Cushioned Oscillation

Cushioned Oscillation
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.

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:

$(x(t\mathrm{f}),v(t_{\mathrm{f}}))=(0,0)$,

with the objective function:

$\min {}_{t_{\mathrm{f}}}{t}_{\mathrm{f}}$

Optimal Control Problem Formulation

The above results in the following OCP

$\begin{array}{llll}\min {}_{x,v,u,t_{\mathrm{f}}}& t_{\mathrm{f}}& & \\ s.t.& \dot{x}& =v,\\ & \dot{v}& =\frac{1}{m}(u-c\cdot x),\\ \\ & x(0)& =x_0,\\ & v(0)& =v_0,\\ & x(t_{\mathrm{f}})& =0,\\ & v(t_{\mathrm{f}})& =0,\\ & |u|& \le u_{\mathrm{m}\mathrm{m}}.\\ \end{array}$

Parameters and Reference Solution

The following parameters were used, to create the reference solution below, with an almost optimal final time $t_{\mathrm{f}}=8.98s$:

$m=5,$ $c=10,$ $x_0=2,$ $v_0=5,$ $u_{\mathrm{m}\mathrm{m}}=5.$

Reference Solution

• 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

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