Category:ODE model: Difference between revisions

Revision as of 02:14, 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

$\begin{array}{llcl} \min_{x (\cdot), u (\cdot), v (\cdot), q, ρ} & ϕ (x (t_{f}), q, ρ) \\ s.t. & \dot{x} (t) & = & f (x (t), u (t), v (t), q, ρ), \\ 0 & \leq & c (x (t), u (t), v (t), q, ρ), \\ 0 & = & r^{eq} (x (t_{0}), x (t_{1}), \dots, x (t_{m}), q, ρ), \\ 0 & \leq & r^{ieq} (x (t_{0}), x (t_{1}), \dots, x (t_{m}), q, ρ), \\ v (t) & \in & Ω : = {v^{1}, v^{2}, \dots, v^{n_{ω}}}, \\ ρ & \in & P : = {ρ^{1}, ρ^{2}, \dots, ρ^{n_{P}}}, \end{array}$

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, $ρ$ denotes the integer (constant-in-time) control values.

The multipoint constraints $r^{\cdot} (\cdot)$ are defined on a time grid $t_{0} \leq t_{1} \leq \dots \leq t_{m} = t_{f}$ . The Mayer term functional $ϕ : ℝ^{n_{x} + n_{q}} \to ℝ$ , the path- and control constraints $c : ℝ^{n_{x} \times n_{u} + n_{v} + n_{q}} \to ℝ^{n_{c}}$ and the constraint functions $r^{\cdot} : ℝ^{(m + 1) n_{x} + n_{q}} \to ℝ^{n_{r \cdot}}$ are assumed to be sufficiently often differentiable.

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

Extensions

For some problems the functions may as well depend explicitely on the time $t$ .

The differential equations might depend on state-dependent switches.

Note that a Lagrange term $\int_{t_{0}}^{t_{f}} L (x (t), u (t), v (t), q, ρ)$ can be transformed into a Mayer-type objective functional.

Subcategories

This category has only the following subcategory.

S

Simulated moving bed

Pages in category "ODE model"

The following 69 pages are in this category, out of 69 total.

A

B

C

D

E

F

G

H

I

J

L

M

O

P

Particle steering problem

Q

R

S

T

U

Urethane

V

@@ 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
@@ Line 5: / Line 5: @@
 <math>
 \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}
 </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]]