Jan-17-2019, 11:30 AM
Because I didn't know that initial conditions are known
I considered extended system, so R0 = [x0_approx, y0_appros, p0_approx, a_approx, B_approx, G_approx];
Therefore, dO_k -- residual between known measurement at time t_k and measurement at t_k obtained for trajectory started at (R0, t0).
From now, we know that (x0, y0, p0) are exactly known
(I wrote x0, y0, p0 because we have 3 coupled differential equations as you denoted that in the first post),
so our iterative process should be slightly changed:
There are several ways we can adopt our iterative process:
1) don't change x0, y0, p0 when iterating; apply changes to a, B, G only;
2) If you have measurement at t0, you can consider weighted least squares and assign very big weight to this measurement. Since measurements are raw coordinates
big weight for t0-measurement will account that x0 and y0 are very important measurements and shouldn't be changed over iterations;
Finally, everything is depending on the system of ODEs you are trying to solve. Is your system really hard to solve analytically?
You are likely needing to construct problem-specific algorithm to get estimations for a, B, G.
Have you some a priory information regarding a, B, G values...?
If you have fast numerical procedure for solving the ODE, you can try, e.g. simulated annealing approach.
At last, did you try the way proposed by Gribouillis?
I considered extended system, so R0 = [x0_approx, y0_appros, p0_approx, a_approx, B_approx, G_approx];
Therefore, dO_k -- residual between known measurement at time t_k and measurement at t_k obtained for trajectory started at (R0, t0).
From now, we know that (x0, y0, p0) are exactly known
(I wrote x0, y0, p0 because we have 3 coupled differential equations as you denoted that in the first post),
so our iterative process should be slightly changed:
There are several ways we can adopt our iterative process:
1) don't change x0, y0, p0 when iterating; apply changes to a, B, G only;
2) If you have measurement at t0, you can consider weighted least squares and assign very big weight to this measurement. Since measurements are raw coordinates
big weight for t0-measurement will account that x0 and y0 are very important measurements and shouldn't be changed over iterations;
Finally, everything is depending on the system of ODEs you are trying to solve. Is your system really hard to solve analytically?
You are likely needing to construct problem-specific algorithm to get estimations for a, B, G.
Have you some a priory information regarding a, B, G values...?
If you have fast numerical procedure for solving the ODE, you can try, e.g. simulated annealing approach.
At last, did you try the way proposed by Gribouillis?