Add files using upload-large-folder tool

.gitattributes

@@ -334,3 +334,4 @@ parrot/lib/python3.10/site-packages/transformers/models/speecht5/__pycache__/mod
 parrot/lib/python3.10/site-packages/transformers/utils/__pycache__/dummy_tf_objects.cpython-310.pyc filter=lfs diff=lfs merge=lfs -text
 parrot/lib/python3.10/site-packages/transformers/__pycache__/modeling_tf_utils.cpython-310.pyc filter=lfs diff=lfs merge=lfs -text
 parrot/lib/python3.10/site-packages/transformers/__pycache__/trainer.cpython-310.pyc filter=lfs diff=lfs merge=lfs -text
+moondream/lib/python3.10/site-packages/numpy.libs/libopenblas64_p-r0-0cf96a72.3.23.dev.so filter=lfs diff=lfs merge=lfs -text

llava_next/lib/python3.10/site-packages/fsspec-2024.10.0.dist-info/INSTALLER

@@ -0,0 +1 @@
+pip

llava_next/lib/python3.10/site-packages/fsspec-2024.10.0.dist-info/METADATA

@@ -0,0 +1,278 @@
+Metadata-Version: 2.3
+Name: fsspec
+Version: 2024.10.0
+Summary: File-system specification
+Project-URL: Changelog, https://filesystem-spec.readthedocs.io/en/latest/changelog.html
+Project-URL: Documentation, https://filesystem-spec.readthedocs.io/en/latest/
+Project-URL: Homepage, https://github.com/fsspec/filesystem_spec
+Maintainer-email: Martin Durant <mdurant@anaconda.com>
+License: BSD 3-Clause License
+        
+        Copyright (c) 2018, Martin Durant
+        All rights reserved.
+        
+        Redistribution and use in source and binary forms, with or without
+        modification, are permitted provided that the following conditions are met:
+        
+        * Redistributions of source code must retain the above copyright notice, this
+          list of conditions and the following disclaimer.
+        
+        * Redistributions in binary form must reproduce the above copyright notice,
+          this list of conditions and the following disclaimer in the documentation
+          and/or other materials provided with the distribution.
+        
+        * Neither the name of the copyright holder nor the names of its
+          contributors may be used to endorse or promote products derived from
+          this software without specific prior written permission.
+        
+        THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
+        AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
+        IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
+        DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
+        FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
+        DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
+        SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
+        CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
+        OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
+        OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+License-File: LICENSE
+Keywords: file
+Classifier: Development Status :: 4 - Beta
+Classifier: Intended Audience :: Developers
+Classifier: License :: OSI Approved :: BSD License
+Classifier: Operating System :: OS Independent
+Classifier: Programming Language :: Python :: 3.8
+Classifier: Programming Language :: Python :: 3.9
+Classifier: Programming Language :: Python :: 3.10
+Classifier: Programming Language :: Python :: 3.11
+Classifier: Programming Language :: Python :: 3.12
+Requires-Python: >=3.8
+Provides-Extra: abfs
+Requires-Dist: adlfs; extra == 'abfs'
+Provides-Extra: adl
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+Requires-Dist: requests; extra == 'dropbox'
+Provides-Extra: entrypoints
+Provides-Extra: full
+Requires-Dist: adlfs; extra == 'full'
+Requires-Dist: aiohttp!=4.0.0a0,!=4.0.0a1; extra == 'full'
+Requires-Dist: dask; extra == 'full'
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+Requires-Dist: dropboxdrivefs; extra == 'full'
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+Provides-Extra: http
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+Provides-Extra: s3
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+Provides-Extra: tqdm
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+Description-Content-Type: text/markdown
+
+# filesystem_spec
+
+[![PyPI version](https://badge.fury.io/py/fsspec.svg)](https://pypi.python.org/pypi/fsspec/)
+[![Anaconda-Server Badge](https://anaconda.org/conda-forge/fsspec/badges/version.svg)](https://anaconda.org/conda-forge/fsspec)
+![Build](https://github.com/fsspec/filesystem_spec/workflows/CI/badge.svg)
+[![Docs](https://readthedocs.org/projects/filesystem-spec/badge/?version=latest)](https://filesystem-spec.readthedocs.io/en/latest/?badge=latest)
+
+A specification for pythonic filesystems.
+
+## Install
+
+```bash
+pip install fsspec
+```
+
+would install the base fsspec. Various optionally supported features might require specification of custom
+extra require, e.g. `pip install fsspec[ssh]` will install dependencies for `ssh` backends support.
+Use `pip install fsspec[full]` for installation of all known extra dependencies.
+
+Up-to-date package also provided through conda-forge distribution:
+
+```bash
+conda install -c conda-forge fsspec
+```
+
+
+## Purpose
+
+To produce a template or specification for a file-system interface, that specific implementations should follow,
+so that applications making use of them can rely on a common behaviour and not have to worry about the specific
+internal implementation decisions with any given backend. Many such implementations are included in this package,
+or in sister projects such as `s3fs` and `gcsfs`.
+
+In addition, if this is well-designed, then additional functionality, such as a key-value store or FUSE
+mounting of the file-system implementation may be available for all implementations "for free".
+
+## Documentation
+
+Please refer to [RTD](https://filesystem-spec.readthedocs.io/en/latest/?badge=latest)
+
+## Develop
+
+fsspec uses GitHub Actions for CI. Environment files can be found
+in the "ci/" directory. Note that the main environment is called "py38",
+but it is expected that the version of python installed be adjustable at
+CI runtime. For local use, pick a version suitable for you.
+
+```bash
+# For a new environment (mamba / conda).
+mamba create -n fsspec -c conda-forge  python=3.9 -y
+conda activate fsspec
+
+# Standard dev install with docs and tests.
+pip install -e ".[dev,doc,test]"
+
+# Full tests except for downstream
+pip install s3fs
+pip uninstall s3fs
+pip install -e .[dev,doc,test_full]
+pip install s3fs --no-deps
+pytest -v
+
+# Downstream tests.
+sh install_s3fs.sh
+# Windows powershell.
+install_s3fs.sh
+```
+
+### Testing
+
+Tests can be run in the dev environment, if activated, via ``pytest fsspec``.
+
+The full fsspec suite requires a system-level docker, docker-compose, and fuse
+installation. If only making changes to one backend implementation, it is
+not generally necessary to run all tests locally.
+
+It is expected that contributors ensure that any change to fsspec does not
+cause issues or regressions for either other fsspec-related packages such
+as gcsfs and s3fs, nor for downstream users of fsspec. The "downstream" CI
+run and corresponding environment file run a set of tests from the dask
+test suite, and very minimal tests against pandas and zarr from the
+test_downstream.py module in this repo.
+
+### Code Formatting
+
+fsspec uses [Black](https://black.readthedocs.io/en/stable) to ensure
+a consistent code format throughout the project.
+Run ``black fsspec`` from the root of the filesystem_spec repository to
+auto-format your code. Additionally, many editors have plugins that will apply
+``black`` as you edit files. ``black`` is included in the ``tox`` environments.
+
+Optionally, you may wish to setup [pre-commit hooks](https://pre-commit.com) to
+automatically run ``black`` when you make a git commit.
+Run ``pre-commit install --install-hooks`` from the root of the
+filesystem_spec repository to setup pre-commit hooks. ``black`` will now be run
+before you commit, reformatting any changed files. You can format without
+committing via ``pre-commit run`` or skip these checks with ``git commit
+--no-verify``.

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llava_next/lib/python3.10/site-packages/fsspec-2024.10.0.dist-info/WHEEL

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+Wheel-Version: 1.0
+Generator: hatchling 1.25.0
+Root-Is-Purelib: true
+Tag: py3-none-any

llava_next/lib/python3.10/site-packages/fsspec-2024.10.0.dist-info/licenses/LICENSE

@@ -0,0 +1,29 @@
+BSD 3-Clause License
+
+Copyright (c) 2018, Martin Durant
+All rights reserved.
+
+Redistribution and use in source and binary forms, with or without
+modification, are permitted provided that the following conditions are met:
+
+* Redistributions of source code must retain the above copyright notice, this
+  list of conditions and the following disclaimer.
+
+* Redistributions in binary form must reproduce the above copyright notice,
+  this list of conditions and the following disclaimer in the documentation
+  and/or other materials provided with the distribution.
+
+* Neither the name of the copyright holder nor the names of its
+  contributors may be used to endorse or promote products derived from
+  this software without specific prior written permission.
+
+THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
+AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
+IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
+DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
+FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
+DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
+SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
+CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
+OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
+OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

llava_next/lib/python3.10/site-packages/latex2mathml-3.77.0.dist-info/INSTALLER

@@ -0,0 +1 @@
+pip

llava_next/lib/python3.10/site-packages/latex2mathml-3.77.0.dist-info/LICENSE

@@ -0,0 +1,21 @@
+MIT License
+
+Copyright (c) 2016 Ronie Martinez
+
+Permission is hereby granted, free of charge, to any person obtaining a copy
+of this software and associated documentation files (the "Software"), to deal
+in the Software without restriction, including without limitation the rights
+to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+copies of the Software, and to permit persons to whom the Software is
+furnished to do so, subject to the following conditions:
+
+The above copyright notice and this permission notice shall be included in all
+copies or substantial portions of the Software.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
+AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+SOFTWARE.

llava_next/lib/python3.10/site-packages/latex2mathml-3.77.0.dist-info/METADATA

@@ -0,0 +1,172 @@
+Metadata-Version: 2.1
+Name: latex2mathml
+Version: 3.77.0
+Summary: Pure Python library for LaTeX to MathML conversion
+Home-page: https://github.com/roniemartinez/latex2mathml
+License: MIT
+Keywords: latex,mathml
+Author: Ronie Martinez
+Author-email: ronmarti18@gmail.com
+Requires-Python: >=3.8.1,<4.0.0
+Classifier: Development Status :: 5 - Production/Stable
+Classifier: License :: OSI Approved :: MIT License
+Classifier: Programming Language :: Python :: 3
+Classifier: Programming Language :: Python :: 3.9
+Classifier: Programming Language :: Python :: 3.10
+Classifier: Programming Language :: Python :: 3.11
+Classifier: Programming Language :: Python :: 3.12
+Classifier: Programming Language :: Python :: 3.8
+Classifier: Programming Language :: Python :: Implementation :: CPython
+Classifier: Topic :: Scientific/Engineering :: Mathematics
+Classifier: Topic :: Software Development :: Libraries :: Python Modules
+Classifier: Topic :: Text Processing :: Markup :: HTML
+Classifier: Topic :: Text Processing :: Markup :: LaTeX
+Project-URL: Donate, https://www.buymeacoffee.com/roniemartinez
+Project-URL: Repository, https://github.com/roniemartinez/latex2mathml
+Description-Content-Type: text/markdown
+
+<table>
+    <tr>
+        <td>License</td>
+        <td><img src='https://img.shields.io/pypi/l/latex2mathml.svg?style=for-the-badge' alt="License"></td>
+        <td>Version</td>
+        <td><img src='https://img.shields.io/pypi/v/latex2mathml.svg?logo=pypi&style=for-the-badge' alt="Version"></td>
+    </tr>
+    <tr>
+        <td>Github Actions</td>
+        <td><img src='https://img.shields.io/github/actions/workflow/status/roniemartinez/latex2mathml/python.yml?branch=master&label=actions&logo=github%20actions&style=for-the-badge' alt="Github Actions"></td>
+        <td>Coverage</td>
+        <td><img src='https://img.shields.io/codecov/c/github/roniemartinez/latex2mathml/master?label=codecov&logo=codecov&style=for-the-badge' alt="CodeCov"></td>
+    </tr>
+    <tr>
+        <td>Supported versions</td>
+        <td><img src='https://img.shields.io/pypi/pyversions/latex2mathml.svg?logo=python&style=for-the-badge' alt="Python Versions"></td>
+        <td>Wheel</td>
+        <td><img src='https://img.shields.io/pypi/wheel/latex2mathml.svg?style=for-the-badge' alt="Wheel"></td>
+    </tr>
+    <tr>
+        <td>Status</td>
+        <td><img src='https://img.shields.io/pypi/status/latex2mathml.svg?style=for-the-badge' alt="Status"></td>
+        <td>Downloads</td>
+        <td><img src='https://img.shields.io/pypi/dm/latex2mathml.svg?style=for-the-badge' alt="Downloads"></td>
+    </tr>
+    <tr>
+        <td>All Contributors</td>
+        <td><a href="#contributors-"><img src='https://img.shields.io/github/all-contributors/roniemartinez/latex2mathml?style=for-the-badge' alt="All Contributors"></a></td>
+    </tr>
+</table>
+
+# latex2mathml
+
+Pure Python library for LaTeX to MathML conversion
+
+## Installation
+
+```bash
+pip install latex2mathml
+```
+
+## Usage
+
+### Python
+
+```python
+import latex2mathml.converter
+
+latex_input = "<your_latex_string>"
+mathml_output = latex2mathml.converter.convert(latex_input)
+```
+
+### Command-line
+
+```shell
+% latex2mathml -h
+usage: latex2mathml [-h] [-V] [-b] [-t TEXT | -f FILE | -s]
+
+Pure Python library for LaTeX to MathML conversion
+
+options:
+  -h, --help            show this help message and exit
+  -V, --version         Show version
+  -b, --block           Display block
+
+required arguments:
+  -t TEXT, --text TEXT  Text
+  -f FILE, --file FILE  File
+  -s, --stdin           Stdin
+```
+
+## References
+### LaTeX
+
+- https://en.wikibooks.org/wiki/LaTeX/Mathematics
+- http://artofproblemsolving.com/wiki/index.php?title=Main_Page
+- http://milde.users.sourceforge.net/LUCR/Math/
+- https://math-linux.com/latex-26/faq/latex-faq/article/latex-derivatives-limits-sums-products-and-integrals
+- https://www.tutorialspoint.com/tex_commands
+- https://www.giss.nasa.gov/tools/latex/ltx-86.html
+- https://ftp.gwdg.de/pub/ctan/info/l2tabu/english/l2tabuen.pdf
+
+### MathML
+
+- http://www.xmlmind.com/tutorials/MathML/
+
+
+## Author
+
+- [Ronie Martinez](mailto:ronmarti18@gmail.com)
+
+## Contributors ✨
+
+Thanks goes to these wonderful people ([emoji key](https://allcontributors.org/docs/en/emoji-key)):
+
+<!-- ALL-CONTRIBUTORS-LIST:START - Do not remove or modify this section -->
+<!-- prettier-ignore-start -->
+<!-- markdownlint-disable -->
+<table>
+  <tbody>
+    <tr>
+      <td align="center" valign="top" width="14.28%"><a href="https://ron.sh"><img src="https://avatars.githubusercontent.com/u/2573537?v=4?s=100" width="100px;" alt="Ronie Martinez"/><br /><sub><b>Ronie Martinez</b></sub></a><br /><a href="#maintenance-roniemartinez" title="Maintenance">🚧</a> <a href="https://github.com/roniemartinez/latex2mathml/commits?author=roniemartinez" title="Code">💻</a> <a href="#infra-roniemartinez" title="Infrastructure (Hosting, Build-Tools, etc)">🚇</a></td>
+      <td align="center" valign="top" width="14.28%"><a href="https://anwen.cc/"><img src="https://avatars.githubusercontent.com/u/1472850?v=4?s=100" width="100px;" alt="askender"/><br /><sub><b>askender</b></sub></a><br /><a href="https://github.com/roniemartinez/latex2mathml/commits?author=askender" title="Documentation">📖</a></td>
+      <td align="center" valign="top" width="14.28%"><a href="https://github.com/06180339"><img src="https://avatars.githubusercontent.com/u/25408501?v=4?s=100" width="100px;" alt="06180339"/><br /><sub><b>06180339</b></sub></a><br /><a href="https://github.com/roniemartinez/latex2mathml/commits?author=06180339" title="Code">💻</a></td>
+      <td align="center" valign="top" width="14.28%"><a href="https://github.com/chaihahaha"><img src="https://avatars.githubusercontent.com/u/24356676?v=4?s=100" width="100px;" alt="chaihahaha"/><br /><sub><b>chaihahaha</b></sub></a><br /><a href="https://github.com/roniemartinez/latex2mathml/commits?author=chaihahaha" title="Code">💻</a></td>
+      <td align="center" valign="top" width="14.28%"><a href="https://github.com/huangradio"><img src="https://avatars.githubusercontent.com/u/63624395?v=4?s=100" width="100px;" alt="HQY"/><br /><sub><b>HQY</b></sub></a><br /><a href="https://github.com/roniemartinez/latex2mathml/issues?q=author%3Ahuangradio" title="Bug reports">🐛</a></td>
+      <td align="center" valign="top" width="14.28%"><a href="https://github.com/Sun-ZhenXing"><img src="https://avatars.githubusercontent.com/u/44517244?v=4?s=100" width="100px;" alt="鸭梨"/><br /><sub><b>鸭梨</b></sub></a><br /><a href="https://github.com/roniemartinez/latex2mathml/issues?q=author%3ASun-ZhenXing" title="Bug reports">🐛</a></td>
+      <td align="center" valign="top" width="14.28%"><a href="https://github.com/oliverstefanov"><img src="https://avatars.githubusercontent.com/u/33491656?v=4?s=100" width="100px;" alt="oliverstefanov"/><br /><sub><b>oliverstefanov</b></sub></a><br /><a href="https://github.com/roniemartinez/latex2mathml/issues?q=author%3Aoliverstefanov" title="Bug reports">🐛</a></td>
+    </tr>
+    <tr>
+      <td align="center" valign="top" width="14.28%"><a href="https://github.com/ghost"><img src="https://avatars.githubusercontent.com/u/10137?v=4?s=100" width="100px;" alt="Deleted user"/><br /><sub><b>Deleted user</b></sub></a><br /><a href="https://github.com/roniemartinez/latex2mathml/issues?q=author%3Aghost" title="Bug reports">🐛</a></td>
+      <td align="center" valign="top" width="14.28%"><a href="https://github.com/cesaryuan"><img src="https://avatars.githubusercontent.com/u/35998162?v=4?s=100" width="100px;" alt="Cesaryuan"/><br /><sub><b>Cesaryuan</b></sub></a><br /><a href="https://github.com/roniemartinez/latex2mathml/issues?q=author%3Acesaryuan" title="Bug reports">🐛</a></td>
+      <td align="center" valign="top" width="14.28%"><a href="https://github.com/tonystank3000"><img src="https://avatars.githubusercontent.com/u/6315974?v=4?s=100" width="100px;" alt="TonyStank"/><br /><sub><b>TonyStank</b></sub></a><br /><a href="https://github.com/roniemartinez/latex2mathml/issues?q=author%3Atonystank3000" title="Bug reports">🐛</a></td>
+      <td align="center" valign="top" width="14.28%"><a href="https://polarwinkel.de"><img src="https://avatars.githubusercontent.com/u/1512713?v=4?s=100" width="100px;" alt="Dirk Winkel"/><br /><sub><b>Dirk Winkel</b></sub></a><br /><a href="https://github.com/roniemartinez/latex2mathml/issues?q=author%3Apolarwinkel" title="Bug reports">🐛</a></td>
+      <td align="center" valign="top" width="14.28%"><a href="https://github.com/sinslu"><img src="https://avatars.githubusercontent.com/u/12248270?v=4?s=100" width="100px;" alt="sinslu"/><br /><sub><b>sinslu</b></sub></a><br /><a href="https://github.com/roniemartinez/latex2mathml/issues?q=author%3Asinslu" title="Bug reports">🐛</a></td>
+      <td align="center" valign="top" width="14.28%"><a href="https://ubavic.rs"><img src="https://avatars.githubusercontent.com/u/53820106?v=4?s=100" width="100px;" alt="Nikola Ubavić"/><br /><sub><b>Nikola Ubavić</b></sub></a><br /><a href="https://github.com/roniemartinez/latex2mathml/issues?q=author%3Aubavic" title="Bug reports">🐛</a></td>
+      <td align="center" valign="top" width="14.28%"><a href="https://github.com/abhisheksia"><img src="https://avatars.githubusercontent.com/u/68808662?v=4?s=100" width="100px;" alt="abhisheksia"/><br /><sub><b>abhisheksia</b></sub></a><br /><a href="https://github.com/roniemartinez/latex2mathml/issues?q=author%3Aabhisheksia" title="Bug reports">🐛</a></td>
+    </tr>
+    <tr>
+      <td align="center" valign="top" width="14.28%"><a href="http://denissalem.tuxfamily.org"><img src="https://avatars.githubusercontent.com/u/4476506?v=4?s=100" width="100px;" alt="Denis Salem"/><br /><sub><b>Denis Salem</b></sub></a><br /><a href="https://github.com/roniemartinez/latex2mathml/issues?q=author%3ADenisSalem" title="Bug reports">🐛</a> <a href="https://github.com/roniemartinez/latex2mathml/commits?author=DenisSalem" title="Code">💻</a></td>
+      <td align="center" valign="top" width="14.28%"><a href="https://clontz.org"><img src="https://avatars.githubusercontent.com/u/1559632?v=4?s=100" width="100px;" alt="Steven Clontz"/><br /><sub><b>Steven Clontz</b></sub></a><br /><a href="https://github.com/roniemartinez/latex2mathml/issues?q=author%3AStevenClontz" title="Bug reports">🐛</a></td>
+      <td align="center" valign="top" width="14.28%"><a href="https://github.com/yuwenjun1"><img src="https://avatars.githubusercontent.com/u/43265090?v=4?s=100" width="100px;" alt="空白"/><br /><sub><b>空白</b></sub></a><br /><a href="https://github.com/roniemartinez/latex2mathml/issues?q=author%3Ayuwenjun1" title="Bug reports">🐛</a></td>
+      <td align="center" valign="top" width="14.28%"><a href="https://github.com/amuramatsu"><img src="https://avatars.githubusercontent.com/u/6500918?v=4?s=100" width="100px;" alt="MURAMATSU Atshshi"/><br /><sub><b>MURAMATSU Atshshi</b></sub></a><br /><a href="https://github.com/roniemartinez/latex2mathml/issues?q=author%3Aamuramatsu" title="Bug reports">🐛</a></td>
+      <td align="center" valign="top" width="14.28%"><a href="https://github.com/leingang"><img src="https://avatars.githubusercontent.com/u/570942?v=4?s=100" width="100px;" alt="leingang"/><br /><sub><b>leingang</b></sub></a><br /><a href="https://github.com/roniemartinez/latex2mathml/issues?q=author%3Aleingang" title="Bug reports">🐛</a></td>
+      <td align="center" valign="top" width="14.28%"><a href="https://github.com/Nigel-Amers"><img src="https://avatars.githubusercontent.com/u/14248498?v=4?s=100" width="100px;" alt="Nigel Amers"/><br /><sub><b>Nigel Amers</b></sub></a><br /><a href="https://github.com/roniemartinez/latex2mathml/issues?q=author%3ANigel-Amers" title="Bug reports">🐛</a></td>
+      <td align="center" valign="top" width="14.28%"><a href="https://github.com/retsyo"><img src="https://avatars.githubusercontent.com/u/7960913?v=4?s=100" width="100px;" alt="retsyo"/><br /><sub><b>retsyo</b></sub></a><br /><a href="https://github.com/roniemartinez/latex2mathml/issues?q=author%3Aretsyo" title="Bug reports">🐛</a></td>
+    </tr>
+    <tr>
+      <td align="center" valign="top" width="14.28%"><a href="https://github.com/miakramer"><img src="https://avatars.githubusercontent.com/u/16845265?v=4?s=100" width="100px;" alt="Mia Kramer"/><br /><sub><b>Mia Kramer</b></sub></a><br /><a href="https://github.com/roniemartinez/latex2mathml/commits?author=miakramer" title="Code">💻</a> <a href="https://github.com/roniemartinez/latex2mathml/commits?author=miakramer" title="Documentation">📖</a></td>
+      <td align="center" valign="top" width="14.28%"><a href="http://cnx.gdn"><img src="https://avatars.githubusercontent.com/u/13689192?v=4?s=100" width="100px;" alt="Nguyễn Gia Phong"/><br /><sub><b>Nguyễn Gia Phong</b></sub></a><br /><a href="https://github.com/roniemartinez/latex2mathml/issues?q=author%3AMcSinyx" title="Bug reports">🐛</a> <a href="https://github.com/roniemartinez/latex2mathml/commits?author=McSinyx" title="Code">💻</a> <a href="https://github.com/roniemartinez/latex2mathml/commits?author=McSinyx" title="Tests">⚠️</a></td>
+      <td align="center" valign="top" width="14.28%"><a href="https://cdelker.github.io"><img src="https://avatars.githubusercontent.com/u/44102190?v=4?s=100" width="100px;" alt="Collin Delker"/><br /><sub><b>Collin Delker</b></sub></a><br /><a href="https://github.com/roniemartinez/latex2mathml/issues?q=author%3Acdelker" title="Bug reports">🐛</a></td>
+      <td align="center" valign="top" width="14.28%"><a href="https://github.com/jiotv321"><img src="https://avatars.githubusercontent.com/u/118644533?v=4?s=100" width="100px;" alt="jiotv321"/><br /><sub><b>jiotv321</b></sub></a><br /><a href="https://github.com/roniemartinez/latex2mathml/issues?q=author%3Ajiotv321" title="Bug reports">🐛</a></td>
+      <td align="center" valign="top" width="14.28%"><a href="https://github.com/felixonmars"><img src="https://avatars.githubusercontent.com/u/1006477?v=4?s=100" width="100px;" alt="Felix Yan"/><br /><sub><b>Felix Yan</b></sub></a><br /><a href="https://github.com/roniemartinez/latex2mathml/commits?author=felixonmars" title="Code">💻</a></td>
+      <td align="center" valign="top" width="14.28%"><a href="https://rob-blackbourn.github.io/blog/"><img src="https://avatars.githubusercontent.com/u/2880305?v=4?s=100" width="100px;" alt="Rob Blackbourn"/><br /><sub><b>Rob Blackbourn</b></sub></a><br /><a href="https://github.com/roniemartinez/latex2mathml/commits?author=rob-blackbourn" title="Code">💻</a></td>
+    </tr>
+  </tbody>
+</table>
+
+<!-- markdownlint-restore -->
+<!-- prettier-ignore-end -->
+
+<!-- ALL-CONTRIBUTORS-LIST:END -->
+
+This project follows the [all-contributors](https://github.com/all-contributors/all-contributors) specification. Contributions of any kind welcome!
+

llava_next/lib/python3.10/site-packages/latex2mathml-3.77.0.dist-info/RECORD

@@ -0,0 +1,25 @@
+../../../bin/l2m,sha256=tiRsFHuNF2Y1lraq2gGaQg1Zkg_6YLd9DEP9CMfowBM,237
+../../../bin/latex2mathml,sha256=tiRsFHuNF2Y1lraq2gGaQg1Zkg_6YLd9DEP9CMfowBM,237
+latex2mathml-3.77.0.dist-info/INSTALLER,sha256=zuuue4knoyJ-UwPPXg8fezS7VCrXJQrAP7zeNuwvFQg,4
+latex2mathml-3.77.0.dist-info/LICENSE,sha256=SsHh2gf2s0PVQZHF4XFoQFNOCFkbP56uyQ1oAM3EdUM,1071
+latex2mathml-3.77.0.dist-info/METADATA,sha256=8I_5snYGZtFTF_ZdrISzE_x4L1PcEc9WFhHJfwkOUhA,14844
+latex2mathml-3.77.0.dist-info/RECORD,,
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llava_next/lib/python3.10/site-packages/pydantic-2.9.2.dist-info/WHEEL

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+Generator: hatchling 1.25.0
+Root-Is-Purelib: true
+Tag: py3-none-any

llava_next/lib/python3.10/site-packages/pydantic-2.9.2.dist-info/licenses/LICENSE

@@ -0,0 +1,21 @@
+The MIT License (MIT)
+
+Copyright (c) 2017 to present Pydantic Services Inc. and individual contributors.
+
+Permission is hereby granted, free of charge, to any person obtaining a copy
+of this software and associated documentation files (the "Software"), to deal
+in the Software without restriction, including without limitation the rights
+to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+copies of the Software, and to permit persons to whom the Software is
+furnished to do so, subject to the following conditions:
+
+The above copyright notice and this permission notice shall be included in all
+copies or substantial portions of the Software.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
+AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+SOFTWARE.

moondream/lib/python3.10/site-packages/numpy.libs/libopenblas64_p-r0-0cf96a72.3.23.dev.so

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+oid sha256:9254d0854dd7615e11de28d771ae408878ca8123a7ac204f21e4cc7a376cc2e5
+size 35123345

