Optimal control problem

Consider the optimal control problem for a cart system taken from Ref. ¹. The variables in problem are defined as follows.

1. $z_1$ - position of the cart
2. $z_2$ - velocity of the cart
3. $f$ - force applied to the cart

The cart starts from rest and the final position is dependent on the velocity of the cart at final time. There is also a drag force which slows the cart and the optimal control problem seeks the control action which minimizes control effort while respecting the constraints.

$$ \begin{gathered} \text{min.}~~\int_{0}^{2}f^2dt\\ \frac{d}{dt}\begin{bmatrix} z_1\\ z_2 \end{bmatrix}= \begin{bmatrix} z_2\\ -z_2+f \end{bmatrix}\\ \begin{bmatrix} z_1\\ z_2 \end{bmatrix}(0)= \begin{bmatrix} 0\\ 0 \end{bmatrix}\\ z_1(2)-2.694528z_2(2)+1.155356=0 \end{gathered} $$

Analytical solution

Pontrygin's minimization principle leads to the following analytical solution for the objective, states and control.

$$ \begin{aligned} \mathrm{Obj.}&=0.577678\\ z_{1}(t)&=\frac{-3}{8}e^{-t}+\frac{1}{8}e^{t}-\frac{t}{2}+\frac{1}{4}\\ z_{2}(t)&=\frac{3}{8}e^{-t}+\frac{1}{8}e^{t}-\frac{1}{2}\\ f(t)&=\frac{1}{4}e^{t}-\frac{1}{2}\\ \end{aligned} $$

Numerical methods for optimal control

The sample codes demonstrate direct methods ², a class of methods used to solve optimal control problems numerically. The time domain, objective functional and the constraints are discretized and transformed to a nonlinear optimization problem. The solution of the optimization problem comprise the optimal discrete state and control vector.

Numerical methods implemented

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

Results

Numerical and analytical results compared side by side.

Implementation

The code are available in both python, octave or matlab for use.

MATLAB/OCTAVE

• MATLAB/OCTAVE-6.1.0
• Casadi 3.5.5

python

Requirements

• python ^3.11
• casadi 3.6.5
• matplotlib
• numpy ^1.26.4

• The jupyter notebooks for all the methods are available in the python folder.
• lglpsmethods.py- module which generates the differentiation matrix, quadrature and nodes for pseudospectral methods.

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. 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. ↩