Category:ODE model - Revision history

SebastianSager: Added Outer convexification

2010-11-20T11:02:47Z

Added Outer convexification

← Older revision		Revision as of 11:02, 20 November 2010
Line 34:		Line 34:
	* The dynamics might be [[:Category:Unstable \| unstable]].		* The dynamics might be [[:Category:Unstable \| unstable]].
	* There might be an underlying [[:Category:Network topology \| network topology]].		* There might be an underlying [[:Category:Network topology \| network topology]].
			* The integer control functions might have been (re)formulated by means of an [[:Category:Outer convexification\|outer convexification]].

	Note that a Lagrange term <math>\int_{t_0}^{t_f} L( x(t), u(t), v(t), q, \rho) \; \mathrm{d} t</math> can be transformed into a Mayer-type objective functional.		Note that a Lagrange term <math>\int_{t_0}^{t_f} L( x(t), u(t), v(t), q, \rho) \; \mathrm{d} t</math> can be transformed into a Mayer-type objective functional.

	[[Category:Model characterization]]		[[Category:Model characterization]]

SebastianSager: Added extensions

2010-11-18T16:38:59Z

Added extensions

← Older revision		Revision as of 16:38, 18 November 2010
Line 1:		Line 1:
	This category includes all problems constrained by the solution of [http://en.wikipedia.org/wiki/Ordinary_differential_equation ordinary differential equations] (ODE). In particular, no algebraic variables and derivatives with respect to one independent variable only (typically time) are present in the ~~mathematical~~ model.		This category includes all problems constrained by the solution of [http://en.wikipedia.org/wiki/Ordinary_differential_equation ordinary differential equations] (ODE). In particular, no algebraic variables and derivatives with respect to one independent variable only (typically time) are present in the [[:Category:Model characterization\|model equations]] for <math>F(\cdot)</math>.

	The mixed-integer optimal control problem is of the form		The mixed-integer optimal control problem is of the form
Line 28:		Line 28:

	== Extensions ==		== Extensions ==
	For some problems the functions may as well depend explicitely on the time <math>t</math>.		* For some problems the functions may as well depend explicitely on the time <math>t</math>.
			* The differential equations might depend on [[:Category:State dependent switches \| state-dependent switches]].
	The differential equations might depend on [[:Category:State dependent switches \| state-dependent switches]].		* The variables may include [[:Category:Boolean variables \| boolean variables]].
			* The underlying process might be a [[:Category:Multistage process \| multistage process]].
			* The dynamics might be [[:Category:Unstable \| unstable]].
			* There might be an underlying [[:Category:Network topology \| network topology]].

	Note that a Lagrange term <math>\int_{t_0}^{t_f} L( x(t), u(t), v(t), q, \rho) \; \mathrm{d} t</math> can be transformed into a Mayer-type objective functional.		Note that a Lagrange term <math>\int_{t_0}^{t_f} L( x(t), u(t), v(t), q, \rho) \; \mathrm{d} t</math> can be transformed into a Mayer-type objective functional.

	[[Category:Model characterization]]		[[Category:Model characterization]]

SebastianSager at 05:20, 29 November 2008

2008-11-29T05:20:21Z

← Older revision		Revision as of 05:20, 29 November 2008
Line 30:		Line 30:
	For some problems the functions may as well depend explicitely on the time <math>t</math>.		For some problems the functions may as well depend explicitely on the time <math>t</math>.

	The differential equations might depend on [[:Category:State-dependent switches \| state-dependent switches]].		The differential equations might depend on [[:Category:State dependent switches \| state-dependent switches]].

	Note that a Lagrange term <math>\int_{t_0}^{t_f} L( x(t), u(t), v(t), q, \rho) \; \mathrm{d} t</math> can be transformed into a Mayer-type objective functional.		Note that a Lagrange term <math>\int_{t_0}^{t_f} L( x(t), u(t), v(t), q, \rho) \; \mathrm{d} t</math> can be transformed into a Mayer-type objective functional.

	[[Category:Model characterization]]		[[Category:Model characterization]]

SebastianSager at 05:05, 29 November 2008

2008-11-29T05:05:42Z

← Older revision		Revision as of 05:05, 29 November 2008
Line 32:		Line 32:
	The differential equations might depend on [[:Category:State-dependent switches \| state-dependent switches]].		The differential equations might depend on [[:Category:State-dependent switches \| state-dependent switches]].

