Cushioned Oscillation

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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\limits_{t_f} t_f

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}

Parameters and Reference Solution

The following parameters were used, to create the reference solution below, with an almost optimal final time  t_f = 16.66 s:

 m=10,

 c=10,

 x_0=2,

 v_0=5,

 u_{mm}=5



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).



Source Code

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

References