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 | + | 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> | ||
| − | + | 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 | + | <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]] | ||
Revision as of 03:14, 29 November 2008
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 mathematical model.
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}](https://mintoc.de/images/math/c/2/e/c2ea6ba320c6ca71260f7eeb98105969.png)
for ![t \in [t_0, t_f]](https://mintoc.de/images/math/5/5/8/55823791d9100bcb5461801aff4f6edd.png)
almost everywhere.
denotes the differential states,
denotes the continuous control functions,
denotes the integer control functions,
denotes the continuous (constant-in-time) control values,
denotes the integer (constant-in-time) control values.
The multipoint constraints 
are defined on a time grid 
. The Mayer term functional 
, the path- and control constraints 
and the constraint functions 
are assumed to be sufficiently often differentiable.
The equality constraints 
will often fix the initial values, i.e., 
, or impose a periodicity constraint.
Extensions
For some problems the functions may as well depend explicitely on the time 
.
The differential equations might depend on state-dependent switches.
Note that a Lagrange term 
can be transformed into a Mayer-type objective functional.
Pages in category "ODE model"
The following 52 pages are in this category, out of 52 total.
