Difference between revisions of "Cushioned Oscillation"

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(Resetting an oscillating object in shortest time)
 
 
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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.
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{{Dimensions
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|nd        = 1
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|nx        = 2
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|nu        = 1
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|nc        = 2
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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.
  
 
== Model formulation ==  
 
== Model formulation ==  
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An object with mass <math> m </math> is attached to a spring with stiffness constant <math> c </math>.
 
An object with mass <math> m </math> is attached to a spring with stiffness constant <math> c </math>.
  
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If the resetting spring force is proportional to the deviation <math>x=x(t)</math>, an oscillation induced by an external force <math>u(t)</math> satisfies:
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If the resetting spring force is proportional to the deviation <math>x=x(t)</math>, an oscillation, induced by an external force <math>u(t)</math>, satisfies:
  
  
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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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== Optimal Control Problem Formulation ==
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== Optimal Control Problem (OCP) Formulation ==
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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}{llll}
  
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<math> \begin{array}{llr}
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\min\limits_{x,v,u,t_f}  & t_f & & \\  
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\min\limits_{x,v,u,t_f}  & t_f\\  
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s.t. & \dot x (t) = v(t), & \forall t \in [0,t_f]\\
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s.t. & \dot x & = v,\\
  
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& \dot v (t)= \frac{1}{m}(u(t) - cx(t)),  & \forall t \in [0,t_f]\\
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& \dot v & =   \frac{1}{m}(u - c \cdot x),\\
 
\\
 
\\
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& x(0) & = x_0,\\
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& x(0)=x_0,\\
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& v(0) & = v_0,\\
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& x(t_f) & = 0,\\
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& v(0)=v_0,\\
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& v(t_f) & = 0,\\
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& |u| & \le u_{mm}.\\
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& x(t_f)=0,\\
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& 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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  \end{array}</math>
 
  \end{array}</math>
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== 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>:
  
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The following parameters were used, to create the reference solution below, with an almost optimal final time <math> t_f = 16.66 s</math>:
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<math> m=5, </math>
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<math> m=10, </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 ==
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<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="1">
 
<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="1">
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  Image:ref_sol_plot_cushioned_oscillation.png| States and Controls
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  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 approache and automatically finds a suiting solver included in the TOMLAB Optimization Package (in this case, SNOPT was used).
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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).
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== Source Code ==
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* A MATLAB script using [[:Category:TomDyn/PROPT | PROPT]] can be found in: [[Cushioned Oscillation (PROPT)]]
  
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== References ==
  
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[[Category:OCP]]
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[[Category:MIOCP]]
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[[Category:Bang bang]]
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[[Category:ODE model]]
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[[Category: Minimum time]]

Latest revision as of 10:06, 30 June 2016

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_f),v(t_f)) = (0,0),

with the objective function:

\min\limits_{t_f} t_f

Optimal Control Problem Formulation

The above results in the following OCP

\begin{array}{llll} \min\limits_{x,v,u,t_f} & t_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_f) & = 0,\\ & v(t_f) & = 0,\\ & |u| & \le u_{mm}.\\ \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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