parrot/lib/python3.10/site-packages/sympy/physics/continuum_mechanics/cable.py

@@ -0,0 +1,587 @@
+"""
+This module can be used to solve problems related
+to 2D Cables.
+"""
+
+from sympy.core.sympify import sympify
+from sympy.core.symbol import Symbol
+from sympy import sin, cos, pi, atan, diff
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.solvers.solveset import linsolve
+from sympy.matrices import Matrix
+
+
+class Cable:
+    """
+    Cables are structures in engineering that support
+    the applied transverse loads through the tensile
+    resistance developed in its members.
+
+    Cables are widely used in suspension bridges, tension
+    leg offshore platforms, transmission lines, and find
+    use in several other engineering applications.
+
+    Examples
+    ========
+    A cable is supported at (0, 10) and (10, 10). Two point loads
+    acting vertically downwards act on the cable, one with magnitude 3 kN
+    and acting 2 meters from the left support and 3 meters below it, while
+    the other with magnitude 2 kN is 6 meters from the left support and
+    6 meters below it.
+
+    >>> from sympy.physics.continuum_mechanics.cable import Cable
+    >>> c = Cable(('A', 0, 10), ('B', 10, 10))
+    >>> c.apply_load(-1, ('P', 2, 7, 3, 270))
+    >>> c.apply_load(-1, ('Q', 6, 4, 2, 270))
+    >>> c.loads
+    {'distributed': {}, 'point_load': {'P': [3, 270], 'Q': [2, 270]}}
+    >>> c.loads_position
+    {'P': [2, 7], 'Q': [6, 4]}
+    """
+    def __init__(self, support_1, support_2):
+        """
+        Initializes the class.
+
+        Parameters
+        ==========
+
+        support_1 and support_2 are tuples of the form
+        (label, x, y), where
+
+        label : String or symbol
+            The label of the support
+
+        x : Sympifyable
+            The x coordinate of the position of the support
+
+        y : Sympifyable
+            The y coordinate of the position of the support
+        """
+        self._left_support = []
+        self._right_support = []
+        self._supports = {}
+        self._support_labels = []
+        self._loads = {"distributed": {}, "point_load": {}}
+        self._loads_position = {}
+        self._length = 0
+        self._reaction_loads = {}
+        self._tension = {}
+        self._lowest_x_global = sympify(0)
+
+        if support_1[0] == support_2[0]:
+            raise ValueError("Supports can not have the same label")
+
+        elif support_1[1] == support_2[1]:
+            raise ValueError("Supports can not be at the same location")
+
+        x1 = sympify(support_1[1])
+        y1 = sympify(support_1[2])
+        self._supports[support_1[0]] = [x1, y1]
+
+        x2 = sympify(support_2[1])
+        y2 = sympify(support_2[2])
+        self._supports[support_2[0]] = [x2, y2]
+
+        if support_1[1] < support_2[1]:
+            self._left_support.append(x1)
+            self._left_support.append(y1)
+            self._right_support.append(x2)
+            self._right_support.append(y2)
+            self._support_labels.append(support_1[0])
+            self._support_labels.append(support_2[0])
+
+        else:
+            self._left_support.append(x2)
+            self._left_support.append(y2)
+            self._right_support.append(x1)
+            self._right_support.append(y1)
+            self._support_labels.append(support_2[0])
+            self._support_labels.append(support_1[0])
+
+        for i in self._support_labels:
+            self._reaction_loads[Symbol("R_"+ i +"_x")] = 0
+            self._reaction_loads[Symbol("R_"+ i +"_y")] = 0
+
+    @property
+    def supports(self):
+        """
+        Returns the supports of the cable along with their
+        positions.
+        """
+        return self._supports
+
+    @property
+    def left_support(self):
+        """
+        Returns the position of the left support.
+        """
+        return self._left_support
+
+    @property
+    def right_support(self):
+        """
+        Returns the position of the right support.
+        """
+        return self._right_support
+
+    @property
+    def loads(self):
+        """
+        Returns the magnitude and direction of the loads
+        acting on the cable.
+        """
+        return self._loads
+
+    @property
+    def loads_position(self):
+        """
+        Returns the position of the point loads acting on the
+        cable.
+        """
+        return self._loads_position
+
+    @property
+    def length(self):
+        """
+        Returns the length of the cable.
+        """
+        return self._length
+
+    @property
+    def reaction_loads(self):
+        """
+        Returns the reaction forces at the supports, which are
+        initialized to 0.
+        """
+        return self._reaction_loads
+
+    @property
+    def tension(self):
+        """
+        Returns the tension developed in the cable due to the loads
+        applied.
+        """
+        return self._tension
+
+    def tension_at(self, x):
+        """
+        Returns the tension at a given value of x developed due to
+        distributed load.
+        """
+        if 'distributed' not in self._tension.keys():
+            raise ValueError("No distributed load added or solve method not called")
+
+        if x > self._right_support[0] or x < self._left_support[0]:
+            raise ValueError("The value of x should be between the two supports")
+
+        A = self._tension['distributed']
+        X = Symbol('X')
+
+        return A.subs({X:(x-self._lowest_x_global)})
+
+    def apply_length(self, length):
+        """
+        This method specifies the length of the cable
+
+        Parameters
+        ==========
+
+        length : Sympifyable
+            The length of the cable
+
+        Examples
+        ========
+
+        >>> from sympy.physics.continuum_mechanics.cable import Cable
+        >>> c = Cable(('A', 0, 10), ('B', 10, 10))
+        >>> c.apply_length(20)
+        >>> c.length
+        20
+        """
+        dist = ((self._left_support[0] - self._right_support[0])**2
+                - (self._left_support[1] - self._right_support[1])**2)**(1/2)
+
+        if length < dist:
+            raise ValueError("length should not be less than the distance between the supports")
+
+        self._length = length
+
+    def change_support(self, label, new_support):
+        """
+        This method changes the mentioned support with a new support.
+
+        Parameters
+        ==========
+        label: String or symbol
+            The label of the support to be changed
+
+        new_support: Tuple of the form (new_label, x, y)
+            new_label: String or symbol
+                The label of the new support
+
+            x: Sympifyable
+                The x-coordinate of the position of the new support.
+
+            y: Sympifyable
+                The y-coordinate of the position of the new support.
+
+        Examples
+        ========
+
+        >>> from sympy.physics.continuum_mechanics.cable import Cable
+        >>> c = Cable(('A', 0, 10), ('B', 10, 10))
+        >>> c.supports
+        {'A': [0, 10], 'B': [10, 10]}
+        >>> c.change_support('B', ('C', 5, 6))
+        >>> c.supports
+        {'A': [0, 10], 'C': [5, 6]}
+        """
+        if label not in self._supports:
+            raise ValueError("No support exists with the given label")
+
+        i = self._support_labels.index(label)
+        rem_label = self._support_labels[(i+1)%2]
+        x1 = self._supports[rem_label][0]
+        y1 = self._supports[rem_label][1]
+
+        x = sympify(new_support[1])
+        y = sympify(new_support[2])
+
+        for l in self._loads_position:
+            if l[0] >= max(x, x1) or l[0] <= min(x, x1):
+                raise ValueError("The change in support will throw an existing load out of range")
+
+        self._supports.pop(label)
+        self._left_support.clear()
+        self._right_support.clear()
+        self._reaction_loads.clear()
+        self._support_labels.remove(label)
+
+        self._supports[new_support[0]] = [x, y]
+
+        if x1 < x:
+            self._left_support.append(x1)
+            self._left_support.append(y1)
+            self._right_support.append(x)
+            self._right_support.append(y)
+            self._support_labels.append(new_support[0])
+
+        else:
+            self._left_support.append(x)
+            self._left_support.append(y)
+            self._right_support.append(x1)
+            self._right_support.append(y1)
+            self._support_labels.insert(0, new_support[0])
+
+        for i in self._support_labels:
+            self._reaction_loads[Symbol("R_"+ i +"_x")] = 0
+            self._reaction_loads[Symbol("R_"+ i +"_y")] = 0
+
+    def apply_load(self, order, load):
+        """
+        This method adds load to the cable.
+
+        Parameters
+        ==========
+
+        order : Integer
+            The order of the applied load.
+
+                - For point loads, order = -1
+                - For distributed load, order = 0
+
+        load : tuple
+
+            * For point loads, load is of the form (label, x, y, magnitude, direction), where:
+
+            label : String or symbol
+                The label of the load
+
+            x : Sympifyable
+                The x coordinate of the position of the load
+
+            y : Sympifyable
+                The y coordinate of the position of the load
+
+            magnitude : Sympifyable
+                The magnitude of the load. It must always be positive
+
+            direction : Sympifyable
+                The angle, in degrees, that the load vector makes with the horizontal
+                in the counter-clockwise direction. It takes the values 0 to 360,
+                inclusive.
+
+
+            * For uniformly distributed load, load is of the form (label, magnitude)
+
+            label : String or symbol
+                The label of the load
+
+            magnitude : Sympifyable
+                The magnitude of the load. It must always be positive
+
+        Examples
+        ========
+
+        For a point load of magnitude 12 units inclined at 30 degrees with the horizontal:
+
+        >>> from sympy.physics.continuum_mechanics.cable import Cable
+        >>> c = Cable(('A', 0, 10), ('B', 10, 10))
+        >>> c.apply_load(-1, ('Z', 5, 5, 12, 30))
+        >>> c.loads
+        {'distributed': {}, 'point_load': {'Z': [12, 30]}}
+        >>> c.loads_position
+        {'Z': [5, 5]}
+
+
+        For a uniformly distributed load of magnitude 9 units:
+
+        >>> from sympy.physics.continuum_mechanics.cable import Cable
+        >>> c = Cable(('A', 0, 10), ('B', 10, 10))
+        >>> c.apply_load(0, ('X', 9))
+        >>> c.loads
+        {'distributed': {'X': 9}, 'point_load': {}}
+        """
+        if order == -1:
+            if len(self._loads["distributed"]) != 0:
+                raise ValueError("Distributed load already exists")
+
+            label = load[0]
+            if label in self._loads["point_load"]:
+                raise ValueError("Label already exists")
+
+            x = sympify(load[1])
+            y = sympify(load[2])
+
+            if x > self._right_support[0] or x < self._left_support[0]:
+                raise ValueError("The load should be positioned between the supports")
+
+            magnitude = sympify(load[3])
+            direction = sympify(load[4])
+
+            self._loads["point_load"][label] = [magnitude, direction]
+            self._loads_position[label] = [x, y]
+
+        elif order == 0:
+            if len(self._loads_position) != 0:
+                raise ValueError("Point load(s) already exist")
+
+            label = load[0]
+            if label in self._loads["distributed"]:
+                raise ValueError("Label already exists")
+
+            magnitude = sympify(load[1])
+
+            self._loads["distributed"][label] = magnitude
+
+        else:
+            raise ValueError("Order should be either -1 or 0")
+
+    def remove_loads(self, *args):
+        """
+        This methods removes the specified loads.
+
+        Parameters
+        ==========
+        This input takes multiple label(s) as input
+        label(s): String or symbol
+            The label(s) of the loads to be removed.
+
+        Examples
+        ========
+
+        >>> from sympy.physics.continuum_mechanics.cable import Cable
+        >>> c = Cable(('A', 0, 10), ('B', 10, 10))
+        >>> c.apply_load(-1, ('Z', 5, 5, 12, 30))
+        >>> c.loads
+        {'distributed': {}, 'point_load': {'Z': [12, 30]}}
+        >>> c.remove_loads('Z')
+        >>> c.loads
+        {'distributed': {}, 'point_load': {}}
+        """
+        for i in args:
+            if len(self._loads_position) == 0:
+                if i not in self._loads['distributed']:
+                    raise ValueError("Error removing load " + i + ": no such load exists")
+
+                else:
+                    self._loads['disrtibuted'].pop(i)
+
+            else:
+                if i not in self._loads['point_load']:
+                    raise ValueError("Error removing load " + i + ": no such load exists")
+
+                else:
+                    self._loads['point_load'].pop(i)
+                    self._loads_position.pop(i)
+
+    def solve(self, *args):
+        """
+        This method solves for the reaction forces at the supports, the tension developed in
+        the cable, and updates the length of the cable.
+
+        Parameters
+        ==========
+        This method requires no input when solving for point loads
+        For distributed load, the x and y coordinates of the lowest point of the cable are
+        required as
+
+        x: Sympifyable
+            The x coordinate of the lowest point
+
+        y: Sympifyable
+            The y coordinate of the lowest point
+
+        Examples
+        ========
+        For point loads,
+
+        >>> from sympy.physics.continuum_mechanics.cable import Cable
+        >>> c = Cable(("A", 0, 10), ("B", 10, 10))
+        >>> c.apply_load(-1, ('Z', 2, 7.26, 3, 270))
+        >>> c.apply_load(-1, ('X', 4, 6, 8, 270))
+        >>> c.solve()
+        >>> c.tension
+        {A_Z: 8.91403453669861, X_B: 19*sqrt(13)/10, Z_X: 4.79150773600774}
+        >>> c.reaction_loads
+        {R_A_x: -5.25547445255474, R_A_y: 7.2, R_B_x: 5.25547445255474, R_B_y: 3.8}
+        >>> c.length
+        5.7560958484519 + 2*sqrt(13)
+
+        For distributed load,
+
+        >>> from sympy.physics.continuum_mechanics.cable import Cable
+        >>> c=Cable(("A", 0, 40),("B", 100, 20))
+        >>> c.apply_load(0, ("X", 850))
+        >>> c.solve(58.58, 0)
+        >>> c.tension
+        {'distributed': 36456.8485*sqrt(0.000543529004799705*(X + 0.00135624381275735)**2 + 1)}
+        >>> c.tension_at(0)
+        61709.0363315913
+        >>> c.reaction_loads
+        {R_A_x: 36456.8485, R_A_y: -49788.5866682485, R_B_x: 44389.8401587246, R_B_y: 42866.621696333}
+        """
+
+        if len(self._loads_position) != 0:
+            sorted_position = sorted(self._loads_position.items(), key = lambda item : item[1][0])
+
+            sorted_position.append(self._support_labels[1])
+            sorted_position.insert(0, self._support_labels[0])
+
+            self._tension.clear()
+            moment_sum_from_left_support = 0
+            moment_sum_from_right_support = 0
+            F_x = 0
+            F_y = 0
+            self._length = 0
+
+            for i in range(1, len(sorted_position)-1):
+                if i == 1:
+                    self._length+=sqrt((self._left_support[0] - self._loads_position[sorted_position[i][0]][0])**2 + (self._left_support[1] - self._loads_position[sorted_position[i][0]][1])**2)
+
+                else:
+                    self._length+=sqrt((self._loads_position[sorted_position[i-1][0]][0] - self._loads_position[sorted_position[i][0]][0])**2 + (self._loads_position[sorted_position[i-1][0]][1] - self._loads_position[sorted_position[i][0]][1])**2)
+
+                if i == len(sorted_position)-2:
+                    self._length+=sqrt((self._right_support[0] - self._loads_position[sorted_position[i][0]][0])**2 + (self._right_support[1] - self._loads_position[sorted_position[i][0]][1])**2)
+
+                moment_sum_from_left_support += self._loads['point_load'][sorted_position[i][0]][0] * cos(pi * self._loads['point_load'][sorted_position[i][0]][1] / 180) * abs(self._left_support[1] - self._loads_position[sorted_position[i][0]][1])
+                moment_sum_from_left_support += self._loads['point_load'][sorted_position[i][0]][0] * sin(pi * self._loads['point_load'][sorted_position[i][0]][1] / 180) * abs(self._left_support[0] - self._loads_position[sorted_position[i][0]][0])
+
+                F_x += self._loads['point_load'][sorted_position[i][0]][0] * cos(pi * self._loads['point_load'][sorted_position[i][0]][1] / 180)
+                F_y += self._loads['point_load'][sorted_position[i][0]][0] * sin(pi * self._loads['point_load'][sorted_position[i][0]][1] / 180)
+
+                label = Symbol(sorted_position[i][0]+"_"+sorted_position[i+1][0])
+                y2 = self._loads_position[sorted_position[i][0]][1]
+                x2 = self._loads_position[sorted_position[i][0]][0]
+                y1 = 0
+                x1 = 0
+
+                if i == len(sorted_position)-2:
+                    x1 = self._right_support[0]
+                    y1 = self._right_support[1]
+
+                else:
+                    x1 = self._loads_position[sorted_position[i+1][0]][0]
+                    y1 = self._loads_position[sorted_position[i+1][0]][1]
+
+                angle_with_horizontal = atan((y1 - y2)/(x1 - x2))
+
+                tension = -(moment_sum_from_left_support)/(abs(self._left_support[1] - self._loads_position[sorted_position[i][0]][1])*cos(angle_with_horizontal) + abs(self._left_support[0] - self._loads_position[sorted_position[i][0]][0])*sin(angle_with_horizontal))
+                self._tension[label] = tension
+                moment_sum_from_right_support += self._loads['point_load'][sorted_position[i][0]][0] * cos(pi * self._loads['point_load'][sorted_position[i][0]][1] / 180) * abs(self._right_support[1] - self._loads_position[sorted_position[i][0]][1])
+                moment_sum_from_right_support += self._loads['point_load'][sorted_position[i][0]][0] * sin(pi * self._loads['point_load'][sorted_position[i][0]][1] / 180) * abs(self._right_support[0] - self._loads_position[sorted_position[i][0]][0])
+
+            label = Symbol(sorted_position[0][0]+"_"+sorted_position[1][0])
+            y2 = self._loads_position[sorted_position[1][0]][1]
+            x2 = self._loads_position[sorted_position[1][0]][0]
+            x1 = self._left_support[0]
+            y1 = self._left_support[1]
+
+            angle_with_horizontal = -atan((y2 - y1)/(x2 - x1))
+            tension = -(moment_sum_from_right_support)/(abs(self._right_support[1] - self._loads_position[sorted_position[1][0]][1])*cos(angle_with_horizontal) + abs(self._right_support[0] - self._loads_position[sorted_position[1][0]][0])*sin(angle_with_horizontal))
+            self._tension[label] = tension
+
+            angle_with_horizontal = pi/2 - angle_with_horizontal
+            label = self._support_labels[0]
+            self._reaction_loads[Symbol("R_"+label+"_x")] = -sin(angle_with_horizontal) * tension
+            F_x += -sin(angle_with_horizontal) * tension
+            self._reaction_loads[Symbol("R_"+label+"_y")] = cos(angle_with_horizontal) * tension
+            F_y += cos(angle_with_horizontal) * tension
+
+            label = self._support_labels[1]
+            self._reaction_loads[Symbol("R_"+label+"_x")] = -F_x
+            self._reaction_loads[Symbol("R_"+label+"_y")] = -F_y
+
+        elif len(self._loads['distributed']) != 0 :
+
+            if len(args) == 0:
+                raise ValueError("Provide the lowest point of the cable")
+
+            lowest_x = sympify(args[0])
+            lowest_y = sympify(args[1])
+            self._lowest_x_global = lowest_x
+
+            a = Symbol('a')
+            b = Symbol('b')
+            c = Symbol('c')
+            # augmented matrix form of linsolve
+
+            M = Matrix(
+                [[self._left_support[0]**2, self._left_support[0], 1, self._left_support[1]],
+                [self._right_support[0]**2, self._right_support[0], 1, self._right_support[1]],
+                [lowest_x**2, lowest_x, 1, lowest_y]  ]
+                       )
+
+            coefficient_solution = list(linsolve(M, (a, b, c)))
+
+            if len(coefficient_solution) == 0:
+                raise ValueError("The lowest point is inconsistent with the supports")
+
+            A = coefficient_solution[0][0]
+            B = coefficient_solution[0][1]
+            C = coefficient_solution[0][2]
+
+
+            # y = A*x**2 + B*x + C
+            # shifting origin to lowest point
+            X = Symbol('X')
+            Y = Symbol('Y')
+            Y = A*(X + lowest_x)**2 + B*(X + lowest_x) + C - lowest_y
+
+            temp_list = list(self._loads['distributed'].values())
+            applied_force = temp_list[0]
+
+            horizontal_force_constant = (applied_force * (self._right_support[0] - lowest_x)**2) / (2 * (self._right_support[1] - lowest_y))
+
+            self._tension.clear()
+            tangent_slope_to_curve = diff(Y, X)
+            self._tension['distributed'] = horizontal_force_constant / (cos(atan(tangent_slope_to_curve)))
+
+            label = self._support_labels[0]
+            self._reaction_loads[Symbol("R_"+label+"_x")] = self.tension_at(self._left_support[0]) * cos(atan(tangent_slope_to_curve.subs(X, self._left_support[0] - lowest_x)))
+            self._reaction_loads[Symbol("R_"+label+"_y")] = self.tension_at(self._left_support[0]) * sin(atan(tangent_slope_to_curve.subs(X, self._left_support[0] - lowest_x)))
+
+            label = self._support_labels[1]
+            self._reaction_loads[Symbol("R_"+label+"_x")] = self.tension_at(self._left_support[0]) * cos(atan(tangent_slope_to_curve.subs(X, self._right_support[0] - lowest_x)))
+            self._reaction_loads[Symbol("R_"+label+"_y")] = self.tension_at(self._left_support[0]) * sin(atan(tangent_slope_to_curve.subs(X, self._right_support[0] - lowest_x)))

parrot/lib/python3.10/site-packages/sympy/physics/continuum_mechanics/tests/test_beam.py

@@ -0,0 +1,802 @@
+from sympy.core.function import expand
+from sympy.core.numbers import (Rational, pi)
+from sympy.core.singleton import S
+from sympy.core.symbol import (Symbol, symbols)
+from sympy.sets.sets import Interval
+from sympy.simplify.simplify import simplify
+from sympy.physics.continuum_mechanics.beam import Beam
+from sympy.functions import SingularityFunction, Piecewise, meijerg, Abs, log
+from sympy.testing.pytest import raises
+from sympy.physics.units import meter, newton, kilo, giga, milli
+from sympy.physics.continuum_mechanics.beam import Beam3D
+from sympy.geometry import Circle, Polygon, Point2D, Triangle
+from sympy.core.sympify import sympify
+
+x = Symbol('x')
+y = Symbol('y')
+R1, R2 = symbols('R1, R2')
+
+
+def test_Beam():
+    E = Symbol('E')
+    E_1 = Symbol('E_1')
+    I = Symbol('I')
+    I_1 = Symbol('I_1')
+    A = Symbol('A')
+
+    b = Beam(1, E, I)
+    assert b.length == 1
+    assert b.elastic_modulus == E
+    assert b.second_moment == I
+    assert b.variable == x
+
+    # Test the length setter
+    b.length = 4
+    assert b.length == 4
+
+    # Test the E setter
+    b.elastic_modulus = E_1
+    assert b.elastic_modulus == E_1
+
+    # Test the I setter
+    b.second_moment = I_1
+    assert b.second_moment is I_1
+
+    # Test the variable setter
+    b.variable = y
+    assert b.variable is y
+
+    # Test for all boundary conditions.
+    b.bc_deflection = [(0, 2)]
+    b.bc_slope = [(0, 1)]
+    assert b.boundary_conditions == {'deflection': [(0, 2)], 'slope': [(0, 1)]}
+
+    # Test for slope boundary condition method
+    b.bc_slope.extend([(4, 3), (5, 0)])
+    s_bcs = b.bc_slope
+    assert s_bcs == [(0, 1), (4, 3), (5, 0)]
+
+    # Test for deflection boundary condition method
+    b.bc_deflection.extend([(4, 3), (5, 0)])
+    d_bcs = b.bc_deflection
+    assert d_bcs == [(0, 2), (4, 3), (5, 0)]
+
+    # Test for updated boundary conditions
+    bcs_new = b.boundary_conditions
+    assert bcs_new == {
+        'deflection': [(0, 2), (4, 3), (5, 0)],
+        'slope': [(0, 1), (4, 3), (5, 0)]}
+
+    b1 = Beam(30, E, I)
+    b1.apply_load(-8, 0, -1)
+    b1.apply_load(R1, 10, -1)
+    b1.apply_load(R2, 30, -1)
+    b1.apply_load(120, 30, -2)
+    b1.bc_deflection = [(10, 0), (30, 0)]
+    b1.solve_for_reaction_loads(R1, R2)
+
+    # Test for finding reaction forces
+    p = b1.reaction_loads
+    q = {R1: 6, R2: 2}
+    assert p == q
+
+    # Test for load distribution function.
+    p = b1.load
+    q = -8*SingularityFunction(x, 0, -1) + 6*SingularityFunction(x, 10, -1) \
+    + 120*SingularityFunction(x, 30, -2) + 2*SingularityFunction(x, 30, -1)
+    assert p == q
+
+    # Test for shear force distribution function
+    p = b1.shear_force()
+    q = 8*SingularityFunction(x, 0, 0) - 6*SingularityFunction(x, 10, 0) \
+    - 120*SingularityFunction(x, 30, -1) - 2*SingularityFunction(x, 30, 0)
+    assert p == q
+
+    # Test for shear stress distribution function
+    p = b1.shear_stress()
+    q = (8*SingularityFunction(x, 0, 0) - 6*SingularityFunction(x, 10, 0) \
+    - 120*SingularityFunction(x, 30, -1) \
+    - 2*SingularityFunction(x, 30, 0))/A
+    assert p==q
+
+    # Test for bending moment distribution function
+    p = b1.bending_moment()
+    q = 8*SingularityFunction(x, 0, 1) - 6*SingularityFunction(x, 10, 1) \
+    - 120*SingularityFunction(x, 30, 0) - 2*SingularityFunction(x, 30, 1)
+    assert p == q
+
+    # Test for slope distribution function
+    p = b1.slope()
+    q = -4*SingularityFunction(x, 0, 2) + 3*SingularityFunction(x, 10, 2) \
+    + 120*SingularityFunction(x, 30, 1) + SingularityFunction(x, 30, 2) \
+    + Rational(4000, 3)
+    assert p == q/(E*I)
+
+    # Test for deflection distribution function
+    p = b1.deflection()
+    q = x*Rational(4000, 3) - 4*SingularityFunction(x, 0, 3)/3 \
+    + SingularityFunction(x, 10, 3) + 60*SingularityFunction(x, 30, 2) \
+    + SingularityFunction(x, 30, 3)/3 - 12000
+    assert p == q/(E*I)
+
+    # Test using symbols
+    l = Symbol('l')
+    w0 = Symbol('w0')
+    w2 = Symbol('w2')
+    a1 = Symbol('a1')
+    c = Symbol('c')
+    c1 = Symbol('c1')
+    d = Symbol('d')
+    e = Symbol('e')
+    f = Symbol('f')
+
+    b2 = Beam(l, E, I)
+
+    b2.apply_load(w0, a1, 1)
+    b2.apply_load(w2, c1, -1)
+
+    b2.bc_deflection = [(c, d)]
+    b2.bc_slope = [(e, f)]
+
+    # Test for load distribution function.
+    p = b2.load
+    q = w0*SingularityFunction(x, a1, 1) + w2*SingularityFunction(x, c1, -1)
+    assert p == q
+
+    # Test for shear force distribution function
+    p = b2.shear_force()
+    q = -w0*SingularityFunction(x, a1, 2)/2 \
+    - w2*SingularityFunction(x, c1, 0)
+    assert p == q
+
+    # Test for shear stress distribution function
+    p = b2.shear_stress()
+    q = (-w0*SingularityFunction(x, a1, 2)/2 \
+    - w2*SingularityFunction(x, c1, 0))/A
+    assert p == q
+
+    # Test for bending moment distribution function
+    p = b2.bending_moment()
+    q = -w0*SingularityFunction(x, a1, 3)/6 - w2*SingularityFunction(x, c1, 1)
+    assert p == q
+
+    # Test for slope distribution function
+    p = b2.slope()
+    q = (w0*SingularityFunction(x, a1, 4)/24 + w2*SingularityFunction(x, c1, 2)/2)/(E*I) + (E*I*f - w0*SingularityFunction(e, a1, 4)/24 - w2*SingularityFunction(e, c1, 2)/2)/(E*I)
+    assert expand(p) == expand(q)
+
+    # Test for deflection distribution function
+    p = b2.deflection()
+    q = x*(E*I*f - w0*SingularityFunction(e, a1, 4)/24 \
+    - w2*SingularityFunction(e, c1, 2)/2)/(E*I) \
+    + (w0*SingularityFunction(x, a1, 5)/120 \
+    + w2*SingularityFunction(x, c1, 3)/6)/(E*I) \
+    + (E*I*(-c*f + d) + c*w0*SingularityFunction(e, a1, 4)/24 \
+    + c*w2*SingularityFunction(e, c1, 2)/2 \
+    - w0*SingularityFunction(c, a1, 5)/120 \
+    - w2*SingularityFunction(c, c1, 3)/6)/(E*I)
+    assert simplify(p - q) == 0
+
+    b3 = Beam(9, E, I, 2)
+    b3.apply_load(value=-2, start=2, order=2, end=3)
+    b3.bc_slope.append((0, 2))
+    C3 = symbols('C3')
+    C4 = symbols('C4')
+
+    p = b3.load
+    q = -2*SingularityFunction(x, 2, 2) + 2*SingularityFunction(x, 3, 0) \
+    + 4*SingularityFunction(x, 3, 1) + 2*SingularityFunction(x, 3, 2)
+    assert p == q
+
+    p = b3.shear_force()
+    q = 2*SingularityFunction(x, 2, 3)/3 - 2*SingularityFunction(x, 3, 1) \
+    - 2*SingularityFunction(x, 3, 2) - 2*SingularityFunction(x, 3, 3)/3
+    assert p == q
+
+    p = b3.shear_stress()
+    q = SingularityFunction(x, 2, 3)/3 - 1*SingularityFunction(x, 3, 1) \
+    - 1*SingularityFunction(x, 3, 2) - 1*SingularityFunction(x, 3, 3)/3
+    assert p == q
+
+    p = b3.slope()
+    q = 2 - (SingularityFunction(x, 2, 5)/30 - SingularityFunction(x, 3, 3)/3 \
+    - SingularityFunction(x, 3, 4)/6 - SingularityFunction(x, 3, 5)/30)/(E*I)
+    assert p == q
+
+    p = b3.deflection()
+    q = 2*x - (SingularityFunction(x, 2, 6)/180 \
+    - SingularityFunction(x, 3, 4)/12 - SingularityFunction(x, 3, 5)/30 \
+    - SingularityFunction(x, 3, 6)/180)/(E*I)
+    assert p == q + C4
+
+    b4 = Beam(4, E, I, 3)
+    b4.apply_load(-3, 0, 0, end=3)
+
+    p = b4.load
+    q = -3*SingularityFunction(x, 0, 0) + 3*SingularityFunction(x, 3, 0)
+    assert p == q
+
+    p = b4.shear_force()
+    q = 3*SingularityFunction(x, 0, 1) \
+    - 3*SingularityFunction(x, 3, 1)
+    assert p == q
+
+    p = b4.shear_stress()
+    q = SingularityFunction(x, 0, 1) - SingularityFunction(x, 3, 1)
+    assert p == q
+
+    p = b4.slope()
+    q = -3*SingularityFunction(x, 0, 3)/6 + 3*SingularityFunction(x, 3, 3)/6
+    assert p == q/(E*I) + C3
+
+    p = b4.deflection()
+    q = -3*SingularityFunction(x, 0, 4)/24 + 3*SingularityFunction(x, 3, 4)/24
+    assert p == q/(E*I) + C3*x + C4
+
+    # can't use end with point loads
+    raises(ValueError, lambda: b4.apply_load(-3, 0, -1, end=3))
+    with raises(TypeError):
+        b4.variable = 1
+
+
+def test_insufficient_bconditions():
+    # Test cases when required number of boundary conditions
+    # are not provided to solve the integration constants.
+    L = symbols('L', positive=True)
+    E, I, P, a3, a4 = symbols('E I P a3 a4')
+
+    b = Beam(L, E, I, base_char='a')
+    b.apply_load(R2, L, -1)
+    b.apply_load(R1, 0, -1)
+    b.apply_load(-P, L/2, -1)
+    b.solve_for_reaction_loads(R1, R2)
+
+    p = b.slope()
+    q = P*SingularityFunction(x, 0, 2)/4 - P*SingularityFunction(x, L/2, 2)/2 + P*SingularityFunction(x, L, 2)/4
+    assert p == q/(E*I) + a3
+
+    p = b.deflection()
+    q = P*SingularityFunction(x, 0, 3)/12 - P*SingularityFunction(x, L/2, 3)/6 + P*SingularityFunction(x, L, 3)/12
+    assert p == q/(E*I) + a3*x + a4
+
+    b.bc_deflection = [(0, 0)]
+    p = b.deflection()
+    q = a3*x + P*SingularityFunction(x, 0, 3)/12 - P*SingularityFunction(x, L/2, 3)/6 + P*SingularityFunction(x, L, 3)/12
+    assert p == q/(E*I)
+
+    b.bc_deflection = [(0, 0), (L, 0)]
+    p = b.deflection()
+    q = -L**2*P*x/16 + P*SingularityFunction(x, 0, 3)/12 - P*SingularityFunction(x, L/2, 3)/6 + P*SingularityFunction(x, L, 3)/12
+    assert p == q/(E*I)
+
+
+def test_statically_indeterminate():
+    E = Symbol('E')
+    I = Symbol('I')
+    M1, M2 = symbols('M1, M2')
+    F = Symbol('F')
+    l = Symbol('l', positive=True)
+
+    b5 = Beam(l, E, I)
+    b5.bc_deflection = [(0, 0),(l, 0)]
+    b5.bc_slope = [(0, 0),(l, 0)]
+
+    b5.apply_load(R1, 0, -1)
+    b5.apply_load(M1, 0, -2)
+    b5.apply_load(R2, l, -1)
+    b5.apply_load(M2, l, -2)
+    b5.apply_load(-F, l/2, -1)
+
+    b5.solve_for_reaction_loads(R1, R2, M1, M2)
+    p = b5.reaction_loads
+    q = {R1: F/2, R2: F/2, M1: -F*l/8, M2: F*l/8}
+    assert p == q
+
+
+def test_beam_units():
+    E = Symbol('E')
+    I = Symbol('I')
+    R1, R2 = symbols('R1, R2')
+
+    kN = kilo*newton
+    gN = giga*newton
+
+    b = Beam(8*meter, 200*gN/meter**2, 400*1000000*(milli*meter)**4)
+    b.apply_load(5*kN, 2*meter, -1)
+    b.apply_load(R1, 0*meter, -1)
+    b.apply_load(R2, 8*meter, -1)
+    b.apply_load(10*kN/meter, 4*meter, 0, end=8*meter)
+    b.bc_deflection = [(0*meter, 0*meter), (8*meter, 0*meter)]
+    b.solve_for_reaction_loads(R1, R2)
+    assert b.reaction_loads == {R1: -13750*newton, R2: -31250*newton}
+
+    b = Beam(3*meter, E*newton/meter**2, I*meter**4)
+    b.apply_load(8*kN, 1*meter, -1)
+    b.apply_load(R1, 0*meter, -1)
+    b.apply_load(R2, 3*meter, -1)
+    b.apply_load(12*kN*meter, 2*meter, -2)
+    b.bc_deflection = [(0*meter, 0*meter), (3*meter, 0*meter)]
+    b.solve_for_reaction_loads(R1, R2)
+    assert b.reaction_loads == {R1: newton*Rational(-28000, 3), R2: newton*Rational(4000, 3)}
+    assert b.deflection().subs(x, 1*meter) == 62000*meter/(9*E*I)
+
+
+def test_variable_moment():
+    E = Symbol('E')
+    I = Symbol('I')
+
+    b = Beam(4, E, 2*(4 - x))
+    b.apply_load(20, 4, -1)
+    R, M = symbols('R, M')
+    b.apply_load(R, 0, -1)
+    b.apply_load(M, 0, -2)
+    b.bc_deflection = [(0, 0)]
+    b.bc_slope = [(0, 0)]
+    b.solve_for_reaction_loads(R, M)
+    assert b.slope().expand() == ((10*x*SingularityFunction(x, 0, 0)
+        - 10*(x - 4)*SingularityFunction(x, 4, 0))/E).expand()
+    assert b.deflection().expand() == ((5*x**2*SingularityFunction(x, 0, 0)
+        - 10*Piecewise((0, Abs(x)/4 < 1), (x**2*meijerg(((-1, 1), ()), ((), (-2, 0)), x/4), True))
+        + 40*SingularityFunction(x, 4, 1))/E).expand()
+
+    b = Beam(4, E - x, I)
+    b.apply_load(20, 4, -1)
+    R, M = symbols('R, M')
+    b.apply_load(R, 0, -1)
+    b.apply_load(M, 0, -2)
+    b.bc_deflection = [(0, 0)]
+    b.bc_slope = [(0, 0)]
+    b.solve_for_reaction_loads(R, M)
+    assert b.slope().expand() == ((-80*(-log(-E) + log(-E + x))*SingularityFunction(x, 0, 0)
+        + 80*(-log(-E + 4) + log(-E + x))*SingularityFunction(x, 4, 0) + 20*(-E*log(-E)
+        + E*log(-E + x) + x)*SingularityFunction(x, 0, 0) - 20*(-E*log(-E + 4) + E*log(-E + x)
+        + x - 4)*SingularityFunction(x, 4, 0))/I).expand()
+
+
+def test_composite_beam():
+    E = Symbol('E')
+    I = Symbol('I')
+    b1 = Beam(2, E, 1.5*I)
+    b2 = Beam(2, E, I)
+    b = b1.join(b2, "fixed")
+    b.apply_load(-20, 0, -1)
+    b.apply_load(80, 0, -2)
+    b.apply_load(20, 4, -1)
+    b.bc_slope = [(0, 0)]
+    b.bc_deflection = [(0, 0)]
+    assert b.length == 4
+    assert b.second_moment == Piecewise((1.5*I, x <= 2), (I, x <= 4))
+    assert b.slope().subs(x, 4) == 120.0/(E*I)
+    assert b.slope().subs(x, 2) == 80.0/(E*I)
+    assert int(b.deflection().subs(x, 4).args[0]) == -302  # Coefficient of 1/(E*I)
+
+    l = symbols('l', positive=True)
+    R1, M1, R2, R3, P = symbols('R1 M1 R2 R3 P')
+    b1 = Beam(2*l, E, I)
+    b2 = Beam(2*l, E, I)
+    b = b1.join(b2,"hinge")
+    b.apply_load(M1, 0, -2)
+    b.apply_load(R1, 0, -1)
+    b.apply_load(R2, l, -1)
+    b.apply_load(R3, 4*l, -1)
+    b.apply_load(P, 3*l, -1)
+    b.bc_slope = [(0, 0)]
+    b.bc_deflection = [(0, 0), (l, 0), (4*l, 0)]
+    b.solve_for_reaction_loads(M1, R1, R2, R3)
+    assert b.reaction_loads == {R3: -P/2, R2: P*Rational(-5, 4), M1: -P*l/4, R1: P*Rational(3, 4)}
+    assert b.slope().subs(x, 3*l) == -7*P*l**2/(48*E*I)
+    assert b.deflection().subs(x, 2*l) == 7*P*l**3/(24*E*I)
+    assert b.deflection().subs(x, 3*l) == 5*P*l**3/(16*E*I)
+
+    # When beams having same second moment are joined.
+    b1 = Beam(2, 500, 10)
+    b2 = Beam(2, 500, 10)
+    b = b1.join(b2, "fixed")
+    b.apply_load(M1, 0, -2)
+    b.apply_load(R1, 0, -1)
+    b.apply_load(R2, 1, -1)
+    b.apply_load(R3, 4, -1)
+    b.apply_load(10, 3, -1)
+    b.bc_slope = [(0, 0)]
+    b.bc_deflection = [(0, 0), (1, 0), (4, 0)]
+    b.solve_for_reaction_loads(M1, R1, R2, R3)
+    assert b.slope() == -2*SingularityFunction(x, 0, 1)/5625 + SingularityFunction(x, 0, 2)/1875\
+                - 133*SingularityFunction(x, 1, 2)/135000 + SingularityFunction(x, 3, 2)/1000\
+                - 37*SingularityFunction(x, 4, 2)/67500
+    assert b.deflection() == -SingularityFunction(x, 0, 2)/5625 + SingularityFunction(x, 0, 3)/5625\
+                    - 133*SingularityFunction(x, 1, 3)/405000 + SingularityFunction(x, 3, 3)/3000\
+                    - 37*SingularityFunction(x, 4, 3)/202500
+
+
+def test_point_cflexure():
+    E = Symbol('E')
+    I = Symbol('I')
+    b = Beam(10, E, I)
+    b.apply_load(-4, 0, -1)
+    b.apply_load(-46, 6, -1)
+    b.apply_load(10, 2, -1)
+    b.apply_load(20, 4, -1)
+    b.apply_load(3, 6, 0)
+    assert b.point_cflexure() == [Rational(10, 3)]
+
+
+def test_remove_load():
+    E = Symbol('E')
+    I = Symbol('I')
+    b = Beam(4, E, I)
+
+    try:
+        b.remove_load(2, 1, -1)
+    # As no load is applied on beam, ValueError should be returned.
+    except ValueError:
+        assert True
+    else:
+        assert False
+
+    b.apply_load(-3, 0, -2)
+    b.apply_load(4, 2, -1)
+    b.apply_load(-2, 2, 2, end = 3)
+    b.remove_load(-2, 2, 2, end = 3)
+    assert b.load == -3*SingularityFunction(x, 0, -2) + 4*SingularityFunction(x, 2, -1)
+    assert b.applied_loads == [(-3, 0, -2, None), (4, 2, -1, None)]
+
+    try:
+        b.remove_load(1, 2, -1)
+    # As load of this magnitude was never applied at
+    # this position, method should return a ValueError.
+    except ValueError:
+        assert True
+    else:
+        assert False
+
+    b.remove_load(-3, 0, -2)
+    b.remove_load(4, 2, -1)
+    assert b.load == 0
+    assert b.applied_loads == []
+
+
+def test_apply_support():
+    E = Symbol('E')
+    I = Symbol('I')
+
+    b = Beam(4, E, I)
+    b.apply_support(0, "cantilever")
+    b.apply_load(20, 4, -1)
+    M_0, R_0 = symbols('M_0, R_0')
+    b.solve_for_reaction_loads(R_0, M_0)
+    assert simplify(b.slope()) == simplify((80*SingularityFunction(x, 0, 1) - 10*SingularityFunction(x, 0, 2)
+                + 10*SingularityFunction(x, 4, 2))/(E*I))
+    assert simplify(b.deflection()) == simplify((40*SingularityFunction(x, 0, 2) - 10*SingularityFunction(x, 0, 3)/3
+                + 10*SingularityFunction(x, 4, 3)/3)/(E*I))
+
+    b = Beam(30, E, I)
+    p0 = b.apply_support(10, "pin")
+    p1 = b.apply_support(30, "roller")
+    b.apply_load(-8, 0, -1)
+    b.apply_load(120, 30, -2)
+    b.solve_for_reaction_loads(p0, p1)
+    assert b.slope() == (-4*SingularityFunction(x, 0, 2) + 3*SingularityFunction(x, 10, 2)
+            + 120*SingularityFunction(x, 30, 1) + SingularityFunction(x, 30, 2) + Rational(4000, 3))/(E*I)
+    assert b.deflection() == (x*Rational(4000, 3) - 4*SingularityFunction(x, 0, 3)/3 + SingularityFunction(x, 10, 3)
+            + 60*SingularityFunction(x, 30, 2) + SingularityFunction(x, 30, 3)/3 - 12000)/(E*I)
+    R_10 = Symbol('R_10')
+    R_30 = Symbol('R_30')
+    assert p0 == R_10
+    assert b.reaction_loads == {R_10: 6, R_30: 2}
+    assert b.reaction_loads[p0] == 6
+
+    b = Beam(8, E, I)
+    p0, m0 = b.apply_support(0, "fixed")
+    p1 = b.apply_support(8, "roller")
+    b.apply_load(-5, 0, 0, 8)
+    b.solve_for_reaction_loads(p0, m0, p1)
+    R_0 = Symbol('R_0')
+    M_0 = Symbol('M_0')
+    R_8 = Symbol('R_8')
+    assert p0 == R_0
+    assert m0 == M_0
+    assert p1 == R_8
+    assert b.reaction_loads == {R_0: 25, M_0: -40, R_8: 15}
+    assert b.reaction_loads[m0] == -40
+
+    P = Symbol('P', positive=True)
+    L = Symbol('L', positive=True)
+    b = Beam(L, E, I)
+    b.apply_support(0, type='fixed')
+    b.apply_support(L, type='fixed')
+    b.apply_load(-P, L/2, -1)
+    R_0, R_L, M_0, M_L = symbols('R_0, R_L, M_0, M_L')
+    b.solve_for_reaction_loads(R_0, R_L, M_0, M_L)
+    assert b.reaction_loads == {R_0: P/2, R_L: P/2, M_0: -L*P/8, M_L: L*P/8}
+
+
+def test_max_shear_force():
+    E = Symbol('E')
+    I = Symbol('I')
+
+    b = Beam(3, E, I)
+    R, M = symbols('R, M')
+    b.apply_load(R, 0, -1)
+    b.apply_load(M, 0, -2)
+    b.apply_load(2, 3, -1)
+    b.apply_load(4, 2, -1)
+    b.apply_load(2, 2, 0, end=3)
+    b.solve_for_reaction_loads(R, M)
+    assert b.max_shear_force() == (Interval(0, 2), 8)
+
+    l = symbols('l', positive=True)
+    P = Symbol('P')
+    b = Beam(l, E, I)
+    R1, R2 = symbols('R1, R2')
+    b.apply_load(R1, 0, -1)
+    b.apply_load(R2, l, -1)
+    b.apply_load(P, 0, 0, end=l)
+    b.solve_for_reaction_loads(R1, R2)
+    max_shear = b.max_shear_force()
+    assert max_shear[0] == 0
+    assert simplify(max_shear[1] - (l*Abs(P)/2)) == 0
+
+
+def test_max_bmoment():
+    E = Symbol('E')
+    I = Symbol('I')
+    l, P = symbols('l, P', positive=True)
+
+    b = Beam(l, E, I)
+    R1, R2 = symbols('R1, R2')
+    b.apply_load(R1, 0, -1)
+    b.apply_load(R2, l, -1)
+    b.apply_load(P, l/2, -1)
+    b.solve_for_reaction_loads(R1, R2)
+    b.reaction_loads
+    assert b.max_bmoment() == (l/2, P*l/4)
+
+    b = Beam(l, E, I)
+    R1, R2 = symbols('R1, R2')
+    b.apply_load(R1, 0, -1)
+    b.apply_load(R2, l, -1)
+    b.apply_load(P, 0, 0, end=l)
+    b.solve_for_reaction_loads(R1, R2)
+    assert b.max_bmoment() == (l/2, P*l**2/8)
+
+
+def test_max_deflection():
+    E, I, l, F = symbols('E, I, l, F', positive=True)
+    b = Beam(l, E, I)
+    b.bc_deflection = [(0, 0),(l, 0)]
+    b.bc_slope = [(0, 0),(l, 0)]
+    b.apply_load(F/2, 0, -1)
+    b.apply_load(-F*l/8, 0, -2)
+    b.apply_load(F/2, l, -1)
+    b.apply_load(F*l/8, l, -2)
+    b.apply_load(-F, l/2, -1)
+    assert b.max_deflection() == (l/2, F*l**3/(192*E*I))
+
+
+def test_Beam3D():
+    l, E, G, I, A = symbols('l, E, G, I, A')
+    R1, R2, R3, R4 = symbols('R1, R2, R3, R4')
+
+    b = Beam3D(l, E, G, I, A)
+    m, q = symbols('m, q')
+    b.apply_load(q, 0, 0, dir="y")
+    b.apply_moment_load(m, 0, 0, dir="z")
+    b.bc_slope = [(0, [0, 0, 0]), (l, [0, 0, 0])]
+    b.bc_deflection = [(0, [0, 0, 0]), (l, [0, 0, 0])]
+    b.solve_slope_deflection()
+
+    assert b.polar_moment() == 2*I
+    assert b.shear_force() == [0, -q*x, 0]
+    assert b.shear_stress() == [0, -q*x/A, 0]
+    assert b.axial_stress() == 0
+    assert b.bending_moment() == [0, 0, -m*x + q*x**2/2]
+    expected_deflection = (x*(A*G*q*x**3/4 + A*G*x**2*(-l*(A*G*l*(l*q - 2*m) +
+        12*E*I*q)/(A*G*l**2 + 12*E*I)/2 - m) + 3*E*I*l*(A*G*l*(l*q - 2*m) +
+        12*E*I*q)/(A*G*l**2 + 12*E*I) + x*(-A*G*l**2*q/2 +
+        3*A*G*l**2*(A*G*l*(l*q - 2*m) + 12*E*I*q)/(A*G*l**2 + 12*E*I)/4 +
+        A*G*l*m*Rational(3, 2) - 3*E*I*q))/(6*A*E*G*I))
+    dx, dy, dz = b.deflection()
+    assert dx == dz == 0
+    assert simplify(dy - expected_deflection) == 0
+
+    b2 = Beam3D(30, E, G, I, A, x)
+    b2.apply_load(50, start=0, order=0, dir="y")
+    b2.bc_deflection = [(0, [0, 0, 0]), (30, [0, 0, 0])]
+    b2.apply_load(R1, start=0, order=-1, dir="y")
+    b2.apply_load(R2, start=30, order=-1, dir="y")
+    b2.solve_for_reaction_loads(R1, R2)
+    assert b2.reaction_loads == {R1: -750, R2: -750}
+
+    b2.solve_slope_deflection()
+    assert b2.slope() == [0, 0, 25*x**3/(3*E*I) - 375*x**2/(E*I) + 3750*x/(E*I)]
+    expected_deflection = 25*x**4/(12*E*I) - 125*x**3/(E*I) + 1875*x**2/(E*I) - \
+        25*x**2/(A*G) + 750*x/(A*G)
+    dx, dy, dz = b2.deflection()
+    assert dx == dz == 0
+    assert dy == expected_deflection
+
+    # Test for solve_for_reaction_loads
+    b3 = Beam3D(30, E, G, I, A, x)
+    b3.apply_load(8, start=0, order=0, dir="y")
+    b3.apply_load(9*x, start=0, order=0, dir="z")
+    b3.apply_load(R1, start=0, order=-1, dir="y")
+    b3.apply_load(R2, start=30, order=-1, dir="y")
+    b3.apply_load(R3, start=0, order=-1, dir="z")
+    b3.apply_load(R4, start=30, order=-1, dir="z")
+    b3.solve_for_reaction_loads(R1, R2, R3, R4)
+    assert b3.reaction_loads == {R1: -120, R2: -120, R3: -1350, R4: -2700}
+
+
+def test_polar_moment_Beam3D():
+    l, E, G, A, I1, I2 = symbols('l, E, G, A, I1, I2')
+    I = [I1, I2]
+
+    b = Beam3D(l, E, G, I, A)
+    assert b.polar_moment() == I1 + I2
+
+
+def test_parabolic_loads():
+
+    E, I, L = symbols('E, I, L', positive=True, real=True)
+    R, M, P = symbols('R, M, P', real=True)
+
+    # cantilever beam fixed at x=0 and parabolic distributed loading across
+    # length of beam
+    beam = Beam(L, E, I)
+
+    beam.bc_deflection.append((0, 0))
+    beam.bc_slope.append((0, 0))
+    beam.apply_load(R, 0, -1)
+    beam.apply_load(M, 0, -2)
+
+    # parabolic load
+    beam.apply_load(1, 0, 2)
+
+    beam.solve_for_reaction_loads(R, M)
+
+    assert beam.reaction_loads[R] == -L**3/3
+
+    # cantilever beam fixed at x=0 and parabolic distributed loading across
+    # first half of beam
+    beam = Beam(2*L, E, I)
+
+    beam.bc_deflection.append((0, 0))
+    beam.bc_slope.append((0, 0))
+    beam.apply_load(R, 0, -1)
+    beam.apply_load(M, 0, -2)
+
+    # parabolic load from x=0 to x=L
+    beam.apply_load(1, 0, 2, end=L)
+
+    beam.solve_for_reaction_loads(R, M)
+
+    # result should be the same as the prior example
+    assert beam.reaction_loads[R] == -L**3/3
+
+    # check constant load
+    beam = Beam(2*L, E, I)
+    beam.apply_load(P, 0, 0, end=L)
+    loading = beam.load.xreplace({L: 10, E: 20, I: 30, P: 40})
+    assert loading.xreplace({x: 5}) == 40
+    assert loading.xreplace({x: 15}) == 0
+
+    # check ramp load
+    beam = Beam(2*L, E, I)
+    beam.apply_load(P, 0, 1, end=L)
+    assert beam.load == (P*SingularityFunction(x, 0, 1) -
+                         P*SingularityFunction(x, L, 1) -
+                         P*L*SingularityFunction(x, L, 0))
+
+    # check higher order load: x**8 load from x=0 to x=L
+    beam = Beam(2*L, E, I)
+    beam.apply_load(P, 0, 8, end=L)
+    loading = beam.load.xreplace({L: 10, E: 20, I: 30, P: 40})
+    assert loading.xreplace({x: 5}) == 40*5**8
+    assert loading.xreplace({x: 15}) == 0
+
+
+def test_cross_section():
+    I = Symbol('I')
+    l = Symbol('l')
+    E = Symbol('E')
+    C3, C4 = symbols('C3, C4')
+    a, c, g, h, r, n = symbols('a, c, g, h, r, n')
+
+    # test for second_moment and cross_section setter
+    b0 = Beam(l, E, I)
+    assert b0.second_moment == I
+    assert b0.cross_section == None
+    b0.cross_section = Circle((0, 0), 5)
+    assert b0.second_moment == pi*Rational(625, 4)
+    assert b0.cross_section == Circle((0, 0), 5)
+    b0.second_moment = 2*n - 6
+    assert b0.second_moment == 2*n-6
+    assert b0.cross_section == None
+    with raises(ValueError):
+        b0.second_moment = Circle((0, 0), 5)
+
+    # beam with a circular cross-section
+    b1 = Beam(50, E, Circle((0, 0), r))
+    assert b1.cross_section == Circle((0, 0), r)
+    assert b1.second_moment == pi*r*Abs(r)**3/4
+
+    b1.apply_load(-10, 0, -1)
+    b1.apply_load(R1, 5, -1)
+    b1.apply_load(R2, 50, -1)
+    b1.apply_load(90, 45, -2)
+    b1.solve_for_reaction_loads(R1, R2)
+    assert b1.load == (-10*SingularityFunction(x, 0, -1) + 82*SingularityFunction(x, 5, -1)/S(9)
+                         + 90*SingularityFunction(x, 45, -2) + 8*SingularityFunction(x, 50, -1)/9)
+    assert b1.bending_moment() == (10*SingularityFunction(x, 0, 1) - 82*SingularityFunction(x, 5, 1)/9
+                                     - 90*SingularityFunction(x, 45, 0) - 8*SingularityFunction(x, 50, 1)/9)
+    q = (-5*SingularityFunction(x, 0, 2) + 41*SingularityFunction(x, 5, 2)/S(9)
+           + 90*SingularityFunction(x, 45, 1) + 4*SingularityFunction(x, 50, 2)/S(9))/(pi*E*r*Abs(r)**3)
+    assert b1.slope() == C3 + 4*q
+    q = (-5*SingularityFunction(x, 0, 3)/3 + 41*SingularityFunction(x, 5, 3)/27 + 45*SingularityFunction(x, 45, 2)
+           + 4*SingularityFunction(x, 50, 3)/27)/(pi*E*r*Abs(r)**3)
+    assert b1.deflection() == C3*x + C4 + 4*q
+
+    # beam with a recatangular cross-section
+    b2 = Beam(20, E, Polygon((0, 0), (a, 0), (a, c), (0, c)))
+    assert b2.cross_section == Polygon((0, 0), (a, 0), (a, c), (0, c))
+    assert b2.second_moment == a*c**3/12
+    # beam with a triangular cross-section
+    b3 = Beam(15, E, Triangle((0, 0), (g, 0), (g/2, h)))
+    assert b3.cross_section == Triangle(Point2D(0, 0), Point2D(g, 0), Point2D(g/2, h))
+    assert b3.second_moment == g*h**3/36
+
+    # composite beam
+    b = b2.join(b3, "fixed")
+    b.apply_load(-30, 0, -1)
+    b.apply_load(65, 0, -2)
+    b.apply_load(40, 0, -1)
+    b.bc_slope = [(0, 0)]
+    b.bc_deflection = [(0, 0)]
+
+    assert b.second_moment == Piecewise((a*c**3/12, x <= 20), (g*h**3/36, x <= 35))
+    assert b.cross_section == None
+    assert b.length == 35
+    assert b.slope().subs(x, 7) == 8400/(E*a*c**3)
+    assert b.slope().subs(x, 25) == 52200/(E*g*h**3) + 39600/(E*a*c**3)
+    assert b.deflection().subs(x, 30) == -537000/(E*g*h**3) - 712000/(E*a*c**3)
+
+def test_max_shear_force_Beam3D():
+    x = symbols('x')
+    b = Beam3D(20, 40, 21, 100, 25)
+    b.apply_load(15, start=0, order=0, dir="z")
+    b.apply_load(12*x, start=0, order=0, dir="y")
+    b.bc_deflection = [(0, [0, 0, 0]), (20, [0, 0, 0])]
+    assert b.max_shear_force() == [(0, 0), (20, 2400), (20, 300)]
+
+def test_max_bending_moment_Beam3D():
+    x = symbols('x')
+    b = Beam3D(20, 40, 21, 100, 25)
+    b.apply_load(15, start=0, order=0, dir="z")
+    b.apply_load(12*x, start=0, order=0, dir="y")
+    b.bc_deflection = [(0, [0, 0, 0]), (20, [0, 0, 0])]
+    assert b.max_bmoment() == [(0, 0), (20, 3000), (20, 16000)]
+
+def test_max_deflection_Beam3D():
+    x = symbols('x')
+    b = Beam3D(20, 40, 21, 100, 25)
+    b.apply_load(15, start=0, order=0, dir="z")
+    b.apply_load(12*x, start=0, order=0, dir="y")
+    b.bc_deflection = [(0, [0, 0, 0]), (20, [0, 0, 0])]
+    b.solve_slope_deflection()
+    c = sympify("495/14")
+    p = sympify("-10 + 10*sqrt(10793)/43")
+    q = sympify("(10 - 10*sqrt(10793)/43)**3/160 - 20/7 + (10 - 10*sqrt(10793)/43)**4/6400 + 20*sqrt(10793)/301 + 27*(10 - 10*sqrt(10793)/43)**2/560")
+    assert b.max_deflection() == [(0, 0), (10, c), (p, q)]
+
+def test_torsion_Beam3D():
+    x = symbols('x')
+    b = Beam3D(20, 40, 21, 100, 25)
+    b.apply_moment_load(15, 5, -2, dir='x')
+    b.apply_moment_load(25, 10, -2, dir='x')
+    b.apply_moment_load(-5, 20, -2, dir='x')
+    b.solve_for_torsion()
+    assert b.angular_deflection().subs(x, 3) == sympify("1/40")
+    assert b.angular_deflection().subs(x, 9) == sympify("17/280")
+    assert b.angular_deflection().subs(x, 12) == sympify("53/840")
+    assert b.angular_deflection().subs(x, 17) == sympify("2/35")
+    assert b.angular_deflection().subs(x, 20) == sympify("3/56")

