Optimal Control Problem - Revision history

TobiasWeber: /* Constraints */

2016-05-17T15:30:15Z

Constraints

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	where one hast to state <math>x_0</math>. More complicated expressions like ~~periodicity conditions~~:		where one hast to state <math>x_0</math>. More complicated expressions like [[:Category:Periodic\|periodic constraints]]:

	<p>		<p>
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	restricting hyperplanes:		and restricting hyperplanes:

	<p>		<p>

TobiasWeber at 15:27, 17 May 2016

2016-05-17T15:27:38Z

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	Here the weighting matrix <math>S</math> has to be given. For an tracking type ~~OCP~~ with vector valued state and inputs an quadratic objective would be:		Here the weighting matrix <math>S</math> has to be given. For an [[:Category:Tracking objective\|tracking type objective]] with vector valued state and inputs an quadratic objective would be:

	<p>		<p>

TobiasWeber at 15:26, 17 May 2016

2016-05-17T15:26:34Z

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	=== Objective ===		=== Objective ===

	To specify the [[:Category:Objective characterization\|objective}} the Mayer and/or Lagrange term has to be described. For a parameter estimation problem the objective would be a finite sum over the squared difference between the measurements and the model response:		To specify the [[:Category:Objective characterization\|objective]] the Mayer and/or Lagrange term has to be described. For a parameter estimation problem the objective would be a finite sum over the squared difference between the measurements and the model response:

	<p>		<p>

TobiasWeber at 15:25, 17 May 2016

2016-05-17T15:25:03Z

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	=== Objective ===		=== Objective ===

	To specify the objective the Mayer and/or Lagrange term has to be described. For a parameter estimation problem the objective would be a finite sum over the squared difference between the measurements and the model response:		To specify the [[:Category:Objective characterization\|objective}} the Mayer and/or Lagrange term has to be described. For a parameter estimation problem the objective would be a finite sum over the squared difference between the measurements and the model response:

	<p>		<p>

TobiasWeber at 15:21, 17 May 2016

2016-05-17T15:21:43Z

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	This is a template page to show how a general Optimal Control Problem should be described on this website. In this first paragraph a short description of the problem is given. What is the practical or mathematical context that motivates to solve the problem. What do the free and dependent variables of the optimization problem describe. What kind of model (equality constraints) fix the dependent variables (ODE, RDE, DAE, ...). What is the objective? Also one can cite a paper that introduced or solved this problem just by saying:		This is a template page to show how a general Optimal Control Problem should be described on this website. In this first paragraph a short description of the problem is given. What is the [[:Category:Application\|practical]] or mathematical context that motivates to solve the problem. What do the free and dependent variables of the optimization problem describe. What kind of model (equality constraints) fix the dependent variables (ODE, RDE, DAE, ...). What is the objective? Also one can cite a paper that introduced or solved this problem just by saying:

	Max Mustermann, "This paper doesn't exist!",		Max Mustermann, "This paper doesn't exist!",
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	== Model Formulation ==		== Model Formulation ==

	In this section the dynamic model, that describes the physical or technical process of interest, should be described. First the differential equation should be written down. Then the states, inputs, and parameters should be declared (Dimension, Type (integer, real, binary, ...)). Finally the parameters that are not free in the optimization and the coefficients should be stated in an extra table or list. All equations and functions are defined over an interval <math>I \subset \R</math>. Initial values and free parameters are taken into account later in the Optimal Control Problem.		In this section the [[:Category:Model characterization\|dynamic model]], that describes the physical or technical process of interest, should be described. First the differential equation should be written down. Then the states, inputs, and parameters should be declared (Dimension, Type (integer, real, binary, ...)). Finally the parameters that are not free in the optimization and the coefficients should be stated in an extra table or list. All equations and functions are defined over an interval <math>I \subset \R</math>. Initial values and free parameters are taken into account later in the Optimal Control Problem.

