solution = solve_ivp(system, t_span, x0, args=(u1_func, u2_func), dense_output=True)
# Partition of the time interval
t = np.linspace(t_span[0], t_span[1], int((tf-t0)//dt+1))
y = solution.sol(t)
# Define the symbolic variables
x1, x2 = sp.symbols('x1 x2')
x = sp.Matrix([x1, x2])
# Define the system symbolically
g = sp.transpose(sp.Matrix([[-x1*x2, x1*x2], [x1, 0], [0, - x2]]))
# Define the output symbolically
h = x1
# The truncation of the length of the words that index the Chen-Fliess series
Ntrunc = 4
# Coefficients of the Chen-Fliess series evaluated at the initial state
Ceta = np.array(iter_lie(h,g,x,Ntrunc).subs([(x[0], 1/3),(x[1], 2/3)]))
# inputs as arrays
u1 = np.sin(t)
u2 = np.cos(t)
# input array
u = np.vstack([u1, u2])
# List of iterated integral
Eu = iter_int(u,t0, tf, dt, Ntrunc)
# Chen-Fliess series
F_cu = x0[0]+np.sum(Ceta*Eu, axis = 0)
# Graph of the output and the Chen-Fliess series
plt.figure(figsize = (12,5))
plt.plot(t, y[0].T)
plt.plot(t, F_cu, color='red', linewidth=5, linestyle = '--', alpha = 0.5)
plt.xlabel('$t$')
plt.ylabel('$x_1$')
plt.legend(['Output of the system','Chen-Fliess series'])
plt.grid()
plt.show()
```
<img src="https://raw.githubusercontent.com/iperezav/CFSpy/main/examples/output_chenfliess.png" alt="iter_int(), iter_lie()" />
For more examples, see the [CFSpy demos](https://github.com/iperezav/CFSpy/blob/main/examples/)
# Resources
- [**PyPi**](https://pypi.org/project/CFSpy/)
- [**Documentation**](https://github.com/iperezav/CFSpy/blob/main/README.md)
- [**Issue tracking**](https://github.com/iperezav/CFSpy/issues)
# Contributing
All feedback is welcome.
# Asking for help
Please reach out if you have any questions:
1. [Github CFSpy discussions](https://github.com/iperezav/CFSpy/discussions/).
2. [Github CFSpy issues](https://github.com/iperezav/CFSpy/issues).
# License
CFSpy is open-source and released under the [MIT License](LICENSE).
# BibTeX
Feel free to cite my work:
```bibtex
@article{iperezave,
title={CFSpy},
author={Perez Avellaneda, Ivan},
journal={GitHub. Note: https://github.com/iperezav/CFSpy},
volume={1},
year={2024}
}
```
[issues]: https://github.com/iperezav/CFSpy/issues
[demos]: https://github.com/iperezav/CFSpy/blob/main/examples/
[downloads]: https://pepy.tech/projects/cfspy
[py-versions]: https://pypi.org/project/cfspy/
[pypi-latest-version]: https://pypi.org/project/cfspy/
[license]: https://github.com/iperezav/CFSpy/blob/main/LICENSE
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"description": "<div align=\"center\">\r\n<img src=\"https://raw.githubusercontent.com/iperezav/CFSpy/main/logo/CFSpy_logo.png\" alt=\"CFSpy\" / >\r\n\r\n---\r\n\r\n[][py-versions]\r\n[][pypi-latest-version]\r\n\r\n[][downloads]\r\n[][license]\r\n\r\n</div>\r\n\r\n# CFSpy\r\n\r\nCFSpy is a package to simulate the output of a control system by means of the Chen-Fliess series.\r\n\r\nIt provides:\r\n\r\n- The list of iterated integrals indexed by words of a certain length or less. \r\n- The list of Lie derivatives indexed by words of a certain length or less.\r\n- A single iterated integral indexed by a given word.\r\n- A single Lie derivative indexed by a given word.\r\n\r\n\r\n## Overview\r\n\r\nCFSpy is a Python library that contains the following functions:\r\n\r\n| Function | Description |\r\n| ---- | --- |\r\n| [**iter_int**](https://github.com/iperezav/CFSpy/blob/main/build/lib/CFS/iter_int.py) | A function for the numerical computation of a list of iterated integrals |\r\n| [**iter_lie**](https://github.com/iperezav/CFSpy/blob/main/build/lib/CFS/iter_lie.py) | A function for the analytical computation of a list of Lie derivatives |\r\n| [**single_iter_int**](https://github.com/iperezav/CFSpy/blob/main/build/lib/CFS/single_iter_int.py) | A function for the numerical computation of a single iterated integral |\r\n| [**single_iter_lie**](https://github.com/iperezav/CFSpy/blob/main/build/lib/CFS/single_iter_lie.py) | A function for the analytical computation of a single Lie derivative |\r\n\r\nCFSpy is used for:\r\n\r\n- Simulation of the output of a control systems.