parrot/lib/python3.10/site-packages/sympy/physics/continuum_mechanics/tests/test_cable.py

@@ -0,0 +1,83 @@
+from sympy.physics.continuum_mechanics.cable import Cable
+from sympy.core.symbol import Symbol
+
+
+def test_cable():
+    c = Cable(('A', 0, 10), ('B', 10, 10))
+    assert c.supports == {'A': [0, 10], 'B': [10, 10]}
+    assert c.left_support == [0, 10]
+    assert c.right_support == [10, 10]
+    assert c.loads == {'distributed': {}, 'point_load': {}}
+    assert c.loads_position == {}
+    assert c.length == 0
+    assert c.reaction_loads == {Symbol("R_A_x"): 0, Symbol("R_A_y"): 0, Symbol("R_B_x"): 0, Symbol("R_B_y"): 0}
+
+    # tests for change_support method
+    c.change_support('A', ('C', 12, 3))
+    assert c.supports == {'B': [10, 10], 'C': [12, 3]}
+    assert c.left_support == [10, 10]
+    assert c.right_support == [12, 3]
+    assert c.reaction_loads == {Symbol("R_B_x"): 0, Symbol("R_B_y"): 0, Symbol("R_C_x"): 0, Symbol("R_C_y"): 0}
+
+    c.change_support('C', ('A', 0, 10))
+
+    # tests for apply_load method for point loads
+    c.apply_load(-1, ('X', 2, 5, 3, 30))
+    c.apply_load(-1, ('Y', 5, 8, 5, 60))
+    assert c.loads == {'distributed': {}, 'point_load': {'X': [3, 30], 'Y': [5, 60]}}
+    assert c.loads_position == {'X': [2, 5], 'Y': [5, 8]}
+    assert c.length == 0
+    assert c.reaction_loads == {Symbol("R_A_x"): 0, Symbol("R_A_y"): 0, Symbol("R_B_x"): 0, Symbol("R_B_y"): 0}
+
+    # tests for remove_loads method
+    c.remove_loads('X')
+    assert c.loads == {'distributed': {}, 'point_load': {'Y': [5, 60]}}
+    assert c.loads_position == {'Y': [5, 8]}
+    assert c.length == 0
+    assert c.reaction_loads == {Symbol("R_A_x"): 0, Symbol("R_A_y"): 0, Symbol("R_B_x"): 0, Symbol("R_B_y"): 0}
+
+    c.remove_loads('Y')
+
+    #tests for apply_load method for distributed load
+    c.apply_load(0, ('Z', 9))
+    assert c.loads == {'distributed': {'Z': 9}, 'point_load': {}}
+    assert c.loads_position == {}
+    assert c.length == 0
+    assert c.reaction_loads == {Symbol("R_A_x"): 0, Symbol("R_A_y"): 0, Symbol("R_B_x"): 0, Symbol("R_B_y"): 0}
+
+    # tests for apply_length method
+    c.apply_length(20)
+    assert c.length == 20
+
+    del c
+    # tests for solve method
+    # for point loads
+    c = Cable(("A", 0, 10), ("B", 5.5, 8))
+    c.apply_load(-1, ('Z', 2, 7.26, 3, 270))
+    c.apply_load(-1, ('X', 4, 6, 8, 270))
+    c.solve()
+    #assert c.tension == {Symbol("Z_X"): 4.79150773600774, Symbol("X_B"): 6.78571428571429, Symbol("A_Z"): 6.89488895397307}
+    assert abs(c.tension[Symbol("A_Z")] - 6.89488895397307) < 10e-12
+    assert abs(c.tension[Symbol("Z_X")] - 4.79150773600774) < 10e-12
+    assert abs(c.tension[Symbol("X_B")] - 6.78571428571429) < 10e-12
+    #assert c.reaction_loads == {Symbol("R_A_x"): -4.06504065040650, Symbol("R_A_y"): 5.56910569105691, Symbol("R_B_x"): 4.06504065040650, Symbol("R_B_y"): 5.43089430894309}
+    assert abs(c.reaction_loads[Symbol("R_A_x")] + 4.06504065040650) < 10e-12
+    assert abs(c.reaction_loads[Symbol("R_A_y")] - 5.56910569105691) < 10e-12
+    assert abs(c.reaction_loads[Symbol("R_B_x")] - 4.06504065040650) < 10e-12
+    assert abs(c.reaction_loads[Symbol("R_B_y")] - 5.43089430894309) < 10e-12
+    assert abs(c.length - 8.25609584845190) < 10e-12
+
+    del c
+    # tests for solve method
+    # for distributed loads
+    c=Cable(("A", 0, 40),("B", 100, 20))
+    c.apply_load(0, ("X", 850))
+    c.solve(58.58, 0)
+
+    # assert c.tension['distributed'] == 36456.8485*sqrt(0.000543529004799705*(X + 0.00135624381275735)**2 + 1)
+    assert abs(c.tension_at(0) - 61709.0363315913) < 10e-11
+    assert abs(c.tension_at(40) - 39729.7316969361) < 10e-11
+    assert abs(c.reaction_loads[Symbol("R_A_x")] - 36456.8485000000) < 10e-11
+    assert abs(c.reaction_loads[Symbol("R_A_y")] + 49788.5866682486) < 10e-11
+    assert abs(c.reaction_loads[Symbol("R_B_x")] - 44389.8401587246) < 10e-11
+    assert abs(c.reaction_loads[Symbol("R_B_y")] - 42866.6216963330) < 10e-11

parrot/lib/python3.10/site-packages/sympy/physics/continuum_mechanics/tests/test_truss.py

@@ -0,0 +1,100 @@
+from sympy.core.symbol import Symbol, symbols
+from sympy.physics.continuum_mechanics.truss import Truss
+from sympy import sqrt
+
+
+def test_truss():
+    A = Symbol('A')
+    B = Symbol('B')
+    C = Symbol('C')
+    AB, BC, AC = symbols('AB, BC, AC')
+    P = Symbol('P')
+
+    t = Truss()
+    assert t.nodes == []
+    assert t.node_labels == []
+    assert t.node_positions == []
+    assert t.members == {}
+    assert t.loads == {}
+    assert t.supports == {}
+    assert t.reaction_loads == {}
+    assert t.internal_forces == {}
+
+    # testing the add_node method
+    t.add_node((A, 0, 0), (B, 2, 2), (C, 3, 0))
+    assert t.nodes == [(A, 0, 0), (B, 2, 2), (C, 3, 0)]
+    assert t.node_labels == [A, B, C]
+    assert t.node_positions == [(0, 0), (2, 2), (3, 0)]
+    assert t.loads == {}
+    assert t.supports == {}
+    assert t.reaction_loads == {}
+
+    # testing the remove_node method
+    t.remove_node(C)
+    assert t.nodes == [(A, 0, 0), (B, 2, 2)]
+    assert t.node_labels == [A, B]
+    assert t.node_positions == [(0, 0), (2, 2)]
+    assert t.loads == {}
+    assert t.supports == {}
+
+    t.add_node((C, 3, 0))
+
+    # testing the add_member method
+    t.add_member((AB, A, B), (BC, B, C), (AC, A, C))
+    assert t.members == {AB: [A, B], BC: [B, C], AC: [A, C]}
+    assert t.internal_forces == {AB: 0, BC: 0, AC: 0}
+
+    # testing the remove_member method
+    t.remove_member(BC)
+    assert t.members == {AB: [A, B], AC: [A, C]}
+    assert t.internal_forces == {AB: 0, AC: 0}
+
+    t.add_member((BC, B, C))
+
+    D, CD = symbols('D, CD')
+
+    # testing the change_label methods
+    t.change_node_label((B, D))
+    assert t.nodes == [(A, 0, 0), (D, 2, 2), (C, 3, 0)]
+    assert t.node_labels == [A, D, C]
+    assert t.loads == {}
+    assert t.supports == {}
+    assert t.members == {AB: [A, D], BC: [D, C], AC: [A, C]}
+
+    t.change_member_label((BC, CD))
+    assert t.members == {AB: [A, D], CD: [D, C], AC: [A, C]}
+    assert t.internal_forces == {AB: 0, CD: 0, AC: 0}
+
+
+    # testing the apply_load method
+    t.apply_load((A, P, 90), (A, P/4, 90), (A, 2*P,45), (D, P/2, 90))
+    assert t.loads == {A: [[P, 90], [P/4, 90], [2*P, 45]], D: [[P/2, 90]]}
+    assert t.loads[A] == [[P, 90], [P/4, 90], [2*P, 45]]
+
+    # testing the remove_load method
+    t.remove_load((A, P/4, 90))
+    assert t.loads == {A: [[P, 90], [2*P, 45]], D: [[P/2, 90]]}
+    assert t.loads[A] == [[P, 90], [2*P, 45]]
+
+    # testing the apply_support method
+    t.apply_support((A, "pinned"), (D, "roller"))
+    assert t.supports == {A: 'pinned', D: 'roller'}
+    assert t.reaction_loads == {}
+    assert t.loads == {A: [[P, 90], [2*P, 45], [Symbol('R_A_x'), 0], [Symbol('R_A_y'), 90]],  D: [[P/2, 90], [Symbol('R_D_y'), 90]]}
+
+    # testing the remove_support method
+    t.remove_support(A)
+    assert t.supports == {D: 'roller'}
+    assert t.reaction_loads == {}
+    assert t.loads == {A: [[P, 90], [2*P, 45]], D: [[P/2, 90], [Symbol('R_D_y'), 90]]}
+
+    t.apply_support((A, "pinned"))
+
+    # testing the solve method
+    t.solve()
+    assert t.reaction_loads['R_A_x'] == -sqrt(2)*P
+    assert t.reaction_loads['R_A_y'] == -sqrt(2)*P - P
+    assert t.reaction_loads['R_D_y'] == -P/2
+    assert t.internal_forces[AB]/P == 0
+    assert t.internal_forces[CD] == 0
+    assert t.internal_forces[AC] == 0

