MATLAB: Is this the way to solve algebraic loops

Question

So far, I understand that to solve common algebraic loop problems, it´s enough to decouple the input/output direct feedthrough of every direct feedthrough block by adding a state variable between the input and the output. And, as far as I understand, this can be easily done with an IC block. Inserting this block is also equivalent to equip the direct feedthrough block with a state x that evolves like x(k+1)=x(k) (or \dot x=0), and to consider a switch of the output so that we have y=x at t=t0 and y=f(u) for t>t0, where x,u and y represent the state, the input and the output respectively). This is equivalent to force the output at the value of the IC block at t=t0, and the algebraic loop would be solved, isn’t it? But at this point, I would argue that it is possible to use any block for which you can set a initial condition/output (e.g. rate transition blocks for example), isn’t it? So, to solve an algebraic loop is enough to break it in just one point, right? Or there is more?

Answer 1

I agree that using a delay block or a memory block is a solution. Actually you can use a delay block with inherited sample time even for continuous-time systems. The result is the same. However, by doing that, you are modifying the system behavior, and my challenge is to find a way to solve algebraic loops without changing the system dynamics... is it possible or it is necessary to insert this small time delay? :-)