How to obtain a stabilization problem in linear system with controller?

Question

The scheme of system:

The equasion after Laplace transform: $$Y(p) = \frac{PID(p)\cdot H(p)}{1 + PID(p)\cdot H(p)} Y^d(p)$$ Now I want to make inverse Laplace transform and then plot $y(t)$, but $y^d(t) = 0 $ in my case. What should I do?

Answer 1

This is really confusing and your question is not well-posed. I assume $p$ is $s$, "PID" is a PID controller, and you are using inverting feedback. If your input is frequency dependent, nothing says this compensator is going to be able to regulate the plant, if your input is a constant setpoint set at zero on the other hand, you have a good chance with a PID.

Now: you have no business inverting the Laplace transform at this point. First I don't think you can, and even if there is some way it certainly wont help you do anything. What you need is an actual model for your plant transfer function $H$. This will be an $s$ polynomial. When you have that you can use it to design the PID gains using some linear control technique--that is, you need some requirements which make sense in terms of the physical thing you are trying to control. After you get good gains for your PID you can look at the response in the time domain by solving or (more likely) simulating the system with PID forcing which uses the error feedback you've setup. This will be the plot you want but again your problem is far too general to solve as is.