Van der Pol Oscillator (Jump) - Revision history

FelixMueller at 11:35, 23 February 2016

2016-02-23T11:35:32Z

← Older revision		Revision as of 11:35, 23 February 2016
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	\begin{array}{llclr}		\begin{array}{llclr}
	\displaystyle \max_{x, u} & x_3(t_f) \\[1.5ex]		\displaystyle \max_{x, u} & x_3(t_f) \\[1.5ex]
	\mbox{s.t.} & \dot{x}_1~~(t)~~ & = & (1 - x_2~~(t)~~^2) x_1~~(t)~~ - x_2~~(t)~~ + u~~(t) \qquad & \forall t \in [t_0, t_f]~~, \\		\mbox{s.t.} & \dot{x}_1 & = & (1 - x_2^2) x_1 - x_2 + u, \\
	& \dot{x}_2~~(t)~~ & = & x_1~~(t) \qquad & \forall t \in [t_0, t_f]~~, \\		& \dot{x}_2 & = & x_1, \\
	& \dot{x}_3~~(t)~~ & = & x_1~~(t)~~^2 + x_2~~(t)~~^2 + u~~(t)~~^2 ~~\qquad & \forall t \in [t_0, t_f]~~, \\[1.5ex]		& \dot{x}_3 & = & x_1^2 + x_2^2 + u^2, \\[1.5ex]
	& x(0) &=& (0, 1, 0)^T, \\		& x(0) &=& (0, 1, 0)^T, \\
	& u(t) &\in& [-0.3, 1~~] \qquad & \forall t \in [t_0, t_f~~].		& u(t) &\in& [-0.3, 1].
	\end{array}		\end{array}
	</math>		</math>

2016-01-19T13:59:29Z

← Older revision		Revision as of 13:59, 19 January 2016
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	This is an implementation of a slightly modified form of the [[Van der Pol ~~oscillator~~]] problem using [[:category: Julia/JuMP \| JuMP]].		This is an implementation of a slightly modified form of the [[Van der Pol Oscillator]] problem using [[:category: Julia/JuMP \| JuMP]].

	The problem in question can be stated as follows:		The problem in question can be stated as follows:

2016-01-19T13:59:10Z

← Older revision		Revision as of 13:59, 19 January 2016
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	This is an implementation of a slightly modified form of the Van der Pol oscillator problem using [[:category: Julia/JuMP \| JuMP]].		This is an implementation of a slightly modified form of the [[Van der Pol oscillator]] problem using [[:category: Julia/JuMP \| JuMP]].

	The problem in question can be stated as follows:		The problem in question can be stated as follows:

2016-01-19T13:35:36Z

Reference solutions

← Older revision		Revision as of 13:35, 19 January 2016
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	== Reference solutions ==		== Reference solutions ==
	The optimal objective value is 2.902143.		This solution was computed using JuMP with a collocation method and 300 discretization points (using the code above). The differential equations were solved using the explicit Euler Method. The optimal objective value is 2.902143.

	<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">		<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
	File:Vdposc states plot.PNG\| Optimal states x1(t), x2(t) and x3(t).		File:Vdposc states plot.PNG\| Optimal states x1(t), x2(t) and x3(t).

2016-01-19T13:34:55Z

← Older revision		Revision as of 13:34, 19 January 2016
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	Although not necessary in JuMP the code was divided into three parts (following AMPL) - model file, data file and run file. The run file calls the other files and performs additional tasks such as printing results.		Although not necessary in JuMP the code was divided into three parts (following AMPL) - model file, data file and run file. The run file calls the other files and performs additional tasks such as printing results.

	A reference solution including plots done with JuMP ~~will~~ be ~~added shortly~~!		A reference solution including plots done with JuMP can be found below the code!

