# Principle and generalization of proportional-resonance controller

simulate this circuit – Schematic created using CircuitLab

We can introduce the model of $$\u(s)\$$ into the controller $$\K(s)\$$ to realize high accuracy control.

• $$\u(t) = 1\$$, i.e. $$\u(s) = 1/s\$$. We use $$\K(s) = K_p + \frac{K_i}{s}\$$ (PI controller) to realize zero static error.
• $$\u(t) = \sin(\omega t)\$$, i.e. $$\u(s) = \frac{\omega}{s^2+\omega^2}\$$. We use $$\K(s) = K_p + \frac{K}{s^2+\omega^2}\$$ to realize zero static error.

Now my problem is that, does this work for any other $$\u(t)\$$? Such as $$\u(t) = \exp(\alpha t)\$$?

P.S. I learnt that the high accuracy of PI or proportional-resonance controller is high gain. I can understand this for DC input ($$\u(t) = 1\$$) or sine input. But if this kind of controller works for other input, how to understand the gain here?

• Put a "\" before each instance of "\$" to get Latex working properly. Why do you call a PI controller a "proportional-resonance" controller? – Andy aka Nov 29 '18 at 12:56
• @Andyaka proportional-resonance controller (PR controller) is the controller shown in the 2nd bullet – Alexander Zhang Nov 29 '18 at 13:00