How to improve speed simulation results with ODE45/23/23s/15s

Question

I want to simulate an ODE system, very fast. I usually use Euler-Method because it's fast, but not the most stable nor accurate. But I want to use a more standardlized method to simulate ODEs.

The problem with this code is that this way to simulate model is very slow and all depends on how large the input vector u is. If u is short, then the simulation will be fast. Also if stepTime is large, e.g 1 or 2, then the simulation will also be slow.

How can I improve the simulation speed by just modify options variable?

My idea is that if I could have fewer steps in the simulation, that would be great.

If we look at this code below.

% User inputs 
model = @(t, x, u) [x(2); -0.7*x(1) - 0.2*x(2) + 2*u(1)]; % Second order state space model
u = linspace(5, 5, 100);                                  % Input signal
stepTime = 0.5;                                           % 0.5 seconds step time 
x0 = [0;0];                                               % Initial states
options = odeset();                                       % Default values

% Savings 
L = length(u);
x = zeros(size(x0, 1), L);
t = zeros(1, L);
 
% Loop
for i = 1:L
  % Save time and output 
  x(:, i) = x0;
  t(i) = stepTime*(i-1);
    
  % Simulate ode45
  tspan = [stepTime*(i-1) stepTime*i];
  [time, output] = ode45(@(t, x) model(t, x, u(:, i)), tspan, x0, options);

  % Next state 
  x0 = output(end, :)';
end

plot(t, x(1, :))
title('Position')
grid on 
xlabel('Time')
ylabel('y0');

figure
plot(t, x(2, :))
title('Velocity')
grid on 
xlabel('Time')
ylabel('y1');

It will give out this output signals. One is position and the other one is velocity.

The most interesting part is

% Next state 
x0 = output(end, :)';

I'm taking the last row of output and call it next initial state. Here is the problem. If I could reduce output from ode45 etc, then the simulation will become much faster.

How can I do that? Here is my thoughts:

• Is fixed step a solution? Is there any drawbacks from that? What about stability?
• Is it possible to only simulate the last value and avoid the rest of the values in output?