Difference between revisions of "Category:ODE model"

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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.
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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
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<math>
 
<math>
 
\begin{array}{llcl}
 
\begin{array}{llcl}
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  \displaystyle \min_{x(\cdot), u(\cdot), v(\cdot)} & \phi(x(t_f)) \\[1.5ex]
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  \displaystyle \min_{x(\cdot), u(\cdot), v(\cdot), q, \rho} & \phi(x(t_f), q, \rho) \\[1.5ex]
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  \mbox{s.t.} & \dot{x}(t) & = & f(x(t), u(t), v(t)), \\
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  \mbox{s.t.} & \dot{x}(t) & = & f(x(t), u(t), v(t), q, \rho), \\
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  & 0 &\le& c(x(t),u(t),v(t)), \\[1.5ex]
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  & 0 &\le& c(x(t),u(t),v(t), q, \rho), \\[1.5ex]
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  & 0 &=& r^{\text{eq}}(x(t_0),x(t_1), \dots, x(t_m)), \\
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  & 0 &=& r^{\text{eq}}(x(t_0),x(t_1), \dots, x(t_m), q, \rho), \\
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  & 0 &\le& r^{\text{ieq}}(x(t_0),x(t_1), \dots, x(t_m)), \\
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  & 0 &\le& r^{\text{ieq}}(x(t_0),x(t_1), \dots, x(t_m), q, \rho), \\[1.5ex]
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  & v(t) &\in& \Omega := \{v^1, v^2, \dots, v^{n_\omega} \}.
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  & v(t) &\in& \Omega := \{v^1, v^2, \dots, v^{n_\omega} \},\\
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& \rho &\in& \Rho := \{\rho^1, \rho^2, \dots, \rho^{n_\Rho} \},
 
\end{array}  
 
\end{array}  
 
</math>
 
</math>
  
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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.
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for <math>t \in [t_0, t_f]</math> almost everywhere.
  
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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.
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<math>x(\cdot)</math> denotes the differential states,
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<math>u(\cdot)</math> denotes the continuous control functions,
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<math>v(\cdot)</math> denotes the integer control functions,
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<math>q</math> denotes the continuous (constant-in-time) control values,
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<math>\rho</math> denotes the integer (constant-in-time) control values.
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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.
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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.
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== Extensions ==
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* For some problems the functions may as well depend explicitely on the time <math>t</math>.
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* The differential equations might depend on [[:Category:State dependent switches | state-dependent switches]].
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* The variables may include [[:Category:Boolean variables | boolean variables]].
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* The underlying process might be a [[:Category:Multistage process | multistage process]].
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* The dynamics might be [[:Category:Unstable | unstable]].
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* There might be an underlying [[:Category:Network topology | network topology]].
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* The integer control functions might have been (re)formulated by means of an [[:Category:Outer convexification|outer convexification]].
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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]]

Latest revision as of 12:02, 20 November 2010

This category includes all problems constrained by the solution of ordinary differential equations (ODE). In particular, no algebraic variables and derivatives with respect to one independent variable only (typically time) are present in the model equations for F(\cdot).

The mixed-integer optimal control problem is of the form

\begin{array}{llcl} \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), q, \rho), \\ & 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), q, \rho), \\ & 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} \},\\ & \rho &\in& \Rho := \{\rho^1, \rho^2, \dots, \rho^{n_\Rho} \}, \end{array}

for t \in [t_0, t_f] almost everywhere.

x(\cdot) denotes the differential states, u(\cdot) denotes the continuous control functions, v(\cdot) denotes the integer control functions, q denotes the continuous (constant-in-time) control values, \rho denotes the integer (constant-in-time) control values.

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

The equality constraints r^{\text{eq}}(\cdot) will often fix the initial values, i.e., x(0) = x_0, or impose a periodicity constraint.

Extensions

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

Subcategories

This category has only the following subcategory.

S

Pages in category "ODE model"

The following 52 pages are in this category, out of 52 total.

A

B

C

D

E

F

G

H

I

L

M

O

P

Q

R

S

T

U

V

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