Control Engineering › System Behaviors 2

PT1 Behavior in the Control Loop

Next we look at PT1 behaviour in a control loop. If parameter A shows PT1 behaviour, we obtain the following block diagram:

Control loop with PT1 plant k_PS / (1 + τ_S · s)

\[ \begin{gathered} A = \frac{k_{PS}}{1 + \tau_S \cdot s} \quad \text{(subscript S: system without control)} \\[8pt] H_{\mathrm{FÜ}} = \frac{A}{1 + A} = \frac{\frac{k_{PS}}{1 + \tau_S \cdot s}}{1 + \frac{k_{PS}}{1 + \tau_S \cdot s}}\text{, expand by } (1 + \tau_S \cdot s) \\[8pt] = \frac{k_{PS}}{1 + \tau_S \cdot s + k_{PS}} = \frac{k_{PS}}{(1 + k_{PS}) + \tau_S \cdot s}\text{, divide by } (1 + k_{PS}) \\[8pt] = \frac{\frac{k_{PS}}{1 + k_{PS}}}{\frac{1 + k_{PS}}{1 + k_{PS}} + \frac{\tau_S}{1 + k_{PS}}\,s} = \frac{\frac{k_{PS}}{1 + k_{PS}}}{1 + \frac{\tau_S}{1 + k_{PS}}\,s} = \frac{k_{P,\mathrm{FÜ}}}{1 + \tau_{\mathrm{FÜ}} \cdot s} \\[8pt] \text{Subscript FÜ: controlled system in the control loop} \\[8pt] H_{\mathrm{FÜ}} = \frac{k_{P,\mathrm{FÜ}}}{1 + \tau_{\mathrm{FÜ}} \cdot s} \text{ with } k_{P,\mathrm{FÜ}} = \frac{k_{PS}}{1 + k_{PS}} \text{ and } \tau_{\mathrm{FÜ}} = \frac{\tau_S}{1 + k_{PS}} \end{gathered} \]

The reference response also shows PT1 behaviour. However, its characteristic parameters differ from those of the PT1 behaviour of block A. Above all, τ_FÜ changes. It becomes smaller the larger k_PS is. So with k_PS we can make the controlled system faster than A.

In the steady state:

\[ H_{\mathrm{FÜ}} = \frac{\frac{k_{PS}}{1 + k_{PS}}}{1 + \frac{\tau_S}{1 + k_{PS}} \cdot 0} = \frac{k_{PS}}{1 + k_{PS}} \neq 1 \]

We have the same problem as with P behaviour in the control loop: for real values of k_PS, the reference response is always less than 1. So there is a control error even in the steady state. This is because PT1 behaviour behaves like time-delayed P behaviour. In the steady state, A takes finite values if we insert PT1 behaviour for A. So a control error e must remain in the system.

PT1 behaviour in the control loop is even worse than P behaviour in the control loop. There is a steady-state control error. In addition, after a step there is a large control error in the transition range. P behaviour at least reacts immediately to the step at the output.