mintOC - User contributions [en]

Category:JModelica

2016-04-15T12:25:17Z

MadeleineSchroeter:

JModelica.org is an extensible Modelica-based open source platform for optimization, simulation and analysis of
complex dynamic systems. JModelica supports the modeling language Modelica and it's extension Optimica. It was developed by the Modelon AB and the Department of Automatic Control at Lund University, Sweden.

== Availibility ==
JModelica is an open source platform and comes with all third party requirements. Installation on Windows can be done via binarys, on Linux systems a compilation from source is required.

== Technical Details ==

The software package relies on the modeling languange Modelica and supports it's extension Optimica. The software uses the Phyton language to run problems and the Phyton packages Numpy and Scipy provide support for numerical computation. The software is extensible in several ways.

== Supported Problem Classes ==

* NLP, LP, QP
* Optimal Control Problems with ODEs or DAEs
* Minimum Time Problems
* Optimal Design
* Model Calibration
* Parameter Estimation

== Algorithms ==

JModelica uses two different Algorithms. The Dynamic optimization of DAEs using direct collocation with CasADI is the default Algorithm and is used for optimization purposes. The Derivative free calibration and optimization of ODEs with FMUs is used for simulation puproses.

== References ==
<http://www.jmodelica.org/>


[[Category: Implementation]]

F-8 aircraft (JModelica)

2016-03-15T12:47:37Z

MadeleineSchroeter:

This page contains the model formulation of the MIOCP [[F-8 aircraft]] in [http://jmodelica.org JModelica] format.

=== JModelica ===

The model in JModelica code.

<source lang="optimica">
// The F-8 aircraft control problem is based on a very simple aircraft model.
// (c) Sebastian Sager, 2005-2009
// More info on http://mintoc.de/index.php/F-8_aircraft

optimization F8_aircraft_opt (objective = tf,
startTime = 0,
finalTime = 1)

// Parameters
parameter Real ksi=0.05236;
parameter Real tf(free=true,min=0.1);

// The states
Real x0(start=0.4655, fixed=true); //angle of attack
Real x1(start=0, fixed=true); //pitch angle
Real x2(start=0, fixed=true); //pitch rate

// The control signal
input Real w(min=0,max=1);

equation

der(x0)= 1*tf*(-0.877*x0 + x2 - 0.088*x0*x2 + 0.47*x0^2 - 0.019*x1^2 -x0^2*x2 + 3.846*x0^3-(0.215*ksi-0.28*x0^2*ksi -0.47*x0*ksi^2 - 0.63*ksi^3)*w-(-0.215*ksi + 0.28*x0^2 -0.47*x0*ksi^2 + 0.63*ksi^3)*(1-w));
der(x1) = 1*tf*x2;
der(x2) = 1*tf*(-4.208*x0 - 0.396*x2 - 0.47*x0^2 - 3.564*x0^3+20.967*ksi - 6.265*x0^2*ksi + 46*x0*ksi^2 -61.4*ksi^3-(20.967*ksi - 6.265*x0^2*ksi - 61.4*ksi^3)*2*w);

constraint
x0(finalTime)=0;
x1(finalTime)=0;
x2(finalTime)=0;

end F8_aircraft_opt;

</source>

[[Category:JModelica]]

Fuller's problem

2016-03-15T12:46:50Z

MadeleineSchroeter:

{{Dimensions
|nd = 1
|nx = 2
|nw = 1
|nre = 4
}}

The first control problem with an optimal [[:Category:Chattering|chattering]] solution was given by <bib id="Fuller1963" />. An optimal trajectory does exist for all initial and terminal values in a vicinity of the origin. As Fuller showed, this optimal trajectory contains a bang-bang control function that switches infinitely often.

The mathematical equations form a small-scale [[:Category:ODE model|ODE model]]. The interior point equality conditions fix initial and terminal values of the differential states.

== Mathematical formulation ==

For <math>t \in [t_0, t_f]</math> almost everywhere the mixed-integer optimal control problem is given by

<p>
<math>
\begin{array}{llcl}
\displaystyle \min_{x, w} & \int_{0}^{1} x_0^2 & \; \mathrm{d} t \\[1.5ex]
\mbox{s.t.} & \dot{x}_0 & = & x_1, \\
& \dot{x}_1 & = & 1 - 2 \; w, \\[1.5ex]
& x(0) &=& x_S, \\
& x(t_f) &=& x_T, \\
& w(t) &\in& \{0, 1\}.
\end{array}
</math>
</p>

== Parameters ==

We use <math>x_S = x_T = (0.01, 0)^T</math>.

== Reference Solutions ==

===Solutions obtained with optimica===

The solution found for the relaxed Fuller's problem with optimica using the solver Ipopt (with the linear solver MA27) is obtained with 12 iterations and the objective is 1.5296058259296967e-05.
[[File:Fullerspng.png|left|200px|thumb|alt=a graph with the optimal solution of the Fuller's Problem with Optimica and Ipopt|Solution of the Fuller's Problem with Optimica and Ipopt]]
<br />
<br style="clear:both;">



== Source Code ==

* [[:Category:Muscod | Muscod code]] at [[Fuller's Problem (Muscod)]]
* [[:Category:optimica | optimica]] at [[Fuller's Problem (optimica)]]
* [[:Category:JModelica | JModelica]] at [[Fuller's Problem (JModelica)]]

== Miscellaneous and further reading ==

An extensive analytical investigation of this problem and a discussion of the ubiquity of Fuller's problem can be found in <bib id="Zelikin1994" />, a recent investigation of chattering controls in relay feedback systems in <bib id="Johansson2002" />.

== References ==
<biblist />


[[Category:MIOCP]]
[[Category:ODE model]]
[[Category:Tracking objective]]
[[Category:Chattering]]
[[Category:Bang bang]]

Fuller's Problem (JModelica)

2016-03-15T12:45:36Z

MadeleineSchroeter: Created page with "This page contains the model formulation of the MIOCP Fuller's problem in [http://jmodelica.org JModelica] format. === JModelica === The model in JModelica code. <sour..."

This page contains the model formulation of the MIOCP [[Fuller's problem]] in [http://jmodelica.org JModelica] format.

=== JModelica ===

The model in JModelica code.

<source lang="optimica">
// Fuller's problem for an optimal chattering solution.
// (c) Sebastian Sager, 2005-2009
// More info on http://mintoc.de/index.php/Fuller's_Problem

optimization fuller_opt (objective = cost(finalTime),
startTime = 0,
finalTime = 1)

// Differential states
Real x0(start=0.01, fixed=true);
Real x1(start=0, fixed=true);
Real cost(start=0, fixed=true);

// The control signal
input Real u(free=true);

equation
der(x0) = x1;
der(x1) = 1-2*u;
der(cost) = x0^2;
constraint
u<=1;
u>=0;
x0(finalTime)=0.01;
x1(finalTime)=0;
end fuller_opt;
</source>

[[Category:JModelica]]

F-8 aircraft

2016-03-15T12:38:45Z

MadeleineSchroeter:

{{Dimensions
|nd = 1
|nx = 3
|nw = 1
|nre = 6
}}

The F-8 aircraft control problem is based on a very simple aircraft model. The control problem was introduced by Kaya and Noakes and aims at controlling an aircraft in a time-optimal way from an initial state to a terminal state.

The mathematical equations form a small-scale [[:Category:ODE model|ODE model]]. The interior point equality conditions fix both initial and terminal values of the differential states.

The optimal, relaxed control function shows [[:Category:Bang bang|bang bang]] behavior. The problem is furthermore interesting as it should be reformulated equivalently.