2016-01-19T13:33:55Z

Reference solutions

← Older revision		Revision as of 13:33, 19 January 2016
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	The optimal objective value is 2.902143.		The optimal objective value is 2.902143.
	<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">		<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
	File:Vdposc states plot.PNG\| Optimal states ~~x_1~~(t), ~~x_2~~(t) and ~~x_3~~(t).		File:Vdposc states plot.PNG\| Optimal states x1(t), x2(t) and x3(t).
	File:Vdposc control plot.~~png~~\| Optimal control u(t).		File:Vdposc control plot.PNG\| Optimal control u(t).
	</gallery>		</gallery>

	[[Category: Julia/JuMP]]		[[Category: Julia/JuMP]]

2016-01-19T13:33:36Z

Reference solutions

← Older revision		Revision as of 13:33, 19 January 2016
Line 117:		Line 117:
	The optimal objective value is 2.902143.		The optimal objective value is 2.902143.
	<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">		<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
	File:Vdposc states plot.~~png~~\| Optimal states x_1(t), x_2(t) and x_3(t).		File:Vdposc states plot.PNG\| Optimal states x_1(t), x_2(t) and x_3(t).
	File:Vdposc control plot.png\| Optimal control u(t).		File:Vdposc control plot.png\| Optimal control u(t).
	</gallery>		</gallery>

	[[Category: Julia/JuMP]]		[[Category: Julia/JuMP]]

2016-01-19T13:32:28Z

Reference solutions

← Older revision		Revision as of 13:32, 19 January 2016
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	The optimal objective value is 2.902143.		The optimal objective value is 2.902143.
	<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">		<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
	File:~~Vdposc_states_plot~~.png\| Optimal states x_1(t), x_2(t) and x_3(t).		File:Vdposc states plot.png\| Optimal states x_1(t), x_2(t) and x_3(t).
	File:~~Vdposc_control_plot~~.png\| Optimal control u(t).		File:Vdposc control plot.png\| Optimal control u(t).
	</gallery>		</gallery>

	[[Category: Julia/JuMP]]		[[Category: Julia/JuMP]]

2016-01-19T13:31:56Z

← Older revision		Revision as of 13:31, 19 January 2016
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	println("Optimal objective value is: ", getObjectiveValue(m))		println("Optimal objective value is: ", getObjectiveValue(m))
	println("Optimal Solution is: \n", getValue(x), getValue(u))</source>		println("Optimal Solution is: \n", getValue(x), getValue(u))</source>

			== Reference solutions ==
			The optimal objective value is 2.902143.
			<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
			File:Vdposc_states_plot.png\| Optimal states x_1(t), x_2(t) and x_3(t).
			File:Vdposc_control_plot.png\| Optimal control u(t).
			</gallery>

	[[Category: Julia/JuMP]]		[[Category: Julia/JuMP]]

2016-01-19T13:01:14Z

← Older revision		Revision as of 13:01, 19 January 2016
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	#ODE - solved with explicit euler method (i.e. x_k+1 = x_k + stepsize * f(x_k, t_k))		#ODE - solved with explicit euler method (i.e. x_k+1 = x_k + stepsize * f(x_k, t_k))
	@addNLConstraint(m, ODE_nonlin[ii=1:1, tt=1:N-1], x[ii,tt+1] - x[ii,tt] - step_size * ((1 - x[2,tt]^2) * x[1,tt] - x[2,tt] + u[1,tt]) == 0)		@addNLConstraint(m, ODE_nonlin[ii=1:1, tt=1:N-1], x[ii,tt+1] - x[ii,tt] - step_size * ((1 - x[2,tt]^2) * x[1,tt]
			- x[2,tt] + u[1,tt]) == 0)
	@addConstraint(m, ODE[ii=2:n_x, tt=1:N-1], x[ii,tt+1] - x[ii,tt] - step_size * ode_rhs(time_disc[tt], x[:,tt], u[:,tt])[ii] >= 0)		@addConstraint(m, ODE[ii=2:n_x, tt=1:N-1], x[ii,tt+1] - x[ii,tt] - step_size * ode_rhs(time_disc[tt], x[:,tt], u[:,tt])[ii] >= 0)
	@addConstraint(m, ODE[ii=2:n_x, tt=1:N-1], x[ii,tt+1] - x[ii,tt] - step_size * ode_rhs(time_disc[tt], x[:,tt], u[:,tt])[ii] <= 0)		@addConstraint(m, ODE[ii=2:n_x, tt=1:N-1], x[ii,tt+1] - x[ii,tt] - step_size * ode_rhs(time_disc[tt], x[:,tt], u[:,tt])[ii] <= 0)