parrot/lib/python3.10/site-packages/sympy/physics/continuum_mechanics/truss.py

@@ -0,0 +1,1108 @@
+"""
+This module can be used to solve problems related
+to 2D Trusses.
+"""
+
+
+from cmath import atan, inf
+from sympy.core.add import Add
+from sympy.core.evalf import INF
+from sympy.core.mul import Mul
+from sympy.core.symbol import Symbol
+from sympy.core.sympify import sympify
+from sympy import Matrix, pi
+from sympy.external.importtools import import_module
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.matrices.dense import zeros
+import math
+from sympy.physics.units.quantities import Quantity
+from sympy.plotting import plot
+from sympy.utilities.decorator import doctest_depends_on
+from sympy import sin, cos
+
+
+__doctest_requires__ = {('Truss.draw'): ['matplotlib']}
+
+
+numpy = import_module('numpy', import_kwargs={'fromlist':['arange']})
+
+
+class Truss:
+    """
+    A Truss is an assembly of members such as beams,
+    connected by nodes, that create a rigid structure.
+    In engineering, a truss is a structure that
+    consists of two-force members only.
+
+    Trusses are extremely important in engineering applications
+    and can be seen in numerous real-world applications like bridges.
+
+    Examples
+    ========
+
+    There is a Truss consisting of four nodes and five
+    members connecting the nodes. A force P acts
+    downward on the node D and there also exist pinned
+    and roller joints on the nodes A and B respectively.
+
+    .. image:: truss_example.png
+
+    >>> from sympy.physics.continuum_mechanics.truss import Truss
+    >>> t = Truss()
+    >>> t.add_node(("node_1", 0, 0), ("node_2", 6, 0), ("node_3", 2, 2), ("node_4", 2, 0))
+    >>> t.add_member(("member_1", "node_1", "node_4"), ("member_2", "node_2", "node_4"), ("member_3", "node_1", "node_3"))
+    >>> t.add_member(("member_4", "node_2", "node_3"), ("member_5", "node_3", "node_4"))
+    >>> t.apply_load(("node_4", 10, 270))
+    >>> t.apply_support(("node_1", "pinned"), ("node_2", "roller"))
+    """
+
+    def __init__(self):
+        """
+        Initializes the class
+        """
+        self._nodes = []
+        self._members = {}
+        self._loads = {}
+        self._supports = {}
+        self._node_labels = []
+        self._node_positions = []
+        self._node_position_x = []
+        self._node_position_y = []
+        self._nodes_occupied = {}
+        self._member_lengths = {}
+        self._reaction_loads = {}
+        self._internal_forces = {}
+        self._node_coordinates = {}
+
+    @property
+    def nodes(self):
+        """
+        Returns the nodes of the truss along with their positions.
+        """
+        return self._nodes
+
+    @property
+    def node_labels(self):
+        """
+        Returns the node labels of the truss.
+        """
+        return self._node_labels
+
+    @property
+    def node_positions(self):
+        """
+        Returns the positions of the nodes of the truss.
+        """
+        return self._node_positions
+
+    @property
+    def members(self):
+        """
+        Returns the members of the truss along with the start and end points.
+        """
+        return self._members
+
+    @property
+    def member_lengths(self):
+        """
+        Returns the length of each member of the truss.
+        """
+        return self._member_lengths
+
+    @property
+    def supports(self):
+        """
+        Returns the nodes with provided supports along with the kind of support provided i.e.
+        pinned or roller.
+        """
+        return self._supports
+
+    @property
+    def loads(self):
+        """
+        Returns the loads acting on the truss.
+        """
+        return self._loads
+
+    @property
+    def reaction_loads(self):
+        """
+        Returns the reaction forces for all supports which are all initialized to 0.
+        """
+        return self._reaction_loads
+
+    @property
+    def internal_forces(self):
+        """
+        Returns the internal forces for all members which are all initialized to 0.
+        """
+        return self._internal_forces
+
+    def add_node(self, *args):
+        """
+        This method adds a node to the truss along with its name/label and its location.
+        Multiple nodes can be added at the same time.
+
+        Parameters
+        ==========
+        The input(s) for this method are tuples of the form (label, x, y).
+
+        label:  String or a Symbol
+            The label for a node. It is the only way to identify a particular node.
+
+        x: Sympifyable
+            The x-coordinate of the position of the node.
+
+        y: Sympifyable
+            The y-coordinate of the position of the node.
+
+        Examples
+        ========
+
+        >>> from sympy.physics.continuum_mechanics.truss import Truss
+        >>> t = Truss()
+        >>> t.add_node(('A', 0, 0))
+        >>> t.nodes
+        [('A', 0, 0)]
+        >>> t.add_node(('B', 3, 0), ('C', 4, 1))
+        >>> t.nodes
+        [('A', 0, 0), ('B', 3, 0), ('C', 4, 1)]
+        """
+
+        for i in args:
+            label = i[0]
+            x = i[1]
+            x = sympify(x)
+            y=i[2]
+            y = sympify(y)
+            if label in self._node_coordinates:
+                raise ValueError("Node needs to have a unique label")
+
+            elif [x, y] in self._node_coordinates.values():
+                raise ValueError("A node already exists at the given position")
+
+            else :
+                self._nodes.append((label, x, y))
+                self._node_labels.append(label)
+                self._node_positions.append((x, y))
+                self._node_position_x.append(x)
+                self._node_position_y.append(y)
+                self._node_coordinates[label] = [x, y]
+
+
+
+    def remove_node(self, *args):
+        """
+        This method removes a node from the truss.
+        Multiple nodes can be removed at the same time.
+
+        Parameters
+        ==========
+        The input(s) for this method are the labels of the nodes to be removed.
+
+        label:  String or Symbol
+            The label of the node to be removed.
+
+        Examples
+        ========
+
+        >>> from sympy.physics.continuum_mechanics.truss import Truss
+        >>> t = Truss()
+        >>> t.add_node(('A', 0, 0), ('B', 3, 0), ('C', 5, 0))
+        >>> t.nodes
+        [('A', 0, 0), ('B', 3, 0), ('C', 5, 0)]
+        >>> t.remove_node('A', 'C')
+        >>> t.nodes
+        [('B', 3, 0)]
+        """
+        for label in args:
+            for i in range(len(self.nodes)):
+                if self._node_labels[i] == label:
+                    x = self._node_position_x[i]
+                    y = self._node_position_y[i]
+
+            if label not in self._node_coordinates:
+                raise ValueError("No such node exists in the truss")
+
+            else:
+                members_duplicate = self._members.copy()
+                for member in members_duplicate:
+                    if label == self._members[member][0] or label == self._members[member][1]:
+                        raise ValueError("The given node already has member attached to it")
+                self._nodes.remove((label, x, y))
+                self._node_labels.remove(label)
+                self._node_positions.remove((x, y))
+                self._node_position_x.remove(x)
+                self._node_position_y.remove(y)
+                if label in self._loads:
+                    self._loads.pop(label)
+                if label in self._supports:
+                    self._supports.pop(label)
+                self._node_coordinates.pop(label)
+
+
+
+    def add_member(self, *args):
+        """
+        This method adds a member between any two nodes in the given truss.
+
+        Parameters
+        ==========
+        The input(s) of the method are tuple(s) of the form (label, start, end).
+
+        label: String or Symbol
+            The label for a member. It is the only way to identify a particular member.
+
+        start: String or Symbol
+            The label of the starting point/node of the member.
+
+        end: String or Symbol
+            The label of the ending point/node of the member.
+
+        Examples
+        ========
+
+        >>> from sympy.physics.continuum_mechanics.truss import Truss
+        >>> t = Truss()
+        >>> t.add_node(('A', 0, 0), ('B', 3, 0), ('C', 2, 2))
+        >>> t.add_member(('AB', 'A', 'B'), ('BC', 'B', 'C'))
+        >>> t.members
+        {'AB': ['A', 'B'], 'BC': ['B', 'C']}
+        """
+        for i in args:
+            label = i[0]
+            start = i[1]
+            end = i[2]
+
+            if start not in self._node_coordinates or end not in self._node_coordinates or start==end:
+                raise ValueError("The start and end points of the member must be unique nodes")
+
+            elif label in self._members:
+                raise ValueError("A member with the same label already exists for the truss")
+
+            elif self._nodes_occupied.get((start, end)):
+                raise ValueError("A member already exists between the two nodes")
+
+            else:
+                self._members[label] = [start, end]
+                self._member_lengths[label] = sqrt((self._node_coordinates[end][0]-self._node_coordinates[start][0])**2 + (self._node_coordinates[end][1]-self._node_coordinates[start][1])**2)
+                self._nodes_occupied[start, end] = True
+                self._nodes_occupied[end, start] = True
+                self._internal_forces[label] = 0
+
+    def remove_member(self, *args):
+        """
+        This method removes members from the given truss.
+
+        Parameters
+        ==========
+        labels: String or Symbol
+            The label for the member to be removed.
+
+        Examples
+        ========
+
+        >>> from sympy.physics.continuum_mechanics.truss import Truss
+        >>> t = Truss()
+        >>> t.add_node(('A', 0, 0), ('B', 3, 0), ('C', 2, 2))
+        >>> t.add_member(('AB', 'A', 'B'), ('AC', 'A', 'C'), ('BC', 'B', 'C'))
+        >>> t.members
+        {'AB': ['A', 'B'], 'AC': ['A', 'C'], 'BC': ['B', 'C']}
+        >>> t.remove_member('AC', 'BC')
+        >>> t.members
+        {'AB': ['A', 'B']}
+        """
+        for label in args:
+            if label not in self._members:
+                raise ValueError("No such member exists in the Truss")
+
+            else:
+                self._nodes_occupied.pop((self._members[label][0], self._members[label][1]))
+                self._nodes_occupied.pop((self._members[label][1], self._members[label][0]))
+                self._members.pop(label)
+                self._member_lengths.pop(label)
+                self._internal_forces.pop(label)
+
+    def change_node_label(self, *args):
+        """
+        This method changes the label(s) of the specified node(s).
+
+        Parameters
+        ==========
+        The input(s) of this method are tuple(s) of the form (label, new_label).
+
+        label: String or Symbol
+            The label of the node for which the label has
+            to be changed.
+
+        new_label: String or Symbol
+            The new label of the node.
+
+        Examples
+        ========
+
+        >>> from sympy.physics.continuum_mechanics.truss import Truss
+        >>> t = Truss()
+        >>> t.add_node(('A', 0, 0), ('B', 3, 0))
+        >>> t.nodes
+        [('A', 0, 0), ('B', 3, 0)]
+        >>> t.change_node_label(('A', 'C'), ('B', 'D'))
+        >>> t.nodes
+        [('C', 0, 0), ('D', 3, 0)]
+        """
+        for i in args:
+            label = i[0]
+            new_label = i[1]
+            if label not in self._node_coordinates:
+                raise ValueError("No such node exists for the Truss")
+            elif new_label in self._node_coordinates:
+                raise ValueError("A node with the given label already exists")
+            else:
+                for node in self._nodes:
+                    if node[0] == label:
+                        self._nodes[self._nodes.index((label, node[1], node[2]))] = (new_label, node[1], node[2])
+                        self._node_labels[self._node_labels.index(node[0])] = new_label
+                        self._node_coordinates[new_label] = self._node_coordinates[label]
+                        self._node_coordinates.pop(label)
+                        if node[0] in self._supports:
+                            self._supports[new_label] = self._supports[node[0]]
+                            self._supports.pop(node[0])
+                        if new_label in self._supports:
+                            if self._supports[new_label] == 'pinned':
+                                if 'R_'+str(label)+'_x' in self._reaction_loads and 'R_'+str(label)+'_y' in self._reaction_loads:
+                                    self._reaction_loads['R_'+str(new_label)+'_x'] = self._reaction_loads['R_'+str(label)+'_x']
+                                    self._reaction_loads['R_'+str(new_label)+'_y'] = self._reaction_loads['R_'+str(label)+'_y']
+                                    self._reaction_loads.pop('R_'+str(label)+'_x')
+                                    self._reaction_loads.pop('R_'+str(label)+'_y')
+                                self._loads[new_label] = self._loads[label]
+                                for load in self._loads[new_label]:
+                                    if load[1] == 90:
+                                        load[0] -= Symbol('R_'+str(label)+'_y')
+                                        if load[0] == 0:
+                                            self._loads[label].remove(load)
+                                        break
+                                for load in self._loads[new_label]:
+                                    if load[1] == 0:
+                                        load[0] -= Symbol('R_'+str(label)+'_x')
+                                        if load[0] == 0:
+                                            self._loads[label].remove(load)
+                                        break
+                                self.apply_load(new_label, Symbol('R_'+str(new_label)+'_x'), 0)
+                                self.apply_load(new_label, Symbol('R_'+str(new_label)+'_y'), 90)
+                                self._loads.pop(label)
+                            elif self._supports[new_label] == 'roller':
+                                self._loads[new_label] = self._loads[label]
+                                for load in self._loads[label]:
+                                    if load[1] == 90:
+                                        load[0] -= Symbol('R_'+str(label)+'_y')
+                                        if load[0] == 0:
+                                            self._loads[label].remove(load)
+                                        break
+                                self.apply_load(new_label, Symbol('R_'+str(new_label)+'_y'), 90)
+                                self._loads.pop(label)
+                        else:
+                            if label in self._loads:
+                                self._loads[new_label] = self._loads[label]
+                                self._loads.pop(label)
+                        for member in self._members:
+                            if self._members[member][0] == node[0]:
+                                self._members[member][0] = new_label
+                                self._nodes_occupied[(new_label, self._members[member][1])] = True
+                                self._nodes_occupied[(self._members[member][1], new_label)] = True
+                                self._nodes_occupied.pop((label, self._members[member][1]))
+                                self._nodes_occupied.pop((self._members[member][1], label))
+                            elif self._members[member][1] == node[0]:
+                                self._members[member][1] = new_label
+                                self._nodes_occupied[(self._members[member][0], new_label)] = True
+                                self._nodes_occupied[(new_label, self._members[member][0])] = True
+                                self._nodes_occupied.pop((self._members[member][0], label))
+                                self._nodes_occupied.pop((label, self._members[member][0]))
+
+    def change_member_label(self, *args):
+        """
+        This method changes the label(s) of the specified member(s).
+
+        Parameters
+        ==========
+        The input(s) of this method are tuple(s) of the form (label, new_label)
+
+        label: String or Symbol
+            The label of the member for which the label has
+            to be changed.
+
+        new_label: String or Symbol
+            The new label of the member.
+
+        Examples
+        ========
+
+        >>> from sympy.physics.continuum_mechanics.truss import Truss
+        >>> t = Truss()
+        >>> t.add_node(('A', 0, 0), ('B', 3, 0), ('D', 5, 0))
+        >>> t.nodes
+        [('A', 0, 0), ('B', 3, 0), ('D', 5, 0)]
+        >>> t.change_node_label(('A', 'C'))
+        >>> t.nodes
+        [('C', 0, 0), ('B', 3, 0), ('D', 5, 0)]
+        >>> t.add_member(('BC', 'B', 'C'), ('BD', 'B', 'D'))
+        >>> t.members
+        {'BC': ['B', 'C'], 'BD': ['B', 'D']}
+        >>> t.change_member_label(('BC', 'BC_new'), ('BD', 'BD_new'))
+        >>> t.members
+        {'BC_new': ['B', 'C'], 'BD_new': ['B', 'D']}
+        """
+        for i in args:
+            label = i[0]
+            new_label = i[1]
+            if label not in self._members:
+                raise ValueError("No such member exists for the Truss")
+            else:
+                members_duplicate = list(self._members).copy()
+                for member in members_duplicate:
+                    if member == label:
+                        self._members[new_label] = [self._members[member][0], self._members[member][1]]
+                        self._members.pop(label)
+                        self._member_lengths[new_label] = self._member_lengths[label]
+                        self._member_lengths.pop(label)
+                        self._internal_forces[new_label] = self._internal_forces[label]
+                        self._internal_forces.pop(label)
+
+    def apply_load(self, *args):
+        """
+        This method applies external load(s) at the specified node(s).
+
+        Parameters
+        ==========
+        The input(s) of the method are tuple(s) of the form (location, magnitude, direction).
+
+        location: String or Symbol
+            Label of the Node at which load is applied.
+
+        magnitude: Sympifyable
+            Magnitude of the load applied. It must always be positive and any changes in
+            the direction of the load are not reflected here.
+
+        direction: Sympifyable
+            The angle, in degrees, that the load vector makes with the horizontal
+            in the counter-clockwise direction. It takes the values 0 to 360,
+            inclusive.
+
+        Examples
+        ========
+
+        >>> from sympy.physics.continuum_mechanics.truss import Truss
+        >>> from sympy import symbols
+        >>> t = Truss()
+        >>> t.add_node(('A', 0, 0), ('B', 3, 0))
+        >>> P = symbols('P')
+        >>> t.apply_load(('A', P, 90), ('A', P/2, 45), ('A', P/4, 90))
+        >>> t.loads
+        {'A': [[P, 90], [P/2, 45], [P/4, 90]]}
+        """
+        for i in args:
+            location = i[0]
+            magnitude = i[1]
+            direction = i[2]
+            magnitude = sympify(magnitude)
+            direction = sympify(direction)
+
+            if location not in self._node_coordinates:
+                raise ValueError("Load must be applied at a known node")
+
+            else:
+                if location in self._loads:
+                    self._loads[location].append([magnitude, direction])
+                else:
+                    self._loads[location] = [[magnitude, direction]]
+
+    def remove_load(self, *args):
+        """
+        This method removes already
+        present external load(s) at specified node(s).
+
+        Parameters
+        ==========
+        The input(s) of this method are tuple(s) of the form (location, magnitude, direction).
+
+        location: String or Symbol
+            Label of the Node at which load is applied and is to be removed.
+
+        magnitude: Sympifyable
+            Magnitude of the load applied.
+
+        direction: Sympifyable
+            The angle, in degrees, that the load vector makes with the horizontal
+            in the counter-clockwise direction. It takes the values 0 to 360,
+            inclusive.
+
+        Examples
+        ========
+
+        >>> from sympy.physics.continuum_mechanics.truss import Truss
+        >>> from sympy import symbols
+        >>> t = Truss()
+        >>> t.add_node(('A', 0, 0), ('B', 3, 0))
+        >>> P = symbols('P')
+        >>> t.apply_load(('A', P, 90), ('A', P/2, 45), ('A', P/4, 90))
+        >>> t.loads
+        {'A': [[P, 90], [P/2, 45], [P/4, 90]]}
+        >>> t.remove_load(('A', P/4, 90), ('A', P/2, 45))
+        >>> t.loads
+        {'A': [[P, 90]]}
+        """
+        for i in args:
+            location = i[0]
+            magnitude = i[1]
+            direction = i[2]
+            magnitude = sympify(magnitude)
+            direction = sympify(direction)
+
+            if location not in self._node_coordinates:
+                raise ValueError("Load must be removed from a known node")
+
+            else:
+                if [magnitude, direction] not in self._loads[location]:
+                    raise ValueError("No load of this magnitude and direction has been applied at this node")
+                else:
+                    self._loads[location].remove([magnitude, direction])
+            if self._loads[location] == []:
+                self._loads.pop(location)
+
+    def apply_support(self, *args):
+        """
+        This method adds a pinned or roller support at specified node(s).
+
+        Parameters
+        ==========
+        The input(s) of this method are of the form (location, type).
+
+        location: String or Symbol
+            Label of the Node at which support is added.
+
+        type: String
+            Type of the support being provided at the node.
+
+        Examples
+        ========
+
+        >>> from sympy.physics.continuum_mechanics.truss import Truss
+        >>> t = Truss()
+        >>> t.add_node(('A', 0, 0), ('B', 3, 0))
+        >>> t.apply_support(('A', 'pinned'), ('B', 'roller'))
+        >>> t.supports
+        {'A': 'pinned', 'B': 'roller'}
+        """
+        for i in args:
+            location = i[0]
+            type = i[1]
+            if location not in self._node_coordinates:
+                raise ValueError("Support must be added on a known node")
+
+            else:
+                if location not in self._supports:
+                    if type == 'pinned':
+                        self.apply_load((location, Symbol('R_'+str(location)+'_x'), 0))
+                        self.apply_load((location, Symbol('R_'+str(location)+'_y'), 90))
+                    elif type == 'roller':
+                        self.apply_load((location, Symbol('R_'+str(location)+'_y'), 90))
+                elif self._supports[location] == 'pinned':
+                    if type == 'roller':
+                        self.remove_load((location, Symbol('R_'+str(location)+'_x'), 0))
+                elif self._supports[location] == 'roller':
+                    if type == 'pinned':
+                        self.apply_load((location, Symbol('R_'+str(location)+'_x'), 0))
+                self._supports[location] = type
+
+    def remove_support(self, *args):
+        """
+        This method removes support from specified node(s.)
+
+        Parameters
+        ==========
+
+        locations: String or Symbol
+            Label of the Node(s) at which support is to be removed.
+
+        Examples
+        ========
+
+        >>> from sympy.physics.continuum_mechanics.truss import Truss
+        >>> t = Truss()
+        >>> t.add_node(('A', 0, 0), ('B', 3, 0))
+        >>> t.apply_support(('A', 'pinned'), ('B', 'roller'))
+        >>> t.supports
+        {'A': 'pinned', 'B': 'roller'}
+        >>> t.remove_support('A','B')
+        >>> t.supports
+        {}
+        """
+        for location in args:
+
+            if location not in self._node_coordinates:
+                raise ValueError("No such node exists in the Truss")
+
+            elif location not in self._supports:
+                raise ValueError("No support has been added to the given node")
+
+            else:
+                if self._supports[location] == 'pinned':
+                    self.remove_load((location, Symbol('R_'+str(location)+'_x'), 0))
+                    self.remove_load((location, Symbol('R_'+str(location)+'_y'), 90))
+                elif self._supports[location] == 'roller':
+                    self.remove_load((location, Symbol('R_'+str(location)+'_y'), 90))
+                self._supports.pop(location)
+
+    def solve(self):
+        """
+        This method solves for all reaction forces of all supports and all internal forces
+        of all the members in the truss, provided the Truss is solvable.
+
+        A Truss is solvable if the following condition is met,
+
+        2n >= r + m
+
+        Where n is the number of nodes, r is the number of reaction forces, where each pinned
+        support has 2 reaction forces and each roller has 1, and m is the number of members.
+
+        The given condition is derived from the fact that a system of equations is solvable
+        only when the number of variables is lesser than or equal to the number of equations.
+        Equilibrium Equations in x and y directions give two equations per node giving 2n number
+        equations. However, the truss needs to be stable as well and may be unstable if 2n > r + m.
+        The number of variables is simply the sum of the number of reaction forces and member
+        forces.
+
+        .. note::
+           The sign convention for the internal forces present in a member revolves around whether each
+           force is compressive or tensile. While forming equations for each node, internal force due
+           to a member on the node is assumed to be away from the node i.e. each force is assumed to
+           be compressive by default. Hence, a positive value for an internal force implies the
+           presence of compressive force in the member and a negative value implies a tensile force.
+
+        Examples
+        ========
+
+        >>> from sympy.physics.continuum_mechanics.truss import Truss
+        >>> t = Truss()
+        >>> t.add_node(("node_1", 0, 0), ("node_2", 6, 0), ("node_3", 2, 2), ("node_4", 2, 0))
+        >>> t.add_member(("member_1", "node_1", "node_4"), ("member_2", "node_2", "node_4"), ("member_3", "node_1", "node_3"))
+        >>> t.add_member(("member_4", "node_2", "node_3"), ("member_5", "node_3", "node_4"))
+        >>> t.apply_load(("node_4", 10, 270))
+        >>> t.apply_support(("node_1", "pinned"), ("node_2", "roller"))
+        >>> t.solve()
+        >>> t.reaction_loads
+        {'R_node_1_x': 0, 'R_node_1_y': 20/3, 'R_node_2_y': 10/3}
+        >>> t.internal_forces
+        {'member_1': 20/3, 'member_2': 20/3, 'member_3': -20*sqrt(2)/3, 'member_4': -10*sqrt(5)/3, 'member_5': 10}
+        """
+        count_reaction_loads = 0
+        for node in self._nodes:
+            if node[0] in self._supports:
+                if self._supports[node[0]]=='pinned':
+                    count_reaction_loads += 2
+                elif self._supports[node[0]]=='roller':
+                    count_reaction_loads += 1
+        if 2*len(self._nodes) != len(self._members) + count_reaction_loads:
+            raise ValueError("The given truss cannot be solved")
+        coefficients_matrix = [[0 for i in range(2*len(self._nodes))] for j in range(2*len(self._nodes))]
+        load_matrix = zeros(2*len(self.nodes), 1)
+        load_matrix_row = 0
+        for node in self._nodes:
+            if node[0] in self._loads:
+                for load in self._loads[node[0]]:
+                    if load[0]!=Symbol('R_'+str(node[0])+'_x') and load[0]!=Symbol('R_'+str(node[0])+'_y'):
+                        load_matrix[load_matrix_row] -= load[0]*cos(pi*load[1]/180)
+                        load_matrix[load_matrix_row + 1] -= load[0]*sin(pi*load[1]/180)
+            load_matrix_row += 2
+        cols = 0
+        row = 0
+        for node in self._nodes:
+            if node[0] in self._supports:
+                if self._supports[node[0]]=='pinned':
+                    coefficients_matrix[row][cols] += 1
+                    coefficients_matrix[row+1][cols+1] += 1
+                    cols += 2
+                elif self._supports[node[0]]=='roller':
+                    coefficients_matrix[row+1][cols] += 1
+                    cols += 1
+            row += 2
+        for member in self._members:
+            start = self._members[member][0]
+            end = self._members[member][1]
+            length = sqrt((self._node_coordinates[start][0]-self._node_coordinates[end][0])**2 + (self._node_coordinates[start][1]-self._node_coordinates[end][1])**2)
+            start_index = self._node_labels.index(start)
+            end_index = self._node_labels.index(end)
+            horizontal_component_start = (self._node_coordinates[end][0]-self._node_coordinates[start][0])/length
+            vertical_component_start = (self._node_coordinates[end][1]-self._node_coordinates[start][1])/length
+            horizontal_component_end = (self._node_coordinates[start][0]-self._node_coordinates[end][0])/length
+            vertical_component_end = (self._node_coordinates[start][1]-self._node_coordinates[end][1])/length
+            coefficients_matrix[start_index*2][cols] += horizontal_component_start
+            coefficients_matrix[start_index*2+1][cols] += vertical_component_start
+            coefficients_matrix[end_index*2][cols] += horizontal_component_end
+            coefficients_matrix[end_index*2+1][cols] += vertical_component_end
+            cols += 1
+        forces_matrix = (Matrix(coefficients_matrix)**-1)*load_matrix
+        self._reaction_loads = {}
+        i = 0
+        min_load = inf
+        for node in self._nodes:
+            if node[0] in self._loads:
+                for load in self._loads[node[0]]:
+                    if type(load[0]) not in [Symbol, Mul, Add]:
+                        min_load = min(min_load, load[0])
+        for j in range(len(forces_matrix)):
+            if type(forces_matrix[j]) not in [Symbol, Mul, Add]:
+                if abs(forces_matrix[j]/min_load) <1E-10:
+                    forces_matrix[j] = 0
+        for node in self._nodes:
+            if node[0] in self._supports:
+                if self._supports[node[0]]=='pinned':
+                    self._reaction_loads['R_'+str(node[0])+'_x'] = forces_matrix[i]
+                    self._reaction_loads['R_'+str(node[0])+'_y'] = forces_matrix[i+1]
+                    i += 2
+                elif self._supports[node[0]]=='roller':
+                    self._reaction_loads['R_'+str(node[0])+'_y'] = forces_matrix[i]
+                    i += 1
+        for member in self._members:
+            self._internal_forces[member] = forces_matrix[i]
+            i += 1
+        return
+
+    @doctest_depends_on(modules=('numpy',))
+    def draw(self, subs_dict=None):
+        """
+        Returns a plot object of the Truss with all its nodes, members,
+        supports and loads.
+
+        .. note::
+            The user must be careful while entering load values in their
+            directions. The draw function assumes a sign convention that
+            is used for plotting loads.
+
+            Given a right-handed coordinate system with XYZ coordinates,
+            the supports are assumed to be such that the reaction forces of a
+            pinned support is in the +X and +Y direction while those of a
+            roller support is in the +Y direction. For the load, the range
+            of angles, one can input goes all the way to 360 degrees which, in the
+            the plot is the angle that the load vector makes with the positive x-axis in the anticlockwise direction.
+
+            For example, for a 90-degree angle, the load will be a vertically
+            directed along +Y while a 270-degree angle denotes a vertical
+            load as well but along -Y.
+
+        Examples
+        ========
+
+        .. plot::
+            :context: close-figs
+            :format: doctest
+            :include-source: True
+
+            >>> from sympy.physics.continuum_mechanics.truss import Truss
+            >>> import math
+            >>> t = Truss()
+            >>> t.add_node(("A", -4, 0), ("B", 0, 0), ("C", 4, 0), ("D", 8, 0))
+            >>> t.add_node(("E", 6, 2/math.sqrt(3)))
+            >>> t.add_node(("F", 2, 2*math.sqrt(3)))
+            >>> t.add_node(("G", -2, 2/math.sqrt(3)))
+            >>> t.add_member(("AB","A","B"), ("BC","B","C"), ("CD","C","D"))
+            >>> t.add_member(("AG","A","G"), ("GB","G","B"), ("GF","G","F"))
+            >>> t.add_member(("BF","B","F"), ("FC","F","C"), ("CE","C","E"))
+            >>> t.add_member(("FE","F","E"), ("DE","D","E"))
+            >>> t.apply_support(("A","pinned"), ("D","roller"))
+            >>> t.apply_load(("G", 3, 90), ("E", 3, 90), ("F", 2, 90))
+            >>> p = t.draw()
+            >>> p  # doctest: +ELLIPSIS
+            Plot object containing:
+            [0]: cartesian line: 1 for x over (1.0, 1.0)
+            ...
+            >>> p.show()
+        """
+        if not numpy:
+            raise ImportError("To use this function numpy module is required")
+
+        x = Symbol('x')
+
+        markers = []
+        annotations = []
+        rectangles = []
+
+        node_markers = self._draw_nodes(subs_dict)
+        markers += node_markers
+
+        member_rectangles = self._draw_members()
+        rectangles += member_rectangles
+
+        support_markers = self._draw_supports()
+        markers += support_markers
+
+        load_annotations = self._draw_loads()
+        annotations += load_annotations
+
+        xmax = -INF
+        xmin = INF
+        ymax = -INF
+        ymin = INF
+
+        for node in self._node_coordinates:
+            xmax = max(xmax, self._node_coordinates[node][0])
+            xmin = min(xmin, self._node_coordinates[node][0])
+            ymax = max(ymax, self._node_coordinates[node][1])
+            ymin = min(ymin, self._node_coordinates[node][1])
+
+        lim = max(xmax*1.1-xmin*0.8+1, ymax*1.1-ymin*0.8+1)
+
+        if lim==xmax*1.1-xmin*0.8+1:
+            sing_plot = plot(1, (x, 1, 1), markers=markers, show=False, annotations=annotations, xlim=(xmin-0.05*lim, xmax*1.1), ylim=(xmin-0.05*lim, xmax*1.1), axis=False, rectangles=rectangles)
+
+        else:
+            sing_plot = plot(1, (x, 1, 1), markers=markers, show=False, annotations=annotations, xlim=(ymin-0.05*lim, ymax*1.1), ylim=(ymin-0.05*lim, ymax*1.1), axis=False, rectangles=rectangles)
+
+        return sing_plot
+
+
+    def _draw_nodes(self, subs_dict):
+        node_markers = []
+
+        for node in self._node_coordinates:
+            if (type(self._node_coordinates[node][0]) in (Symbol, Quantity)):
+                if self._node_coordinates[node][0] in subs_dict:
+                    self._node_coordinates[node][0] = subs_dict[self._node_coordinates[node][0]]
+                else:
+                    raise ValueError("provided substituted dictionary is not adequate")
+            elif (type(self._node_coordinates[node][0]) == Mul):
+                objects = self._node_coordinates[node][0].as_coeff_Mul()
+                for object in objects:
+                    if type(object) in (Symbol, Quantity):
+                        if subs_dict==None or object not in subs_dict:
+                            raise ValueError("provided substituted dictionary is not adequate")
+                        else:
+                            self._node_coordinates[node][0] /= object
+                            self._node_coordinates[node][0] *= subs_dict[object]
+
+            if (type(self._node_coordinates[node][1]) in (Symbol, Quantity)):
+                if self._node_coordinates[node][1] in subs_dict:
+                    self._node_coordinates[node][1] = subs_dict[self._node_coordinates[node][1]]
+                else:
+                    raise ValueError("provided substituted dictionary is not adequate")
+            elif (type(self._node_coordinates[node][1]) == Mul):
+                objects = self._node_coordinates[node][1].as_coeff_Mul()
+                for object in objects:
+                    if type(object) in (Symbol, Quantity):
+                        if subs_dict==None or object not in subs_dict:
+                            raise ValueError("provided substituted dictionary is not adequate")
+                        else:
+                            self._node_coordinates[node][1] /= object
+                            self._node_coordinates[node][1] *= subs_dict[object]
+
+        for node in self._node_coordinates:
+            node_markers.append(
+                {
+                    'args':[[self._node_coordinates[node][0]], [self._node_coordinates[node][1]]],
+                    'marker':'o',
+                    'markersize':5,
+                    'color':'black'
+                }
+            )
+        return node_markers
+
+    def _draw_members(self):
+
+        member_rectangles = []
+
+        xmax = -INF
+        xmin = INF
+        ymax = -INF
+        ymin = INF
+
+        for node in self._node_coordinates:
+            xmax = max(xmax, self._node_coordinates[node][0])
+            xmin = min(xmin, self._node_coordinates[node][0])
+            ymax = max(ymax, self._node_coordinates[node][1])
+            ymin = min(ymin, self._node_coordinates[node][1])
+
+        if abs(1.1*xmax-0.8*xmin)>abs(1.1*ymax-0.8*ymin):
+            max_diff = 1.1*xmax-0.8*xmin
+        else:
+            max_diff = 1.1*ymax-0.8*ymin
+
+        for member in self._members:
+            x1 = self._node_coordinates[self._members[member][0]][0]
+            y1 = self._node_coordinates[self._members[member][0]][1]
+            x2 = self._node_coordinates[self._members[member][1]][0]
+            y2 = self._node_coordinates[self._members[member][1]][1]
+            if x2!=x1 and y2!=y1:
+                if x2>x1:
+                    member_rectangles.append(
+                        {
+                            'xy':(x1-0.005*max_diff*cos(pi/4+atan((y2-y1)/(x2-x1)))/2, y1-0.005*max_diff*sin(pi/4+atan((y2-y1)/(x2-x1)))/2),
+                            'width':sqrt((x1-x2)**2+(y1-y2)**2)+0.005*max_diff/math.sqrt(2),
+                            'height':0.005*max_diff,
+                            'angle':180*atan((y2-y1)/(x2-x1))/pi,
+                            'color':'brown'
+                        }
+                    )
+                else:
+                    member_rectangles.append(
+                        {
+                            'xy':(x2-0.005*max_diff*cos(pi/4+atan((y2-y1)/(x2-x1)))/2, y2-0.005*max_diff*sin(pi/4+atan((y2-y1)/(x2-x1)))/2),
+                            'width':sqrt((x1-x2)**2+(y1-y2)**2)+0.005*max_diff/math.sqrt(2),
+                            'height':0.005*max_diff,
+                            'angle':180*atan((y2-y1)/(x2-x1))/pi,
+                            'color':'brown'
+                        }
+                    )
+            elif y2==y1:
+                if x2>x1:
+                    member_rectangles.append(
+                        {
+                            'xy':(x1-0.005*max_diff/2, y1-0.005*max_diff/2),
+                            'width':sqrt((x1-x2)**2+(y1-y2)**2),
+                            'height':0.005*max_diff,
+                            'angle':90*(1-math.copysign(1, x2-x1)),
+                            'color':'brown'
+                        }
+                    )
+                else:
+                    member_rectangles.append(
+                        {
+                            'xy':(x1-0.005*max_diff/2, y1-0.005*max_diff/2),
+                            'width':sqrt((x1-x2)**2+(y1-y2)**2),
+                            'height':-0.005*max_diff,
+                            'angle':90*(1-math.copysign(1, x2-x1)),
+                            'color':'brown'
+                        }
+                    )
+            else:
+                if y1<y2:
+                    member_rectangles.append(
+                        {
+                            'xy':(x1-0.005*max_diff/2, y1-0.005*max_diff/2),
+                            'width':sqrt((x1-x2)**2+(y1-y2)**2)+0.005*max_diff/2,
+                            'height':0.005*max_diff,
+                            'angle':90*math.copysign(1, y2-y1),
+                            'color':'brown'
+                        }
+                    )
+                else:
+                    member_rectangles.append(
+                        {
+                            'xy':(x2-0.005*max_diff/2, y2-0.005*max_diff/2),
+                            'width':-(sqrt((x1-x2)**2+(y1-y2)**2)+0.005*max_diff/2),
+                            'height':0.005*max_diff,
+                            'angle':90*math.copysign(1, y2-y1),
+                            'color':'brown'
+                        }
+                    )
+
+        return member_rectangles
+
+    def _draw_supports(self):
+        support_markers = []
+
+        xmax = -INF
+        xmin = INF
+        ymax = -INF
+        ymin = INF
+
+        for node in self._node_coordinates:
+            xmax = max(xmax, self._node_coordinates[node][0])
+            xmin = min(xmin, self._node_coordinates[node][0])
+            ymax = max(ymax, self._node_coordinates[node][1])
+            ymin = min(ymin, self._node_coordinates[node][1])
+        if abs(1.1*xmax-0.8*xmin)>abs(1.1*ymax-0.8*ymin):
+            max_diff = 1.1*xmax-0.8*xmin
+        else:
+            max_diff = 1.1*ymax-0.8*ymin
+
+        for node in self._supports:
+            if self._supports[node]=='pinned':
+                support_markers.append(
+                    {
+                        'args':[
+                            [self._node_coordinates[node][0]],
+                            [self._node_coordinates[node][1]]
+                        ],
+                        'marker':6,
+                        'markersize':15,
+                        'color':'black',
+                        'markerfacecolor':'none'
+                    }
+                )
+                support_markers.append(
+                    {
+                        'args':[
+                            [self._node_coordinates[node][0]],
+                            [self._node_coordinates[node][1]-0.035*max_diff]
+                        ],
+                        'marker':'_',
+                        'markersize':14,
+                        'color':'black'
+                    }
+                )
+
+            elif self._supports[node]=='roller':
+                support_markers.append(
+                    {
+                        'args':[
+                            [self._node_coordinates[node][0]],
+                            [self._node_coordinates[node][1]-0.02*max_diff]
+                        ],
+                        'marker':'o',
+                        'markersize':11,
+                        'color':'black',
+                        'markerfacecolor':'none'
+                    }
+                )
+                support_markers.append(
+                    {
+                        'args':[
+                            [self._node_coordinates[node][0]],
+                            [self._node_coordinates[node][1]-0.0375*max_diff]
+                        ],
+                        'marker':'_',
+                        'markersize':14,
+                        'color':'black'
+                    }
+                )
+        return support_markers
+
+    def _draw_loads(self):
+        load_annotations = []
+
+        xmax = -INF
+        xmin = INF
+        ymax = -INF
+        ymin = INF
+
+        for node in self._node_coordinates:
+            xmax = max(xmax, self._node_coordinates[node][0])
+            xmin = min(xmin, self._node_coordinates[node][0])
+            ymax = max(ymax, self._node_coordinates[node][1])
+            ymin = min(ymin, self._node_coordinates[node][1])
+
+        if abs(1.1*xmax-0.8*xmin)>abs(1.1*ymax-0.8*ymin):
+            max_diff = 1.1*xmax-0.8*xmin+5
+        else:
+            max_diff = 1.1*ymax-0.8*ymin+5
+
+        for node in self._loads:
+            for load in self._loads[node]:
+                if load[0] in [Symbol('R_'+str(node)+'_x'), Symbol('R_'+str(node)+'_y')]:
+                    continue
+                x = self._node_coordinates[node][0]
+                y = self._node_coordinates[node][1]
+                load_annotations.append(
+                    {
+                        'text':'',
+                        'xy':(
+                            x-math.cos(pi*load[1]/180)*(max_diff/100),
+                            y-math.sin(pi*load[1]/180)*(max_diff/100)
+                        ),
+                        'xytext':(
+                            x-(max_diff/100+abs(xmax-xmin)+abs(ymax-ymin))*math.cos(pi*load[1]/180)/20,
+                            y-(max_diff/100+abs(xmax-xmin)+abs(ymax-ymin))*math.sin(pi*load[1]/180)/20
+                        ),
+                        'arrowprops':{'width':1.5, 'headlength':5, 'headwidth':5, 'facecolor':'black'}
+                    }
+                )
+        return load_annotations

parrot/lib/python3.10/site-packages/sympy/physics/control/__init__.py

@@ -0,0 +1,16 @@
+from .lti import (TransferFunction, Series, MIMOSeries, Parallel, MIMOParallel,
+    Feedback, MIMOFeedback, TransferFunctionMatrix, StateSpace, gbt, bilinear, forward_diff,
+    backward_diff, phase_margin, gain_margin)
+from .control_plots import (pole_zero_numerical_data, pole_zero_plot, step_response_numerical_data,
+    step_response_plot, impulse_response_numerical_data, impulse_response_plot, ramp_response_numerical_data,
+    ramp_response_plot, bode_magnitude_numerical_data, bode_phase_numerical_data, bode_magnitude_plot,
+    bode_phase_plot, bode_plot)
+
+__all__ = ['TransferFunction', 'Series', 'MIMOSeries', 'Parallel',
+    'MIMOParallel', 'Feedback', 'MIMOFeedback', 'TransferFunctionMatrix', 'StateSpace',
+    'gbt', 'bilinear', 'forward_diff', 'backward_diff', 'phase_margin', 'gain_margin',
+    'pole_zero_numerical_data', 'pole_zero_plot', 'step_response_numerical_data',
+    'step_response_plot', 'impulse_response_numerical_data', 'impulse_response_plot',
+    'ramp_response_numerical_data', 'ramp_response_plot',
+    'bode_magnitude_numerical_data', 'bode_phase_numerical_data',
+    'bode_magnitude_plot', 'bode_phase_plot', 'bode_plot']

