# 3. Controllers and controller types

Block diagram of a PID controller

Just like the controlled system, the controller shows a specific transmission behaviour. Unlike the controlled system, here the **transmission behaviour can be deliberately adjusted according to the desired function**.

The manipulating variable y from the frequently used **PID controller** can be described as the sum of the output signals from different transmission blocks each with a different time response.

The same input signal is present at the inputs of all transmission blocks, namely the control difference e.

Step responses of the PID controller: **P component**

The transmission behaviour of the** PID controller** is defined by the following parameters:

**control gain K**_{p}**(proportional behaviour)**

Step responses of the PID controller:

**I component**

**reset time T**_{i}**(integral behaviour)**

Step responses of the PID controller: **D component**

**rate time T**_{d}**(****derivative****behaviour)**

Depending on the parameter settings, the controller can demonstrate P, PI, PD or PID behaviour.

The properties of the different controller types are set out below.

In a unity feedback configuration, an ideal PID compensator is used to enhance the step response of a system with the transfer function

The PID controller has the standard form K_{p}+K_{i}/s+K_{d}s. Therefore, the closed loop transfer function of the whole system becomes

As the proportional, integral and derivative terms of the PID compensator are increased, the evolution of the step response, from the initial uncompensated response (corresponding to PID coefficents Kp=1, Ki=0, Kd=0) to the final desired response is depicted. The following effects of PID compensation can be readily observed:

The proportional term increases the speed of the system. It also decreases the residual steady state error of the step response, but can not eliminate it completely.

The integral term eliminates the residual steady state error of the step response, but adds undesired "oscillations" to the transient response (overshoot).

The derivative term "damps out" the undesired oscillations in the transient response.