Difference between revisions of "Cushioned Oscillation"
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Revision as of 00:29, 13 January 2016
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 
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 (OCP) Formulation
The above results in the following OCP
![\begin{array}{llr}
\min\limits_{x,v,u,t_f} & t_f\\
s.t. & \dot x (t) = v(t), & \forall t \in [0,t_f]\\
& \dot v (t)= \frac{1}{m}(u(t) - cx(t)), & \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}, & \forall t \in [0,t_f]\\
\end{array}](https://mintoc.de/images/math/2/f/e/2fe76b06cb3939ec900a852d9e7b9bfe.png)
Parameters and Reference Solution
The following parameters were used, to create the reference solution below, with an almost optimal final time 
:
- 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 approache and automatically finds a suiting solver included in the TOMLAB Optimization Package (in this case, SNOPT was used).