== Mathematical formulation ==

For <math>t \in [0, T]</math> almost everywhere the mixed-integer optimal control problem is given by

<math>
\begin{array}{llcl}
\displaystyle \min_{x, w, T} & T \\[1.5ex]
\mbox{s.t.} & \dot{x}_0 &=& -0.877 \; x_0 + x_2 - 0.088 \; x_0 \; x_2 + 0.47 \; x_0^2 - 0.019 \; x_1^2 - x_0^2 \; x_2 + 3.846 \; x_0^3 \\
&&& - 0.215 \; w + 0.28 \; x_0^2 \; w + 0.47 \; x_0 \; w^2 + 0.63 \; w^3 \\
& \dot{x}_1 &=& x_2 \\
& \dot{x}_2 &=& -4.208 \; x_0 - 0.396 \; x_2 - 0.47 \; x_0^2 - 3.564 \; x_0^3 \\
&&& - 20.967 \; w + 6.265 \; x_0^2 \; w + 46 \; x_0 \; w^2 + 61.4 \; w^3 \\
& x(0) &=& (0.4655,0,0)^T, \\
& x(T) &=& (0,0,0)^T, \\
& w(t) &\in& \{-0.05236,0.05236\}.
\end{array}
</math>

<math>x_0</math> is the angle of attack in radians, <math>x_1</math> is the pitch angle, <math>x_2</math> is the pitch rate in rad/s, and the control function <math>w = w(t)</math> is the tail deflection angle in radians. This model goes back to Garrard<bib id="Garrard1977" />.

In the control problem, both initial and terminal values of the differential states are fixed.

== Reformulation ==

The control w(t) is restricted to take values from a finite set only. Hence, the control problem can be reformulated equivalently to

<math>
\begin{array}{llcl}
\displaystyle \min_{x, w, T} & T \\[1.5ex]
\mbox{s.t.} & \dot{x}_0 &=& -0.877 \; x_0 + x_2 - 0.088 \; x_0 \; x_2 + 0.47 \; x_0^2 - 0.019 \; x_1^2 - x_0^2 \; x_2 + 3.846 \; x_0^3 \\
&&& - \left( 0.215 \; \xi - 0.28 \; x_0^2 \; \xi - 0.47 \; x_0 \; \xi^2 - 0.63 \; \xi^3 \right) \; w \\
&&& - \left( - 0.215 \; \xi + 0.28 \; x_0^2 \; \xi - 0.47 \; x_0 \; \xi^2 + 0.63 \; \xi^3 \right) \; (1 - w) \\
& &=& -0.877 \; x_0 + x_2 - 0.088 \; x_0 \; x_2 + 0.47 \; x_0^2 - 0.019 \; x_1^2 - x_0^2 \; x_2 + 3.846 \; x_0^3 \\
&&& + 0.215 \; \xi - 0.28 \; x_0^2 \; \xi + 0.47 \; x_0 \; \xi^2 - 0.63 \; \xi^3 \\
&&& - \left( 0.215 \; \xi - 0.28 \; x_0^2 \; \xi - 0.63 \; \xi^3 \right) \; 2 w \\
& \dot{x}_1 &=& x_2 \\
& \dot{x}_2 &=& -4.208 \; x_0 - 0.396 \; x_2 - 0.47 \; x_0^2 - 3.564 \; x_0^3 \\
&&& - \left( 20.967 \; \xi - 6.265 \; x_0^2 \; \xi -46 \; x_0 \; \xi^2 - 61.4 \; \xi^3 \right) \; w \\
&&& - \left( - 20.967 \; \xi + 6.265 \; x_0^2 \; \xi -46 \; x_0 \; \xi^2 + 61.4 \; \xi^3 \right) \; (1 - w) \\
& &=& -4.208 \; x_0 - 0.396 \; x_2 - 0.47 \; x_0^2 - 3.564 \; x_0^3 \\
&&& + 20.967 \; \xi - 6.265 \; x_0^2 \; \xi + 46 \; x_0 \; \xi^2 - 61.4 \; \xi^3 \\
&&& - \left( 20.967 \; \xi - 6.265 \; x_0^2 \; \xi - 61.4 \; \xi^3 \right) \; 2 w \\
& x(0) &=& (0.4655,0,0)^T, \\
& x(T) &=& (0,0,0)^T, \\
& w(t) &\in& \{0,1\},
\end{array}
</math>

with <math>\xi = 0.05236</math>. Note that there is a bijection between optimal solutions of the two problems.

== Reference solutions ==

We provide here a comparison of different solutions reported in the literature. The numbers show the respective lengths <math>t_i - t_{i-1}</math> of the switching arcs with the value of <math>w(t)</math> on the upper or lower bound (given in the second column). ''Claim'' denotes what is stated in the respective publication, ''Simulation'' shows values obtained by a simulation with a Runge-Kutta-Fehlberg method of 4th/5th order and an integration tolerance of <math>10^{-8}</math>.

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Arc !! w(t) !! Lee et al.<bib id="Lee1997a" /> !! Kaya<bib id="Kaya2003" /> !! Sager<bib id="Sager2005" /> !! [[User:MartinSchlueter | Schlueter]]/[[User:Matthias.gerdts | Gerdts]] !! Sager
|-
| align=center | 1 || align=center | 1 || 0.00000 || 0.10292 || 0.10235 || 0.0 || 1.13492
|-
| align=center | 2 || align=center | 0 || 2.18800 || 1.92793 || 1.92812 || 0.608750 || 0.34703
|-
| align=center | 3 || align=center | 1 || 0.16400 || 0.16687 || 0.16645 || 3.136514 || 1.60721
|-
| align=center | 4 || align=center | 0 || 2.88100 || 2.74338 || 2.73071 || 0.654550 || 0.69169
|-
| align=center | 5 || align=center | 1 || 0.33000 || 0.32992 || 0.32994 || 0.0 || 0.0
|-
| align=center | 6 || align=center | 0 || 0.47200 || 0.47116 || 0.47107 || 0.0 || 0.0
|-
| Claim: Infeasibility || align=center | - || 1.00E-10 || 7.30E-06 || 5.90E-06 || 3.29169e-06 || 2.21723e-07
|-
| Claim: Objective || align=center | - || 6.03500 || 5.74217 || 5.72864 || 4.39981 || 3.78086
|- style="font-style:italic;background-color:#ffffcc;"
! Simulation: Infeasibility || align=center | - || 1.75E-03 || 1.64E-03 || 5.90E-06 || 3.29169e-06 || 2.21723e-07
|- style="font-style:italic;background-color:#ffffcc;"
! Simulation: Objective || align=center | - || 6.03500 || 5.74218 || 5.72864 || 4.39981 || 3.78086
|}

The best known optimal objective value of this problem given is given by <math>T = 3.78086</math>. The corresponding solution is shown in the rightmost plot. The solution of bang-bang type switches three times, starting with <math>w(t) = 1</math>.

<gallery caption="Reference solution plots" widths="240px" heights="167px" perrow="3">
Image:f8aircraftRelaxedCoarse.png| Locally optimal relaxed control on a coarse control discretization grid and corresponding differential states.
Image:f8aircraftRelaxedFine.png| Locally optimal relaxed control on a fine, adaptively chosen control discretization grid and corresponding differential states.
Image:f8aircraftInteger.png| Integer control and corresponding differential states from Sager<bib id="Sager2005" />.

Image:f8aircraftSchlueter.png| Integer control and corresponding differential states from [[User:MartinSchlueter | Schlueter]]/[[User:Matthias.gerdts | Gerdts]] solution.
Image:f8aircraftSager2009.png| Best known integer control and corresponding differential states from Sager solution.
</gallery>

=== Optimica ===

Objective : 5.12799232
infeasibility : 6.2235588037251599e-10

<gallery caption="Obtained solution plots" widths="240px" heights="167px" perrow="3">
Image:f8aircraft1.png| (Probably sub-)Optimal control.
Image:f8aircraft2.png| Corresponding differential states.

</gallery>

== Source Code ==

Model descriptions are available in

* [[:Category:AMPL | AMPL]] at [[F-8 aircraft (AMPL)]]
* [[:Category:Muscod | Muscod code]] at [[F-8 aircraft (Muscod)]]
* [[:Category:optimica | optimica]] at [[F-8 aircraft (optimica)]]
* [[:Category:JModelica | JModelica]] at [[F-8 aircraft (JModelica)]]

== Variants ==

* a prescribed time grid for the control function, see [[F-8 aircraft (AMPL)]],

== Miscellaneous and Further Reading ==
See <bib id="Kaya2003" /> and <bib id="Sager2005" /> for details.

