Euler method with state space model

Hello,

Im trying solve an equation in form of a state space model with Eulers (forward) method.

My input u is a vector of accelerations that I have given on discrete points (ax). As output I expect a scalar for each discrete point on which eulers method is calculated.

What I did so far:

generate a PTD2 with those values:

s=tf('s');
pt2 = s/((s+0.1)*(s+4));
SYS = ss(pt2);

A = SYS.A; 
B = SYS.B; 
C = SYS.C;
D = SYS.D;

I created a linspaced timevector t_i that is equally distributed over all my discrete points. To make it easier to read I extracted some values of the real data:

% ax = traj.ax(1:10);
% you can use this vector of accelerations if you want
ax = [-0.7412; -0.749 ;-0.7525;-0.7508;-0.7443;-0.7331;-0.7178;-0.699;-0.6779;-0.6540];

t_i = linspace(1, 5, 10);
h = diff(t_i(1:2));  % calc step size

x = t_i(1) : h : t_i(end);  % the range of x - not sure about this 

y = zeros(length(x),2);  % allocate the result y 
u= ax; %make clear that ax is the input  

Now I want to solve Eulers method as state space model:

for i=1:10-1
    
    dx = A*x(i) + B*u(i);
    x_k1 = x_k + dx*h; %x_k1 is x_k+1 
    x_k = x_k1;
    % y should be a scalar output in my case, since I want to get weighted accelerations as output - but I can only compile if I put the (i,:) right now.
    y(i,:) = C*x_k;
    
end

Is it possible to use x as a state vector of the ptd2 inside the eulers method? how should I define the range for x in this case, do I even need it?

I wrote this with the help of some code I found in the forum to differential equation problems, and tried to expand it to a state space model but im a bit lost right now. I tricked with the matrices for y, since I expect an output that is a scalar for each input but I have no idea how to achieve this.

Is it even possible to solve a state space model with this euler method?

Answer 1

[Michael Brehmer]

Hello, I figured it out by my own.

I had some major errors in the definition of x_k and y.

You have to initialize x_k as standing vector, that is updated in each step of the iteration

x_k = [0;0];

for i=1:N
    
    
    dx = A*x_k + B*u(i);
    x_k1 = x_k + dx*h;

    y(i) = C*x_k1;
    x_k = x_k1;
    
end