	Note that a Lagrange term <math>\int_{t_0}^{t_f} L( x(t), u(t), v(t), q, \rho)</math> can be transformed into a Mayer-type objective functional.		Note that a Lagrange term <math>\int_{t_0}^{t_f} L( x(t), u(t), v(t), q, \rho) \; \mathrm{d} t</math> can be transformed into a Mayer-type objective functional.

	[[Category:Model characterization]]		[[Category:Model characterization]]

SebastianSager at 02:14, 29 November 2008

2008-11-29T02:14:23Z

← Older revision		Revision as of 02:14, 29 November 2008
Line 1:		Line 1:
	This category includes all problems constrained by the solution of [http://en.wikipedia.org/wiki/Ordinary_differential_equation ordinary differential equations] (ODE). In particular, no algebraic variables and derivatives with respect to one independent variable only are present in the mathematical model.		This category includes all problems constrained by the solution of [http://en.wikipedia.org/wiki/Ordinary_differential_equation ordinary differential equations] (ODE). In particular, no algebraic variables and derivatives with respect to one independent variable only (typically time) are present in the mathematical model.

	The mixed-integer optimal control problem is of the form		The mixed-integer optimal control problem is of the form
Line 5:		Line 5:
	<math>		<math>
	\begin{array}{llcl}		\begin{array}{llcl}
	\displaystyle \min_{x(\cdot), u(\cdot), v(\cdot)} & \phi(x(t_f)) \\[1.5ex]		\displaystyle \min_{x(\cdot), u(\cdot), v(\cdot), q, \rho} & \phi(x(t_f), q, \rho) \\[1.5ex]
	\mbox{s.t.} & \dot{x}(t) & = & f(x(t), u(t), v(t)), \\		\mbox{s.t.} & \dot{x}(t) & = & f(x(t), u(t), v(t), q, \rho), \\
	& 0 &\le& c(x(t),u(t),v(t)), \\[1.5ex]		& 0 &\le& c(x(t),u(t),v(t), q, \rho), \\[1.5ex]
	& 0 &=& r^{\text{eq}}(x(t_0),x(t_1), \dots, x(t_m)), \\		& 0 &=& r^{\text{eq}}(x(t_0),x(t_1), \dots, x(t_m), q, \rho), \\
	& 0 &\le& r^{\text{ieq}}(x(t_0),x(t_1), \dots, x(t_m)), \\		& 0 &\le& r^{\text{ieq}}(x(t_0),x(t_1), \dots, x(t_m), q, \rho), \\[1.5ex]
	& v(t) &\in& \Omega := \{v^1, v^2, \dots, v^{n_\omega} \}.		& v(t) &\in& \Omega := \{v^1, v^2, \dots, v^{n_\omega} \},\\
			& \rho &\in& \Rho := \{\rho^1, \rho^2, \dots, \rho^{n_\Rho} \},
	\end{array}		\end{array}
	</math>		</math>

	~~The multipoint constraints~~ <math>r^\~~cdot(\cdot)</math> are defined on a time grid <math>~~t_0 ~~\le t_1 \le \dots \le t_m =~~ t_f </math>. The Mayer term functional <math>\phi: \mathbb{R}^{n_x} \rightarrow \mathbb{R}</math>, the path- and control constraints <math>c: \mathbb{R}^{n_x \times n_u \times n_v} \rightarrow \mathbb{R}^{n_c}</math> and the constraint functions <math>r^\cdot: \mathbb{R}^{(m+1) n_x} \rightarrow \mathbb{R}^{n_{r\cdot}}</math> are assumed to be sufficiently often differentiable.		for <math>t \in [t_0, t_f]</math> almost everywhere.

	The equality constraints <math>r^{\text{eq}}(\cdot)</math> will often fix the initial values, i.e., <math>x(0) = x_0</math>, or impose of [[:Category:Periodic \| periodicity]] constraint.		<math>x(\cdot)</math> denotes the differential states,
			<math>u(\cdot)</math> denotes the continuous control functions,
			<math>v(\cdot)</math> denotes the integer control functions,
			<math>q</math> denotes the continuous (constant-in-time) control values,
			<math>\rho</math> denotes the integer (constant-in-time) control values.

