Superposition

Disturbances and reference variable act as inputs on the system. The controlled variable is no longer influenced only by the reference variable, but additionally by the disturbances. For simplicity, we reduce the problem to a single disturbance.

In systems with several input quantities, we use the mathematics of the superposition principle. It only applies to linear time-invariant systems. The output quantity is calculated as follows:

1. Set the disturbance to 0.

2. Calculate, as intermediate result 1, the output quantity from the reference variable.

3. Set the reference variable to 0.

4. Calculate, as intermediate result 2, the output quantity from the disturbance.

5. Add the two intermediate results.

In this way we obtain two transfer functions. The transfer function of the output quantity as a function of the reference variable is called the “reference response”; you already know it. The transfer function of the output quantity as a function of the disturbance is called the “disturbance response” (German: Störunterdrückung, SU); this is new for you.

To calculate the reference response as transfer function H_FÜ, we set the disturbance to 0 and calculate the controlled variable as a function of the reference variable. To calculate the disturbance response as transfer function H_SU, we set the reference variable to 0 and calculate the controlled variable as a function of the disturbance d. At the end we add both solutions. As a formula:

\[ \begin{gathered} y = H_{\mathrm{FÜ}} \cdot w\,\big|_{d=0} \rightarrow H_{\mathrm{FÜ}} = \frac{y}{w}\Big|_{d=0} \\[6pt] y = H_{\mathrm{SU}} \cdot d\,\big|_{w=0} \rightarrow H_{\mathrm{SU}} = \frac{y}{d}\Big|_{w=0} \\[6pt] y = H_{\mathrm{FÜ}} \cdot w + H_{\mathrm{SU}} \cdot d \end{gathered} \]

The controlled variable should be independent of the disturbance. That is the case for H_SU = 0. Then it does not matter how large the disturbance is; it has no effect at all on the controlled variable.

\[ \begin{gathered} \text{Goals of control engineering:} \\[6pt] H_{\mathrm{FÜ}} = \frac{y}{w}\Big|_{d=0} = 1 \\[6pt] H_{\mathrm{SU}} = \frac{y}{d}\Big|_{w=0} = 0 \\[6pt] \text{Substitute: } y = H_{\mathrm{FÜ}} \cdot w + H_{\mathrm{SU}} \cdot d = 1 \cdot w + 0 \cdot d = w \end{gathered} \]