== References ==
<biblist />


[[Category:Bang bang]]
[[Category:MIOCP]]
[[Category:ODE model]]
[[Category:Aeronautics]]
[[Category:Outer convexification]]
[[Category: Minimum time]]

F-8 aircraft (JModelica)

2016-03-15T12:29:10Z

MadeleineSchroeter: Created page with "This page contains the model formulation of the MIOCP F-8 aircraft in [http://jmodelica.org JModelica] format. === JModelica === The model in JModelica code. <source l..."

Lotka Volterra fishing problem (JModelica)

2016-01-18T10:17:55Z

MadeleineSchroeter:

This page contains the model formulation of the MIOCP Lotka Volterra fishing problem in JModelica format.

== JModelica ==

The model for compilation with JModelica.

<source lang="optimica">
//--------------------------------------------------------------------
//Lotka Volterra Fishing Problem
// (c) Madeleine Schroeter
//--------------------------------------------------------------------

optimization lotka(objective = cost(finalTime), startTime = 0, finalTime = 12)
"Steady State Solution with u=0"

Real cost(start=0, min=0, max=25) "Integrated Deviation";

// Differential state variables
Real x0(min=0, max=20.0) "Biomass of Prey";
Real x1(min=0, max=20.0) "Biomass of Predator";

// Control functions
input Real u(free=true, min=0,max=1);

constant Real ref0 = 1.0 "Steady State Prey";
constant Real ref1 = 1.0 "Steady State Predator";

equation
der(x0) = x0 - x0*x1 - 0.4*x0*u;
der(x1) = -x1 + x0*x1 - 0.2*x1*u;
der(cost) = (x0 - ref0)*(x0 - ref0) + (x1 - ref1)*(x1 - ref1); // Quadratic deviation

constraint
x0(0)=0.5;
x1(0)=0.7;
end lotka;

</source>

The Python run file.

<source lang="python">
#Import the function for transfering a model to CasADiInterface
from pyjmi import transfer_optimization_problem

op=transfer_optimization_problem("lotka", "lotka.mop")

res=op.optimize()

</source>

[[Category:JModelica]]

Lotka Volterra fishing problem (JModelica)

2016-01-18T10:17:00Z

MadeleineSchroeter:

This page contains the model formulation of the MIOCP Lotka Volterra fishing problem in JModelica format.

== JModelica ==

The model for compilation with JModelica.

<source lang="optimica">
//--------------------------------------------------------------------
//Lotka Volterra Fishing Problem
// (c) Madeleine Schroeter
//--------------------------------------------------------------------

optimization lotka(objective = cost(finalTime), startTime = 0, finalTime = 12)
"Steady State Solution with u=0"

Real cost(start=0, min=0, max=25) "Integrated Deviation";

// Differential state variables
Real x0(min=0, max=20.0) "Biomass of Prey";
Real x1(min=0, max=20.0) "Biomass of Predator";

// Control functions
input Real u(free=true, min=0,max=1);

constant Real ref0 = 1.0 "Steady State Prey";
constant Real ref1 = 1.0 "Steady State Predator";

equation
der(x0) = x0 - x0*x1 - 0.4*x0*u;
der(x1) = -x1 + x0*x1 - 0.2*x1*u;
der(cost) = (x0 - ref0)*(x0 - ref0) + (x1 - ref1)*(x1 - ref1); // Quadratic deviation

constraint
x0(0)=0.5;
x1(0)=0.7;
end lotka;

</source>

The run file.

<source lang="python">
#Import the function for transfering a model to CasADiInterface
from pyjmi import transfer_optimization_problem

op=transfer_optimization_problem("lotka", "lotka.mop")

res=op.optimize()

</source>

[[Category:JModelica]]

Lotka Volterra fishing problem (JModelica)

2016-01-18T10:15:00Z

MadeleineSchroeter: Created page with "This page contains the model formulation of the MIOCP Lotka Volterra fishing problem in JModelica format. == JModelica == The model for compilation with JModelica. <source..."

Van der Pol Oscillator (JModelica)

2016-01-12T19:57:01Z

MadeleineSchroeter:

This page contains the source code to solve the Van der Pol Oscillator problem with JModelica. The automatic differentiation tool CasADI and the solver IPOPT were used to solve the problem.

Model file (VDP_Opt.mop)
<source lang="Optimica">
//-------------------------------------------------------------------------
//Van der Pol Oscillator with direct collocation using JModelica
//(c) Madeleine Schroter
//--------------------------------------------------------------------------

optimization VDP_Opt(objectiveIntegrand = y^2+x^2+u^2, startTime = 0, finalTime = 20)

//The states
Real x(start=1, fixed=true); //position coordinate
Real y(start=0, fixed=true);

//The control signal
input Real u; //damping of the oscillation

equation
der(y) = (1-x^2) * y - x +u;
der(x) = y;

constraint
u<=0.75;

end VDP_Opt;

</source>

Run file
<source lang ="Python">
#Import the function for transfering a model to CasADiInterface
from pyjmi import transfer_optimization_problem

op=transfer_optimization_problem("VDP_Opt", "VDP_Opt.mop")

res=op.optimize()

</source>

[[Category:JModelica]]

Van der Pol Oscillator (JModelica)

2016-01-12T19:56:11Z

MadeleineSchroeter:

This page contains the source code to solve the Van der Pol Oscillator problem with JModelica. The automatic differentiation tool CasADI and the solver IPOPT were used to solve the problem.

Model file (VDP_Opt.mop)
<source lang="Python">
//-------------------------------------------------------------------------
//Van der Pol Oscillator with direct collocation using JModelica
//(c) Madeleine Schroter
//--------------------------------------------------------------------------

optimization VDP_Opt(objectiveIntegrand = y^2+x^2+u^2, startTime = 0, finalTime = 20)

//The states
Real x(start=1, fixed=true); //position coordinate
Real y(start=0, fixed=true);

//The control signal
input Real u; //damping of the oscillation

equation
der(y) = (1-x^2) * y - x +u;
der(x) = y;

constraint
u<=0.75;

end VDP_Opt;

</source>

Run file
<source lang ="Python">
#Import the function for transfering a model to CasADiInterface
from pyjmi import transfer_optimization_problem

op=transfer_optimization_problem("VDP_Opt", "VDP_Opt.mop")

res=op.optimize()

</source>

[[Category:JModelica]]

Van der Pol Oscillator

2016-01-12T18:40:11Z

MadeleineSchroeter:

The Van der Pol Oscillator is an oscillating system with non-linear damping and self regulation. The System was first introduced by the Dutch physician Balthasar Van der Pol in 1927. The aim is to control the oscillation such that the system stays in a mean position.

== Model formulation ==

The Van der Pol Oscillator evolves over time according to the second order differential equation:

<math>{d^2x \over dt^2}-u(1-x^2){dx \over dt}+x= 0</math>

where <math>x</math> is the position coordinate, which is a function of the time <math>t</math>, and <math>u</math> is a scalar parameter indicating the non-linearity and the strength of the damping.

For <math>u>>1</math> the oscillator is being damped, whereas for <math>u<<1</math> energy is added to the system.

Based on the transformation <math>y = \dot x</math> the problem can be reformulated:

<math> \dot x = y</math>

<math>\dot y = u(1-x^2) y-x</math>

The Optimal Control Problem arises by adding the objective function:

<math>\min\limits_{u}\int\limits_{t_0}^{t_f}(x(t)^2+y(t)^2+u(t)^2) dt</math>

== Optimal Control Problem ==
The Optimal Control Problem with the aim to minimize the deflection can be formulated as follows:

<math> \begin{array}{ll}
\min\limits_{u} & \int\limits_{t_0}^{t_f}(x(t)^2+y(t)^2+u(t)^2) dt\\
s.t. & \dot x = y\\
& \dot y = u(1-x^2) y-x\\
& x(0)=1\\
& y(0)=0\\
& u\le 0.75\\

\end{array}</math>

== Parameters ==
These fixed values are used within the model:

<math>[t_0,t_f]=[0,20]</math>

== Reference Solution ==

The following reference solution was generated using JModelica with the automatic differentiation tool CasADI and the solver IPOPT. The Optimica code used to solve the problem can be found under [[Van der Pol Oscillator (JModelica)]] The optimal value of the objective function is 3.1762.

