Cushioned Oscillation

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 no velocity.

Model formulation

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

If the resetting spring force is proportional to the deviation $x=x(t)$, an oscillation, induced by an external force $u(t)$, satisfies:

$m\dot{v}(t)+cx(t)=u(t)$ (which is equivalent to $\dot{v}(t)=\frac{1}{m}(u(t)-cx(t))$)

where $x(t)$ denotes the deviation to the relaxed position and $v(t)=\dot{x}(t)$ the velocity of the oscillating object.

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

$(x(0),v(0))=(x_0,v_0)$

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

$(x(t_f),v(t_f))=(0,0)$,

with the objective function:

$\min {}_{t_f}{t}_f$

Optimal Control Problem (OCP) Formulation

The above results in the following OCP

$\begin{array}{ll}\min {}_{x,v,u,t_f}& t_f\\ s.t.& \dot{x}(t)=v(t),& \mathrm{\forall }t\in [0,t_f]\\ & \dot{v}(t)=\frac{1}{m}(u(t)-cx(t)),& \mathrm{\forall }t\in [0,t_f]\\ \\ & x(0)=x_0,\\ & v(0)=v_0,\\ & x(t_f)=0,\\ & v(t_f)=0,\\ \\ & -u_{mm}\le u(t)\le u_{mm},& \mathrm{\forall }t\in [0,t_f]\\ \end{array}$

Parameters and Reference Solution

The following parameters were used, to create the reference solution below, with an almost optimal final time $t_f=8.98s$:

$m=5,$

$c=10,$

$x_0=2,$

$v_0=5,$

$u_{mm}=5$

• Reference solution plots
• 
  States and Controls

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

The MATLAB script can be found in: Cushioned Oscillation (PROPT)