Electronic – PID controller modelling

controlpid controller

According to the following simple PID control system:

PID Controller
(source: umich.edu)

We could create the equation:

$$(R(s)-Y(s))C(s)P(s) = Y(s)$$

Wich will lead us to

$$\frac{Y}{R}=\frac{CP}{1+CP}$$

So, if we want to minimize error (which is \$R-Y\$), then it makes sense to think that ideally we want to make \$Y=R\$ (which means the output is following the input at best).
Although, this would lead us to:

$$\frac{Y}{R}=1=\frac{CP}{1+CP}$$
$$1+CP=CP$$
$$1=0$$

Which is obviously wrong. I believe that I did not miss anything stupid in the mathematical steps. So, what does this contradiction mean in the control context? And why did this appeared in the equations?

Best Answer

"C" and "P" aren't constants or fixed gains - but functions of time (or by Laplace transf. -> on complex "s"). In other words, are dynamical systems - where the output depends not only on actual input, but also on its past values. To be short: A input change not reflects immediately on output (instead, need to pass through "C" and "P" blocks) in order to be sensed and compared. This "lag" is reflected by the poles and zeros of transfer functions. The controller "C" is created by design, but the plant "P" does not depend on you - it has its own features. If you want Y = R all the time, then you do not need a control system.