<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
Image:VDP_Plot_control.png| Control u over time (damping).
Image:VDP_Plot_states.png| Position coordinate x and it's derivative y.
Image:VDP_Plot_derivatives.png| Derivative <math>\dot x</math> and second derivative <math>\dot y</math>.
</gallery>

== Source Code ==
Model descriptions are available in:

[[Category:JModelica]] [[Van der Pol Oscillator (JModelica)]]

== References ==
The Problem can be found under the following [https://en.wikipedia.org/wiki/Van_der_Pol_oscillator link] or in the [http://www.jmodelica.org/api-docs/usersguide/1.4.0/ch08s02.html JModelica Users Guide].

[[Category:MIOCP]]

Van der Pol Oscillator

2016-01-12T18:37:00Z

MadeleineSchroeter:

The Van der Pol Oscillator is an oscillating system with non-linear damping and self regulation. The System was first introduced by the Dutch physician Balthasar Van der Pol in 1927. The aim is to control the oscillation such that the system stays in a mean position.

== Model formulation ==

The Van der Pol Oscillator evolves over time according to the second order differential equation:

<math>{d^2x \over dt^2}-u(1-x^2){dx \over dt}+x= 0</math>

where <math>x</math> is the position coordinate, which is a function of the time <math>t</math>, and <math>u</math> is a scalar parameter indicating the non-linearity and the strength of the damping.

For <math>u>>1</math> the oscillator is being damped, whereas for <math>u<<1</math> energy is added to the system.

Based on the transformation <math>y = \dot x</math> the problem can be reformulated:

<math> \dot x = y</math>

<math>\dot y = u(1-x^2) y-x</math>

The Optimal Control Problem arises by adding the objective function:

<math>\min\limits_{u}\int\limits_{t_0}^{t_f}(x(t)^2+y(t)^2+u(t)^2) dt</math>

== Optimal Control Problem ==
The Optimal Control Problem with the aim to minimize the deflection can be formulated as follows:

<math> \begin{array}{ll}
\min\limits_{u} & \int\limits_{t_0}^{t_f}(x(t)^2+y(t)^2+u(t)^2) dt\\
s.t. & \dot x = y\\
& \dot y = u(1-x^2) y-x\\
& x(0)=1\\
& y(0)=0\\
& u\le 0.75\\

\end{array}</math>

== Parameters ==
These fixed values are used within the model:

<math>[t_0,t_f]=[0,20]</math>

== Reference Solution ==

The following reference solution was generated using JModelica with the automatic differentiation tool CasADI and the solver IPOPT. The Optimica code used to solve the problem can be found under [[Van der Pol Oscillator (JModelica)]] The optimal value of the objective function is 3.1762.

<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
Image:VDP_Plot_control.png| Control u over time (damping).
Image:VDP_Plot_states.png| Position coordinate x and it's derivative y.
Image:VDP_Plot_derivatives.png| Derivative <math>\dot x</math> and second derivative <math>\dot y</math>.
</gallery>

== Source Code ==
Model descriptions are available in:

[[Category:JModelica]] [[Van der Pol Oscillator (JModelica)]]

== References ==
[https://en.wikipedia.org/wiki/Van_der_Pol_oscillator]

[[Category:MIOCP]]

Van der Pol Oscillator

2016-01-12T15:09:42Z

MadeleineSchroeter:

Van der Pol Oscillator (JModelica)

2016-01-12T13:29:41Z

MadeleineSchroeter:

This page contains the source code to solve the Van der Pol Oscillator problem with JModelica. The automatic differentiation tool CasADI and the solver IPOPT were used to solve the problem.

<source lang="Python">
//-------------------------------------------------------------------------
//Van der Pol Oscillator with direct collocation using JModelica
//(c) Madeleine Schroter
//--------------------------------------------------------------------------

optimization VDP_Opt(objectiveIntegrand = y^2+x^2+u^2, startTime = 0, finalTime = 20)

//The states
Real x(start=1, fixed=true); //position coordinate
Real y(start=0, fixed=true);

//The control signal
input Real u; //damping of the oscillation

equation
der(y) = (1-x^2) * y - x +u;
der(x) = y;

constraint
u<=0.75;

end VDP_Opt;

</source>

[[Category:JModelica]]

Category:JModelica

2016-01-12T13:25:20Z

MadeleineSchroeter:

JModelica.org is an extensible Modelica-based open source platform for optimization, simulation and analysis of
complex dynamic systems. JModelica supports the modeling language Modelica and it's extension Optimica, for more information see [http://www.jmodelica.org/]

== References ==
<bibreferences/>


[[Category:Problem characterization]]

Category:JModelica

2016-01-12T13:24:53Z

MadeleineSchroeter:

JModelica.org is an extensible Modelica-based open source platform for optimization, simulation and analysis of
complex dynamic systems. JModelica supports the modeling language Modelica and it's extension Optimica, for more inforation see [http://www.jmodelica.org/]

== References ==
<bibreferences/>


[[Category:Problem characterization]]

Category:JModelica

2016-01-12T13:24:40Z

MadeleineSchroeter:

[JModelica.org is an extensible Modelica-based open source platform for optimization, simulation and analysis of
complex dynamic systems. JModelica supports the modeling language Modelica and it's extension Optimica, for more inforation see [http://www.jmodelica.org/]]

== References ==
<bibreferences/>


[[Category:Problem characterization]]

Van der Pol Oscillator

2016-01-12T13:19:50Z

MadeleineSchroeter:

Van der Pol Oscillator

2016-01-12T13:19:23Z

MadeleineSchroeter:

Van der Pol Oscillator (JModelica)

2016-01-12T13:18:27Z

MadeleineSchroeter:

This page contains the source code to solve the Van der Pol Oscillator problem with JModelica. The automatic differentiation tool CasADI and the solver IPOPT were used to solve the problem.

<source lang="Python">
#-------------------------------------------------------------------------
#Van der Pol Oscillator with direct collocation using JModelica
#(c) Madeleine Schroter
#--------------------------------------------------------------------------

optimization VDP_Opt(objectiveIntegrand = y^2+x^2+u^2, startTime = 0, finalTime = 20)

//The states
Real x(start=1, fixed=true); //position coordinate
Real y(start=0, fixed=true);

//The control signal
input Real u; //damping of the oscillation

equation
der(y) = (1-x^2) * y - x +u;
der(x) = y;

constraint
u<=0.75;

end VDP_Opt;

</source>

[[Category:JModelica]]

Van der Pol Oscillator (JModelica)

2016-01-12T12:53:14Z

MadeleineSchroeter: Created page with "This page contains the source code to solve the Van der Pol Oscillator problem with JModelica. The automatic differentiation tool CasADI and the solver IPOPT were used to solv..."

This page contains the source code to solve the Van der Pol Oscillator problem with JModelica. The automatic differentiation tool CasADI and the solver IPOPT were used to solve the problem.

<source lang="Python">
#-------------------------------------------------------------------------
#Van der Pol Oscillator with direct collocation using JModelica
#(c) Madeleine Schroter
#--------------------------------------------------------------------------

optimization VDP_Opt(objectiveIntegrand = y^2+x^2+u^2, startTime = 0, finalTime = 20)

//The states
Real x(start=1, fixed=true); //position coordinate
Real y(start=0, fixed=true);

//The control signal
input Real u; //damping of the oscillation

equation
der(y) = (1-x^2) * y - x +u;
der(x) = y;

constraint
u<=0.75;

end VDP_Opt;

</source>

Van der Pol Oscillator

2016-01-12T12:35:31Z

MadeleineSchroeter:

The Van der Pol Oscillator is an oscillating system with non-linear damping and self regulation. The System was first introduced by the Dutch physician Balthasar Van der Pol in 1927. The aim is to control the oscillation such that the system stays in a mean position.