	=== Ordinary Differential Equation (ODE) ===		=== Ordinary Differential Equation (ODE) ===
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	\|}		\|}

	This would result in an right hand side function <math>f(x,u)</math> in this case, because there are no free parameters. Of course one could also specify an implicit DAE system (<math>F(\dot{x},x,u,p,t)=0</math>), where it just has to be said which states are differential and which are algebraic.		This would result in an right hand side function <math>f(x,u)</math> in this case, because there are no free parameters. Of course one could also specify an [[:Category:DAE model\|implicit DAE]] system (<math>F(\dot{x},x,u,p,t)=0</math>), where it just has to be said which states are differential and which are algebraic.
	The states and inputs can also be vector valued functions requiring matrix valued coefficients.		The states and inputs can also be vector valued functions requiring matrix valued coefficients.

TobiasWeber at 14:41, 17 May 2016

2016-05-17T14:41:20Z

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	In this section the dynamic model, that describes the physical or technical process of interest, should be described. First the differential equation should be written down. Then the states, inputs, and parameters should be declared (Dimension, Type (integer, real, binary, ...)). Finally the parameters that are not free in the optimization and the coefficients should be stated in an extra table or list. All equations and functions are defined over an interval <math>I \subset \R</math>. Initial values and free parameters are taken into account later in the Optimal Control Problem.		In this section the dynamic model, that describes the physical or technical process of interest, should be described. First the differential equation should be written down. Then the states, inputs, and parameters should be declared (Dimension, Type (integer, real, binary, ...)). Finally the parameters that are not free in the optimization and the coefficients should be stated in an extra table or list. All equations and functions are defined over an interval <math>I \subset \R</math>. Initial values and free parameters are taken into account later in the Optimal Control Problem.

	=== Ordinary Differential Equation ===		=== Ordinary Differential Equation (ODE) ===

	For a simple scalar [[:Category:ODE model\|ODE model]] this would look like the following:		For a simple scalar [[:Category:ODE model\|ODE model]] this would look like the following:
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	<p>		<p>
	<math>		<math>
	\dot{x}=px+u,\; \forall \, t \in I;\; x: I \rightarrow \R,\; u: I \rightarrow \R, \; p \in \R.		\frac{dx(t)}{dt} = \dot{x}(t)=px(t)+u(t),\; \forall \, t \in I;\; x: I \rightarrow \R,\; u: I \rightarrow \R, \; p \in \R.
	</math>		</math>
	</p>		</p>
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	\|}		\|}

	This would result in an right hand side function <math>f(x,u)</math> in this case, because there are no free parameters. Of course one could also specify an implicit DAE system (<math>F(\dot{x},x,u,p,t)=0</math>), where it just ~~hast~~ to be said which states are differential and which are algebraic.		This would result in an right hand side function <math>f(x,u)</math> in this case, because there are no free parameters. Of course one could also specify an implicit DAE system (<math>F(\dot{x},x,u,p,t)=0</math>), where it just has to be said which states are differential and which are algebraic.
			The states and inputs can also be vector valued functions requiring matrix valued coefficients.

	=== Matrix Differential Equation ===		=== Matrix Differential Equation (MDE) ===

	For an Lyapunov differential equation (with constant coefficient) one could write:		For an Lyapunov differential equation (with constant coefficient) one could write:
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	<p>		<p>
	<math>		<math>
	\dot{X}=AX+XA^T, \forall \, t \in I; \; X: I \rightarrow \R^{n \times n}, A \in \R^{n \times n}.		\dot{X}(t)=AX(t)+X(t)A^T, \forall \, t \in I; \; X: I \rightarrow \R^{n \times n}, A \in \R^{n \times n}.
	</math>		</math>
	</p>		</p>
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	* <math>a_{21} = 1, \; a_{22} = 0 </math>		* <math>a_{21} = 1, \; a_{22} = 0 </math>

	=== Variational Differential Equation ===		=== Variational Differential Equation (VDE) ===

