NOSNOC

NOSNOC is an open-source MATLAB software package for NOnSmooth Numerical Optimal Control. The Python package nosnoc_py with similar functionality is available as well.

A detailed overview of the theory and methods behind NOSNOC can be found in the course material of the Summer School on Direct Methods for Optimal Control of Nonsmooth Systems.

General

NOSNOC is a tool for numerically solving optimal control problems with nonsmooth dynamical systems with switches and/or state jumps. It supports:

1. Automatic discretization via the FESD method - high accuracy and correct sensitivities. Note that classical time-stepping methods only have first-order accuracy and wrong sensitivities even when they appear to be differentiable.

2. Automatic reformulations of systems with state jumps (e.g. contact problems) via time-freezing into Filippov systems/PSS. (enables high accuracy even for systems with state jumps)

3. Solving the nonsmooth nonlinear programs via homotopy methods. Enables the use of off-the-shelf solvers like IPOPT and SNOPT.

NOSNOC relies on the recently introduced Finite Elements with Switch Detection (FESD) which enables high accuracy optimal control of systems with switches and jumps. It enables the treatment of a broad class of nonsmooth systems in a unified way.

NOSNOC offers several ways to treat switched systems, piecewise smooth systems, Filippov systems, hybrid systems, rigid body models with impacts and friction in simulation, and optimal control. It discretizes a Dynamic Complementarity System (DCS) with the FESD method and solves the resulting mathematical program with complementarity constraints (MPCCs). The MPCCs are solved in a homotopy loop with a standard solver like IPOPT or SNOPT. The user may directly provide a DCS or define the different modes of a Filippov system and the reformulation is automated.

With NOSNOC one can simulate and solve optimal control problems subject to different kinds of nonsmooth systems, by declaring the dcs_mode:

1. settings.dcs_mode = 'Stewart' - for treating Filippov systems via Stewart's reformulation
2. settings.dcs_mode = 'Step' - for treating nonsmooth systems via set valued step functions (covers also all Filippov systems that are treated with Stewart's).
3. settings.dcs_mode = 'CLS' - for rigid bodies with friction and impact (also called complementarity Lagrangian systems (CLS)) - uses FESD tailored to CLS.
4. settings.time_freezing = 1 - and using for the dcs_mode = Step, the CLS is reformulated into an equivalent Filippov system and treated with FESD.

Installation

NOSNOC requires CasADi version 3.5.5 and Matlab version R2021b or later.

Installation for MATLAB

1. Install CasADi and make sure it is added to your MATLAB path. For CasADi installation instructions follow visit: https://web.casadi.org/get/

2. Clone this repository

git clone --recursive https://github.com/nurkanovic/nosnoc.git

3. Open the NOSNOC folder in Matlab and run the install_nosnoc script

>> install_nosnoc

Note that IPOPT is shipped with CasADi. More information including detailed documentation can be found on its homepage

Installation for python

Go to the nosnoc_py repository and clone it.

Afterward,

1. Setup virtual environment:

virtualenv env --python=python3

2. Source environment:

source env/bin/activate

3. Install

pip install -e .

Using NOSNOC

The interface of NOSNOC is based on the symbolic modeling framework CasADi. User inputs should be given as CasADi expressions.

To get started we recommend you look into our example libraries for MATLAB or python.

Literature - theory and algorithms

FESD

Finite Elements with Switch Detection for Direct Optimal Control of Nonsmooth Systems
A.Nurkanović, M. Sperl, S. Albrecht, M. Diehl
arXiv preprint 2022

@article{Nurkanovic2022,
  title={Finite elements with switch detection for direct optimal control of nonsmooth systems},
  author={Nurkanovi{\'c}, Armin and Sperl, Mario and Albrecht, Sebastian and Diehl, Moritz},
  journal={arXiv preprint arXiv:2205.05337},
  year={2022}
}

Time - Freezing

A Time-Freezing Approach for Numerical Optimal Control of Nonsmooth Differential Equations with State Jumps
A. Nurkanović, T. Sartor, S. Albrecht, M. Diehl
IEEE Control Systems Letters 2021

The Time-Freezing Reformulation for Numerical Optimal Control of Complementarity Lagrangian Systems with State Jumps
A. Nurkanović, S. Albrecht, B. Brogliato, M. Diehl
Automatica 2023 (accepted for publication)

Continuous Optimization for Control of Hybrid Systems with Hysteresis via Time-Freezing
A.Nurkanović , M. Diehl
IEEE Control Systems Letters 2022

Literature - Software

NOSNOC

NOSNOC: A Software Package for Numerical Optimal Control of Nonsmooth Systems
A.Nurkanović , M. Diehl
IEEE Control Systems Letters 2022

@article{Nurkanovic2022,
  title={NOSNOC: A software package for numerical optimal control of nonsmooth systems},
  author={Nurkanovi{\'c}, Armin and Diehl, Moritz},
  journal={IEEE Control Systems Letters},
  volume={6},
  pages={3110--3115},
  year={2022},
  publisher={IEEE}
}

CasADi

CasADi - A software framework for nonlinear optimization and optimal control
J.A.E. Andersson, J. Gillis, G. Horn, J.B. Rawlings, M. Diehl
Mathematical Programming Computation, 2019

IPOPT

On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming
A. Wächter, L. T. Biegler Mathematical programming, 2006

Contact

Feel free to contact one of the main developers directly: Armin Nurkanović, armin.nurkanovic@imtek.uni-freiburg.de Anton Pozharskiy anton.pozharskiy@imtek.uni-freiburg.de Jonathan Frey jonathan.frey@imtek.uni-freiburg.de If you have questions, remarks, or comments, you are strongly encouraged to report them by creating a new issue on this Github page. Success stories and source code contributions are very welcome.