# Help with fminsearch to maximise non-analytic function via specific parameters

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Brantosaurus on 22 Nov 2022
Commented: Brantosaurus on 23 Nov 2022
I have a simple trajectory model. Let's pretend its non-analytic as it would be with drag. Key variables of interest are launch elevation, theta, and impact range z(:,1).
Is it possible to parameterise the objective function to allow fminsearch to maximise (negative minimise) the function value for range with theta as the independent variable?
A plot of it would look similar to below but with theta (0 to 90 deg) replacing range, and range replacing height ( 0 to max range at theta=45 deg).
I'm struggling to set up theta as the driver and range as the function output to drive such a calculation. Basically finding theta = 45 deg is the maximum range.
Something like [x,fval] = fminsearch(@traj, x0) ???
Any help setting this up would be very much appreciated.
clear,clc
global rho d m g
g = 9.81; rho = 1.2; m = 0.145; d = 0.0732; v = 100;
tspan = 0:0.001:20;
for theta = 35:10:55
z0 = [0 0 v*cosd(theta) v*sind(theta)];
options = odeset('Events',@trajevents);
[~,z,~,ze,ie] = ode45(@traj,tspan,z0,options);
hold on, plot(z(:,1),z(:,2))
plot(ze(ie(1),1),ze(ie(1),2),'k+')
xlabel('range'), ylabel('height')
axis equal, xlim([0 1100]); ylim([0 400]); grid
end
function zdot = traj(~,z)
global g
zdot = [z(3); z(4); 0; -g];
end
function [check,stop,direction] = trajevents(~,z)
check = z(2); stop = 1; direction = -1;
end

Torsten on 23 Nov 2022
Here is the optimization solution:
theta0 = 10;
sol = fminsearch(@fun,theta0)
sol = 45
function range = fun(theta)
global rho d m g
g = 9.81; rho = 1.2; m = 0.145; d = 0.0732; v = 100;
tspan = 0:0.001:20;
z0 = [0 0 v*cosd(theta) v*sind(theta)];
options = odeset('Events',@trajevents);
[~,z,~,ze,~] = ode45(@traj,tspan,z0,options);
range = -z(end,1) ;
end
function zdot = traj(~,z)
global g
zdot = [z(3); z(4); 0; -g];
end
function [check,stop,direction] = trajevents(~,z)
check = z(2); stop = 1; direction = -1;
end
Brantosaurus on 23 Nov 2022
I had struggled with this for some time.
Much appreciated :)

Torsten on 22 Nov 2022
Edited: Torsten on 22 Nov 2022
Of course, you could use fminsearch for optimization. But using it, you will only get a single point of the below graph.
Theta = 1:89;
global rho d m g
g = 9.81; rho = 1.2; m = 0.145; d = 0.0732; v = 100;
tspan = 0:0.001:20;
range = zeros(size(Theta));
for i=1:numel(Theta)
theta = Theta(i);
z0 = [0 0 v*cosd(theta) v*sind(theta)];
options = odeset('Events',@trajevents);
[~,z,~,ze,~] = ode45(@traj,tspan,z0,options);
range(i) = z(end,1) ;
end
plot(Theta,range)
grid on
function zdot = traj(~,z)
global g
zdot = [z(3); z(4); 0; -g];
end
function [check,stop,direction] = trajevents(~,z)
check = z(2); stop = 1; direction = -1;
end

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