MATLAB: Solve differnetial equation of state space

differential equationsspace-state

Hello everyone,
I want to solve a differential equation of state space dx/dt = A*x + B*u, so I have a simple model with a Ground movement.
My equation is y'' = -d/m * y' – k/m * y + d/m * u' + k/m * u. I have the y (Output) and I want to find u, thats the Input to my system.
To have a first order equation i wrote dy(1) = y(2) and than dy(2) = -(d/m)*y(1)-(k/m)*y + (d/m)*u(1) + (k/m)*u(2)
m = 50;
d = 1000;
k = 30000;
A = [0,1;-k/m,-d/m];
B = [k/m;d/m];
C = [1,0];
D = [0];
Thats the code I've tried but it doesent function. 
function du = fun(u,y)
dy = zeros(2,1)
dy(1) = y(2)
du = zeros(1,1)
du(1) = u2
%dy(2) = -(d/m)*y(1)-(k/m)*y + (d/m)*u(1) + (k/m)*u(2)
du(1) = -(m/d)*dy(2)-y(1)-(k/d)*y + (k/d)*u(1)
 end
tspan = [0 10]
y0 = 1;
[u,y] = ode45(@(u,y) tspan, u0);

Best Answer

try this code. It assumes that y is know and u is unknown. I set y(t) to sin(t) as an example.
time = linspace(0,10,1000);
y = sin(time);
dy = gradient(y,time);
ddy = gradient(dy,time);
u0 = 0;
[t, u] = ode45(@(t,u) odeFun(t,u,time,y,dy,ddy), time, u0);
plot(t,u);
function dudt = odeFun(t, u, time, y, dy, ddy)
    m = 50;
    d = 1000;
    k = 30000;
    
    y = interp1(time, y, t);
    dy = interp1(time, dy, t);
    ddy = interp1(time, ddy, t);
    
    dudt = m/d*(ddy + d/m*dy + k/m*y - k/m*u);
end
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