== Model formulation ==

The Van der Pol Oscillator evolves over time according to the second order differential equation:

<math>{d^2x \over dt^2}-u(1-x^2){dx \over dt}+x= 0</math>

where <math>x</math> is the position coordinate, which is a function of the time <math>t</math>, and <math>u</math> is a scalar parameter indicating the non-linearity and the strength of the damping.

For <math>u>>1</math> the oscillator is being damped, whereas for <math>u<<1</math> energy is added to the system.

Based on the transformation <math>y = \dot x</math> the problem can be reformulated:

<math> \dot x = y</math>

<math>\dot y = u(1-x^2) y-x</math>

The Optimal Control Problem arises by adding the objective function:

<math>\min\limits_{u}\int\limits_{t_0}^{t_f}(x(t)^2+y(t)^2+u(t)^2) dt</math>

== Optimal Control Problem ==
The Optimal Control Problem with the aim to minimize the deflection can be formulated as follows:

<math> \begin{array}{ll}
\min\limits_{u} & \int\limits_{t_0}^{t_f}(x(t)^2+y(t)^2+u(t)^2) dt\\
s.t. & \dot x = y\\
& \dot y = u(1-x^2) y-x\\
& x(0)=1\\
& y(0)=0\\
& u\le 0.75\\

\end{array}</math>

== Parameters ==
These fixed values are used within the model:

<math>[t_0,t_f]=[0,20]</math>

== Reference Solution ==

The following reference solution was generated using JModelica with the automatic differentiation tool CasADI and the solver IPOPT. The Optimica code used to solve the problem can be found under [[Van der Pol Oscillator (JModelica)]] The optimal value of the objective function is 3.1762.

<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
Image:VDP_Plot_control.png| Control u over time (damping).
Image:VDP_Plot_states.png| Position coordinate x and it's derivative y.
Image:VDP_Plot_derivatives.png| Derivative <math>\dot x</math> and second derivative <math>\dot y</math>.
</gallery>

== Source Code ==
Model descriptions are available in:

[[JModelica]] at [[Van der Pol Oscillator (JModelica)]]

Van der Pol Oscillator

2016-01-12T12:27:34Z

MadeleineSchroeter:

The Van der Pol Oscillator is an oscillating system with non-linear damping and self regulation. The System was first introduced by the Dutch physician Balthasar Van der Pol in 1927. The aim is to control the oscillation such that the system stays in a mean position.

== Model formulation ==

The Van der Pol Oscillator evolves over time according to the second order differential equation:

<math>{d^2x \over dt^2}-u(1-x^2){dx \over dt}+x= 0</math>

where <math>x</math> is the position coordinate, which is a function of the time <math>t</math>, and <math>u</math> is a scalar parameter indicating the non-linearity and the strength of the damping.

For <math>u>>1</math> the oscillator is being damped, whereas for <math>u<<1</math> energy is added to the system.

Based on the transformation <math>y = \dot x</math> the problem can be reformulated:

<math> \dot x = y</math>

<math>\dot y = u(1-x^2) y-x</math>

The Optimal Control Problem arises by adding the objective function:

<math>\min\limits_{u}\int\limits_{t_0}^{t_f}(x(t)^2+y(t)^2+u(t)^2) dt</math>

== Optimal Control Problem ==
The Optimal Control Problem with the aim to minimize the deflection can be formulated as follows:

<math> \begin{array}{ll}
\min\limits_{u} & \int\limits_{t_0}^{t_f}(x(t)^2+y(t)^2+u(t)^2) dt\\
s.t. & \dot x = y\\
& \dot y = u(1-x^2) y-x\\
& x(0)=1\\
& y(0)=0\\
& u\le 0.75\\

\end{array}</math>

== Parameters ==
These fixed values are used within the model:

<math>[t_0,t_f]=[0,20]</math>

== Reference Solution ==

The following reference solution was generated using JModelica with the automatic differentiation tool CasADI and the solver IPOPT. The Optimica code used to solve the problem can be found under [[Van der Pol Oscillator (JModelica)]] The optimal value of the objective function is 3.1762.
[[File:VDP_Plot_control.png|200px|thumb|left|control u]][[File:VDP_Plot_states.png|200px|thumb|left|states x and y]] [[File:VDP_Plot_derivatives.png|200px|thumb|right|derivatives <math>\dot x</math> and <math>\dot y</math> ]]

== Source Code ==
Model descriptions are available in:

[[JModelica]] at [[Van der Pol Oscillator (JModelica)]]

Van der Pol Oscillator

2016-01-12T12:25:32Z

MadeleineSchroeter:

The Van der Pol Oscillator is an oscillating system with non-linear damping and self regulation. The System was first introduced by the Dutch physician Balthasar Van der Pol in 1927. The aim is to control the oscillation such that the system stays in a mean position.

== Model formulation ==

The Van der Pol Oscillator evolves over time according to the second order differential equation:

<math>{d^2x \over dt^2}-u(1-x^2){dx \over dt}+x= 0</math>

where <math>x</math> is the position coordinate, which is a function of the time <math>t</math>, and <math>u</math> is a scalar parameter indicating the non-linearity and the strength of the damping.

For <math>u>>1</math> the oscillator is being damped, whereas for <math>u<<1</math> energy is added to the system.

Based on the transformation <math>y = \dot x</math> the problem can be reformulated:

<math> \dot x = y</math>

<math>\dot y = u(1-x^2) y-x</math>

The Optimal Control Problem arises by adding the objective function:

<math>\min\limits_{u}\int\limits_{t_0}^{t_f}(x(t)^2+y(t)^2+u(t)^2) dt</math>

== Optimal Control Problem ==
The Optimal Control Problem with the aim to minimize the deflection can be formulated as follows:

<math> \begin{array}{ll}
\min\limits_{u} & \int\limits_{t_0}^{t_f}(x(t)^2+y(t)^2+u(t)^2) dt\\
s.t. & \dot x = y\\
& \dot y = u(1-x^2) y-x\\
& x(0)=1\\
& y(0)=0\\
& u\le 0.75\\

\end{array}</math>

== Parameters ==
These fixed values are used within the model:

<math>[t_0,t_f]=[0,20]</math>

== Reference Solution ==

The following reference solution was generated using JModelica with the automatic differentiation tool CasADI and the solver IPOPT. The Optimica code used to solve the problem can be found under [[Van der Pol Oscillator (JModelica)]] The optimal value of the objective function is 3.1762.
[[File:VDP_Plot_control.png|200px|thumb|left|control u]][[File:VDP_Plot_states.png|200px|thumb|left|states x and y]] [[File:VDP_Plot_derivatives.png|200px|thumb|left|derivatives <math>\dot x</math> and <math>\dot y</math> ]]

== Source Code ==
Model descriptions are available in:

[[JModelica]] at [[Van der Pol Oscillator (JModelica)]]

Van der Pol Oscillator

2016-01-12T12:25:08Z

MadeleineSchroeter:

The Van der Pol Oscillator is an oscillating system with non-linear damping and self regulation. The System was first introduced by the Dutch physician Balthasar Van der Pol in 1927. The aim is to control the oscillation such that the system stays in a mean position.

== Model formulation ==

The Van der Pol Oscillator evolves over time according to the second order differential equation:

<math>{d^2x \over dt^2}-u(1-x^2){dx \over dt}+x= 0</math>

where <math>x</math> is the position coordinate, which is a function of the time <math>t</math>, and <math>u</math> is a scalar parameter indicating the non-linearity and the strength of the damping.

For <math>u>>1</math> the oscillator is being damped, whereas for <math>u<<1</math> energy is added to the system.

