Control Engineering › System Behaviors 2

P and I Behavior

P behaviour and I behaviour can be combined in two ways.

P∙I behaviour

If blocks with P behaviour and I behaviour are connected in series, the transfer functions are multiplied. The overall behaviour then acts like I behaviour.

Series connection of P element k_P and I element (k_I, 1/s) combined into k_I' and 1/s

\[ k_I' = k_I \cdot k_P \]

The two gain blocks can simply be combined. The type of behaviour remains I behaviour. Only the intensity of the behaviour changes.

P+I behaviour

If P behaviour is to be added to I behaviour, we have to adapt the block diagram:

Parallel connection of P element k_P and I element (k_I, 1/s): PI element

\[ \begin{gathered} y = k_P \cdot x + k_I \cdot \frac{1}{s} \cdot x \\[6pt] H = \frac{y}{x} = k_P + k_I \cdot \frac{1}{s} \end{gathered} \]

Let us look at the step response of this behaviour: through the P behaviour, the output quantity jumps up immediately at the step time. Then the I behaviour begins to take effect.

Step response of the PI element: jump to k_P, then linear rise with slope k_I — Eingang = input · Ausgang = output

\[ \text{In this example: } k_P = 1 \text{ and } k_I = 1 \]

We do not consider P+I behaviour further in this tutorial, because the calculation in a control loop is difficult.