Particle steering problem: Difference between revisions

Particle steering problem
State dimension:	1
Differential states:	2
Discrete control functions:	1
Path constraints:	2
Interior point equalities:	7

Latest revision as of 08:33, 27 July 2016

The Particle steering problem minimizes "the time taken for a particle, acted upon by a thrust of constant magnitude, to achieve a given altitude and terminal velocity." (Cite and problem taken from the COPS library)

Mathematical formulation

The problem is given by

$\begin{array}{llcl} \min_{x, u, t_{f}} & t_{f} \\ s.t. & {\ddot{x}}_{1} & = & a \cos (u), \\ {\ddot{x}}_{2} & = & a \sin (u), \\ x (0) & = & (0, 0)^{T}, \\ \dot{x} (0) & = & (0, 0)^{T}, \\ x_{2} (t_{f}) & = & 5, \\ \dot{x} (t_{f}) & = & (45, 0)^{T}, \\ u (t) & \in & [- \frac{π}{2}, \frac{π}{2}] . \end{array}$

where $(x_{1}, x_{2})$ is the position of the particle, $u$ is the control angle and $a$ is the constant magnitude of thrust.

Source Code

Model descriptions are available in

AMPL/TACO code at Particle steering problem (TACO)

@@ Line 3: / Line 3: @@
 |nx        = 2
 |nw        = 1
+|nc        = 2
 |nre       = 7
 }}<!-- Do not insert line break here or Dimensions Box moves up in the layout...
@@ Line 21: / Line 22: @@
   & \ddot{x}_1 & = & a \cos (u), \\
   & \ddot{x}_2 & = & a \sin (u),  \\
-  & x(t_0) &=& (0, 0)^T, \\
+  & x(0) &=& (0, 0)^T, \\
   & \dot{x}(0) &=& (0, 0)^T, \\
   & x_2 (t_f) &=& 5, \\
@@ Line 29: / Line 30: @@
 </math>
 </p>
 where <math> (x_1, x_2) </math> is the position of the particle, <math> u </math> is the control angle and <math> a </math> is the constant magnitude of thrust.
 == Source Code ==
@@ Line 43: / Line 44: @@
 [[Category:MIOCP]]
 [[Category:ODE model]]
+[[Category:Minimum time]]