parrot/lib/python3.10/site-packages/sympy/physics/control/control_plots.py

@@ -0,0 +1,978 @@
+from sympy.core.numbers import I, pi
+from sympy.functions.elementary.exponential import (exp, log)
+from sympy.polys.partfrac import apart
+from sympy.core.symbol import Dummy
+from sympy.external import import_module
+from sympy.functions import arg, Abs
+from sympy.integrals.laplace import _fast_inverse_laplace
+from sympy.physics.control.lti import SISOLinearTimeInvariant
+from sympy.plotting.series import LineOver1DRangeSeries
+from sympy.polys.polytools import Poly
+from sympy.printing.latex import latex
+
+__all__ = ['pole_zero_numerical_data', 'pole_zero_plot',
+    'step_response_numerical_data', 'step_response_plot',
+    'impulse_response_numerical_data', 'impulse_response_plot',
+    'ramp_response_numerical_data', 'ramp_response_plot',
+    'bode_magnitude_numerical_data', 'bode_phase_numerical_data',
+    'bode_magnitude_plot', 'bode_phase_plot', 'bode_plot']
+
+matplotlib = import_module(
+        'matplotlib', import_kwargs={'fromlist': ['pyplot']},
+        catch=(RuntimeError,))
+
+numpy = import_module('numpy')
+
+if matplotlib:
+    plt = matplotlib.pyplot
+
+if numpy:
+    np = numpy  # Matplotlib already has numpy as a compulsory dependency. No need to install it separately.
+
+
+def _check_system(system):
+    """Function to check whether the dynamical system passed for plots is
+    compatible or not."""
+    if not isinstance(system, SISOLinearTimeInvariant):
+        raise NotImplementedError("Only SISO LTI systems are currently supported.")
+    sys = system.to_expr()
+    len_free_symbols = len(sys.free_symbols)
+    if len_free_symbols > 1:
+        raise ValueError("Extra degree of freedom found. Make sure"
+            " that there are no free symbols in the dynamical system other"
+            " than the variable of Laplace transform.")
+    if sys.has(exp):
+        # Should test that exp is not part of a constant, in which case
+        # no exception is required, compare exp(s) with s*exp(1)
+        raise NotImplementedError("Time delay terms are not supported.")
+
+
+def pole_zero_numerical_data(system):
+    """
+    Returns the numerical data of poles and zeros of the system.
+    It is internally used by ``pole_zero_plot`` to get the data
+    for plotting poles and zeros. Users can use this data to further
+    analyse the dynamics of the system or plot using a different
+    backend/plotting-module.
+
+    Parameters
+    ==========
+
+    system : SISOLinearTimeInvariant
+        The system for which the pole-zero data is to be computed.
+
+    Returns
+    =======
+
+    tuple : (zeros, poles)
+        zeros = Zeros of the system. NumPy array of complex numbers.
+        poles = Poles of the system. NumPy array of complex numbers.
+
+    Raises
+    ======
+
+    NotImplementedError
+        When a SISO LTI system is not passed.
+
+        When time delay terms are present in the system.
+
+    ValueError
+        When more than one free symbol is present in the system.
+        The only variable in the transfer function should be
+        the variable of the Laplace transform.
+
+    Examples
+    ========
+
+    >>> from sympy.abc import s
+    >>> from sympy.physics.control.lti import TransferFunction
+    >>> from sympy.physics.control.control_plots import pole_zero_numerical_data
+    >>> tf1 = TransferFunction(s**2 + 1, s**4 + 4*s**3 + 6*s**2 + 5*s + 2, s)
+    >>> pole_zero_numerical_data(tf1)   # doctest: +SKIP
+    ([-0.+1.j  0.-1.j], [-2. +0.j        -0.5+0.8660254j -0.5-0.8660254j -1. +0.j       ])
+
+    See Also
+    ========
+
+    pole_zero_plot
+
+    """
+    _check_system(system)
+    system = system.doit()  # Get the equivalent TransferFunction object.
+
+    num_poly = Poly(system.num, system.var).all_coeffs()
+    den_poly = Poly(system.den, system.var).all_coeffs()
+
+    num_poly = np.array(num_poly, dtype=np.complex128)
+    den_poly = np.array(den_poly, dtype=np.complex128)
+
+    zeros = np.roots(num_poly)
+    poles = np.roots(den_poly)
+
+    return zeros, poles
+
+
+def pole_zero_plot(system, pole_color='blue', pole_markersize=10,
+    zero_color='orange', zero_markersize=7, grid=True, show_axes=True,
+    show=True, **kwargs):
+    r"""
+    Returns the Pole-Zero plot (also known as PZ Plot or PZ Map) of a system.
+
+    A Pole-Zero plot is a graphical representation of a system's poles and
+    zeros. It is plotted on a complex plane, with circular markers representing
+    the system's zeros and 'x' shaped markers representing the system's poles.
+
+    Parameters
+    ==========
+
+    system : SISOLinearTimeInvariant type systems
+        The system for which the pole-zero plot is to be computed.
+    pole_color : str, tuple, optional
+        The color of the pole points on the plot. Default color
+        is blue. The color can be provided as a matplotlib color string,
+        or a 3-tuple of floats each in the 0-1 range.
+    pole_markersize : Number, optional
+        The size of the markers used to mark the poles in the plot.
+        Default pole markersize is 10.
+    zero_color : str, tuple, optional
+        The color of the zero points on the plot. Default color
+        is orange. The color can be provided as a matplotlib color string,
+        or a 3-tuple of floats each in the 0-1 range.
+    zero_markersize : Number, optional
+        The size of the markers used to mark the zeros in the plot.
+        Default zero markersize is 7.
+    grid : boolean, optional
+        If ``True``, the plot will have a grid. Defaults to True.
+    show_axes : boolean, optional
+        If ``True``, the coordinate axes will be shown. Defaults to False.
+    show : boolean, optional
+        If ``True``, the plot will be displayed otherwise
+        the equivalent matplotlib ``plot`` object will be returned.
+        Defaults to True.
+
+    Examples
+    ========
+
+    .. plot::
+        :context: close-figs
+        :format: doctest
+        :include-source: True
+
+        >>> from sympy.abc import s
+        >>> from sympy.physics.control.lti import TransferFunction
+        >>> from sympy.physics.control.control_plots import pole_zero_plot
+        >>> tf1 = TransferFunction(s**2 + 1, s**4 + 4*s**3 + 6*s**2 + 5*s + 2, s)
+        >>> pole_zero_plot(tf1)   # doctest: +SKIP
+
+    See Also
+    ========
+
+    pole_zero_numerical_data
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Pole%E2%80%93zero_plot
+
+    """
+    zeros, poles = pole_zero_numerical_data(system)
+
+    zero_real = np.real(zeros)
+    zero_imag = np.imag(zeros)
+
+    pole_real = np.real(poles)
+    pole_imag = np.imag(poles)
+
+    plt.plot(pole_real, pole_imag, 'x', mfc='none',
+        markersize=pole_markersize, color=pole_color)
+    plt.plot(zero_real, zero_imag, 'o', markersize=zero_markersize,
+        color=zero_color)
+    plt.xlabel('Real Axis')
+    plt.ylabel('Imaginary Axis')
+    plt.title(f'Poles and Zeros of ${latex(system)}$', pad=20)
+
+    if grid:
+        plt.grid()
+    if show_axes:
+        plt.axhline(0, color='black')
+        plt.axvline(0, color='black')
+    if show:
+        plt.show()
+        return
+
+    return plt
+
+
+def step_response_numerical_data(system, prec=8, lower_limit=0,
+    upper_limit=10, **kwargs):
+    """
+    Returns the numerical values of the points in the step response plot
+    of a SISO continuous-time system. By default, adaptive sampling
+    is used. If the user wants to instead get an uniformly
+    sampled response, then ``adaptive`` kwarg should be passed ``False``
+    and ``n`` must be passed as additional kwargs.
+    Refer to the parameters of class :class:`sympy.plotting.series.LineOver1DRangeSeries`
+    for more details.
+
+    Parameters
+    ==========
+
+    system : SISOLinearTimeInvariant
+        The system for which the unit step response data is to be computed.
+    prec : int, optional
+        The decimal point precision for the point coordinate values.
+        Defaults to 8.
+    lower_limit : Number, optional
+        The lower limit of the plot range. Defaults to 0.
+    upper_limit : Number, optional
+        The upper limit of the plot range. Defaults to 10.
+    kwargs :
+        Additional keyword arguments are passed to the underlying
+        :class:`sympy.plotting.series.LineOver1DRangeSeries` class.
+
+    Returns
+    =======
+
+    tuple : (x, y)
+        x = Time-axis values of the points in the step response. NumPy array.
+        y = Amplitude-axis values of the points in the step response. NumPy array.
+
+    Raises
+    ======
+
+    NotImplementedError
+        When a SISO LTI system is not passed.
+
+        When time delay terms are present in the system.
+
+    ValueError
+        When more than one free symbol is present in the system.
+        The only variable in the transfer function should be
+        the variable of the Laplace transform.
+
+        When ``lower_limit`` parameter is less than 0.
+
+    Examples
+    ========
+
+    >>> from sympy.abc import s
+    >>> from sympy.physics.control.lti import TransferFunction
+    >>> from sympy.physics.control.control_plots import step_response_numerical_data
+    >>> tf1 = TransferFunction(s, s**2 + 5*s + 8, s)
+    >>> step_response_numerical_data(tf1)   # doctest: +SKIP
+    ([0.0, 0.025413462339411542, 0.0484508722725343, ... , 9.670250533855183, 9.844291913708725, 10.0],
+    [0.0, 0.023844582399907256, 0.042894276802320226, ..., 6.828770759094287e-12, 6.456457160755703e-12])
+
+    See Also
+    ========
+
+    step_response_plot
+
+    """
+    if lower_limit < 0:
+        raise ValueError("Lower limit of time must be greater "
+            "than or equal to zero.")
+    _check_system(system)
+    _x = Dummy("x")
+    expr = system.to_expr()/(system.var)
+    expr = apart(expr, system.var, full=True)
+    _y = _fast_inverse_laplace(expr, system.var, _x).evalf(prec)
+    return LineOver1DRangeSeries(_y, (_x, lower_limit, upper_limit),
+        **kwargs).get_points()
+
+
+def step_response_plot(system, color='b', prec=8, lower_limit=0,
+    upper_limit=10, show_axes=False, grid=True, show=True, **kwargs):
+    r"""
+    Returns the unit step response of a continuous-time system. It is
+    the response of the system when the input signal is a step function.
+
+    Parameters
+    ==========
+
+    system : SISOLinearTimeInvariant type
+        The LTI SISO system for which the Step Response is to be computed.
+    color : str, tuple, optional
+        The color of the line. Default is Blue.
+    show : boolean, optional
+        If ``True``, the plot will be displayed otherwise
+        the equivalent matplotlib ``plot`` object will be returned.
+        Defaults to True.
+    lower_limit : Number, optional
+        The lower limit of the plot range. Defaults to 0.
+    upper_limit : Number, optional
+        The upper limit of the plot range. Defaults to 10.
+    prec : int, optional
+        The decimal point precision for the point coordinate values.
+        Defaults to 8.
+    show_axes : boolean, optional
+        If ``True``, the coordinate axes will be shown. Defaults to False.
+    grid : boolean, optional
+        If ``True``, the plot will have a grid. Defaults to True.
+
+    Examples
+    ========
+
+    .. plot::
+        :context: close-figs
+        :format: doctest
+        :include-source: True
+
+        >>> from sympy.abc import s
+        >>> from sympy.physics.control.lti import TransferFunction
+        >>> from sympy.physics.control.control_plots import step_response_plot
+        >>> tf1 = TransferFunction(8*s**2 + 18*s + 32, s**3 + 6*s**2 + 14*s + 24, s)
+        >>> step_response_plot(tf1)   # doctest: +SKIP
+
+    See Also
+    ========
+
+    impulse_response_plot, ramp_response_plot
+
+    References
+    ==========
+
+    .. [1] https://www.mathworks.com/help/control/ref/lti.step.html
+
+    """
+    x, y = step_response_numerical_data(system, prec=prec,
+        lower_limit=lower_limit, upper_limit=upper_limit, **kwargs)
+    plt.plot(x, y, color=color)
+    plt.xlabel('Time (s)')
+    plt.ylabel('Amplitude')
+    plt.title(f'Unit Step Response of ${latex(system)}$', pad=20)
+
+    if grid:
+        plt.grid()
+    if show_axes:
+        plt.axhline(0, color='black')
+        plt.axvline(0, color='black')
+    if show:
+        plt.show()
+        return
+
+    return plt
+
+
+def impulse_response_numerical_data(system, prec=8, lower_limit=0,
+    upper_limit=10, **kwargs):
+    """
+    Returns the numerical values of the points in the impulse response plot
+    of a SISO continuous-time system. By default, adaptive sampling
+    is used. If the user wants to instead get an uniformly
+    sampled response, then ``adaptive`` kwarg should be passed ``False``
+    and ``n`` must be passed as additional kwargs.
+    Refer to the parameters of class :class:`sympy.plotting.series.LineOver1DRangeSeries`
+    for more details.
+
+    Parameters
+    ==========
+
+    system : SISOLinearTimeInvariant
+        The system for which the impulse response data is to be computed.
+    prec : int, optional
+        The decimal point precision for the point coordinate values.
+        Defaults to 8.
+    lower_limit : Number, optional
+        The lower limit of the plot range. Defaults to 0.
+    upper_limit : Number, optional
+        The upper limit of the plot range. Defaults to 10.
+    kwargs :
+        Additional keyword arguments are passed to the underlying
+        :class:`sympy.plotting.series.LineOver1DRangeSeries` class.
+
+    Returns
+    =======
+
+    tuple : (x, y)
+        x = Time-axis values of the points in the impulse response. NumPy array.
+        y = Amplitude-axis values of the points in the impulse response. NumPy array.
+
+    Raises
+    ======
+
+    NotImplementedError
+        When a SISO LTI system is not passed.
+
+        When time delay terms are present in the system.
+
+    ValueError
+        When more than one free symbol is present in the system.
+        The only variable in the transfer function should be
+        the variable of the Laplace transform.
+
+        When ``lower_limit`` parameter is less than 0.
+
+    Examples
+    ========
+
+    >>> from sympy.abc import s
+    >>> from sympy.physics.control.lti import TransferFunction
+    >>> from sympy.physics.control.control_plots import impulse_response_numerical_data
+    >>> tf1 = TransferFunction(s, s**2 + 5*s + 8, s)
+    >>> impulse_response_numerical_data(tf1)   # doctest: +SKIP
+    ([0.0, 0.06616480200395854,... , 9.854500743565858, 10.0],
+    [0.9999999799999999, 0.7042848373025861,...,7.170748906965121e-13, -5.1901263495547205e-12])
+
+    See Also
+    ========
+
+    impulse_response_plot
+
+    """
+    if lower_limit < 0:
+        raise ValueError("Lower limit of time must be greater "
+            "than or equal to zero.")
+    _check_system(system)
+    _x = Dummy("x")
+    expr = system.to_expr()
+    expr = apart(expr, system.var, full=True)
+    _y = _fast_inverse_laplace(expr, system.var, _x).evalf(prec)
+    return LineOver1DRangeSeries(_y, (_x, lower_limit, upper_limit),
+        **kwargs).get_points()
+
+
+def impulse_response_plot(system, color='b', prec=8, lower_limit=0,
+    upper_limit=10, show_axes=False, grid=True, show=True, **kwargs):
+    r"""
+    Returns the unit impulse response (Input is the Dirac-Delta Function) of a
+    continuous-time system.
+
+    Parameters
+    ==========
+
+    system : SISOLinearTimeInvariant type
+        The LTI SISO system for which the Impulse Response is to be computed.
+    color : str, tuple, optional
+        The color of the line. Default is Blue.
+    show : boolean, optional
+        If ``True``, the plot will be displayed otherwise
+        the equivalent matplotlib ``plot`` object will be returned.
+        Defaults to True.
+    lower_limit : Number, optional
+        The lower limit of the plot range. Defaults to 0.
+    upper_limit : Number, optional
+        The upper limit of the plot range. Defaults to 10.
+    prec : int, optional
+        The decimal point precision for the point coordinate values.
+        Defaults to 8.
+    show_axes : boolean, optional
+        If ``True``, the coordinate axes will be shown. Defaults to False.
+    grid : boolean, optional
+        If ``True``, the plot will have a grid. Defaults to True.
+
+    Examples
+    ========
+
+    .. plot::
+        :context: close-figs
+        :format: doctest
+        :include-source: True
+
+        >>> from sympy.abc import s
+        >>> from sympy.physics.control.lti import TransferFunction
+        >>> from sympy.physics.control.control_plots import impulse_response_plot
+        >>> tf1 = TransferFunction(8*s**2 + 18*s + 32, s**3 + 6*s**2 + 14*s + 24, s)
+        >>> impulse_response_plot(tf1)   # doctest: +SKIP
+
+    See Also
+    ========
+
+    step_response_plot, ramp_response_plot
+
+    References
+    ==========
+
+    .. [1] https://www.mathworks.com/help/control/ref/dynamicsystem.impulse.html
+
+    """
+    x, y = impulse_response_numerical_data(system, prec=prec,
+        lower_limit=lower_limit, upper_limit=upper_limit, **kwargs)
+    plt.plot(x, y, color=color)
+    plt.xlabel('Time (s)')
+    plt.ylabel('Amplitude')
+    plt.title(f'Impulse Response of ${latex(system)}$', pad=20)
+
+    if grid:
+        plt.grid()
+    if show_axes:
+        plt.axhline(0, color='black')
+        plt.axvline(0, color='black')
+    if show:
+        plt.show()
+        return
+
+    return plt
+
+
+def ramp_response_numerical_data(system, slope=1, prec=8,
+    lower_limit=0, upper_limit=10, **kwargs):
+    """
+    Returns the numerical values of the points in the ramp response plot
+    of a SISO continuous-time system. By default, adaptive sampling
+    is used. If the user wants to instead get an uniformly
+    sampled response, then ``adaptive`` kwarg should be passed ``False``
+    and ``n`` must be passed as additional kwargs.
+    Refer to the parameters of class :class:`sympy.plotting.series.LineOver1DRangeSeries`
+    for more details.
+
+    Parameters
+    ==========
+
+    system : SISOLinearTimeInvariant
+        The system for which the ramp response data is to be computed.
+    slope : Number, optional
+        The slope of the input ramp function. Defaults to 1.
+    prec : int, optional
+        The decimal point precision for the point coordinate values.
+        Defaults to 8.
+    lower_limit : Number, optional
+        The lower limit of the plot range. Defaults to 0.
+    upper_limit : Number, optional
+        The upper limit of the plot range. Defaults to 10.
+    kwargs :
+        Additional keyword arguments are passed to the underlying
+        :class:`sympy.plotting.series.LineOver1DRangeSeries` class.
+
+    Returns
+    =======
+
+    tuple : (x, y)
+        x = Time-axis values of the points in the ramp response plot. NumPy array.
+        y = Amplitude-axis values of the points in the ramp response plot. NumPy array.
+
+    Raises
+    ======
+
+    NotImplementedError
+        When a SISO LTI system is not passed.
+
+        When time delay terms are present in the system.
+
+    ValueError
+        When more than one free symbol is present in the system.
+        The only variable in the transfer function should be
+        the variable of the Laplace transform.
+
+        When ``lower_limit`` parameter is less than 0.
+
+        When ``slope`` is negative.
+
+    Examples
+    ========
+
+    >>> from sympy.abc import s
+    >>> from sympy.physics.control.lti import TransferFunction
+    >>> from sympy.physics.control.control_plots import ramp_response_numerical_data
+    >>> tf1 = TransferFunction(s, s**2 + 5*s + 8, s)
+    >>> ramp_response_numerical_data(tf1)   # doctest: +SKIP
+    (([0.0, 0.12166980856813935,..., 9.861246379582118, 10.0],
+    [1.4504508011325967e-09, 0.006046440489058766,..., 0.12499999999568202, 0.12499999999661349]))
+
+    See Also
+    ========
+
+    ramp_response_plot
+
+    """
+    if slope < 0:
+        raise ValueError("Slope must be greater than or equal"
+            " to zero.")
+    if lower_limit < 0:
+        raise ValueError("Lower limit of time must be greater "
+            "than or equal to zero.")
+    _check_system(system)
+    _x = Dummy("x")
+    expr = (slope*system.to_expr())/((system.var)**2)
+    expr = apart(expr, system.var, full=True)
+    _y = _fast_inverse_laplace(expr, system.var, _x).evalf(prec)
+    return LineOver1DRangeSeries(_y, (_x, lower_limit, upper_limit),
+        **kwargs).get_points()
+
+
+def ramp_response_plot(system, slope=1, color='b', prec=8, lower_limit=0,
+    upper_limit=10, show_axes=False, grid=True, show=True, **kwargs):
+    r"""
+    Returns the ramp response of a continuous-time system.
+
+    Ramp function is defined as the straight line
+    passing through origin ($f(x) = mx$). The slope of
+    the ramp function can be varied by the user and
+    the default value is 1.
+
+    Parameters
+    ==========
+
+    system : SISOLinearTimeInvariant type
+        The LTI SISO system for which the Ramp Response is to be computed.
+    slope : Number, optional
+        The slope of the input ramp function. Defaults to 1.
+    color : str, tuple, optional
+        The color of the line. Default is Blue.
+    show : boolean, optional
+        If ``True``, the plot will be displayed otherwise
+        the equivalent matplotlib ``plot`` object will be returned.
+        Defaults to True.
+    lower_limit : Number, optional
+        The lower limit of the plot range. Defaults to 0.
+    upper_limit : Number, optional
+        The upper limit of the plot range. Defaults to 10.
+    prec : int, optional
+        The decimal point precision for the point coordinate values.
+        Defaults to 8.
+    show_axes : boolean, optional
+        If ``True``, the coordinate axes will be shown. Defaults to False.
+    grid : boolean, optional
+        If ``True``, the plot will have a grid. Defaults to True.
+
+    Examples
+    ========
+
+    .. plot::
+        :context: close-figs
+        :format: doctest
+        :include-source: True
+
+        >>> from sympy.abc import s
+        >>> from sympy.physics.control.lti import TransferFunction
+        >>> from sympy.physics.control.control_plots import ramp_response_plot
+        >>> tf1 = TransferFunction(s, (s+4)*(s+8), s)
+        >>> ramp_response_plot(tf1, upper_limit=2)   # doctest: +SKIP
+
+    See Also
+    ========
+
+    step_response_plot, impulse_response_plot
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Ramp_function
+
+    """
+    x, y = ramp_response_numerical_data(system, slope=slope, prec=prec,
+        lower_limit=lower_limit, upper_limit=upper_limit, **kwargs)
+    plt.plot(x, y, color=color)
+    plt.xlabel('Time (s)')
+    plt.ylabel('Amplitude')
+    plt.title(f'Ramp Response of ${latex(system)}$ [Slope = {slope}]', pad=20)
+
+    if grid:
+        plt.grid()
+    if show_axes:
+        plt.axhline(0, color='black')
+        plt.axvline(0, color='black')
+    if show:
+        plt.show()
+        return
+
+    return plt
+
+
+def bode_magnitude_numerical_data(system, initial_exp=-5, final_exp=5, freq_unit='rad/sec', **kwargs):
+    """
+    Returns the numerical data of the Bode magnitude plot of the system.
+    It is internally used by ``bode_magnitude_plot`` to get the data
+    for plotting Bode magnitude plot. Users can use this data to further
+    analyse the dynamics of the system or plot using a different
+    backend/plotting-module.
+
+    Parameters
+    ==========
+
+    system : SISOLinearTimeInvariant
+        The system for which the data is to be computed.
+    initial_exp : Number, optional
+        The initial exponent of 10 of the semilog plot. Defaults to -5.
+    final_exp : Number, optional
+        The final exponent of 10 of the semilog plot. Defaults to 5.
+    freq_unit : string, optional
+        User can choose between ``'rad/sec'`` (radians/second) and ``'Hz'`` (Hertz) as frequency units.
+
+    Returns
+    =======
+
+    tuple : (x, y)
+        x = x-axis values of the Bode magnitude plot.
+        y = y-axis values of the Bode magnitude plot.
+
+    Raises
+    ======
+
+    NotImplementedError
+        When a SISO LTI system is not passed.
+
+        When time delay terms are present in the system.
+
+    ValueError
+        When more than one free symbol is present in the system.
+        The only variable in the transfer function should be
+        the variable of the Laplace transform.
+
+        When incorrect frequency units are given as input.
+
+    Examples
+    ========
+
+    >>> from sympy.abc import s
+    >>> from sympy.physics.control.lti import TransferFunction
+    >>> from sympy.physics.control.control_plots import bode_magnitude_numerical_data
+    >>> tf1 = TransferFunction(s**2 + 1, s**4 + 4*s**3 + 6*s**2 + 5*s + 2, s)
+    >>> bode_magnitude_numerical_data(tf1)   # doctest: +SKIP
+    ([1e-05, 1.5148378120533502e-05,..., 68437.36188804005, 100000.0],
+    [-6.020599914256786, -6.0205999155219505,..., -193.4117304087953, -200.00000000260573])
+
+    See Also
+    ========
+
+    bode_magnitude_plot, bode_phase_numerical_data
+
+    """
+    _check_system(system)
+    expr = system.to_expr()
+    freq_units = ('rad/sec', 'Hz')
+    if freq_unit not in freq_units:
+        raise ValueError('Only "rad/sec" and "Hz" are accepted frequency units.')
+
+    _w = Dummy("w", real=True)
+    if freq_unit == 'Hz':
+        repl = I*_w*2*pi
+    else:
+        repl = I*_w
+    w_expr = expr.subs({system.var: repl})
+
+    mag = 20*log(Abs(w_expr), 10)
+
+    x, y = LineOver1DRangeSeries(mag,
+        (_w, 10**initial_exp, 10**final_exp), xscale='log', **kwargs).get_points()
+
+    return x, y
+
+
+def bode_magnitude_plot(system, initial_exp=-5, final_exp=5,
+    color='b', show_axes=False, grid=True, show=True, freq_unit='rad/sec', **kwargs):
+    r"""
+    Returns the Bode magnitude plot of a continuous-time system.
+
+    See ``bode_plot`` for all the parameters.
+    """
+    x, y = bode_magnitude_numerical_data(system, initial_exp=initial_exp,
+        final_exp=final_exp, freq_unit=freq_unit)
+    plt.plot(x, y, color=color, **kwargs)
+    plt.xscale('log')
+
+
+    plt.xlabel('Frequency (%s) [Log Scale]' % freq_unit)
+    plt.ylabel('Magnitude (dB)')
+    plt.title(f'Bode Plot (Magnitude) of ${latex(system)}$', pad=20)
+
+    if grid:
+        plt.grid(True)
+    if show_axes:
+        plt.axhline(0, color='black')
+        plt.axvline(0, color='black')
+    if show:
+        plt.show()
+        return
+
+    return plt
+
+
+def bode_phase_numerical_data(system, initial_exp=-5, final_exp=5, freq_unit='rad/sec', phase_unit='rad', phase_unwrap = True, **kwargs):
+    """
+    Returns the numerical data of the Bode phase plot of the system.
+    It is internally used by ``bode_phase_plot`` to get the data
+    for plotting Bode phase plot. Users can use this data to further
+    analyse the dynamics of the system or plot using a different
+    backend/plotting-module.
+
+    Parameters
+    ==========
+
+    system : SISOLinearTimeInvariant
+        The system for which the Bode phase plot data is to be computed.
+    initial_exp : Number, optional
+        The initial exponent of 10 of the semilog plot. Defaults to -5.
+    final_exp : Number, optional
+        The final exponent of 10 of the semilog plot. Defaults to 5.
+    freq_unit : string, optional
+        User can choose between ``'rad/sec'`` (radians/second) and '``'Hz'`` (Hertz) as frequency units.
+    phase_unit : string, optional
+        User can choose between ``'rad'`` (radians) and ``'deg'`` (degree) as phase units.
+    phase_unwrap : bool, optional
+        Set to ``True`` by default.
+
+    Returns
+    =======
+
+    tuple : (x, y)
+        x = x-axis values of the Bode phase plot.
+        y = y-axis values of the Bode phase plot.
+
+    Raises
+    ======
+
+    NotImplementedError
+        When a SISO LTI system is not passed.
+
+        When time delay terms are present in the system.
+
+    ValueError
+        When more than one free symbol is present in the system.
+        The only variable in the transfer function should be
+        the variable of the Laplace transform.
+
+        When incorrect frequency or phase units are given as input.
+
+    Examples
+    ========
+
+    >>> from sympy.abc import s
+    >>> from sympy.physics.control.lti import TransferFunction
+    >>> from sympy.physics.control.control_plots import bode_phase_numerical_data
+    >>> tf1 = TransferFunction(s**2 + 1, s**4 + 4*s**3 + 6*s**2 + 5*s + 2, s)
+    >>> bode_phase_numerical_data(tf1)   # doctest: +SKIP
+    ([1e-05, 1.4472354033813751e-05, 2.035581932165858e-05,..., 47577.3248186011, 67884.09326036123, 100000.0],
+    [-2.5000000000291665e-05, -3.6180885085e-05, -5.08895483066e-05,...,-3.1415085799262523, -3.14155265358979])
+
+    See Also
+    ========
+
+    bode_magnitude_plot, bode_phase_numerical_data
+
+    """
+    _check_system(system)
+    expr = system.to_expr()
+    freq_units = ('rad/sec', 'Hz')
+    phase_units = ('rad', 'deg')
+    if freq_unit not in freq_units:
+        raise ValueError('Only "rad/sec" and "Hz" are accepted frequency units.')
+    if phase_unit not in phase_units:
+        raise ValueError('Only "rad" and "deg" are accepted phase units.')
+
+    _w = Dummy("w", real=True)
+    if freq_unit == 'Hz':
+        repl = I*_w*2*pi
+    else:
+        repl = I*_w
+    w_expr = expr.subs({system.var: repl})
+
+    if phase_unit == 'deg':
+        phase = arg(w_expr)*180/pi
+    else:
+        phase = arg(w_expr)
+
+    x, y = LineOver1DRangeSeries(phase,
+        (_w, 10**initial_exp, 10**final_exp), xscale='log', **kwargs).get_points()
+
+    half = None
+    if phase_unwrap:
+        if(phase_unit == 'rad'):
+            half = pi
+        elif(phase_unit == 'deg'):
+            half = 180
+    if half:
+        unit = 2*half
+        for i in range(1, len(y)):
+            diff = y[i] - y[i - 1]
+            if diff > half:      # Jump from -half to half
+                y[i] = (y[i] - unit)
+            elif diff < -half:   # Jump from half to -half
+                y[i] = (y[i] + unit)
+
+    return x, y
+
+
+def bode_phase_plot(system, initial_exp=-5, final_exp=5,
+    color='b', show_axes=False, grid=True, show=True, freq_unit='rad/sec', phase_unit='rad', phase_unwrap=True, **kwargs):
+    r"""
+    Returns the Bode phase plot of a continuous-time system.
+
+    See ``bode_plot`` for all the parameters.
+    """
+    x, y = bode_phase_numerical_data(system, initial_exp=initial_exp,
+        final_exp=final_exp, freq_unit=freq_unit, phase_unit=phase_unit, phase_unwrap=phase_unwrap)
+    plt.plot(x, y, color=color, **kwargs)
+    plt.xscale('log')
+
+    plt.xlabel('Frequency (%s) [Log Scale]' % freq_unit)
+    plt.ylabel('Phase (%s)' % phase_unit)
+    plt.title(f'Bode Plot (Phase) of ${latex(system)}$', pad=20)
+
+    if grid:
+        plt.grid(True)
+    if show_axes:
+        plt.axhline(0, color='black')
+        plt.axvline(0, color='black')
+    if show:
+        plt.show()
+        return
+
+    return plt
+
+
+def bode_plot(system, initial_exp=-5, final_exp=5,
+    grid=True, show_axes=False, show=True, freq_unit='rad/sec', phase_unit='rad', phase_unwrap=True, **kwargs):
+    r"""
+    Returns the Bode phase and magnitude plots of a continuous-time system.
+
+    Parameters
+    ==========
+
+    system : SISOLinearTimeInvariant type
+        The LTI SISO system for which the Bode Plot is to be computed.
+    initial_exp : Number, optional
+        The initial exponent of 10 of the semilog plot. Defaults to -5.
+    final_exp : Number, optional
+        The final exponent of 10 of the semilog plot. Defaults to 5.
+    show : boolean, optional
+        If ``True``, the plot will be displayed otherwise
+        the equivalent matplotlib ``plot`` object will be returned.
+        Defaults to True.
+    prec : int, optional
+        The decimal point precision for the point coordinate values.
+        Defaults to 8.
+    grid : boolean, optional
+        If ``True``, the plot will have a grid. Defaults to True.
+    show_axes : boolean, optional
+        If ``True``, the coordinate axes will be shown. Defaults to False.
+    freq_unit : string, optional
+        User can choose between ``'rad/sec'`` (radians/second) and ``'Hz'`` (Hertz) as frequency units.
+    phase_unit : string, optional
+        User can choose between ``'rad'`` (radians) and ``'deg'`` (degree) as phase units.
+
+    Examples
+    ========
+
+    .. plot::
+        :context: close-figs
+        :format: doctest
+        :include-source: True
+
+        >>> from sympy.abc import s
+        >>> from sympy.physics.control.lti import TransferFunction
+        >>> from sympy.physics.control.control_plots import bode_plot
+        >>> tf1 = TransferFunction(1*s**2 + 0.1*s + 7.5, 1*s**4 + 0.12*s**3 + 9*s**2, s)
+        >>> bode_plot(tf1, initial_exp=0.2, final_exp=0.7)   # doctest: +SKIP
+
+    See Also
+    ========
+
+    bode_magnitude_plot, bode_phase_plot
+
+    """
+    plt.subplot(211)
+    mag = bode_magnitude_plot(system, initial_exp=initial_exp, final_exp=final_exp,
+        show=False, grid=grid, show_axes=show_axes,
+        freq_unit=freq_unit, **kwargs)
+    mag.title(f'Bode Plot of ${latex(system)}$', pad=20)
+    mag.xlabel(None)
+    plt.subplot(212)
+    bode_phase_plot(system, initial_exp=initial_exp, final_exp=final_exp,
+        show=False, grid=grid, show_axes=show_axes, freq_unit=freq_unit, phase_unit=phase_unit, phase_unwrap=phase_unwrap, **kwargs).title(None)
+
+    if show:
+        plt.show()
+        return
+
+    return plt

parrot/lib/python3.10/site-packages/sympy/physics/control/tests/test_control_plots.py

@@ -0,0 +1,299 @@
+from math import isclose
+from sympy.core.numbers import I
+from sympy.core.symbol import Dummy
+from sympy.functions.elementary.complexes import (Abs, arg)
+from sympy.functions.elementary.exponential import log
+from sympy.abc import s, p, a
+from sympy.external import import_module
+from sympy.physics.control.control_plots import \
+    (pole_zero_numerical_data, pole_zero_plot, step_response_numerical_data,
+    step_response_plot, impulse_response_numerical_data,
+    impulse_response_plot, ramp_response_numerical_data,
+    ramp_response_plot, bode_magnitude_numerical_data,
+    bode_phase_numerical_data, bode_plot)
+from sympy.physics.control.lti import (TransferFunction,
+    Series, Parallel, TransferFunctionMatrix)
+from sympy.testing.pytest import raises, skip
+
+matplotlib = import_module(
+        'matplotlib', import_kwargs={'fromlist': ['pyplot']},
+        catch=(RuntimeError,))
+
+numpy = import_module('numpy')
+
+tf1 = TransferFunction(1, p**2 + 0.5*p + 2, p)
+tf2 = TransferFunction(p, 6*p**2 + 3*p + 1, p)
+tf3 = TransferFunction(p, p**3 - 1, p)
+tf4 = TransferFunction(10, p**3, p)
+tf5 = TransferFunction(5, s**2 + 2*s + 10, s)
+tf6 = TransferFunction(1, 1, s)
+tf7 = TransferFunction(4*s*3 + 9*s**2 + 0.1*s + 11, 8*s**6 + 9*s**4 + 11, s)
+tf8 = TransferFunction(5, s**2 + (2+I)*s + 10, s)
+
+ser1 = Series(tf4, TransferFunction(1, p - 5, p))
+ser2 = Series(tf3, TransferFunction(p, p + 2, p))
+
+par1 = Parallel(tf1, tf2)
+
+
+def _to_tuple(a, b):
+    return tuple(a), tuple(b)
+
+def _trim_tuple(a, b):
+    a, b = _to_tuple(a, b)
+    return tuple(a[0: 2] + a[len(a)//2 : len(a)//2 + 1] + a[-2:]), \
+        tuple(b[0: 2] + b[len(b)//2 : len(b)//2 + 1] + b[-2:])
+
+def y_coordinate_equality(plot_data_func, evalf_func, system):
+    """Checks whether the y-coordinate value of the plotted
+    data point is equal to the value of the function at a
+    particular x."""
+    x, y = plot_data_func(system)
+    x, y = _trim_tuple(x, y)
+    y_exp = tuple(evalf_func(system, x_i) for x_i in x)
+    return all(Abs(y_exp_i - y_i) < 1e-8 for y_exp_i, y_i in zip(y_exp, y))
+
+
+def test_errors():
+    if not matplotlib:
+        skip("Matplotlib not the default backend")
+
+    # Invalid `system` check
+    tfm = TransferFunctionMatrix([[tf6, tf5], [tf5, tf6]])
+    expr = 1/(s**2 - 1)
+    raises(NotImplementedError, lambda: pole_zero_plot(tfm))
+    raises(NotImplementedError, lambda: pole_zero_numerical_data(expr))
+    raises(NotImplementedError, lambda: impulse_response_plot(expr))
+    raises(NotImplementedError, lambda: impulse_response_numerical_data(tfm))
+    raises(NotImplementedError, lambda: step_response_plot(tfm))
+    raises(NotImplementedError, lambda: step_response_numerical_data(expr))
+    raises(NotImplementedError, lambda: ramp_response_plot(expr))
+    raises(NotImplementedError, lambda: ramp_response_numerical_data(tfm))
+    raises(NotImplementedError, lambda: bode_plot(tfm))
+
+    # More than 1 variables
+    tf_a = TransferFunction(a, s + 1, s)
+    raises(ValueError, lambda: pole_zero_plot(tf_a))
+    raises(ValueError, lambda: pole_zero_numerical_data(tf_a))
+    raises(ValueError, lambda: impulse_response_plot(tf_a))
+    raises(ValueError, lambda: impulse_response_numerical_data(tf_a))
+    raises(ValueError, lambda: step_response_plot(tf_a))
+    raises(ValueError, lambda: step_response_numerical_data(tf_a))
+    raises(ValueError, lambda: ramp_response_plot(tf_a))
+    raises(ValueError, lambda: ramp_response_numerical_data(tf_a))
+    raises(ValueError, lambda: bode_plot(tf_a))
+
+    # lower_limit > 0 for response plots
+    raises(ValueError, lambda: impulse_response_plot(tf1, lower_limit=-1))
+    raises(ValueError, lambda: step_response_plot(tf1, lower_limit=-0.1))
+    raises(ValueError, lambda: ramp_response_plot(tf1, lower_limit=-4/3))
+
+    # slope in ramp_response_plot() is negative
+    raises(ValueError, lambda: ramp_response_plot(tf1, slope=-0.1))
+
+    # incorrect frequency or phase unit
+    raises(ValueError, lambda: bode_plot(tf1,freq_unit = 'hz'))
+    raises(ValueError, lambda: bode_plot(tf1,phase_unit = 'degree'))
+
+
+def test_pole_zero():
+    if not numpy:
+        skip("NumPy is required for this test")
+
+    def pz_tester(sys, expected_value):
+        z, p = pole_zero_numerical_data(sys)
+        z_check = numpy.allclose(z, expected_value[0])
+        p_check = numpy.allclose(p, expected_value[1])
+        return p_check and z_check
+
+    exp1 = [[], [-0.24999999999999994+1.3919410907075054j, -0.24999999999999994-1.3919410907075054j]]
+    exp2 = [[0.0], [-0.25+0.3227486121839514j, -0.25-0.3227486121839514j]]
+    exp3 = [[0.0], [-0.5000000000000004+0.8660254037844395j,
+        -0.5000000000000004-0.8660254037844395j, 0.9999999999999998+0j]]
+    exp4 = [[], [5.0, 0.0, 0.0, 0.0]]
+    exp5 = [[-5.645751311064592, -0.5000000000000008, -0.3542486889354093],
+        [-0.24999999999999986+1.3919410907075052j,
+        -0.24999999999999986-1.3919410907075052j, -0.2499999999999998+0.32274861218395134j,
+        -0.2499999999999998-0.32274861218395134j]]
+    exp6 = [[], [-1.1641600331447917-3.545808351896439j,
+          -0.8358399668552097+2.5458083518964383j]]
+
+    assert pz_tester(tf1, exp1)
+    assert pz_tester(tf2, exp2)
+    assert pz_tester(tf3, exp3)
+    assert pz_tester(ser1, exp4)
+    assert pz_tester(par1, exp5)
+    assert pz_tester(tf8, exp6)
+
+
+def test_bode():
+    if not numpy:
+        skip("NumPy is required for this test")
+
+    def bode_phase_evalf(system, point):
+        expr = system.to_expr()
+        _w = Dummy("w", real=True)
+        w_expr = expr.subs({system.var: I*_w})
+        return arg(w_expr).subs({_w: point}).evalf()
+
+    def bode_mag_evalf(system, point):
+        expr = system.to_expr()
+        _w = Dummy("w", real=True)
+        w_expr = expr.subs({system.var: I*_w})
+        return 20*log(Abs(w_expr), 10).subs({_w: point}).evalf()
+
+    def test_bode_data(sys):
+        return y_coordinate_equality(bode_magnitude_numerical_data, bode_mag_evalf, sys) \
+            and y_coordinate_equality(bode_phase_numerical_data, bode_phase_evalf, sys)
+
+    assert test_bode_data(tf1)
+    assert test_bode_data(tf2)
+    assert test_bode_data(tf3)
+    assert test_bode_data(tf4)
+    assert test_bode_data(tf5)
+
+
+def check_point_accuracy(a, b):
+    return all(isclose(*_, rel_tol=1e-1, abs_tol=1e-6
+        ) for _ in zip(a, b))
+
+
+def test_impulse_response():
+    if not numpy:
+        skip("NumPy is required for this test")
+
+    def impulse_res_tester(sys, expected_value):
+        x, y = _to_tuple(*impulse_response_numerical_data(sys,
+            adaptive=False, n=10))
+        x_check = check_point_accuracy(x, expected_value[0])
+        y_check = check_point_accuracy(y, expected_value[1])
+        return x_check and y_check
+
+    exp1 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445,
+        5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0),
+        (0.0, 0.544019738507865, 0.01993849743234938, -0.31140243360893216, -0.022852779906491996, 0.1778306498155759,
+        0.01962941084328499, -0.1013115194573652, -0.014975541213105696, 0.0575789724730714))
+    exp2 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555,
+        6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (0.1666666675, 0.08389223412935855,
+        0.02338051973475047, -0.014966807776379383, -0.034645954223054234, -0.040560075735512804,
+        -0.037658628907103885, -0.030149507719590022, -0.021162090730736834, -0.012721292737437523))
+    exp3 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555,
+        6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (4.369893391586999e-09, 1.1750333000630964,
+        3.2922404058312473, 9.432290008148343, 28.37098083007151, 86.18577464367974, 261.90356653762115,
+        795.6538758627842, 2416.9920942096983, 7342.159505206647))
+    exp4 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555,
+        6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (0.0, 6.17283950617284, 24.69135802469136,
+        55.555555555555564, 98.76543209876544, 154.320987654321, 222.22222222222226, 302.46913580246917,
+        395.0617283950618, 500.0))
+    exp5 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555,
+        6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (0.0, -0.10455606138085417,
+        0.06757671513476461, -0.03234567568833768, 0.013582514927757873, -0.005273419510705473,
+        0.0019364083003354075, -0.000680070134067832, 0.00022969845960406913, -7.476094359583917e-05))
+    exp6 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445,
+        5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0),
+        (-6.016699583000218e-09, 0.35039802056107394, 3.3728423827689884, 12.119846079276684,
+        25.86101014293389, 29.352480635282088, -30.49475907497664, -273.8717189554019, -863.2381702029659,
+        -1747.0262164682233))
+    exp7 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335,
+        4.444444444444445, 5.555555555555555, 6.666666666666667, 7.777777777777779,
+        8.88888888888889, 10.0), (0.0, 18.934638095560974, 5346.93244680907, 1384609.8718249386,
+        358161126.65801865, 92645770015.70108, 23964739753087.42, 6198974342083139.0, 1.603492601616059e+18,
+        4.147764422869658e+20))
+
+    assert impulse_res_tester(tf1, exp1)
+    assert impulse_res_tester(tf2, exp2)
+    assert impulse_res_tester(tf3, exp3)
+    assert impulse_res_tester(tf4, exp4)
+    assert impulse_res_tester(tf5, exp5)
+    assert impulse_res_tester(tf7, exp6)
+    assert impulse_res_tester(ser1, exp7)
+
+
+def test_step_response():
+    if not numpy:
+        skip("NumPy is required for this test")
+
+    def step_res_tester(sys, expected_value):
+        x, y = _to_tuple(*step_response_numerical_data(sys,
+            adaptive=False, n=10))
+        x_check = check_point_accuracy(x, expected_value[0])
+        y_check = check_point_accuracy(y, expected_value[1])
+        return x_check and y_check
+
+    exp1 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445,
+        5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0),
+        (-1.9193285738516863e-08, 0.42283495488246126, 0.7840485977945262, 0.5546841805655717,
+        0.33903033806932087, 0.4627251747410237, 0.5909907598988051, 0.5247213989553071,
+        0.4486997874319281, 0.4839358435839171))
+    exp2 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445,
+        5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0),
+        (0.0, 0.13728409095645816, 0.19474559355325086, 0.1974909129243011, 0.16841657696573073,
+        0.12559777736159378, 0.08153828016664713, 0.04360471317348958, 0.015072994568868221,
+        -0.003636420058445484))
+    exp3 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445,
+        5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0),
+        (0.0, 0.6314542141914303, 2.9356520038101035, 9.37731009663807, 28.452300356688376,
+        86.25721933273988, 261.9236645044672, 795.6435410577224, 2416.9786984578764, 7342.154119725917))
+    exp4 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445,
+        5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0),
+        (0.0, 2.286236899862826, 18.28989519890261, 61.72839629629631, 146.31916159122088, 285.7796124828532,
+        493.8271703703705, 784.1792566529494, 1170.553292729767, 1666.6667))
+    exp5 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445,
+        5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0),
+        (-3.999999997894577e-09, 0.6720357068882895, 0.4429938256137113, 0.5182010838004518,
+        0.4944139147159695, 0.5016379853883338, 0.4995466896527733, 0.5001154784851325,
+        0.49997448824584123, 0.5000039745919259))
+    exp6 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445,
+        5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0),
+        (-1.5433688493882158e-09, 0.3428705539937336, 1.1253619102202777, 3.1849962651016517,
+        9.47532757182671, 28.727231099148135, 87.29426924860557, 265.2138681048606, 805.6636260007757,
+        2447.387582370878))
+
+    assert step_res_tester(tf1, exp1)
+    assert step_res_tester(tf2, exp2)
+    assert step_res_tester(tf3, exp3)
+    assert step_res_tester(tf4, exp4)
+    assert step_res_tester(tf5, exp5)
+    assert step_res_tester(ser2, exp6)
+
+
+def test_ramp_response():
+    if not numpy:
+        skip("NumPy is required for this test")
+
+    def ramp_res_tester(sys, num_points, expected_value, slope=1):
+        x, y = _to_tuple(*ramp_response_numerical_data(sys,
+            slope=slope, adaptive=False, n=num_points))
+        x_check = check_point_accuracy(x, expected_value[0])
+        y_check = check_point_accuracy(y, expected_value[1])
+        return x_check and y_check
+
+    exp1 = ((0.0, 2.0, 4.0, 6.0, 8.0, 10.0), (0.0, 0.7324667795033895, 1.9909720978650398,
+        2.7956587704217783, 3.9224897567931514, 4.85022655284895))
+    exp2 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445,
+        5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0),
+        (2.4360213402019326e-08, 0.10175320182493253, 0.33057612497658406, 0.5967937263298935,
+        0.8431511866718248, 1.0398805391471613, 1.1776043125035738, 1.2600994825747305, 1.2981042689274653,
+        1.304684417610106))
+    exp3 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555,
+        6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (-3.9329040468771836e-08,
+        0.34686634635794555, 2.9998828170537903, 12.33303690737476, 40.993913948137795, 127.84145222317912,
+        391.41713691996, 1192.0006858708389, 3623.9808672503405, 11011.728034546572))
+    exp4 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555,
+        6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (0.0, 1.9051973784484078, 30.483158055174524,
+        154.32098765432104, 487.7305288827924, 1190.7483615302544, 2469.1358024691367, 4574.3789056546275,
+        7803.688462124678, 12500.0))
+    exp5 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555,
+        6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (0.0, 3.8844361856975635, 9.141792069209865,
+        14.096349157657231, 19.09783068994694, 24.10179770390321, 29.09907319114121, 34.10040420185154,
+        39.09983919254265, 44.10006013058409))
+    exp6 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555,
+        6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (0.0, 1.1111111111111112, 2.2222222222222223,
+        3.3333333333333335, 4.444444444444445, 5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0))
+
+    assert ramp_res_tester(tf1, 6, exp1)
+    assert ramp_res_tester(tf2, 10, exp2, 1.2)
+    assert ramp_res_tester(tf3, 10, exp3, 1.5)
+    assert ramp_res_tester(tf4, 10, exp4, 3)
+    assert ramp_res_tester(tf5, 10, exp5, 9)
+    assert ramp_res_tester(tf6, 10, exp6)

