Control Engineering › System Behaviors 2

Summary

I behaviour

I element: blocks k_I and 1/s from x to y

\[ H(s) = \frac{1}{s} \cdot k_I \]

Step at input x and linearly rising output y of the I element — Eingang = input · Ausgang = output

I behaviour in the control loop

\[ \begin{gathered} A = k_I \cdot \frac{1}{s} \\[6pt] H_{\mathrm{FÜ}} = \frac{1}{1 + \frac{1}{k_I} \cdot s} \\[6pt] H_{\mathrm{FÜ}} = \frac{1}{1 + \tau_{\mathrm{FÜ}} \cdot s} \text{ with } \tau_{\mathrm{FÜ}} = \frac{1}{k_I} \\[6pt] \text{I behaviour in the control loop} \rightarrow \text{PT1 behaviour of the controlled system} \end{gathered} \]

PT1 behaviour

\[ H(s) = \frac{k_P}{1 + \tau s} \]

Step at input x(t) and PT1 step response y(t), which reaches 63 % of the final value after the time constant τ — Eingang = input · Ausgang = output

PT1 behaviour in the control loop

\[ \begin{gathered} A = \frac{k_P}{1 + \tau s} \\[6pt] H_{\mathrm{FÜ}} = \frac{\frac{k_P}{1 + k_P}}{1 + \frac{\tau}{1 + k_P}\,s} = \frac{k_{P,\mathrm{FÜ}}}{1 + \tau_{\mathrm{FÜ}} \cdot s} \\[6pt] k_{P,\mathrm{FÜ}} = \frac{k_P}{1 + k_P} \\[6pt] \tau_{\mathrm{FÜ}} = \frac{\tau}{1 + k_P} \\[6pt] \text{PT1 behaviour in the control loop} \rightarrow \text{PT1 behaviour of the controlled system} \end{gathered} \]

D behaviour

\[ H_D = k_D \cdot s \]

D element (k_D and s) with step at input x and pulse-shaped response at output y — Eingang = input · Ausgang = output

Dead time

\[ \text{Dead time } t_T = 1\,\mathrm{min} \text{ for a system with P behaviour and } k_P = 0.5 \]

In general: the behaviour describes a type of behaviour of a function block or system. We treat the mathematical description of all systems with the same behaviour in the same way. They differ only in their characteristic parameters. With the characteristic parameters we describe something like the intensity of the behaviour.