Difference between revisions of "Cushioned Oscillation"

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|nu        = 1
 
|nu        = 1
 
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|nre      = 4
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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 zero velocity.
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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 zero velocity.
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== Model formulation ==  
 
== Model formulation ==  
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where <math>x(t)</math> denotes the deviation to the relaxed position and <math> v(t)=\dot x (t) </math> the velocity of the oscillating object.
 
where <math>x(t)</math> denotes the deviation to the relaxed position and <math> v(t)=\dot x (t) </math> the velocity of the oscillating object.
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Through external force, the object has been put into an initial state :
 
Through external force, the object has been put into an initial state :
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<math>(x(0),v(0)) = (x_0,v_0)</math>
 
<math>(x(0),v(0)) = (x_0,v_0)</math>
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The goal is to reset position and velocity of the object as fast as possible, meaning:
 
The goal is to reset position and velocity of the object as fast as possible, meaning:
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<math>(x(t_f),v(t_f)) = (0,0)</math>,
 
<math>(x(t_f),v(t_f)) = (0,0)</math>,
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with the objective function:
 
with the objective function:
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<math>\min\limits_{t_f} t_f</math>
 
<math>\min\limits_{t_f} t_f</math>
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The above results in the following OCP  
 
The above results in the following OCP  
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<math> \begin{array}{llr}
 
<math> \begin{array}{llr}
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& \dot v (t)= \frac{1}{m}(u(t) - cx(t)),  & \forall t \in [0,t_f]\\
 
& \dot v (t)= \frac{1}{m}(u(t) - cx(t)),  & \forall t \in [0,t_f]\\
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\\
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\\                                                
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& x(0)=x_0,\\
 
& x(0)=x_0,\\
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& v(0)=v_0,\\
 
& v(0)=v_0,\\
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& x(t_f)=0,\\
 
& x(t_f)=0,\\
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& v(t_f)=0,\\
 
& v(t_f)=0,\\
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\\
 
\\
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& -u_{mm} \le u(t) \le u_{mm},  & \forall t \in [0,t_f]\\
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& -u_{mm} \le u(t) \le u_{mm},  & \forall t \in [0,t_f]\\
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  \end{array}</math>
 
  \end{array}</math>
  
 
== Parameters and Reference Solution ==
 
== Parameters and Reference Solution ==
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The following parameters were used, to create the reference solution below, with an almost optimal final time <math> t_f = 8.98 s</math>:
 
The following parameters were used, to create the reference solution below, with an almost optimal final time <math> t_f = 8.98 s</math>:
  
 
<math> m=5, </math>
 
<math> m=5, </math>
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<math> c=10, </math>
 
<math> c=10, </math>
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<math> x_0=2, </math>
 
<math> x_0=2, </math>
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<math> v_0=5, </math>
 
<math> v_0=5, </math>
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<math> u_{mm}=5.</math>
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<math> u_{mm}=5</math>
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== Reference Solution ==
 
== Reference Solution ==
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  Image:Ref_sol_plot_cushioned_oscillation_m5.png| States and Controls
 
  Image:Ref_sol_plot_cushioned_oscillation_m5.png| States and Controls
 
</gallery>
 
</gallery>
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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).
 
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).
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== Source Code ==
 
== Source Code ==
  
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The MATLAB script can be found in:
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* A MATLAB script using [[Category:TomDyn/PROPT]] can be found in: [[Cushioned Oscillation (PROPT)]]
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[[Category:TomDyn/PROPT]] [[Cushioned Oscillation (PROPT)]]
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== References ==
 
== References ==
  
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[[Category:MIOCP]] [[Category: Minimum time]]
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[[Category:MIOCP]]  
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[[Category: Minimum time]]

Revision as of 10:40, 28 January 2016

Cushioned Oscillation
State dimension: 1
Differential states: 2
Continuous control functions: 1
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_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 = 8.98 s:

m=5, c=10, x_0=2, v_0=5, u_{mm}=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

References

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