parrot/lib/python3.10/site-packages/sympy/physics/control/tests/test_lti.py

@@ -0,0 +1,1750 @@
+from sympy.core.add import Add
+from sympy.core.function import Function
+from sympy.core.mul import Mul
+from sympy.core.numbers import (I, pi, Rational, oo)
+from sympy.core.power import Pow
+from sympy.core.singleton import S
+from sympy.core.symbol import symbols
+from sympy.functions.elementary.exponential import (exp, log)
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import atan
+from sympy.matrices.dense import eye
+from sympy.polys.polytools import factor
+from sympy.polys.rootoftools import CRootOf
+from sympy.simplify.simplify import simplify
+from sympy.core.containers import Tuple
+from sympy.matrices import ImmutableMatrix, Matrix, ShapeError
+from sympy.physics.control import (TransferFunction, Series, Parallel,
+    Feedback, TransferFunctionMatrix, MIMOSeries, MIMOParallel, MIMOFeedback,
+    StateSpace, gbt, bilinear, forward_diff, backward_diff, phase_margin, gain_margin)
+from sympy.testing.pytest import raises
+
+a, x, b, c, s, g, d, p, k, tau, zeta, wn, T = symbols('a, x, b, c, s, g, d, p, k,\
+    tau, zeta, wn, T')
+a0, a1, a2, a3, b0, b1, b2, b3, c0, c1, c2, c3, d0, d1, d2, d3 = symbols('a0:4,\
+    b0:4, c0:4, d0:4')
+TF1 = TransferFunction(1, s**2 + 2*zeta*wn*s + wn**2, s)
+TF2 = TransferFunction(k, 1, s)
+TF3 = TransferFunction(a2*p - s, a2*s + p, s)
+
+
+def test_TransferFunction_construction():
+    tf = TransferFunction(s + 1, s**2 + s + 1, s)
+    assert tf.num == (s + 1)
+    assert tf.den == (s**2 + s + 1)
+    assert tf.args == (s + 1, s**2 + s + 1, s)
+
+    tf1 = TransferFunction(s + 4, s - 5, s)
+    assert tf1.num == (s + 4)
+    assert tf1.den == (s - 5)
+    assert tf1.args == (s + 4, s - 5, s)
+
+    # using different polynomial variables.
+    tf2 = TransferFunction(p + 3, p**2 - 9, p)
+    assert tf2.num == (p + 3)
+    assert tf2.den == (p**2 - 9)
+    assert tf2.args == (p + 3, p**2 - 9, p)
+
+    tf3 = TransferFunction(p**3 + 5*p**2 + 4, p**4 + 3*p + 1, p)
+    assert tf3.args == (p**3 + 5*p**2 + 4, p**4 + 3*p + 1, p)
+
+    # no pole-zero cancellation on its own.
+    tf4 = TransferFunction((s + 3)*(s - 1), (s - 1)*(s + 5), s)
+    assert tf4.den == (s - 1)*(s + 5)
+    assert tf4.args == ((s + 3)*(s - 1), (s - 1)*(s + 5), s)
+
+    tf4_ = TransferFunction(p + 2, p + 2, p)
+    assert tf4_.args == (p + 2, p + 2, p)
+
+    tf5 = TransferFunction(s - 1, 4 - p, s)
+    assert tf5.args == (s - 1, 4 - p, s)
+
+    tf5_ = TransferFunction(s - 1, s - 1, s)
+    assert tf5_.args == (s - 1, s - 1, s)
+
+    tf6 = TransferFunction(5, 6, s)
+    assert tf6.num == 5
+    assert tf6.den == 6
+    assert tf6.args == (5, 6, s)
+
+    tf6_ = TransferFunction(1/2, 4, s)
+    assert tf6_.num == 0.5
+    assert tf6_.den == 4
+    assert tf6_.args == (0.500000000000000, 4, s)
+
+    tf7 = TransferFunction(3*s**2 + 2*p + 4*s, 8*p**2 + 7*s, s)
+    tf8 = TransferFunction(3*s**2 + 2*p + 4*s, 8*p**2 + 7*s, p)
+    assert not tf7 == tf8
+
+    tf7_ = TransferFunction(a0*s + a1*s**2 + a2*s**3, b0*p - b1*s, s)
+    tf8_ = TransferFunction(a0*s + a1*s**2 + a2*s**3, b0*p - b1*s, s)
+    assert tf7_ == tf8_
+    assert -(-tf7_) == tf7_ == -(-(-(-tf7_)))
+
+    tf9 = TransferFunction(a*s**3 + b*s**2 + g*s + d, d*p + g*p**2 + g*s, s)
+    assert tf9.args == (a*s**3 + b*s**2 + d + g*s, d*p + g*p**2 + g*s, s)
+
+    tf10 = TransferFunction(p**3 + d, g*s**2 + d*s + a, p)
+    tf10_ = TransferFunction(p**3 + d, g*s**2 + d*s + a, p)
+    assert tf10.args == (d + p**3, a + d*s + g*s**2, p)
+    assert tf10_ == tf10
+
+    tf11 = TransferFunction(a1*s + a0, b2*s**2 + b1*s + b0, s)
+    assert tf11.num == (a0 + a1*s)
+    assert tf11.den == (b0 + b1*s + b2*s**2)
+    assert tf11.args == (a0 + a1*s, b0 + b1*s + b2*s**2, s)
+
+    # when just the numerator is 0, leave the denominator alone.
+    tf12 = TransferFunction(0, p**2 - p + 1, p)
+    assert tf12.args == (0, p**2 - p + 1, p)
+
+    tf13 = TransferFunction(0, 1, s)
+    assert tf13.args == (0, 1, s)
+
+    # float exponents
+    tf14 = TransferFunction(a0*s**0.5 + a2*s**0.6 - a1, a1*p**(-8.7), s)
+    assert tf14.args == (a0*s**0.5 - a1 + a2*s**0.6, a1*p**(-8.7), s)
+
+    tf15 = TransferFunction(a2**2*p**(1/4) + a1*s**(-4/5), a0*s - p, p)
+    assert tf15.args == (a1*s**(-0.8) + a2**2*p**0.25, a0*s - p, p)
+
+    omega_o, k_p, k_o, k_i = symbols('omega_o, k_p, k_o, k_i')
+    tf18 = TransferFunction((k_p + k_o*s + k_i/s), s**2 + 2*omega_o*s + omega_o**2, s)
+    assert tf18.num == k_i/s + k_o*s + k_p
+    assert tf18.args == (k_i/s + k_o*s + k_p, omega_o**2 + 2*omega_o*s + s**2, s)
+
+    # ValueError when denominator is zero.
+    raises(ValueError, lambda: TransferFunction(4, 0, s))
+    raises(ValueError, lambda: TransferFunction(s, 0, s))
+    raises(ValueError, lambda: TransferFunction(0, 0, s))
+
+    raises(TypeError, lambda: TransferFunction(Matrix([1, 2, 3]), s, s))
+
+    raises(TypeError, lambda: TransferFunction(s**2 + 2*s - 1, s + 3, 3))
+    raises(TypeError, lambda: TransferFunction(p + 1, 5 - p, 4))
+    raises(TypeError, lambda: TransferFunction(3, 4, 8))
+
+
+def test_TransferFunction_functions():
+    # classmethod from_rational_expression
+    expr_1 = Mul(0, Pow(s, -1, evaluate=False), evaluate=False)
+    expr_2 = s/0
+    expr_3 = (p*s**2 + 5*s)/(s + 1)**3
+    expr_4 = 6
+    expr_5 = ((2 + 3*s)*(5 + 2*s))/((9 + 3*s)*(5 + 2*s**2))
+    expr_6 = (9*s**4 + 4*s**2 + 8)/((s + 1)*(s + 9))
+    tf = TransferFunction(s + 1, s**2 + 2, s)
+    delay = exp(-s/tau)
+    expr_7 = delay*tf.to_expr()
+    H1 = TransferFunction.from_rational_expression(expr_7, s)
+    H2 = TransferFunction(s + 1, (s**2 + 2)*exp(s/tau), s)
+    expr_8 = Add(2,  3*s/(s**2 + 1), evaluate=False)
+
+    assert TransferFunction.from_rational_expression(expr_1) == TransferFunction(0, s, s)
+    raises(ZeroDivisionError, lambda: TransferFunction.from_rational_expression(expr_2))
+    raises(ValueError, lambda: TransferFunction.from_rational_expression(expr_3))
+    assert TransferFunction.from_rational_expression(expr_3, s) == TransferFunction((p*s**2 + 5*s), (s + 1)**3, s)
+    assert TransferFunction.from_rational_expression(expr_3, p) == TransferFunction((p*s**2 + 5*s), (s + 1)**3, p)
+    raises(ValueError, lambda: TransferFunction.from_rational_expression(expr_4))
+    assert TransferFunction.from_rational_expression(expr_4, s) == TransferFunction(6, 1, s)
+    assert TransferFunction.from_rational_expression(expr_5, s) == \
+        TransferFunction((2 + 3*s)*(5 + 2*s), (9 + 3*s)*(5 + 2*s**2), s)
+    assert TransferFunction.from_rational_expression(expr_6, s) == \
+        TransferFunction((9*s**4 + 4*s**2 + 8), (s + 1)*(s + 9), s)
+    assert H1 == H2
+    assert TransferFunction.from_rational_expression(expr_8, s) == \
+        TransferFunction(2*s**2 + 3*s + 2, s**2 + 1, s)
+
+    # classmethod from_coeff_lists
+    tf1 = TransferFunction.from_coeff_lists([1, 2], [3, 4, 5], s)
+    num2 = [p**2, 2*p]
+    den2 = [p**3, p + 1, 4]
+    tf2 = TransferFunction.from_coeff_lists(num2, den2, s)
+    num3 = [1, 2, 3]
+    den3 = [0, 0]
+
+    assert tf1 == TransferFunction(s + 2, 3*s**2 + 4*s + 5, s)
+    assert tf2 == TransferFunction(p**2*s + 2*p, p**3*s**2 + s*(p + 1) + 4, s)
+    raises(ZeroDivisionError, lambda: TransferFunction.from_coeff_lists(num3, den3, s))
+
+    # classmethod from_zpk
+    zeros = [4]
+    poles = [-1+2j, -1-2j]
+    gain = 3
+    tf1 = TransferFunction.from_zpk(zeros, poles, gain, s)
+
+    assert tf1 == TransferFunction(3*s - 12, (s + 1.0 - 2.0*I)*(s + 1.0 + 2.0*I), s)
+
+    # explicitly cancel poles and zeros.
+    tf0 = TransferFunction(s**5 + s**3 + s, s - s**2, s)
+    a = TransferFunction(-(s**4 + s**2 + 1), s - 1, s)
+    assert tf0.simplify() == simplify(tf0) == a
+
+    tf1 = TransferFunction((p + 3)*(p - 1), (p - 1)*(p + 5), p)
+    b = TransferFunction(p + 3, p + 5, p)
+    assert tf1.simplify() == simplify(tf1) == b
+
+    # expand the numerator and the denominator.
+    G1 = TransferFunction((1 - s)**2, (s**2 + 1)**2, s)
+    G2 = TransferFunction(1, -3, p)
+    c = (a2*s**p + a1*s**s + a0*p**p)*(p**s + s**p)
+    d = (b0*s**s + b1*p**s)*(b2*s*p + p**p)
+    e = a0*p**p*p**s + a0*p**p*s**p + a1*p**s*s**s + a1*s**p*s**s + a2*p**s*s**p + a2*s**(2*p)
+    f = b0*b2*p*s*s**s + b0*p**p*s**s + b1*b2*p*p**s*s + b1*p**p*p**s
+    g = a1*a2*s*s**p + a1*p*s + a2*b1*p*s*s**p + b1*p**2*s
+    G3 = TransferFunction(c, d, s)
+    G4 = TransferFunction(a0*s**s - b0*p**p, (a1*s + b1*s*p)*(a2*s**p + p), p)
+
+    assert G1.expand() == TransferFunction(s**2 - 2*s + 1, s**4 + 2*s**2 + 1, s)
+    assert tf1.expand() == TransferFunction(p**2 + 2*p - 3, p**2 + 4*p - 5, p)
+    assert G2.expand() == G2
+    assert G3.expand() == TransferFunction(e, f, s)
+    assert G4.expand() == TransferFunction(a0*s**s - b0*p**p, g, p)
+
+    # purely symbolic polynomials.
+    p1 = a1*s + a0
+    p2 = b2*s**2 + b1*s + b0
+    SP1 = TransferFunction(p1, p2, s)
+    expect1 = TransferFunction(2.0*s + 1.0, 5.0*s**2 + 4.0*s + 3.0, s)
+    expect1_ = TransferFunction(2*s + 1, 5*s**2 + 4*s + 3, s)
+    assert SP1.subs({a0: 1, a1: 2, b0: 3, b1: 4, b2: 5}) == expect1_
+    assert SP1.subs({a0: 1, a1: 2, b0: 3, b1: 4, b2: 5}).evalf() == expect1
+    assert expect1_.evalf() == expect1
+
+    c1, d0, d1, d2 = symbols('c1, d0:3')
+    p3, p4 = c1*p, d2*p**3 + d1*p**2 - d0
+    SP2 = TransferFunction(p3, p4, p)
+    expect2 = TransferFunction(2.0*p, 5.0*p**3 + 2.0*p**2 - 3.0, p)
+    expect2_ = TransferFunction(2*p, 5*p**3 + 2*p**2 - 3, p)
+    assert SP2.subs({c1: 2, d0: 3, d1: 2, d2: 5}) == expect2_
+    assert SP2.subs({c1: 2, d0: 3, d1: 2, d2: 5}).evalf() == expect2
+    assert expect2_.evalf() == expect2
+
+    SP3 = TransferFunction(a0*p**3 + a1*s**2 - b0*s + b1, a1*s + p, s)
+    expect3 = TransferFunction(2.0*p**3 + 4.0*s**2 - s + 5.0, p + 4.0*s, s)
+    expect3_ = TransferFunction(2*p**3 + 4*s**2 - s + 5, p + 4*s, s)
+    assert SP3.subs({a0: 2, a1: 4, b0: 1, b1: 5}) == expect3_
+    assert SP3.subs({a0: 2, a1: 4, b0: 1, b1: 5}).evalf() == expect3
+    assert expect3_.evalf() == expect3
+
+    SP4 = TransferFunction(s - a1*p**3, a0*s + p, p)
+    expect4 = TransferFunction(7.0*p**3 + s, p - s, p)
+    expect4_ = TransferFunction(7*p**3 + s, p - s, p)
+    assert SP4.subs({a0: -1, a1: -7}) == expect4_
+    assert SP4.subs({a0: -1, a1: -7}).evalf() == expect4
+    assert expect4_.evalf() == expect4
+
+    # evaluate the transfer function at particular frequencies.
+    assert tf1.eval_frequency(wn) == wn**2/(wn**2 + 4*wn - 5) + 2*wn/(wn**2 + 4*wn - 5) - 3/(wn**2 + 4*wn - 5)
+    assert G1.eval_frequency(1 + I) == S(3)/25 + S(4)*I/25
+    assert G4.eval_frequency(S(5)/3) == \
+        a0*s**s/(a1*a2*s**(S(8)/3) + S(5)*a1*s/3 + 5*a2*b1*s**(S(8)/3)/3 + S(25)*b1*s/9) - 5*3**(S(1)/3)*5**(S(2)/3)*b0/(9*a1*a2*s**(S(8)/3) + 15*a1*s + 15*a2*b1*s**(S(8)/3) + 25*b1*s)
+
+    # Low-frequency (or DC) gain.
+    assert tf0.dc_gain() == 1
+    assert tf1.dc_gain() == Rational(3, 5)
+    assert SP2.dc_gain() == 0
+    assert expect4.dc_gain() == -1
+    assert expect2_.dc_gain() == 0
+    assert TransferFunction(1, s, s).dc_gain() == oo
+
+    # Poles of a transfer function.
+    tf_ = TransferFunction(x**3 - k, k, x)
+    _tf = TransferFunction(k, x**4 - k, x)
+    TF_ = TransferFunction(x**2, x**10 + x + x**2, x)
+    _TF = TransferFunction(x**10 + x + x**2, x**2, x)
+    assert G1.poles() == [I, I, -I, -I]
+    assert G2.poles() == []
+    assert tf1.poles() == [-5, 1]
+    assert expect4_.poles() == [s]
+    assert SP4.poles() == [-a0*s]
+    assert expect3.poles() == [-0.25*p]
+    assert str(expect2.poles()) == str([0.729001428685125, -0.564500714342563 - 0.710198984796332*I, -0.564500714342563 + 0.710198984796332*I])
+    assert str(expect1.poles()) == str([-0.4 - 0.66332495807108*I, -0.4 + 0.66332495807108*I])
+    assert _tf.poles() == [k**(Rational(1, 4)), -k**(Rational(1, 4)), I*k**(Rational(1, 4)), -I*k**(Rational(1, 4))]
+    assert TF_.poles() == [CRootOf(x**9 + x + 1, 0), 0, CRootOf(x**9 + x + 1, 1), CRootOf(x**9 + x + 1, 2),
+        CRootOf(x**9 + x + 1, 3), CRootOf(x**9 + x + 1, 4), CRootOf(x**9 + x + 1, 5), CRootOf(x**9 + x + 1, 6),
+        CRootOf(x**9 + x + 1, 7), CRootOf(x**9 + x + 1, 8)]
+    raises(NotImplementedError, lambda: TransferFunction(x**2, a0*x**10 + x + x**2, x).poles())
+
+    # Stability of a transfer function.
+    q, r = symbols('q, r', negative=True)
+    t = symbols('t', positive=True)
+    TF_ = TransferFunction(s**2 + a0 - a1*p, q*s - r, s)
+    stable_tf = TransferFunction(s**2 + a0 - a1*p, q*s - 1, s)
+    stable_tf_ = TransferFunction(s**2 + a0 - a1*p, q*s - t, s)
+
+    assert G1.is_stable() is False
+    assert G2.is_stable() is True
+    assert tf1.is_stable() is False   # as one pole is +ve, and the other is -ve.
+    assert expect2.is_stable() is False
+    assert expect1.is_stable() is True
+    assert stable_tf.is_stable() is True
+    assert stable_tf_.is_stable() is True
+    assert TF_.is_stable() is False
+    assert expect4_.is_stable() is None   # no assumption provided for the only pole 's'.
+    assert SP4.is_stable() is None
+
+    # Zeros of a transfer function.
+    assert G1.zeros() == [1, 1]
+    assert G2.zeros() == []
+    assert tf1.zeros() == [-3, 1]
+    assert expect4_.zeros() == [7**(Rational(2, 3))*(-s)**(Rational(1, 3))/7, -7**(Rational(2, 3))*(-s)**(Rational(1, 3))/14 -
+        sqrt(3)*7**(Rational(2, 3))*I*(-s)**(Rational(1, 3))/14, -7**(Rational(2, 3))*(-s)**(Rational(1, 3))/14 + sqrt(3)*7**(Rational(2, 3))*I*(-s)**(Rational(1, 3))/14]
+    assert SP4.zeros() == [(s/a1)**(Rational(1, 3)), -(s/a1)**(Rational(1, 3))/2 - sqrt(3)*I*(s/a1)**(Rational(1, 3))/2,
+        -(s/a1)**(Rational(1, 3))/2 + sqrt(3)*I*(s/a1)**(Rational(1, 3))/2]
+    assert str(expect3.zeros()) == str([0.125 - 1.11102430216445*sqrt(-0.405063291139241*p**3 - 1.0),
+        1.11102430216445*sqrt(-0.405063291139241*p**3 - 1.0) + 0.125])
+    assert tf_.zeros() == [k**(Rational(1, 3)), -k**(Rational(1, 3))/2 - sqrt(3)*I*k**(Rational(1, 3))/2,
+        -k**(Rational(1, 3))/2 + sqrt(3)*I*k**(Rational(1, 3))/2]
+    assert _TF.zeros() == [CRootOf(x**9 + x + 1, 0), 0, CRootOf(x**9 + x + 1, 1), CRootOf(x**9 + x + 1, 2),
+        CRootOf(x**9 + x + 1, 3), CRootOf(x**9 + x + 1, 4), CRootOf(x**9 + x + 1, 5), CRootOf(x**9 + x + 1, 6),
+        CRootOf(x**9 + x + 1, 7), CRootOf(x**9 + x + 1, 8)]
+    raises(NotImplementedError, lambda: TransferFunction(a0*x**10 + x + x**2, x**2, x).zeros())
+
+    # negation of TF.
+    tf2 = TransferFunction(s + 3, s**2 - s**3 + 9, s)
+    tf3 = TransferFunction(-3*p + 3, 1 - p, p)
+    assert -tf2 == TransferFunction(-s - 3, s**2 - s**3 + 9, s)
+    assert -tf3 == TransferFunction(3*p - 3, 1 - p, p)
+
+    # taking power of a TF.
+    tf4 = TransferFunction(p + 4, p - 3, p)
+    tf5 = TransferFunction(s**2 + 1, 1 - s, s)
+    expect2 = TransferFunction((s**2 + 1)**3, (1 - s)**3, s)
+    expect1 = TransferFunction((p + 4)**2, (p - 3)**2, p)
+    assert (tf4*tf4).doit() == tf4**2 == pow(tf4, 2) == expect1
+    assert (tf5*tf5*tf5).doit() == tf5**3 == pow(tf5, 3) == expect2
+    assert tf5**0 == pow(tf5, 0) == TransferFunction(1, 1, s)
+    assert Series(tf4).doit()**-1 == tf4**-1 == pow(tf4, -1) == TransferFunction(p - 3, p + 4, p)
+    assert (tf5*tf5).doit()**-1 == tf5**-2 == pow(tf5, -2) == TransferFunction((1 - s)**2, (s**2 + 1)**2, s)
+
+    raises(ValueError, lambda: tf4**(s**2 + s - 1))
+    raises(ValueError, lambda: tf5**s)
+    raises(ValueError, lambda: tf4**tf5)
+
+    # SymPy's own functions.
+    tf = TransferFunction(s - 1, s**2 - 2*s + 1, s)
+    tf6 = TransferFunction(s + p, p**2 - 5, s)
+    assert factor(tf) == TransferFunction(s - 1, (s - 1)**2, s)
+    assert tf.num.subs(s, 2) == tf.den.subs(s, 2) == 1
+    # subs & xreplace
+    assert tf.subs(s, 2) == TransferFunction(s - 1, s**2 - 2*s + 1, s)
+    assert tf6.subs(p, 3) == TransferFunction(s + 3, 4, s)
+    assert tf3.xreplace({p: s}) == TransferFunction(-3*s + 3, 1 - s, s)
+    raises(TypeError, lambda: tf3.xreplace({p: exp(2)}))
+    assert tf3.subs(p, exp(2)) == tf3
+
+    tf7 = TransferFunction(a0*s**p + a1*p**s, a2*p - s, s)
+    assert tf7.xreplace({s: k}) == TransferFunction(a0*k**p + a1*p**k, a2*p - k, k)
+    assert tf7.subs(s, k) == TransferFunction(a0*s**p + a1*p**s, a2*p - s, s)
+
+    # Conversion to Expr with to_expr()
+    tf8 = TransferFunction(a0*s**5 + 5*s**2 + 3, s**6 - 3, s)
+    tf9 = TransferFunction((5 + s), (5 + s)*(6 + s), s)
+    tf10 = TransferFunction(0, 1, s)
+    tf11 = TransferFunction(1, 1, s)
+    assert tf8.to_expr() == Mul((a0*s**5 + 5*s**2 + 3), Pow((s**6 - 3), -1, evaluate=False), evaluate=False)
+    assert tf9.to_expr() == Mul((s + 5), Pow((5 + s)*(6 + s), -1, evaluate=False), evaluate=False)
+    assert tf10.to_expr() == Mul(S(0), Pow(1, -1, evaluate=False), evaluate=False)
+    assert tf11.to_expr() == Pow(1, -1, evaluate=False)
+
+def test_TransferFunction_addition_and_subtraction():
+    tf1 = TransferFunction(s + 6, s - 5, s)
+    tf2 = TransferFunction(s + 3, s + 1, s)
+    tf3 = TransferFunction(s + 1, s**2 + s + 1, s)
+    tf4 = TransferFunction(p, 2 - p, p)
+
+    # addition
+    assert tf1 + tf2 == Parallel(tf1, tf2)
+    assert tf3 + tf1 == Parallel(tf3, tf1)
+    assert -tf1 + tf2 + tf3 == Parallel(-tf1, tf2, tf3)
+    assert tf1 + (tf2 + tf3) == Parallel(tf1, tf2, tf3)
+
+    c = symbols("c", commutative=False)
+    raises(ValueError, lambda: tf1 + Matrix([1, 2, 3]))
+    raises(ValueError, lambda: tf2 + c)
+    raises(ValueError, lambda: tf3 + tf4)
+    raises(ValueError, lambda: tf1 + (s - 1))
+    raises(ValueError, lambda: tf1 + 8)
+    raises(ValueError, lambda: (1 - p**3) + tf1)
+
+    # subtraction
+    assert tf1 - tf2 == Parallel(tf1, -tf2)
+    assert tf3 - tf2 == Parallel(tf3, -tf2)
+    assert -tf1 - tf3 == Parallel(-tf1, -tf3)
+    assert tf1 - tf2 + tf3 == Parallel(tf1, -tf2, tf3)
+
+    raises(ValueError, lambda: tf1 - Matrix([1, 2, 3]))
+    raises(ValueError, lambda: tf3 - tf4)
+    raises(ValueError, lambda: tf1 - (s - 1))
+    raises(ValueError, lambda: tf1 - 8)
+    raises(ValueError, lambda: (s + 5) - tf2)
+    raises(ValueError, lambda: (1 + p**4) - tf1)
+
+
+def test_TransferFunction_multiplication_and_division():
+    G1 = TransferFunction(s + 3, -s**3 + 9, s)
+    G2 = TransferFunction(s + 1, s - 5, s)
+    G3 = TransferFunction(p, p**4 - 6, p)
+    G4 = TransferFunction(p + 4, p - 5, p)
+    G5 = TransferFunction(s + 6, s - 5, s)
+    G6 = TransferFunction(s + 3, s + 1, s)
+    G7 = TransferFunction(1, 1, s)
+
+    # multiplication
+    assert G1*G2 == Series(G1, G2)
+    assert -G1*G5 == Series(-G1, G5)
+    assert -G2*G5*-G6 == Series(-G2, G5, -G6)
+    assert -G1*-G2*-G5*-G6 == Series(-G1, -G2, -G5, -G6)
+    assert G3*G4 == Series(G3, G4)
+    assert (G1*G2)*-(G5*G6) == \
+        Series(G1, G2, TransferFunction(-1, 1, s), Series(G5, G6))
+    assert G1*G2*(G5 + G6) == Series(G1, G2, Parallel(G5, G6))
+
+    # division - See ``test_Feedback_functions()`` for division by Parallel objects.
+    assert G5/G6 == Series(G5, pow(G6, -1))
+    assert -G3/G4 == Series(-G3, pow(G4, -1))
+    assert (G5*G6)/G7 == Series(G5, G6, pow(G7, -1))
+
+    c = symbols("c", commutative=False)
+    raises(ValueError, lambda: G3 * Matrix([1, 2, 3]))
+    raises(ValueError, lambda: G1 * c)
+    raises(ValueError, lambda: G3 * G5)
+    raises(ValueError, lambda: G5 * (s - 1))
+    raises(ValueError, lambda: 9 * G5)
+
+    raises(ValueError, lambda: G3 / Matrix([1, 2, 3]))
+    raises(ValueError, lambda: G6 / 0)
+    raises(ValueError, lambda: G3 / G5)
+    raises(ValueError, lambda: G5 / 2)
+    raises(ValueError, lambda: G5 / s**2)
+    raises(ValueError, lambda: (s - 4*s**2) / G2)
+    raises(ValueError, lambda: 0 / G4)
+    raises(ValueError, lambda: G7 / (1 + G6))
+    raises(ValueError, lambda: G7 / (G5 * G6))
+    raises(ValueError, lambda: G7 / (G7 + (G5 + G6)))
+
+
+def test_TransferFunction_is_proper():
+    omega_o, zeta, tau = symbols('omega_o, zeta, tau')
+    G1 = TransferFunction(omega_o**2, s**2 + p*omega_o*zeta*s + omega_o**2, omega_o)
+    G2 = TransferFunction(tau - s**3, tau + p**4, tau)
+    G3 = TransferFunction(a*b*s**3 + s**2 - a*p + s, b - s*p**2, p)
+    G4 = TransferFunction(b*s**2 + p**2 - a*p + s, b - p**2, s)
+    assert G1.is_proper
+    assert G2.is_proper
+    assert G3.is_proper
+    assert not G4.is_proper
+
+
+def test_TransferFunction_is_strictly_proper():
+    omega_o, zeta, tau = symbols('omega_o, zeta, tau')
+    tf1 = TransferFunction(omega_o**2, s**2 + p*omega_o*zeta*s + omega_o**2, omega_o)
+    tf2 = TransferFunction(tau - s**3, tau + p**4, tau)
+    tf3 = TransferFunction(a*b*s**3 + s**2 - a*p + s, b - s*p**2, p)
+    tf4 = TransferFunction(b*s**2 + p**2 - a*p + s, b - p**2, s)
+    assert not tf1.is_strictly_proper
+    assert not tf2.is_strictly_proper
+    assert tf3.is_strictly_proper
+    assert not tf4.is_strictly_proper
+
+
+def test_TransferFunction_is_biproper():
+    tau, omega_o, zeta = symbols('tau, omega_o, zeta')
+    tf1 = TransferFunction(omega_o**2, s**2 + p*omega_o*zeta*s + omega_o**2, omega_o)
+    tf2 = TransferFunction(tau - s**3, tau + p**4, tau)
+    tf3 = TransferFunction(a*b*s**3 + s**2 - a*p + s, b - s*p**2, p)
+    tf4 = TransferFunction(b*s**2 + p**2 - a*p + s, b - p**2, s)
+    assert tf1.is_biproper
+    assert tf2.is_biproper
+    assert not tf3.is_biproper
+    assert not tf4.is_biproper
+
+
+def test_Series_construction():
+    tf = TransferFunction(a0*s**3 + a1*s**2 - a2*s, b0*p**4 + b1*p**3 - b2*s*p, s)
+    tf2 = TransferFunction(a2*p - s, a2*s + p, s)
+    tf3 = TransferFunction(a0*p + p**a1 - s, p, p)
+    tf4 = TransferFunction(1, s**2 + 2*zeta*wn*s + wn**2, s)
+    inp = Function('X_d')(s)
+    out = Function('X')(s)
+
+    s0 = Series(tf, tf2)
+    assert s0.args == (tf, tf2)
+    assert s0.var == s
+
+    s1 = Series(Parallel(tf, -tf2), tf2)
+    assert s1.args == (Parallel(tf, -tf2), tf2)
+    assert s1.var == s
+
+    tf3_ = TransferFunction(inp, 1, s)
+    tf4_ = TransferFunction(-out, 1, s)
+    s2 = Series(tf, Parallel(tf3_, tf4_), tf2)
+    assert s2.args == (tf, Parallel(tf3_, tf4_), tf2)
+
+    s3 = Series(tf, tf2, tf4)
+    assert s3.args == (tf, tf2, tf4)
+
+    s4 = Series(tf3_, tf4_)
+    assert s4.args == (tf3_, tf4_)
+    assert s4.var == s
+
+    s6 = Series(tf2, tf4, Parallel(tf2, -tf), tf4)
+    assert s6.args == (tf2, tf4, Parallel(tf2, -tf), tf4)
+
+    s7 = Series(tf, tf2)
+    assert s0 == s7
+    assert not s0 == s2
+
+    raises(ValueError, lambda: Series(tf, tf3))
+    raises(ValueError, lambda: Series(tf, tf2, tf3, tf4))
+    raises(ValueError, lambda: Series(-tf3, tf2))
+    raises(TypeError, lambda: Series(2, tf, tf4))
+    raises(TypeError, lambda: Series(s**2 + p*s, tf3, tf2))
+    raises(TypeError, lambda: Series(tf3, Matrix([1, 2, 3, 4])))
+
+
+def test_MIMOSeries_construction():
+    tf_1 = TransferFunction(a0*s**3 + a1*s**2 - a2*s, b0*p**4 + b1*p**3 - b2*s*p, s)
+    tf_2 = TransferFunction(a2*p - s, a2*s + p, s)
+    tf_3 = TransferFunction(1, s**2 + 2*zeta*wn*s + wn**2, s)
+
+    tfm_1 = TransferFunctionMatrix([[tf_1, tf_2, tf_3], [-tf_3, -tf_2, tf_1]])
+    tfm_2 = TransferFunctionMatrix([[-tf_2], [-tf_2], [-tf_3]])
+    tfm_3 = TransferFunctionMatrix([[-tf_3]])
+    tfm_4 = TransferFunctionMatrix([[TF3], [TF2], [-TF1]])
+    tfm_5 = TransferFunctionMatrix.from_Matrix(Matrix([1/p]), p)
+
+    s8 = MIMOSeries(tfm_2, tfm_1)
+    assert s8.args == (tfm_2, tfm_1)
+    assert s8.var == s
+    assert s8.shape == (s8.num_outputs, s8.num_inputs) == (2, 1)
+
+    s9 = MIMOSeries(tfm_3, tfm_2, tfm_1)
+    assert s9.args == (tfm_3, tfm_2, tfm_1)
+    assert s9.var == s
+    assert s9.shape == (s9.num_outputs, s9.num_inputs) == (2, 1)
+
+    s11 = MIMOSeries(tfm_3, MIMOParallel(-tfm_2, -tfm_4), tfm_1)
+    assert s11.args == (tfm_3, MIMOParallel(-tfm_2, -tfm_4), tfm_1)
+    assert s11.shape == (s11.num_outputs, s11.num_inputs) == (2, 1)
+
+    # arg cannot be empty tuple.
+    raises(ValueError, lambda: MIMOSeries())
+
+    # arg cannot contain SISO as well as MIMO systems.
+    raises(TypeError, lambda: MIMOSeries(tfm_1, tf_1))
+
+    # for all the adjacent transfer function matrices:
+    # no. of inputs of first TFM must be equal to the no. of outputs of the second TFM.
+    raises(ValueError, lambda: MIMOSeries(tfm_1, tfm_2, -tfm_1))
+
+    # all the TFMs must use the same complex variable.
+    raises(ValueError, lambda: MIMOSeries(tfm_3, tfm_5))
+
+    # Number or expression not allowed in the arguments.
+    raises(TypeError, lambda: MIMOSeries(2, tfm_2, tfm_3))
+    raises(TypeError, lambda: MIMOSeries(s**2 + p*s, -tfm_2, tfm_3))
+    raises(TypeError, lambda: MIMOSeries(Matrix([1/p]), tfm_3))
+
+
+def test_Series_functions():
+    tf1 = TransferFunction(1, s**2 + 2*zeta*wn*s + wn**2, s)
+    tf2 = TransferFunction(k, 1, s)
+    tf3 = TransferFunction(a2*p - s, a2*s + p, s)
+    tf4 = TransferFunction(a0*p + p**a1 - s, p, p)
+    tf5 = TransferFunction(a1*s**2 + a2*s - a0, s + a0, s)
+
+    assert tf1*tf2*tf3 == Series(tf1, tf2, tf3) == Series(Series(tf1, tf2), tf3) \
+        == Series(tf1, Series(tf2, tf3))
+    assert tf1*(tf2 + tf3) == Series(tf1, Parallel(tf2, tf3))
+    assert tf1*tf2 + tf5 == Parallel(Series(tf1, tf2), tf5)
+    assert tf1*tf2 - tf5 == Parallel(Series(tf1, tf2), -tf5)
+    assert tf1*tf2 + tf3 + tf5 == Parallel(Series(tf1, tf2), tf3, tf5)
+    assert tf1*tf2 - tf3 - tf5 == Parallel(Series(tf1, tf2), -tf3, -tf5)
+    assert tf1*tf2 - tf3 + tf5 == Parallel(Series(tf1, tf2), -tf3, tf5)
+    assert tf1*tf2 + tf3*tf5 == Parallel(Series(tf1, tf2), Series(tf3, tf5))
+    assert tf1*tf2 - tf3*tf5 == Parallel(Series(tf1, tf2), Series(TransferFunction(-1, 1, s), Series(tf3, tf5)))
+    assert tf2*tf3*(tf2 - tf1)*tf3 == Series(tf2, tf3, Parallel(tf2, -tf1), tf3)
+    assert -tf1*tf2 == Series(-tf1, tf2)
+    assert -(tf1*tf2) == Series(TransferFunction(-1, 1, s), Series(tf1, tf2))
+    raises(ValueError, lambda: tf1*tf2*tf4)
+    raises(ValueError, lambda: tf1*(tf2 - tf4))
+    raises(ValueError, lambda: tf3*Matrix([1, 2, 3]))
+
+    # evaluate=True -> doit()
+    assert Series(tf1, tf2, evaluate=True) == Series(tf1, tf2).doit() == \
+        TransferFunction(k, s**2 + 2*s*wn*zeta + wn**2, s)
+    assert Series(tf1, tf2, Parallel(tf1, -tf3), evaluate=True) == Series(tf1, tf2, Parallel(tf1, -tf3)).doit() == \
+        TransferFunction(k*(a2*s + p + (-a2*p + s)*(s**2 + 2*s*wn*zeta + wn**2)), (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2)**2, s)
+    assert Series(tf2, tf1, -tf3, evaluate=True) == Series(tf2, tf1, -tf3).doit() == \
+        TransferFunction(k*(-a2*p + s), (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), s)
+    assert not Series(tf1, -tf2, evaluate=False) == Series(tf1, -tf2).doit()
+
+    assert Series(Parallel(tf1, tf2), Parallel(tf2, -tf3)).doit() == \
+        TransferFunction((k*(s**2 + 2*s*wn*zeta + wn**2) + 1)*(-a2*p + k*(a2*s + p) + s), (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), s)
+    assert Series(-tf1, -tf2, -tf3).doit() == \
+        TransferFunction(k*(-a2*p + s), (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), s)
+    assert -Series(tf1, tf2, tf3).doit() == \
+        TransferFunction(-k*(a2*p - s), (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), s)
+    assert Series(tf2, tf3, Parallel(tf2, -tf1), tf3).doit() == \
+        TransferFunction(k*(a2*p - s)**2*(k*(s**2 + 2*s*wn*zeta + wn**2) - 1), (a2*s + p)**2*(s**2 + 2*s*wn*zeta + wn**2), s)
+
+    assert Series(tf1, tf2).rewrite(TransferFunction) == TransferFunction(k, s**2 + 2*s*wn*zeta + wn**2, s)
+    assert Series(tf2, tf1, -tf3).rewrite(TransferFunction) == \
+        TransferFunction(k*(-a2*p + s), (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), s)
