https://github.com/sandeep026/numerical-optimal-control
Direct methods to solve optimal control problems in python and MATLAB
https://github.com/sandeep026/numerical-optimal-control
casadi direct-methods ipopt optimal-control optimization
Last synced: 4 days ago
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Direct methods to solve optimal control problems in python and MATLAB
- Host: GitHub
- URL: https://github.com/sandeep026/numerical-optimal-control
- Owner: sandeep026
- License: mit
- Created: 2024-06-28T01:42:36.000Z (almost 2 years ago)
- Default Branch: main
- Last Pushed: 2026-04-29T21:09:15.000Z (about 1 month ago)
- Last Synced: 2026-04-29T23:15:49.563Z (about 1 month ago)
- Topics: casadi, direct-methods, ipopt, optimal-control, optimization
- Language: Jupyter Notebook
- Homepage:
- Size: 154 KB
- Stars: 1
- Watchers: 1
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- License: LICENSE
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README
## Cart System: Optimal Control Benchmark
This repository explores the optimal control of a cart system subject to drag [^1]. It provides a comprehensive comparison between the analytical solution (derived via Pontryagin's Minimum Principle) and various numerical direct methods, including shooting, collocation and pseudospectral techniques.
IPOPT is used for numerical optimization.
### Implementation & Requirements
The code is available for both MATLAB/Octave and Python.
#### Python Setup
Ensure you have Python 3.11+ installed. You can install the dependencies via pip:
``` bash
pip install casadi numpy matplotlib
```
Notebooks: Interactive .ipynb files for all methods are located in the /python folder.
Utility: lglpsmethods.py contains the logic for differentiation matrices and LGL node generation.
#### MATLAB / Octave Setup
Environment: MATLAB or Octave 6.1.0+
Dependency: CasADi 3.5.5+ must be added to your path
### OCP
We aim to find the force $f(t)$ that minimizes the control effort required to move a cart to a state-dependent target position within a fixed time $T=2$.
The cart starts from rest and the final position is dependent on the velocity of the cart at final time.
#### System variables
1. $z_1$ - position of the cart
2. $z_2$ - velocity of the cart
3. $f$ - force applied to the cart
#### Mathematical Formulation
$$
\begin{gathered}
\text{min.}~~\int_{0}^{2}f^2dt\\
\frac{d}{dt}\begin{bmatrix}
z_1\\
z_2
\end{bmatrix}=
\begin{bmatrix}
z_2\\
-z_2+f
\end{bmatrix}\\
\begin{bmatrix}
z_1\\
z_2
\end{bmatrix}(0)=
\begin{bmatrix}
0\\
0
\end{bmatrix}\\
z_1(2)-2.694528z_2(2)+1.155356=0
\end{gathered}
$$
#### Analytical solution
Using Pontryagin's Minimization Principle, the exact solution for the state trajectories and control law is determined to be:
$$
\begin{aligned}
\mathrm{Obj.}&=0.577678\\
z_{1}(t)&=\frac{-3}{8}e^{-t}+\frac{1}{8}e^{t}-\frac{t}{2}+\frac{1}{4}\\
z_{2}(t)&=\frac{3}{8}e^{-t}+\frac{1}{8}e^{t}-\frac{1}{2}\\
f(t)&=\frac{1}{4}e^{t}-\frac{1}{2}\\
\end{aligned}
$$
### Numerical methods for optimal control
This project implements several Direct Methods to transform the continuous optimal control problem (OCP) into a nonlinear programming (NLP) problem.
### Numerical methods implemented
#### MATLAB/Octave
|method|control parameterization|files|
|-|-|-|
|trapezoidal piecewise | piecewise constant control|```trapezoidal_constant.m```|
|Hermite Simpson |piecewise linear control|```simpson.m```|
|Runge Kutta 4 (single shooting) | piecewise constant control|```single_shooting.m```|
|Runge Kutta 4 (multiple shooting) | piecewise constant control|```multiple_shooting.m```|
|Legendre Gauss Lobatto [^3]|global polynomial|```LGL pseudospectral.m, legslb.m, legslbdiff.m, lepoly.m, lepolym.m```|
#### Python
All numerical methods in `directmethods.py`
### Results
The numerical methods closely approximate the analytical curves. Below are the comparisons of the phase plot and control effort.
### References
[^1]: Conway, B. A. and K. Larson (1998). Collocation versus differential inclusion in direct optimization. Journal of Guidance, Control, and Dynamics, 21(5), 780–785
[^2]: Diehl, Moritz, and Sébastien Gros. "Numerical optimal control." Optimization in Engineering Center (OPTEC) (2011).
[^3]: Shen, Jie, Tao Tang, and Li-Lian Wang. Spectral methods: algorithms, analysis and applications. Vol. 41. Springer Science & Business Media, 2011.