			The multipoint constraints <math>r^\cdot(\cdot)</math> are defined on a time grid <math>t_0 \le t_1 \le \dots \le t_m = t_f </math>. The Mayer term functional <math>\phi: \mathbb{R}^{n_x + n_q} \rightarrow \mathbb{R}</math>, the path- and control constraints <math>c: \mathbb{R}^{n_x \times n_u + n_v + n_q} \rightarrow \mathbb{R}^{n_c}</math> and the constraint functions <math>r^\cdot: \mathbb{R}^{(m+1) n_x + n_q} \rightarrow \mathbb{R}^{n_{r\cdot}}</math> are assumed to be sufficiently often differentiable.

			The equality constraints <math>r^{\text{eq}}(\cdot)</math> will often fix the initial values, i.e., <math>x(0) = x_0</math>, or impose a [[:Category:Periodic \| periodicity]] constraint.

			== Extensions ==
			For some problems the functions may as well depend explicitely on the time <math>t</math>.

			The differential equations might depend on [[:Category:State-dependent switches \| state-dependent switches]].

			Note that a Lagrange term <math>\int_{t_0}^{t_f} L( x(t), u(t), v(t), q, \rho)</math> can be transformed into a Mayer-type objective functional.

	[[Category:Model characterization]]		[[Category:Model characterization]]

SebastianSager: Added mathematical problem definition

2008-11-29T01:51:03Z

Added mathematical problem definition

← Older revision		Revision as of 01:51, 29 November 2008
Line 1:		Line 1:
	This category includes all problems constrained by the solution of [http://en.wikipedia.org/wiki/Ordinary_differential_equation ordinary differential equations] (ODE). In particular, no algebraic variables and derivatives with respect to one independent variable only are present in the mathematical model.		This category includes all problems constrained by the solution of [http://en.wikipedia.org/wiki/Ordinary_differential_equation ordinary differential equations] (ODE). In particular, no algebraic variables and derivatives with respect to one independent variable only are present in the mathematical model.

			The mixed-integer optimal control problem is of the form

			<math>
			\begin{array}{llcl}
			\displaystyle \min_{x(\cdot), u(\cdot), v(\cdot)} & \phi(x(t_f)) \\[1.5ex]
			\mbox{s.t.} & \dot{x}(t) & = & f(x(t), u(t), v(t)), \\
			& 0 &\le& c(x(t),u(t),v(t)), \\[1.5ex]
			& 0 &=& r^{\text{eq}}(x(t_0),x(t_1), \dots, x(t_m)), \\
			& 0 &\le& r^{\text{ieq}}(x(t_0),x(t_1), \dots, x(t_m)), \\
			& v(t) &\in& \Omega := \{v^1, v^2, \dots, v^{n_\omega} \}.
			\end{array}
			</math>

			The multipoint constraints <math>r^\cdot(\cdot)</math> are defined on a time grid <math>t_0 \le t_1 \le \dots \le t_m = t_f </math>. The Mayer term functional <math>\phi: \mathbb{R}^{n_x} \rightarrow \mathbb{R}</math>, the path- and control constraints <math>c: \mathbb{R}^{n_x \times n_u \times n_v} \rightarrow \mathbb{R}^{n_c}</math> and the constraint functions <math>r^\cdot: \mathbb{R}^{(m+1) n_x} \rightarrow \mathbb{R}^{n_{r\cdot}}</math> are assumed to be sufficiently often differentiable.

			The equality constraints <math>r^{\text{eq}}(\cdot)</math> will often fix the initial values, i.e., <math>x(0) = x_0</math>, or impose of [[:Category:Periodic \| periodicity]] constraint.

	[[Category:Model characterization]]		[[Category:Model characterization]]

SebastianSager: New page: This category includes all problems constrained by the solution of [http://en.wikipedia.org/wiki/Ordinary_differential_equation ordinary differential equations] (ODE). In particular, no al...

2008-11-09T13:19:24Z

New page: This category includes all problems constrained by the solution of [http://en.wikipedia.org/wiki/Ordinary_differential_equation ordinary differential equations] (ODE). In particular, no al...

New page

This category includes all problems constrained by the solution of [http://en.wikipedia.org/wiki/Ordinary_differential_equation ordinary differential equations] (ODE). In particular, no algebraic variables and derivatives with respect to one independent variable only are present in the mathematical model.

[[Category:Model characterization]]