At the beginning of this course we introduced block diagram representations for controlled physical processes. In this lecture we use the laplace transforms and transfer functions to represent a control system by a block diagram.

Assume we have a process with input u and output y . If the transfer function of the process is equal to P(s) , then we can write

The block diagram representation of this process is as follows:

The output of the system is equal to

In practice different systems are interconnected. If each system is represented by a block then from the block diagram of the overall system we can determine the transfer functions from any signal (variable) to any other signal in the system.


Example 1: Consider the block diagram shown below. The system represented by the the transfer function C(s) has the input E(s) and the output U(s). The system represented by the the transfer function P(s) has the input U(s) and the output Y(s). We would like to find the transfer function from E to Y. In this case the input will be E and the output will be Y .

We have

Therefore,

In control applications we usually deal with processes, controllers, sensors, actuators, reference (or desired) signal sources. We will now give a block diagram representation of the overall control system. We will combine the actuator and process in one block. The following is the block diagram representation of a control system.

Let us find the transfer function T(s) of the above closed loop (feedback) system. This will be the transfer function from the input to the system R to the output of the system Y .

Rearranging terms we get

We next find the transfer function J(s) from R to E . We have

PID Controllers

In feedback control systems the controller takes the error signal (difference between the desired and measured signals) and processes it. The output of the controller is passed as an input to the process. One type of controller which is widely used in industrial applications is the PID (proportional integral derivative) controller. The proportional part of this controller multiplies the error by a constant. The integral part of the PID controller integrates the error. Finally the derivative part differentiates the error. The output of the controller is the sum of the previous three signals. We have

where are the proportional, integral and derivative gains, respectively.
Take Laplace transform of each side of the above equation.

C(s) is the transfer function of the PID controller.

Click here to view Dr. L. K. Wong pdf Notes on PID Controllers (pages 1-24)

The student Ra3ed Kattoura found these useful notes.