Genetic algorithm for discrete trajectory design

Hi, I have to optimize an interplanetary trajectory. I thought to follow this procedure: 1- divide the trajectory in N step 2- define the control vector u = [T1,alpha1,...,TN,alphaN] 3- define the objective function J = -mf +w1Dr+w2Dv (where Dr and Dv are the errors on position and velocity respectively) 4- integrate the motion equations (considering the thrust constant for each segment)

The control vector has an upper bound and a lower bound for T and alpha.

How can I implement this problem using the "ga" of Matlab? In particular, can I pass to "ga" a population given by discrete set of values? And how can I define the bounds on the control vector?

Thank you.

Answer 1

[Alan Weiss]

You can set bound for your control vector u in the lb and ub arguments; see the ga function reference page.

You should probably sum the absolute values or the squares of the errors, not just sum the errors, in your objective function.

Good luck,

Alan Weiss

MATLAB mathematical toolbox documentation

[fe ri]

Thank you Alan. But I have two question. 1. Is "Nvar" the number of the elements of the control vector or of the fitness function? 2. How can I pass the N elements of the control vector to the ode45?

I think I'm making a little bit of confusion because on the reference page there aren't example similar to my case.

[Alan Weiss]

As the function reference page states, nvars is the number of design variables, also called control variables, which are the variables that ga passes to your fitness function. In your case nvars might be 2N, because you seem to have alpha_i and T_i for i ranging from 1 to N.

Generally your fitness function accepts a row vector x of length nvars and returns the value of the fitness function. If you need to call ode45 inside your fitness function, just do so. For example, (this just solves a simple ODE starting from y(0) = x(1) then adds the squares of the solution minus a fixed function):

 function fval = myobj(x)
 odefun = @(t,y)(2*y-t);
 tspan = 0:0.2:5;
 y0 = x(1);
 [t,y] = ode45(@(t,y) 2*t, tspan, y0);
 fval = sum((2*y-t).^2);

Of course, you probably would do better to pass extra parameters for some arguments.

Good luck,

Alan Weiss

MATLAB mathematical toolbox documentation

[fe ri]

Thank you again, Alan. Since this morning (when I saw your answer) I'm trying to implement my problem defining my objfunction in this way:

function y = objmass(x)
tof = 2;
tstep = linspace(0,tof,11);
options = odeset('RelTol', 1.0e-10, 'AbsTol', 1.0e-10);
initcond = [1;0;0;1;1];
i = 1;
for n = 1:1:10
       t1 = tstep(i);
       t2 = tstep(i+1);
       T = x(n);
       alpha = x(10+n);
       [twrk, ysol] = ode45(@motioneq,[t1 t2],initcond,options,T,alpha);
       initcond = ysol(length(twrk), 1:5);
       i = i+1;
end

w1 = 2.5;
w2 = 2;
dr = abs(ysol(length(twrk), 1)- 1.523);
dvr = abs(ysol(length(twrk), 3));
dvth = abs(0.81 - ysol(length(twrk), 4));
y = -ysol(length(twrk),5)+w1*dr^2+w2*(dvr^2+dvth^2);

But the solution doesn't match the optimal one. I think that it could depend on the initial population. Do you think so?

Thank you.