Based on the transformation <math>y = \dot x</math> the problem can be reformulated:

<math> \dot x = y</math>

<math>\dot y = u(1-x^2) y-x</math>

The Optimal Control Problem arises by adding the objective function:

<math>\min\limits_{u}\int\limits_{t_0}^{t_f}(x(t)^2+y(t)^2+u(t)^2) dt</math>

== Optimal Control Problem ==
The Optimal Control Problem with the aim to minimize the deflection can be formulated as follows:

<math> \begin{array}{ll}
\min\limits_{u} & \int\limits_{t_0}^{t_f}(x(t)^2+y(t)^2+u(t)^2) dt\\
s.t. & \dot x = y\\
& \dot y = u(1-x^2) y-x\\
& x(0)=1\\
& y(0)=0\\
& u\le 0.75\\

\end{array}</math>

== Parameters ==
These fixed values are used within the model:

<math>[t_0,t_f]=[0,20]</math>

== Reference Solution ==

The following reference solution was generated using JModelica with the automatic differentiation tool CasADI and the solver IPOPT. The Optimica code used to solve the problem can be found under [[Van der Pol Oscillator (JModelica)]] The optimal value of the objective function is 3.1762.
[[File:VDP_Plot_control.png|200px|thumb|left|control u]] [[File:VDP_Plot_states.png|200px|thumb|left|states x and y]] [[File:VDP_Plot_derivatives.png|200px|thumb|left|derivatives <math>\dot x</math> and <math>\dot y</math> ]]

== Source Code ==
Model descriptions are available in:

[[JModelica]] at [[Van der Pol Oscillator (JModelica)]]

Van der pol oscillator

2016-01-12T12:23:03Z

MadeleineSchroeter: MadeleineSchroeter moved page Van der pol oscillator to Van der Pol Oscillator

#REDIRECT [[Van der Pol Oscillator]]

Van der Pol Oscillator

2016-01-12T12:23:03Z

MadeleineSchroeter: MadeleineSchroeter moved page Van der pol oscillator to Van der Pol Oscillator

The Van der Pol Oscillator is an oscillating system with non-linear damping and self regulation. The System was first introduced by the Dutch physician Balthasar Van der Pol in 1927. The aim is to control the oscillation such that the system stays in a mean position.

== Model formulation ==

The Van der Pol Oscillator evolves over time according to the second order differential equation:

<math>{d^2x \over dt^2}-u(1-x^2){dx \over dt}+x= 0</math>

where <math>x</math> is the position coordinate, which is a function of the time <math>t</math>, and <math>u</math> is a scalar parameter indicating the non-linearity and the strength of the damping.

For <math>u>>1</math> the oscillator is being damped, whereas for <math>u<<1</math> energy is added to the system.

Based on the transformation <math>y = \dot x</math> the problem can be reformulated:

<math> \dot x = y</math>

<math>\dot y = u(1-x^2) y-x</math>

The optimal control problem arises by adding the objective function:

<math>\min\limits_{u}\int\limits_{t_0}^{t_f}(x(t)^2+y(t)^2+u(t)^2) dt</math>

== Optimal Control Problem ==
The Optimal Control Problem with the aim to minimize the deflection can be formulated as follows:

<math> \begin{array}{ll}
\min\limits_{u} & \int\limits_{t_0}^{t_f}(x(t)^2+y(t)^2+u(t)^2) dt\\
s.t. & \dot x = y\\
& \dot y = u(1-x^2) y-x\\
& x(0)=1\\
& y(0)=0\\
& u\le 0.75\\

\end{array}</math>

== Parameters ==
These fixed values are used within the model:

<math>[t_0,t_f]=[0,20]</math>

== Reference Solution ==

The following reference solution was generated using JModelica with the automatic differentiation tool CasADI and the solver IPOPT. The Optimica code used to solve the problem can be found under.... The optimal value of the objective function is 3.1762.
[[File:VDP_Plot_control.png|200px|thumb|left|control u]] [[File:VDP_Plot_states.png|200px|thumb|left|states x and y]] [[File:VDP_Plot_derivatives.png|200px|thumb|left|derivatives <math>\dot x</math> and <math>\dot y</math> ]]

== Source Code ==
Model descriptions are available in:

[[JModelica]] at [[Van der Pol oscillator (JModelica)]]

Van der Pol Oscillator

2016-01-12T12:16:25Z

MadeleineSchroeter:

The Van der Pol Oscillator is an oscillating system with non-linear damping and self regulation. The System was first introduced by the Dutch physician Balthasar Van der Pol in 1927. The aim is to control the oscillation such that the system stays in a mean position.

== Model formulation ==

The Van der Pol Oscillator evolves over time according to the second order differential equation:

<math>{d^2x \over dt^2}-u(1-x^2){dx \over dt}+x= 0</math>

where <math>x</math> is the position coordinate, which is a function of the time <math>t</math>, and <math>u</math> is a scalar parameter indicating the non-linearity and the strength of the damping.

For <math>u>>1</math> the oscillator is being damped, whereas for <math>u<<1</math> energy is added to the system.

Based on the transformation <math>y = \dot x</math> the problem can be reformulated:

<math> \dot x = y</math>

<math>\dot y = u(1-x^2) y-x</math>

The optimal control problem arises by adding the objective function:

<math>\min\limits_{u}\int\limits_{t_0}^{t_f}(x(t)^2+y(t)^2+u(t)^2) dt</math>

== Optimal Control Problem ==
The Optimal Control Problem with the aim to minimize the deflection can be formulated as follows:

<math> \begin{array}{ll}
\min\limits_{u} & \int\limits_{t_0}^{t_f}(x(t)^2+y(t)^2+u(t)^2) dt\\
s.t. & \dot x = y\\
& \dot y = u(1-x^2) y-x\\
& x(0)=1\\
& y(0)=0\\
& u\le 0.75\\

\end{array}</math>

== Parameters ==
These fixed values are used within the model:

<math>[t_0,t_f]=[0,20]</math>

== Reference Solution ==

The following reference solution was generated using JModelica with the automatic differentiation tool CasADI and the solver IPOPT. The Optimica code used to solve the problem can be found under.... The optimal value of the objective function is 3.1762.
[[File:VDP_Plot_control.png|200px|thumb|left|control u]] [[File:VDP_Plot_states.png|200px|thumb|left|states x and y]] [[File:VDP_Plot_derivatives.png|200px|thumb|left|derivatives <math>\dot x</math> and <math>\dot y</math> ]]

== Source Code ==
Model descriptions are available in:

[[JModelica]] at [[Van der Pol oscillator (JModelica)]]

Van der Pol Oscillator

2016-01-12T12:14:37Z

MadeleineSchroeter:

The Van der Pol Oscillator is an oscillating system with non-linear damping and self regulation. The System was first introduced by the Dutch physician Balthasar Van der Pol in 1927. The aim is to control the oscillation such that the system stays in a mean position.

== Model formulation ==

The Van der Pol Oscillator evolves over time according to the second order differential equation:

<math>{d^2x \over dt^2}-u(1-x^2){dx \over dt}+x= 0</math>

where <math>x</math> is the position coordinate, which is a function of the time <math>t</math>, and <math>u</math> is a scalar parameter indicating the non-linearity and the strength of the damping.

For <math>u>>1</math> the oscillator is being damped, whereas for <math>u<<1</math> energy is added to the system.

Based on the transformation <math>y = \dot x</math> the problem can be reformulated:

<math> \dot x = y</math>

<math>\dot y = u(1-x^2) y-x</math>

The optimal control problem arises by adding the objective function:

<math>\min\limits_{u}\int\limits_{t_0}^{t_f}(x(t)^2+y(t)^2+u(t)^2) dt</math>

== Optimal Control Problem ==
The Optimal Control Problem with the aim to minimize the deflection can be formulated as follows:

<math> \begin{array}{ll}
\min\limits_{u} & \int\limits_{t_0}^{t_f}(x(t)^2+y(t)^2+u(t)^2) dt\\
s.t. & \dot x = y\\
& \dot y = u(1-x^2) y-x\\
& x(0)=1\\
& y(0)=0\\
& u\le 0.75\\

\end{array}</math>

== Parameters ==
These fixed values are used within the model:

<math>[t_0,t_f]=[0,20]</math>

== Reference Solution ==

The following reference solution was generated using JModelica with the automatic differentiation tool CasADI and the solver IPOPT. The Optimica code used to solve the problem can be found under.... The optimal value of the objective function is 3.1762.
[[File:VDP_Plot_control.png|200px|thumb|left|control u]] [[File:VDP_Plot_states.png|200px|thumb|left|states x and y]] [[File:VDP_Plot_derivatives.png|200px|thumb|left|derivatives <math>\dot x</math> and <math>\dot y</math> ]]

== Source Code ==
Model descriptions are available in:
\begin{itemize}
\item [[JModelica]] at [[Van der Pol oscillator (JModelica)]]
\end{itemize}

Van der Pol Oscillator

2016-01-12T12:09:05Z

MadeleineSchroeter:

The Van der Pol Oscillator is an oscillating system with non-linear damping and self regulation. The System was first introduced by the Dutch physician Balthasar Van der Pol in 1927. The aim is to control the oscillation such that the system stays in a mean position.

