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:

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 2	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

Footnotes

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. Shen, Jie, Tao Tang, and Li-Lian Wang. Spectral methods: algorithms, analysis and applications. Vol. 41. Springer Science & Business Media, 2011. ↩