+
+    S1 = Series(Parallel(tf1, tf2), Parallel(tf2, -tf3))
+    assert S1.is_proper
+    assert not S1.is_strictly_proper
+    assert S1.is_biproper
+
+    S2 = Series(tf1, tf2, tf3)
+    assert S2.is_proper
+    assert S2.is_strictly_proper
+    assert not S2.is_biproper
+
+    S3 = Series(tf1, -tf2, Parallel(tf1, -tf3))
+    assert S3.is_proper
+    assert S3.is_strictly_proper
+    assert not S3.is_biproper
+
+
+def test_MIMOSeries_functions():
+    tfm1 = TransferFunctionMatrix([[TF1, TF2, TF3], [-TF3, -TF2, TF1]])
+    tfm2 = TransferFunctionMatrix([[-TF1], [-TF2], [-TF3]])
+    tfm3 = TransferFunctionMatrix([[-TF1]])
+    tfm4 = TransferFunctionMatrix([[-TF2, -TF3], [-TF1, TF2]])
+    tfm5 = TransferFunctionMatrix([[TF2, -TF2], [-TF3, -TF2]])
+    tfm6 = TransferFunctionMatrix([[-TF3], [TF1]])
+    tfm7 = TransferFunctionMatrix([[TF1], [-TF2]])
+
+    assert tfm1*tfm2 + tfm6 == MIMOParallel(MIMOSeries(tfm2, tfm1), tfm6)
+    assert tfm1*tfm2 + tfm7 + tfm6 == MIMOParallel(MIMOSeries(tfm2, tfm1), tfm7, tfm6)
+    assert tfm1*tfm2 - tfm6 - tfm7 == MIMOParallel(MIMOSeries(tfm2, tfm1), -tfm6, -tfm7)
+    assert tfm4*tfm5 + (tfm4 - tfm5) == MIMOParallel(MIMOSeries(tfm5, tfm4), tfm4, -tfm5)
+    assert tfm4*-tfm6 + (-tfm4*tfm6) == MIMOParallel(MIMOSeries(-tfm6, tfm4), MIMOSeries(tfm6, -tfm4))
+
+    raises(ValueError, lambda: tfm1*tfm2 + TF1)
+    raises(TypeError, lambda: tfm1*tfm2 + a0)
+    raises(TypeError, lambda: tfm4*tfm6 - (s - 1))
+    raises(TypeError, lambda: tfm4*-tfm6 - 8)
+    raises(TypeError, lambda: (-1 + p**5) + tfm1*tfm2)
+
+    # Shape criteria.
+
+    raises(TypeError, lambda: -tfm1*tfm2 + tfm4)
+    raises(TypeError, lambda: tfm1*tfm2 - tfm4 + tfm5)
+    raises(TypeError, lambda: tfm1*tfm2 - tfm4*tfm5)
+
+    assert tfm1*tfm2*-tfm3 == MIMOSeries(-tfm3, tfm2, tfm1)
+    assert (tfm1*-tfm2)*tfm3 == MIMOSeries(tfm3, -tfm2, tfm1)
+
+    # Multiplication of a Series object with a SISO TF not allowed.
+
+    raises(ValueError, lambda: tfm4*tfm5*TF1)
+    raises(TypeError, lambda: tfm4*tfm5*a1)
+    raises(TypeError, lambda: tfm4*-tfm5*(s - 2))
+    raises(TypeError, lambda: tfm5*tfm4*9)
+    raises(TypeError, lambda: (-p**3 + 1)*tfm5*tfm4)
+
+    # Transfer function matrix in the arguments.
+    assert (MIMOSeries(tfm2, tfm1, evaluate=True) == MIMOSeries(tfm2, tfm1).doit()
+        == TransferFunctionMatrix(((TransferFunction(-k**2*(a2*s + p)**2*(s**2 + 2*s*wn*zeta + wn**2)**2 + (-a2*p + s)*(a2*p - s)*(s**2 + 2*s*wn*zeta + wn**2)**2 - (a2*s + p)**2,
+        (a2*s + p)**2*(s**2 + 2*s*wn*zeta + wn**2)**2, s),),
+        (TransferFunction(k**2*(a2*s + p)**2*(s**2 + 2*s*wn*zeta + wn**2)**2 + (-a2*p + s)*(a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2) + (a2*p - s)*(a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2),
+        (a2*s + p)**2*(s**2 + 2*s*wn*zeta + wn**2)**2, s),))))
+
+    # doit() should not cancel poles and zeros.
+    mat_1 = Matrix([[1/(1+s), (1+s)/(1+s**2+2*s)**3]])
+    mat_2 = Matrix([[(1+s)], [(1+s**2+2*s)**3/(1+s)]])
+    tm_1, tm_2 = TransferFunctionMatrix.from_Matrix(mat_1, s), TransferFunctionMatrix.from_Matrix(mat_2, s)
+    assert (MIMOSeries(tm_2, tm_1).doit()
+        == TransferFunctionMatrix(((TransferFunction(2*(s + 1)**2*(s**2 + 2*s + 1)**3, (s + 1)**2*(s**2 + 2*s + 1)**3, s),),)))
+    assert MIMOSeries(tm_2, tm_1).doit().simplify() == TransferFunctionMatrix(((TransferFunction(2, 1, s),),))
+
+    # calling doit() will expand the internal Series and Parallel objects.
+    assert (MIMOSeries(-tfm3, -tfm2, tfm1, evaluate=True)
+        == MIMOSeries(-tfm3, -tfm2, tfm1).doit()
+        == TransferFunctionMatrix(((TransferFunction(k**2*(a2*s + p)**2*(s**2 + 2*s*wn*zeta + wn**2)**2 + (a2*p - s)**2*(s**2 + 2*s*wn*zeta + wn**2)**2 + (a2*s + p)**2,
+        (a2*s + p)**2*(s**2 + 2*s*wn*zeta + wn**2)**3, s),),
+        (TransferFunction(-k**2*(a2*s + p)**2*(s**2 + 2*s*wn*zeta + wn**2)**2 + (-a2*p + s)*(a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2) + (a2*p - s)*(a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2),
+        (a2*s + p)**2*(s**2 + 2*s*wn*zeta + wn**2)**3, s),))))
+    assert (MIMOSeries(MIMOParallel(tfm4, tfm5), tfm5, evaluate=True)
+        == MIMOSeries(MIMOParallel(tfm4, tfm5), tfm5).doit()
+        == TransferFunctionMatrix(((TransferFunction(-k*(-a2*s - p + (-a2*p + s)*(s**2 + 2*s*wn*zeta + wn**2)), (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), s), TransferFunction(k*(-a2*p - \
+            k*(a2*s + p) + s), a2*s + p, s)), (TransferFunction(-k*(-a2*s - p + (-a2*p + s)*(s**2 + 2*s*wn*zeta + wn**2)), (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), s), \
+            TransferFunction((-a2*p + s)*(-a2*p - k*(a2*s + p) + s), (a2*s + p)**2, s)))) == MIMOSeries(MIMOParallel(tfm4, tfm5), tfm5).rewrite(TransferFunctionMatrix))
+
+
+def test_Parallel_construction():
+    tf = TransferFunction(a0*s**3 + a1*s**2 - a2*s, b0*p**4 + b1*p**3 - b2*s*p, s)
+    tf2 = TransferFunction(a2*p - s, a2*s + p, s)
+    tf3 = TransferFunction(a0*p + p**a1 - s, p, p)
+    tf4 = TransferFunction(1, s**2 + 2*zeta*wn*s + wn**2, s)
+    inp = Function('X_d')(s)
+    out = Function('X')(s)
+
+    p0 = Parallel(tf, tf2)
+    assert p0.args == (tf, tf2)
+    assert p0.var == s
+
+    p1 = Parallel(Series(tf, -tf2), tf2)
+    assert p1.args == (Series(tf, -tf2), tf2)
+    assert p1.var == s
+
+    tf3_ = TransferFunction(inp, 1, s)
+    tf4_ = TransferFunction(-out, 1, s)
+    p2 = Parallel(tf, Series(tf3_, -tf4_), tf2)
+    assert p2.args == (tf, Series(tf3_, -tf4_), tf2)
+
+    p3 = Parallel(tf, tf2, tf4)
+    assert p3.args == (tf, tf2, tf4)
+
+    p4 = Parallel(tf3_, tf4_)
+    assert p4.args == (tf3_, tf4_)
+    assert p4.var == s
+
+    p5 = Parallel(tf, tf2)
+    assert p0 == p5
+    assert not p0 == p1
+
+    p6 = Parallel(tf2, tf4, Series(tf2, -tf4))
+    assert p6.args == (tf2, tf4, Series(tf2, -tf4))
+
+    p7 = Parallel(tf2, tf4, Series(tf2, -tf), tf4)
+    assert p7.args == (tf2, tf4, Series(tf2, -tf), tf4)
+
+    raises(ValueError, lambda: Parallel(tf, tf3))
+    raises(ValueError, lambda: Parallel(tf, tf2, tf3, tf4))
+    raises(ValueError, lambda: Parallel(-tf3, tf4))
+    raises(TypeError, lambda: Parallel(2, tf, tf4))
+    raises(TypeError, lambda: Parallel(s**2 + p*s, tf3, tf2))
+    raises(TypeError, lambda: Parallel(tf3, Matrix([1, 2, 3, 4])))
+
+
+def test_MIMOParallel_construction():
+    tfm1 = TransferFunctionMatrix([[TF1], [TF2], [TF3]])
+    tfm2 = TransferFunctionMatrix([[-TF3], [TF2], [TF1]])
+    tfm3 = TransferFunctionMatrix([[TF1]])
+    tfm4 = TransferFunctionMatrix([[TF2], [TF1], [TF3]])
+    tfm5 = TransferFunctionMatrix([[TF1, TF2], [TF2, TF1]])
+    tfm6 = TransferFunctionMatrix([[TF2, TF1], [TF1, TF2]])
+    tfm7 = TransferFunctionMatrix.from_Matrix(Matrix([[1/p]]), p)
+
+    p8 = MIMOParallel(tfm1, tfm2)
+    assert p8.args == (tfm1, tfm2)
+    assert p8.var == s
+    assert p8.shape == (p8.num_outputs, p8.num_inputs) == (3, 1)
+
+    p9 = MIMOParallel(MIMOSeries(tfm3, tfm1), tfm2)
+    assert p9.args == (MIMOSeries(tfm3, tfm1), tfm2)
+    assert p9.var == s
+    assert p9.shape == (p9.num_outputs, p9.num_inputs) == (3, 1)
+
+    p10 = MIMOParallel(tfm1, MIMOSeries(tfm3, tfm4), tfm2)
+    assert p10.args == (tfm1, MIMOSeries(tfm3, tfm4), tfm2)
+    assert p10.var == s
+    assert p10.shape == (p10.num_outputs, p10.num_inputs) == (3, 1)
+
+    p11 = MIMOParallel(tfm2, tfm1, tfm4)
+    assert p11.args == (tfm2, tfm1, tfm4)
+    assert p11.shape == (p11.num_outputs, p11.num_inputs) == (3, 1)
+
+    p12 = MIMOParallel(tfm6, tfm5)
+    assert p12.args == (tfm6, tfm5)
+    assert p12.shape == (p12.num_outputs, p12.num_inputs) == (2, 2)
+
+    p13 = MIMOParallel(tfm2, tfm4, MIMOSeries(-tfm3, tfm4), -tfm4)
+    assert p13.args == (tfm2, tfm4, MIMOSeries(-tfm3, tfm4), -tfm4)
+    assert p13.shape == (p13.num_outputs, p13.num_inputs) == (3, 1)
+
+    # arg cannot be empty tuple.
+    raises(TypeError, lambda: MIMOParallel(()))
+
+    # arg cannot contain SISO as well as MIMO systems.
+    raises(TypeError, lambda: MIMOParallel(tfm1, tfm2, TF1))
+
+    # all TFMs must have same shapes.
+    raises(TypeError, lambda: MIMOParallel(tfm1, tfm3, tfm4))
+
+    # all TFMs must be using the same complex variable.
+    raises(ValueError, lambda: MIMOParallel(tfm3, tfm7))
+
+    # Number or expression not allowed in the arguments.
+    raises(TypeError, lambda: MIMOParallel(2, tfm1, tfm4))
+    raises(TypeError, lambda: MIMOParallel(s**2 + p*s, -tfm4, tfm2))
+
+
+def test_Parallel_functions():
+    tf1 = TransferFunction(1, s**2 + 2*zeta*wn*s + wn**2, s)
+    tf2 = TransferFunction(k, 1, s)
+    tf3 = TransferFunction(a2*p - s, a2*s + p, s)
+    tf4 = TransferFunction(a0*p + p**a1 - s, p, p)
+    tf5 = TransferFunction(a1*s**2 + a2*s - a0, s + a0, s)
+
+    assert tf1 + tf2 + tf3 == Parallel(tf1, tf2, tf3)
+    assert tf1 + tf2 + tf3 + tf5 == Parallel(tf1, tf2, tf3, tf5)
+    assert tf1 + tf2 - tf3 - tf5 == Parallel(tf1, tf2, -tf3, -tf5)
+    assert tf1 + tf2*tf3 == Parallel(tf1, Series(tf2, tf3))
+    assert tf1 - tf2*tf3 == Parallel(tf1, -Series(tf2,tf3))
+    assert -tf1 - tf2 == Parallel(-tf1, -tf2)
+    assert -(tf1 + tf2) == Series(TransferFunction(-1, 1, s), Parallel(tf1, tf2))
+    assert (tf2 + tf3)*tf1 == Series(Parallel(tf2, tf3), tf1)
+    assert (tf1 + tf2)*(tf3*tf5) == Series(Parallel(tf1, tf2), tf3, tf5)
+    assert -(tf2 + tf3)*-tf5 == Series(TransferFunction(-1, 1, s), Parallel(tf2, tf3), -tf5)
+    assert tf2 + tf3 + tf2*tf1 + tf5 == Parallel(tf2, tf3, Series(tf2, tf1), tf5)
+    assert tf2 + tf3 + tf2*tf1 - tf3 == Parallel(tf2, tf3, Series(tf2, tf1), -tf3)
+    assert (tf1 + tf2 + tf5)*(tf3 + tf5) == Series(Parallel(tf1, tf2, tf5), Parallel(tf3, tf5))
+    raises(ValueError, lambda: tf1 + tf2 + tf4)
+    raises(ValueError, lambda: tf1 - tf2*tf4)
+    raises(ValueError, lambda: tf3 + Matrix([1, 2, 3]))
+
+    # evaluate=True -> doit()
+    assert Parallel(tf1, tf2, evaluate=True) == Parallel(tf1, tf2).doit() == \
+        TransferFunction(k*(s**2 + 2*s*wn*zeta + wn**2) + 1, s**2 + 2*s*wn*zeta + wn**2, s)
+    assert Parallel(tf1, tf2, Series(-tf1, tf3), evaluate=True) == \
+        Parallel(tf1, tf2, Series(-tf1, tf3)).doit() == TransferFunction(k*(a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2)**2 + \
+            (-a2*p + s)*(s**2 + 2*s*wn*zeta + wn**2) + (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), (a2*s + p)*(s**2 + \
+                2*s*wn*zeta + wn**2)**2, s)
+    assert Parallel(tf2, tf1, -tf3, evaluate=True) == Parallel(tf2, tf1, -tf3).doit() == \
+        TransferFunction(a2*s + k*(a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2) + p + (-a2*p + s)*(s**2 + 2*s*wn*zeta + wn**2) \
+            , (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), s)
+    assert not Parallel(tf1, -tf2, evaluate=False) == Parallel(tf1, -tf2).doit()
+
+    assert Parallel(Series(tf1, tf2), Series(tf2, tf3)).doit() == \
+        TransferFunction(k*(a2*p - s)*(s**2 + 2*s*wn*zeta + wn**2) + k*(a2*s + p), (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), s)
+    assert Parallel(-tf1, -tf2, -tf3).doit() == \
+        TransferFunction(-a2*s - k*(a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2) - p + (-a2*p + s)*(s**2 + 2*s*wn*zeta + wn**2), \
+            (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), s)
+    assert -Parallel(tf1, tf2, tf3).doit() == \
+        TransferFunction(-a2*s - k*(a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2) - p - (a2*p - s)*(s**2 + 2*s*wn*zeta + wn**2), \
+            (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), s)
+    assert Parallel(tf2, tf3, Series(tf2, -tf1), tf3).doit() == \
+        TransferFunction(k*(a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2) - k*(a2*s + p) + (2*a2*p - 2*s)*(s**2 + 2*s*wn*zeta \
+            + wn**2), (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), s)
+
+    assert Parallel(tf1, tf2).rewrite(TransferFunction) == \
+        TransferFunction(k*(s**2 + 2*s*wn*zeta + wn**2) + 1, s**2 + 2*s*wn*zeta + wn**2, s)
+    assert Parallel(tf2, tf1, -tf3).rewrite(TransferFunction) == \
+        TransferFunction(a2*s + k*(a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2) + p + (-a2*p + s)*(s**2 + 2*s*wn*zeta + \
+             wn**2), (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), s)
+
+    assert Parallel(tf1, Parallel(tf2, tf3)) == Parallel(tf1, tf2, tf3) == Parallel(Parallel(tf1, tf2), tf3)
+
+    P1 = Parallel(Series(tf1, tf2), Series(tf2, tf3))
+    assert P1.is_proper
+    assert not P1.is_strictly_proper
+    assert P1.is_biproper
+
+    P2 = Parallel(tf1, -tf2, -tf3)
+    assert P2.is_proper
+    assert not P2.is_strictly_proper
+    assert P2.is_biproper
+
+    P3 = Parallel(tf1, -tf2, Series(tf1, tf3))
+    assert P3.is_proper
+    assert not P3.is_strictly_proper
+    assert P3.is_biproper
+
+
+def test_MIMOParallel_functions():
+    tf4 = TransferFunction(a0*p + p**a1 - s, p, p)
+    tf5 = TransferFunction(a1*s**2 + a2*s - a0, s + a0, s)
+
+    tfm1 = TransferFunctionMatrix([[TF1], [TF2], [TF3]])
+    tfm2 = TransferFunctionMatrix([[-TF2], [tf5], [-TF1]])
+    tfm3 = TransferFunctionMatrix([[tf5], [-tf5], [TF2]])
+    tfm4 = TransferFunctionMatrix([[TF2, -tf5], [TF1, tf5]])
+    tfm5 = TransferFunctionMatrix([[TF1, TF2], [TF3, -tf5]])
+    tfm6 = TransferFunctionMatrix([[-TF2]])
+    tfm7 = TransferFunctionMatrix([[tf4], [-tf4], [tf4]])
+
+    assert tfm1 + tfm2 + tfm3 == MIMOParallel(tfm1, tfm2, tfm3) == MIMOParallel(MIMOParallel(tfm1, tfm2), tfm3)
+    assert tfm2 - tfm1 - tfm3 == MIMOParallel(tfm2, -tfm1, -tfm3)
+    assert tfm2 - tfm3 + (-tfm1*tfm6*-tfm6) == MIMOParallel(tfm2, -tfm3, MIMOSeries(-tfm6, tfm6, -tfm1))
+    assert tfm1 + tfm1 - (-tfm1*tfm6) == MIMOParallel(tfm1, tfm1, -MIMOSeries(tfm6, -tfm1))
+    assert tfm2 - tfm3 - tfm1 + tfm2 == MIMOParallel(tfm2, -tfm3, -tfm1, tfm2)
+    assert tfm1 + tfm2 - tfm3 - tfm1 == MIMOParallel(tfm1, tfm2, -tfm3, -tfm1)
+    raises(ValueError, lambda: tfm1 + tfm2 + TF2)
+    raises(TypeError, lambda: tfm1 - tfm2 - a1)
+    raises(TypeError, lambda: tfm2 - tfm3 - (s - 1))
+    raises(TypeError, lambda: -tfm3 - tfm2 - 9)
+    raises(TypeError, lambda: (1 - p**3) - tfm3 - tfm2)
+    # All TFMs must use the same complex var. tfm7 uses 'p'.
+    raises(ValueError, lambda: tfm3 - tfm2 - tfm7)
+    raises(ValueError, lambda: tfm2 - tfm1 + tfm7)
+    # (tfm1 +/- tfm2) has (3, 1) shape while tfm4 has (2, 2) shape.
+    raises(TypeError, lambda: tfm1 + tfm2 + tfm4)
+    raises(TypeError, lambda: (tfm1 - tfm2) - tfm4)
+
+    assert (tfm1 + tfm2)*tfm6 == MIMOSeries(tfm6, MIMOParallel(tfm1, tfm2))
+    assert (tfm2 - tfm3)*tfm6*-tfm6 == MIMOSeries(-tfm6, tfm6, MIMOParallel(tfm2, -tfm3))
+    assert (tfm2 - tfm1 - tfm3)*(tfm6 + tfm6) == MIMOSeries(MIMOParallel(tfm6, tfm6), MIMOParallel(tfm2, -tfm1, -tfm3))
+    raises(ValueError, lambda: (tfm4 + tfm5)*TF1)
+    raises(TypeError, lambda: (tfm2 - tfm3)*a2)
+    raises(TypeError, lambda: (tfm3 + tfm2)*(s - 6))
+    raises(TypeError, lambda: (tfm1 + tfm2 + tfm3)*0)
+    raises(TypeError, lambda: (1 - p**3)*(tfm1 + tfm3))
+
+    # (tfm3 - tfm2) has (3, 1) shape while tfm4*tfm5 has (2, 2) shape.
+    raises(ValueError, lambda: (tfm3 - tfm2)*tfm4*tfm5)
+    # (tfm1 - tfm2) has (3, 1) shape while tfm5 has (2, 2) shape.
+    raises(ValueError, lambda: (tfm1 - tfm2)*tfm5)
+
+    # TFM in the arguments.
+    assert (MIMOParallel(tfm1, tfm2, evaluate=True) == MIMOParallel(tfm1, tfm2).doit()
+    == MIMOParallel(tfm1, tfm2).rewrite(TransferFunctionMatrix)
+    == TransferFunctionMatrix(((TransferFunction(-k*(s**2 + 2*s*wn*zeta + wn**2) + 1, s**2 + 2*s*wn*zeta + wn**2, s),), \
+        (TransferFunction(-a0 + a1*s**2 + a2*s + k*(a0 + s), a0 + s, s),), (TransferFunction(-a2*s - p + (a2*p - s)* \
+        (s**2 + 2*s*wn*zeta + wn**2), (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), s),))))
+
+
+def test_Feedback_construction():
+    tf1 = TransferFunction(1, s**2 + 2*zeta*wn*s + wn**2, s)
+    tf2 = TransferFunction(k, 1, s)
+    tf3 = TransferFunction(a2*p - s, a2*s + p, s)
+    tf4 = TransferFunction(a0*p + p**a1 - s, p, p)
+    tf5 = TransferFunction(a1*s**2 + a2*s - a0, s + a0, s)
+    tf6 = TransferFunction(s - p, p + s, p)
+
+    f1 = Feedback(TransferFunction(1, 1, s), tf1*tf2*tf3)
+    assert f1.args == (TransferFunction(1, 1, s), Series(tf1, tf2, tf3), -1)
+    assert f1.sys1 == TransferFunction(1, 1, s)
+    assert f1.sys2 == Series(tf1, tf2, tf3)
+    assert f1.var == s
+
+    f2 = Feedback(tf1, tf2*tf3)
+    assert f2.args == (tf1, Series(tf2, tf3), -1)
+    assert f2.sys1 == tf1
+    assert f2.sys2 == Series(tf2, tf3)
+    assert f2.var == s
+
+    f3 = Feedback(tf1*tf2, tf5)
+    assert f3.args == (Series(tf1, tf2), tf5, -1)
+    assert f3.sys1 == Series(tf1, tf2)
+
+    f4 = Feedback(tf4, tf6)
+    assert f4.args == (tf4, tf6, -1)
+    assert f4.sys1 == tf4
+    assert f4.var == p
+
+    f5 = Feedback(tf5, TransferFunction(1, 1, s))
+    assert f5.args == (tf5, TransferFunction(1, 1, s), -1)
+    assert f5.var == s
+    assert f5 == Feedback(tf5)  # When sys2 is not passed explicitly, it is assumed to be unit tf.
+
+    f6 = Feedback(TransferFunction(1, 1, p), tf4)
+    assert f6.args == (TransferFunction(1, 1, p), tf4, -1)
+    assert f6.var == p
+
+    f7 = -Feedback(tf4*tf6, TransferFunction(1, 1, p))
+    assert f7.args == (Series(TransferFunction(-1, 1, p), Series(tf4, tf6)), -TransferFunction(1, 1, p), -1)
+    assert f7.sys1 == Series(TransferFunction(-1, 1, p), Series(tf4, tf6))
+
+    # denominator can't be a Parallel instance
+    raises(TypeError, lambda: Feedback(tf1, tf2 + tf3))
+    raises(TypeError, lambda: Feedback(tf1, Matrix([1, 2, 3])))
+    raises(TypeError, lambda: Feedback(TransferFunction(1, 1, s), s - 1))
+    raises(TypeError, lambda: Feedback(1, 1))
+    # raises(ValueError, lambda: Feedback(TransferFunction(1, 1, s), TransferFunction(1, 1, s)))
+    raises(ValueError, lambda: Feedback(tf2, tf4*tf5))
+    raises(ValueError, lambda: Feedback(tf2, tf1, 1.5))  # `sign` can only be -1 or 1
+    raises(ValueError, lambda: Feedback(tf1, -tf1**-1))  # denominator can't be zero
+    raises(ValueError, lambda: Feedback(tf4, tf5))  # Both systems should use the same `var`
+
+
+def test_Feedback_functions():
+    tf = TransferFunction(1, 1, s)
+    tf1 = TransferFunction(1, s**2 + 2*zeta*wn*s + wn**2, s)
+    tf2 = TransferFunction(k, 1, s)
+    tf3 = TransferFunction(a2*p - s, a2*s + p, s)
+    tf4 = TransferFunction(a0*p + p**a1 - s, p, p)
+    tf5 = TransferFunction(a1*s**2 + a2*s - a0, s + a0, s)
+    tf6 = TransferFunction(s - p, p + s, p)
+
+    assert (tf1*tf2*tf3 / tf3*tf5) == Series(tf1, tf2, tf3, pow(tf3, -1), tf5)
+    assert (tf1*tf2*tf3) / (tf3*tf5) == Series((tf1*tf2*tf3).doit(), pow((tf3*tf5).doit(),-1))
+    assert tf / (tf + tf1) == Feedback(tf, tf1)
+    assert tf / (tf + tf1*tf2*tf3) == Feedback(tf, tf1*tf2*tf3)
+    assert tf1 / (tf + tf1*tf2*tf3) == Feedback(tf1, tf2*tf3)
+    assert (tf1*tf2) / (tf + tf1*tf2) == Feedback(tf1*tf2, tf)
+    assert (tf1*tf2) / (tf + tf1*tf2*tf5) == Feedback(tf1*tf2, tf5)
+    assert (tf1*tf2) / (tf + tf1*tf2*tf5*tf3) in (Feedback(tf1*tf2, tf5*tf3), Feedback(tf1*tf2, tf3*tf5))
+    assert tf4 / (TransferFunction(1, 1, p) + tf4*tf6) == Feedback(tf4, tf6)
+    assert tf5 / (tf + tf5) == Feedback(tf5, tf)
+
+    raises(TypeError, lambda: tf1*tf2*tf3 / (1 + tf1*tf2*tf3))
+    raises(ValueError, lambda: tf2*tf3 / (tf + tf2*tf3*tf4))
+
+    assert Feedback(tf, tf1*tf2*tf3).doit() == \
+        TransferFunction((a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), k*(a2*p - s) + \
+        (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), s)
+    assert Feedback(tf, tf1*tf2*tf3).sensitivity == \
+        1/(k*(a2*p - s)/((a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2)) + 1)
+    assert Feedback(tf1, tf2*tf3).doit() == \
+        TransferFunction((a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), (k*(a2*p - s) + \
+        (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2))*(s**2 + 2*s*wn*zeta + wn**2), s)
+    assert Feedback(tf1, tf2*tf3).sensitivity == \
+        1/(k*(a2*p - s)/((a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2)) + 1)
+    assert Feedback(tf1*tf2, tf5).doit() == \
+        TransferFunction(k*(a0 + s)*(s**2 + 2*s*wn*zeta + wn**2), (k*(-a0 + a1*s**2 + a2*s) + \
+        (a0 + s)*(s**2 + 2*s*wn*zeta + wn**2))*(s**2 + 2*s*wn*zeta + wn**2), s)
+    assert Feedback(tf1*tf2, tf5, 1).sensitivity == \
+        1/(-k*(-a0 + a1*s**2 + a2*s)/((a0 + s)*(s**2 + 2*s*wn*zeta + wn**2)) + 1)
+    assert Feedback(tf4, tf6).doit() == \
+        TransferFunction(p*(p + s)*(a0*p + p**a1 - s), p*(p*(p + s) + (-p + s)*(a0*p + p**a1 - s)), p)
+    assert -Feedback(tf4*tf6, TransferFunction(1, 1, p)).doit() == \
+        TransferFunction(-p*(-p + s)*(p + s)*(a0*p + p**a1 - s), p*(p + s)*(p*(p + s) + (-p + s)*(a0*p + p**a1 - s)), p)
+    assert Feedback(tf, tf).doit() == TransferFunction(1, 2, s)
+
+    assert Feedback(tf1, tf2*tf5).rewrite(TransferFunction) == \
+        TransferFunction((a0 + s)*(s**2 + 2*s*wn*zeta + wn**2), (k*(-a0 + a1*s**2 + a2*s) + \
+        (a0 + s)*(s**2 + 2*s*wn*zeta + wn**2))*(s**2 + 2*s*wn*zeta + wn**2), s)
+    assert Feedback(TransferFunction(1, 1, p), tf4).rewrite(TransferFunction) == \
+        TransferFunction(p, a0*p + p + p**a1 - s, p)
+
+
+def test_Feedback_as_TransferFunction():
+    # Solves issue https://github.com/sympy/sympy/issues/26161
+    tf1 = TransferFunction(s+1, 1, s)
+    tf2 = TransferFunction(s+2, 1, s)
+    fd1 = Feedback(tf1, tf2, -1) # Negative Feedback system
+    fd2 = Feedback(tf1, tf2, 1) # Positive Feedback system
+    unit = TransferFunction(1, 1, s)
+
+    # Checking the type
+    assert isinstance(fd1, TransferFunction)
+    assert isinstance(fd1, Feedback)
+
+    # Testing the numerator and denominator
+    assert fd1.num == tf1
+    assert fd2.num == tf1
+    assert fd1.den == Parallel(unit, Series(tf2, tf1))
+    assert fd2.den == Parallel(unit, -Series(tf2, tf1))
+
+    # Testing the Series and Parallel Combination with Feedback and TransferFunction
+    s1 = Series(tf1, fd1)
+    p1 = Parallel(tf1, fd1)
+    assert tf1 * fd1 == s1
+    assert tf1 + fd1 == p1
+    assert s1.doit() == TransferFunction((s + 1)**2, (s + 1)*(s + 2) + 1, s)
+    assert p1.doit() == TransferFunction(s + (s + 1)*((s + 1)*(s + 2) + 1) + 1, (s + 1)*(s + 2) + 1, s)
+
+    # Testing the use of Feedback and TransferFunction with Feedback
+    fd3 = Feedback(tf1*fd1, tf2, -1)
+    assert fd3 == Feedback(Series(tf1, fd1), tf2)
+    assert fd3.num == tf1 * fd1
+    assert fd3.den == Parallel(unit, Series(tf2, Series(tf1, fd1)))
+
+    # Testing the use of Feedback and TransferFunction with TransferFunction
+    tf3 = TransferFunction(tf1*fd1, tf2, s)
+    assert tf3 == TransferFunction(Series(tf1, fd1), tf2, s)
+    assert tf3.num == tf1*fd1
+
+def test_issue_26161():
+    # Issue https://github.com/sympy/sympy/issues/26161
+    Ib, Is, m, h, l2, l1 = symbols('I_b, I_s, m, h, l2, l1',
+                                            real=True, nonnegative=True)
+    KD, KP, v = symbols('K_D, K_P, v', real=True)
+
+    tau1_sq = (Ib + m * h ** 2) / m / g / h
+    tau2 = l2 / v
+    tau3 = v / (l1 + l2)
+    K = v ** 2 / g / (l1 + l2)
+
+    Gtheta = TransferFunction(-K * (tau2 * s + 1), tau1_sq * s ** 2 - 1, s)
+    Gdelta = TransferFunction(1, Is * s ** 2 + c * s, s)
+    Gpsi = TransferFunction(1, tau3 * s, s)
+    Dcont = TransferFunction(KD * s, 1, s)
+    PIcont = TransferFunction(KP, s, s)
+    Gunity = TransferFunction(1, 1, s)
+
+    Ginner = Feedback(Dcont * Gdelta, Gtheta)
+    Gouter = Feedback(PIcont * Ginner * Gpsi, Gunity)
+    assert Gouter == Feedback(Series(PIcont, Series(Ginner, Gpsi)), Gunity)
+    assert Gouter.num == Series(PIcont, Series(Ginner, Gpsi))
+    assert Gouter.den == Parallel(Gunity, Series(Gunity, Series(PIcont, Series(Ginner, Gpsi))))
+    expr = (KD*KP*g*s**3*v**2*(l1 + l2)*(Is*s**2 + c*s)**2*(-g*h*m + s**2*(Ib + h**2*m))*(-KD*g*h*m*s*v**2*(l2*s + v) + \
+            g*v*(l1 + l2)*(Is*s**2 + c*s)*(-g*h*m + s**2*(Ib + h**2*m))))/((s**2*v*(Is*s**2 + c*s)*(-KD*g*h*m*s*v**2* \
+            (l2*s + v) + g*v*(l1 + l2)*(Is*s**2 + c*s)*(-g*h*m + s**2*(Ib + h**2*m)))*(KD*KP*g*s*v*(l1 + l2)**2* \
+            (Is*s**2 + c*s)*(-g*h*m + s**2*(Ib + h**2*m)) + s**2*v*(Is*s**2 + c*s)*(-KD*g*h*m*s*v**2*(l2*s + v) + \
+            g*v*(l1 + l2)*(Is*s**2 + c*s)*(-g*h*m + s**2*(Ib + h**2*m))))/(l1 + l2)))
+
+    assert (Gouter.to_expr() - expr).simplify() == 0
+
+
+def test_MIMOFeedback_construction():
+    tf1 = TransferFunction(1, s, s)
+    tf2 = TransferFunction(s, s**3 - 1, s)
+    tf3 = TransferFunction(s, s + 1, s)
+    tf4 = TransferFunction(s, s**2 + 1, s)
+
+    tfm_1 = TransferFunctionMatrix([[tf1, tf2], [tf3, tf4]])
+    tfm_2 = TransferFunctionMatrix([[tf2, tf3], [tf4, tf1]])
+    tfm_3 = TransferFunctionMatrix([[tf3, tf4], [tf1, tf2]])
+
+    f1 = MIMOFeedback(tfm_1, tfm_2)
+    assert f1.args == (tfm_1, tfm_2, -1)
+    assert f1.sys1 == tfm_1
+    assert f1.sys2 == tfm_2
+    assert f1.var == s
+    assert f1.sign == -1
+    assert -(-f1) == f1
+
+    f2 = MIMOFeedback(tfm_2, tfm_1, 1)
+    assert f2.args == (tfm_2, tfm_1, 1)
+    assert f2.sys1 == tfm_2
+    assert f2.sys2 == tfm_1
+    assert f2.var == s
+    assert f2.sign == 1
+
+    f3 = MIMOFeedback(tfm_1, MIMOSeries(tfm_3, tfm_2))
+    assert f3.args == (tfm_1, MIMOSeries(tfm_3, tfm_2), -1)
+    assert f3.sys1 == tfm_1
+    assert f3.sys2 == MIMOSeries(tfm_3, tfm_2)
+    assert f3.var == s
+    assert f3.sign == -1
+
+    mat = Matrix([[1, 1/s], [0, 1]])
+    sys1 = controller = TransferFunctionMatrix.from_Matrix(mat, s)
+    f4 = MIMOFeedback(sys1, controller)
+    assert f4.args == (sys1, controller, -1)
+    assert f4.sys1 == f4.sys2 == sys1
+
+
+def test_MIMOFeedback_errors():
+    tf1 = TransferFunction(1, s, s)
+    tf2 = TransferFunction(s, s**3 - 1, s)
+    tf3 = TransferFunction(s, s - 1, s)
+    tf4 = TransferFunction(s, s**2 + 1, s)
+    tf5 = TransferFunction(1, 1, s)
+    tf6 = TransferFunction(-1, s - 1, s)
+
+    tfm_1 = TransferFunctionMatrix([[tf1, tf2], [tf3, tf4]])
+    tfm_2 = TransferFunctionMatrix([[tf2, tf3], [tf4, tf1]])
+    tfm_3 = TransferFunctionMatrix.from_Matrix(eye(2), var=s)
+    tfm_4 = TransferFunctionMatrix([[tf1, tf5], [tf5, tf5]])
+    tfm_5 = TransferFunctionMatrix([[-tf3, tf3], [tf3, tf6]])
+    # tfm_4 is inverse of tfm_5. Therefore tfm_5*tfm_4 = I
+    tfm_6 = TransferFunctionMatrix([[-tf3]])
+    tfm_7 = TransferFunctionMatrix([[tf3, tf4]])
+
+    # Unsupported Types
+    raises(TypeError, lambda: MIMOFeedback(tf1, tf2))
+    raises(TypeError, lambda: MIMOFeedback(MIMOParallel(tfm_1, tfm_2), tfm_3))
+    # Shape Errors
+    raises(ValueError, lambda: MIMOFeedback(tfm_1, tfm_6, 1))
+    raises(ValueError, lambda: MIMOFeedback(tfm_7, tfm_7))
+    # sign not 1/-1
+    raises(ValueError, lambda: MIMOFeedback(tfm_1, tfm_2, -2))
+    # Non-Invertible Systems
+    raises(ValueError, lambda: MIMOFeedback(tfm_5, tfm_4, 1))
+    raises(ValueError, lambda: MIMOFeedback(tfm_4, -tfm_5))
+    raises(ValueError, lambda: MIMOFeedback(tfm_3, tfm_3, 1))
+    # Variable not same in both the systems
+    tfm_8 = TransferFunctionMatrix.from_Matrix(eye(2), var=p)
+    raises(ValueError, lambda: MIMOFeedback(tfm_1, tfm_8, 1))
+
+
+def test_MIMOFeedback_functions():
+    tf1 = TransferFunction(1, s, s)
+    tf2 = TransferFunction(s, s - 1, s)
+    tf3 = TransferFunction(1, 1, s)
+    tf4 = TransferFunction(-1, s - 1, s)
+
+    tfm_1 = TransferFunctionMatrix.from_Matrix(eye(2), var=s)
+    tfm_2 = TransferFunctionMatrix([[tf1, tf3], [tf3, tf3]])
+    tfm_3 = TransferFunctionMatrix([[-tf2, tf2], [tf2, tf4]])
+    tfm_4 = TransferFunctionMatrix([[tf1, tf2], [-tf2, tf1]])