== Model formulation ==

The Van der Pol Oscillator evolves over time according to the second order differential equation:

<math>{d^2x \over dt^2}-u(1-x^2){dx \over dt}+x= 0</math>

where <math>x</math> is the position coordinate, which is a function of the time <math>t</math>, and <math>u</math> is a scalar parameter indicating the non-linearity and the strength of the damping.

For <math>u>>1</math> the oscillator is being damped, whereas for <math>u<<1</math> energy is added to the system.

Based on the transformation <math>y = \dot x</math> the problem can be reformulated:

<math> \dot x = y</math>

<math>\dot y = u(1-x^2) y-x</math>

The optimal control problem arises by adding the objective function:

<math>\min\limits_{u}\int\limits_{t_0}^{t_f}(x(t)^2+y(t)^2+u(t)^2) dt</math>

== Optimal Control Problem ==
The Optimal Control Problem with the aim to minimize the deflection can be formulated as follows:

<math> \begin{array}{ll}
\min\limits_{u} & \int\limits_{t_0}^{t_f}(x(t)^2+y(t)^2+u(t)^2) dt\\
s.t. & \dot x = y\\
& \dot y = u(1-x^2) y-x\\
& x(0)=1\\
& y(0)=0\\
& u\le 0.75\\

\end{array}</math>

== Parameters ==
These fixed values are used within the model:

<math>[t_0,t_f]=[0,20]</math>

== Reference Solution ==

The following reference solution was generated using JModelica with the automatic differentiation tool CasADI and the solver IPOPT. The Optimica code used to solve the problem can be found under.... The optimal value of the objective function is 3.1762.
[[File:VDP_Plot_control.png|200px|thumb|left|control u]] [[File:VDP_Plot_states.png|200px|thumb|left|states x and y]] [[File:VDP_Plot_derivatives.png|200px|thumb|left|derivatives <math>\dot x</math> and <math>\dot y</math> ]]

Van der Pol Oscillator

2016-01-12T12:07:28Z

MadeleineSchroeter:

The Van der Pol Oscillator is an oscillating system with non-linear damping and self regulation. The System was first introduced by the Dutch physician Balthasar Van der Pol in 1927. The aim is to control the oscillation such that the system stays in a mean position.

== Model formulation ==

The Van der Pol Oscillator evolves over time according to the second order differential equation:

<math>{d^2x \over dt^2}-u(1-x^2){dx \over dt}+x= 0</math>

where <math>x</math> is the position coordinate, which is a function of the time <math>t</math>, and <math>u</math> is a scalar parameter indicating the non-linearity and the strength of the damping.

For <math>u>>1</math> the oscillator is being damped, whereas for <math>u<<1</math> energy is added to the system.

Based on the transformation <math>y = \dot x</math> the problem can be reformulated:

<math> \dot x = y</math>

<math>\dot y = u(1-x^2) y-x</math>

The optimal control problem arises by adding the objective function:

<math>\min\limits_{u}\int\limits_{t_0}^{t_f}(x(t)^2+y(t)^2+u(t)^2) dt</math>

== Optimal Control Problem ==
The Optimal Control Problem with the aim to minimize the deflection can be formulated as follows:

<math> \begin{array}{ll}
\min\limits_{u} & \int\limits_{t_0}^{t_f}(x(t)^2+y(t)^2+u(t)^2) dt\\
s.t. & \dot x = y\\
& \dot y = u(1-x^2) y-x\\
& x(0)=1\\
& y(0)=0\\
& u\le 0.75\\

\end{array}</math>

== Parameters ==
These fixed values are used within the model:

<math>[t_0,t_f]=[0,20]</math>

== Reference Solution ==

The following reference solution was generated using JModelica with the automatic differentiation tool CasADI and the solver IPOPT. The Optimica code used to solve the problem can be found under.... The optimal value of the objective function is 3.1762.
[[File:VDP_Plot_control.png|200px|thumb|left|alt text]] [[File:VDP_Plot_states.png|200px|thumb|left|alt text]] [[File:VDP_Plot_derivatives.png|200px|thumb|left|alt text]]

File:VDP Plot derivatives.png

2016-01-12T12:06:24Z

MadeleineSchroeter: der(x): derivative of the position coordinate x der(y): second derivatie of x

der(x): derivative of the position coordinate x
der(y): second derivatie of x

Van der Pol Oscillator

2016-01-12T12:02:52Z

MadeleineSchroeter:

The Van der Pol Oscillator is an oscillating system with non-linear damping and self regulation. The System was first introduced by the Dutch physician Balthasar Van der Pol in 1927. The aim is to control the oscillation such that the system stays in a mean position.

== Model formulation ==

The Van der Pol Oscillator evolves over time according to the second order differential equation:

<math>{d^2x \over dt^2}-u(1-x^2){dx \over dt}+x= 0</math>

where <math>x</math> is the position coordinate, which is a function of the time <math>t</math>, and <math>u</math> is a scalar parameter indicating the non-linearity and the strength of the damping.

For <math>u>>1</math> the oscillator is being damped, whereas for <math>u<<1</math> energy is added to the system.

Based on the transformation <math>y = \dot x</math> the problem can be reformulated:

<math> \dot x = y</math>

<math>\dot y = u(1-x^2) y-x</math>

The optimal control problem arises by adding the objective function:

<math>\min\limits_{u}\int\limits_{t_0}^{t_f}(x(t)^2+y(t)^2+u(t)^2) dt</math>

== Optimal Control Problem ==
The Optimal Control Problem with the aim to minimize the deflection can be formulated as follows:

<math> \begin{array}{ll}
\min\limits_{u} & \int\limits_{t_0}^{t_f}(x(t)^2+y(t)^2+u(t)^2) dt\\
s.t. & \dot x = y\\
& \dot y = u(1-x^2) y-x\\
& x(0)=1\\
& y(0)=0\\
& u\le 0.75\\

\end{array}</math>

== Parameters ==
These fixed values are used within the model:

<math>[t_0,t_f]=[0,20]</math>

== Reference Solution ==

The following reference solution was generated using JModelica with the automatic differentiation tool CasADI and the solver IPOPT. The Optimica code used to solve the problem can be found under.... The optimal value of the objective function is 3.1762.
[[File:VDP_Plot_control.png|200px|thumb|left|alt text]] [[File:VDP_Plot_states.png|200px|thumb|left|alt text]]

Van der Pol Oscillator

2016-01-12T12:02:25Z

MadeleineSchroeter:

The Van der Pol Oscillator is an oscillating system with non-linear damping and self regulation. The System was first introduced by the Dutch physician Balthasar Van der Pol in 1927. The aim is to control the oscillation such that the system stays in a mean position.

== Model formulation ==

The Van der Pol Oscillator evolves over time according to the second order differential equation:

<math>{d^2x \over dt^2}-u(1-x^2){dx \over dt}+x= 0</math>

where <math>x</math> is the position coordinate, which is a function of the time <math>t</math>, and <math>u</math> is a scalar parameter indicating the non-linearity and the strength of the damping.

For <math>u>>1</math> the oscillator is being damped, whereas for <math>u<<1</math> energy is added to the system.