	For an VDE with respect to the parameters the equations would be:		For a VDE with respect to the parameters the equations would be:

	<p>		<p>
	<math>		<math>
	\dot{G}_{p}=f_x(x,p)G_{p}+f_p(x,p), \forall \, t \in I; \; G_{p}: I \rightarrow \R^{n_x \times n_p}, \; x: I \rightarrow \R^{n_x}.		\dot{G}_{p}(t)=f_x(x,p)G_{p}(t)+f_p(x,p), \forall \, t \in I; \; G_{p}: I \rightarrow \R^{n_x \times n_p}, \; x: I \rightarrow \R^{n_x}.
	</math>		</math>
	</p>		</p>
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	<math>		<math>
	\dot{x}=f(x,u,p,t) \forall \, t \in I.		\dot{x}(t)=f(x(t),u(t),p,t) \forall \, t \in I.
	</math>		</math>
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	are also possible. General expressions could also be used. Constraints working on the whole interval <math>I</math> can be also very general or just as box constraints.		are also possible. General expressions could also be used. Constraints working on the whole interval <math>I</math> can be also very general or just box constraints.

	The set for inputs and states should be used to define the function spaces these functions live in and other set constraints that can not be expressed as nonlinear functions. One could say that the inputs should be piecewise constant or piecewise linear continuous functions and the states should then be <math>\mathcal{C}^1</math> or <math>\mathcal{C}^2</math>. The set constraint on the parameters should be used to define all constraints on the parameters, if they are free to the optimization, otherwise it doesn't appear. This could look like:		The set for inputs and states should be used to define the function spaces these functions live in and other set constraints that can not be expressed as nonlinear functions. One could say that the inputs should be piecewise constant or piecewise linear continuous functions and the states should then be <math>\mathcal{C}^1</math> or <math>\mathcal{C}^2</math>. The set constraint on the parameters should be used to define all constraints on the parameters, if they are free to the optimization, otherwise it doesn't appear. This could look like:
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	== Reference Solution ==		== Reference Solution ==

	In this section one should provide reference solutions. First the discretization, the initial guess, the solution algorithm and the used hardware should be stated, as well as the solving time. Then the optimal solution should be given, at least as a ~~.png~~ of the plots for the optimal trajectories.		In this section one should provide reference solutions. First the discretization, the initial guess, the solution algorithm and the used hardware should be stated, as well as the solving time. Then the optimal solution should be given, at least as a PNG of the plots for the optimal trajectories.
	One should provide the optimal objective value, the solution time for the OCP and the trajectories which where used to initialize the solver. One could for example say that one used a zero input trajectory and just integrated the ODE to get a feasible initialization for the solver.		One should provide the optimal objective value.

	== Source Code ==		== Source Code ==

FelixJost: /* Optimal Control Problem */

2015-12-04T10:17:34Z

Optimal Control Problem

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	== Reference Solution ==		== Reference Solution ==

TobiasWeber at 12:25, 16 October 2015

2015-10-16T12:25:22Z

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	<math>		<math>
	P=\{p \in \R^{n_p} \| p_{min} \leq p \leq p_{max}\}		P=\{p \in \R^{n_p} \| p_{min} \leq p \leq p_{max}\}.
	</math>		</math>
	</p>		</p>

TobiasWeber at 12:24, 16 October 2015

2015-10-16T12:24:46Z

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	<math>		<math>
	P=\{p \in \R^{n_p} \\| p_{min} \leq p \leq p_{max}\}		P=\{p \in \R^{n_p} \| p_{min} \leq p \leq p_{max}\}
	</math>		</math>
	</p>		</p>

TobiasWeber at 12:24, 16 October 2015

2015-10-16T12:24:24Z

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	<math>		<math>
	P=\{p \in \R^{n_p \\| p_{min} \leq p \leq p_{max}}\}		P=\{p \in \R^{n_p} \\| p_{min} \leq p \leq p_{max}\}
	</math>		</math>
	</p>		</p>