+
+    # sensitivity, doit(), rewrite()
+    F_1 = MIMOFeedback(tfm_2, tfm_3)
+    F_2 = MIMOFeedback(tfm_2, MIMOSeries(tfm_4, -tfm_1), 1)
+
+    assert F_1.sensitivity == Matrix([[S.Half, 0], [0, S.Half]])
+    assert F_2.sensitivity == Matrix([[(-2*s**4 + s**2)/(s**2 - s + 1),
+        (2*s**3 - s**2)/(s**2 - s + 1)], [-s**2, s]])
+
+    assert F_1.doit() == \
+        TransferFunctionMatrix(((TransferFunction(1, 2*s, s),
+        TransferFunction(1, 2, s)), (TransferFunction(1, 2, s),
+        TransferFunction(1, 2, s)))) == F_1.rewrite(TransferFunctionMatrix)
+    assert F_2.doit(cancel=False, expand=True) == \
+        TransferFunctionMatrix(((TransferFunction(-s**5 + 2*s**4 - 2*s**3 + s**2, s**5 - 2*s**4 + 3*s**3 - 2*s**2 + s, s),
+        TransferFunction(-2*s**4 + 2*s**3, s**2 - s + 1, s)), (TransferFunction(0, 1, s), TransferFunction(-s**2 + s, 1, s))))
+    assert F_2.doit(cancel=False) == \
+        TransferFunctionMatrix(((TransferFunction(s*(2*s**3 - s**2)*(s**2 - s + 1) + \
+        (-2*s**4 + s**2)*(s**2 - s + 1), s*(s**2 - s + 1)**2, s), TransferFunction(-2*s**4 + 2*s**3, s**2 - s + 1, s)),
+        (TransferFunction(0, 1, s), TransferFunction(-s**2 + s, 1, s))))
+    assert F_2.doit() == \
+        TransferFunctionMatrix(((TransferFunction(s*(-2*s**2 + s*(2*s - 1) + 1), s**2 - s + 1, s),
+        TransferFunction(-2*s**3*(s - 1), s**2 - s + 1, s)), (TransferFunction(0, 1, s), TransferFunction(s*(1 - s), 1, s))))
+    assert F_2.doit(expand=True) == \
+        TransferFunctionMatrix(((TransferFunction(-s**2 + s, s**2 - s + 1, s), TransferFunction(-2*s**4 + 2*s**3, s**2 - s + 1, s)),
+        (TransferFunction(0, 1, s), TransferFunction(-s**2 + s, 1, s))))
+
+    assert -(F_1.doit()) == (-F_1).doit()  # First negating then calculating vs calculating then negating.
+
+
+def test_TransferFunctionMatrix_construction():
+    tf5 = TransferFunction(a1*s**2 + a2*s - a0, s + a0, s)
+    tf4 = TransferFunction(a0*p + p**a1 - s, p, p)
+
+    tfm3_ = TransferFunctionMatrix([[-TF3]])
+    assert tfm3_.shape == (tfm3_.num_outputs, tfm3_.num_inputs) == (1, 1)
+    assert tfm3_.args == Tuple(Tuple(Tuple(-TF3)))
+    assert tfm3_.var == s
+
+    tfm5 = TransferFunctionMatrix([[TF1, -TF2], [TF3, tf5]])
+    assert tfm5.shape == (tfm5.num_outputs, tfm5.num_inputs) == (2, 2)
+    assert tfm5.args == Tuple(Tuple(Tuple(TF1, -TF2), Tuple(TF3, tf5)))
+    assert tfm5.var == s
+
+    tfm7 = TransferFunctionMatrix([[TF1, TF2], [TF3, -tf5], [-tf5, TF2]])
+    assert tfm7.shape == (tfm7.num_outputs, tfm7.num_inputs) == (3, 2)
+    assert tfm7.args == Tuple(Tuple(Tuple(TF1, TF2), Tuple(TF3, -tf5), Tuple(-tf5, TF2)))
+    assert tfm7.var == s
+
+    # all transfer functions will use the same complex variable. tf4 uses 'p'.
+    raises(ValueError, lambda: TransferFunctionMatrix([[TF1], [TF2], [tf4]]))
+    raises(ValueError, lambda: TransferFunctionMatrix([[TF1, tf4], [TF3, tf5]]))
+
+    # length of all the lists in the TFM should be equal.
+    raises(ValueError, lambda: TransferFunctionMatrix([[TF1], [TF3, tf5]]))
+    raises(ValueError, lambda: TransferFunctionMatrix([[TF1, TF3], [tf5]]))
+
+    # lists should only support transfer functions in them.
+    raises(TypeError, lambda: TransferFunctionMatrix([[TF1, TF2], [TF3, Matrix([1, 2])]]))
+    raises(TypeError, lambda: TransferFunctionMatrix([[TF1, Matrix([1, 2])], [TF3, TF2]]))
+
+    # `arg` should strictly be nested list of TransferFunction
+    raises(ValueError, lambda: TransferFunctionMatrix([TF1, TF2, tf5]))
+    raises(ValueError, lambda: TransferFunctionMatrix([TF1]))
+
+def test_TransferFunctionMatrix_functions():
+    tf5 = TransferFunction(a1*s**2 + a2*s - a0, s + a0, s)
+
+    #  Classmethod (from_matrix)
+
+    mat_1 = ImmutableMatrix([
+        [s*(s + 1)*(s - 3)/(s**4 + 1), 2],
+        [p, p*(s + 1)/(s*(s**1 + 1))]
+        ])
+    mat_2 = ImmutableMatrix([[(2*s + 1)/(s**2 - 9)]])
+    mat_3 = ImmutableMatrix([[1, 2], [3, 4]])
+    assert TransferFunctionMatrix.from_Matrix(mat_1, s) == \
+        TransferFunctionMatrix([[TransferFunction(s*(s - 3)*(s + 1), s**4 + 1, s), TransferFunction(2, 1, s)],
+        [TransferFunction(p, 1, s), TransferFunction(p, s, s)]])
+    assert TransferFunctionMatrix.from_Matrix(mat_2, s) == \
+        TransferFunctionMatrix([[TransferFunction(2*s + 1, s**2 - 9, s)]])
+    assert TransferFunctionMatrix.from_Matrix(mat_3, p) == \
+        TransferFunctionMatrix([[TransferFunction(1, 1, p), TransferFunction(2, 1, p)],
+        [TransferFunction(3, 1, p), TransferFunction(4, 1, p)]])
+
+    #  Negating a TFM
+
+    tfm1 = TransferFunctionMatrix([[TF1], [TF2]])
+    assert -tfm1 == TransferFunctionMatrix([[-TF1], [-TF2]])
+
+    tfm2 = TransferFunctionMatrix([[TF1, TF2, TF3], [tf5, -TF1, -TF3]])
+    assert -tfm2 == TransferFunctionMatrix([[-TF1, -TF2, -TF3], [-tf5, TF1, TF3]])
+
+    # subs()
+
+    H_1 = TransferFunctionMatrix.from_Matrix(mat_1, s)
+    H_2 = TransferFunctionMatrix([[TransferFunction(a*p*s, k*s**2, s), TransferFunction(p*s, k*(s**2 - a), s)]])
+    assert H_1.subs(p, 1) == TransferFunctionMatrix([[TransferFunction(s*(s - 3)*(s + 1), s**4 + 1, s), TransferFunction(2, 1, s)], [TransferFunction(1, 1, s), TransferFunction(1, s, s)]])
+    assert H_1.subs({p: 1}) == TransferFunctionMatrix([[TransferFunction(s*(s - 3)*(s + 1), s**4 + 1, s), TransferFunction(2, 1, s)], [TransferFunction(1, 1, s), TransferFunction(1, s, s)]])
+    assert H_1.subs({p: 1, s: 1}) == TransferFunctionMatrix([[TransferFunction(s*(s - 3)*(s + 1), s**4 + 1, s), TransferFunction(2, 1, s)], [TransferFunction(1, 1, s), TransferFunction(1, s, s)]]) # This should ignore `s` as it is `var`
+    assert H_2.subs(p, 2) == TransferFunctionMatrix([[TransferFunction(2*a*s, k*s**2, s), TransferFunction(2*s, k*(-a + s**2), s)]])
+    assert H_2.subs(k, 1) == TransferFunctionMatrix([[TransferFunction(a*p*s, s**2, s), TransferFunction(p*s, -a + s**2, s)]])
+    assert H_2.subs(a, 0) == TransferFunctionMatrix([[TransferFunction(0, k*s**2, s), TransferFunction(p*s, k*s**2, s)]])
+    assert H_2.subs({p: 1, k: 1, a: a0}) == TransferFunctionMatrix([[TransferFunction(a0*s, s**2, s), TransferFunction(s, -a0 + s**2, s)]])
+
+    # eval_frequency()
+    assert H_2.eval_frequency(S(1)/2 + I) == Matrix([[2*a*p/(5*k) - 4*I*a*p/(5*k), I*p/(-a*k - 3*k/4 + I*k) + p/(-2*a*k - 3*k/2 + 2*I*k)]])
+
+    # transpose()
+
+    assert H_1.transpose() == TransferFunctionMatrix([[TransferFunction(s*(s - 3)*(s + 1), s**4 + 1, s), TransferFunction(p, 1, s)], [TransferFunction(2, 1, s), TransferFunction(p, s, s)]])
+    assert H_2.transpose() == TransferFunctionMatrix([[TransferFunction(a*p*s, k*s**2, s)], [TransferFunction(p*s, k*(-a + s**2), s)]])
+    assert H_1.transpose().transpose() == H_1
+    assert H_2.transpose().transpose() == H_2
+
+    # elem_poles()
+
+    assert H_1.elem_poles() == [[[-sqrt(2)/2 - sqrt(2)*I/2, -sqrt(2)/2 + sqrt(2)*I/2, sqrt(2)/2 - sqrt(2)*I/2, sqrt(2)/2 + sqrt(2)*I/2], []],
+        [[], [0]]]
+    assert H_2.elem_poles() == [[[0, 0], [sqrt(a), -sqrt(a)]]]
+    assert tfm2.elem_poles() == [[[wn*(-zeta + sqrt((zeta - 1)*(zeta + 1))), wn*(-zeta - sqrt((zeta - 1)*(zeta + 1)))], [], [-p/a2]],
+        [[-a0], [wn*(-zeta + sqrt((zeta - 1)*(zeta + 1))), wn*(-zeta - sqrt((zeta - 1)*(zeta + 1)))], [-p/a2]]]
+
+    # elem_zeros()
+
+    assert H_1.elem_zeros() == [[[-1, 0, 3], []], [[], []]]
+    assert H_2.elem_zeros() == [[[0], [0]]]
+    assert tfm2.elem_zeros() == [[[], [], [a2*p]],
+        [[-a2/(2*a1) - sqrt(4*a0*a1 + a2**2)/(2*a1), -a2/(2*a1) + sqrt(4*a0*a1 + a2**2)/(2*a1)], [], [a2*p]]]
+
+    # doit()
+
+    H_3 = TransferFunctionMatrix([[Series(TransferFunction(1, s**3 - 3, s), TransferFunction(s**2 - 2*s + 5, 1, s), TransferFunction(1, s, s))]])
+    H_4 = TransferFunctionMatrix([[Parallel(TransferFunction(s**3 - 3, 4*s**4 - s**2 - 2*s + 5, s), TransferFunction(4 - s**3, 4*s**4 - s**2 - 2*s + 5, s))]])
+
+    assert H_3.doit() == TransferFunctionMatrix([[TransferFunction(s**2 - 2*s + 5, s*(s**3 - 3), s)]])
+    assert H_4.doit() == TransferFunctionMatrix([[TransferFunction(1, 4*s**4 - s**2 - 2*s + 5, s)]])
+
+    # _flat()
+
+    assert H_1._flat() == [TransferFunction(s*(s - 3)*(s + 1), s**4 + 1, s), TransferFunction(2, 1, s), TransferFunction(p, 1, s), TransferFunction(p, s, s)]
+    assert H_2._flat() == [TransferFunction(a*p*s, k*s**2, s), TransferFunction(p*s, k*(-a + s**2), s)]
+    assert H_3._flat() == [Series(TransferFunction(1, s**3 - 3, s), TransferFunction(s**2 - 2*s + 5, 1, s), TransferFunction(1, s, s))]
+    assert H_4._flat() == [Parallel(TransferFunction(s**3 - 3, 4*s**4 - s**2 - 2*s + 5, s), TransferFunction(4 - s**3, 4*s**4 - s**2 - 2*s + 5, s))]
+
+    # evalf()
+
+    assert H_1.evalf() == \
+        TransferFunctionMatrix(((TransferFunction(s*(s - 3.0)*(s + 1.0), s**4 + 1.0, s), TransferFunction(2.0, 1, s)), (TransferFunction(1.0*p, 1, s), TransferFunction(p, s, s))))
+    assert H_2.subs({a:3.141, p:2.88, k:2}).evalf() == \
+        TransferFunctionMatrix(((TransferFunction(4.5230399999999999494093572138808667659759521484375, s, s),
+        TransferFunction(2.87999999999999989341858963598497211933135986328125*s, 2.0*s**2 - 6.282000000000000028421709430404007434844970703125, s)),))
+
+    # simplify()
+
+    H_5 = TransferFunctionMatrix([[TransferFunction(s**5 + s**3 + s, s - s**2, s),
+        TransferFunction((s + 3)*(s - 1), (s - 1)*(s + 5), s)]])
+
+    assert H_5.simplify() == simplify(H_5) == \
+        TransferFunctionMatrix(((TransferFunction(-s**4 - s**2 - 1, s - 1, s), TransferFunction(s + 3, s + 5, s)),))
+
+    # expand()
+
+    assert (H_1.expand()
+            == TransferFunctionMatrix(((TransferFunction(s**3 - 2*s**2 - 3*s, s**4 + 1, s), TransferFunction(2, 1, s)),
+            (TransferFunction(p, 1, s), TransferFunction(p, s, s)))))
+    assert H_5.expand() == \
+        TransferFunctionMatrix(((TransferFunction(s**5 + s**3 + s, -s**2 + s, s), TransferFunction(s**2 + 2*s - 3, s**2 + 4*s - 5, s)),))
+
+def test_TransferFunction_gbt():
+    # simple transfer function, e.g. ohms law
+    tf = TransferFunction(1, a*s+b, s)
+    numZ, denZ = gbt(tf, T, 0.5)
+    # discretized transfer function with coefs from tf.gbt()
+    tf_test_bilinear = TransferFunction(s * numZ[0] + numZ[1], s * denZ[0] + denZ[1], s)
+    # corresponding tf with manually calculated coefs
+    tf_test_manual = TransferFunction(s * T/(2*(a + b*T/2)) + T/(2*(a + b*T/2)), s + (-a + b*T/2)/(a + b*T/2), s)
+
+    assert S.Zero == (tf_test_bilinear.simplify()-tf_test_manual.simplify()).simplify().num
+
+    tf = TransferFunction(1, a*s+b, s)
+    numZ, denZ = gbt(tf, T, 0)
+    # discretized transfer function with coefs from tf.gbt()
+    tf_test_forward = TransferFunction(numZ[0], s*denZ[0]+denZ[1], s)
+    # corresponding tf with manually calculated coefs
+    tf_test_manual = TransferFunction(T/a, s + (-a + b*T)/a, s)
+
+    assert S.Zero == (tf_test_forward.simplify()-tf_test_manual.simplify()).simplify().num
+
+    tf = TransferFunction(1, a*s+b, s)
+    numZ, denZ = gbt(tf, T, 1)
+    # discretized transfer function with coefs from tf.gbt()
+    tf_test_backward = TransferFunction(s*numZ[0], s*denZ[0]+denZ[1], s)
+    # corresponding tf with manually calculated coefs
+    tf_test_manual = TransferFunction(s * T/(a + b*T), s - a/(a + b*T), s)
+
+    assert S.Zero == (tf_test_backward.simplify()-tf_test_manual.simplify()).simplify().num
+
+    tf = TransferFunction(1, a*s+b, s)
+    numZ, denZ = gbt(tf, T, 0.3)
+    # discretized transfer function with coefs from tf.gbt()
+    tf_test_gbt = TransferFunction(s*numZ[0]+numZ[1], s*denZ[0]+denZ[1], s)
+    # corresponding tf with manually calculated coefs
+    tf_test_manual = TransferFunction(s*3*T/(10*(a + 3*b*T/10)) + 7*T/(10*(a + 3*b*T/10)), s + (-a + 7*b*T/10)/(a + 3*b*T/10), s)
+
+    assert S.Zero == (tf_test_gbt.simplify()-tf_test_manual.simplify()).simplify().num
+
+def test_TransferFunction_bilinear():
+    # simple transfer function, e.g. ohms law
+    tf = TransferFunction(1, a*s+b, s)
+    numZ, denZ = bilinear(tf, T)
+    # discretized transfer function with coefs from tf.bilinear()
+    tf_test_bilinear = TransferFunction(s*numZ[0]+numZ[1], s*denZ[0]+denZ[1], s)
+    # corresponding tf with manually calculated coefs
+    tf_test_manual = TransferFunction(s * T/(2*(a + b*T/2)) + T/(2*(a + b*T/2)), s + (-a + b*T/2)/(a + b*T/2), s)
+
+    assert S.Zero == (tf_test_bilinear.simplify()-tf_test_manual.simplify()).simplify().num
+
+def test_TransferFunction_forward_diff():
+    # simple transfer function, e.g. ohms law
+    tf = TransferFunction(1, a*s+b, s)
+    numZ, denZ = forward_diff(tf, T)
+    # discretized transfer function with coefs from tf.forward_diff()
+    tf_test_forward = TransferFunction(numZ[0], s*denZ[0]+denZ[1], s)
+    # corresponding tf with manually calculated coefs
+    tf_test_manual = TransferFunction(T/a, s + (-a + b*T)/a, s)
+
+    assert S.Zero == (tf_test_forward.simplify()-tf_test_manual.simplify()).simplify().num
+
+def test_TransferFunction_backward_diff():
+    # simple transfer function, e.g. ohms law
+    tf = TransferFunction(1, a*s+b, s)
+    numZ, denZ = backward_diff(tf, T)
+    # discretized transfer function with coefs from tf.backward_diff()
+    tf_test_backward = TransferFunction(s*numZ[0]+numZ[1], s*denZ[0]+denZ[1], s)
+    # corresponding tf with manually calculated coefs
+    tf_test_manual = TransferFunction(s * T/(a + b*T), s - a/(a + b*T), s)
+
+    assert S.Zero == (tf_test_backward.simplify()-tf_test_manual.simplify()).simplify().num
+
+def test_TransferFunction_phase_margin():
+    # Test for phase margin
+    tf1 = TransferFunction(10, p**3 + 1, p)
+    tf2 = TransferFunction(s**2, 10, s)
+    tf3 = TransferFunction(1, a*s+b, s)
+    tf4 = TransferFunction((s + 1)*exp(s/tau), s**2 + 2, s)
+    tf_m = TransferFunctionMatrix([[tf2],[tf3]])
+
+    assert phase_margin(tf1) == -180 + 180*atan(3*sqrt(11))/pi
+    assert phase_margin(tf2) == 0
+
+    raises(NotImplementedError, lambda: phase_margin(tf4))
+    raises(ValueError, lambda: phase_margin(tf3))
+    raises(ValueError, lambda: phase_margin(MIMOSeries(tf_m)))
+
+def test_TransferFunction_gain_margin():
+    # Test for gain margin
+    tf1 = TransferFunction(s**2, 5*(s+1)*(s-5)*(s-10), s)
+    tf2 = TransferFunction(s**2 + 2*s + 1, 1, s)
+    tf3 = TransferFunction(1, a*s+b, s)
+    tf4 = TransferFunction((s + 1)*exp(s/tau), s**2 + 2, s)
+    tf_m = TransferFunctionMatrix([[tf2],[tf3]])
+
+    assert gain_margin(tf1) == -20*log(S(7)/540)/log(10)
+    assert gain_margin(tf2) == oo
+
+    raises(NotImplementedError, lambda: gain_margin(tf4))
+    raises(ValueError, lambda: gain_margin(tf3))
+    raises(ValueError, lambda: gain_margin(MIMOSeries(tf_m)))
+
+
+def test_StateSpace_construction():
+    # using different numbers for a SISO system.
+    A1 = Matrix([[0, 1], [1, 0]])
+    B1 = Matrix([1, 0])
+    C1 = Matrix([[0, 1]])
+    D1 = Matrix([0])
+    ss1 = StateSpace(A1, B1, C1, D1)
+
+    assert ss1.state_matrix == Matrix([[0, 1], [1, 0]])
+    assert ss1.input_matrix == Matrix([1, 0])
+    assert ss1.output_matrix == Matrix([[0, 1]])
+    assert ss1.feedforward_matrix == Matrix([0])
+    assert ss1.args == (Matrix([[0, 1], [1, 0]]), Matrix([[1], [0]]), Matrix([[0, 1]]), Matrix([[0]]))
+
+    # using different symbols for a SISO system.
+    ss2 = StateSpace(Matrix([a0]), Matrix([a1]),
+                    Matrix([a2]), Matrix([a3]))
+
+    assert ss2.state_matrix == Matrix([[a0]])
+    assert ss2.input_matrix == Matrix([[a1]])
+    assert ss2.output_matrix == Matrix([[a2]])
+    assert ss2.feedforward_matrix == Matrix([[a3]])
+    assert ss2.args == (Matrix([[a0]]), Matrix([[a1]]), Matrix([[a2]]), Matrix([[a3]]))
+
+    # using different numbers for a MIMO system.
+    ss3 = StateSpace(Matrix([[-1.5, -2], [1, 0]]),
+                    Matrix([[0.5, 0], [0, 1]]),
+                    Matrix([[0, 1], [0, 2]]),
+                    Matrix([[2, 2], [1, 1]]))
+
+    assert ss3.state_matrix == Matrix([[-1.5, -2], [1,  0]])
+    assert ss3.input_matrix == Matrix([[0.5, 0], [0, 1]])
+    assert ss3.output_matrix == Matrix([[0, 1], [0, 2]])
+    assert ss3.feedforward_matrix == Matrix([[2, 2], [1, 1]])
+    assert ss3.args == (Matrix([[-1.5, -2],
+                                [1,  0]]),
+                        Matrix([[0.5, 0],
+                                [0, 1]]),
+                        Matrix([[0, 1],
+                                [0, 2]]),
+                        Matrix([[2, 2],
+                                [1, 1]]))
+
+    # using different symbols for a MIMO system.
+    A4 = Matrix([[a0, a1], [a2, a3]])
+    B4 = Matrix([[b0, b1], [b2, b3]])
+    C4 = Matrix([[c0, c1], [c2, c3]])
+    D4 = Matrix([[d0, d1], [d2, d3]])
+    ss4 = StateSpace(A4, B4, C4, D4)
+
+    assert ss4.state_matrix == Matrix([[a0, a1], [a2, a3]])
+    assert ss4.input_matrix == Matrix([[b0, b1], [b2, b3]])
+    assert ss4.output_matrix == Matrix([[c0, c1], [c2, c3]])
+    assert ss4.feedforward_matrix == Matrix([[d0, d1], [d2, d3]])
+    assert ss4.args == (Matrix([[a0, a1],
+                                [a2, a3]]),
+                        Matrix([[b0, b1],
+                                [b2, b3]]),
+                        Matrix([[c0, c1],
+                                [c2, c3]]),
+                        Matrix([[d0, d1],
+                                [d2, d3]]))
+
+    # using less matrices. Rest will be filled with a minimum of zeros.
+    ss5 = StateSpace()
+    assert ss5.args == (Matrix([[0]]), Matrix([[0]]), Matrix([[0]]), Matrix([[0]]))
+
+    A6 = Matrix([[0, 1], [1, 0]])
+    B6 = Matrix([1, 1])
+    ss6 = StateSpace(A6, B6)
+
+    assert ss6.state_matrix == Matrix([[0, 1], [1, 0]])
+    assert ss6.input_matrix ==  Matrix([1, 1])
+    assert ss6.output_matrix == Matrix([[0, 0]])
+    assert ss6.feedforward_matrix == Matrix([[0]])
+    assert ss6.args == (Matrix([[0, 1],
+                                [1, 0]]),
+                        Matrix([[1],
+                                [1]]),
+                        Matrix([[0, 0]]),
+                        Matrix([[0]]))
+
+    # Check if the system is SISO or MIMO.
+    # If system is not SISO, then it is definitely MIMO.
+
+    assert ss1.is_SISO == True
+    assert ss2.is_SISO == True
+    assert ss3.is_SISO == False
+    assert ss4.is_SISO == False
+    assert ss5.is_SISO == True
+    assert ss6.is_SISO == True
+
+    # ShapeError if matrices do not fit.
+    raises(ShapeError, lambda: StateSpace(Matrix([s, (s+1)**2]), Matrix([s+1]),
+                                          Matrix([s**2 - 1]), Matrix([2*s])))
+    raises(ShapeError, lambda: StateSpace(Matrix([s]), Matrix([s+1, s**3 + 1]),
+                                          Matrix([s**2 - 1]), Matrix([2*s])))
+    raises(ShapeError, lambda: StateSpace(Matrix([s]), Matrix([s+1]),
+                                          Matrix([[s**2 - 1], [s**2 + 2*s + 1]]), Matrix([2*s])))
+    raises(ShapeError, lambda: StateSpace(Matrix([[-s, -s], [s, 0]]),
+                                                Matrix([[s/2, 0], [0, s]]),
+                                                Matrix([[0, s]]),
+                                                Matrix([[2*s, 2*s], [s, s]])))
+
+    # TypeError if arguments are not sympy matrices.
+    raises(TypeError, lambda: StateSpace(s**2, s+1, 2*s, 1))
+    raises(TypeError, lambda: StateSpace(Matrix([2, 0.5]), Matrix([-1]),
+                                         Matrix([1]), 0))
+def test_StateSpace_add():
+    A1 = Matrix([[4, 1],[2, -3]])
+    B1 = Matrix([[5, 2],[-3, -3]])
+    C1 = Matrix([[2, -4],[0, 1]])
+    D1 = Matrix([[3, 2],[1, -1]])
+    ss1 = StateSpace(A1, B1, C1, D1)
+
+    A2 = Matrix([[-3, 4, 2],[-1, -3, 0],[2, 5, 3]])
+    B2 = Matrix([[1, 4],[-3, -3],[-2, 1]])
+    C2 = Matrix([[4, 2, -3],[1, 4, 3]])
+    D2 = Matrix([[-2, 4],[0, 1]])
+    ss2 = StateSpace(A2, B2, C2, D2)
+    ss3 = StateSpace()
+    ss4 = StateSpace(Matrix([1]), Matrix([2]), Matrix([3]), Matrix([4]))
+
+    expected_add = \
+        StateSpace(
+        Matrix([
+        [4,  1,  0,  0, 0],
+        [2, -3,  0,  0, 0],
+        [0,  0, -3,  4, 2],
+        [0,  0, -1, -3, 0],
+        [0,  0,  2,  5, 3]]),
+        Matrix([
+        [ 5,  2],
+        [-3, -3],
+        [ 1,  4],
+        [-3, -3],
+        [-2,  1]]),
+        Matrix([
+        [2, -4, 4, 2, -3],
+        [0,  1, 1, 4,  3]]),
+        Matrix([
+        [1, 6],
+        [1, 0]]))
+
+    expected_mul = \
+        StateSpace(
+        Matrix([
+        [ -3,   4,  2, 0,  0],
+        [ -1,  -3,  0, 0,  0],
+        [  2,   5,  3, 0,  0],
+        [ 22,  18, -9, 4,  1],
+        [-15, -18,  0, 2, -3]]),
+        Matrix([
+        [  1,   4],
+        [ -3,  -3],
+        [ -2,   1],
+        [-10,  22],
+        [  6, -15]]),
+        Matrix([
+        [14, 14, -3, 2, -4],
+        [ 3, -2, -6, 0,  1]]),
+        Matrix([
+        [-6, 14],
+        [-2,  3]]))
+
+    assert ss1 + ss2 == expected_add
+    assert ss1*ss2 == expected_mul
+    assert ss3 + 1/2 == StateSpace(Matrix([[0]]), Matrix([[0]]), Matrix([[0]]), Matrix([[0.5]]))
+    assert ss4*1.5 == StateSpace(Matrix([[1]]), Matrix([[2]]), Matrix([[4.5]]), Matrix([[6.0]]))
+    assert 1.5*ss4 == StateSpace(Matrix([[1]]), Matrix([[3.0]]), Matrix([[3]]), Matrix([[6.0]]))
+    raises(ShapeError, lambda: ss1 + ss3)
+    raises(ShapeError, lambda: ss2*ss4)
+
+def test_StateSpace_negation():
+    A = Matrix([[a0, a1], [a2, a3]])
+    B = Matrix([[b0, b1], [b2, b3]])
+    C = Matrix([[c0, c1], [c1, c2], [c2, c3]])
+    D = Matrix([[d0, d1], [d1, d2], [d2, d3]])
+    SS = StateSpace(A, B, C, D)
+    SS_neg = -SS
+
+    state_mat = Matrix([[-1, 1], [1, -1]])
+    input_mat = Matrix([1, -1])
+    output_mat = Matrix([[-1, 1]])
+    feedforward_mat = Matrix([1])
+    system = StateSpace(state_mat, input_mat, output_mat, feedforward_mat)
+
+    assert SS_neg == \
+        StateSpace(Matrix([[a0, a1],
+                           [a2, a3]]),
+                   Matrix([[b0, b1],
+                           [b2, b3]]),
+                   Matrix([[-c0, -c1],
+                           [-c1, -c2],
+                           [-c2, -c3]]),
+                   Matrix([[-d0, -d1],
+                           [-d1, -d2],
+                           [-d2, -d3]]))
+    assert -system == \
+        StateSpace(Matrix([[-1,  1],
+                           [ 1, -1]]),
+                   Matrix([[ 1],[-1]]),
+                   Matrix([[1, -1]]),
+                   Matrix([[-1]]))
+    assert -SS_neg == SS
+    assert -(-(-(-system))) == system
+
+def test_SymPy_substitution_functions():
+    # subs
+    ss1 = StateSpace(Matrix([s]), Matrix([(s + 1)**2]), Matrix([s**2 - 1]), Matrix([2*s]))
+    ss2 = StateSpace(Matrix([s + p]), Matrix([(s + 1)*(p - 1)]), Matrix([p**3 - s**3]), Matrix([s - p]))
+
+    assert ss1.subs({s:5}) == StateSpace(Matrix([[5]]), Matrix([[36]]), Matrix([[24]]), Matrix([[10]]))
+    assert ss2.subs({p:1}) == StateSpace(Matrix([[s + 1]]), Matrix([[0]]), Matrix([[1 - s**3]]), Matrix([[s - 1]]))
+
+    # xreplace
+    assert ss1.xreplace({s:p}) == \
+        StateSpace(Matrix([[p]]), Matrix([[(p + 1)**2]]), Matrix([[p**2 - 1]]), Matrix([[2*p]]))
+    assert ss2.xreplace({s:a, p:b}) == \
+        StateSpace(Matrix([[a + b]]), Matrix([[(a + 1)*(b - 1)]]), Matrix([[-a**3 + b**3]]), Matrix([[a - b]]))
+
+    # evalf
+    p1 = a1*s + a0
+    p2 = b2*s**2 + b1*s + b0
+    G = StateSpace(Matrix([p1]), Matrix([p2]))
+    expect = StateSpace(Matrix([[2*s + 1]]), Matrix([[5*s**2 + 4*s + 3]]), Matrix([[0]]), Matrix([[0]]))
+    expect_ = StateSpace(Matrix([[2.0*s + 1.0]]), Matrix([[5.0*s**2 + 4.0*s + 3.0]]), Matrix([[0]]), Matrix([[0]]))
+    assert G.subs({a0: 1, a1: 2, b0: 3, b1: 4, b2: 5}) == expect
+    assert G.subs({a0: 1, a1: 2, b0: 3, b1: 4, b2: 5}).evalf() == expect_
+    assert expect.evalf() == expect_
+
+def test_conversion():
+    # StateSpace to TransferFunction for SISO
+    A1 = Matrix([[-5, -1], [3, -1]])
+    B1 = Matrix([2, 5])
+    C1 = Matrix([[1, 2]])
+    D1 = Matrix([0])
+    H1 = StateSpace(A1, B1, C1, D1)
+    tm1 = H1.rewrite(TransferFunction)
+    tm2 = (-H1).rewrite(TransferFunction)
+
+    tf1 = tm1[0][0]
+    tf2 = tm2[0][0]
+
+    assert tf1 == TransferFunction(12*s + 59, s**2 + 6*s + 8, s)
+    assert tf2.num == -tf1.num
+    assert tf2.den == tf1.den
+
+    # StateSpace to TransferFunction for MIMO
+    A2 = Matrix([[-1.5, -2, 3], [1, 0, 1], [2, 1, 1]])
+    B2 = Matrix([[0.5, 0, 1], [0, 1, 2], [2, 2, 3]])
+    C2 = Matrix([[0, 1, 0], [0, 2, 1], [1, 0, 2]])
+    D2 = Matrix([[2, 2, 0], [1, 1, 1], [3, 2, 1]])
+    H2 = StateSpace(A2, B2, C2, D2)
+    tm3 = H2.rewrite(TransferFunction)
+
+    # outputs for input i obtained at Index i-1. Consider input 1
+    assert tm3[0][0] == TransferFunction(2.0*s**3 + 1.0*s**2 - 10.5*s + 4.5, 1.0*s**3 + 0.5*s**2 - 6.5*s - 2.5, s)
+    assert tm3[0][1] == TransferFunction(2.0*s**3 + 2.0*s**2 - 10.5*s - 3.5, 1.0*s**3 + 0.5*s**2 - 6.5*s - 2.5, s)
+    assert tm3[0][2] == TransferFunction(2.0*s**2 + 5.0*s - 0.5, 1.0*s**3 + 0.5*s**2 - 6.5*s - 2.5, s)
+
+    # TransferFunction to StateSpace
+    SS = TF1.rewrite(StateSpace)
+    assert SS == \
+        StateSpace(Matrix([[     0,          1],
+                           [-wn**2, -2*wn*zeta]]),
+                   Matrix([[0],
+                           [1]]),
+                   Matrix([[1, 0]]),
+                   Matrix([[0]]))
+    assert SS.rewrite(TransferFunction)[0][0] == TF1
+
+    # Transfer function has to be proper
+    raises(ValueError, lambda: TransferFunction(b*s**2 + p**2 - a*p + s, b - p**2, s).rewrite(StateSpace))
+
+
+def test_StateSpace_functions():
+    # https://in.mathworks.com/help/control/ref/statespacemodel.obsv.html
+
+    A_mat = Matrix([[-1.5, -2], [1, 0]])
+    B_mat = Matrix([0.5, 0])
+    C_mat = Matrix([[0, 1]])
+    D_mat = Matrix([1])
+    SS1 = StateSpace(A_mat, B_mat, C_mat, D_mat)
+    SS2 = StateSpace(Matrix([[1, 1], [4, -2]]),Matrix([[0, 1], [0, 2]]),Matrix([[-1, 1], [1, -1]]))
+    SS3 = StateSpace(Matrix([[1, 1], [4, -2]]),Matrix([[1, -1], [1, -1]]))
+
+    # Observability
+    assert SS1.is_observable() == True
+    assert SS2.is_observable() == False
+    assert SS1.observability_matrix() == Matrix([[0, 1], [1, 0]])
+    assert SS2.observability_matrix() == Matrix([[-1,  1], [ 1, -1], [ 3, -3], [-3,  3]])
+    assert SS1.observable_subspace() == [Matrix([[0], [1]]), Matrix([[1], [0]])]
+    assert SS2.observable_subspace() == [Matrix([[-1], [ 1], [ 3], [-3]])]
+
+    # Controllability
+    assert SS1.is_controllable() == True
+    assert SS3.is_controllable() == False
+    assert SS1.controllability_matrix() ==  Matrix([[0.5, -0.75], [  0,   0.5]])
+    assert SS3.controllability_matrix() == Matrix([[1, -1, 2, -2], [1, -1, 2, -2]])
+    assert SS1.controllable_subspace() == [Matrix([[0.5], [  0]]), Matrix([[-0.75], [  0.5]])]
+    assert SS3.controllable_subspace() == [Matrix([[1], [1]])]
+
+    # Append
+    A1 = Matrix([[0, 1], [1, 0]])
+    B1 = Matrix([[0], [1]])
+    C1 = Matrix([[0, 1]])
+    D1 = Matrix([[0]])
+    ss1 = StateSpace(A1, B1, C1, D1)
+    ss2 = StateSpace(Matrix([[1, 0], [0, 1]]), Matrix([[1], [0]]), Matrix([[1, 0]]), Matrix([[1]]))
+    ss3 = ss1.append(ss2)
+
+    assert ss3.num_states == ss1.num_states + ss2.num_states
+    assert ss3.num_inputs == ss1.num_inputs + ss2.num_inputs
+    assert ss3.num_outputs == ss1.num_outputs + ss2.num_outputs
+    assert ss3.state_matrix == Matrix([[0, 1, 0, 0], [1, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]])
+    assert ss3.input_matrix == Matrix([[0, 0], [1, 0], [0, 1], [0, 0]])
+    assert ss3.output_matrix == Matrix([[0, 1, 0, 0], [0, 0, 1, 0]])
+    assert ss3.feedforward_matrix == Matrix([[0, 0], [0, 1]])