Based on the transformation <math>y = \dot x</math> the problem can be reformulated:

<math> \dot x = y</math>

<math>\dot y = u(1-x^2) y-x</math>

The optimal control problem arises by adding the objective function:

<math>\min\limits_{u}\int\limits_{t_0}^{t_f}(x(t)^2+y(t)^2+u(t)^2) dt</math>

== Optimal Control Problem ==
The Optimal Control Problem with the aim to minimize the deflection can be formulated as follows:

<math> \begin{array}{ll}
\min\limits_{u} & \int\limits_{t_0}^{t_f}(x(t)^2+y(t)^2+u(t)^2) dt\\
s.t. & \dot x = y\\
& \dot y = u(1-x^2) y-x\\
& x(0)=1\\
& y(0)=0\\
& u\le 0.75\\

\end{array}</math>

== Parameters ==
These fixed values are used within the model:

<math>[t_0,t_f]=[0,20]</math>

== Reference Solution ==

The following reference solution was generated using JModelica with the automatic differentiation tool CasADI and the solver IPOPT. The Optimica code used to solve the problem can be found under.... The optimal value of the objective function is 3.1762.
[[File:VDP_Plot_control.png|200px|thumb|left|alt text]]] [[File:VDP_Plot_states.png|200px|thumb|left|alt text]]]

Van der Pol Oscillator

2016-01-12T12:01:38Z

MadeleineSchroeter:

The Van der Pol Oscillator is an oscillating system with non-linear damping and self regulation. The System was first introduced by the Dutch physician Balthasar Van der Pol in 1927. The aim is to control the oscillation such that the system stays in a mean position.

== Model formulation ==

The Van der Pol Oscillator evolves over time according to the second order differential equation:

<math>{d^2x \over dt^2}-u(1-x^2){dx \over dt}+x= 0</math>

where <math>x</math> is the position coordinate, which is a function of the time <math>t</math>, and <math>u</math> is a scalar parameter indicating the non-linearity and the strength of the damping.

For <math>u>>1</math> the oscillator is being damped, whereas for <math>u<<1</math> energy is added to the system.

Based on the transformation <math>y = \dot x</math> the problem can be reformulated:

<math> \dot x = y</math>

<math>\dot y = u(1-x^2) y-x</math>

The optimal control problem arises by adding the objective function:

<math>\min\limits_{u}\int\limits_{t_0}^{t_f}(x(t)^2+y(t)^2+u(t)^2) dt</math>

== Optimal Control Problem ==
The Optimal Control Problem with the aim to minimize the deflection can be formulated as follows:

<math> \begin{array}{ll}
\min\limits_{u} & \int\limits_{t_0}^{t_f}(x(t)^2+y(t)^2+u(t)^2) dt\\
s.t. & \dot x = y\\
& \dot y = u(1-x^2) y-x\\
& x(0)=1\\
& y(0)=0\\
& u\le 0.75\\

\end{array}</math>

== Parameters ==
These fixed values are used within the model:

<math>[t_0,t_f]=[0,20]</math>

== Reference Solution ==

The following reference solution was generated using JModelica with the automatic differentiation tool CasADI and the solver IPOPT. The Optimica code used to solve the problem can be found under.... The optimal value of the objective function is 3.1762.
[[File:VDP_Plot_control.png|200px|thumb|left|alt text]]]

File:VDP Plot states.png

2016-01-12T12:00:43Z

MadeleineSchroeter:

States of the Van der Pol oscillator on a time horizon from 0 to 20. (position coordinate x and it's derivative y)

File:VDP Plot states.png

2016-01-12T11:58:49Z

MadeleineSchroeter: States of the Van der Pol oscillator on a time horizon from 0 to 20.

States of the Van der Pol oscillator on a time horizon from 0 to 20.

Van der Pol Oscillator

2016-01-12T11:56:20Z

MadeleineSchroeter:

The Van der Pol Oscillator is an oscillating system with non-linear damping and self regulation. The System was first introduced by the Dutch physician Balthasar Van der Pol in 1927. The aim is to control the oscillation such that the system stays in a mean position.

== Model formulation ==

The Van der Pol Oscillator evolves over time according to the second order differential equation:

<math>{d^2x \over dt^2}-u(1-x^2){dx \over dt}+x= 0</math>

where <math>x</math> is the position coordinate, which is a function of the time <math>t</math>, and <math>u</math> is a scalar parameter indicating the non-linearity and the strength of the damping.

For <math>u>>1</math> the oscillator is being damped, whereas for <math>u<<1</math> energy is added to the system.

Based on the transformation <math>y = \dot x</math> the problem can be reformulated:

<math> \dot x = y</math>

<math>\dot y = u(1-x^2) y-x</math>

The optimal control problem arises by adding the objective function:

<math>\min\limits_{u}\int\limits_{t_0}^{t_f}(x(t)^2+y(t)^2+u(t)^2) dt</math>

== Optimal Control Problem ==
The Optimal Control Problem with the aim to minimize the deflection can be formulated as follows:

<math> \begin{array}{ll}
\min\limits_{u} & \int\limits_{t_0}^{t_f}(x(t)^2+y(t)^2+u(t)^2) dt\\
s.t. & \dot x = y\\
& \dot y = u(1-x^2) y-x\\
& x(0)=1\\
& y(0)=0\\
& u\le 0.75\\

\end{array}</math>

== Parameters ==
These fixed values are used within the model:

<math>[t_0,t_f]=[0,20]</math>

== Reference Solution ==

The following reference solution was generated using JModelica with the automatic differentiation tool CasADI and the solver IPOPT. The Optimica code used to solve the problem can be found under.... The optimal value of the objective function is 3.1762.
[[File:VDP_Plot_control.png]]

File:VDP Plot control.png

2016-01-12T11:55:18Z

MadeleineSchroeter:

Optimal Control Function of the Van der Pol Oscillator on a time horizon from 0 to 20.

File:VDP Plot control.png

2016-01-12T11:53:59Z

MadeleineSchroeter: Optimal Control of the Van der Pol Oscillator on a time horizon from 0 to 20.

Optimal Control of the Van der Pol Oscillator on a time horizon from 0 to 20.

Van der Pol Oscillator

2016-01-12T11:50:45Z

MadeleineSchroeter: Created page with "The Van der Pol Oscillator is an oscillating system with non-linear damping and self regulation. The System was first introduced by the Dutch physician Balthasar Van der Pol i..."

The Van der Pol Oscillator is an oscillating system with non-linear damping and self regulation. The System was first introduced by the Dutch physician Balthasar Van der Pol in 1927. The aim is to control the oscillation such that the system stays in a mean position.

== Model formulation ==

The Van der Pol Oscillator evolves over time according to the second order differential equation:

<math>{d^2x \over dt^2}-u(1-x^2){dx \over dt}+x= 0</math>

where <math>x</math> is the position coordinate, which is a function of the time <math>t</math>, and <math>u</math> is a scalar parameter indicating the non-linearity and the strength of the damping.

For <math>u>>1</math> the oscillator is being damped, whereas for <math>u<<1</math> energy is added to the system.

Based on the transformation <math>y = \dot x</math> the problem can be reformulated:

<math> \dot x = y</math>

<math>\dot y = u(1-x^2) y-x</math>

The optimal control problem arises by adding the objective function:

<math>\min\limits_{u}\int\limits_{t_0}^{t_f}(x(t)^2+y(t)^2+u(t)^2) dt</math>

== Optimal Control Problem ==
The Optimal Control Problem with the aim to minimize the deflection can be formulated as follows:

<math> \begin{array}{ll}
\min\limits_{u} & \int\limits_{t_0}^{t_f}(x(t)^2+y(t)^2+u(t)^2) dt\\
s.t. & \dot x = y\\
& \dot y = u(1-x^2) y-x\\
& x(0)=1\\
& y(0)=0\\
& u\le 0.75\\

\end{array}</math>

== Parameters ==
These fixed values are used within the model:

<math>[t_0,t_f]=[0,20]</math>

== Reference Solution ==

The following reference solution was generated using JModelica with the automatic differentiation tool CasADI and the solver IPOPT. The Optimica code used to solve the problem can be found under.... The optimal value of the objective function is 3.1762.
[[File:VDP_Plotcontrolu.png]]