\r\n- Reachability analysis of a control system.\r\n\r\n\r\n# Installation \r\nCurrently, `CFSpy` supports releases of Python 3.12.4 onwards.\r\nTo install the current release:\r\n\r\n```shell\r\n$ pip install --upgrade CFSpy\r\n```\r\n\r\n\r\n# Getting Started\r\n\r\n## Minimal Example\r\n```python\r\nfrom CFS import iter_int, iter_lie, single_iter_int, single_iter_lie\r\n\r\nimport numpy as np\r\nfrom scipy.integrate import solve_ivp\r\nimport matplotlib.pyplot as plt\r\nimport sympy as sp\r\n\r\n# Define the Lotka-Volterra system\r\ndef system(t, x, u1_func, u2_func):\r\n x1, x2 = x\r\n u1 = u1_func(t)\r\n u2 = u2_func(t)\r\n dx1 = -x1*x2 + x1 * u1\r\n dx2 = x1*x2 - x2* u2\r\n return [dx1, dx2]\r\n\r\n# Input 1\r\ndef u1_func(t):\r\n return np.sin(t)\r\n\r\n# Input 2\r\ndef u2_func(t):\r\n return np.cos(t)\r\n\r\n# Initial condition\r\nx0 = [1/3,2/3]\r\n\r\n# Time range\r\nt0 = 0\r\ntf = 3\r\ndt = 0.001\r\nt_span = (t0, tf)\r\n\r\n# Simulation of the system\r\nsolution = solve_ivp(system, t_span, x0, args=(u1_func, u2_func), dense_output=True)\r\n\r\n# Partition of the time interval\r\nt = np.linspace(t_span[0], t_span[1], int((tf-t0)//dt+1))\r\ny = solution.sol(t)\r\n\r\n# Define the symbolic variables\r\nx1, x2 = sp.symbols('x1 x2')\r\nx = sp.Matrix([x1, x2])\r\n\r\n\r\n# Define the system symbolically\r\ng = sp.transpose(sp.Matrix([[-x1*x2, x1*x2], [x1, 0], [0, - x2]]))\r\n\r\n# Define the output symbolically\r\nh = x1\r\n\r\n# The truncation of the length of the words that index the Chen-Fliess series\r\nNtrunc = 4\r\n\r\n# Coefficients of the Chen-Fliess series evaluated at the initial state\r\nCeta = np.array(iter_lie(h,g,x,Ntrunc).subs([(x[0], 1/3),(x[1], 2/3)]))\r\n\r\n# inputs as arrays\r\nu1 = np.sin(t)\r\nu2 = np.cos(t)\r\n\r\n# input array\r\nu = np.vstack([u1, u2])\r\n\r\n# List of iterated integral\r\nEu = iter_int(u,t0, tf, dt, Ntrunc)\r\n\r\n# Chen-Fliess series\r\nF_cu = x0[0]+np.sum(Ceta*Eu, axis = 0)\r\n\r\n# Graph of the output and the Chen-Fliess series\r\nplt.figure(figsize = (12,5))\r\nplt.plot(t, y[0].T)\r\nplt.plot(t, F_cu, color='red', linewidth=5, linestyle = '--', alpha = 0.5)\r\nplt.xlabel('$t$')\r\nplt.ylabel('$x_1$')\r\nplt.legend(['Output of the system','Chen-Fliess series'])\r\nplt.grid()\r\nplt.show()\r\n```\r\n<img src=\"https://raw.githubusercontent.com/iperezav/CFSpy/main/examples/output_chenfliess.png\" alt=\"iter_int(), iter_lie()\" />\r\n\r\nFor more examples, see the [CFSpy demos](https://github.com/iperezav/CFSpy/blob/main/examples/)\r\n\r\n\r\n# Resources\r\n\r\n- [**PyPi**](https://pypi.org/project/CFSpy/)\r\n- [**Documentation**](https://github.com/iperezav/CFSpy/blob/main/README.md)\r\n- [**Issue tracking**](https://github.com/iperezav/CFSpy/issues)\r\n\r\n\r\n# Contributing\r\n\r\nAll feedback is welcome. \r\n\r\n\r\n# Asking for help\r\nPlease reach out if you have any questions:\r\n1. [Github CFSpy discussions](https://github.com/iperezav/CFSpy/discussions/).\r\n2. [Github CFSpy issues](https://github.com/iperezav/CFSpy/issues).\r\n\r\n\r\n# License\r\n\r\nCFSpy is open-source and released under the [MIT License](LICENSE).\r\n\r\n\r\n# BibTeX\r\nFeel free to cite my work:\r\n\r\n```bibtex\r\n@article{iperezave,\r\n title={CFSpy},\r\n author={Perez Avellaneda, Ivan},\r\n journal={GitHub. Note: https://github.com/iperezav/CFSpy},\r\n volume={1},\r\n year={2024}\r\n}\r\n```\r\n\r\n[issues]: https://github.com/iperezav/CFSpy/issues\r\n[demos]: https://github.com/iperezav/CFSpy/blob/main/examples/\r\n\r\n[downloads]: https://pepy.tech/projects/cfspy\r\n[py-versions]: https://pypi.org/project/cfspy/\r\n[pypi-latest-version]: https://pypi.org/project/cfspy/\r\n[license]: https://github.com/iperezav/CFSpy/blob/main/LICENSE\r\n",
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