I have the following system: $$\dot{x} = A(t)x+B(t)u$$ $$y = x$$
$A(t)$ and $B(t)$ are actually scalar, but time-dependent. If they would be constant, I could simulate the system in Matlab using:
sys = ss(A,B,C,0);
lsim(sys,u,t,x0);
However, it would be nice to simulate the system with dynamic state and input matrix. The matrices are based on measurement data, this means I would have for each discrete time step $t_i$ another matrix $A(t_i)$. Any suggestions how to do that?
When you say the matrices are based on measurement data, do you mean that $A(t)$ varies with your state space output $y$, which should be configured to provide your outputs (measurements), or that you want to actually provide a hard-coded series of parameters (field measurements) to the $A$ matrix?
If you are trying to do the former, have essentially $A=f(y)$, where $y=f(t)$ (because actually $y=f(x)$, but $x=f(t)$ by definition), then you can use ode45. In the function you want to use you can define y = x, then use that to update the parameters in $A$.
If you want the latter, or the former really, you can always brute force it. Setting up some standards real quick, let's say you have:
nSamples, which would be the number of time points in your simulation.
Your time step is dT.
You have nStates, so $size(A) = [nStates,nStates]$. Now assume that your A matrix evolves over time based of field samples (testing, etc.), you should have a bunch of $A$ matrices, such that you actually have $size(A) = [nStates,nStates,nSamples]$.
Now, try this:
x = Initial Conditions;
for i = 1:nSamples
currentA = A[: , : , i];
xDot = currentA * x + B * u;
x = x + xDot * dT;
y = x;
end
As before, if you just want to swap out parameters in $A$ based on $y$ then that's easy to implement too.
x = Initial Conditions;
y = f(x);
for i = 1:nSamples
A = f(y);
xDot = Ax + Bu;
x = x + xDot*dT;
y = f(x);
end
Again, this is just a brute-force method to implement what lsim does for you. It's just numeric integration.
Hope this helps!
