Cascade control systems are special cases of control systems with auxiliary controlled variables. Here, as shown in Figure 11.6, the main controller with the transfer function does not directly effect the actuator, but provides the reference value for the underlying auxiliary controller with the transfer function . This auxiliary controller forms together with the first plant section the auxiliary control loop, which is inside the main control loop. Disturbances in the first plant section will be already controlled by the auxiliary controller such that they have less influence on the second section. The main controller has then only to act slightly.

When multiple auxiliary variables are measured, multiple cascade control systems can be built. For the cascade control system of Figure 11.6 the controlled variable is given by

which gives

and can be written as

with

The characteristic equation with regard to the disturbance behaviour is

and with regard to the reference behaviour

It is obvious that the stability is influenced by the underlying auxiliary control loop. If the reference behaviour of the auxiliary control loop in Figure 11.6 is summarised as

then Eq. (11.21) can be represented by

From this equation the block diagram of a single-loop control system can be drawn as shown in Figure 11.7, which describes the same system as Figure 11.6. The auxiliary control system with the transfer function is an element of the basic control loop. Therefore, it is possible to design a cascade control system in the following steps:

- Design of the auxiliary control system, i.e. parametrisation
of the controller transfer function
for the given section
of the plant for disturbances. The auxiliary
control system must be fast (high bandwidth) and therefore mostly
a P or PD controller is chosen for
.
- Design of the main controller transfer function for the plant transfer function . has the task to remove steady-state errors in the controlled variable. Therefore, it is expedient to use a PI controller as long as the plant has behaviour.

Figure 11.8 shows two examples of cascade control systems.