Bang-bang free horizon (bang_free_tf)

Question. How fast can a double integrator move from rest to a distant rest when the horizon is free and thrust bounds are asymmetric?

Formulation

Symbol	Role	Meaning
\(x,v\)	states	position, velocity
\(u\)	control	acceleration

Dynamics. \(\dot x=v\), \(\dot v=u\).

Objective. \(\min T\).

Constraints. \(-2\le u\le 1\), \((x,v)(0)=(0,0)\), \((x,v)(T)=(300,0)\), \(T\ge 0\).

Horizon. Free.

Analytic: \(u=+1\) on \([0,20]\), \(u=-2\) on \([20,30]\), \(T^*=30\).

What was done

ocp.min(t_f(t)) with a free upper bound on \(T\). Legendre collocation.

What we found

Numerical \(T\) agrees with \(30\) to about 1%. States meet the terminal rest condition.

Interpretation

Unequal bounds on \(u\) shift the switch away from the midpoint. The horizon is a decision: the problem is finished when the target is reached, not at a prescribed clock time.

Public API

Bare dx, no Namespace (examples/bang_free_tf/bang_free_tf.py):

import eco_opt


def bang_model(t, x, u):
    return [x[1], u[0]]


sys = eco_opt.System(states=2, controls=1, model=bang_model)
ocp = eco_opt.Ocp()
ocp.min(eco_opt.t_f(sys.t))
ocp.st(system=sys, constraints=[...])

Run and tests

cd examples/bang_free_tf
python bang_free_tf.py
python plot_bang_free_tf.py
python -m pytest ../../tests/test_bang_free_tf.py -q

References

Transcription of the Yop example Linear problem with bang-bang control. The problem is the same; the text and code here are ours.