# Cushioned Oscillation

Cushioned Oscillation | |
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State dimension: | 1 |

Differential states: | 2 |

Continuous control functions: | 1 |

Path constraints: | 2 |

Interior point equalities: | 4 |

The Cushioned Oscillation is a simplified model of time optimal "stopping" of an oscillating object attached to a spring by applying a control and moving it back into the relaxed position and zero velocity.

## Contents

## Model formulation

An object with mass is attached to a spring with stiffness constant .

If the resetting spring force is proportional to the deviation , an oscillation, induced by an external force , satisfies:

(which is equivalent to )

where denotes the deviation to the relaxed position and the velocity of the oscillating object.

Through external force, the object has been put into an initial state :

The goal is to reset position and velocity of the object as fast as possible, meaning:

,

with the objective function:

## Optimal Control Problem Formulation

The above results in the following OCP

## Parameters and Reference Solution

The following parameters were used, to create the reference solution below, with an almost optimal final time :

## Reference Solution

The OCP was solved within MATLAB R2015b, using the TOMLAB Optimization Package. PROPT reformulates such problems with the direct collocation approach (n=80 collocation points) and automatically finds a suiting solver included in the TOMLAB Optimization Package (in this case, SNOPT was used).

## Source Code

- A MATLAB script using PROPT can be found in: Cushioned Oscillation (PROPT)